The role of wild reindeer as a flagship species in new management models in Nrway

1 Copyright © 2013 by ICFD11

Proceedings of ICFD11: Eleventh International Conference of Fluid Dynamics

December 19-21, 2013, Alexandria, Egypt

ICFD11-EG-4096

Aerodynamic Shape-Optimization of Axial-Compressor Blades

Prof. Dr. Atef M. Alm-Eldien Mech. Power Eng. Dept., Faculty of

Eng. Vice-president for Students Affairs

Port-Said University, Egypt [email protected]

Prof. Dr. Ahmed F. Abdel Gawad Mech. Power Eng. Dept., Faculty of Eng.

Zagazig Univ., Zagazig, Egypt Currently: Mech. Eng. Dept.

Umm Al-Qura Univ., Saudi Arabia Fellow IEF, Assoc. Fellow AIAA

Member ASME, ACS, SIAM, AAAS Marquis Who is Who, IBC, ABI Biographee ICFDP8, ICFDP9, ICFD10, ICFD11 General

Coordinator [email protected]

Prof. Dr. Gamal Hafaz Mech. Power Eng. Dept. Faculty of Engineering

Port-Said University Egypt

Eng. Mohamed G. Abd El Kreim M.Sc. in Mech. Power Eng.

Gas Turbine Maintenance Engineer Al Shaba Simple Gas Turbine Power Plant East

Delta Electricity Production Company Egyptian Electricity Holding Company, Egypt

[email protected]

ABSTRACT

The purpose of this work is to develop and evaluate an inverse optimization algorithm, which designs two-dimensional compressor-blade shapes based on a prescribed pressure or velocity distribution. Thus, the design meets prescribed constraints imposed by the aerodynamic performance requirements that are suggested by the designer/stakeholder. The algorithm is coded using "Visual Basic". Its three main components are: (i) Surface-panel-method flow solver, (ii) Bezier curve-surface-definition routine, (iii) Optimization method. The procedure starts with a given compressor-blade shape C4 as a base-profile. Interesting findings and beneficial conclusions are presented.

KEYWORDS:

Compressor, Inverse design, 2-D airfoil, Bezier curve

1. INTRODUCTION

Optimization and design are of great interest in any industrial activity. Aerodynamic shape-optimization involves designing the best shapes of bodies that move through fluids. For example, one might require a compressor blade that will maximize lift, minimize the drag. In the aerodynamic design problem, the designer aims to have control over the aerodynamic characteristics such as lift, drag and pressure distributions and the geometry as well. Also, he aims to be sure, for instance, that the aerodynamic shape is acceptable from the point of view of the structural engineer. Another aspect where the aerodynamic designer is faced with is the question whether the design problem as formulated has a solution. The approaches taken to solve the problem of design optimization vary widely. Most of these approaches can be categorized into two types: Direct and Indirect design methods.

In Direct design techniques, the computing task concerns of determining the aerodynamic performance of a given blade-section shape. Then, it is analyzed with a computer code to define its performance. Finally, based on the results, the designer modifies the blade shape in accordance with his experience. The Indirect, or inverse, method uses an optimization function to design a shape that will exhibit a prescribed surface pressure or velocity distribution. The design objective in this case is to minimize the deviation between the target distribution and the distribution corresponding to the current geometry. The quality of the optimized shape depends on how well the distribution is defined. Unrealistic geometries may be obtained unless logical constraints on the geometry are imposed.

The inverse-design method was studied by many researchers for different airfoil applications, Ref. [1-6]. Also, the optimization techniques were adopted by other investigators, Ref. [7-12].

In this paper, an inverse-design optimization program, Fig.1, is developed to design the section profile/shape of a compressor blade. The input parameters include a desired pressure/velocity distribution over the blade, an initial compressor blade shape C4 as a base profile. It begins with potential flow calculations of a given starting shape using the panel method and successfully modifies the shape by an iterative process so that the calculated velocity distribution approaches the desired one. Every iteration is performed in two main steps:

- Step 1: Flow calculation.

