AIAA 2003–1068 Aerodynamic Design of Cascades by Using an Adjoint Equation Method Shuchi Yang, Hsiao-Yuan Wu, and Feng Liu Department of Mechanical and Aerospace Engineering University of California, Irvine, CA 92697-3975 Her-Mann Tsai Temasek Laboratories National University of Singapore Kent Ridge Crescent, Singapore 119260 41st AIAA Aerospace Sciences Meeting and Exhibit Jan 6–9, 2003/Reno, Nevada For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344
Transcript
AIAA 2003–1068Aerodynamic Design of Cascades byUsing an Adjoint Equation MethodShuchi Yang, Hsiao-Yuan Wu, and Feng LiuDepartment of Mechanical and Aerospace EngineeringUniversity of California, Irvine, CA 92697-3975
Her-Mann TsaiTemasek LaboratoriesNational University of SingaporeKent Ridge Crescent, Singapore 119260
41st AIAA Aerospace SciencesMeeting and Exhibit
Jan 6–9, 2003/Reno, NevadaFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344
Aerodynamic Design of Cascades by Using anAdjoint Equation Method
Shuchi Yang,∗ Hsiao-Yuan Wu,† and Feng Liu‡
Department of Mechanical and Aerospace EngineeringUniversity of California, Irvine, CA 92697-3975
Her-Mann Tsai§
Temasek LaboratoriesNational University of Singapore
Kent Ridge Crescent, Singapore 119260
A continuous adjoint equation method is developed for the aerodynamic design ofcascade blades in a two-dimensional, inviscid, and compressible flow. A cost functionbased on a prescribed target pressure distribution is defined and the purpose of design isto minimize the value of the cost function. The adjoint equations and the correspondingboundary conditions are derived based on the Euler equations, the flow boundary condi-tions, and the definition of the cost function. Gradient information is obtained by solvingthe adjoint equations. A one-dimensional search algorithm is used to perform the opti-mization in the calculated gradient direction. Multigrid method is applied to acceleratethe computation for both the Euler and the adjoint equations. Three transonic cascadeblade design cases are tested. The results show that the method is effective and efficientfor turbomachinery blade design. The effect of shape functions on the performance ofthe design method is discussed.
I. Introduction
CFD-based design methods can be broadly classifiedinto two basic categories: inverse methods and numer-ical optimization methods. Inverse methods solve theinverse problem of determining the shape given a pres-sure distribution. A classical example of inverse designwas the conformal mapping method by Lighthill1 fortwo-dimensional incompressible flows. Bauer, Garabe-dian and Korn designed supercritical wing sections us-ing inverse hodograph methods.2 Hobson designed twoshock free impulse cascades by hodograph method.3
Inverse design for turbomachinery cascades were stud-ied by Tan et al.4 and Giles et al.5 Dang et al.6,7
developed an inverse method where only the pressuredifference on the blade surfaces are specified instead ofvelocity distributions.
Inverse methods are relatively fast in computation.However, the user must be highly experienced to pre-scribe the appropriate surface flow distributions andinitial guesses of the geometry. Even with that given,there is in general no guarantee that such an objectiveis attainable. And if it is, it may not be the optimal.
Aerodynamic optimization has been a topic stud-ied since the early days of aerodynamics. It was notuntil recently, however, that systematic computer op-timization procedures were developed to optimize thedesign of airfoils and wings involving nonlinear flows.This was only made possible by the new capabilityof calculating complex nonlinear flows within shortcomputational time and the advances in nonlinear op-timization techniques. The basic idea of these methodsis to couple a fast CFD analysis code with a numericaloptimization scheme. The CFD analysis code is usedto evaluate an aerodynamic measure of merit, whichthe optimization method uses to extremize by modify-ing a given set of design variables. Hicks et al.8 usednumerical optimization for transonic airfoil shape de-sign with a transonic potential code. Reuther et al.9
used an optimization method with the Euler equationsfor the design of supersonic wing-body configurations.Hager et al.10 considered transonic airfoil optimizationusing the Navier-Stokes equations. With these meth-ods any reasonable aerodynamic quantity can be usedas a design objective. If a surface pressure distribu-tion is specified as a target function, the method willalso mimic an inverse method to find out the particu-lar shape that gives the desired pressure distribution.If this pressure distribution is not attainable, the op-timization method will not fail as an inverse methodwould, but rather it will provide a best approximation
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to the given distribution.The numerical optimization methods used by the
authors cited in the above paragraph use direct per-turbation and finite-differences to obtain the gradientof the cost function with respect to the design vari-ables. This makes such methods extremely expensivefor relatively large number of design variables becausethe large number of flow solutions needed to determinethe gradient information. An alternative way is the ad-joint equation method. Gradient information can beobtained by solving the adjoint equations of the flowgoverning equations. The computational time of solv-ing the adjoint equations is approximately the same asthat of solving the flow equations, and it is indepen-dent of the number of design parameters. Therefore,this method is far more efficient than the direct pertur-bation method when the number of design parametersis large.
