Universität der Bundeswehr München Fakultät für Luft- und Raumfahrttechnik Institut für Strahlantriebe Aerodynamic Optimisation of Highly Loaded Turbine Cascade Blades for Heavy Duty Gas Turbine Applications Dipl.-Ing. Pasquale Cardamone Vollständiger Abdruck der bei der Fakultät für Luft- und Raumfahrttechnik der Universität der Bundeswehr München zur Erlangung des akademischen Grades eines Doktor Ingenieurs (Dr.-Ing.) genehmigten Dissertation Vorsitzender: Prof. Dr. sc. math. Kurt Marti 1. Berichterstatter: Prof. Dr. rer. nat. Michael Pfitzner 2. Berichterstatter: Prof. Dr.-Ing. Francesco Martelli Tag der Einreichung: 04.10.2005 Tag der Annahme: 25.01.2006 Tag der Promotion: 03.02.2006
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Universität der Bundeswehr München Fakultät für Luft- und Raumfahrttechnik
Institut für Strahlantriebe
Aerodynamic Optimisation of Highly Loaded Turbine Cascade Blades
for Heavy Duty Gas Turbine Applications
Dipl.-Ing. Pasquale Cardamone
Vollständiger Abdruck der bei der Fakultät für Luft- und Raumfahrttechnik
der Universität der Bundeswehr München zur Erlangung des akademischen Grades eines
Doktor Ingenieurs (Dr.-Ing.)
genehmigten Dissertation
Vorsitzender: Prof. Dr. sc. math. Kurt Marti 1. Berichterstatter: Prof. Dr. rer. nat. Michael Pfitzner 2. Berichterstatter: Prof. Dr.-Ing. Francesco Martelli
Tag der Einreichung: 04.10.2005 Tag der Annahme: 25.01.2006 Tag der Promotion: 03.02.2006
I
Preface This thesis is based on the investigations carried out during my activity as research engineer at the “Institute of Jet Propulsion” of the “University of the German Armed Forces Munich”.
I am truly grateful to Prof. Dr.-Ing. Leonhard Fottner, head of the “Institute of Jet Propulsion” until June 2002, because he gave me the opportunity to carry out this work. I will never forget his professional competence, guidance and support as well as the familiar atmosphere that he established as chief professor at the Institute. He unexpectedly passed away on June 21, 2002.
Prof. Dr. rer. nat. Michael Pfitzner, head of the “Institute of Thermodynamics” of the “University of the German Armed Forces Munich” is gratefully acknowledged for taking over the supervision of the present work after the death of Prof. Fottner. The beneficial suggestions of Prof. Pfitzner are thankfully acknowledged. I would like to thank Prof. Dr.-Ing. Francesco Martelli, chief of the “Turbomachinery Energy and Environment Group” of the Department of Energetics “Sergio Stecco” of the University of Florence, who kindly agreed to be part of the board of examiners of the present thesis. His valuable advices are thankfully acknowledged. Prof. Dr. Sc. Math. Kurt Marti is kindly acknowledged for being the chairman of the examination board of this thesis. Dr.-Ing. Michael Lötzerich of ALSTOM Power, is thankfully acknowledged for the valuable discussions and his precious suggestions during the development of the present work.
I would like to thank all the colleagues of the Institute of Jet Propulsion for the many interesting discussions (not only) on turbomachinery and for the years spent together in a motivating and friendly atmosphere. In particular I would like to thank Dr.-Ing. Peter Müller for the detailed work of revision and for the precious suggestion, which contributed to the successful conclusion of this work. For their support I would like to thank all the students who worked intensively within their thesis at the development of the present work.
The present investigations were mainly performed within the German research project “CO2 armes Kraftwerk. 500 MW auf einer Welle” of the research programme “AGTURBO II”. The funding by the German Federal Ministry of Economics and Labour (BMWA) and ALSTOM Power is gratefully acknowledged. For supporting the printing of this work (see Cardamone, 2006) the University of the German Armed Forces Munich is thankfully acknowledged as well.
Last but not least I would like to thank my family: my parents, who encouraged my choice to do this experience in Germany and my wife Katrin for her encouragement and her patience also in difficult moments and for being always at my side during these beautiful years spent in Munich. I am sure that without her support, this work would not have been possible.
eM [m] Distance of the wake traversing plane from the cascade exit plane
E [-] Internal energy
ε [m2/s3] Dissipation rate
f Generic function
ft Auxiliary function for the transition model
F Objective function
g Probability density (generating function of ASA-algorithms)
γ [°] blade wedge angle
h Probability density (acceptance function of ASA-algorithms)
H12 [-] Form factor ( )1 2δ δ=
l, L [m] Chord
κ [-] Isentropic exponent
k [-]; [m2/s2] Profile curvature; Turbulent Kinetic Energy
VI Nomenclature
Ma [-] Mach Number
ν [m2/s] Kinematical viscosity
νT [m2/s] Eddy viscosity
p [Pa] Pressure
p Probability
r [m] Radius
ρ [-]; [kg/m3] Aspect ratio of the profile curvature; Density
Re [-] Reynolds number
σ [-] Aspect ratio of the profile slope
t [m] Pitch
T [K]; [-] Temperature; ASA algorithm parameter “temperature”
Tu [-] Turbulence intensity
U [m/s] Flow velocity
x Generic design variable
X Design variables vector
y+ [-] Dimensionless wall distance
w [°]; [m/s] blade tangent angle; flow velocity
ω [s-1]; [s-1] Specific dissipation rate; Vorticity
ζ [-] Total pressure loss coefficient
Subscripts, Superscripts
* Sonic state
1 Inlet
2 Exit
ax Axial
crit Critical
E Boundary layer edge
hk, H Trailing edge
i Generic quantity
Nomenclature VII
Infl Inflection
Is Isentropic
K Tank (Kammer)
met Metal angle
N Nose
r clockwise direction on the blade
Ref Reference
s, SG Stagger
t Total
th theoretic
u local tangential position in the wake measurement plane
Umg Environment (Umgebung)
v counter-clockwise direction on the blade
Abbreviations
AG TURBO German National Research Association on Turbomachines
(Arbeitsgemeinschaft Turbomaschinen)
ANN Artificial Neural Network
ASA Adaptive Simulated Annealing
AVDR Axial Velocity Density Ratio (ρ2w2sinβ2/ρ1w1sinβ1)
BMWA Federal Ministry of Economics and Labour (Bundesministerium für
Wirtschaft und Arbeit)
CFD Computational Fluid Dynamics
CTA Constant Temperature Anemometry
DLR German Aerospace Center (Deutsches Zentrum für Luft- und
Raumfahrt)
DM Turbulence model destruction term by Menter
HFA Hot Film Anemometer
HGK High Speed Cascade Wind Tunnel (Hochgeschwindigkeits-
Gitterwindkanal)
VIII Nomenclature
HP High Pressure
IEA International Energy Agency
FSTI Free Stream Turbulence Intensity
GA Genetic Algorithm
LE Leading Edge
MIGA Multi Island Genetic Algorithm
MUSCL Monotone Upstream Scheme for Conservation Laws
NAG Numerical Algorithm Group
OEM Original Equipment Manufacturer
PS Pressure Side
RAM Reliability, Availability, Maintainability
RANS Reynolds Averaged Navier Stokes
RMS Root Mean Square
SA Simulated Annealing, Spalart-Allmaras turbulence model
SA2 Two layers version of the Spalart-Allmaras turbulence model
SAL Low Reynolds version of the Spalart-Allmaras turbulence model
SKE Secondary Kinetic Energy
SQP Sequential Quadratic Programming
SS Suction Side
TBC Thermal Barrier Coating
TE Trailing Edge
TRACE Turbomachinery Research Aerodynamic Computational Environment
TVD Total Variation Diminishing
UniBwM University of the German Armed Forces Munich (Universität der
Bundeswehr München)
VKI Von Karman Institute
WINPANDA Windows Software for the automatic measurement and evaluation of
the cascade wake and profile pressure distribution (WINdows
Programm zur Automatisierung von Nachlauf- und
Druckverteilungsmessungen inkl. Auswertung)
Nomenclature IX
WINSMASH Windows Software for the measurement and evaluation of hot-wire
anemometry signals (WINdows Software zur Messung und Auswertung
von Signalen der Heißfühler-Anemometrie)
X Abstract
Aerodynamic Optimisation of Highly Loaded Turbine Cascade Blades for Heavy Duty Gas Turbine Applications
Abstract
The present work deals with the development and validation of a method for the automatic
aerodynamic optimisation of turbine cascade blades for high pressure stages of heavy duty
gas turbines. This class of profiles features aerodynamic and geometric properties which can
strongly depart from typical conditions of turbine profiles for aero engines applications. In
fact, the Reynolds number and the trailing edge thickness of these profiles can be an order of
magnitude higher than the corresponding values of aeronautical gas turbines. In order to gain
better insight into these major differences, extensive experimental investigations were
performed at the High Speed Cascade Wind Tunnel of the University of the German Armed
Forces Munich on various turbine cascade blades designed by ALSTOM. These reference
profiles feature characteristics typical for high pressure turbine blades for heavy duty gas
turbines. The experimental results furnish an exhaustive database for the validation of the
flow solver applied within the developed design method. Furthermore, a comparison of the
optimisation results and the reference turbine cascades attests the high potential of the newly
developed procedure for the aerodynamic design of highly loaded turbine cascade blades.
The developed tool is conceived for the application in an industrial framework and design
time scales compatible with industrial requirements have to be considered as well. In this
context a method consisting of a two-dimensional RANS flow simulation approach combined
with a parametric geometry generator and an optimisation algorithm is proposed. For the
simulation of the turbine cascade flow a quasi three dimensional version of the Navier-Stokes
solver TRACE from the DLR in Cologne is applied. The parametrical representation of the
turbine profiles is realised using the blade geometry generator PROGEN, which is a tool
applied successfully for industrial blade design today. Various stochastic global optimisation
techniques were tested. The Adaptive Simulated Annealing algorithm demonstrated best
properties for a detailed investigation of wide parameter ranges in reduced timeframes.
Furthermore, this optimisation algorithm showed best capabilities in handling highly non-
linear objective functions, like the scalar objective function used within the present
investigations.
The main optimisation target in this work was the reduction of the cascade total pressure
losses by imposing a fixed operating point. Additional requirements on the profile pressure
distribution were introduced as well in order to allow optimal conditions for an efficient
cooling of the blade. This is a fundamental aspect for the generation of optimal blade profiles
which are of relevance for practical applications. In fact, a major goal of the present work was
Abstract XI
the development of an aerodynamic design method which does not merely optimise the
location of the transition zone on the blade suction surface, but also ensures profile velocity
distributions satisfying major aerodynamic requirements for the optimal cooling of the blade
(e.g. smooth acceleration on the suction and pressure surface). All these requirements were
integrated in a single value objective function. The form of the various components of the
scalar function was tailored ad hoc in extensive preliminary studies. Furthermore, some major
mechanic and geometric constraints were specified in order to restrict the search to a sub set
of realistic geometries. In this way the optimisation task was reduced to a single-objective,
constrained approach.
The results of the proposed numerical design system indicate that the present method is able
to generate automatically blade geometries with reduced losses and featuring profile velocity
distributions which ensure favourable conditions for the cooling of the blade. The reliability
of the method at changed geometric and mechanical boundary conditions was demonstrated
as well. In fact this is an aspect of major importance considering that the aerodynamic design
method has to be integrated into a more complex design process where various disciplines
with contrasting aims interact and modifications to the basic mechanical and geometrical
blade constraints often occur during an iterative blade design process.
1. Introduction 1
1. Introduction
Today the earth’s global warming is an undeniable phenomenon, confirmed by studies of
several independent scientific organisations. Over the last 140 years the global average earth
surface temperature has increased by about half a degree, as illustrated below in Figure 1.1.
Figure 1.1 Combined annual land-surface, air and sea-surface temperature anomalies (°C)
1861 to 2000, relative to 1961 to 1990 (Folland et al., 2001)
At present there is stronger evidence that most of the warming observed over the last 50 years
is of anthropogenic nature (Folland et al., 2001), deriving from increased emissions of
greenhouse gases.
This evidence has launched major intergovernmental efforts over the last two decades in order
to address the problem. At the beginning of the nineties the United Nations General Assembly
initiated negotiations leading to a framework convention on climate change. The convention
was opened at the UN Conference on Environment and Development, the so called “Earth-
Summit”, held at Rio de Janeiro, Brazil, in June 1992. The parties of the convention adopted a
protocol in December 1997, during a session held in Kyoto, Japan. This protocol outlines
clear objectives to limit the concentration of greenhouse gases in the atmosphere
(United Nations, 1997). The key point of the protocol is the reduction of the anthropogenic
carbon dioxide equivalent emissions of six major greenhouse gases1 to at least 5 per cent
below the 1990 level in the commitment period between 2008 and 2012. This goal represents
a worldwide challenge, that can only be met by new policies and approaches for a common
technological innovation. 1 The six gases addressed by the Kyoto protocol are carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), hydrofluorocarbons (HFCs), perfluorocarbons PFCs and sulphur hexafluoride (SF6)
2 1. Introduction
At the moment only two large major industrialised countries like the United States and
Australia, which together account for over one third of the greenhouse gases emitted by the
industrialised world, have not yet ratified the protocol. In November 2004, however, the
ratification of the treaty by the Russian Federation opened the door for the protocol to enter
into force in February 2005, thus becoming legally binding for its actual 128 parties.
The European union as a group has to reduce its equivalent CO2 emissions by 8% in the
commitment period. Internal European agreements establish specific national rates and a
Kyoto target of 21% for Germany represents one of the most challenging worldwide.
Germany, as the only country in the EU apart from the United Kingdom, can point to
considerable success in reducing emissions. In the period from 1990 to 2000 a reduction of
19% was achieved (BMWA, 2003).
Figure 1.2 World electricity generation (IEA, 2002)
In spite of these efforts, changing global boundary conditions have to be taken into account.
During the last two decades about three quarters of the anthropogenic carbon dioxide
emissions to the atmosphere were due to fossil fuel burning (Folland et al., 2001). At present
about 64% of the worldwide energy production arises from fossil fuelled power plants
(BMWA, 2003). For the first two decades of this century the International Energy Agency
IEA estimates an increase of the world demand for electricity by more than 70% (see
Figure 1.2). According to the IEA estimations, the use of conventional energy resources like
gas, oil and coal will remain constant or even increase over the examined period. The limited
increase in the use of renewable energy sources (wind, biomass, geothermal power, solar
energy etc.) is associated with high electricity generation costs. Furthermore, at present there
are still strong locality and time restrictions on the supply technologies associated with
renewable energy sources (BMWA, 2003).
1. Introduction 3
Other factors like the estimated increase in the dependence of Europe from energy imports
and, for countries like Germany, the planned nuclear power phase out must be considered as
well. In the actual political scenario, the high fluctuations in the gas and oil prices can assume
a principal role for the electricity generation costs. The large German reserves of lignite are of
particular significance for the development of advanced coal-fired power plants.
This background points out the importance of extensive research activities for the reduction of
CO2 emissions of thermal power plant facilities. Thus major efforts are being made in the
development of the so called zero-emissions fossil fuelled power plants2. This concept can
only be achieved by the introduction of CO2 capture/storage technologies together with
increased energy conversion efficiencies. In Germany extensive research programmes have
been started and successfully carried out in the last decade in order to meet this challenge. The
present investigations were performed within a sub-research-project of the German national
cooperative project “CO2-armes Kraftwerk. 500 MW auf einer Welle” (AG Turbo Phase II).
Over time various innovative technologies can be listed, which allow major improvements in
the energy conversion efficiency. Among these the so-called hybrid-processes, consisting of a
combination of high temperature fuel cells and combined gas cycle power plants seem to be
the most promising today. Combined cycles on coal basis like pressurised fluidised-bed
combustion with partial gasification, externally fired combined cycles, pressurised pulverised
combustion and integrated gasification combined cycles are being currently tested and appear
to be a possible solution for the medium term. In a short period perspective, however,
substantial progresses can be only achieved by improving existing advanced technologies like
steam and gas combined power plants. In the nineties, in fact, extensive research efforts have
been performed in this field and great advances have been achieved already. Moreover the
components used within steam and gas combined cycle power plants present reliability,
availability and maintainability (RAM) which are in line with the market expectations for
energy production (Steel et al., 2004).
Today combined gas cycle processes show efficiencies slightly below 60%. By 2010 an
efficiency increase of the combined gas and steam turbine power plants to 62% is expected
(BMWA, 2003). This is a very challenging task which requires significant multidisciplinary
efforts both in the fluid mechanics research field and in the field of material technology.
Major improvement potential for increasing the efficiency is to be found in the gas turbine.
One of the key factors is the turbine inlet temperature. Figure 1.3 presents the development
trend of the maximum turbine inlet temperature as reported today by a leading original
equipment manufacturer (OEM) like Siemens Power Generation (Thien et al., 2004).
2 This denotation indicates power plants which release less than 0,1 kgCO2/kWh in the atmosphere by applying CO2 retention measures (BMWA, 2003)
4 1. Introduction
Figure 1.3 Development trend of the firing and material temperature over the last decades
in heavy duty gas turbines (Thien et al., 2004)
This diagram shows that firing temperatures higher than 1400°C are only possible through the
combined use of film cooling and thermal barrier coatings. Today flame temperatures above
1550°C are strived, but this challenge can be met only by developing materials with near gas
temperature capabilities. Additionally it must be mentioned that today typical life time
expectations for industrial gas turbines are higher than 100,000 hours (Madfeld, 2004). This
requirement limits the maximal admissible temperature in comparison to military or even civil
aero engine applications. Therefore a successful market placement of heavy duty gas turbines
requires advanced high temperature materials (nickel based alloys, nickel aluminides, ceramic
matrix composites) and optimised casting and machining processes (single crystal blades) for
the front turbine stages. Furthermore, protective coatings with improved temperature
capabilities and reliable life time prediction models have to be developed. All this has to be
done together with the development of more efficient cooling techniques. An idea of the
complexity of the technologies applied is shown in Figure 1.4, where a schematic
representation of the cooling system of a modern gas turbine TBC-coated vane and blade for a
heavy duty gas turbine is illustrated.
On the other hand it must be kept in mind that such an approach is associated with higher
manufacturing and operating costs. Therefore an obvious solution for decreasing the
manufacturing costs seems to be the reduction of the number of expensive parts in the high
pressure turbine. At the same time the resulting reduced wetted surface allows a saving of
compressed air flow for the cooling of the turbine components. Furthermore the necessary
high lift design approach has to deal with different challenges, like stronger shock structures,
higher secondary flow losses, higher tip clearance losses and higher profile diffusion
gradients. These undesirable effects, which are also negative with regard to the heat transfer
taking place on the profile, can be kept under control only through a better knowledge of the
1. Introduction 5
aerodynamic behaviour of the components operating at conditions typical for industrial gas
turbines. The challenge of a high lift blade/stage design methodology is confirmed by the fact
that at present important OEMs (IPG, 2004) prefer more conservative approaches for the
upgrade of modern industrial gas turbines even if higher reliability is obtained with a certain
loss of performance.
Figure 1.4 Cooling system for the first turbine row vane and blade of the
MITSUBISHI M501G gas turbine (Tsukagoshi et al., 2004)
A fundamental condition for the development of highly loaded turbine components is the
existence of appropriate design methods supported by reliable design tools which have been
extensively validated with experimental data. However, while over the last decade various
research activities have been successfully carried out with the aim of the development of
high-lift and ultra-high lift blading concepts for the application in low pressure turbines of gas
turbine engines, as described by Haselbach et al. (2001), there is still a certain information
deficit in the area of the components for modern heavy duty gas turbines. In fact, the
mechanical, geometrical and aerodynamic boundary conditions of heavy duty gas turbine
blade profiles differ considerably from those encountered in gas turbine aero engines. The
trailing edge thickness can be an order of magnitude higher than typical values encountered in
gas turbine aero engines. The operating Reynolds numbers are about ten times higher than the
Reynolds numbers of gas turbine for aeronautical applications and the resulting thinner
6 1. Introduction
boundary layers must be considered in appropriate form within the aerodynamic design
process. In order to address these major differences extensive experimental investigations
have to be carried out at typical boundary conditions of industrial gas turbines.
In this complex technological field automatic design procedures are gaining more and more
relevance, as they offer a high potential for the reduction of the design time and can
consequently contribute to a significant reduction of the design costs. A fundamental
condition that automatic design procedures have to fulfil is the possibility to react rapidly to
boundary condition changes deriving from other disciplines by adapting the design without
modifying the performance level. Various works in the literature give evidence of the huge
potential gains offered by automatic methods for the aerodynamic design of innovative, non-
standardised compressor and turbine blade profiles (Köller et al., 2000). However, an
essential requirement for the application of these procedures is a detailed validation work
supported by experimental data representative for realistic turbo machinery Mach and
Reynolds numbers. Furthermore, it must be kept in mind that in spite of the dramatic
evolution progress undergone by the computational resources over the last decades, today a
fully three dimensional aerodynamic optimisation analysis cannot be performed within
computational times reasonable for industrial applications yet.