- Step 2: Geometrical marching.

The calculated velocity distribution is compared with the desired one and a transpiration model is applied to modify the


current shape towards another one more close to the target shape. The geometrical marching is conducted by varying the panel slopes as a function of the normal velocity excess induced by the difference between the required and calculated velocities.

2. INVERSE-DESIGN PROBLEM FORMULATION

This paper deals with the construction of a blade shape/profile which has a specified pressure/velocity distribution that meets certain aerodynamic features. An iterative inverse method using geometry-modification algorithm to modify the blade shape is used, Fig. 2. The process starts with the analysis of a given blade shape C4 base profile using a flow solver and the difference between the calculated velocity and the required one is used to calculate flow distortion in the algorithm vortices is used to calculate the flow distortion. The initial geometry can be modified after convergence of the flow calculation which has been performed by the solver. The blade modification is performed using a physical model which relates the displacement of each point of the geometry to the difference between the current pressure/velocity distribution and the required one. The obvious advantage of inverse method is the control the designer has over the force characteristics of the blade profile and over the boundary layer development on its surface; a control is gained through the pressure/velocity distribution that is specified. However the inverse problem doesn't necessarily have a solution. A solution to the inverse problem exists only if a certain constraints are satisfied by the prescribed characteristics and by the profile itself. These constraints will be explained, as well as the means by which they might be satisfied to produce a practical blade shape. Sophisticated CFD flow solutions take run-time in calculating flow for a single shape, which is unacceptable in an optimization program in which many design parameters must be compared. Despite the imitations of the panel method “cannot predict flow separation“, panel method is effective as a filter to allow candidate solution for the optimization part of the program.

3. GEOMETRICAL AND SURFACE PRESSURE RESTRICTIONS IN BLADE DESIGN

3.1 Preview

In order to design an acceptable blade shape/profile that corresponds to a prescribed surface pressure/velocity distribution, various restrictions have to be met by the imposed target pressure/velocity distribution and by the blade contour. 3.2 Geometrical Constraints 3.2.1 Airfoil Contour Considerations

A closed contour is described by a continuous line whose end points coincide. The end points of the blade line are the points corresponding to the lower and upper trailing edge points. Thus, a closed contour is the one in which the two

trailing edge points coincide. Consequently, the airfoil in Fig.3a is closed and the other one in Fig.3b is not.

Fig.1 Operating window of the "Inverse Program".

Fig.2 Flow chart of the algorithm of the inverse method.

In practice, some trailing-edge thickness, usually of the order of one percent of the chord, is desirable for structural integrity. In such a case, the definition of closure is expanded to include the case in which the trailing edge points are separated by some small distance. None re-entrant airfoil is the one for


which the line describing the contour never crossover itself. An airfoil may be closed but not necessarily none re-entrant. Thus, the airfoil depicted in Fig.3c is closed but re-entrant and the one in Fig.3d is open and re-entrant. Re-entrant airfoils are non-physical. None re-entrant exhibits a positive thickness from leading edge to trailing edge. Structural requirements impose some limitations on the minimum acceptable thickness though the airfoil. The blade that is shown in Fig.3e is likely undesirable from practical point of view since the thickness upstream of the trailing edge is very small. Thus, structural and operational requirements place restrictions on the acceptable thickness distribution on the airfoil [13].

a) Closed trailing edge.

b) Open trailing edge.

c) Closed re-entrant airfoil.

d) Open re-entrant airfoil.

e) Structurally unfit airfoil.

Fig.3 Possibilities of airfoil contours [13].

3.2.2 Leading/Trailing-Edge Region Considerations

There is a relation between the leading-edge and trailing-edge regions of an airfoil and the velocity distribution. Most airfoils have rounded leading edges. A rounded leading-edge

allows operation over a wide range of angle of attack. While, a sharp leading-edge would cause the flow to separate at the corner and it allows a limited range of flow incidence. Since both the upper and lower surfaces of the airfoil must be streamlines of the flow, a stagnation point must be present on the surface in the leading edge region, Fig.4. That stagnation point is also a branch point since the flow splits into two downstream of it. Thus, when designing for a prescribed velocity distribution, it is a must to include a stagnation point at the leading-edge region.

