Numerical Modelling of a Radial Inflow Turbine with and without Nozzle Ring at Design and Off-Design Conditions Filippo Valentini Master of Science Thesis EGI_2016-094 MSC EKV1169 KTH School of Industrial Engineering and Management Machine Design SE-100 44 STOCKHOLM
77
Embed
Numerical Modelling of a Radial Inflow Turbine with and ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Numerical Modelling of a Radial Inflow Turbine with and without Nozzle Ring at Design and Off-Design Conditions
Filippo Valentini
Master of Science Thesis EGI_2016-094 MSC EKV1169
KTH School of Industrial Engineering and Management
Machine Design
SE-100 44 STOCKHOLM
Master of Science Thesis
EGI_2016-094 MSC EKV1169
Numerical Modelling of a Radial Inflow Turbine with and without Nozzle Ring at Design and Off-
Design Conditions
Filippo Valentini
Approved
Examiner
Paul Petrie-Repar
Supervisor
Jens Fridh
Commissioner
Contact person
Abstract
The design of a radial turbine working at peak efficiency over a wide range of operating
conditions is nowadays an active topic of research, as this constitutes a target feature for
applications on turbochargers. To this purpose many solutions have been suggested, including
the use of devices for better flow guidance, namely the nozzle ring, which are reported to boost
the performance of a radial turbine at both design and off-design points. However the majority of
performance evaluations available in literature are based on one-dimensional meanline analysis,
hence loss terms related to the three-dimensional nature of real flows inside a radial turbine are
either approximated through empirical relations or simply neglected.
In this thesis a three-dimensional approach to the design of a radial turbine is implemented, and
two configurations, with and without fixed nozzle ring, are generated. The turbine is designed for
a turbocharging system of a typical six-cylinder diesel truck engine, of which exhaust gas
thermodynamic properties are known. The models are studied by means of a CFD commercial
software, and their performance at steady design and off-design conditions are compared.
Results show that, at design point, the addition of a static nozzle ring leads to non negligible
increments, with respect to the vaneless case, of both efficiency and power output: such
increments are estimated in +1.5% and +3.5% respectively, despite these data should be
compared with the uncertainty of the numerical model. On the other hand both turbine
configurations are found to be very sensitive to variations of pressure and temperature of the
incoming fluid, hence off-design performances are dependent on the particular off-design point
considered and a “best” configuration within all the combustion cycle does not exist.
ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor, Jens Fridh, and my examiner, Paul Petrie-
Repar, for their availability and support, and without whom this work would have never been
realised.
Sincere gratitude to my Italian examiner Alessandro Talamelli, who followed the project of Dual
Degree between UniBo and KTH since its inception: it is also thanks to him that I ended up at
KTH, a circumstance which I will never regret.
I would like to thank my parents, who believed in me and allowed me to complete my academic
iter, providing full moral and material support.
A special mention goes to my Italian friends: Pietro, who was always ready to support me with a
Skype call and welcome me back during my short visits in Italy, and Simone, who, no matter
where he is, always finds a way to be present in the most crucial moments.
Thank you Danilo, you proved to be not only a respectful roommate but also a real valuable
person from all sides. And many thanks to all friends that I met in Stockholm, who turned my
stay abroad into an exciting and enjoyable experience, because “there’s no point in living if you
can’t feel alive”.
iii
TABLE OF CONTENTS
List of Figures.................................................................................................................................v
List of Tables.................................................................................................................................vii
The work per unit mass exerted by the flow on the turbine shaft is linked to the rate of change of
angular momentum that the flow itself undergoes inside the rotor, and may be expressed by the
so called “Euler turbine equation”, here in differential form (for the exact derivation see Mora
[14])
𝑑𝑊 = 𝑑 𝑈𝑐𝜃 (1.2)
The combination of eqn.(1.1) and eqn.(1.2) yields to eqn.(1.3):
𝑑(0 − 𝑈𝑐𝜃) 𝑟𝑜𝑡𝑎𝑙𝑝𝑦
= 0 (1.3)
which states that in the thermodynamic process through the impeller the rotational total enthalpy
𝐼 ≝ 0 − 𝑈𝑐𝜃 (also called rothalpy) is constant. The same expression is re-arranged according
to the velocity triangle in Fig.3, which relates the velocity vectors in the fixed and the rotating
frame of reference:
𝐼 = +1
2𝑐2 −𝑈𝑐𝜃 = +
1
2 𝑤2 + 𝑈2 + 2𝑈𝑤𝜃 − 𝑈 𝑤𝜃 + 𝑈 = +
1
2𝑤2 −
1
2𝑈2 (1.4)
Figure 3: flow velocity triangles within a radial turbine (Dixon, [6])
Through eqn.(1.4) it is possible to express the variation of static enthalpy between inlet and
outlet of the rotor (denoted by indexes 2 and 3, Fig.4), thus from the combination of eqn.(1.1)
with eqn.(1.4) the total specific work on the turbine is:
∆𝑊2−3 = +𝑐2
2
2−3=
1
2[ 𝑈2
2 − 𝑈32 − 𝑤2
2 − 𝑤32 + 𝑐2
2 − 𝑐32 ] (1.5)
Eqn.(1.5) leads to the following considerations:
4
a) In the case of an IFR turbine the term 1
2 𝑈2
2 − 𝑈32 is always positive because 𝑈2 > 𝑈3; the
same does not hold for an axial turbine, where inlet and outlet radiuses are equal. As
direct consequence, the work per unit mass that can be extracted with a single turbine
stage is higher in the former case, and so the efficiency, which justifies the exclusive
employment of radial turbines in turbochargers.
b) A positive contribution to the specific work is obtained when 𝑤3 > 𝑤2. This is achieved
if the channels between blades are convergent.
Eqn.(1.4) shows that the higher is 𝑤3 the higher is the static enthalpy jump 2 − 3 across the rotor (and consequently the specific work) but the lower is the static pressure 𝑝3 at rotor outlet (see Fig.4). However 𝑝3 cannot be lower than the atmospheric pressure,
otherwise the discharge would not be possible. The diffuser is used to allow 𝑝3 < 𝑝𝑎𝑡𝑚 (with a benefit for efficiency) and to recover part of the static pressure afterwards so that
at turbine outlet the condition 𝑝4 ≥ 𝑝𝑎𝑡𝑚 is fulfilled.
Moreover “accelerating the relative velocity through the rotor is a most useful aim of the
designer as this is conducive to achieving a low loss flow” (Dixon, [6]).
c) Since the specific work is proportional to the quantity 1
2(𝑐2
2 − 𝑐32) the absolute velocity
should be large at impeller inlet, which is achieved by means of the volute.
The volute (as well as the diffuser) is a static component (𝑑𝑊 = 0) and from eqn. (1.1)
the total enthalpy is conserved (see also Fig.4) which implies that the higher is the drop in
static enthalpy 1 − 2 the higher is the absolute velocity seen by the rotor at its inlet.