The present work deals with the development and application of an automatic design
procedure for the aerodynamic optimisation of two-dimensional turbine blade profiles. This
method consists fundamentally of a Navier-Stokes solver (TRACE, DLR in Cologne), a
parametric geometry generator and an optimisation algorithm. The whole procedure was set
up within the commercial software package iSIGHT. This enabled a comparison between
different optimisation techniques and reduced the programming efforts for interfacing the
various components. The validation of the design method represents a fundamental point for
the successful application of the procedure itself. Thereby extensive experimental
investigations were performed on different turbine cascade blades representative for front
stage profiles of industrial gas turbines. The experiments were carried out in the High Speed
Cascade Wind Tunnel of the University of the German Armed Forces Munich. This allowed
the reproduction of similar operating conditions typical for turbo machinery and in particular
high Reynolds numbers, which were indispensable for the representation of this class of
profiles. The experimental data gave the opportunity to quantify the aerodynamic behaviour
of different blade design strategies at such operating conditions. The database was then used
both for the validation of the RANS solver TRACE and of the newly developed aerodynamic
optimisation method. A fundamental aspect distinguishing the present design procedure from
conventional two-dimensional optimisation methods is the direct control of the profile
velocity distribution. This major feature was introduced in order to ensure velocity
distributions on the optimised profiles featuring optimal characteristics for an efficient
cooling of the blade.
2. Scientific background and motivation 7
2. Scientific background and motivation
Today the components of modern gas turbines feature high aerodynamic efficiencies, so that a
further improvement represents a very challenging task for the turbo machinery aerodynamics
designer. The introduction of multidisciplinary aspects and of a three dimensional way of
thinking in the design process of the blading is fundamental for better engineered and more
clearly understood components. Moreover advanced methods are required which permit
higher levels of automation within the design process in order to reduce the development time
and costs. Over the last decades, the extraordinary development and improvement of
computing resources has been offering an ideal terrain for the design process to develop in
this direction. The use of Computational Fluid Dynamics (CFD) has been gaining even more
importance and covers a fundamental part within the aerodynamic design process today. A
fundamental aspect for the successful application of Navier-Stokes codes within the
aerodynamic blade design process is an extensive validation work based on experimental data
for operating conditions and aerodynamic loadings similar to those encountered in modern
turbo machinery blades. Moreover, the development of advanced optimisation techniques
handling large numbers of design parameters and eventually contrasting objectives could be
observed and a large amount of applications of these methods can be found in the literature.
The invention of splines in the 1960s followed by the introduction of more advanced
geometrical description systems based on spline- and Bezier-curves in the 1980s should not
be underestimated as this opened the way for the use of non conventional profile forms for
gas turbine blades.
On the other hand it must be kept in mind that in spite of the encouraging progress made in
the area of automatic aerodynamic optimisation of turbomachinery blade profiles, the
designer still plays a central role at present. Today, in fact the actual computational resources
do not permit the industrial use of three dimensional automatic methods for the aerodynamic
design of turbo machinery blades. This is due to the still high computational costs required by
three dimensional Navier-Stokes flow calculations, which restrict strongly the search space
for the optimisation and reduce the maximum number of design parameters.
The aim of the present chapter is firstly to give a theoretical background about the boundary
layer development in turbo machinery blades. In fact the knowledge of the boundary layer
development is a fundamental aspect for the design of turbine blade profiles with reduced
losses. Furthermore reliable numerical tools predicting with a reasonable accuracy the
transition phenomena and the turbulent boundary layer are the backbone for an accurate
assessment of the results obtained within the optimisation cycle. The question of the optimal
profile velocity distribution on high pressure turbine blade profiles for heavy duty gas turbine
applications and the question of the optimal turbine blade spacing will be discussed as well. In
fact, the former aspect is strictly associated to both the boundary layer development and to an
8 2. Scientific background and motivation
efficient blade cooling. The choice of the blade spacing is of major importance for the
improvement of the aerodynamic design of gas turbines components, featuring a reduced
number of parts. The blade spacing influences the profile Mach number distribution as well.
After the discussion of these fundamental physical aspects, a survey of recent progresses in
the area of automatic optimisation methods for aerodynamic blading design is illustrated. The
application of these procedures within the industrial design process is discussed as well.
Finally, a short overview about conventional aerodynamic design systems is given for a better
identification of the potentials offered by automatic methods for increasing the efficiency of
the design process.
2.1 Boundary layer development on turbo machinery blade profiles
Over the last few years the large amount of experiments performed on full- and large scale
compressor and turbine test facilities supported by cascade tests together with the
development of new high frequency measurement techniques and the introduction of
innovative calculation methods have produced extensive data for a better understanding of the
boundary layer behaviour in gas turbine engines. The state of the boundary layer influences in
a major way the loss development as well as the heat transfer to the gas turbine components.
Therefore, the detailed knowledge of the transition mechanisms and the distribution of
laminar and turbulent flow regimes over the blade profile is of fundamental importance for
the designer and the improvement of the aerodynamic design can only be achieved using
numerical solvers which ensure an accurate prediction of the profile boundary layer
behaviour.
The concept of a viscous boundary layer was first introduced by Prandtl (1904). When a real
fluid flows past a solid wall, the influence of the viscosity is confined to a relatively thin layer
in the immediate neighbourhood of the wall, called the boundary layer. At the wall the flow
velocity is zero in all directions (no slip condition) and it is the frictional, or viscous forces
acting within the boundary layer which reduce the fluid velocity from its free-stream value to
zero. The boundary layer flow can either be laminar or turbulent. In a laminar boundary layer,
fluid layers parallel to the blade surface slide over each other. The local fluctuations are
sufficiently damped so that they have negligible influence on the smoothness of the overall
flow. The initial portion of the boundary layer on turbine cascade blades is always laminar.
When increasing the Reynolds number the momentum exchange between the flow layers
increases and the laminar flow breaks down into a turbulent flow. Turbulent motion consists
of rapid random fluctuations which are superimposed on the mean motion. The zone over
which this change takes place is called transition region. The turbulent fluctuations cause high
momentum flow to be transported from the outer part of the flow field towards the wall. This
causes turbulent boundary layers to feature higher velocity gradients near the wall than
laminar boundary layers. The increased fluid exchange near the wall is associated with higher
2. Scientific background and motivation 9
wall shear stresses, higher friction losses and higher heat transfer coefficients (three to five
times higher) for a turbulent boundary layer in comparison to a laminar boundary layer. On
the other hand, the properties of a turbulent boundary layer lead to an increased stability and
to a higher diffusion capability, so that in a region of adverse pressure gradients turbulent
boundary layers have a lower tendency to separate than laminar boundary layers.
Figure 2.1 Reynolds effects on the boundary layer development on turbine cascade blades
(Hourmouziadis, 1989)
Figure 2.1 presents the qualitative distribution of the profile losses for the suction side of
typical turbine blades versus the Reynolds number as given by Hourmouziadis (1989). The
lower curve in the diagram (curve no. 2) represents the shear layer losses. At high Reynolds
numbers (configuration “a” in the figure) a turbulent boundary layer forms on the suction side
near the leading edge and turbulent separation at the trailing edge may occur, producing
further mixing losses (curve no. 3). Reducing the Reynolds number the transition region
moves further downstream on the profile suction side and the turbulent separation near the
trailing edge disappears (see configuration “b”). A further reduction of the Reynolds number
generates a short laminar separation bubble on the suction side (see configuration “c”). In this
case the losses are generated in the wall shear layer and the wake of the profile trailing edge.
Proceeding further to the left of the diagram, the transition region moves so far downstream
that no turbulent reattachment of the boundary layer is possible and the major part of the
losses is generated by mixing effects in the wake (curve no. 1).
The transition behaviour is affected by various parameters. The most important parameters are
Reynolds number, pressure gradient on the blade surface, free stream turbulence level, surface
curvature, surface roughness, temperature gradient between wall and fluid flow and the
history of these parameters as well. In real turbo machinery the transition position and
10 2. Scientific background and motivation
character is influenced by further parameters like steady and unsteady inhomogeneity of the
inlet flow, film cooling, acoustic disturbances and wall surface vibrations.
Mayle (1991), in an excellent review publication about transition in components of turbo
machines, distinguishes between three principal modes of transition: natural, bypass and
separated-flow transition. The development of the boundary layer and the transition process in
through-flow components of gas turbine engines, however, are stochastic, three dimensional
and unsteady phenomena so that they can hardly be described using a steady, two dimensional
approximation of this kind. On the other hand, only time-averaged loss and heat load
distributions are of relevance for the designer in the light of the actual computational
resources. The laminar-turbulent transition can take place in a reverse direction too, as usually
found on the pressure side of turbine blades, where the main flow accelerates under very high
velocity gradients. Furthermore, in gas turbine engines the transition may occur through
different transition modes at different locations at the same time on the same surface under
the influence of periodic unsteady wakes generated by an upstream blade row. This kind of
transition is indicated in the literature as periodic-unsteady transition (Mayle, 1991).
The natural transition process occurs in three main stages. In an initial phase the laminar
boundary layer is susceptible to small disturbances, a critical value of the momentum
thickness Reynolds number is reached and as consequence two dimensional Tollmien-
Schlichting waves develop (Schlichting et al., 1997). These instabilities grow and develop
into three dimensional highly fluctuating structures. Finally these structures develop into
turbulent spots which convect downstream into the transition region and merge together to
form the turbulent boundary layer. This mode of transition is of high relevance for external
aerodynamic applications, where low free stream turbulence levels are found.
For transition at free stream turbulence levels higher than 5-10 percent, like those encountered
in gas turbine engines, the first and second stage of the natural transition are completely
bypassed. This mode of transition, consisting exclusively of the production, growth and
convection of the turbulent spots is known in the literature as bypass transition.
Emmons (1951) developed a theory, introducing the concept of intermittency. This concept is
associated with the alternating appearance of turbulent spots within the transition region.
According to this idea, the transition zone is composed exclusively of fully turbulent spots
and laminar flow regions. In the transition zone the boundary layer presents therefore an
intermittent character, depending on the occurrence of laminar or turbulent regions of the
flow.
In adverse pressure gradients, if the laminar boundary layer separates, transition may take
place in a shear layer over the separation bubble. Since a turbulent shear layer has much
higher diffusion capability than a laminar one, the flow reattaches usually shortly after
transition. This mode of transition is known as separated-flow transition. For turbine blade
profiles of gas turbine engines a laminar separation bubble may occur near the leading edge
2. Scientific background and motivation 11
on the suction or pressure side or at the location on the suction side where the pressure
minimum is found. The separated flow transition is a very effective method to force the
development of a turbulent boundary layer. This method has been widely used for the design
of compressor blades and of low pressure turbine profiles (Hourmouziadis, 1989). Thereby
the idea consists in delaying the transition further downstream on the suction side through
continuous acceleration up to the pressure minimum, where transition takes place over a short
separation bubble. The challenge for this design task is the ability to predict the extension of
the bubble within the design phase, thus ensuring that a turbulent boundary layer reattaches in
the rear part of the suction side. If this does not happen, then the losses increase dramatically
(see Figure 2.1) and the prescribed flow deflection within the passage is not realised.
Figure 2.2 Topology of the different modes of transition in a Reynolds number,
acceleration plane (adapted from Mayle, 1991)
The different character of the fundamental modes of transition is shown in Figure 2.2. This
diagram presents the zones in which each mode of transition is expected to occur. Thereby an
acceleration parameter-momentum thickness Reynolds number plane is used. The
acceleration parameter accounts for the effects of free-stream acceleration and thus of the
profile velocity distribution on the boundary layer. This parameter is defined as
(ν/U2)/(dU/dx), while the momentum thickness Reynolds number is defined as (Uδ2/ν), where
U is the free-stream flow velocity, x is the surface coordinate in streamwise direction, ν the
kinematic viscosity and δ2 the boundary layer momentum thickness. The acceleration
parameter and the momentum thickness represented in the diagram are intended as the values
at the beginning of the transition zone. The curves at constant turbulence level represent the
momentum thickness Reynolds numbers for which transition takes place at the corresponding
value of the acceleration parameter for the present turbulence level. Natural transition can be
found only at low turbulence levels for negative acceleration parameters, which are associated
12 2. Scientific background and motivation
with adverse pressure gradients. The line indicated as “Stability criterion” represents in fact
the boundary above which Tollmien-Schlichting waves are possible. For reduced acceleration
parameters, as long as the turbulence level is not high enough, a separated-flow transition can
be encountered as well. The diagram shows that as long as the turbulence level is low,
increased acceleration parameters are associated with higher momentum thickness Reynolds
numbers at the beginning of transition. At higher turbulence levels, typically encountered in
turbomachinery components, the acceleration parameter has low influence on the momentum
thickness Reynolds number and thus the dependence of transition on the velocity distribution
on the profile is reduced.
Two further modes of transition are known from the literature. The first one indicated as
periodic-unsteady transition and also known as wake-induced transition
(Mayle et al., 1989), is caused by the periodic impingement of the wakes of upstream
aerofoils on the blade surface. As different modes of transition coexist at the same time on the
same surface this process is also called multimode transition. Over the last decade extensive
experimental investigations have been performed on compressor and turbine blade profiles in
appropriate test facilities (Tiedemann, 1998) and cascade wind tunnels (Schulte, 1995 and
Stadtmüller, 2001) for a better understanding of this complex phenomenon. A detailed
overview about blade row interference features and their effect on transition is given by
Hodson (1998). As in the present work steady aerodynamic investigations on turbine cascade
blades are carried out, this kind of transition will not be further considered.
A further mode of transition is represented by the transition process from turbulent to laminar
boundary layer, the so-called reverse transition or relaminarisation. It occurs in
turbomachinery components as well and is of particular importance for the gas turbine
designer, since it usually takes place on the pressure side of most profiles near the trailing
edge and may occur on the suction side near the leading edge in presence of strong
acceleration gradients. Mayle (1991) explains this process as the stretching of the streamwise
vortex lines associated with the turbulence in the boundary layer under the effect of a large
acceleration, so that the vorticity is dissipated through viscous effects. The relaminarisation
process is expected to occur at moderate turbulence levels if the acceleration parameter
exceeds a value of 3·106. This indirectly means that forward transition cannot take place as
long as the acceleration parameter does not fall below this value. The blade profiles class
examined within this work features reverse transition on the rear part of the pressure surface.
Some profiles feature reverse transition on the front part of the suction surface as well.
At the end of this section a short description of the possible boundary layer development on
high pressure turbine blades will be presented as described in the literature (Mayle, 1991).
Figure 2.3 presents the essential features of the profile boundary layer for this application on
the suction and on the pressure side. On the suction side it is usually expected that
downstream of an initial laminar part the boundary layer becomes turbulent (right side). The
2. Scientific background and motivation 13
length of the transition zone depends on whether the onset of transition is upstream or
downstream of the minimum pressure location. In the first case the transition zone will be
more extended. If a laminar separation bubble occurs in the front part of the suction side (left
side), then, in presence of extremely favourable pressure gradients, the boundary layer may
become laminar-like again and only marginally downstream a forward transition takes place.
Different authors show that reverse transition may occur on the suction surface (Hodson, 1984
and Warren et al., 1972). For film cooled blades the transition is expected at the injection
location. Downstream, however, a reverse transition process cannot be excluded. This
circumstance could strongly influence the heat transfer distribution on film cooled turbine
blades. On the profile pressure side two probable scenarios are illustrated as well. If a
separation bubble occurs the reattached turbulent boundary layer may become laminar like
again (right side). If no separation bubble is featured, a forward transition zone followed by a
reverse one, in the rear part of the profile, is expected (left side).
Figure 2.3 Boundary layer development on high pressure turbine blades (Mayle, 1991)
2.2 The choice of the optimal turbine blade profile velocity distribution
The form of the profile velocity distribution is essential for a satisfactory development of the
profile boundary layer and thus for the cascade loss and deflection as well as for the heat
transfer development on the blade. Moreover, various requirements on the profile pressure
distribution have to be taken into account in order to ensure optimal aerodynamic conditions
for the efficient action of the film cooling air on the blade profile. The form of the profile
velocity distribution is strictly dependent on the blade spacing. This parameter plays a
14 2. Scientific background and motivation
fundamental role within the design process since it influences directly the blade loading level
and the characteristics of the cooling as well.
Since the task of turbine cascades is to accelerate the flow from the inlet to the outlet, one
would imagine that an optimal turbine profile velocity distribution shows an ideal shape
without any deceleration region as presented in Figure 2.4. This distribution realises the
maximal enthalpy conversion. However, this distribution can not be realised not only because
it would lead, through the excessive length of the constant velocity region on the suction side,
to an unsatisfactory development of the boundary layer but also because it would require
blade shapes which do not satisfy major mechanical constraints.
Figure 2.4 Ideal velocity distribution for accelerating cascades (Hoheisel et al., 1987)
Figure 2.5 Mach number distribution for a subsonic turbine blade and boundary layer
development (Casey, 1994)
2. Scientific background and motivation 15
The essential features of the Mach number distribution on a turbine blade operating at
subsonic conditions are shown in Figure 2.5. The shape shown in this picture is a typical “aft-
loaded” profile velocity distribution. This means that the aerodynamic loading of the profile,
quantifiable through the area between the Mach number curves on the suction and the
pressure surface, is concentrated in the rear part of the blade. The flow on the suction surface
accelerates rapidly over the first third of the surface. This is followed by a more gradual
acceleration to the maximum velocity near to the passage throat and a subsequent deceleration
to the exit Mach number. A fully developed turbulent boundary layer is expected to form
slightly downstream of the blade passage throat. The deceleration and the amount of flow
turning in the rear part of the suction surface, which can be convex for subsonic applications,
can be limited by means of conventional diffusion criteria. The adverse velocity gradients in
the rear part of the suction surface contribute to the production of thicker boundary layers at
the trailing edge. When using transonic profiles the profile curvature beyond the throat should
be carefully controlled. On the pressure surface, instead, the flow accelerates almost
continuously from the leading to the trailing edge. The favourable pressure gradients lead to
the development of thin boundary layers which can eventually relaminarise. The small
diffusion region near the leading edge on the pressure surface, as a consequence of the change
of the blade curvature, is often found on turbine aerofoils and occurs on either the pressure or
suction surface or both.
In an excellent work about the influence of free stream turbulence and profile pressure
distribution on the boundary layer and loss development of turbine cascades,
Hoheisel et al., (1987) present a comparison between the front- and aft-loaded design
strategies, for low pressure turbine applications. The profile velocity distributions and blade
shapes of the three turbine profiles used within this study T104, T105 and T106 are shown in
Figure 2.6. The three profiles are designed for the same operating conditions (β1=127.7°,
Ma2th=0.59 and Re2th=500 000) and present the same aerodynamic loading. While T105 and
T106 feature an aft-loaded profile velocity distribution, the velocity maximum on the suction
side of T104 is shifted towards the leading edge as for a front loaded profile. The deceleration
taking place over the suction side of T104 is limited. T106 presents a lower peak velocity than
T105. This produces more limited pressure gradients in the rear part of the suction surface.
The pressure side distributions of T105 and T106 are almost identical. T104 features a
different velocity distribution on the pressure side. Since the boundary layer on the pressure
side is expected to be laminar for an extended region, the authors present the results of
detailed boundary layer investigations on the suction side. In this way a relation between
profile velocity distribution and boundary layer development as well as loss behaviour is
found.
16 2. Scientific background and motivation
Figure 2.6 Design velocity distributions and cascade geometries for the low pressure
turbine cascades T104, T105 and T106 (Hoheisel et al., 1987)
Figure 2.7 Effect of the deceleration factor wmax/w2th and Reynolds number Re2 on the
aerodynamic behaviour of the low pressure turbine cascades T104, T105 and
T106 (adapted from Hoheisel et al., 1987)
2. Scientific background and motivation 17
Figure 2.7 presents some of the results from these investigations. On the left hand side the
predicted influence of the suction side deceleration factor (defined as the ratio between the
suction side peak velocity wmax and exit isentropic velocity w2th) on the momentum thickness
at the suction side trailing edge is presented. The central diagram shows the effect of the
deceleration factor wmax/w2th on the cascade total pressure losses. In the right hand diagram the
effect of the Reynolds number Re2th on the boundary layer momentum thickness at the suction
side trailing edge is displayed. Both theory and experiments indicate that at the design
conditions and for free stream turbulence levels Tu1 higher than 0.8 %, an aft-loaded profile
velocity distribution with limited deceleration in the rear part of the suction side such as T106
leads to a lower momentum thickness at the suction side trailing edge and produces lower
losses. Furthermore, it must be mentioned that the front-loaded velocity profile featured by
T104 is associated with a boundary layer thickness lower than for the cascade T105. This lets
conclude that a front loaded design strategy like T104 cannot be excluded a priori within the
design process under a merely aerodynamic point of view. However, it must be pointed out
that both in the literature and in the practical design process there is a certain lack of
knowledge about the effectiveness of film cooling in presence of such velocity distributions.