Fig.4 Flow characteristics near airfoil leading-edge [13].

3.2.3 Considerations of Trailing-Edge Included-Angle

The included-angle of the trailing-edge has an effect on the velocity distribution. In the case of an inviscid stream, two possibilities arise for the flow at the trailing-edge. If the included-angle of the trailing-edge is zero (a cusp) the pressure at the upper and lower surfaces will have the same magnitude at that point. For an isentropic flow, the velocities on the two sides are also identical. In such a case, the velocity has a non-zero value, usually slightly less than the free-stream value, Fig.5a. If the included-angle is not zero, the only way for the pressure and the total velocity at the upper and lower trailing-edge points to match is for the velocity to be zero. In this case, the trailing-edge point is a stagnation point, Fig.5b. It should be noted that the gradients of the velocity distribution in the vicinity of the trailing-edge are dependent on the magnitude of the trailing-edge angle (Figs.5b and 5c). In viscous flows, Fig.5d, if one were to design for a velocity distribution to be achieved outside boundary layer, one has to demand only that the velocities on opposite sides of the trailing edge and outside boundary layer match, they do not vanish even for a non-zero include-angle. As a consequence, the shape of the trailing-edge region is highly-dependent on the local distribution of the velocity in the region [13].


Fig.5a Cusp trailing-edge.

Fig.5b Finite-angle at trailing edge.

Fig.5c Large Finite-angle at trailing edge.

Fig.5d Trailing-edge for viscous flow.

Fig.5 Possible flow conditions near trailing-edge [13].

4. OPTIMIZATION OF THE TARGET PRESSURE DISTIBUTION

In order to obtain a practical blade shape, certain characteristics of the pressure distribution on the blade such as the presence of a stagnation point at the leading-edge and possibly another at the trailing-edge are to be considered. Other features depend on the designer's objective. In all cases, the objective is to obtain a certain lift-coefficient and a level of drag as low as possible. All these goals can be achieved by tailoring the surface pressure/velocity distribution.

The lift-coefficient of an airfoil can be determined once the surface velocity is prescribed. The lift coefficient, CL, of an airfoil is given by

Cu

LC L

2

21

∞∞

=ρ

(1)

Where, L is the lift, C is the airfoil chord and ρ∞ and u∞ are the free-stream values of the density and velocity, respectively. In potential flows, the lift can be expressed as a function of the circulation, Γ, around the airfoil

uL ∞∞Γ= ρ (2) Circulation,Γ , is obtained by integration around the contour

∫=Γ dssu )( (3) Hence, the lift-coefficient is given by

∫=∞

dsu

suc

C L)(12 (4)

As a result, CL is known once u

su∞

)( is specified.

In incompressible potential flow, the pressure drag will be zero for all possible surface velocity distributions. At supercritical speeds wave drag will be present if a shock wave is present in the flow field. In the absence of shock waves, drag is due to viscous effects and with some amount of control over these viscous effects by proper design of the imposed surface pressure/velocity distribution; one can control the growth of the boundary layer, delay transition to turbulence and hopefully, avoid flow separation. All of these effects tend to lower drag levels.

Flow separation is a disastrously flow feature. It occurs due to high adverse-pressure gradients as they might occur on the upper surface of an airfoil as the flow is recompressed to stagnation or near free-stream values at the trailing-edge for sharp or cusped geometries. Separation can be avoided by imposing pressure distributions with gentle gradients, Fig.6.


Fig.6 Types of pressure distributions [13].