Figure 4: thermodynamic diagram of the process through a 90° IFR turbine (Dixon, [6])
With reference to point c) above, a deeper analysis is needed. Eqn.(1.2) can be evaluated
between points 2 and 3, leading to eqn.(1.6)
∆𝑊2−3 = 𝑈2𝑐𝜃2 − 𝑈3𝑐𝜃3 (1.6)
which shows that:
5
a high value of 𝑐2 is not the only requirement because what matters is the projection of 𝑐2
along the circumferential direction, 𝑐𝜃2. The ideal condition would be 𝑐2 ≡ 𝑐𝜃2 but this is
not physical since there would be no inflow at rotor inlet: the absolute flow angle is
chosen such that the relative velocity has only radial component.
A good volute design must therefore take into account the orientation of the absolute
velocity at rotor inlet, which may be accomplished by using vaned stators.
the specific work is increased if the absolute velocity of the flow at rotor outlet is axial,
i.e. 𝑐𝜃3 = 0.
The two conditions mentioned above constitute the so called nominal design (Fig.5).
Five equidistant spanwise locations, namely layers, are identified (the 1st being the hub, the 5
th
being the shroud) and for each of them the wrap angle distribution is modelled by means of a
Bezier curve. This operation is done efficiently in BLADEGEN®, where it is possible to drag, add
or delete control points so to determine the Bezier curve in terms of order and shape.
For the specific case a number of control points between 26 and 31 was used for each layer. High
degree polynomials were chosen in order to locally have control over the curvature, which is not
constant streamwise (it was mentioned that for mechanical reasons the blade is required to
develop radially in its first segment).
Given a wrap angle distribution BLADEGEN® numerically evaluates the tangent at each point.
The resultant plot corresponds to the flow angle distribution, since 𝜃 and 𝛽 are connected by the
relation 𝛽 =𝑑𝜃
𝑑𝑚 (see Fig.19 – right): hence the 𝜃-distribution is designed so that the associated 𝛽
fulfils the condition of axial flow at rotor outlet (Tab.3).
29
Figure 21: Rotor: θ-distribution (top), β-distribution (middle), thickness distribution (down)
30
For each layer BLADEGEN® must create the blade profile. The latter is defined by a mean line,
which is the wrap angle line, and a streamwise distribution of thickness, which is modelled by
the designer with a Bezier curve. Profiles of adjacent layers are connected together by
streamwise lofting and the result is the generation of the blade surface.
Distributions of 𝜃, 𝛽 and thickness are shown in Fig.21, while the result of such design on the
meridional plane is reported in Fig.22.
Figure 22: Rotor: distribution of wrap angle (top left), flow angle (top right) and thickness
(bottom) in the meridional plane
As seen in the introductory chapter, in order to increase the work exchange the flow must
accelerate inside the rotor, i.e. 𝑤3 > 𝑤2, which implies that for subsonic flows the channel
between rotor blades is convergent. Fig.23 – left shows that this is the case for the present
design, as the cross section area of the channel reduces from inlet to outlet.
The 3D CAD model of the impeller, which results from the design above, is presented in Fig.24.
31
Figure 23: Rotor: variation from inlet to outlet of channel cross-section area (left) and lean
angle (right)
Figure 24: 3D geometrical model of the rotor
5.3.4 Supplementary issues on 3D design of the rotor
So far the design of the impeller was made from a geometrical point of view, without due
consideration of the influence on the internal flow pattern. This relationship is impossible to
investigate analytically because it will be seen that, no matter how “good” the design is, the
presence of secondary flows cannot be avoided, thus the problem is fully 3D. CFD analysis is the
most common tool to have an insight on the internal flow, because the only way to have a
comprehensive description of it is by numerically solving the full set of NS equations: however
in this section an analytical approach is presented, whose purpose is to highlight the influence of
some design choices on the development of secondary flows, and possibly suggest improvements
at subsequent phases of the deign process.
32
This analysis is an adaptation of the study presented by Van den Braembussche [28] and
originally performed on the impeller of a radial compressor.
Blade lean is defined as the variation of 𝜃 of the blade from hub to shroud. Under the assumption
that the blade lean is null, i.e. 𝜕𝜃
𝜕𝑛= 0 (this is almost satisfied, as the lean angle lies within the
interval [0°,4°] everywhere from inlet to outlet, see Fig.23 – right) the components of velocity
which induce centrifugal accelerations in the meridional plane are:
Meridional velocity 𝑊𝑚 , whose radius of curvature is the one of the streamline, namely 𝑅𝑛 (the
subscript 𝑛 denotes that this radius is aligned with the spanwise direction)
Tangential velocity 𝑉𝑡 = 𝑊𝑡 − 𝛺𝑅, whose radius of curvature is 𝑅. Notice that this vector is
orthogonal to the meridional plane, but the resultant acceleration is in plane
Figure 25: geometrical definition of the problem
At all streamwise sections the overall centrifugal acceleration is balanced by a pressure gradient
orthogonal to the streamsurface (Fig.25). The equilibrium is expressed by eqn.(4.16)
1
𝜌
𝜕𝑝
𝜕𝑛= 𝑊𝑢 − 𝛺𝑅
2
𝑅cos 𝜆 −
𝑊𝑚2
𝑅𝑛 (4.16)
where 𝜆 is the angle between the meridional component of the streamline and the axis of
rotation. At inlet cos λ = 0 and the acceleration due to the curvature of the streamline increases
from hub to shroud because of decreasing 𝑅𝑛 , hence the pressure gradient decreases. At outlet
the direction of the pressure gradient depends on the values of 𝑉𝑡 and 𝑊𝑚, but with increasing 𝑅
from hub to shroud the spanwise acceleration due to the tangential velocity component tends to increase
as well (𝑎𝑡 ∝ 𝛺2𝑅), thus the pressure gradient is still negative.
The above considerations suggest that, together with the main motion of the flow from inlet to
outlet, there exist also a secondary flow moving from hub to shroud, i.e. against the spanwise
pressure gradient.
Imposing 𝜕𝑝
𝜕𝑛= 0 eqn.(4.16) gives the value of 𝑅𝑛 for zero pressure gradient
33
𝑅𝑛 =𝑊𝑚
2𝑅
𝑉𝑡2 cos 𝜆
(4.17)
However 𝑊𝑚 is higher at the suction side of the blade (where the flow is accelerated) and lower
at the pressure side, and from eqn.(4.17) two different values of 𝑅𝑛 should exist at the same
point, thus there is no design which can eliminate the spanwise pressure gradient, and the
correspondent secondary flow, at both sides of the blade. The hub-to-shroud pressure gradient
can be reduced by increasing the curvature radius of the meridional contour or by reducing the
blade height (hub-to-shroud distance).
Moreover there also exists a pressure gradient in the blade-to-blade plane because of the flow
moving between pressure side and suction side of adjacent blades. Inside the channel the flow
feels the simultaneous effect of both pressure gradients. The situation is depicted in Fig.26.
Figure 26: pressure distribution in a crosswise section. Effect of spanwise pressure gradient
(left), effect of blade-to-blade pressure gradient (middle), ensemble (right) (Van den
Braembussche, [28])
A possible strategy to reduce the effect of the secondary flow is the employment of splitter
blades.
5.4 Design of the diffuser
The diffuser must convert part of the kinetic energy of the flow into pressure in order to reach
the condition 𝑃4 > 𝑃𝑎𝑡𝑚 at the outlet of the turbine and allow the flow to be discharged.