This is the reason of the strong concerns of the designer to use front-loaded velocity
distributions for rotor blades and only extensive measurements in presence of film cooling
performed at realistic turbomachinery conditions can give a better insight (Lötzerich, 2004a).
A major advantage of a front loaded configuration is the possibility to reduce the maximal
suction side Mach number (see profile T104 in figure 2.6), which is of particular significance
for turbine vane applications, where higher blade spacing but lower deflection are featured
than for rotor blade applications. These considerations were integrated into the optimisation
process by providing the single value objective function with a supplementary term
accounting for the location of the maximal suction side Mach number with respect to the
passage throat. A comparison of the two formulations of the objective function is presented in
chapter 5.
All these considerations illustrate the key role of the profile velocity distribution for the
development of turbine blades with minimal losses. Hourmouziadis (1989) discusses the
optimal profile velocity distribution and specifies detailed rules for the aerodynamic design of
optimal turbine blades for low pressure turbine applications:
- Minimise boundary layer thickness at the trailing edge
- Minimise trailing edge thickness
- Avoid separation upstream of the trailing edge
- Select the highest pitch possible avoiding separation upstream of the trailing edge
- Select high acceleration on the pressure side upstream of the trailing edge
- Delay transition on the suction surface as far downstream as possible
18 2. Scientific background and motivation
- Use suction side acceleration to control transition
- Force suction side transition early enough to ensure reattachment
- Limit trailing edge diffusion to keep the flow attached
These rules apply for the design of high pressure turbine blades as well. However, some
additional aspects have to be considered. The reduction of the trailing edge thickness is not
imperative for these kinds of application for different reasons. First of all the necessary space
for the allocation of the cooling channels within the blades must be ensured. Moreover, an
excessive reduction of the trailing edge thickness leads to bigger exit passage areas and
consequently to reduced exit velocities. In this way the gradients between suction side peak
Mach number and exit Mach number become higher. This leads to increased back diffusion
and could make the cooling of the rear part of the blade more difficult. Moreover the losses
across a possible suction side profile shock increase, thus reducing the benefits of thinner
trailing edges. The use of aft-loaded profiles in high pressure turbine blades has benefits as
well, since the delay of transition on the suction side is associated with an increased extension
of the laminar boundary layer. However, this aspect should be realised ensuring adequate
acceleration gradients for an efficient action of the shower head cooling in the front part of the
profile suction side as well. Therefore, the acceleration to the peak suction side Mach number
should feature gradients high enough to ensure the cooling film to be uniformly distributed
over the surface. “Fuller” profile velocity distributions in this area are more beneficial for this
scope than distributions accelerating to the peak Mach number under low gradients.
Figure 2.8 Cooling effectiveness distribution from cascade tests (Yoshida et al., 1982)
Any kind of diffusion on the profile velocity distribution should be avoided, because it affects
the film cooling effectiveness negatively. A typical distribution of the film cooling
2. Scientific background and motivation 19
effectiveness on a turbine cascade blade is shown in Figure 2.8. The curves correspond to
different coolant mass flows. The distributions feature a rapid decrease of the cooling
effectiveness in the rear part of the suction side, independent on the coolant mass flow. The
diagram shows that as long as the flow accelerates, an increase of the cooling effectiveness is
achieved. Even though the best behaviour is observed at higher coolant mass flows, it must be
considered that the air mass flow used for cooling is limited by different factors. First of all it
does not experience the increase of enthalpy occurring in the combustion process and
therefore contributes only slightly to the enthalpy conversion in the turbine. Furthermore
aerodynamic losses due to mixing processes of cooling air and mainstream flow have to be
considered.
While for low pressure turbines in gas turbine aero engines excellent publications exist
(Hoheisel et al., 1987), dealing with the question of optimal profile velocity distributions and
where detailed rules are specified for the optimal design of these components
(Hourmouziadis, 1989), there is still a lack of information for high pressure turbine profiles in
heavy duty gas turbines. In fact, for these applications some geometrical and mechanical
properties of the profiles like leading and trailing edge thickness, due to cooling requirements,
assume values which cannot be classified as optimal from an aerodynamic point of view.
Furthermore, the boundary layer development at Reynolds numbers which are an order of
magnitude higher than in low pressure components of gas turbine aero engines have to be
investigated in more detail. Additional requirements associated with the need for an efficient
heat transfer rate and film cooling action on the blade surface also have to be taken into
account in the design process.
The choice of the blade spacing is another fundamental point within the design process of
turbine blades. Pioneer publications by Zweifel (1945) and some years later by
Ainley et al. (1951) specify rules for the choice of an optimal blade spacing, realising minimal
losses. In this context two main contrasting phenomena have to be considered. On the one
hand, a low blade spacing is associated with increasing profile losses due to high values of the
blade wetted surface. On the other hand, high blade spacing is associated with higher
aerodynamic loading which could lead to flow separation. Zweifel (1945) introduced an
aerodynamic loading coefficient, which is indicated in the literature with his name. The idea
behind this coefficient is to quantify the maximal opposite pressure gradient realisable within
the cascade. The Zweifel-loading-coefficient is defined as the ratio between the peripheral
force actually obtainable from the momentum theory, if the deflection diagram is given, and
the force obtainable from an ideal pressure distribution, featuring the value of the cascade
inlet total pressure on the pressure surface and the outlet static pressure on the suction surface.
Such a profile pressure distribution would be represented by a rectangle and it would feature
no deceleration region. The evaluation of various measurements on turbine cascades
(Scholz, 1977) indicates that for minimal drag-to-lift ratios this coefficient assumes a constant
20 2. Scientific background and motivation
value between 0.9 and 1.0 for accelerating profiles featuring high deflections. This empirical
loading limit has been widely used for the choice of the optimal spacing for turbine blades in
recent times as well. Ainley et al. (1951) presents a method for the determination of the
optimal blade spacing. The procedure is based on the interpolation of experimental data
obtained from tests performed on turbine blade profiles of “conventional” shape (e.g. RAF27
and C7 profile shapes) with circular and parabolic camber line, featuring thickness to chord
ratios between 10 and 25 percent. Figure 2.9 shows the profile-loss distribution versus the
blade spacing for different cascade deflections at a Reynolds number of 200 000 and a Mach
number lower than 0.5. The left hand diagram shows the family of curves obtained for
conventional nozzle blades, while on the right hand side a family of curves obtained from
conventional impulse blades are displayed. In order to obtain the profile losses for different
blade shapes the authors propose an interpolation formula using the results of nozzle and
impulse blades.
Figure 2.9 Profile-loss coefficients for conventional turbine blades (Ainley et al., 1951)
Obviously these experimental data cannot be used for the design of innovative turbine blade
profiles, since they are based on dated turbine profiles, featuring moderate aerodynamic
loading. Moreover, it must be considered that today a major trend in the gas turbine industry
is the design of high-lift blading. In fact, increasing the blade spacing and the deflection
reduces the number of parts and is thus associated with reduced manufacturing and
maintenance costs. This has particular benefits for high pressure turbine stages, where a
reduction of the number of blades means a reduction of the components for cooling and
therefore saves expensive compressed cooling air. Furthermore, the increase of the stage
loading is associated with higher temperature drops and the cooling of the following stages
can potentially be avoided. However, a remarkable increase of the stage loading can only be
achieved by introducing new design concepts (Haller et al., 2002) in order to address the
various challenges deriving from increased stage loading. An increase of the stage loading can
produce supersonic flow regions and therefore strong shock losses and increased trailing edge
2. Scientific background and motivation 21
losses. Therefore particular attention should be given to the curvature in the rear part of the
suction surface. Higher pressure gradients and higher deflections lead to the development of
more important secondary flow losses. This aspect is of particular importance for high
pressure turbine profiles, where low aspect ratio blades are found. Higher pressure ratios
across the stage increase tip leakage flows as well. When increasing the stage loading one has
to consider these challenges to ensure a high level of efficiency within the gas turbine.
Finally, it must be kept in mind that even if the existence of modern flow simulation tools can
reduce the experimental efforts and make the families of curves introduced by Ainley and
Mathieson or the Zweifel loading criteria superfluous, extensive experimental investigations
at typical heavy duty gas turbine operating conditions are needed in order to validate the
prediction tools applied.
2.3 Automation possibilities of the aerodynamic blade design process
In this section a short overview of the conventional aerodynamic design process of turbo
machinery blades is presented in order to facilitate a better understanding of possible
application area for automatic blade design methods. Even though in the last years three
dimensional analysis methods have been gaining more and more importance, the backbone of
turbo machinery aerodynamic design systems remains still the quasi-three dimensional
approach introduced by Wu (1952). This general scheme assumes that the three dimensional
flow field can be approximated using two families of intersecting stream surfaces, as shown
in Figure 2.10.
Figure 2.10 Representation of the S1-S2 stream surfaces in the scheme by Wu (1952)
The stream surfaces indicated as S1 in the figure are the so-called blade-to-blade surfaces.
The S2-surfaces are the through-flow surfaces. The present technique is based on independent
22 2. Scientific background and motivation
flow calculations on each of the surface families. However, additional linkage terms between
the two surfaces, such as blade forces, are considered as well. The scheme introduced by
Wu (1952) ensures a quasi three dimensional description of the flow field within the blade
rows. The scheme in its original form is based on the assumption that the flow develops axis-
symmetrically within the machine. Over the last 50 years various modifications to the original
scheme have been developed and implemented in through-flow codes in order to account for
the mixing of flow properties occurring within a real machine.
The schematic representation given in Figure 2.11 contains the main elements used within the
aerodynamic design process as given by Jennions (1994). Links to other disciplines are
presented in the flux diagram as well. At the beginning of the design process basic
relationships, one-dimensional stream line methods, existing experience charts (Smith,
Swindell) and the designer experience are used to obtain a one-dimensional description of the
machine. The aerodynamic requirements prescribed in this introductory phase are passed to
the next design step, where a meridional (S2) through-flow code is used to compute a radial
distribution of the main aerodynamic properties within each row. The resulting model will
represent the desired flow conditions throughout the turbine. Developing this model the
designer must decide which aerodynamic conditions he wants to achieve in each blade row.
Thus at this point it can be decided if new aerodynamic concepts will be considered. For
example if previous rig tests have shown that a particular axial distance between aerofoils
reduces the blade row interaction, this has to be considered at this stage of the design process.
Figure 2.11 Elements of the turbo machinery aerodynamic design process (Jennions, 1994)
As indicated in figure 2.11 the through-flow code can be supplemented by information about
spanwise mixing effects occurring in the machine in order to consider secondary flow
phenomena in the basic physical model (Jennions, 1994). The design step following the
2. Scientific background and motivation 23
through-flow analysis is an iterative one, also known in the literature as direct design
problem. In this case the blade geometry is iteratively prescribed and a flow simulation tool
predicts the resulting aerodynamic data until the specified aerodynamic conditions are
fulfilled. The objective of this step is therefore to find −for each streamline− a blade shape,
which fulfils the aerodynamic boundary conditions of the S2 calculation. As a further
requirement the specified flow turning must be fulfilled while the goal of low aerodynamic
losses is pursued. The blade-to-blade (S1) code predicts loading, exit angle and losses for a
given blade geometry. This part of the design process can be quite time-consuming. Therefore
the application of automated reliable design procedure is particularly interesting for this task,
because of its potential to significantly reduce time and cost of the design. Additionally, it
must be mentioned that, since blade blockages and flow asymmetries will affect the position
of the stream lines and the target design values, the through-flow model has to be modified
iteratively. The two-dimensional blade shapes obtained using this quasi-three dimensional
process (with or without spanwise model extensions) are then stacked to a three dimensional
blade form. At this point of the process a full three dimensional CFD analysis is usually
performed. The resulting information is passed eventually to the through-flow model in order
to update loss and turning throughout the blade rows. However, it must be pointed out that
three dimensional calculations even using today’s available computational resources are very
time-consuming. Therefore a large part of the design task is still concerned with a two-
dimensional approach.
More automation within the design process is thus fundamental to reduce the development
costs by allowing the engineer to react rapidly to changes in functional requirements of the
gas turbine components. In this context, the present work aims to the development of a design
procedure for the aerodynamic optimisation of two-dimensional turbine cascade blades.
Aerodynamic design optimisation is distinguished in the literature into inverse and direct
design. In the following a comparison of the main features of inverse and direct optimisation
strategies is given, in order to underline the reasons which led to the exclusion of a pure
inverse method for the present task.
While the direct design considers the blade shape optimisation striving for low losses, the
inverse design approach consists of prescribing the profile velocity distribution and searching
for a blade shape which satisfies this condition. The main benefit of inverse design methods is
the low computational cost required. Usually few iterations are needed to obtain convergence
(Büche et al., 2003). The computational cost of direct design optimisation instead is a
multiple of a single flow calculation and could be very time consuming. On the other hand,
for inverse design methods, an iteration process for the prescribed profile velocity distribution
is needed in order to obtain an acceptable blade shape. Demeulenaere (1997) shows the
benefits of the application of inverse design strategies for the improvement of existing blade
designs. The shape of a given profile pressure distribution is locally improved using different
24 2. Scientific background and motivation
compressor and turbine cascade blades. The author points out that when prescribing the
pressure distribution only on a limited part of the suction and/or the pressure surface the new
design usually meets the specified mechanical constraints. Furthermore, a degree of freedom
in the target has to be introduced, in order to control the trailing edge thickness and the
prescribed outlet flow angle. Shahpar (2000) points out that the question on how to integrate
the mechanical and geometrical constraints in an inverse design approach is still open. The
success of this methodology is therefore strongly associated with the designer experience and
does not represent an optimal solution for automating processes which have to be applied for
innovative design tasks.
On the other hand, direct design methods aim to optimise the blade shape with respect to the
objectives (e.g. aerodynamic losses) specified by the design task. Mechanical and geometrical
constraints can be considered in various ways within the process. An originally constrained
minimisation problem can be modified into an unconstrained one by transforming the
constraints into penalty terms. This procedure is somewhat diffused in the literature
(Demeulenaere et al., 2004) and used within the present work for the treatment of some
specific requirements on the profile velocity distribution. Furthermore, dedicated
mathematical publications (Dennis et al., 1996) point out that the attempt to solve constrained
problems is always reduced either to solve a related unconstrained problem or to finding a
non-linear system of equations whose solution is the same as that of the constrained problem.
The main disadvantage of direct design methods is that the computational cost is represented
by the number of simulations needed by the optimisation algorithm until an optimal blade
shape is found. Time-consuming simulations with a high degree of modelling detail should
therefore be avoided.
The method developed within this work presents the same structure as common aerodynamic
optimisation methods for turbomachinery components consisting of three main components:
flow solver, parametric geometry generator and optimisation algorithm. A detailed description
of the specific components used within the present work is given in Chapter 4. At this point,
however, it must be pointed out that while the reliability of the flow solver and the flexibility
of the parametric geometry generator for reproducing a wide range of blade geometries can be
well assessed in a preliminary phase, the success of the optimisation algorithm depends
strongly not only on the specific problem but also on the interaction between the various
components and the form of the objective function. Furthermore, eventual misleading
information deriving from problems like network interruptions (with consequent erroneous
evaluation of the actual blade geometry) or erroneous interpretation of the actual results
caused by the evaluation procedure itself3 have to be taken into account for the choice of the
optimisation technique. Therefore stable algorithms are needed which perform well in 3 Shahpar (2000) points out how optimisation techniques are particularly suited to detect “weak points” of the design procedure producing geometries which are handled in erroneous way by the evaluation tools. This implies an extensive preliminary work, in order to debug “online” the design method.
2. Scientific background and motivation 25
presence of highly non-linear objective functions and are less sensitive to occasional
misleading information occurring during the optimisation loop. Optimisation algorithms for a
direct design can be divided into two main classes: gradient-based methods and stochastic
algorithms. Gradient-based methods rely on derivative information of all objectives and all
constraints for determining the search direction of the optimisation. Stochastic optimisation
algorithms comprise techniques like genetic algorithms, evolution strategies and simulated
annealing methods. This second class of algorithms is characterised by excellent properties in
coupling with “noisy” objective functions, thus ensuring a high degree of stability in the
design process. However, the computational effort associated with stochastic algorithms is
usually higher than gradient-based methods. On the other hand, results from the literature
show that for a number of variables higher than 14 some gradient-based methods become as
expensive as genetic algorithms (Shahpar, 2000). In contrast to the gradient-based methods,
stochastic techniques avoid focusing on local regions of the search space, evaluating designs
throughout the parameter space in search of global optima. These major considerations led to
the application of stochastic techniques for the present investigations.
2.4 Recent progress in the field of the automatic aerodynamic blade design methods
Over the last decade the scientific community has been striving for the development and
usage of optimisation techniques within the aerodynamic design process of turbo machinery
components. Turbo machinery design is in fact a mature field where the efficiencies featured
by the single components are already quite high and improvements are hard to achieve. In
addition there is a need for the designer to rely on dependable automatic design tools. These
have to ensure a reliable reaction in ever decreasing time scales to changes in the design
objectives and constraints. An extensive validation work of the simulation tools applied
within the process is essential as well, to ensure that consistent results are achieved. Recently
a large number of scientific publications highlight the potential of automatic optimisation
methods within the turbo machinery blade design. Köller et al. (2000) apply an automatic
design process for the development of a new family of subsonic compressor aerofoils for
heavy-duty gas turbine applications. In this publication the inviscid/viscous Q3D flow solver
MISES (Giles, 1985 and Drela, 1986) is coupled together with a parametric geometry
generator and an optimisation approach based on the combination of a random search
algorithm and a gradient-based method. The authors point out that apart from the reliability of
the flow solver and the quality of the search algorithm, the formulation of the objective
function is a key aspect for a successful optimisation. They combine the physical targets and
the geometrical/mechanical constraints in a single function using weighting coefficients for
each objective. This procedure is the already discussed method for reducing an initial
constrained direct problem to an unconstrained one. The choice of the weighting coefficients
requires extensive preliminary investigations to ensure that the specified targets are
26 2. Scientific background and motivation
appropriately fulfilled. In their investigation Köller et al. (2000) use an ad hoc formulated
objective function which contains the following elements: cascade total pressure losses at
design conditions, extension of the cascade operating range and stall margin, loss level over
the whole operating range and different geometrical constraints not directly specified in the
publication. The new family of aerofoils features reduced total pressure losses and a
significant extension of the operating range. The resulting profile Mach number distribution is
more front-loaded (Figure 2.12). The authors demonstrate that for the high Reynolds numbers
and turbulence levels characteristic of heavy-duty operating conditions, conventional CDA
aerofoils no longer represent an optimal solution. They associate the movement of the profile
loading towards the leading edge with the high Reynolds number of 2.5·106 and the high inlet
turbulence intensity of 3.5%. These operating conditions force the transition near the leading
edge, as it is expected in heavy duty gas turbine compressors. As a turbulent boundary layer
forms on the suction side very close to the leading edge, the optimisation method tends to
move the diffusion zone near the leading edge as well, where a thinner turbulent boundary
layer is found. The results of the optimisation method are confirmed experimentally
(Küsters et al., 2000).
Figure 2.12 Mach number distribution and total pressure losses for starting (CDA) and
optimised compressor blade profile (Köller et al., 2000)
The high potential offered by a combination of a three dimensional transitional flow
simulation and a gradient-based method are demonstrated by Nagel et al. (2003). Here an
experimental verification of the obtained results is given as well. The approach provides the
basis for a fully three dimensional design control over the whole wetted surface of a blade
passage. Starting from a low loss, low pressure turbine vane with linear diverging side-walls
an innovative geometry is obtained featuring non-axis symmetric end walls and reduced
integral total pressure losses. The main target of these investigations was the reduction of the
integral total pressure losses and secondary flows by fixed cascade deflection. Using a set of
40 design parameters the system converges after about 550 RANS computations, which
2. Scientific background and motivation 27
correspond to a real time of about four days on six 200 MHz processors of an ORIGIN 2000.
The resulting geometry, which is shown on the left hand side of Figure 2.13, underlines the
high potential offered by optimisation methods for exploring non conventional design spaces.
The advantages of the method in terms of total pressure losses is shown on the right side of
figure 2.13 where the spanwise total pressure loss distributions of the starting geometry
(dotted line) and the optimised geometry (continuous line) are compared. A good agreement
with the experimental data (circular symbols) is achieved as well. The total pressure losses
resulting from the optimised geometry are reduced with respect to the starting geometry and
the loss core is shifted towards the endwall. Even though the results obtained with this
optimisation method are very promising, the authors point out that the computational cost of
the three dimensional flow simulation is quite high. To save CPU time a relatively coarse
mesh has to be used within the optimisation process. Therefore further criteria to reduce the
mesh effects on the solution and the reliability of the numerical results have to be considered.