This solution, however, is at odds with the other goal of trying to maximize lift. The latter objective can be achieved by having the flow undergoes the required recompression to the trailing-edge values in the shortest possible distance with still avoiding separation. Liebeck [14] addressed this problem and the solution was found in the use of pressure distributions proposed by Stratford [15]. Such pressure distributions achieve recompression in the shortest possible distance or, alternatively, the maximum possible pressure recovery in a given distance. The pressure distributions in Fig.6 have been drawn with a "rooftop" region. The level and extent of the rooftop region are clearly designed to maximize lift. Lift can also be increased by increasing the prescribed pressure levels on the lower surface of the blade or by equivalently decreasing the velocity distribution there.

Clearly, the maximum lift is obtained if the velocity is zero everywhere on the lower surface. Obviously, such a flow is impossible, and all practical airfoils will have values of the speed on the lower surface well above the stagnation value.

The drag of an airfoil can be significantly lowered by delaying the transition of the flow from laminar to turbulent. Transition can be delayed by prescribing a continuously favorable pressure gradient from the leading-edge stagnation point to the position of minimum pressure. Thus, carful tailoring of the surface pressure distribution is needed to control the growth of instabilities that brings transition of the flow. 5.ANALYSIS

In this paper, an inverse-design optimization algorithm is developed to design the sectional shape/profile of an axial compressor blade. The input parameters include a desired pressure distribution over the blade. Various restrictions of the imposed target pressure/velocity distribution and the blade contour have to be met. In order to obtain an acceptable blade profile, three essential elements characterize the process: First, a method to describe the geometry of the blade and the change of the blade geometry through design variables; Second, a

module implementing the aerodynamic calculations; Third, an optimization method. An objective function, first, calculates the geometry, then, the corresponding pressure distribution, and finally, the deviation of this flow from the target pressure distribution.

5.1 Shape Definition Algorithm 5.1.1 Specification of Blade Profile Geometry

Airfoil profiles are constructed by superimposing a standard half-thickness shape yt normal to and on either side of a camber line yc. The camber-line shape is usually either a circular arc or a parabola. The slope of the camber-line at any arbitrary position x )0( lx ≤≤ is equal to tan θc. Where, θc = tan-1(dyc/dxc) as shown in Fig.7.

Fig. 7 Construction of a blade profile from a camber line and

a base profile [14].

If we first construct a semi-circle of radius l/2, as shown, and divide it into M equal segments M/2πφ =Δ , camber-line Xc,Yc coordinates are then given by:

)cos1(21 φ−=

lxc

θ ccc xy tan=

The half-thickness yt is combined with the camber-line to obtain the coordinates of profile points a and b on the upper and lower surface as follows:

θ ctca yxx sin−=

θ ctca yyy cos+=

(5)

(6)


θ ctcb yxx sin+=

θ ctcb yyy cos−=

5.1.2 Bezier Curves [16]

Bezier curves are a useful tool in optimization as they can be defined using a relatively few parameters. Bezier curves have many properties that make them attractive for defining the airfoil surface during the design procedure. The end-points are automatically fixed at the two end-vertices. The vector joining the end-point and the closest control-point is tangent to the curve at its end-point. The curve always lies within the convex figure defined by the extreme points of the polygon. Finally, the curve of the nth order continues throughout and never oscillates widely away from its defining control points.

A Bezier curve [16] is defined as the following parametric curve P(u) in terms of n+1 control points Pi:

)()( ,0

uBPuP ni

n

ii∑=

= 10 ≤≤ u (8)

Where, Bi,n(u) is the blending function given by: )1(),()(, uuinCuB ini

ni −= − (9) C(n,i) is the binomial coefficient:

)!(!!),(

inininC−

= (10)

Thus, if the two dimensional control points are Pi = (xi, yi), then, the curve P(u) can be found parametrically from:

∑==

n

inii uBxux

0, )()(

∑==

n

inii uByuy

0, )()(

To define a general airfoil shape, two curves are used. One is used to define the upper surface of the airfoil and the other for the lower surface of the airfoil. Fig.8 shows a layout of the program. Fig.9 shows the input parameters of the program.

Fig.8 Layout of "Geometry Program" that draws blade profile.