The constraints on the design of the diffuser are
Inner radius (𝑟3) at diffuser inlet, which must be equal to the hub radius at rotor outlet
Outer radius (𝑟3𝑡) at diffuser inlet, which must be equal to the shroud radius at rotor outlet
Values provided by the meanline design are reported in Tab.5
PARAMETER VALUE
Outer radius at diffuser outlet 𝑟4𝑡 = 38.6 [mm]
Inner radius at diffuser outlet 𝑟4 = 0 [mm]
Axial length 𝑙 = 38.6 [mm]
Table 5: meanline design parameters for the diffuser
The main problem when designing the diffusers is the risk of boundary layer separation at the
wall [8], which depends on the diffusion angle ψ (Fig.27 – left).
34
Figure 27: conical diffuser. Left: 2D sketch. Right: lines of appreciable stall for given
geometrical configuration (Blevins, [4], adapted)
From meanline parameters the non-dimensional length of the diffuser is 𝑙
𝑟3𝑡= 1.21 while the
designed diffusion angle is expressed from Fig.27 – left as tan𝜓 = 𝑟4𝑡−𝑟3𝑡
𝑙 𝑦𝑖𝑒𝑙𝑑𝑠 2𝜓 ≈ 19.4°.
An extrapolation from the graph in Fig.27 – right suggests that this point is below the limit of
appreciable stall for conical diffusers, hence the present design configuration can be
implemented.
The 3D CAD model of the diffuser is presented below (Fig.28)
Figure 28: 3D geometrical model of the diffuser
35
6 – MESH GENERATION
The problem of how to partition the flow domain in order to create a grid on which discretized
NS equations can be solved is of primary importance in CFD, because it directly affects the
quality of the simulation and the computational time. From a practical point of view the ideal
mesh should be able to capture the critical aspects of the flow (presence of regions of separated
flow, recirculation bubbles, evaluation of losses...) with simulations lasting at most a few hours,
which implies that the mesh must be refined only in regions in which it is needed (typically in
the boundary layer, or where high velocity gradients are expected).
The first part of this section focuses on theoretical aspects about mesh generation: how to choose
an appropriate mesh for the problem in exam, how to mesh regions of the domain close to a wall
and the techniques to check if the mesh is “good”. In the second part the solution which was
implemented is illustrated and the mesh quality is assessed.
6.1 Choice of the grid
Two different techniques are employed for the discretization of a geometric domain. In the so
called structured grid the cells are ordered in a I×J×K array so that given whatever grid point
inside the domain this is univocally identified by a set of coordinates, say 𝑃𝑖 ,𝑗 ,𝑘 , and the points in
its neighbourhood are implicitly known (they will be 𝑃𝑖 ,𝑗 ,𝑘−1, 𝑃𝑖 ,𝑗 ,𝑘+1...). This implies that there
exist a regular pattern of connections among grid points which are close to each other. On the
other hand in an unsctuctured grid the above regularity is not present hence neighbouring cells
cannot be directly accessed by their indexes. The different way to build the two grids also affects
the geometrical shape of their cells, i.e. the elements: hexahedra are usually employed in
structured meshes while unstructured meshes are formed by tetrahedral elements or
combinations of different solids (see Fig.29).
Figure 29: elements of a 3D mesh - tetrahedron, hexahedron, prism, pyramid
The choice of the type of mesh should be done according to the following considerations:
Unstructured meshes can easily model every kind of domain because the shape of the
elements which is employed is not constrained to hexahedra. For the same reason also the
element size can vary considerably between adjacent cells. This flexibility is needed
when the geometry to be meshed is complex or when fast variation in the grid spacing is
desirable (for example close to the walls).
For the same amount of cells structured grids based on hexahedra allow the highest
accuracy in the solution. On the contrary unstructured grids tend to generate more
skewed elements, with consequent numerical errors.
The generation of an unstructured grid is much faster than a structured one. The time
strongly depends on the complexity of the problem, but while for the former it is usually
in the order of hours for the latter it can take up to weeks (Khare et al., [10]).
36
Meshing a radial turbine is a challenging task because structured hexahedral mesh would be
desirable for high computational accuracy, especially in regions where losses most probably
occur, but the complexity of the 3D geometry makes the unstructured mesh approach more
suitable. In the present work a hybrid solution is suggested. The mesh is structured in the
impeller because it was seen that complex secondary flows occur inside the blade-to-blade
channels and their accurate evaluation is critical for a correct performance assessment of the
turbine. Achieving such a degree of precision is less crucial for other components, like the volute
or the diffuser, where instead the main source of losses is given by skin friction: for this reason
an unstructured mesh is used, which becomes more regular only close to the walls in order to
model accurately the boundary layer.
6.2 Meshing the boundary layer
Boundary layer is a thin region adjacent to a solid surface in which there exist a strong velocity
gradient orthogonal to the surface itself. The reason is that mechanical equilibrium is achieved
between molecules of the flow and the wall in the contact region so that as a macroscopic effect
the velocities of the flow and the solid surface are equal (this constraint is called no slip
condition). The existence of a sharp velocity gradient suggests that for a “good description” of
the flow the grid inside the boundary layer shall be refined. But what happens qualitatively with
increasing Reynolds number is that the region where viscous effects are relevant gets more and
more confined to the walls and the boundary layer becomes thinner, thus the grid resolution must
increase accordingly, leading to an increase of the computational effort.
A rough evaluation of Reynolds number can be done taking as reference values the ones at
turbine inlet: 𝑈𝑟𝑒𝑓 is the meanline velocity, 𝐿𝑟𝑒𝑓 is the radius of the duct at volute inlet and
𝜈|𝑇=𝑇𝑟𝑒𝑓 is the kinematic viscosity of air at 𝑇 = 𝑇𝑖𝑛𝑙𝑒𝑡 at design point.
𝑅𝑒 =𝑈𝑟𝑒𝑓 𝐿𝑟𝑒𝑓
𝜈|𝑇=𝑇𝑟𝑒𝑓
≈120
𝑚
𝑠 ∗ 30 ∗ 10−3[𝑚]
9.06 ∗ 10−5 𝑚2
𝑠
≈ 4 ∗ 104 (5.1)
Eqn.(5.1) shows that 𝑅𝑒 is high, in the order of 104. The flow regime is turbulent, and in the
boundary layer exchange of momentum takes place not only between adjacent layers, at
molecular scale, but together with an exchange of fluid particles.
Figure 30: velocity profile in a turbulent boundary layer (Bakker, [3])
37
As a consequence the region close to the wall is subject to steep gradients normal to the
boundary (Fig.30 illustrates a typical velocity profile) and directly resolving the flow with a
suitable mesh is computationally demanding.
Experimental investigation showed that the boundary layer can be divided into an inner and an
outer region. The first region is dominated by viscous effects, and the velocity is a function of
the coordinate 𝑦+ which represents the distance from the wall nondimensionalized by the viscous
scale 𝑢∗
𝜈 (𝑢∗ ≝
𝜏0
𝜌 is called friction velocity).
𝑈
𝑢∗= 𝑓
𝑦𝑢∗𝜈 = 𝑓 𝑦+ (5.2)
The second region is dominated by turbulent mixing and the flow seems not to feel the presence
of the wall. The difference of velocity with respect to the reference value (called velocity defect)
is function of the coordinate 𝜉 which represents the distance from the wall nondimensionalized
by the boundary layer thickness 𝛿.