At least the final and initial geometries have to be computed using more refined grids and the
possible changes in flow patterns and efficiency gains have to be considered as well. The high
computational cost of three dimensional direct optimisation hinders the usage of stochastic
optimisation algorithms, thus restricting the search space, and makes an implementation of the
method difficult within an industrial aerodynamic design process. In this work the authors do
not consider constraints on the form of the profile velocity distribution. The resulting profile
Mach number distributions at different blade heights are shown in Figure 2.14. First of all an
excellent agreement between experimental and numerical results can be observed. However,
the profile pressure distributions feature significant intermediate diffusion zones on the
suction surface over the entire blade height. This aspect conflicts with the requirements on
film cooled blade profile Mach number distributions discussed in the previous section. The
investigations presented by Nagel et al. (2003) address two key aspects, which are still open
topics in the area of aerodynamic blade design optimisation. Firstly a three dimensional flow
simulation approach, due to limited computational resources, is hard to couple with global
optimisation searches and is associated with very high computational costs. Secondly,
requirements to the profile velocity distribution for a future cooling of the optimised blade are
not directly considered within the process. This aspect in particular is not yet addressed by
most of the aerodynamic automated blade design systems presented in the literature.
Shahpar (2000) addresses the problem of the large computational effort associated with three
dimensional flow simulations combined with heuristic approaches by introducing a linear
sensitivity matrix. This matrix results from the evaluation of the flow response for a number
of geometrical perturbations. The method gives the possibility to produce a very fast
alternative flow solution without the need to solve the full set of RANS equations. However,
this approach is based on the key assumption of the linearity of the flow response to large
changes of the parameters in the design space. The parameters used by the author at seven
28 2. Scientific background and motivation
different blade heights are the angle of rotation of the profile about the trailing edge (skew),
the circumferential and the axial movement of the profile.
Figure 2.13 Geometry of the optimised turbine cascade (left) and span wise loss
distribution for optimised and start geometry (right). (Nagel et al., 2003)
Figure 2.14 Comparison of experimental and numerical profile pressure distribution at four
blade heights (Nagel et al., 2003)
It must be pointed out that the set of 56 parameters used for this study corresponds to
established engineering modifications and the profile shape remains unaltered. The validity of
2. Scientific background and motivation 29
the linearity assumption for modifications of the profile form has to be proved. In this case
indeed curvature changes could produce flow phenomena like shocks, whose impact on the
flow solution is not expected to correlate in a linear way with the relative geometry
perturbations. The author demonstrates the validity of the linear assumption on the specified
parameter space in a previous publication (Shahpar et al., 1999). The optimisation target is the
minimisation of the secondary flow kinetic energy at the exit plane of a nozzle guide vane
maintaining the mass flow capacity between specified ranges. The results obtained with a
deterministic technique like the Sequential Quadratic Programming (SQP) algorithm are
compared with the results obtained from the application of two different heuristic approaches:
Genetic Algorithm (GA) and Simulated Annealing (SA). The most promising results are
obtained using the Simulated Annealing technique. Another important aspect resulting from
this comparison is that, when increasing the number of parameters to fourteen, the
computational cost required by the SQP-technique is comparable with that required using a
heuristic approach. Different optimisation results obtained using the SA-method are compared
with the results on the datum blade in Figure 2.15. The secondary kinetic energy SKE is used
here as a parameter which quantifies the passage vortex strength. Reduced SKE values
indicate a more uniform radial distribution of the exit flow angle and less overturning in the
region near the wall. The optimised geometries result from calculations performed using only
the section circumferential movement parameters (SA-1), the section circumferential and
axial movement parameters (SA-2) and the skew angles only (SA-3). All the configurations
obtained within the optimisation process, except the geometry SA-3, feature mass flow
capacity within the specified limits. The author points out that the obtained blade shapes do
not permit additional experimental validation, since discontinuities along the blade span occur
using only seven blade sections. No information about the Mach number distribution on the
blade is given.
Another methodology for the reduction of the computational efforts of a direct optimisation
process is given by the use of artificial neural networks (ANN) in combination with heuristic
optimisation algorithms (Pierret et al., 2000 and Demeulenaere et al., 2004). In this approach,
a database of individuals (blade geometries), whose flow solution has already been computed,
is interpolated using an ANN-method, in order to build an approximate model of the original
analysis problem. Thus the aerodynamic behaviour of new geometries is obtained by
evaluating the interpolating surface instead of performing a complete Navier-Stokes
calculation. At each step an intermediate optimum is found and a Navier-Stokes calculation of
this geometry has to be performed in order to actualise the network. The success of these
methods depends mainly on the knowledge of the neural network, which is fed by previous
designs of similar blades. Therefore a large preliminary database of solutions has to exist in
order to ensure a better accuracy in the approximation of the design problem. The results
obtained by Perriet et al. (2000) indicate great potential of this approach in order to speed up
30 2. Scientific background and motivation
the design process. The results of the aerodynamic optimisation of a high pressure turbine
rotor blade are shown in Figure 2.16.
Figure 2.15 Blade geometries and secondary flow kinetic contours for baseline and SA
optimisation results (Shahpar, 2000)
Three sections equally distributed along the span (0%, 50% and 100% blade height) are
represented parametrically and an overall set of 97 parameters is used. The design goal is the
reduction of the cascade enthalpy loss coefficient maintaining a fixed pressure ratio, targeting
a given cascade deflection and considering further basic requirements on the profile pressure
distribution. Mechanical constraints as well as requirements on the profile Mach number
distribution are considered. The objective function is defined as a single value function
containing, in an appropriate form, the weighted contributions of the various objectives and
constraints. The terms related to the Mach number distribution which are considered are the
maximum Mach number on the suction side, the Mach number slope in the leading edge
region and the Mach number deceleration in the rear part of the suction side. The optimisation
procedure starts with a database containing 100 samples of similar rotor blades. The authors
point out that the time required for a design step is about 2.5 times higher than the time
needed for a three dimensional Navier-Stokes computation. A design step consists of the
network “learning process”4, the run time for the optimisation procedure (in this case a GA-
algorithm) and the Navier-Stokes computation performed on the temporary optimum
4 “Learning process” of the neural network is the process which is required to determine the free parameters of the network in order to fit the given database of samples.
2. Scientific background and motivation 31
geometry for the actualisation of the network. While the time required for the RANS-
computation is associated with the mesh size, the time required by the neural network and by
the GA-algorithm is proportional to the number of training samples and independent on the
mesh size.
Figure 2.16 Results of the aerodynamic optimisation of a high pressure turbine rotor blade
using a combined ANN / GA approach (Pierret et al., 2000)
Convergence is obtained after 9 design steps which correspond to a real time of about 24
hours. This represents a major improvement potential in design time of three dimensional
blades. However, for new design tasks preliminary computational efforts should be taken into
account as well, because an initial database has to be built and an additional preliminary
computational time of about 100 hours has to be taken into account. The upper left diagram of
figure 2.16 shows the distribution of the aerodynamic efficiency along the span, represented
as the difference between unity and the enthalpy loss coefficient. In the hub region, where the
original blade geometry features low efficiencies, major improvements are present. Along the
whole span an efficiency improvement of about 1% is obtained. In the upper right part of the
figure the outlet angle distribution along the span is shown. The target outlet angle is
indicated by the vertical dashed-dotted line. Even though the optimised geometry (continuous
32 2. Scientific background and motivation
line) realises a major improvement compared to the original geometry, there are still large
discrepancies (up to 3 degrees) between target and actual distribution. The isentropic profile
Mach number distribution of the initial database sample is shown at the bottom left of
figure 2.16, while the distributions on the optimised blade profile are shown at the bottom
right. An overall improvement of the distributions can be noticed. The diffusion region on the
pressure side near the leading edge at 50% and 95% blade height disappears and the large
deceleration region on the suction side of the initial geometry at 5% blade height is
suppressed as well. However, it must be pointed out that the initial distributions are quite poor
and not difficult to optimise. Further improvements in order to obtain more continuous
acceleration regions over the entire profile might be possible here.
The potential of the combination of ANN and heuristic optimisation techniques for the
reduction of the design time is also illustrated in a publication by Demeulenaere et al. (2004).
This work presents the results of a multipoint optimisation applied to a turbine rotor blade and
to a transonic compressor rotor blade. The multi-objective problem is reduced to a single-
objective task by introducing ad hoc tailored weighting coefficients for the different
objectives and constraints. Even if the ANN / GA combined method ensures a very low
computational requirement, it must be observed that the efficiency improvements presented
are quite moderate and the profile pressure distribution featured by the optimised geometries
can still be improved further. The authors point out that the choice of alternative approaches
such as multi-objective optimisation techniques would require a too large computational
effort. In fact during the search process these techniques generate families of solutions which
are usually indicated as pareto-front in the literature. The main advantage of multi-objective
techniques is the absence of any weighting function. Benini et al. (2002) show the large
potential of evolutionary multi-objective approaches for the development of a new class of
high-performance aerofoils for axial flow compressors. In this case, however, even using a
very fast quasi-three dimensional flow solver like MISES and limiting the number of
objectives to two, a large computational time has to be considered. The authors in fact state
that the final pareto front, compared with the original NACA65 cascades in the paper, is
obtained after 200 generations. For each generation 50 to 100 individuals are necessary.
Considering that the single-value approach developed within the present work requires, in
conjunction with an adaptive simulated annealing technique, 700 to 1000 evaluations for
convergence, the computational effort of a multi-objective approach becomes evident.
In an earlier publication, Goel et al. (1996) present a combined through-flow/blade to blade
quasi three-dimensional method for the aerodynamic design of turbine blades. This work
addresses in particular the question of a convenient profile Mach number distribution on
turbine blades. In this work the aerofoil quality is evaluated considering solely flow diffusion
and uniformity of flow changes of the predicted profile Mach number distribution. No profile
losses are considered, since the calculations are performed with an Euler flow solver. The
2. Scientific background and motivation 33
flow diffusion is computed analytically from the flow solution on the blade. The ratio
between peak Mach number on the suction side and exit Mach number is used as a criterion to
prevent flow separation in the unguided turning region. Moreover, the ratio between the
suction peak leading edge Mach number on the pressure side and the minimal Mach number
on the pressure side is considered in order to avoid the formation of a separation bubble in
this region. The uniformity of the velocity changes is measured by fitting polynomials to the
Mach number distribution on suction and pressure sides and evaluating the “error” between
actual Mach number distribution and fitted data. The authors point out that this approach is
different from an inverse design method, because the target created by fitting the data changes
with the actual Mach number distribution. Mechanical and geometrical constraints as well as
the aerodynamic constraints specified on the profile Mach number distribution form the
objective function. In order to explore a wide design space, a search strategy given by the
combination of a genetic algorithm and a hill-climbing deterministic technique is used. The
procedure is validated on two different turbine cases: the last stage of a high pressure steam
turbine is re-designed and the well known VKI LS59 rotor blade is used as further test case
for validation. For the steam turbine case the hub, mid-span and tip section of the blade are
independently optimised. In a further step the quasi three-dimensional optimised sections are
stacked and optimised in order to obtain a smooth radial geometry. The smoothness is then
measured by fitting splines to the geometry in the radial direction and measuring the
smoothness of the splines. In this second phase information regarding aerodynamic
parameters on the control sections is used as well in order to restrict the search in a region
including the sections which were just obtained. The upper part of Figure 2.17 presents a
comparison between initial and optimised blade geometry and profile Mach number
distribution at the three blade sections considered.
In all cases a more suitable profile velocity distribution is obtained with more uniform flow
acceleration and reduced diffusion. However, it must be pointed out that in absence of any
information about the boundary layer behaviour, more front-loaded solutions like the solution
obtained at mid-span are preferred even though, as discussed previously, this could negatively
influence the boundary layer development on the profile and the cascade total pressure
behaviour. Furthermore, the results obtained at the tip section indicate that the methodology
used does not recognise or classify an increased back diffusion in the rear part of the suction
side in appropriate form. In this case it can be supposed that the diffusion phenomenon taking
place in the leading edge region of the pressure side is rated as predominant by the present
polynomial fit approach. The results obtained at mid-span for the re-design of the VKI LS59
rotor blade are shown in the lower part of Figure 2.17. In this figure the profile Mach number
distribution of initial and optimised geometries are compared. In this case the method
recognises the predominant effect of the strong pressure gradients in the rear part of the
suction side and reacts consequently by reducing the ratio between peak Mach number on the
suction side and exit Mach number from 1.45 to 1.17.
34 2. Scientific background and motivation
Figure 2.17 Initial and optimised section geometries and Mach number distributions at tip,
mid-span and hub section (Goel et al., 1996)
2. Scientific background and motivation 35
The overview regarding automatic optimisation processes for the aerodynamic design of turbo
machinery blades presented in this section indicates that over the last decade large progress
was made in this area. Nevertheless different aspects have been pointed out which need
further improvement in order to be able to integrate these automatic procedures into an
industrial design environment. The computational costs associated with a fully three
dimensional aerodynamic blade optimisation are still too high for the industrial development
time-scales. Today, the aerodynamicist should be able to react rapidly to changes of the
design concept and constraints, as the design of turbomachinery blades is a challenging task
which takes into account different disciplines eventually with contrasting objectives. Fast and
reliable automatic methods are needed, which in addition to an increase of the aerodynamic
efficiency, should yield to good designs which can take into account various constraints and
basic requirements deriving from other disciplines.
In this challenging context an automatic design method for the aerodynamic optimisation of
two-dimensional cascade blades has been developed and is illustrated within the present
work. A method for the suitable evaluation of the profile Mach number distribution has been
developed and tested. Optimal profile velocity distributions for high pressure turbine blades
have been strived for. Cascade loss and deflection have been calculated using the state of the
art Navier-Stokes solver TRACE of the DLR in Cologne (Eulitz, 2000). In order to
investigate large design spaces an Adaptive Simulated Annealing optimisation algorithm has
been used. Extensive validation works have been performed on the basis of large
experimental data on high pressure turbine blades for heavy duty gas turbines obtained on the
High Speed Cascade Wind Tunnel of the University of the German Armed Forces in Munich
(Sturm and Fottner, 1985).
36 3. Experimental investigations
3. Experimental investigations
A fundamental condition for the successful application of automatic optimisation procedures
within the aerodynamic design process of turbomachinery blading is the assessment of the
reliability and application limits of the applied flow simulation tools. In this context extensive
experimental investigations were performed in the High Speed Cascade Wind Tunnel (HGK)
at the University of the German Armed Forces in Munich (UniBwM) on three different high
pressure turbine cascade blades designed by ALSTOM. These reference profiles feature
geometries and operating conditions typical for high pressure rotor blades of modern heavy
duty gas turbines. The High Speed Cascade Wind Tunnel ensures the reproduction of Mach-
and Reynolds numbers typical of heavy duty turbomachinery. The obtained data are used to
build an experimental database for the validation process of the developed aerodynamic
design method.
After a brief introduction of the reference geometries and the background leading to the
design of these cascades, the experimental setup will be presented. The main features of the
High Speed Cascade Wind Tunnel will be outlined as well as the applied measurement
techniques. The results obtained with the reference turbine cascade blades will be discussed in
order to identify detriments and benefits of the different design strategies.
3.1 The reference turbine cascades T150, T151 and T152
The datum profile, named T150, represents the mid-span section of a typical turbine rotor
blade for high pressure stages of large scale stationary gas turbines. The aerodynamic loading
of this profile is moderate. Starting from this reference profile, two further design approaches
were investigated. The resulting turbine cascade blades were indicated as T151 and T152.
Two different design strategies were strived for. While for the design of T151 the objective
was the reduction of the number of parts with a consequent increase of the blade spacing, for
T152 instead an optimisation of the profile Mach number distribution was pursued to enable
favourable conditions for the blade cooling. The reduction of the number of blades within the
stage is associated with reduced wetted surface, friction losses and reduced cooling air mass
flow per stage. Therefore the design strategy for T151 strived for the aerodynamic optimum.
On the other hand it must be kept in mind that increasing the aerodynamic loading produces
an increase of the cooling air mass flow per blade pitch since the increase of the adverse
pressure gradients influences the efficient action of the profile film cooling. This increases the
mixing flow losses. Furthermore, a reduced number of blades is associated with higher blade
sections for mechanical reasons. Thus, the internal cooling channels of the blade have to be
modified as well, with a consequent increase of the number of blade cooling channels
(Lötzerich, 2004b). As a consequence manufacturing costs increase. Furthermore, an
excessive increase of the blade aerodynamic loading is associated with higher pressure
3. Experimental investigations 37
gradients over the blade passage, which would lead to an undesirable increase of secondary
flow structures and leakage losses. These negative aspects deriving from an excessive
increase of the aerodynamic loading were considered for the design of T152, reducing the
blade loading to the level featured by the datum profile. Favourable conditions for the optimal
profile cooling were strived for as well. The gradients in the back diffusion region of the
suction side were reduced, the velocity magnitude on the whole pressure side was increased
and a continuous acceleration over the entire blade surface was realised. The design operating
conditions for the three cascades are quite similar. The design exit Reynolds number exceeds
two million and the design exit Mach number is slightly below 0.80. For the experimental
investigations the nominal reference exit Reynolds number and nominal reference exit Mach
number were fixed at Re2th=1 200 000 and Ma2th=0.75 respectively. The definition of the
theoretical exit Mach and Reynolds number is given by Ladwig (1989).
The geometries of the three cascades are presented in Figure 3.1. Here the profiles are scaled
to the same axial chord. The geometric and aerodynamic data for the reference cascades
T150, T151 and T152 at nominal operating conditions (Re2th=1 200 000 and Ma2th=0.75) are
presented in Table 3.1.
Figure 3.1 Geometries of the reference turbine cascade blades T150, T151 and T152
38 3. Experimental investigations
A parameter is introduced in order to quantify the aerodynamic loading of the profiles. This
coefficient CL, later referred to as compressible aerodynamic blade loading coefficient, is
defined as:
( )( )( ) ( )
2 2 2 2 2 1 1
1 2 1 2
sin cos sin 90
0.5 / 1Lax t
w w wtC
l p p h h
⎛ ⎞⋅ ⋅ ⋅ ⋅ − ⋅ −= ⎜ ⎟⎜ ⎟− ⋅ ⋅ +⎝ ⎠
ρ β β β (3.1)
The subscripts 1 and 2 indicate the inlet and outlet measurement plane respectively, the
symbol w represents the flow velocity, while β corresponds to the flow angle. The ratio h1/h2
corresponds to the axial velocity density ratio. The inlet total pressure is indicated as pt1 and
the exit static pressure as p2. The exit flow density is ρ2, while t and lax indicate respectively
the cascade pitch and axial chord.
Table 3.1 Geometric and aerodynamic data for T150, T151 and T152 at nominal
operating conditions (Re2th=1 200 000 and Ma2th=0.75)
Chord length (l)
Pitch to Chord ratio (t/l)
Trailing edge to Pitch ratio
(rTE/t)
Axial Chord length (lax)
Cascade opening to Pitch ratio
(e/t)
Cascade T150
150 mm 0.7320 0.028 138.98 mm 0.499
Cascade T151
120 mm 0.9572 0.014 94.58 mm 0.407
Cascade T152
140 mm 0.7266 0.016 123.68 mm 0.478
Inlet Flow Angle
(β1) Deflection (∆β)
Lift Coefficient (CL)
Inlet Mach Number (Ma1)
Cascade T150
133.9° 100.6° 0.876 0.42
Cascade T151
135.9° 110.9° 1.084 0.34
Cascade T152
131.5° 101.0° 0.893 0.40
The turbine cascades were manufactured using different chord lengths. Thus a trade-off
between large scale dimensions facilitating high Reynolds numbers and a high number of
blades for periodicity requirements within the test section was realised. For each cascade test
five blades were used.
Turbine cascade T151 features the highest aerodynamic loading among the reference cascades
and the largest blade spacing. The aerodynamic lift coefficient of T151 is about 24% higher
than for T150 and the pitch to chord ratio about 30% higher than for the datum profile T150.
Furthermore, cascade T151 realises about 10° more deflection than the other two cascades,
3. Experimental investigations 39
thus featuring a somewhat lower inlet Mach number in order to respect the specified mass
flow. The investigated cascades are characterised by high trailing edge to pitch ratios typical
for high pressure turbine blades in heavy duty gas turbines. The datum profile, however,
features a higher trailing edge thickness ratio than T151 or T152. This corresponds to a
trailing edge cooling design approach, while for the cascades T151 and T152 a pressure side
bleed approach was followed (Lötzerich, 2004b).
3.2 The High Speed Cascade Wind Tunnel
The aerodynamic investigation of turbine cascade models at Reynolds number levels typical
of heavy duty gas turbines requires appropriate experimental facilities. The High Speed
Cascade Wind Tunnel at the University of the German Armed Forces in Munich with its test
section dimensions and its power supply unit offers optimal conditions for this task.
Figure 3.2 shows a sectional drawing of the whole test facility. The High Speed Cascade
Wind Tunnel is a continuously operating closed loop test facility with an open loop test
section. The Wind Tunnel itself is contained in a cylindrical pressure tank, while its driving
unit is situated outside. The air flow supply is delivered by a six stage axial compressor. The
main feature of this experimental facility is the possibility to vary Mach and Reynolds number
independently from each other. The desired Mach number is obtained by adjusting the
number of revolutions per minute of the axial compressor. The Mach number range in the test
section can be varied between 0.2 and 1.05. By partly evacuating the pressure tank the
Reynolds number can be varied in the range 0.2·106m-1<Re/l<1.6·107m-1, where l is the
cascade chord length. A detailed description of the original configuration of the test facility in
Brunswick can be found in Scholz et al. (1959) while the modifications and extensions
featured by the test facility in Munich are described by Sturm et al. (1985). The following
provides a brief description of the main components of the test facility as shown in Figure 3.2.