Jameson pioneered optimization of airfoils andwings by using the adjoint equation method first forthe potential equation11 and for the Euler equations.12
Reuther and Jameson considered wings and wing-bodycombinations with arbitrary curvilinear grids.13 Therehave been major progresses in the last few years in thedevelopment of adjoint method for external flows.14,15
However, there has been little work done on turboma-chinery blades. The goal of this paper is to developand evaluate the same adjoint equation method as inRef. 12 for cascade blade optimization. Initial applica-tion is restricted to inverse design by using the adjointoptimization method with a cost function based onspecified pressure distribution. Attention is paid to theboundary conditions for cascade flow problems. Theflow solver and the adjoint solver are modified fromFlo52x. The code is validated by comparing the gra-dient calculated by the adjoint equation method withthat calculated by the direct perturbation method.Then three transonic cascade design cases are tested.The target designs are a cascade using the NACA0015airfoil, the Hobson I cascade, and the Hobson II cas-cade. Finally, the effect of the distribution and thenumber of shape functions are discussed in connectionwith the performance of the optimization program.
II. Inverse Design as an OptimizationProblem
Inverse design problems of two-dimensional turbo-machinery cascades are studied based on the Eulerequations. Figure 1 is a sketch of the domain. Periodicboundary conditions are applied on the upper bound-ary and the lower boundary. At the inlet, the stagna-tion pressure, the stagnation density, and the angle ofthe flow are specified. At the exit, the static pressure isspecified. A target pressure distribution pd is given onthe blade. The purpose is to modify the initial blade
Periodic boundary condition
Periodic boundary condition
BladeInlet Exit
Fig. 1 Domain of a cascade blade passage.
shape to obtain the target pressure distribution. Thecost function is defined as:
I =1
2
∫
BW
(p− pd)2dBW (1)
in which BW is the blade surface, p is the pressure. Asmaller value of the cost function means the design iscloser to the target design. The chord length is fixedand normalized to be one.
III. Optimization by an AdjointEquation Method
A. The Adjoint Equations
The continuous adjoint equations are derived basedon the general approach proposed by Jameson.12 Ingeneral, the cost function I is a function of the bladeshape F and the flow field w, i.e.,
I = I(F ,w)
where the flow field w are related to F by the flowgoverning equation
R(F ,w) = 0
subject to appropriate boundary conditions. If theshape is changed, the flow field is also changed so thatthe variation of the cost function is
δI =∂I
∂F δF +∂I
∂wδw
The δF term can be obtained directly from geometry,but the δw term will need flow field evaluations, whichare time consuming. In order to eliminate the explicitdependence on δw, the following procedure is applied.Taking the variations of the flow governing equations,we have
δR =∂R
∂F δF +∂R
∂wδw = 0
Introducing an arbitrary multiplier Φ, we may write
δI =∂I
∂F δF +∂I
∂wδw −ΦT (
∂R
∂F δF +∂R
∂wδw)
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or,
δI = (∂I
∂w−ΦT ∂R
∂w)δw + (
∂I
∂F −ΦT ∂R
∂F )δF
To eliminate the explicit dependence on δw, let
∂I
∂w−ΦT ∂R
∂w= 0
This is the adjoint equation. The solution Φ is calledthe co-state variable. Once Φ is obtained, we haveδI = GδF , where G = ∂I
∂F − ΦT ∂R∂F is the needed
gradient of I with respect to the shape change δF .Once the gradient is known, δF can be chosen to havea negative δI to reduce the cost function. After severaldesign cycles, when the cost function approaches zero,the target design is accomplished.