𝑈 − 𝑈𝑟𝑒𝑓
𝑢∗= 𝑓
𝑦
𝛿 = 𝑓 𝜉 (5.3)
The distance from the wall, y, scales differently in the two regions. However there exist an
intermediate overlap region in which the two expressions (eqn.(5.2) & eqn.(5.3)) are both valid.
By equating their derivatives (for details see Kundu, [11]) it is proved that the overlap region is
described by a logarithmic law, and the so called logarithmic layer becomes wider with
increasing 𝑅𝑒. The situation is illustrated in Fig.31.
Figure 31: non-dimensional velocity as function of 𝑦+ in the inner region (Kundu, [11])
38
Under the assumption that the logarithmic behaviour can be used to model the velocity
distribution near the wall, this provides a suitable law to link distance from the wall to velocity,
hence allows the estimation of the flow shear stress. In this way it is not necessary to resolve the
boundary layer because the velocity in the inner region can be estimated through the log-law,
thus a coarser mesh can be used.
Summarizing, there are two approaches to model the flow in the near-wall region
Wall function method. It is based on empirical formulas which link the velocity of the
flow to the position in the inner region of the boundary layer, thus avoiding to resolve it
and saving computational time. However additional assumptions must be introduced in
order to justify the validity of the wall function in the whole inner region, and this may
lower accuracy of the results.
Low-Reynolds-Number method. This method resolves the details of the boundary layer
profile by using very fine meshes with the first node located at 𝑦+ ∼ 1 or even closer to
the wall.
6.3 Quality of the mesh
There exist many criteria for evaluating the quality of a mesh, and an in-depth discussion on this
topic is beyond the purpose of the present work. However from a practical point of view a mesh
is considered of good quality if its elements are not warped too much with respect to their
nominal shape (tetrahedron, pyramid...): for example if an element is too stretched in a certain
direction (see Fig.32) the variation of any flow characteristic in that direction will be detected
with less accuracy because the grid behaves as if it were coarser.
Figure 32: stretching of a quadrilateral element. Nominal shape (left), deformed shape (right)
In this section the following quality parameters are discussed:
6.3.1 Skewness
Skewness is the measure of how close the shape of a cell is from the ideal shape. A possible way
to calculate it is through the so called normalized angle deviation method
𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 ≝ 𝑚𝑎𝑥 𝜃𝑚𝑎𝑥 − 𝜃𝑒
180 − 𝜃𝑚𝑎𝑥,𝜃𝑒 − 𝜃𝑚𝑖𝑛
𝜃𝑒 (5.4)
Eqn.(5.4) evaluates the deviations of the maximum (𝜃𝑚𝑎𝑥 ) and the minimum (𝜃𝑚𝑖𝑛 ) angle with
respect to the angle relative to an equiangular cell (𝜃𝑒 , which represents the ideal case) and takes
as value for the skewness the maximum between both.
A value of 0 represents an equilateral cell while a value of 1 stands for a degenerated cell (for
example in the case of a 2D cell this would become a 1D segment). Skewness is considered good
for values up to 0.5.
39
6.3.2 Orthogonal quality
Orthogonal quality is another way to evaluate how a cell is close to its ideal shape. Taking the
2D cell in Fig.33 as reference, orthogonal quality is the minimum of eqn.(5.5) computed ∀i
𝑂𝑄 ≝𝐴𝑖 . 𝑒𝑖
𝐴𝑖 𝑒𝑖 (5.5)
where 𝑒𝑖 is the vector joining the centroid of the cell with the centroid of the edge and 𝐴𝑖 is the
vector normal to the edge. Eqn.(5.5) is a measure of how much 𝑒𝑖 and 𝐴𝑖 are aligned, because the
scalar product at the numerator depends on the cosine of the angle between the two vectors.
The range for the orthogonal quality is [0,1], and the closer to 1 the more equilateral is the cell
(in this case, infact, all the vectors are aligned).
Figure 33: orthogonal quality on a 2D quadrilateral cell
6.3.3 Jacobian ratio
The Jacobian matrix describes the properties of the mapping between the computational space
(𝜉1, 𝜉2, 𝜉3), where the NS equations are discretized and solved, and the real domain (𝑋1, 𝑋2, 𝑋3).
In an ideal situation the two domains would coincide, thus the computed solution could be
transferred to the real case without loss of accuracy. For each element of the mesh the
determinant of the Jacobian matrix is computed at some sampling points (for example the corner
nodes, the centroid...).
Figure 34: mapping of an hexahedral element (Bucki, [5])
40
From eqn.(5.6) JR is defined as the maximum to the minimum value among those determinants.
Other definitions presented in literature [5] consider the maximum determinant at denominator,
but here is reported the one used by ANSYS® Meshing.
𝐽𝑅 ≝𝑚𝑎𝑥 𝑑𝑒𝑡 𝐽 𝜉𝑖
𝑚𝑖𝑛 𝑑𝑒𝑡 𝐽 𝜉𝑖 (5.6)
where 𝜉𝑖 , 𝑖 ∈ 1,2,… ,𝑛 is the generic sampling point.
At the end JR is a measure of the maximum distorsion of each element of the mesh. The situation
is sketched if Fig.34. A value close to 1 indicates that the mapping does not lead to distorsion of
the elements, while the higher the Jacobian ratio the worse is the mesh.
6.4 Meshing of components
Volute, nozzle ring and diffuser were meshed using the software 𝐴𝑁𝑆𝑌𝑆®Meshing. The process
of mesh generation is highly automated: the designer specifies general sizing parameters such as
degree of fineness of the grid (relevance), rate at which adjacent elements are allowed to grow
(transition), control over the element quality (smoothing), and the software creates an
unstructured mesh based on those requirements and on constraints about the dimension of the
elements to be used (limits on the size of edges and faces), which are specified as defaults
(defaults may be changed if necessary). Fig.35 shows as an example the setup of
𝐴𝑁𝑆𝑌𝑆®Meshing for mesh generation on the volute.
Figure 35: example of setup of 𝐴𝑁𝑆𝑌𝑆®Meshing (volute)
Generation of unstructured mesh by means of 𝐴𝑁𝑆𝑌𝑆®Meshing is a fast and robust process,
however for a more accurate description of the flow in the boundary layer a structured mesh
close to the walls is needed. This is achieved in 𝐴𝑁𝑆𝑌𝑆®Meshing by using inflation layers: to
this purpose the program requires definition of the surface around which inflation must be
41
performed (named selection) and specifications about inflation option (thickness of the first
layer, total inflation thickness…), number of layers and growth rate between adjacent layers. An
example is reported in Fig.35.
The main parameter to be set at this stage is the first layer thickness, i.e. the distance of the first
node of the mesh from the wall expressed in terms of 𝑦+. A target 𝑦+ is set and the
correspondent 𝑦 is determined through eqn.(5.2), but in order to do so the friction velocity 𝑢∗ must be estimated. Literature (White, [30]) reports a graph, namely Moody’s diagram, which
relates 𝑅𝑒 to 𝐶𝑓 (skin friction coefficient) for circular pipes with smooth walls in turbulent
regime. Many empirical relations were also formulated, among which the 1/7th
law is mentioned.