The driving unit consists of an a.c. electric motor of about 1.3 MW power, a hydraulic
coupling and a gear box which is connected to the axial compressor. The axial compressor
working range and specifications are illustrated in the figure. Since each cascade works like a
throttle and corresponds therefore to a different working point in the compressor working
map, a variable bypass is used to prevent stalling phenomena. For reducing the air
temperature after compression, a system of lamella coolers for main flow and bypass air are
positioned downstream of the compressor. In order to increase the cooling efficiency a
diffusor precedes the main flow cooler. Although the cooler straightens the flow, a settling
chamber is located downstream to mix out the temperature and pressure non-uniformities.
The flow is then accelerated in a nozzle to the Mach- and Reynolds-number specified for the
test section.
40 3. Experimental investigations
The desired level of turbulence is achieved by locating a turbulence generator grid of proper
form at the inlet of the nozzle (Acton, 1994). Turbulence levels between 0.4% and 7.5% can
be achieved.
Figure 3.2 The High Speed Cascade Wind Tunnel (HGK) of the University of the German
Armed Forces Munich
3. Experimental investigations 41
3.3 Measurement section set up
A sectional view of the wind tunnel measurement section is given in Figure 3.3. The
measuring positions and the fundamental instrumentation for one of the investigated turbine
cascades are shown in Figure 3.4. The test section height assumes values between 250 and
500 mm. This height corresponds to the distance between the horizontal test section side
walls. The test section height has to be adjusted in accordance with the desired inlet flow
conditions. The distance between the vertical test section side walls is fixed to 300 mm for
steady measurements. In order to achieve best homogeneity of the inlet flow variable guide
vanes are mounted on the upper and lower test section side walls. These vanes reproduce the
form of the blade camber line and are placed a half pitch above or below the blades at the
extremities of the cascade. During preliminary tests their position is adjusted until the
measured inlet static pressure (p1) features a uniform distribution over the test section height.
The inlet static pressure is measured using an array of pressure tappings aligned 140 mm
upstream of the cascade inlet plane, as displayed in Figure 3.4. The value p1 is an average of
the three tappings in the middle of this array. The inlet total pressure pt1 is measured using a
pitot probe 300 mm upstream of the cascade inlet plane on the same sidewall, where the inlet
pressure tappings are located. The distance of the pitot probe from the test section side wall is
50 mm. Additional measurements of the inlet flow conditions performed with a bent-head five
hole probe (cobra probe) for the three cascades indicate only slight differences (affecting the
third decimal place of the inlet Mach number) in the static and total pressure distribution
along the blade span at reference conditions. The results of these investigations are outlined in
detail in the reports of the reference cascades (Cardamone, 2002, 2003, 2004). The total inlet
temperature Tt1 is measured as an average of four PT 100 resistance thermometers, which are
located in the settling chamber. Assuming an adiabatic flow acceleration inside the nozzle and
the cascade this temperature is equal to the total temperature in the cascade measurement
plane. The pressure inside the tank pK is measured at a location within the tank where the flow
is undisturbed and the velocity is minimal. The only absolute pressure measurement concerns
the environment pressure pUmg outside of the tank. All the other absolute pressures are derived
from differential pressure measurements with respect to pUmg.
The described tests were performed positioning the same turbulence grid generator of the
Type IXgK in front of the nozzle. This turbulence grid is shown in Figure 3.5. Earlier
investigations by Kiock et al. (1982) indicate that inlet turbulence intensity values around 4%
are expected using a turbulence generator of this type, depending on Reynolds-, Mach number
and on the acceleration within the nozzle.
42 3. Experimental investigations
Figure 3.3 Sectional view of the turbine cascade built up
Figure 3.4 Turbine cascade as mounted in the test section and instrumentation equipment
3. Experimental investigations 43
Figure 3.5 Turbulence grid generator IXgK
3.4 Measurement techniques and data evaluation
In order to determine the profile pressure distribution the central blade in the test section
(blade number 3 in Figure 3.3) was instrumented with pressure tappings of 0.6 mm diameter
on the suction and the pressure side. The number of tappings and their distribution over the
profile was designed to catch the main gradients of the designed profile Mach number
distributions. The distribution on T152 is given in Figure 3.6. T150 was provided with 76,
T151 with 65 and T152 with 67 profile pressure tappings. The local pressure px, was
measured as a differential value with respect to the pressure within the tank pK using
Scanivalve equipment. The local pressure value is used to calculate the local isentropic profile
Mach number:
1
1,
21
1
k
kt
is xx
pMa
k p
−⎡ ⎤⎛ ⎞⎢ ⎥= ⋅ −⎜ ⎟⎢ ⎥− ⎝ ⎠⎢ ⎥⎣ ⎦
(3.2)
The profile distribution of this parameter was used to derive conclusions about the loading
distribution on the blade.
Wake traverses in a plane located at a distance of 40% chord length from the cascade outlet
plane (Figure 3.3) were performed using a five-hole-probe of 2.5 mm head diameter.
44 3. Experimental investigations
Figure 3.6 Distribution of the profile pressure tappings on the turbine cascade blade T152
in a bi-tangent coordinate system
At each traversing position the five pressure values measured by the probe are converted
using the calibration polynomials (Hoenen et al., 2001) to determine the local exit pressure
p2,u, the local exit flow angle β2,u and the local total pressure pt2,u. This last value is used to
calculate the local cascade total pressure loss coefficient:
1 2,
1
t t uu
t K
p pζ
p p
−=
− (3.3)
The integral cascade performance parameters are then obtained using the conversion
procedure by Amecke (1967), which consists mainly of applying the conservation laws for
mass, momentum and energy in order to obtain an integral constant value from a wake
distribution of the single flow quantities. The data acquisition was carried out using a pressure
scanner of the type 98RK (Esterline Pressure Systems, 2000). The control of the measuring
devices and evaluation was performed using the in-house software programme WINPANDA
(Ganzert et al., 1996).
A particularly interesting method for the flow visualisation over the blade surface is
represented by the oil flow pictures. Secondary flow structures can be investigated using this
technique without excessive effort (Weiß, 1993). Furthermore, this method can give useful
information about the boundary layer development on the blade surface, visualising the
position and extension of the transition zone (Engber, 1996). The surface of the measuring
3. Experimental investigations 45
blade is uniformly covered with a mixture of oil, petroleum and fluorescent powder. At the
desired operating conditions the visualisation of the flow structures on the painted surface is
possible due to the behaviour of the oil mixture depending on the local wall shear stresses. In
regions of high flow velocities, where no separation bubble occurs, the oil mixture is almost
completely removed from the blade surface. On the other hand, the oil mixture accumulates in
regions of local flow separation or in regions where low flow velocities occur (stagnation
point, pressure side), because of the lower wall shear stresses. In turbulent boundary layer
regions, due to the increased momentum and mass exchange near the wall, low quantities of
oil mixture are removed. This allows the recognition of the end of the transition zone on the
profile. The quality of the oil flow pictures depends strongly on the mixture composition. The
mixture has to remain liquid on the blade during the time taken for the tank evacuation (about
15 to 20 minutes) and should then rapidly dry at operating conditions in order to reduce the
experimental costs. The identification of separation bubbles requires particular attention. In
fact, in this case the relative high quantity of paint accumulated in the separation bubble
region has to be carefully removed towards the trailing edge. This is done by increasing the
compressor rotational speed for a short period of time. A possible alternative solution could
be to let the paint carefully flow back over the blade surface by shutting down the power
supply. In this way the two lines delimiting the separation bubble region can also be well
identified, but the flow configuration in the blade region upstream of the bubble is washed
away. Due to the higher gradients near the wall at higher Reynolds numbers the most evident
results were obtained at a lower Reynolds number level.
The inlet flow turbulence intensity was determined using the Constant Temperature
Anemometry (CTA). Assuming isotropic turbulence distribution, a single hot film probe of
the type “DANTEC HF-55R01” is mounted 500 mm upstream of the cascade inlet plane.
Wolff (1999) describes the CTA data acquisition system. The DANTEC anemometer system
“Streamline” is controlled by DANTEC software “Streamware” (DANTEC, 2001). The in-
house software WINSMASH (Wolff, 1999) controls the data acquisition. The probe is
calibrated for each local static pressure. A 4th order polynomial has been used for the
approximation of the calibration curve. The Hot Film Anemometer (HFA) signals were low-
pass filtered with a cut-off frequency of 10 kHz. Considering the generic quantity b, its mean
value is given by:
0
1 N
jj
b bN =
= ∑ (3.4)
where N is the number of samples and bj represents the generic quantity for each sample. The
standard deviation of b is then calculated using the RMS deviation given by:
( )2
0
1
1
N
jj
RMS b bN =
= ⋅ −− ∑ (3.5)
46 3. Experimental investigations
The turbulence level Tu is calculated using the RMS value of the velocity w measured by the
hot film probe and normalising it using the velocity in the cascade inlet plane w1:
1
100%RMS
Tuw
= ⋅ (3.6)
3.5 Measurement programme
The turbine cascades T150, T151 and T152 were investigated in a wide operating range at
various incidences. The effects of Mach- and Reynolds number on the aerodynamic behaviour
of the cascades were investigated as well. The exit Reynolds number Re2th was varied
between 600 000 and 1 200 000, while the exit Mach number Ma2th was varied in a high
subsonic region between 0.65 and 0.85. In this way the aerodynamic characteristics of the
different design strategies were quantified and an extensive database was provided for the
validation of the simulation tools.
Table 3.2 Measurement program for the turbine cascades T150, T151 and T152
Ma2th
Re2th 0.65 0.75 0.85
600 000 0 1 0 1 2 0 1
900 000 0 1 0 1 2 0 1 ∆β1=0°
1 200 000 0 1 2 0 1 2 0 1 2
Ma2th Re2th
0.65 0.75 0.85
600 000 0 1 2 0
900 000 1 2 ∆β1=+/−10°
1 200 000 1 2 0 1 2 0 1 2
Ma2th Re2th
0.65 0.75 0.85
600 000 0 1 2
900 000 1 2 ∆β1=+/−5°
1 200 000 0 1 2
Table 3.2 shows the measurement programme performed on the three turbine cascades. The
numbers indicate the cascade name (e.g. 1 means T151). In the left column information is
presented regarding the incidence at which the tests were performed (e.g. ∆β1 = +10° means
plus ten degrees incidence with respect to the reference cascade inlet flow angle). The inlet
flow angle is defined as in Figure 3.1 and positive incidences correspond to a rotation of the
inlet flow vector towards the pressure side (clockwise direction in Figure 3.1). The boxes
3. Experimental investigations 47
containing a number indicate a combination of incidence angle, Mach- and Reynolds number
at which a profile pressure distribution and wake traverse with a five hole probe were
performed. The wake traverses were performed at mid-span in a plane downstream of the
cascade at a distance of 40% chord from the cascade outlet plane. A shaded box means that
oil flow pictures were taken for the corresponding operating conditions. Bold characters
indicate the existence of inlet turbulence flow measurements.
For the turbine cascade T150 further measurements were performed at a higher exit Mach
number of Ma2th=0.90, a reference exit Reynolds number Re2th=1 200 000 and reference
incidence. These measurements were helpful for an additional numerical analysis of the
influence of the trailing edge thickness on the cascade total pressure losses (see Chapter 5).
3.6 Experimental results and discussion
The integral parameters quantifying the aerodynamic behaviour of the three cascades, the
measured turbulence levels and the extension of the transition zones, as read from the oil flow
pictures, are listed in the annex. The measured turbulence level confirmed the turbulence
range expected from the use of the turbulence grid generator of type XIgK, featuring values
around 4%. In the present section the influence of the inlet flow angle, the Reynolds- and the
Mach number on the cascade aerodynamic behaviour will be discussed.
A comparison of the isentropic profile Mach number distributions for the different turbine
cascades at reference operating conditions is shown in Figure 3.7. In the front part of the
suction surface of the datum-profile T150 the flow accelerates with quite high gradients to a
Mach number slightly below 0.8. In this initial zone, however, a well marked discontinuity in
the velocity gradients, associated with local diffusion phenomena can be identified. This zone
is followed by a second one in which the velocity level remains almost constant. A third
acceleration zone can be detected between 50% axial chord and the position of the peak Mach
number on the suction surface. In the following diffusion region a flow deceleration takes
place towards the exit Mach number. On the pressure surface near the leading edge T150
features a suction peak, indicating the presence of a short separation bubble. Downstream the
flow decelerates down to the pressure surface local minimum. Then a flow acceleration
occurs towards the exit Mach number. The pressure side acceleration features higher gradients
near the trailing edge, starting from 80% axial chord length. Numerical simulations indicate
that a boundary layer relaminarisation is expected to take place in this region on the pressure
side (see Chapter 5). The suction side acceleration for T151 and T152 takes place more
continuously than for the datum profile, presenting thus more advantageous conditions for an
efficient profile film cooling. However, the increased aerodynamic loading of T151 produces
a back diffusion region of higher gradients on the suction side. This results from the higher
suction surface peak Mach number, located as far downstream on the suction surface as for
T150, and from the lower exit Mach number featured by T151. The reduction of the exit
48 3. Experimental investigations
velocity level for T151 is a direct consequence of the increased blade spacing at a fixed
cascade mass flow. On the other hand, the profile velocity distribution on T152 features
improved characteristics with respect to the other two blade profiles. The deceleration to the
exit Mach number in the rear part of the suction surface of T152 occurs under quite moderate
gradients. The continuous acceleration and the higher velocity level featured by T152 on the
pressure surface are particular advantageous for an efficient action of the profile film cooling
in this region as well.
Figure 3.7 Comparison of the measured profile isentropic Mach number distributions for
T150, T151 and T152 at reference conditions (Re2th=1 200 000; Ma2th=0.75
and β1=β1ref)
Even if heavy duty gas turbines operate at fixed incidence for the most part of the working
time, it is usual to take into account the profile behaviour at different incidences within the
design process. In fact during transitional processes like start up and shut down of the turbine,
strong incidence variations occur, which could negatively influence an efficient film cooling
of the profile. This could lead to severe damage within the machine and it is therefore of
major importance to have knowledge of the aerodynamic behaviour of the investigated
profiles at different incidences as well.
The influence of the inlet flow angle on the development of the integral total pressure losses
of the three cascades is displayed in Figure 3.8. Here, the y axis is expressed as the ratio
between the actual total pressure loss coefficient and the measured total pressure loss
coefficient of the datum-profile at reference conditions. This will be used further on as a
3. Experimental investigations 49
reference value. An increase of the inlet flow angle means a rotation of the incidence vector
towards the pressure side. The diagram shows that the higher aerodynamic loading featured
by T151 produces a strong reduction of the level of the total pressure losses. At reference
conditions T151 presents a loss coefficient about 32% lower than the datum profile (T150).
T152 instead presents a loss reduction of about 10% with respect to T150 at reference
conditions. T150 and T151 react more sensitively to an inlet flow angle increase than T152.
Ten degrees more incidence produce about 30% higher total pressure losses for the datum
profile and for T151, while this increase is restricted to about 20% for T152.
Figure 3.8 Influence of the inlet flow angle onto the cascade characteristics of T150, T151
and T152 at reference conditions (Re2th=1 200 000; Ma2th=0.75 and β1=β1ref)
The influence of the Reynolds number on the aerodynamic behaviour of the cascades is
shown in figure 3.9 and figure 3.10.
Figure 3.9 presents the development of the measured cascade total pressure losses versus the
Reynolds number, while Figure 3.10 shows the behaviour of the cascade integral exit flow
angle versus the Reynolds number. In this diagram, because of the lower exit flow angles
featured by T151, the corresponding angles are represented on the right hand side y-axis of
the diagram. The Reynolds number influence in the range between 600 000 and 1 200 000 is
moderate, however the plots show some tendencies which are confirmed by the analysis of
the profile velocity distributions and oil flow pictures. In the following the cascade
aerodynamic behaviour at negative and positive incidences is discussed in order to evidence
these major features.
50 3. Experimental investigations
Figure 3.9 Influence of the Reynolds number on the total pressure losses of cascades
T150, T151 and T152 at reference conditions (Ma2th=0.75 and β1=β1ref)
Figure 3.10 Influence of the Reynolds number on the exit flow angle of cascades T150,
T151 and T152 at reference conditions (Ma2th=0.75 and β1=β1ref)
At negative incidences, where the blade aerodynamic loading is reduced, both T151 and T152
present a local minimum of the total pressure losses at the intermediate Reynolds number
Re2th=900 000. This is the result of two contrasting aspects. In fact, increasing the Reynolds
number to the nominal value of Re2th=1 200 000 moves the suction side transition region
3. Experimental investigations 51
towards the leading edge and therefore the turbulent boundary layer develops further
upstream. This produces a slight increase of the losses and an increase of the cascade exit
flow angle, which corresponds to a reduction of the cascade deflection because a thicker
turbulent boundary layer exists on the suction surface near the blade trailing edge. By
reducing the Reynolds number to Re2th=600 000 the total pressure losses of T151 and T152
increase. This is associated with the development of a short laminar separation bubble in the
region immediately downstream of the suction side Mach number peak. The existence of the
bubble is confirmed both by the measured profile pressure distributions and the related oil
flow pictures. Additional measurements, performed for T152 at Re2th=400 000, elucidate this
phenomenon even better. The presence of the laminar separation bubble does not influence
the cascade deflection behaviour, as confirmed by the curves in figure 3.10.
Figure 3.11 Profile velocity distribution for T152 at negative incidence ∆β1=−10° (upper)
and at positive incidence ∆β1=+10° at reference Mach number Ma2th=0.75
52 3. Experimental investigations
These observations are confirmed by the profile velocity distribution for cascade T152 at
negative incidence, as displayed in the upper part of Figure 3.11. The laminar separation
bubble featured at the lowest Reynolds number Re2th=400 000 is displayed in detail in the
upper right side of the figure. An analogous but less pronounced behaviour can be observed at
Re2th=600 000. At positive incidences, as presented in the lower part of the figure, the
diffusion region featured in the front part of the suction surface forces the transition further
upstream. Therefore the velocity distributions are almost independent on the Reynolds
number as shown in the lower part of figure 3.11. The distributions displayed in figure 3.11
qualitatively feature the same characteristics as T151, so that the same conclusions can be
drawn for this profile. However, for the datum profile T150 the existence of a velocity plateau
on the profile suction surface forces an earlier development of the turbulent boundary layer
even at negative incidences. Therefore the gradients existing in the suction side back diffusion
region do not lead to the development of a laminar separation bubble at Re2th=600 000. At
positive incidences T150 behaves similarly to T151 and T152, featuring a diffusion region in
the front part of the suction surface, which almost fixes the cascade deflection (see
figure 3.10).
The movement of the transition zone towards the front part of the profile suction surface with
increasing Reynolds number at negative incidences is illustrated in Figure 3.12, Figure 3.13
and Figure 3.14. The flow pictures at incidence ∆β1=−10° and reference Mach number
Ma2th=0.75 for the turbine profile T152 are displayed. The figures represent the unrolling on a
plane of the blade surface. The suction surface is shown in the right hand part of each figure
while the pressure surface is shown on the left. On the bottom of the figures the axial co-
ordinate is shown. Few interesting phenomena are revealed from the oil flow visualisation of
the pressure surface. Instead the visualisation of the suction surface reveals both the passage
vortex separation lines (S4-lines according to the nomenclature used by Sieverding, 1984) and
the extension of the transition zones. The transition zone on the suction side is located where
less paint is found and are indicated by the dashed white lines. At the lower Reynolds number
Re2th=400 000 a laminar separation bubble was observed on the suction surface during the
online monitoring. In order to avoid an upstream flow of the paint collected within this
recirculating zone during the shut down procedure of the wind tunnel the compressor was
driven shortly at higher rotational speed at the end of the test, thus pushing the collected oil
flow material towards the blade trailing edge.
A comparison of the profile velocity distributions for the three cascades at a higher operating
Mach number of Ma2th=0.85 is shown in Figure 3.15. From this figure it becomes clear that
the increased aerodynamic loading of T151 produces higher shock intensity. On the other
hand, even if a supersonic region occurs both on the suction surface of the datum-profile and
on the cascade T152, the peak Mach number on these profiles remains lower than for T151,
realising better conditions for the operation at higher Mach numbers. In the analysed Mach
3. Experimental investigations 53
number range, the effect of this phenomenon onto the cascade total pressure losses is quite
moderate. The measured behaviour at higher exit Mach numbers is confirmed by the
computations performed with TRACE (Martinstetter, 2004a).