Fig.9 Blade-profile input data. 5.2 Flow Solver Procedure

The incompressible potential flow is governed by the Laplace equation that is solved numerically with a panel method, Fig.10. The panel method divides the body shape into straight line segments, called panels, in order to approximate the velocity of the potential flow close to the body profile. Modern sophisticated CFD methods also exist, which can accurately compute the flow field. However, CFD methods are still too computationally-intensive to be feasible in an optimization scheme. While, panel methods are able to simulate the flow quickly enough to be effective in schemes for which many iterations are required.

Fig.10 Inviscid flow field.

Fig.11, Panel distribution on an airfoil.

5.2.1 Panel Method

The panel method can be explained in the following points, Fig.11, [17-19]: • The contour defining the blade is partitioned into N number

of straight panels or elements. • A fictitious source of constant density mj is distributed on the

j-th ( j =1 , 2 , 3 , ... , N ) panel.

(11)

(7)


• A fictitious point-vortex of constant density is distributed on all the panels. Note that the source density changes from panel to panel, but the vortex density remains the same.

• The N number of source densities and the vortex density are computed by solving N+1 number of simultaneous equations. These equations are obtained by enforcing the no-penetration condition at one control-point per panel and the Kutta condition at the trailing-edge.

• The velocity potential is constructed by using the principle of superposition with contributions from the free-stream, the source distribution, and the vortex distribution. This potential as seen by an observer in the fluid is written as

( ) ∑ ∫ ⎥⎦

⎤⎢⎣

⎡−++=

=∞

n

j jelement

j dsrmyxu1 , 2

ln2

sincos θπγ

πααφ (12)

On the j-th element, an element of length ds is taken. With r being the distance between this element and the observer, and θ being the angle this r makes with the x-axis; the integrand is the velocity potential due to source and vortex distribution on the element ds. By integrating this elemental potential over panel j and then summing over all the panels, the total potential is obtained.

5.2.2 Panel Parameters

The starting and the ending points of a panel are at (xi , yi) and (xi+1 , yi+1), respectively. Five quantities are associated with each panel, namely, the length of the panel, the coordinate of the control-point on the panel, and the orientation of the panel. These quantities are calculated, respectively, as follows:

( ) ⎥⎦

⎤⎢⎣

⎡ ++= ++

2,

2, 11 yyxxyx pppp

pp (13)

( ) ( )yyxxl ppppp −+−= ++ 12

12 (14)

l

yy

p

ppp

−= +1sin θ ,

lxx

p

ppp

−= +1cos θ (15)

• No-Penetration Condition

The no-penetration condition enforced at the control point on each panel yields the following set of equations:

∑ =+=

+

n

jiniijj bAAm

11,γ for i=1,2,3,…,n (16)

Where,

)cos(21ln)sin(

21 1, θθ

πθθ

π jiijij

jijiij B

rr

A −+⎟⎟⎠

⎞⎜⎜⎝

⎛−= + (17)

))sin(21ln)cos(

21(

1

1,1, ∑ −+⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

=

++

n

jjiij

ij

jijini B

rr

A θθπ

θθπ

(18)

and )sin( αθ −= ∞ ii ub (19)

• Kutta-Condition

The Kutta-Condition states that the tangential components of velocities approaching the trailing edge along the top and bottom surfaces are equal. This equality condition on velocities is imposed at the two nodes, one on the top surface of the airfoil and the other on the bottom surface, adjacent to the trailing edge.

∑ =+=

++++

n

jnnnjnj bAAm

111,1,1 γ (20)

Where,

∑⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

=

++

nk kj

jkjkjkkjjn

rr

BA,1

1,,1 ln)cos(

21)sin(

21

θθπ

θθπ

(21)

∑ ∑⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

= =

+++

nk

n

j kj

jkjkjkkjnn

rr

BA,1 1

1,1,1 ln)sin(

21)cos(

21

θθπ

θθπ

(22)

and )cos()cos( 11 αθαθ −−−−= ∞∞+ nn uub (23)

• rij and βij Calculation

( ) ( )yyxxr jijiij −+−= 22 (24)