Assuming the validity of such relation in more general cases of ducts with non-circular shapes
(volute, diffuser…), once 𝑅𝑒 is calculated from eqn.(5.1) it is possible to estimate 𝐶𝑓 from
eqn.(5.7)
𝐶𝑓 = 0.027𝑅𝑒𝑥−1
7 (5.7)
and 𝑢∗ from eqn.(5.8)
𝑢∗ ≝ 𝜏0
𝜌= 𝐶𝑓
𝑈𝑟𝑒𝑓2
2 (5.8)
In Tab.6 are reported the target 𝑦+ for each sub-domain of the turbine (except for the rotor,
which will be discussed later) and the correspondent 𝑦, which must be set as first layer thickness
in the setup for generating inflation layers with 𝐴𝑁𝑆𝑌𝑆®Meshing. The reference values for the
computation of 𝑅𝑒 in eqn.(5.1) are considered to vary among the components.
Domain 𝑈𝑟𝑒𝑓 [𝑚/𝑠] 𝐿𝑟𝑒𝑓 [𝑚] 𝑅𝑒 [−] 𝑦+ [−] 𝑦 [𝑚]
Volute 𝑈1 ∼ 120 (speed at inlet)
3 ∗ 10−2 (radius at inlet)
4 ∗ 104 30 4 ∗ 10−4
Nozzle ring 𝑈𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 ∼ 250 (speed at inlet)
3.2 ∗ 10−2 (blade chord)
8.8 ∗ 104 10 6 ∗ 10−5
Diffuser 𝑈3 ∼ 210 (speed at inlet)
2.1 ∗ 10−2 (width at inlet)
4.9 ∗ 104 30 2 ∗ 10−4
Table 6: estimation of first layer thickness for turbine components
Notice that for volute and diffuser 𝑦+ is located in the logarithmic layer while for the nozzle ring
it is placed in the buffer layer (𝑦+ ∼ 10). The goal is to achieve higher resolution in the nozzle
ring where the blades may cause separation of the flow, especially when the turbine is working at
off-design points.
DOMAIN NUMBER
OF NODES
NUMBER OF
ELEMENTS
AVERAGE
SKEWNESS
AVERAGE
JR
AVERAGE
OQ
Volute-
vaneless
113587 338479 0.256 1.069 0.872
Volute-
vaned
111830 336893 0.255 1.074 0.871
Nozzle ring 262591 995715 0.310 1.078 0.807
Diffuser 42571 122799 0.202 1.019 0.899
Table 7: mesh statistics for turbine components
42
Once the inflation is defined the setup of 𝐴𝑁𝑆𝑌𝑆®Meshing is complete and meshes are
generated. The result is shown in Tab.7, where the statistics for the final meshes are reported. As
can be seen all meshes fulfil the quality criteria, thus they will be used for the subsequent CFD
analysis.
The implemented solution is illustrated in the figures below, whose goals are to show the effect
of inflation layers (Fig.36), an ensemble mesh (Fig.37) and the presence of local refinements
around “critical points” (Fig.38, at LE and TE of the nozzle ring blade).
Figure 36: mesh of the volute (section)
Figure 37: mesh of the diffuser (ensemble)
43
Figure 38: mesh of the nozzle ring (one blade)
For meshing the rotor TURBOGRID® was used, a dedicated software for rotating machines which
creates a high quality structured hexahedral mesh. The process of mesh generation proceeds
according to the following steps:
1. The geometry of the rotor is imported from a BLADEGEN® file.
2. Based on the geometry of the blade TURBOGRID® automatically chooses the most
suitable topology for the mesh using the function ATM optimized. Topology denotes the
pattern in which the region of space around the blade is divided. Topology is chosen
according to the blade profile, the local curvature, the shape of LE and TE (cut-off or
rounded) and other geometrical factors: in each block the elements of the mesh are
oriented along the local shape of the blade, and interfaces between blocks are “smooth”
so to avoid warped elements which lower the quality of the mesh.
The topology for the rotor is sketched in Fig.39.
Figure 39: topology for the rotor blade. The blade (blue) is surrounded by meshing blocks
44
3. At this step TURBOGRID® requires to specify the fineness of the overall mesh, through the
global size factor, and the target 𝑦+. For the rotor 𝑦+ = 1 is set: the purpose is to directly
resolve the boundary layer (without the use of wall functions), as higher accuracy is
needed in the rotor, where the flow is fully 3D.
4. The final mesh is automatically generated. The final mesh for the rotor has 613788 nodes
and 579535 elements.
Fig.40 shows a portion of the mesh around the LE of the blade. Progressive refinement can be
appreciated close to the wall, due to the specification on 𝑦+, while the topology is seen from the
orientation of the hexahedral elements.
Figure 40: mesh of the rotor (portion)
After mesh generation TURBOGRID® provides tools for assessing the quality of the mesh. The
most significant ones are listed below:
Minimum/Maximum face angle. For each face of an element, the angle between the two
edges that touch a node is calculated. The smallest/largest angle of all faces is returned as
the value for the minimum/maximum face angle. This parameter evaluates how warped
the faces are with respect to the ideal angle (90°), therefore it can be considered a
measure of skewness.
Minimum volume. This is the minimum volume among all the cells of the grid. Its value
must be always positive in order avoid numerical errors.
Maximum element volume ratio. For each node the volume of all the cells touching that
node is computed and the ratio between the maximum and the minimum volume is
returned. This is a measure of the local expansion factor, and it should be low especially
in regions where high gradients are expected in the flow quantities (velocity,
temperature...).
Maximum edge length ratio. For each face of an element, the ratio between the longest
and the shortest edge is computed and the maximum value is returned. This parameter
measures the aspect ratio (see Fig.32).
Mesh statistics for the rotor are reported in Fig.41: after computation TURBOGRID® checks if
each quality parameter lays within the acceptable range, and returns a feedback.
45
Figure 41: mesh statistics for the rotor
The overall mesh for the vaneless configuration contains 769946 nodes and 1040813 elements,
for the vaned configuration 1030780 nodes and 2034942 elements.
46
7 – CFX SETUP
Once generated and meshed, all components of the turbine are assembled in CFX® − Pre.
However, before running a CFD simulation, it is necessary to specify the mathematical model
that must be solved together with boundary conditions to the problem, initial conditions and
interfaces between various components. Such topics are treated in this chapter.
7.1 Mathematical model for turbulence
A turbulent flow is characterized by swirling structures (eddies) which span a wide spectrum of
scales (Johansson & Wallin, [8], 2012). The dimension of the largest eddies is set by the
reference geometrical size of the problem, while the length scale of the smallest eddies is
supposed to only depend on dissipation rate and viscosity: this is, infact, the scale at which the
flow dissipates kinetic energy through viscous mechanisms. Within the two limits there exist a
range of eddies of intermediate dimensions which transfer kinetic energy from the biggest to the
smallest scales in a process called turbulence cascade.