Figure 3.12 Oil flow visualisation for T152 at Re2th=400 000, Ma2th=0.75 and ∆β1=−10°
Figure 3.13 Oil flow visualisation for T152 at Re2th=600 000, Ma2th=0.75 and ∆β1=−10°
54 3. Experimental investigations
Figure 3.14 Oil flow visualisation for T152 at Re2th=1 200 000, Ma2th=0.75 and ∆β1=−10°
Figure 3.15 Comparison of the measured profile isentropic Mach number distributions for
T150, T151 and T152 at operating conditions: Re2th=1 200 000; Ma2th=0.85
and β1=β1ref
3. Experimental investigations 55
In fact, while an increase of the exit Mach number from Ma2th=0.75 to Ma2th=0.85 produces
an increase of the integral total pressure loss coefficient for T150 and T151 of about 10%,
with respect to the reference operating conditions, the related increase for T152 is only 2%.
The sensitivity of the cascades to a reduction of the operating Mach number was investigated
as well performing tests at Ma2th=0.65. Under these operating conditions the profile Mach
number distributions feature an extended region of almost constant velocities, which forms
near the leading edge on the suction side. Thus it is to be expected that transition moves
upstream on the profile suction side and a thicker turbulent boundary layer develops over the
profile.
The influence of the Mach number onto the cascade deflection is discussed in the following.
From the literature (Scholz, 1978) it is well known that the sine rule is satisfied better as the
operating Mach number rises. This means that the deviation angle becomes smaller and the
cascade deflection rises. At choking conditions the sine rule applies almost exactly. This is
shown by applying the momentum theorem to the control volume of Figure 3.16. The
application of the momentum theorem at sonic conditions in the cascade passage throat leads
to the expression given in the figure. Since for sonic conditions in the throat the exit flow
velocity w2 and the sonic velocity of the Laval state a* are equal, the given expression obeys
the sine rule. Therefore rising the cascade operating Mach number leads to outlet flow angles
nearer to the outlet flow angle predicted by the sine rule.
Figure 3.16 Application of the momentum theorem to the rear part of the blade passage
(Scholz, 1978)
56 3. Experimental investigations
Figure 3.17 Influence of the Mach number on the cascade outlet flow angle at reference
∆β1=0° and positive incidence ∆β1=+10° at reference Reynolds number
Re2th=1 200 000
Scholz (1978) discusses another compressibility effect produced by the finite thickness of the
profile and the boundary layer thickness at the trailing edge. At near sonic conditions, in order
to compensate the increased cross-sectional area downstream of the blade trailing edge, the
fluid experiences velocity changes in a tangential direction to maintain the specified mass
flux. Higher velocity changes are required for higher changes in the cross sectional area (e.g.
produced by higher blade trailing edge thickness).
The measured influence of the operating Mach number on the exit flow angle is illustrated in
Figure 3.17. The diagram contains the measured integral exit flow angle at a reference
Reynolds number Re2th=1 200 000 and at a reference ∆β1=0° (continuous lines) and positive
incidence ∆β1=+10° (dashed lines). It can be observed that higher Mach numbers produce an
increase of the cascade deflection, resulting in an approach of the sine rule. Furthermore, this
occurs under higher gradients for the datum profile T150, which presents a thicker trailing
edge than the other two cascades. This aspect confirms the effects of profile trailing edge
thickness and Mach number on the cascade deflection observed in the literature. At increased
incidence, where the turbulent boundary layer forms more upstream and a higher
displacement thickness is expected at the trailing edge, the deflection changes become more
relevant for T151 and T152 even in the lower Mach number range between 0.65 and 0.75.
3. Experimental investigations 57
The experimental data quantifying the aerodynamic behaviour of the investigated cascades are
shown in the annex. In Table 7.1, Table 7.2, Table 7.3 and Table 7.4 the first column of
results refers to the datum cascade blade T150, while the second column contains the data of
T151 and the third column refers to T152. The data contained in Table 7.5 are all obtained at
the reference Mach number Ma2th=0.75.
58 4. Numerical optimisation environment
4. Numerical optimisation environment
The present chapter illustrates the numerical environment where the aerodynamic
optimisation procedure was developed. The structure of the automatic design method is
shown in Figure 4.1. The optimisation loop consists of three major components: parametric
geometry generator, Navier-Stokes flow solver and optimisation algorithm. The aim of the
optimisation process is to find the n-dimensional vector of design variables X=(x1, x2, …, xn)
which minimises a scalar function, indicated as the objective function F(X), and respecting a
set of m constraints expressed by the vector function B(X)=(b1, b2, …, bm). The generic ith
constraint bi(X) is violated, if bi(X) is located outside a specified range. For the present
application, the design variables correspond to the parameters describing the blade profile
geometry. The objective function is set up by combining various aerodynamic performance
coefficients which result from the flow simulation of the actual blade profile. Thereby the
main optimisation target is the reduction of the cascade total pressure losses by imposing a
fixed operating point. Requirements on the profile velocity distribution with regard to cooling
demands were integrated into the objective function as well. Furthermore, some major
mechanical and geometrical constraints were specified in order to restrict the search to a
subset of realistic geometries. In this way the optimisation task is reduced to a single-
objective, constrained approach.
Figure 4.1 Schematic representation of the optimisation loop and connections of the
components
As illustrated in Figure 4.1, the optimisation algorithm represents the core of the whole
process. It modifies the design parameters according to the information obtained by the
already evaluated parameter datasets.
4. Numerical optimisation environment 59
If the parameters satisfy the specified mechanical and geometrical constraints, the
corresponding blade geometry is transferred to the flow simulation process. Otherwise the
present parameter dataset is associated with an appropriately high value of the objective
function. The flow simulation process is performed in a sequence of automatic steps. The first
step is the grid generation. In this work a method was implemented which ensures reduced
dependence of the flow solution on the mesh by maintaining a fixed grid topology and
modifying, mainly, the mesh in the boundary layer region (Niß, 2002). The flow simulation is
performed using the Reynolds averaged Navier-Stokes solver TRACE developed by the DLR
in Cologne (Eulitz, 2000) in a quasi three dimensional version. In order to reduce the code
running time, each simulation restarts from a well converged solution on a reference mesh of
the reference geometry T150. Furthermore, a convergence criterion based on monitoring the
cascade total pressure losses, the exit flow angle and the total pressure ratio inlet/outlet during
the flow simulation was integrated within the solver. The third step of the flow simulation
process consists in the evaluation of the results obtained for the present set of design
parameters. The evaluation process (Jogwitz, 2002b and Groth, 2004) takes place both in a
plane downstream of the cascade and on the blade profile. The results are then used to build
the objective function, whose value is then computed by the optimisation algorithm. The
whole procedure is set up within the commercial software package iSIGHT
(Engineous Software, 2002). This facilitated the use of various optimisation techniques. The
present investigations were carried out using a probabilistic heuristic optimisation approach as
the adaptive simulated annealing algorithm ASA. Further investigations were performed
using the multi island genetic algorithm MIGA. Another major advantage of implementing
the optimisation procedure within iSIGHT was the reduced efforts to interface the single
components, thanks to the file parsing capabilities of this software package.
In the next sections the numerical tools used to set up the aerodynamic optimisation procedure
are described in detail. Furthermore, the results of the Navier-Stokes simulations are
presented, which were performed on the reference cascades.
4.1 The Parametric geometry generator PROGEN
The parametric representation of the blade geometry is realised using the software PROGEN
of ALSTOM. The three dimensional model of the blade is built up by interpolating the
profiles defined at different blade heights in a radial direction. The single blade profiles are
described as closed curves on conical surfaces. The straight lines used to generate the cones
represent the best fit approximation of the quasi three dimensional stream-surfaces on which
the different blade profiles are designed. Each blade profile consists of four curve segments
(suction side, leading edge, pressure side and trailing edge) which are linked by the profile
control points.
60 4. Numerical optimisation environment
Figure 4.2 Definition of the control points and spline parameters within PROGEN
Each curve segment is a Bezier spline of fifth order and continuously differentiable twice at
the control points. The position of the control points is determined by using the geometric
system illustrated in the upper part of Figure 4.2. Specifying the blade chord length L and the
stagger angle βSG, two auxiliary points are defined. These are the central points of two short
auxiliary segments at trailing and leading edge of the blade. The length of these segments
corresponds to the trailing edge thickness dH and the blade nose thickness dN. The inclinations
of these segments are expressed by the blade metal angle at trailing edge and leading edge,
indicated as αH and αN, respectively. Thus the specification of these six parameters fixes the
position of the four control points unequivocally. The form of the Bezier-curves is then
determined by six additional parameters for each control point. These parameters correspond
to the profile tangent in the control point wi, the curvature ki, the aspect ratio parameters for
the profile slope in counter-clockwise direction ρvi and clockwise direction ρri and the aspect
ratio parameters of the profile curvature in counter-clockwise direction σvi and clockwise
direction σri. Therefore a blade profile section is described unequivocally using a complete set
of 30 parameters. In the case that the profile features symmetric leading and trailing edges the
parameter dataset reduces to 28. In this case in fact the blade wedge angles γH and γN can be
used instead of the blade tangents wi. The effects of the variation of the parameters ρ and σ on
the form of the Bezier-curves are illustrated in Figure 4.3. This figure illustrates that while the
parameter ρ influences the rear part of the spline segment, the influence of σ is more evident
in the front part of the curve. The different variation ranges of the two parameters have to be
considered in an appropriate way within the optimisation process.
4. Numerical optimisation environment 61
Figure 4.3 Influence of the design parameters ρ and σ on the form of the Bezier-spline
4.2 Flow computations procedure
4.2.1. The automatic grid generation method GRIDMOD
A major requirement to the design method is the possibility of automation of the grid
generation process. A reduced mesh dependence of the flow solution is strived for as well. In
order to fulfil these requirements an automatic grid adaptation method, named GRIDMOD
(Niß, 2002) was set up. The automatic mesh generation process is based on the modification
of a low-Reynolds template mesh, which is adapted to fit the actual blade geometry. This
mesh corresponds to a grid of the datum profile T150, containing about 12 000 nodes. The
mesh quality represents a trade-off between accuracy of the solution and computational cost.
The template mesh used for the present investigations is shown in Figure 4.4. The inserts
showing the blade leading and the trailing edges indicate the high resolution of the boundary
layer ensured by this mesh. The present template mesh consists of a multi-block structured
grid featuring a standard O-H topology. The advantages of this topology, compared to more
advanced multi-block structured approaches (e.g. the OCGH topologies, Martinstetter, 2004a)
consist mainly of the possibility of parameterising the mesh in a simple way using a reduced
set of characteristic points at the block-boundaries as control points. On the other hand it must
be kept in mind that this topology does not feature the optimum for achieving high mesh
quality in the wake region for high cascade deflections. However, this aspect is not relevant
except for very high deflections, in a range 115°−120°. For the investigated deflection range,
between 100° and 115°, a satisfactory mesh quality could be achieved. The boundary layer
block of the template mesh contains about half of the mesh nodes. At reference operating
conditions the template grid ensured an average wall dimensionless distance y+ over the
62 4. Numerical optimisation environment
suction side of around 2.5. This corresponds to a resolution which is accurate enough for the
one equation turbulence model approach used (Eulitz, 2000).
Figure 4.4 Template mesh for the automatic grid generation process
In a preliminary phase of the optimisation study, extensive investigations were performed to
quantify the mesh effects onto the flow solution and thus derive the most suitable parameters
for setting up the automatic mesh generation method. These investigations showed that the
integral cascade performance parameters (e.g. integral total pressure losses and exit flow
angle) are not significantly influenced by mesh modifications in the passage block and the
wake block. Since these coefficients are directly used to build the objective function, this
result has major significance for the development of the automatic mesh generator. On the
other hand changes within the boundary layer block (e.g. inclination of the mesh lines with
respect to the solid surface, overall number of nodes and distribution in a perpendicular
direction to the wall) showed stronger influence on the integral cascade coefficients. In order
to reduce the influence of the mesh quality in the O-block onto the flow solution, some major
rules have to be respected, as will be shown from the following mesh studies.
The effect of changes of the position of the boundaries of the wake block on the mesh
structure and on the relative total pressure losses is illustrated in Figure 4.5. The meshes on
which this study was performed are compared in the upper part of the figure. The original
mesh is called mesh A while the modified grid is indicated as mesh B. The major difference
between these meshes consists in the movement of the characteristic point P to the position P′, which corresponds to the position of this control point as calculated using the relationships
4. Numerical optimisation environment 63
implemented within GRIDMOD. A comparison of the wake region of these meshes shows
reduced skewness of the cells in the wake block of mesh B. This feature is responsible for the
re-distribution of the calculated total pressure losses. However, no significant change of the
integral aerodynamic coefficients of the cascade was observed.
Figure 4.5 Influence of the mesh quality on the flow solution in the wake block
A comparison of calculated and measured total pressure loss distributions is displayed in the
lower part of Figure 4.5. The cascade integral total pressure loss coefficient for mesh B is
0.0003 points higher than for mesh A, corresponding to a variation of 0.7 %. Instead the
calculated integral exit flow angle is two hundredths of degree lower. These represent
admissible variations for the present investigations. However, it can be observed that the
computed total pressure loss curves of both cascades are shifted more towards the pressure
64 4. Numerical optimisation environment
side branch of the wake (on the right hand side of the figure) than the measured curves. This
indicates that the predicted deflection is somewhat higher than the measured one. The
calculated total pressure losses are somewhat higher than the measured losses as well. These
discrepancies, however, were observed for all the reference cascades and remain
approximately constant over a wide operating range as will be shown in a further section of
this chapter regarding the validation of the flow solver.
Further investigations were performed both on the datum profile T150 and on the highly
loaded reference profile T151 in order to assess the influence of the angle between the j-lines
of the mesh in the boundary layer block and the blade solid wall on the cascade integral
performance parameters. In fact, a previous version of the automatic grid generation system
did not correct the slopes of the j-lines in the boundary layer block according to blade
geometry modifications. Thus the direction of the j-lines could depart strongly from the blade
orthogonal direction. This represented a major source of error for the calculation of the
integral boundary layer parameters, since the applied transition correlation by Drela (1995) is
based on these parameters. The correction of the j-lines slope implemented within
GRIDMOD is based on the modification of the distance between consecutive nodes,
distributed at specific positions over the entire blade suction surface. These modifications are
performed until a specific value, represented by the sum of the differences between the actual
slope and the direction orthogonal to the blade surface at specified locations, is minimised.
The slope correction is implemented only for the suction surface. In order to assess the
capability of the j-lines slope correction, two different meshes for the reference profiles T150
and T151 were analysed by changing the slopes of the mesh lines on the suction side. The
total number of nodes as well as their distribution law in the direction orthogonal to the wall
remained unchanged. While for T151 both meshes are obtained from the template mesh using
GRIDMOD with and without slope correction, for T150 the original mesh corresponds to the
template mesh itself. A comparison of the original and the modified mesh for T151 is
presented in Figure 4.6. The dashed lines in the zoom windows correspond to the mesh
obtained without slope correction. For T150 the differences between the two meshes are
somewhat reduced, because the template mesh presents j-lines which are orthogonal to the
blade surface. The difference between integral total pressure losses for the modified T150
mesh and template mesh is lower than 0.0001 points (about 0.1 %). The differences in terms
of exit flow angle are negligible. For T151 instead the differences are somewhat higher but
still limited. The integral total pressure loss coefficient calculated for the mesh without slope
correction is about 0.0005 points higher than using the mesh correction. This corresponds to a
difference of about 1.3 % of the calculated integral values. The difference in terms of exit
flow angle is limited to one hundredth of a degree. This underlines the importance of this
correction to increase the reliability of the information obtained from the flow simulation
procedure.
4. Numerical optimisation environment 65
Figure 4.6 Correction of the j-lines slope in the boundary layer block for T151
The choice of the optimal number of nodes in the O-block used in the template mesh is based
on additional preliminary computations performed on the turbine cascade T150. A concept
was sought which reduces the number of mesh variables to be controlled in the boundary
layer block. Thereby the nodes distribution law (Robert’s distribution law,
CFD Norway, 2003) and the boundary layer block thickness were fixed and the minimal
number of nodes was determined which is necessary in the O-block to resolve the boundary
layer correctly (Martinstetter, 2004b). The results of these computations showed that at least
thirty nodes in the j-direction are necessary in order to obtain reliable results. The results of
this mesh sensitivity study are shown in Figure 4.7. On the left hand side the influence of the
mesh onto the distribution of the wall shear stress on the suction side is shown. Slight
discrepancies can be observed between 32 and 40 nodes in j-direction, while the distribution
obtained using 20 nodes departs strongly from the other two. The variation in the computed
total pressure losses with changing number of nodes in j-direction is illustrated on the right
hand side of the figure. The curve shows that the template mesh requires at least 30 nodes in
j-direction in order to ensure reproducible results.
66 4. Numerical optimisation environment
Figure 4.7 Mesh sensitivity study for the choice of the number of nodes in j-direction
Based on the information obtained from these preliminary mesh studies, a system for the
automatic mesh generation was set up. This procedure adapts an existing template mesh to the
new blade geometry. The template mesh is represented by an ASCII file containing a list of
commands for the multi-block mesh generator Threemesh (CFD Norway, 2003). These
instructions specify the form of the block-boundary curves and the nodes distribution law
along the boundaries. GRIDMOD modifies the mesh ASCII file in order to adapt the template
mesh to a new arbitrary blade geometry. So that the grid topology remains unchanged and the
overall number of nodes remains constant. The distance of the nodes from the solid wall and
their distribution within the O-block remains unchanged as well. This is achieved by
extruding the profile using the angle information derived from a spline-representation of the
blade geometry itself. In this way the external boundary of the O-block presents a form which
is quite similar to the profile. The blade is represented by four spline segments (trailing edge,
suction side, leading edge and pressure side) obtained by the interpolation of the blade points.
The extrusion process takes place for each of these segments. The external O-block boundary
is divided into four spline segments as well. The interpolation technique implemented within
GRIDMOD for the treatment of these boundary curves is based on appropriate algorithms of
the NAG numerical libraries (Numerical Algorithms Group, 2005). The iterative procedure
for the slope correction of the j-lines follows, which checks the slope of the j-lines within the
O-block at particular locations and changes the distribution of the nodes along the profile (i-
direction) in order to control the slope of the j-lines. In this way the nodes distribution in the
boundary layer block is almost independent from blade geometry modifications. Furthermore,
a quite high grid resolution in the direction along the blade was chosen, so that strong changes
of the profile thickness could be performed. Further parameters (see Figure 4.8) are used to
adapt the position and form of the block boundary curves in order to reduce possible grid
quality changes in the inlet, passage and wake blocks. It must be considered also that a
generic block boundary curve is defined within the mesh generator by specifying the start and
end point as well as derivatives (angles) and influence factors of the curve angles at these
4. Numerical optimisation environment 67
points. However, GRIDMOD considers only the parametric modification of coordinates and
angles at start and end points of the mesh boundary curves.
The schematic representation of the parameters used within GRIDMOD is illustrated in
Figure 4.8. Once the new O-block is generated, the position of the different control points, Pi,
and the form of the boundary curves is determined. The location of P1 is expressed as a
function of the blade point at minimal axial x-coordinate B1, indicated as the cross point in
the leading edge region. P2 is positioned at a fixed distance from P1 and at the same axial
position. The point P3 is located at the lowest y-coordinate on the O-block external boundary.
The control curves P3-P4 and P3-P5 start from this position. The tangents of these curves
form an angle α3b and α3a with the horizontal axis respectively. While the curve P3-P4 ends
with a horizontal tangent, the curve P3-P5 terminates orthogonal to the O-block. While P4 is
located at a fixed distance from the point at the maximal x-coordinate on the blade B2
(indicated in the figure as the cross point in the trailing edge region), the position of the
control point P5 is fixed by the number of nodes on the lower side of the passage block,
which is related, for boundary conditions, to the number of nodes on the suction side segment.
The code requires a mesh featuring an equal number of nodes on the periodic boundaries.
Figure 4.8 Geometric system for the automatic mesh generation process GRIDMOD
68 4. Numerical optimisation environment
Figure 4.9 Application of GRIDMOD for the mesh generation of the turbine blade T152
Finally P6 is located at the position in the front pressure side region of the O-block where a
horizontal tangent is featured. The block boundary curve P6-P2 forms an angle α2 with the O-
block, which remains unchanged during the mesh adaptation process.
Changes of the trailing and leading edge thickness and of the pitch are treated within
GRIDMOD as well. In this case, however, the overall number of nodes changes within the
mesh adaptation process in order to avoid local mesh overlapping in the trailing and leading
edge regions. The position of P5 changes accordingly in order to respect the condition of an
equal number of nodes on the upper and lower periodic boundaries of the mesh
(Martinstetter, 2004a). Even if trailing and leading edge thickness were not considered as
design parameters within the automatic optimisation procedure, these features of GRIDMOD
reduced the mesh generation efforts considerably allowing extensive investigations of the
influence of the trailing edge thickness on the aerodynamic behaviour of the cascades (see
Chapter 5). The results of the application of GRIDMOD for meshing the reference profile
T152 are shown in Figure 4.9. Where the datum-profile geometry T150, on which the
template mesh is based, is shown as well.