( )( ) ( )( )( )( ) ( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

−−+−−

−−−−−=

++

++

xxxxyyyy

yyxxxxyyATANDB

jijijiji

jijjjijiij

11

11 ,2)( (25)

When ji ≠ and π= when i=j

• Pressure Calculation

The pressure at each control-point can be calculated by utilizing the Bernoulli's equation

vpvp tii22

21

21 ρρ +=+ ∞∞ (26)

The tangential component of velocity at each control-point can be calculated from the equation

∑

∑⎥⎥⎦

⎤

⎢⎢⎣

⎡−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−−+−=

=

=

+

+

∞

n

j

n

jjiij

ij

jji

ij

jijjiij

j

iij

Br

r

rr

Bm

uv1

1

1,1

1,1

)cos(ln)sin(2

ln)cos()sin(2)cos(

θθθθπγ

θθθθπαθ

(27)

5.3 Optimization Procedure

In the optimization process, successive modifications are made to an initially prescribed contour in such a way to minimize the difference between the pressure/velocity distribution over the initial profile and a prescribed target distribution, which is the objective function. The design problem is to try to achieve a velocity distribution v equal to a specified speed distribution vt, in which case the objective function could be described by:


( )∫ ≤−=s

t errordsvvI (28)

Where, v is the velocity over the initial contour and vt, is the target velocity and the integral is performed over the contour and the objective function I is function of design variables ai..,In the optimization procedure, new values are assigned to the ai's such that the resulting contour will have an objective function, which minimizes the difference between the pressure/velocity distribution.

6.RESULTS AND DISCUSIONS

The design of a compressor blade-shape begins with an initial blade shape C4 base-profile [20] and a given target pressure distribution. It is required to obtain the shape of the blade that gives this target pressure. In this work, the target pressure distribution is a modification of the pressure distribution computed over the base-profile C4. In order to obtain a realistic blade-shape in such a technique, the changes to the initial shape are derived by the difference between the pressure distribution computed over the given blade-shape and the target pressure distribution. Four different inverse-design cases were considered to test the program. In each case, C4 base-profile is used along with its pressure distribution and the desired target pressure. The target pressure represents mostly modification of the pressure distribution on the upper surface, which is the critical part of blade-shape design. The C4 base-profile design parameters (dimensionless) are: profile-thickness =1.0, camber-line type is circular arc, x/l of maximum camber =0.5, camber angle θ = 60o, stagger angle λ =0.0o, angle of attack α = 23o and inlet velocity W1 = 1.0. The camber-line coordinates and half-thickness dimensions are listed in table.1

Table.1 Camber-line and half-thickness dimensions of C4 base-profile [20].

x/l yc /l Half yt /l

0.007596 0.004341 0.012862

0.030154 0.016723 0.024839

0.066987 0.035362 0.034763

0.116978 0.057714 0.042403

0.178606 0.080920 0.047408

0.250000 0.102220 0.049800

0.328990 0.1192440 0.049969

0.413176 0.1301980 0.048598

0.500000 0.1339750 0.045700

0.586824 0.1301980 0.041300

0.671010 0.1192440 0.035836

0.750000 0.102220 0.029737

0.821394 0.080920 0.023481

0.883022 0.057714 0.017673

0.933013 0.035362 0.012487

0.969846 0.016723 0.007015

0.992404 0.004341 0.001945

6.1 Validation of the Present Technique

To validate the present methodology, the pressure and velocity distributions of the C4 base-profile were considered as the target distributions. The optimization program succeeded in targeting the desired distributions and predicated the same C4 base-profile, Fig.12, with acceptable accuracy. The target-pressure distribution and the resulted target shape are very similar to the initial shape and pressure/velocity distribution.