The above general procedure can be applied to theEuler equations and the cost function defined by Eqn.(1). The detailed formulation used here follows thatin the works of Reuther16 and Wang.17 However, theirworks concern airfoil optimization for external flows.For internal flows such as a cascade flow, the boundaryconditions need to be changed. Consider the phys-ical domain (x1, x2) and the computational domain(ξ1, ξ2). The flow field shown in Figure 1 is mappedto a rectangular area on the (ξ1, ξ2) plane with an O-mesh configuration in which ξ2 = 0 corresponds to theblade surface BW . Let BC be the outer boundary ofthe domain, which consists of the inlet, the exit, andthe two periodic boundaries. The Euler equations inthe (x1, x2) domain can be written as
∂w
∂t+∂fi∂xi
= 0 (i = 1, 2) (2)
in which t is time, w is the conservative variable vec-tor, and fi are the flux vectors. The flow boundaryconditions are described in the previous section.
A variation of the shape will cause a variation δp ofthe pressure and consequently a variation in the costfunction:
δI =
∫
BW
(p− pd)δpdBWdξ1
dξ1
+1
2
∫
BW
(p− pd)2δ(dBWdξ1
)dξ1 (3)
Let Φ = {φ1, φ2, φ3, φ4}T be the co-state vector. If Φsatisfies the following adjoint equations and boundaryconditions, then δI can be expressed as a function ofΦ, the flow variables w, and the geometric variations.It no longer has explicit dependence on variations ofthe flow variables. The adjoint equations are
ATi
∂Φ
∂xi= 0 in D (4)
in which the Ai are the Jacobian matrices of the Eulerequations, and D is the domain.
B. Boundary Conditions
The boundary conditions for the adjoint equationsare
n1φ2 + n2φ3 = (p− pd) on BW (5)∫
BC
niΦTAiδwdBC = 0 (6)
where, BC is the outer boundary of the domain, and(n1, n2) is the unit normal vector pointing out from theflow field. Then δI can be expressed as the following
δI =1
2
∫
BW
(p− pd)2δ(dBWdξ1
)dξ1
+
∫
D{∂ΦT
∂ξ1[δ(
∂y
∂ξ2)f1 − δ(
∂x
∂ξ2)f2]
+∂ΦT
∂ξ2[−δ( ∂y
∂ξ1)f1 + δ(
∂x
∂ξ1)f2]}dξ1dξ2
+
∫
BW
φ2δ(−∂y
∂ξ1) + φ3δ(
∂x
∂ξ1)pdξ1 (7)
Once Φ is solved, Equation (7) can be evaluated with-out additional flow field evaluations.
Reuther16 and Wang17 mentioned that the bound-ary condition (6) may be simplified by setting Φ tozero on the far boundary provided the far boundary isfar enough for external flow problems. For internalflows where BC is close to BW , the above simpli-fied boundary condition should not be used. Instead,the characteristic wave method is applied here. Timederivative terms are added to the adjoint equationsand they become
∂Φ
∂t−AT
i
∂Φ
∂xi= 0 in D (8)
Thus the adjoint equations can be solved by time-marching methods. They have the same characteristicwave paths as the Euler equations, but, because of theminus sign, the wave propagation directions are oppo-site to those of the Euler equations. This property isimportant for assigning boundary values on the outerboundary.
The boundary BC can be divided into three parts:the two periodic boundaries BCP , the inlet BCI , andthe exit BCE . For the periodic boundaries, if Φ satis-fies the periodic boundary condition, then
∫
BCP
niΦTAiδwdBC = 0 (9)
For the inlet and the exit, the integrand of Equation(6) is set to zero.
For the inlet, three flow variables are specified ac-cording to the flow boundary conditions. Only onedegree of freedom is left for the flow variables. In or-der to let the integrand equal to zero everywhere at
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the inlet , the coefficient in front of this free param-eter has to be zero. This provides one constraint onthe co-state variables on the inlet boundary. Three de-grees of freedom are left for the co-state variables andtheir values can be extrapolated from the inner field.
For the exit, similar derivation can be done. Threeconstraints are derived on the co-state variables andone value can be extrapolated from the inner field. Ac-cording to the wave propagation property mentionedabove, the way of assigning boundary values for theadjoint equations is very similar to that of the Eulerequations. The major difference is just that the wavepropagation directions are reversed.