The ratio between the largest and the smallest scales of turbulence depends on the Reynolds
number and can be estimated (for details see reference [8]) as 𝑅𝑒3/4
. Thus, a grid which aims to
describe all the details in a 3D flow domain would have a resolution of 𝑅𝑒9/4
, and considering
that for the present application the Reynolds number is at least in the order of 104 the mesh for
DNS would have 109 nodes and the computational effort would be massive. Alternatively it is
possible to describe a turbulent flow by means of RANS, where it is assumed that turbulence can
be modelled as fluctuations within an average velocity field: however when this assumption is
introduced in the NS equations it originates an extra term, namely the Reynolds stress, which is
unknown hence it must be modelled through a turbulence model.
Common turbulence models are the so called 𝑘 − 휀 and 𝑘 − 𝜔: both solve 2 additional
equations, together with the set of RANS, which account for the transport of turbulent variables.
In the former case those variables are turbulent kinetic energy, k, (representing the variance of
the fluctuations in velocity) and turbulence dissipation, ε, (representing the rate at which velocity
fluctuations dissipate); in the latter case ε is replaced by the specific turbulence dissipation, ω,
(𝜔 ≝휀
𝑘). Both models rely on the assumption that the Reynolds stress term is related to the
gradient of the mean velocity (strain) through the turbulent viscosity, 𝜈𝑡 , according to the
Boussinesq hypothesis (notice that this hypothesis is purely empirical)
−𝑢𝑖′𝑢𝑗′ 𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑠𝑡𝑟𝑒𝑠𝑠
= 𝜈𝑡 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦
∗ 𝑓 𝜕𝑈𝑖𝜕𝑥𝑗
𝑠𝑡𝑟𝑎𝑖𝑛
, 𝑖, 𝑗 ∈ 1,2,3 (6.1)
An in-depth description of 𝑘 − 휀 and 𝑘 − 𝜔 turbulence models is beyond the scope of this work,
however from the point of view of applications 𝑘 − 휀 is reported to have general good
performances but is not applicable to flows under adverse pressure gradients and is not accurate
in forecasting separation, while 𝑘 − 𝜔 has complementary characteristics and is mostly suitable
in the near-wall region. In this thesis a hybrid turbulence model, namely SST 𝑘 − 𝜔, is used, as it
combines the advantages of both models because it allows a shift from 𝑘 − 𝜔 to 𝑘 − 휀 depending
on the distance from the wall.
The SST 𝑘 − 𝜔 formulation is shown below:
47
k-equation: 𝜕 𝜌𝑘
𝜕𝑡+
𝜕 𝜌𝑈𝑗𝑘
𝜕𝑥𝑗=
𝜕
𝜕𝑥𝑗 𝜇 +
𝜇 𝑡
𝜍𝑘3 𝜕𝑘
𝜕𝑥𝑗 + 𝑃𝑘 − 𝛽
∗𝜌𝑘𝜔 (6.2)
ω-equation:
𝜕 𝜌𝜔
𝜕𝑡+
𝜕 𝜌𝑈𝑗𝜔
𝜕𝑥𝑗=
𝜕
𝜕𝑥𝑗 𝜇 +
𝜇 𝑡
𝜍𝜔3 𝜕𝜔
𝜕𝑥𝑗 + 1− 𝐹1 2𝜌
1
𝜍𝜔2𝜔
𝜕𝑘
𝜕𝑥𝑗
𝜕𝜔
𝜕𝑥𝑗+ 𝛼3
𝜔
𝑘𝑃𝑘 − 𝛽3𝜌𝜔
2 6.3
where 𝑃𝑘 is the term responsible for the production of turbulence and depends on the Reynolds
stress (which is modelled as function of 𝜈𝑡).
𝐹1 ∈ [0,1] is called blending function and accounts for the position with respect to the wall. By
tuning 𝐹1 the model shifts from the expression of the standard 𝑘 − 휀 (when 𝐹1 = 0) to the one of
𝑘 − 𝜔 (when 𝐹1 = 1). The constants of the model (𝛽∗, 𝛽3, 𝜍𝑘3, ...) are a linear combination of
the coefficients appearing in the formulations of 𝑘 − 휀 and 𝑘 − 𝜔 through the blending function,
i.e. taken ϕ as a general constant it holds 𝜙𝑆𝑆𝑇 = 𝐹1𝜙𝑘−𝜔 + (1− 𝐹1)𝜙𝑘−휀 . The blending function is formulated empirically.
7.2 Near-wall treatment
In the previous chapter it was seen that, depending on how close the first node of the mesh is
from the wall, the flow in the boundary layer can be either resolved or modelled through wall
functions, and grids for all components of the turbine were designed based on the target 𝑦+.
However an exact estimation of 𝑦+ is difficult to achieve because 𝑦+ depends on 𝑅𝑒 , which
varies inside the turbine. In particular inaccuracies may occur if the wall function method is used
in regions where the first node of the mesh is inside the viscous sublayer, while if the Low
Reynolds Number method were used with the first node of the grid being in the log-law region it
would not be possible at all to describe viscous and buffer layer.
If the wall function is set to automatic, the correct near-wall treatment is automatically chosen by
CFX®. The program calculates 𝑦+ and if the mesh has a local near-wall distance corresponding
to 𝑦+ < 11.06 (default value, [22]) the boundary layer is resolved, otherwise wall functions are
used. In this way automatic wall function allows the highest possible accuracy.
7.3 Boundary conditions and interfaces
Boundary conditions are relative to the design point and are common to both vaneless and vaned
configurations. Boundary conditions are derived from the meanline design and reported below
Inlet total pressure: 2.5636 [bar]
Inlet total temperature: 845 [K]
Outlet static pressure: 1.1277 [bar]
Flow direction at inlet: normal to inlet section
Other combinations of boundary conditions are possible in CFX® (mass flow rate, velocity...) but
their use turned out to give solutions non-monotonically convergent (the residuals of mass and
momentum showed undamped oscillations) or even non convergent at all, while with the
specification of total pressure at inlet and static pressure at outlet the solution was stable. This
choice is also supported by literature ([9], [22])
No slip condition was set at the walls. Walls are modelled as adiabatic and smooth.
48
Interfaces are used to model the contact region between components of the turbine: across them
there can be a discontinuity in the mesh pattern, a variation of frame of reference (for example in
the matching rotor-stator) or a variation of pitch (this is the case when two contact domains have
different angular width).
For the present study the stage model provided by CFX® is used. In the stage model, also called
mixing plane model, the flow properties (i.e. velocity, pressure, temperature...) are averaged
circumferentially upstream of the interface in order to obtain the boundary condition for the
downstream component. Since the flow is averaged, the inlet boundary condition for the
component downstream of the interface is steady, thus this method is suitable for steady state
simulations, which is the case here.
Another type of interface is the rotational periodicity in a rotor SBP, which allows to model the
whole rotor as just one blade passage, thus saving computational time. All interfaces and
boundary conditions are highlighted in the ensemble Fig.42 (only the vaned configuration is
illustrated, as it is the most complete case).
Figure 42: illustration of interfaces and boundary conditions for vaned configuration
7.4 Choice of off-design points
The design point represents average conditions of inlet pressure and temperature, but within the
combustion cycle these parameters vary over a considerable range, as seen from Fig.43.