Figure 12, Initial Pressure, Velocity and shape

6.2 Case 1 (Forward Loading) Fig.13 shows the effect of forward loading. The upper

pressure falls steadily and then diffuses towards the trailing


edge. On the lower surface, the pressure remains almost constant over most of the chord length. The profile of C4 changed slightly to adjust to the target pressure and velocity distributions. 6.3 Case 2 (Afterward Loading)

Fig.14 demonstrates the effect of afterward loading. The pressure falls steadily over most of the upper-surface in the region 0 < x/l < 0.6 and to a lowest value Cpu = -2.6 at x/l = 0.61, followed by a dramatic diffusion towards the trailing edge. On the lower surface, the pressure remains constant over most of the chord length.

In this case, the profile of C4 changed largely, especially the upper-surface, to adjust to the target pressure and velocity distributions.

Fig.13 Effect of forward loading.

6.4 Case 3 (Liebeck Pressure Distribution)

In Fig. 15, Liebeck pressure distribution [14] is taken as a target of the inverse-design problem. This pressure distribution satisfies three criteria. First, the boundary layer does not separate. Second, the corresponding blade-shape is practical and realistic. Third, a maximum lift is obtained. Liebeck pressure distribution utilizes the pressure recovery distribution,

which avoids separation by a specified margin along its entire length. This consists of a flat rooftop followed by a recovery distribution.

As seen in Fig.15, the obtained solution of the pressure distribution agrees quite-well with Liebeck’s target pressure distribution [14] except in the region near the nose 0<x/l<0.22. The reason of this deviation may be attributed to the inability of the panel method to correctly model such type of complicated flow. Even so, the obtained shape is well-streamlined and the flow around the obtained shape was successfully calculated.

Fig.14 Effect of afterward loading.

6.5 Case 4 (Modified Liebeck Pressure Distribution)

Fig. 16 shows a modification to Liebeck pressure distribution [14], where, the upper-surface pressure goes to a very low value Cpu = -3.5, at x/l = 0.35 followed by a dramatic diffusion towards the trailing edge. This may result in flow-separation approaching the trailing edge.

7.CONCLUSIONS The objective of the present investigation is to develop and

validate an inverse optimization algorithm, which designs two-dimensional compressor-blade shapes based on a prescribed pressure or velocity distribution. The calculation starts with a


given compressor-blade shape C4 as a base-profile. Based on the above discussions, the following points can be stated: 1- The adopted inverse-design optimization technique proved

to be very efficient in predicting the blade-shape according to the target pressure/velocity distribution.

2- The discussion shows that the utilization of the panel method is suitable for such optimization technique. While, the traditional CFD techniques are not suitable due to the excessive-iterative nature of the optimization technique.

3- As panel method is suitable for potential flow, the flow separation and drag-coefficient cannot be considered.

4- Although, the optimization is based on potential flow, it is very beneficial in real applications when the lift-coefficient is to be maximized. Also, it is effective when a prescribed pressure distribution is set to avoid/delay flow separation on the upper-surface of the compressor blade.

5- The present technique may be extended to consider viscous effects with a suitable simple computational technique.

6- Much effort is to be needed to extend the present work to three-dimensional calculations.

Fig.15 Liebeck target, initial and calculated pressure

distributions.

Fig.16 Liebeck modified target, initial and calculated pressure

distributions.

Nomenclature Symbol Unit Description

A [m2] Area a [m/s] Speed of sound C [m/s] Absolute velocity CD [-] Drag coefficient CL [-] Lift coefficient Cm [m/s] Meridional velocity Cp [-] Static pressure rise

coefficient Cθ [m/s] Tangential absolute

velocity c [m] Chord i [°] Incidence angle l [m] Chord U [m/s] Free stream velocity V [m/s] Velocity x,y geometry coordinates L [N] Lift force F Objective funcation


Greek

α : Attack angle

β1 : Relative inflow angle

β2 : Relative outflow angle

δ : Deviation angle

Λ : Stagger angle

ρ : Density

γ : Vortex strength

Γ : Circulation

θ : Camber Angle

Acknowledgments

The fourth author wishes to thank his supervisors for their encouragement, advice and help to complete this work.

References [1] Shigemi, M., "A solution of an inverse problem for multi-

element aerofoils through application of panel method", Trans. Japan Soc. Aero. Space Sci., Vol. 28, No. 80, pp. 97-107, 1985.