4.2.2. The Navier-Stokes flow solver TRACE
The flow computations were performed using a two dimensional MPI-parallelised version of
the Reynolds-Averaged Navier-Stokes flow solver TRACE of the DLR in Cologne
(Eulitz, 2000). The flow solver TRACE is widely used in various German research and
4. Numerical optimisation environment 69
industrial departments. Recent publications document the major cooperation activities
between the DLR in Cologne and the University of the German Armed Forces in Munich for
the validation and further development of the RANS solver TRACE for different application
fields (e.g. Acton, 1998, Cardamone et al., 2002, Hilgenfeld et al., 2003). The present section
describes the most important numerical features of the flow solver and the major
characteristics of the turbulence and transition models used for the present investigations. For
a more detailed description of the numerical approaches used for the solution of the Navier-
Stokes equations the reader can refer to Hirsch (1988), while an excellent review on
turbulence modelling can be found in Wilcox (1993).
The Reynolds-averaged Navier-Stokes equations and the turbulence model are discretised on
multi-block structured grids. The space discretisation of the convective fluxes is based on the
TVD (Total Variation Diminishing) scheme by Roe, (1981), which is combined with a
MUSCL extrapolation scheme by Van Leer, (1979) in order to obtain second order accuracy
in space. The viscous derivatives are discretised using a second order central-differences
scheme. The present calculations were performed by neglecting the viscous diffusion in the
direction parallel to the shear surfaces using a thin-layer approximation (Hirsch, 1988). The
flow governing equations are solved in time using an implicit time integration technique as
described by Engel (1997). Non-reflecting boundary conditions formulated by Giles (1992)
are implemented at the inlet and outlet boundaries.
For the prediction of the boundary layer development on the blade surface a turbulence
closure based on the one-equation model by Spalart and Allmaras (1992) was used. Even
though many authors argue that two-equation turbulence models represent the minimum
acceptable level of closure for the Reynolds stress tensor (Speziale et al., 1998), in the
nineties the development of a new generation of one-equation turbulence models represented
a major trend in the scientific community. The goal was a simpler model ensuring an
equivalent level of computational accuracy as established two-equation models and at the
same time higher numerical robustness. Recent successful implementations of these models
attest the progresses achieved in this area (Eulitz, 1999, Arnone et al., 2001).
The major advantage of a one-equation turbulence model based on the formulation of Spalart-
Allmaras is that in this approach a simple transport equation for the eddy viscosity νT is used.
In this way any conceptual difficulty associated with the algebraic specification by empirical
means of a turbulent length scale is avoided. Furthermore, the eddy viscosity features lower
gradients near the wall than typical variables used for two-equation models like the turbulent
kinetic energy, k, or the dissipation rate, ε. This should ensure a mesh independent resolution
of the laminar sub-layer using a reduced number of nodes near the wall. Eulitz (2000)
indicates that while a mesh independent solution for two-equation turbulence models require
y+ values below 1, the usage of a one-equation model based on the transport of the eddy
viscosity requires y+ values below 3. These characteristics render this model an optimal
70 4. Numerical optimisation environment
choice for the application at high Reynolds numbers. For the present investigations a Spalart-
Allmaras turbulence model was used in the modified version by Eulitz (2000). The original
Spalart-Allmaras model can be written in the form:
1 2T T T T
Prod T Destr w Diff T Diffi i i i
Dv v v vC ω v C D C v C
Dt x x x x
⎡ ⎤∂ ∂ ∂ ∂= − + +⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ (4.1)
The transport model for the eddy viscosity is therefore set up of a convection term (on the left
hand side), a production term depending on the vorticity ω, a destruction term and additional
diffusion terms. The modifications introduced by Eulitz concerned the production and the
destruction terms. For the present investigations these modifications were all adopted with
exception of the additional production term introduced for a better modelling of the free
stream turbulence intensity introduced by incoming wakes. In fact, preliminary investigations
on the reference turbine cascade T150 (Jogwitz, 2002a) indicated that this formulation was
responsible for an un-physically large increase of the total pressure losses and was therefore
no further used for the present class of profiles.
Since the Spalart-Allmaras model, as with any other model based on the Boussinesq eddy
viscosity hypothesis, is completely void of any transition physics, the start of transition has to
be indicated. This is done by coupling the turbulence model with a transition correlation by
Abu-Ghannam and Shaw (1980) in the formulation by Drela (1995). In this model the
transition onset is located where the local momentum thickness Reynolds number, Reδ2,
The present section contains the results obtained applying the automatic design procedure at
different aerodynamic loadings. The investigations were run fixing the cascade deflection to
the value of the datum profile T150 and optimising the profile shape for various blade
spacings. The main target of the optimisation process was the reduction of the cascade total
pressure losses considering the discussed requirements on the profile velocity distribution for
an efficient cooling of the blade. The search space was restricted constraining the profile area
and section modulus according to the indications given at the end of the previous section. The
present investigations were performed using a set of 15 parameters for the description of the
blade geometry within the geometry generator PROGEN. The form of the objective function
F and the parameters used are described in the previous chapter as well. The trailing edge
thickness to pitch ratio and the axial chord were fixed at the values of the datum profile T150.
The influence of the trailing edge thickness on the cascade aerodynamic behaviour was
quantified by performing additional investigations presented at the end of this chapter.
Figure 5.1 shows typical convergence behaviour obtained during an optimisation run carried
out using an adaptive simulated annealing algorithm (ASA). The diagram shows that the
optimisation process converges after about 600 evaluations of the objective function F. This
corresponds to a reduction of the objective function of about 98% with respect to the
reference starting value and to a real computational time of about 24 hours on two SGI MIPS-
RISC 14000 processors of the SGI Origin 3800 system6. Preliminary investigations indicated
that an increase of the parameter temperature T and a reduction of the parameter describing
the cooling time with respect to the standard iSIGHT settings of ASA leads to improved
performance of the algorithm for the present application. The diagram refers to investigations
performed with a temperature parameter value of 10 and a halved cooling time parameter with
respect to the iSIGHT standard settings for the algorithm. The diagram shows that the
modified settings ensure that the algorithm still has sufficient energy to explore completely
different regions of the search space even after convergence has been reached. This is
expressed by the high values assumed by the objective function F even after convergence.
The convergence behaviour obtained using a multi island genetic algorithm (MIGA) is
illustrated in Figure 5.2. The investigations were performed using the iSIGHT standard
settings of the algorithm. A comparison with the convergence behaviour of ASA shows the
superiority of a simulated annealing approach for the present application.
6 The reference value of the objective function corresponds to the value assumed by F for the reference geometry used at the particular blade spacing at which the procedure is run (e.g. at the blade spacing of T151, the reference geometry corresponds to T150B−see section 5.1.1)
94 5. Results and discussion
Figure 5.1 Typical convergence behaviour for the design procedure in combination with
an adaptive simulated annealing algorithm
Figure 5.2 Typical convergence behaviour for the design procedure in combination with a
multi island genetic algorithm
5. Results and discussion 95
In fact, at about 5000 evaluations the MIGA results feature a significantly reduced but still
high value of the objective function F, indicating that the optimisation process has not yet
converged. Furthermore the distribution of the individuals in Figure 5.2 reveals the formation
of subpopulation clusters presenting similar values of the objective function.
5.1 Validation of the procedure at different aerodynamic loadings
The reliability of the method is demonstrated at variable aerodynamic loading, comparing the
optimisation results with the experimental and computational data of the reference cascades.
The results obtained from these investigations are illustrated in Figure 5.3. This diagram
shows the computed and measured integral total pressure losses versus the aerodynamic lift
coefficient. Thereby the results of the design method are compared with the reference data
obtained on cascade T150 and T151. Since the aerodynamic loading level of the profile T152
is about the same as for the datum blade T150, the data of T152 are not shown here for
clarity. The integral total pressure losses are referred to the experimental results obtained on
the datum profile T150 at reference operating conditions (Re2th=1 200 000, Ma2th=0.75,
β1=β1,T150). The triangles represent the TRACE computations, while the squares correspond to
the experimental results.
Figure 5.3 Predicted total pressure losses and comparison with the values of the reference
cascades T150 and T151
The results of the optimisation method are represented by circles. The aerodynamic loading
was modified by changing the pitch to chord ratio. The operating conditions were fixed at the
96 5. Results and discussion
reference conditions of the datum profile, maintaining the cascade deflection featured by
T150 at the reference operating point. The trailing edge thickness and the axial chord were
fixed at the values of the datum profile T150. The optimisation data presented in the figure
were obtained by running the design method at an inlet turbulence intensity Tu1 of 3%. The
results shown in the diagram are summarised in Table 7.6 (see Appendix).
The comparison of the experimental and the numerical results of the reference profiles reveals
the discrepancies already discussed in chapter 4. While the difference in vertical direction can
be directly attributed to the over-prediction of the total pressure losses, the discrepancy in the
horizontal direction can be mainly associated with the difference between the predicted and
the measured exit flow angle, since this is directly used for the determination of the lift
coefficient CL (equation 3.1). Nevertheless both the discrepancies in the total pressure losses
and exit flow angles remain nearly constant for all the reference profiles T150, T151 and
T152, independent of the aerodynamic loading, as observed in the previous chapter.
Furthermore this difference remains approximately constant over a wide operating range for
each reference blade geometry. This leads to the conclusion that the computational results on
the three reference turbine cascades represent a reliable reference for the assessment of the
capabilities of the design method.
The optimisation results indicate the high potential of the method in reducing the total
pressure losses at increased aerodynamic loading. Furthermore, it can be observed that at the
examined reference Reynolds number Re2th=1 200 000, the expected increase of the right
branch of the curve could not be detected. This indicates that even at very high lift
coefficients (point 4 in the plot), the method locates blade geometries, which avoid separation
phenomena. The direct comparison of the optimisation results and the reference data shows a
slight reduction of the total pressure losses at the spacing of the datum profile T150 (point 1
in the diagram, further on indicated as ASAT150). The results obtained at the pitch ratio of
turbine blade T151 (point 3, ASAT151) feature slightly higher losses than T151. In this case,
however, a comparison can only be performed considering that the optimised blade geometry
ASAT151 features the same trailing edge ratio as T150, which is higher than the one of T151.
In order to quantify the influence of the trailing edge thickness on the cascade total pressure
losses, additional investigations were performed. These were carried out on an auxiliary blade
geometry obtained from T150 by reducing the trailing edge thickness to the value of T151.
The resulting blade geometry, called T150HK, was re-staggered in order to perform the
comparison at an unchanged cascade deflection. These investigations showed that the
reduction of the total pressure losses between cascade T150 and T151 (indicated by the
segment AC in the diagram) is attributable by 50% to the reduced trailing edge thickness
featured by T151 (segment AB), while the remaining 50% originates from the reduced wetted
surface (segment BC). The results of these investigations are discussed in more detail at the
5. Results and discussion 97
end of this chapter. These considerations point out clearly the high potential of the developed
method for the reduction of the total pressure losses at high aerodynamic loading.
5.1.1. Results at the spacing of turbine cascade blade T151
Firstly the results of the automatic procedure for the re-design of cascade T150 at the blade
spacing of T151 are presented. The profile Mach number distribution obtained in this case is
compared with the reference distributions in Figure 5.4. The velocity distribution on ASAT151
is represented by the continuous line, the Mach number distribution on the turbine cascade
blade T151 is represented by the dashed line and the dash-dotted line corresponds to the
datum profile T150. The results on an additional reference profile called T150B correspond to
the dash-dotted-dotted line. This profile was obtained from T150 by reducing the blade
stagger angle and the exit blade metal angle in order to ensure the same deflection as T150 at
the pitch ratio of T151. Turbine cascade T151 features ten degrees higher deflection than
cascade T150B, but somewhat lower mass flow capacity. Nevertheless its profile velocity
distribution features smooth acceleration both on the suction and on the pressure surface and a
comparison with the optimisation results is useful to assess the efficiency of the present
method. The comparison shown in Figure 5.4 indicates that the optimisation procedure is able
to locate blade geometries featuring smooth profile Mach number distributions and at the
same time reduced total pressure losses. The optimised geometries respect the specified
geometric and mechanic requirements and ensure the required cascade deflection.
The profile Mach number distribution on turbine cascade ASAT151 fulfils the specified
aerodynamic requirements for an optimal cooling of the blade profile. This demonstrates the
capabilities of the method and the successful design of the objective function for this
application. The flow accelerates smoothly under quite high gradients in the front part of the
blade suction side. This region is followed by a second acceleration area with moderate
gradients which terminates at the suction side peak Mach number. The maximal suction side
Mach number is located slightly below 1.0, as prescribed in appropriate form within the
objective function. Geometries featuring supersonic flow regions on the suction surface, like
the reference profile T150B, are excluded during the optimisation process. The flow on the
pressure side accelerates smoothly over the entire profile featuring stronger gradients in the
rear part of the blade, where a relaminarisation region is predicted. The short separation
bubble featured by the datum profile T150 in the leading edge region on the pressure surface
is no longer present on ASAT151, indicating that the method is able to identify and suppress
this feature. The results of preliminary investigations, performed at the present blade spacing
and presented in this section as well, indicate that the separation bubble in this area and the
local diffusion phenomena in the front part of the suction surface can be suppressed only
using an asymmetric representation of the blade leading edge.
98 5. Results and discussion
Figure 5.4 Optimisation results at the blade spacing of turbine cascade T150
Figure 5.5 Comparison of the optimised blade ASAT151 with the profiles T150 and T151
5. Results and discussion 99
Figure 5.5 shows a comparison of the optimised blade geometry ASAT151 and of the datum
profile T150. The geometry T151 is shown for comparison as well, even though the
geometrical and mechanical constraints in this case differ strongly from the conditions of
T150, used as a reference for the optimisation. The optimised blade ASAT151 is represented by
a continuous line, while the datum profile T150 corresponds to the dash-dotted line. The
zoom of the leading edge region reveals the asymmetry of the optimised blade shape in this
region. Moreover, the detail of the rear part of the blade shows that the new design features an
area distribution fulfilling the requirements for the placement of the necessary internal cooling
air channels. This is associated with large wedge angles at the trailing edge (higher than 4
degrees). A further increase of the aerodynamic loading leads to blade geometries, whose
curvature in the rear part of the suction surface can not be further increased because of
undesired flow over-expansions. This produces the generation of blade geometries with low
profile wedge angles at the trailing edge (see section 5.1.3). Using the tangent angles, wi, at
the blade control points 2 and 3 (see section 4.1) instead of a single wedge angle, γ, at the
leading edge ensures that the form of the suction side spline is decoupled from the form on
the pressure surface. A comparison of the present results with the results obtained using a
symmetric leading edge (presented at the end of this section) indicates that the asymmetric
leading edge approach provides better conditions for an optimal distribution of the blade
curvature over the entire suction surface. In addition the suppression of local diffusion zones
in the front part of the suction and the pressure surface becomes possible. In order to obtain a
smooth acceleration in the front part of the suction surface the related spline segment features
high values of the tangent angle, w2. The suppression of local diffusion zones in the front part
of the pressure surface is achieved in combination with nearly horizontal tangents at the blade
control point 3 (corresponding to w3 values of almost 90 degrees). The increase of the overall
thickness of the profile has to be carefully limited. In fact, an increase of the blade thickness
is associated with higher internal cooling air mass flow and complexity of the internal cooling
channels with a consequent increase of the casting costs.
In the following the calculated boundary layer development on the optimised blade ASAT151
is compared with the reference data on T150 and T150B. The upper diagram in Figure 5.6
illustrates the calculated distribution of the wall shear stress Cf. The distribution of the form
factor H12 on the suction and the pressure surface of the optimised blade geometry ASAT151
and of the reference blades T150 and T150B is shown in the central and the lower diagram of
the same figure. The bars shown in the form factor diagrams indicate the extension of the
transition zones. The results show that, in spite of the increased aerodynamic loading, the
transition location on the suction side of ASAT151 is placed approximately at the same position
as for the datum profile T150. The distribution of the wall shear stress in the front part on the
suction side of ASAT151 is somewhat higher than for the two other cascades. This is in
accordance with the increased loading of the front part of the suction side of the optimised
profile ASAT151.
100 5. Results and discussion
Figure 5.6 Boundary layer development on the optimised and the reference blade profiles
(the bars indicate the transition regions)
5. Results and discussion 101
In the rear part of the suction side, where a full turbulent boundary layer exists, ASAT151
features lower wall shear stress values than T150 or T150B. For the cascade blade T150B the
wall shear stress increase in the rear suction side region is related to the shock occurring in
this region. For all the examined profiles the code predicts a relaminarisation in the rear part
of the pressure side. This is associated with the high acceleration gradients featured by the
profile Mach number distributions in this region and is indicated by the strong increase of the
form factor. For cascade T150, the short separation bubble featured in the front part of the
pressure side causes the turbulent boundary layer to develop further upstream than for the
other two cascades, as shown by the bars in the diagram at the bottom of Figure 5.6.
Extensive preliminary investigations were performed at this blade to pitch ratio in order to test
different formulations of the objective function F. These calculations were performed at a
lower inlet turbulence intensity Tu1 = 1.5%. The weighting coefficient for the total pressure
losses in the objective function was reduced from 50 to 1 in order to limit the weight of the
total pressure losses on the optimisation results. This gave the possibility to investigate in
more detail the behaviour of the other terms related to cascade deflection and profile Mach
number distribution.
Figure 5.7 presents the geometries and the related velocity distributions resulting from the
application of different formulations of the objective function, F. The blade geometries
ASAT151-A, ASAT151-B and ASAT151-C are obtained from preliminary investigations carried out
at reduced inlet turbulence level (Tu1 = 1.5%), while the geometry ASAT151 is the optimised
blade, whose features have already been presented in the first part of this section. While for
the description of the blade nose of the profile ASAT151 an asymmetrical approach was used,
the other cascades feature symmetrical leading edges. The relative total pressure losses and
exit flow angles for these blade geometries are shown in the figure as well. The bars represent
the extension of the computed transition zones on the blade suction surface. The total pressure
losses for profile ASAT151-A are slightly lower than for ASAT151-B and ASAT151-C. The increase
of the inlet turbulence level for ASAT151 moves the transition zone towards the leading edge,
with a related increase of the total pressure losses. The optimised blade ASAT151 is the result
of investigations performed using an objective function like in equation 4.7 and repeated here:
{ }{ }
is is is
ζ β
Ma MAX SS AMa MAX SS BD Diff DS LE Diff SS LE AMa INFL SS
F F F
F F F F F F− − − − − − − − − −
= + +
+ + + + + + (5.1)
where the structure and significance of the single components is described in detail in
section 4.5. Instead, the geometry ASAT151-A is the result of an optimisation carried out using
a simplified form of the objective function, which takes into account only the integral total
pressure losses, the cascade deflection and the maximal Mach number on the suction surface.
This is expressed in the form:
- -isβ Ma MAX SSF ζ F F= + + (5.2)
102 5. Results and discussion
where, as already mentioned, the component accounting for the total pressure losses is
expressed as the integral value of the total pressure losses and FMais-MAX-SS is the linear
function (Mais-MAX-SS − 1.0) / 10.0. This term is activated only if on the suction surface the
maximal Mach number is greater than 1.0. Furthermore the term Fβ remains unchanged with
respect to the form used for ASAT151 (see section 4.5).
Figure 5.7 Optimisation results for different formulations of the objective function F
The objective function used for the investigations relative to the optimised blade geometry
ASAT151-B has the form:
- - - -is isβ Ma MAX SS AMa MAX SS BDF ζ F F F F= + + + + (5.3)
5. Results and discussion 103
Thus no information about local diffusion phenomena in the front part of the pressure and the
suction surface is considered in this case. Furthermore, the component accounting for the area
among inflection points on the profile velocity distribution FAMais-INFL-SS is excluded as well.
The objective function used for cascade ASAT151-C features all the terms contained in the
objective function of ASAT151 with the exception of FDiff-SS-LE and FAMais-INFL-SS. Moreover, the
total pressure losses are expressed without using any weighting factor:
- - - - - -is isβ Ma MAX SS AMa MAX SS BD Diff DS LEF ζ F F F F F= + + + + + (5.4)
The profile Mach number distribution featured by ASAT151-A reveals clearly that a linear form
for the term taking into account the Mach number level in the objective function is not a
suitable approach to avoid supersonic regions on the blade profile. For this term a polynomial
of higher order was therefore proposed (see section 4.5) and applied for generating the other
cascades presented in this section. Moreover, as long as the suction peak in the front part of
the suction side is not strong enough to generate turbulent boundary layers or the following
acceleration region leads to a process of boundary layer relaminarisation, the local region is
not negatively rated within the optimisation loop. This circumstance is clearly shown from the
results on cascades ASAT151-A, ASAT151-B and ASAT151-C, where the solver predicts a
relaminarisation in the acceleration region following the local suction peak on the front part of
the suction surface. Therefore, the objective function F used to obtain ASAT151 was provided
with a supplementary term for the quantification of local diffusion phenomena in the front
part of the suction surface, since these phenomena are not advantageous for an efficient action
of the shower head cooling in this area.