The resultant form of the adjoint boundary condi-tions on the inlet and exit boundaries are:
C11φ1+C12φ2+C13φ3+C14φ4 = 0 at the inlet (10)
Di1φ1 +Di2φ2 +Di3φ3 +Di4φ4 = 0 at the exit (11)
in which i=1, 2, 3, and C1j , Dij are functions of flowvariables and geometry parameters. At the inlet, threelinear combinations of φ1, φ2, φ3, φ4 are treated asthree variables and their boundary values are extrap-olated from the inner field. These three combinationsare arbitrary as long as they are linearly independentto one another and linearly independent to the lefthand side of Equation (10). At the exit, a similar ex-trapolation is used.
C. Shape Representation
The Hicks-Henne shape functions16 are used toadjust the shape of the blade. The formulas forthese functions are bi(x1) = sin4(πxmi1 ) , mi =ln(0.5)/ln(xMi). i = 1, 2, ..., 32. Sixteen functions forthe upper surface and sixteen for the lower surface, re-spectively. xMi are pre-selected values correspondingto the locations of maximum bi. Figure 2 shows theshapes of these functions.
The way the gradient is obtained is as follows. Smallperturbations are added to an initial airfoil shape. Theperturbations ( for both upper and lower surfaces ) are
δyβ(x1) = (δcβ) · bβ(x1),
(no summation convention on β is applied.)
in which β = 1, 2, ..., 32 . cβ is the weight of the shapefunction bβ . It acts like a design parameter. Perturbeach cβ and calculate the resulting variation of thecost function δIβ from Equation (7). Then the β-thcomponent of the gradient is obtained as
gβ =δIβδcβ
(no summation convention on β is applied.)
X1
bi(X
1)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Fig. 2 The Hicks-Henne shape functions.
X
XX
X
Step Size along the Gradient Direction
Cos
tFun
ctio
nA
B
C
D
A Parabola
s 2s
Fig. 3 One-D search method.
Each time the blade shape is perturbed, the meshneeds to be adjusted once. Then Equation (7) can beevaluated. The new mesh can be generated from themesh generator or by perturbing the old mesh. Thelatter is faster and hence adopted.
After the gradient is obtained, a 1-D search opti-mization method is applied to determine the step sizealong the gradient direction. Figure 3 is a sketch forthe 1-D search method. For each design cycle, an ini-tial step size s is given to the program. Denote theinitial design by A. Along the gradient direction witha step size s, the program finds a new design, B. Dou-bling the step size, the program finds another newdesign, C. If the cost function value of B is smallerthan that of A and the cost function value of C is
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larger than that of B, then the program uses the threepoints A, B, and C to perform a parabolic fit and findan approximate local minimum point, D. If the costfunction values of A, B, and C do not satisfy the abovecondition, then the program adjusts the value of s un-til the condition is satisfied. The step size between Aand D is used as the initial step size and D becomesthe initial design for the next design cycle. Solutionsof the latest design is also used as initial conditionsfor the next design cycle. At present no constraint isimplemented on the design space.
The flow solver and the adjoint equation solver areboth modified from a version of Flo52x. Three levelsof multigrid are adopted. For this study, the programonly deals with steady-state solutions. Time accuracyis not important and therefore local time stepping isadopted. A CFL number of 9 is used.
IV. Results and DiscussionsThree design cases are tested for this method: a
transonic cascade with the NACA0015 airfoil and theHobson I and Hobson II supercritical shock-free cas-cades. The meshes are 160 × 32 O-grids. They areshown in Figures 4, 5, and 6. For the NACA0015design case, the solvers converge normally. However,for the Hobson I and the Hobson II design cases, thesolvers are sometimes divergent during the design pro-cess. Therefore the dissipation on coarser meshes areincreased to 1.6 times of its original value to enhancesolution convergence. Since the dissipation on thefinest mesh remains the same, the precision of thesolver is not affected. An example of convergencehistories of the Euler solver is shown in Figure 7.The rates of convergence are quite different but themagnitudes of the final maximum residuals are ap-proximately the same. Only single precision (32 bit)operations are used in the current computations.
On the other hand, an example of the convergencehistories of the adjoint equation solver is shown in Fig-ure 8. The magnitudes of the final maximum residualsare slightly different for the three design cases. Theexplanation is that the magnitude of the final maxi-mum residual of the adjoint equation solver is affectedby the flow condition. For the NACA0015 designcase, if the shock wave locations or strengths are dif-ferent for the present design and the target design,then the boundary conditions of the adjoint equationsare discontinuous at the shock wave locations. Thisdiscontinuity possibly causes some difficulties for theadjoint equation solver and therefore the final residualis larger than the other two cases.