[2] Selig, M. S., Maughmer, M. D., "Multipoint inverse airfoil design method based on conformal mapping", AIAA Journal, Vol. 30, No. 5, pp. 1162-1170, 1992a.

[3] Selig, M. S., Maughmer, M. D., "Generalized multipoint inverse airfoil design", AIAA Journal, Vol. 30, No. 11, pp. 2618-2625, 1992b.

[4] Selig, M. S., "Multipoint inverse design of an infinite cascade of airfoils", AIAA Journal, Vol. 32, No 4, pp. 774-782, 1994.

[5] Petrucci, D. R., "Inverse problem for the flow around isolated airfoils and turbomachine cascades", M. Sc. Dissertation, EFEI, Itajubá-MG, Brazil, 1998.

[6] Petrucci, D. R., Manzanares, F. N., Oliveira, W., "A numerical technique for solving the inverse problem of potential flow in turbomachine cascades", Proceedings of ENCIT 1998, 7th Brazilian Congress of Thermal Engineering and Sciences, ABCM, Rio de Janeiro-RJ, Brazil, pp. 1305 – 1310, 1998.

[7] Venkataraman, P., "A design optimization technique for airfoil shapes", AIAA-94-1813-CP, AIAA 12th Applied Aerodynamics Conference, Colorado Springs, 20-23 June 1994.

[8] Venkataraman, P., "A new procedure for airfoil definition", AIAA-95-1875-CP, 13th AIAA Applied

Aerodynamics Conference, San Diego, CA, 19-22 June 1995.

[9] Venkataraman, P., "Optimum airfoil design in viscous flows", AIAA-95-1876-CP, 13th AIAA Applied Aerodynamics Conference, San Diego, CA, 19-22 June 1995.

[10] Venkataraman, P., "Inverse airfoil design using design optimization", AIAA 14th Applied Aerodynamics Conference, AIAA-96-2503-CP, New Orleans, 19-20 June 1996.

[11] Venkataraman, P., "Optimal airfoil design", AIAA-96-2371-CP, AIAA 14th Applied Aerodynamics Conference, New Orleans, 19-20 June 1996.

[12] Venkataraman, P., "Unique solution for optimal airfoil design", AIAA-97-2233, 15th AIAA Applied Aerodynamics Conference, Atlanta, GA, 23-25 June 1997.

[13] Volpe, G., Melnik, R. E., “The role of constraints in the inverse design problem for transonic airfoils,” AIAA-81-1233, 14th Fluid and Plasma Dynamics Conference, Palo Alto, California, 23-25 June 1981.

[14] Liebeck, R. H., "A class of airfoils designed for high lift in incompressible flow", Journal of Aircraft, Vol. 10, No. 10, pp. 610-617, October 1973.

[15] Stratford, B. S., "The prediction of separation of the turbulent boundary layer", Journal of Fluid Mechanics, Vol. 5, pp. 1-16, 1956a.

[16] Bezier, P., "Mathematical and practical possibilities of unisurf", Computer Aided Geometric Design, Barnhill, R. E., Riesenfeld, R. F., Academic Press, New York, 127-152,1974.

[17] Kuethe, A. M., Chow, C.-Y., “Foundations of Aerodynamics - Bases of Aerodynamic Design,” 4th edition, John Wiley & Sons, Inc., 1986.

[18] Ambar K. Mitra http://www.public.iastate.edu/~akmitra/aero361/design_web/panel_method.pdf

[19] Petrucci, D. R., ManzanaresFilho, N., Ramirez, R. G. C., "An efficient panel method based on linear vortex distributions for flow analysis in turbomachinery cascades", Proceedings of COBEM 2001, 16th Brazilian Congress of Mechanical Engineering, ABCM, Uberlândia, MG, Brazil, Vol. 8, pp. 256-265, 2001.

[20] Lewis, R. I., "Turbomachinery performance analysis", 1st Edition, Butterworth-Heinemann, 1996.

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