Even if the presence of local diffusion zones leads to earlier transition of the boundary layer,
the influence of the pressure side boundary layer on the integral value of the total pressure
losses is moderate. Therefore the introduction of an additional term in the objective function
F accounting for the diffusion in the front part of the pressure surface is necessary to control
the formation of suction peaks in this area. This is confirmed by a comparison of the forms of
the profile velocity distribution of ASAT151-C and ASAT151 (both featuring the objective
function term FDiff-DS-LE) with those of turbine cascade blades ASAT151-A and ASAT151-B.
Furthermore an accurate analysis of the velocity distribution on ASAT151-C in the front part of
the pressure side reveals that the flow does not accelerate as smoothly as at the same location
on ASAT151. This leads to the conclusion that only the combination of an asymmetrical
representation of the leading edge with this additional term of the objective function ensures
optimal conditions for finding profiles like ASAT151 featuring smooth acceleration in the front
part of the pressure surface.
Another additional feature of the objective function applied for cascade ASAT151 is
represented by the term FAMais-INFL-SS. This term ensures a smoother acceleration on the suction
surface by penalising the regions between local inflection points in an appropriate way. The
104 5. Results and discussion
successful realisation of the scope of this additional term is demonstrated by the acceleration
behaviour in the front part of the suction surface on ASAT151.
The present results show that at convergence, independent of the specific form of the profile
Mach number distribution, the method is able to produce geometries featuring an almost
constant level of losses and quite similar cascade deflections. The total pressure losses are
considerably reduced if compared with the reference geometries, both using relatively simple
objective functions like for ASAT151-A and more complex formulations of F like for ASAT151-C
or ASAT151. In particular a comparison of the integral aerodynamic performance parameters
for the cascades ASAT151-A, ASAT151-B and ASAT151-C shows that the modifications of the
objective function concerning the requirements on the profile velocity distribution for an
efficient cooling do not influence the optimisation in terms of overall aerodynamic cascade
performance.
5.1.2. Results at the spacing of turbine cascade blade T150
Further investigations were carried out at the blade spacing of the datum turbine cascade
blade T150. Thereby the structure of the objective function and the blade parameters were
derived from the investigations performed at the pitch to chord ratio of turbine blade cascade
T151. Using the results obtained at increased aerodynamic loading, an asymmetrical leading
edge representation was used. Furthermore, the objective function was provided with an
additional term accounting for the axial position of the maximal Mach number on the suction
surface with respect to the location of the throat of the turbine cascade passage:
- - - - - - - -
- -
is is
is
ζ β Ma MAX SS AMa MAX SS BD Diff DS LE Diff SS LE
AMa INFL SS Xe
F F F F F F F F
F F
= + + + + + + +
+ + (5.5)
The importance of this additional term in fulfilling of the requirements regarding the form of
the profile Mach number distribution is discussed in the present section. The results obtained
using this additional term in F are compared with the results derived from an objective
function without this additional component. The profile Mach number distribution of the
optimised profile ASAT150 is compared with the velocity distribution of the reference cascades
T150 and T152 in Figure 5.8. This figure shows that the present design method ensures
smooth acceleration on the suction side and is able to suppress the short separation bubble
featured at the front part of the pressure surface by T150 even at reduced aerodynamic
loading. Furthermore, the maximal Mach number featured by ASAT150 on the suction side is
lower than the suction side peak Mach number on the reference cascades. The deceleration
taking place in the rear part of the suction surface of ASAT150 occurs under moderate
gradients, as well. The reduction of the maximal Mach number on the suction surface depends
greatly on the formulation of the related term of the objective function FMais-MAX-SS. In fact,
preliminary investigations performed using a more aggressive formulation of the term
limiting the peak Mach number on the suction surface showed that a further reduction of the
5. Results and discussion 105
Mach number level is possible. The figure illustrates that the Mach number distribution on the
pressure surface of profile T152 is smoother than for the other two cascades. Since the present
investigations were carried out using a set of only three parameters for the description of the
pressure side spline segment, the necessary parameters for the representation of this part of
the blade should be investigated in more detail in order to produce smoother velocity
distributions on the pressure surface.
Figure 5.8 Optimisation results at the blade spacing of turbine cascade T150
Figure 5.9 Comparison of the optimised blade ASAT150 with the profiles T150 and T152
106 5. Results and discussion
A comparison of the optimised profile geometry ASAT150 and of the reference geometries
T150 and T152 is shown in Figure 5.9. The turbine cascade blade T152 features a lower
stagger angle than T150 or ASAT150. This is necessary in order to ensure the prescribed
deflection (almost the same as for T150) in spite of the increased throat opening associated
with the reduced trailing edge thickness of T152. The enlargement of the leading edge region
shown on the left hand side of the figure illustrates the asymmetrical form of the blade nose of
ASAT150. This ensures the suppression of local diffusion phenomena in this area. The area
distribution in the rear part of the blade ensures the placement of the necessary cooling ducts.
Figure 5.10 Optimisation results for different formulations of the objective function F
5. Results and discussion 107
Figure 5.10 displays the results obtained applying the optimisation method at the pitch to
chord ratio of turbine cascade blade T150. Different formulations of the objective function, F,
were used. The results indicate that the introduction of ad hoc tailored terms in the objective
function ensures a successful control of the lift distribution over the blade surface. The bars
represent the extension of the transition zones on the suction side. The blade geometry
MIGAT150 is the result of the investigations carried out with a multi island genetic algorithm
at the given aerodynamic loading. In this case, even if F is reduced significantly, no full
convergence was reached after about 5000 evaluations, corresponding to a real time of more
than 8 days. This is a quite large time compared with a usual running time (about 24 hours)
for the investigations carried out with an adaptive simulated annealing algorithm. The
comparison of the profile velocity distributions confirms that the ASA approach presents the
most advantageous characteristics for this kind of application. The integral results show that
also at this aerodynamic loading the method is able to reduce the cascade total pressure losses
maintaining the prescribed deflection.
The effects of different formulations of the components of the objective function related to the
Mach number distribution are presented in the following. The turbine cascade ASAT150-A is
obtained using an objective function with the same structure as the one used for cascade
ASAT151 (equation 5.1). However, the term related to the peak Mach number on the suction
side, FMais-MAX-SS, is modified, as described in section 4.5, in order to take into account the
reduced aerodynamic loading and consequently reduced peak Mach numbers. The profile
Mach number distribution on ASAT150-A is more front loaded than the other configurations
and the peak Mach number is located upstream of the cascade passage throat. Even if the lift
distribution on ASAT150-A is more in front loaded than the others, the transition zone is placed
further axially downstream compared to the other configurations. This can be explained
considering that at the present free stream turbulence level (Tu1 = 3.0%) the transition region
is located in the front part of the profile, at a position where visible changes of the
acceleration gradients occur. The flow follows the profile in this zone smoothly to a slightly
greater extent on ASAT150-A than on the other blades. Furthermore, it has to be considered that
an eventual further acceleration taking place on ASAT150-A downstream of 40% axial chord is
limited by the term of the objective function controlling the peak Mach number on the suction
surface. The maximum admissible Mach number is already achieved at 40% axial chord.
Even if for ASAT150-A the acceleration in the front part of the profile suction side and on the
whole profile pressure side is smooth, it has to be considered that due to the limited
knowledge of the efficiency of film cooling on rotor blades in presence of such profile
distributions (see section 2.2), it is common for these applications to place the maximal
suction side Mach number in the vicinity of the passage throat. Considering this aspect and
since one of the major drivers of the present method is its applicability as a reliable tool
within industrial blade design procedures, an additional term for generating aft-loaded profiles
was integrated within the objective function. This term limits the movement of the suction
108 5. Results and discussion
side peak Mach number upstream of the blade throat. After extensive preliminary tests the
following polynomial formulation could lead to the most suitable performance:
( ){ } 3
1 2( - - ) ( )XeC
Xe is Xe XeF x Ma MAX SS x Throat C C= ⎡ − − ⎤ ⋅⎣ ⎦ (5.6)
An appropriate choice of the coefficients ensures that if the position of the velocity maximum
on the suction surface is located downstream of the passage throat, the term FXe becomes
negligible. The blade geometry ASAT150 results from the application of the same objective
function as ASAT150-A but with the additional term FXe as specified in equation 5.6. The
turbine blade ASAT150-B is obtained using the same objective function as for ASAT150 but with
modified coefficients for FXe in order to enlarge the tolerated zone downstream of the cascade
passage throat. The coefficients used for ASAT150 are CXe1=0.12, CXe2=6.0 and CXe3=14.
Instead for ASAT150-B the admissible range was enlarged by reducing CXe1 to 0.08. The
resulting velocity distributions on cascade ASAT150 and ASAT150-B show that this approach
permits the integration of further lift distribution rules within the optimisation process.
5.1.3. Optimisation results at varied blade spacing
The results obtained at the pitch to chord ratio of turbine cascade blades T150 and T151
indicate that the aerodynamic design method generates blade geometries featuring reduced
total pressure losses by respecting the prescribed cascade deflection. The additional terms of
the objective function, introduced for controlling the velocity distribution on the blade profile,
do not alter significantly the minimal level of losses obtainable using the optimisation
method. The present section illustrates the additional results of the optimisation method at two
further blade loadings, indicated in Figure 5.3 as turbine cascade blades 2 and 4. The present
investigations were carried out in order to ascertain the minimal level of losses obtainable
applying the present aerodynamic design method at different lift coefficients.
Figure 5.11 illustrates the velocity distributions and integral cascade performances for the
profiles obtained from the optimisation process. The results obtained at t/l = 0.89 and at
t/l = 1.15 are illustrated respectively on the left and on the right hand side of the figure. Whilst
the investigations performed at t/l = 0.89 were carried out using the same structure of the
objective function and the same parametrical description of the blade as for ASAT151, the
results at t/l = 1.15 were obtained using the same structure of the objective function, F, and
the same parametrical description of the blade (symmetrical leading edge) used for ASAT151-C.
The term of the objective function, FMais-MAX-SS, introduced for controlling the maximal Mach
number on the suction surface is modified according to the different loading levels (see
section 4.5). As shown in Figure 5.11, the blade geometry ASA2 features an asymmetrical
blade nose while ASA4 presents a symmetrical leading edge. The considerations outlined in
section 5.1.1 about the advantages of an asymmetrical description of the blade leading edge
are underlined again comparing the velocity distributions on the present profiles.
5. Results and discussion 109
Figure 5.11 Optimisation results at further blade spacing (left: t/l = 0.89; right: t/l = 1.15)
5.1.4. Modification of the geometrical and mechanical constraints
Additional investigations were carried out at the pitch to chord ratio of the datum profile T150
modifying the geometrical and mechanical constraints which the blade profiles have to
respect. In fact, as already mentioned in the previous sections, the introduction of an
asymmetrical representation of the blade leading edge decouples the suction side from the
pressure side, leading to a class of optimised profiles with high blade section areas.
Figure 5.12 shows a typical distribution of the section area resulting from an optimisation run
performed using an asymmetrical description of the blade leading edge. The diagram shows
that after a limited number of evaluations, the optimisation procedure generates geometries
which all feature high blade section areas, being located on the upper limit for the relative
area (20% higher than the area for T150). The increase of the blade section area is associated
with an increased demand of internal cooling air mass flow and eventually with a new and
more complex design of the internal cooling channels, which may have important
consequences regarding the blade casting costs. In particular the results presented in the figure
refer to an optimisation carried out at the pitch to chord ratio of the datum profile T150. It is
indeed at low pitch ratio where the present circumstance assumes particular relevance. In fact,
at high blade spacing the increase of the necessary cooling air per pitch is associated with
higher blade section areas. At the same time however, high blade spacing means an overall
reduction of the number of blades with a related reduction of the manufacturing costs.
Therefore if the optimisation method leads to high blade section areas already at a low blade
spacing (where no counterbalancing advantage deriving from a reduced number of parts is
110 5. Results and discussion
present) it is necessary to assess the capability of the design procedure by searching for
optimised blade profiles in a new range of reduced section areas at a lower blade spacing.
Figure 5.12 Relative profile area distribution for asymmetrical leading edge description
With this in mind, investigations were carried out using the same settings used for ASAT150
and reducing the range of admissible blade section areas from 0.90 ≤ Area/AreaT150 ≤ 1.20 to
0.70 ≤ Area/AreaT150 ≤ 0.90. The constraints on the section modula were adjusted
accordingly. In order to reduce the blade section area in such a way, the blade nose thickness
was reduced by one third. Thus within the present investigations blade geometries are
obtained, which no longer feature thick blade noses typical of high pressure turbine profiles.
However, the aim of the present investigations is the assessment of the behaviour and
reliability of the numerical design method at changed geometrical and mechanical conditions
and this aspect is thus not irrelevant in this context.
Figure 5.13 shows the optimisation results at the modified geometrical and mechanical
boundary conditions. The resulting geometry is indicated in the figure with ASAT150-C. The
dash-dotted line represents the velocity distribution on the datum profile T150. The results of
the present investigations indicate that even at modified geometrical and mechanical
constraints the method produces blade geometries leading to low losses and presenting the
prescribed aerodynamic features concerning velocity distribution and cascade deflection. The
bar indicates the location of the predicted transition zone on the suction surface of the profile
ASAT150-C. This zone is shifted slightly downstream with respect to ASAT150 and is associated
with a reduction of the level of the total pressure losses.
5. Results and discussion 111
Figure 5.13 Optimisation results at modified boundary conditions
5.2 Influence of the trailing edge on the aerodynamic behaviour
The trailing edge thickness represents a major factor for the assessment of the optimised
turbine profiles. The reference cascade blades T151 and T152 feature a reduced trailing edge
thickness with respect to the datum profile. In order to quantify the influence of this
geometrical parameter on the aerodynamic performances of the examined profiles additional
investigations were carried out. Preliminary studies took place on the auxiliary turbine
cascade blade T150HK. This geometry was obtained by reducing the trailing edge thickness to
the throat ratio of the datum profile T150 to the value of T151 and re-staggering the profile in
order to maintain the same deflection of T150.
The modifications of the profile were performed within the parametrical geometry generator
PROGEN. In this way, the form of the pressure and suction surface is modified according to
the reduced profile thickness at the trailing edge and the continuity of the profile derivatives
up to the second order at the blade control points is preserved. The resulting blade geometry
T150HK is compared with the datum profile T150 in Figure 5.14.
112 5. Results and discussion
Figure 5.14 Comparison of the geometries of the turbine cascades T150HK and T150
Figure 5.15 Calculated wall shear stress on the turbine cascade blades T150 and T150HK
5. Results and discussion 113
The boundary layer development predicted on the suction and the pressure surface of the two
profiles does not reveal significant differences as shown from the skin friction distribution on
the blade profile in Figure 5.15. Therefore a comparison between T150 and T150HK allows a
direct assessment of the effects of the trailing edge thickness on the cascade total pressure
losses. The evaluation of the predicted losses leads to the conclusion that half of the
difference between the total pressure losses featured by cascade T150 and T151 is attributable
to the reduced trailing edge thickness. Using the present approach the difference between the
predicted exit flow angles for the cascades T150 and T150HK is limited to one tenth of degree.
Cecchi (2003) quantified the influence of the trailing edge thickness in separate
investigations. The trailing edge thickness to throat ratio of the datum profile was reduced to
the value of T151 but the profile was closed by rotating the pressure side curve with respect to
the conjunction point between the leading edge and the pressure side curve. The resulting
blade geometry is indicated as T150HK1 in the following. This approach leads to a
discontinuity of the profile both in the leading and in the trailing edge region. However, since
both discontinuities take place on the pressure surface in zones featuring strong acceleration
gradients, their influence on the boundary layer development is not relevant. Calculations
performed by Martinstetter (2004b) with the RANS solver TRACE on the same blade
geometries attest the negligible influence of these geometric discontinuities on the
development of the boundary layer. The numerical simulations by Cecchi (2003) were carried
out using the viscous-inviscid cascade analysis code MISES (Drela et al., 1998). The
beginning of transition was forced at xax/lax = 0.2 on the suction surface both for T150 and for
T150HK1. This corresponds to the position of abrupt change of the velocity gradients in the
accelerating region of the suction surface. The calculations with MISES performed on T151
fixed the beginning of the transition on the suction surface at a location xax/lax = 0.6. Using the
present approach the cascade exit flow angle was not forced. The 40% of the total pressure
loss reduction between T150 and T151 can be attributed to the reduced trailing edge
thickness. The residual 60% is attributable to the reduced wetted surface featured by T151.
This result is similar to those obtained from the calculations performed with TRACE on the
profile T150HK, where the exit flow angle was forced by modifying the profile suction side.
Since the results regarding T150HK1 obtained from calculations performed with TRACE again
indicate that 50% of the integral total pressure losses difference between T150 and T151
depends on the trailing edge thickness, the slight difference between the two approaches is to
be ascribed to the different simulation methods.
In order to assess the sensitivity of the cascades to the trailing edge thickness at different
Mach number regimes, additional investigations were performed. Starting from the datum
profile T150 and modifying the trailing edge to throat ratio, various blade profiles were
obtained. Only the pressure side of these profiles was modified accordingly and the exit flow
angle was not forced. Figure 5.16 shows the integral total pressure losses resulting from the
114 5. Results and discussion
calculations with TRACE. Each curve in the diagram corresponds to a defined trailing edge to
throat ratio. The abscissa corresponds to the operating Mach number and the ordinates
represent the integral total pressure losses. The circles correspond to the experimental (white)
and numerical (grey) results obtained on the baseline T150. At subsonic flow regimes the
sensitivity of the profiles to Mach number variations is quite limited and a higher trailing edge
thickness corresponds to higher losses. At an increased Mach number, where supersonic flow
regions appear on the profile suction surface, the profiles presenting reduced trailing edge
thickness react more sensitively to Mach number changes. This is illustrated by the increased
gradients featured by the curves for a lower trailing edge ratio in the right part of the diagram.
This evidence is associated with a decrease of the shock wave strength for increased trailing
edge blockage. In fact, a higher trailing edge thickness is associated with higher velocities at
the cascade exit plane and thus reduced velocity differences between peak suction surface
Mach number and exit Mach number. These considerations are illustrated in Figure 5.17 by
the stronger shock wave featured by the profile with reduced trailing edge thickness at
transonic flow regimes (Ma2th = 0.90) with respect to the baseline profile T150.
Figure 5.16 Trailing edge effects on the total pressure losses at different Mach number
regimes (TRACE calculations)
5. Results and discussion 115
Figure 5.17 Profile Mach number at Ma2th = 0.90 at changed trailing edge thickness
Figure 5.18 Trailing edge effects on the total pressure losses at different Mach number
regimes (MISES calculations)
116 5. Results and discussion
The distribution of the total pressure losses over the Mach number at a changed trailing edge
thickness predicted with the MISES code (Cecchi 2003) shows a stronger sensitivity for the
trailing edge ratio at higher Mach number regimes. This depends fundamentally on the
stronger shock waves predicted at transonic conditions by MISES with respect to TRACE
(Martinstetter, 2004b). The results obtained by Cecchi are represented in Figure 5.18. In this
case the predicted sensitivity of the turbine cascades to the trailing edge thickness for
transonic flow regimes is so high that the total pressure losses featured by the blades with
reduced profile thickness at the trailing edge exceed the values of the blades with higher
trailing edge blockage.
6. Summary and conclusions 117
6. Summary and conclusions
The design of turbine cascade blades for heavy duty gas turbines has to take into account
various often counteracting aspects deriving from the interaction of different disciplines. A
major aim pursued in the development of modern turbine bladings is a reduced number of
blades. In fact, even if the maximal temperature in heavy duty gas turbines is somewhat lower
than in aero engines, life expectations and maintenance intervals are expressed here in
thousand of hours instead of hours (Madfeld et al., 2004). This corresponds to a demanding
challenge for the materials applied in the high pressure components, which can be met only
making use of advanced materials (e.g. Nickel based super alloys or ceramic composite
matrix) in combination with expensive casting techniques for obtaining single crystal
structures, advanced thermal protection coatings and extensive cooling procedures. All these
aspects contribute to increased manufacturing costs. Thus, the reduction of the number of
blades represent a possible way for limiting costs. Furthermore, this design strategy is
associated with beneficial effects under an aerodynamic point of view like reduced wetted
surface and reduced quantities of cooling air mass flow per stage. Nevertheless, the resulting
increase of the aerodynamic lift coefficients produces unfavourable effects like increased
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Vita
Personal Pasquale Cardamone Born in Catanzaro (Italy) on the 10th of May 1973
Italian nationality
Married, one child
Education
September /1978 – June/1986 Primary and secondary school in Catanzaro
September/1986 – July/1991 Scientific High school in Catanzaro
October/1991 – April 1999 Degree in Mechanical Engineering at the University of
Florence Professional experience April 1999 – January 2000 Research fellow at the institute of energetics of the
University of Florence (Prof. Martelli) February 2000 – June 2005 PhD Student at the Institute of Jet Propulsion of the
German Armed Forces University Since July 2005 Project engineer at E.ON Energy Projects in Munich