Off-design points were determined [14] by measuring the flow properties of the exhausts at
constant intervals of 2.32 milliseconds after the opening of the discharge valve: among them
only the most representative are chosen for this study, and are listed in Tab.8
49
Figure 43: choice of representative off-design points (Mora, [14], adapted)
In particular points 1, 4, 5 are associated with the “furthest” thermodynamic conditions with
respect to the average, while point 7 is the closest to it and is chosen to represent the effect that
small variations around the mean inlet conditions may have on the performance of the turbine.
Notice that the highest temperature of the combustion cycle is not associated with the highest
pressure, hence both points are studied.
POINT TEMPERATURE (total) [K] PRESSURE (total) [bar]
1 762.9 2.0754
4 993.1 2.8438
5 935.1 3.2098
7 829.1 2.6339
Table 8: thermodynamic properties of the studied off-design points
50
8 – RESULTS
This section presents the main results of the steady simulations run with CFX® on the designed
CAD model. For each working condition (design and off-design) the turbine is analysed in both
configurations, with and without nozzle ring, and results are compared. Such results will then be
discussed in Chapter 8 in order to answer to the stated objective.
8.1 Design point
PARAMETER VANELESS VANED
Mass flow [kg/s] 0.396 0.417
Torque on rotor [Nm] (*) -5.11 -5.29
Power [kW] 45.47 47.07
Efficiency T-T [-] 0.751 0.762
Table 9: performance comparison, design point
(*) The negative sign is a convention, as the sense of rotation of the impeller is clockwise.
Performance data relative to the design condition are calculated and reported in Tab.9, while
Tab.10 illustrates the mean velocity triangles at rotor inlet and outlet.
PARAMETER
(rotor inlet,
average)
VANELESS
(red)
VANED
(blue)
(*)
𝑈2 [𝑚/𝑠] 365 358
𝑐2 [𝑚/𝑠] 333 324
𝑤2 [𝑚/𝑠] 176 161
𝛼2 [°] 61.3 63.2
𝛽2 [°] -24.6 -24.7
PARAMETER
(rotor outlet,
average)
VANELESS
(red)
VANED
(blue)
𝑈3 [𝑚/𝑠] 153 161
𝑤3 [𝑚/𝑠] 398 403
𝑐3 [𝑚/𝑠] 311 311
𝛽3 [°] -38.8 -39.9
𝛼3 [°] -15.8 -15.96
Table 10: comparison between mean velocity triangles at rotor inlet (top) and at rotor outlet
(bottom), design point
(*) Sketches represent the projection in the meridional plane (where α and β are defined) but c and w are general
3D vectors and also have an out-of-plane component.
51
The spanwise distribution of 𝛽2 is illustrated in Fig.44, while distribution of 𝛼3 in Fig.45. For
each spanwise position the correspondent value is obtained as a mass average over the
circumferential coordinate.
Figure 44: spanwise distribution of 𝛽2, comparison at design point
Figure 45: spanwise distribution of 𝛼3, comparison at design point
In this section results are analysed: the goal is to investigate the effects of the static nozzle ring
on the performance of the radial turbine. In order to do a “fair” comparison, at design point both
vaned and vaneless configurations must have the same average inflow conditions at rotor inlet.
Mean velocity triangles are reported in Tab.10. It holds 𝑐2_𝑣𝑎𝑛𝑒𝑑
𝑐2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠= 0.973,
𝛽2_𝑣𝑎𝑛𝑒𝑑
𝛽2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠= 1.004,
hence between the two cases the absolute velocity varies by less than 3% and the relative inflow
angle by less than 1%. The condition is fulfilled. Similar situation is present at rotor outlet,
where again the mean velocity triangles show non-appreciable differences (Tab.10, bottom).
According to a meanline analysis there would be no difference in incidence losses or TE losses
between vaned and vaneless configurations.
Fig.44 shows the spanwise distribution of 𝛽2. In both cases the relative inflow angle drops near
hub and shroud, while it is above the average value around mid span. At each spanwise position,
a strong correlation is noticed between the local 𝛽2 and the flow velocity around the blade: this
was an expected behaviour, as the inflow angle affects the position of the stagnation point at the
LE of the local blade profile and consequently the whole pressure distribution. At 90% span it
holds 𝛽2_𝑣𝑎𝑛𝑒𝑑 > 𝛽2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠 (Fig.44): in the vaned turbine the stagnation point is shifted towards
the pressure side of the blade (the flow is more radial, which represents the ideal nominal design
condition), and as can be seen from Fig.50 this causes the flow on the suction side to separate
closer to the TE. This is in accordance with the correspondent blade loading plot (Fig.46,
bottom), where pressure on the suction side keep decreasing in the vaned case (blue curve) while
in the vaneless case the blade profile has already stalled (red curve). From Fig.46 it can be seen
that the rotor blade is more loaded in the vaned turbine: as blade loading determines the moment
on the impeller about the axis of rotation, the higher blade loading justifies the higher power
output of the vaned case (Tab.9). The difference is about 3.5%. Separation not only affects the
blade loading but also the efficiency. Regions of separated flow are turbulent and strong
dissipation of kinetic energy occurs. This is visualized in Fig.47, which represents the entropy of
the flow in the blade-to-blade plane (90% span). Higher entropy in the vaneless case denotes
higher amount of losses with respect to the turbine with nozzle ring: this difference is indeed
confirmed by the performance calculations and is about 1.5% (Tab.9).
In the vaned configuration separation may occur at nozzle ring blades. This is not the case at
design point, except for the blade downstream of the volute tongue. Here the free-vortex law,
which was the underlying assumption for the design of the volute, is not valid [32] and the flow
angle at volute outlet is higher than the design value. However the region subject to separation is
very small (see Fig.51).
At off-design points both configurations undergo substantial modifications of average inflow and
outflow angles with respect to the design value (Tab.11, bottom). This first observation suggests
that the nozzle ring does not constrain 𝛽2 to the design value (or close values) also at off-design
points. Inspection of Fig.52, 55, 58, 61 shows that not only the mean value but also the spanwise
distribution of 𝛽2 (and 𝛼3) vary considerably. Another straightforward consideration comes from
Tab.11: it does not exist a turbine configuration which guarantees better performance (in terms
of efficiency and power output) at all working conditions.
Efficiency seems to be related to the spanwise distribution of 𝛽2. Let’s consider off-design point
4. The blue curve (vaned) is always above the red one (vaneless), which means that 𝛽2 is closer
to 0 (ideal case) through all the span. A plot in the meridional plane (Fig.56) shows that this
condition corresponds to lower amount of static entropy, especially close to hub and shroud
(where, according to Fig.55, left, the difference in inflow angle increases even more). As a
60
consequence the efficiency of the vaned turbine is higher than vaneless for this off-design point
(around 3.1%). The relashionship between rotor inflow angle and entropy should be investigated
in details, but the intuitive explanation is that the blade has been designed under the condition
𝛽2,𝑏 = 𝛽2 = 0 at all spanwise locations and the less this condition is satisfied the more the flow
is subject to detachment or recirculation, which leads to an increase of entropy. The situation is
visualized in Fig.57: close to the shroud (90% span) both configurations separate at TE (the
velocity around the blade drops at streamwise position near 0.7) but the vaned separates after,
hence not only flow entropy is lower (Fig.56, shroud) but also blade loading is higher (Fig.57,
right) and ultimately power output is higher.
The same argumentation can be followed for off-design point 5, but here 𝛽2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠 >𝛽2_𝑣𝑎𝑛𝑒𝑑 everywhere except close to hub and shroud (Fig.58, left). The meridional plot of entropy
shows the forecasted behaviour (Fig.59) and efficiency is higher for the vaneless configuration
(Tab.11). Moreover in this case the region of separated flow at the blade of the nozzle ring is
bigger than at design point (again the difference is more evident for the blade close to volute
tongue).
At off-design point 1 the inflow angle is large and separation occurs both at LE and TE (Fig.54).
The situation regarding 𝛽2 and consequent flow pattern can be interpreted as the previous cases,
however here the difference in 𝛼3 between vaned and vaneless turbine is not negligible and as
𝛼3_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠 < 𝛼3_𝑣𝑎𝑛𝑒𝑑 (mean value) a difference in TE losses must be considered together with
losses due to 𝛽2. Overall efficiency for the vaned case is slightly higher (+1.6%) than vaneless.
Off-design point 7 falls inside the frame used to describe the previous cases.
Regarding mass flow at design point it can be noticed that the value for both turbine
configurations is higher with respect to the meanline value that was used during the design
phase. This difference has already been pointed out in literature [22]. The most plausible reason
is that the meanline code over-predicts the effects of blockage inside the rotor blade channel. As
can be seen from blade-to-blade Mach plots (Fig.48, 49, 50), Mach can be locally 1 (or higher),
especially near the shroud, but the channel is never chocked at all spanwise positions, not even
for off-design points implying the highest values of inlet pressure and temperature (in this case
points 4 and 5). The reason why mass flow in the vaned configuration is slightly higher than
vaneless at all working conditions may be linked to the inflow angle 𝛽2 near the hub: in the
vaneless case this stays well below the average (while in the vaned it tends to increase more
rapidly) and such low values of the inflow angle locally cause recirculation. Somehow chaotic
streamlines can be seen near the hub, and this may support the statement, but as this cannot be
proved a deeper investigation should be done on the topic.
This analysis shows that the reason why a turbine configuration is “better” than another one is
mainly related to the inflow angle at rotor inlet, which should be somehow close to the design
value for the rotor blade (in this case 0°). If the presence of a static nozzle ring guarantees this
condition in a certain working point the vaned configuration is preferable to the vaneless,
otherwise not, but from this analysis it is possible to conclude that the addition of a static nozzle
ring in a 90° IFR turbine does not improve performance in all the combustion cycle.
61
10 – CONCLUSION AND FUTURE WORKS
In this thesis a comparison between vaneless and vaned turbine configurations has been made on
the basis of numerical simulations carried out in steady conditions with a CFD commercial
software. The use of a static nozzle ring increases both efficiency and power output at design
point (increment is roughly estimated in +1.5% and +3.5% respectively). On the other hand, off-
design performances strongly depend on the thermodynamic flow conditions which characterize
the specific point, and a general trend does not exist: however it is proved that the
implementation of a fixed nozzle ring in a radial turbine does not guarantee higher efficiency
through all the engine combustion cycle.
Inputs for possible future works are listed below:
Results presented in this thesis solely rely on a numerical model, whose uncertainties
have been highlighted in Chapter 4 – Limitations. In order to assess the validity of the
results the latter should be compared with experimental data
In this study only steady conditions have been considered, and it is implicitly assumed
that the flow has enough time to adapt to variations of pressure and temperature in the
exhaust gases. This may not be true in general, especially when the engine operates at
high rpm, hence a further study should model the unsteadiness of the flow
Future efforts could be made for improving the 3D geometrical model of the radial
turbine: for example the casing which surrounds the rotor is not modelled here, which
implies a poor evaluation of tip clearance losses
No structural considerations are made regarding the design of the turbine. The rotor, in
particular, is subject to high centrifugal forces, and despite the blade tip speed is limited
to 𝑈2 < 400 𝑚/𝑠 under design parameters (𝑈2 ~ 356 𝑚/𝑠), there is no guarantee that
the blade thickness distribution suggested in this thesis results in mechanical stresses
which do not lead to fracture or plastic deformation of the blade. For this reason a further
study is needed in order to check the feasibility of the present design.
62
11 – BIBLIOGRAPHY
[1] M. Abidat, M. K. Hamidou, M. Hachemi, M. Hamel, “Design and Flow Analysis of Radial
and Mixed Flow Turbine Volutes”, European Conference on Computational Fluid
Dynamics, TU Delft, The Netherlands, 2006
[2] N. C. Baines, M. Lavy, “Flow in Vaned and Vaneless Stators of Radial Inflow
Turbocharger Turbines”, Proceedings Institution Mechanical Engineers, Turbochargers and
Turbocharging Conference, 1969
[3] A. Bakker, “Course Material and Lectures”, Dartmouth, 2002-2006, available at
http://www.bakker.org
[4] R. D. Blevins, “Applied Fluid Dynamics Handbook”, Krieger Publishing Company, 1984
[5] M. Bucki, C. Lobos, Y. Payan, N. Hitschfeld, “Jacobian-Based Repair Method
for Finite Element Meshes after Registration”, Engineering with Computers, Springer
Verlag, pp.285-297, 2011
[6] S. L. Dixon, C. A. Hall, “Fluid Mechanics and Thermodynamics of Turbomachinery”, 6th
Edition, Butterworth-Heinemann, 2010
[7] M. S. Floater, “Beziér Curves and Surfaces”, Lecture Notes, Oslo, 2003
[8] A. V. Johansson, S. Wallin, “Turbulence Lecture Notes”, Stockholm, 2012
[9] M. Khader, “Optimized Radial Turbine Design D1.8”, School of Mathematics, Computer
Science and Engineering, London, 2014
[10] A. Khare, A. Singh, K. Nokam, “Best Practice in Grid Generation for CFD Applications
Using HyperMesh”, available at http://www.altairatc.com
[11] P. K. Kundu, I. M. Cohen, “Fluid Mechanics”, 2nd
Edition, Academic Press, 2002
[12] S. A. MacGregor, A. Whitfield, A. B. Mohd Noor, “Design and Performance of Vaneless
Volutes for Radial Inflow Turbines. Part 3: Experimental Investigation of the Internal Flow
Structure”, Proceedings of the Institution of Mechanical Engineers Part A Journal of
Power and Energy, June 1994
[13] C. A. de Miranda Ventura, “Aerodynamic Design and Performance Estimation of Radial
Inflow Turbines for Renewable Power Generation Applications”, PhD Thesis, University
of Queensland, School of Mechanical and Mining Engineering, 2012
[14] E. C. Mora, “Variable Stator Nozzle Angle Control in a Turbocharger Inlet”, MSc Thesis,
KTH School of Industrial Engineering and Management Energy Technology, 2015
[15] G. Negri di Montenegro, M. Bianchi, A. Peretto, “Sistemi Energetici e Macchine a
Fluido”, Pitagora Editrice, Bologna, 2009
[16] H. Nguyen-Schäfer, “Rotordynamics of Automotive Turbochargers”, Springer Berlin