During the design process, each flow evaluation usesthe result of the previous evaluation as the initial con-dition. Thus the solver can fully converge faster thanit does in the example of Figure 7. This is true for the
X1
X2
-0.5 0 0.5 1 1.5
-0.5
0
0.5
1
Fig. 4 Mesh for the NACA0015 cascade.
X1
X2
-0.5 0 0.5 1 1.5
-1
-0.5
0
0.5
1
Fig. 5 Mesh for the Hobson I cascade.
adjoint solver, too. For the NACA0015 design case,the solvers perform 200 iterations for each evaluation.Each flow evaluation takes approximately 9 seconds,and each adjoint equation evaluation takes approxi-mately 12 seconds on an AMD athlon 1800+ CPU.For the Hobson I and Hobson II design cases, 400 it-erations are performed, and the computational time isapproximately doubled.
Before the adjoint equation method is applied onany design cases, the gradient calculated by thismethod is compared with that obtained by directperturbation and finite-differences to examine its cor-rectness. For the direct perturbation method, the 32design parameters are perturbed one at a time. Foreach perturbation of the shape, one flow evaluation
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X1
X2
-0.5 0 0.5 1 1.5-1
-0.5
0
0.5
1
Fig. 6 Mesh for the Hobson II cascade.
Iteration Cycles
Max
imum
Res
idua
l
100 200 300 40010-5
10-4
10-3
10-2
10-1
100
101
102
NACA0015 Design Case
Hubson I Design Case
Hubson II Design Case
Fig. 7 Convergence history of the Euler solver.
is performed in order to evaluate the cost function.Compared to the adjoint equation method, the directperturbation method takes approximately 32 times ofthe computational time (excluding one basic evalua-tion of the flow field that is needed for both methods)to obtain the gradient information. Figure 9 showsthe comparison of gradients between the adjoint equa-tion method and the direct perturbation method forthe NACA0015 design case for the first design cycle.They match each other well. The locations of largestdeviations coincide with the shock wave locations. De-tails of the three test cases are discussed below.
A. The NACA0015 Cascade
The first case is the cascade of NACA0015 airfoilswith zero stagger angle and a unit space to chord ra-
Iteration Cycles
Max
imum
Res
idua
l
0 100 200 300 400 50010-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
NACA0015 Design Case
Hubson I Design Case
Hubson II Design Case
Fig. 8 Convergence history of the adjoint equationsolver.
Design Parameters
Gra
dien
tCo
mpo
nen
ts
0 10 20 30-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2Direct Perturbation Method
Adjoint Equation Method
Fig. 9 Comparison of gradients for the NACA0015cascade case.
tio. The flow solver is first used to solve the flowthrough this cascade. Once the solution is obtained,the pressure distribution over the airfoil is calculatedand set up as the target pressure. The design code isthen applied to obtain the desired airfoil shape start-ing from the NACA0012 airfoil. The success of thedesign is then measured by the difference between thefinal shape obtained by the design code and that of theoriginal NACA0015 airfoil and also the differences oftheir corresponding pressure distributions. The geom-etry and the grid used in the computation are shownin Figure 4. The inflow angle is 0 degree with re-spect to the x1 axis. A back pressure correspondingto an isentropic exit Mach number of 0.7 is used. The
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Design Cycles
Cos
tFun
ctio
n
0 25 50 75 100
10-3
10-2
10-1
100
Fig. 10 History of cost function for the NACA0015cascade case.
X1
X2
0 0.25 0.5 0.75 1-0.08
-0.04
0
0.04
0.08
Initial Shape
Designed Shape
Target Shape
Fig. 11 Comparison of the blade shapes for theNACA0015 cascade case.
Hicks-Henne shape functions, i.e., the location of xMi,are uniformly distributed along the chord.
Figure 10 shows the history of the cost function. Atthe 98th design cycle, the program is not able to finda better design within a preset amount of computa-tional effort, and the program stops there. The costfunction is reduced by approximately three orders ofmagnitude. Figure 11 shows the final designed shapeas compared to the initial and target shapes. Figure12 shows the comparison of the corresponding pressuredistributions. The designed shape and the pressuredistribution match well with the target shape and tar-get pressure distribution, respectively. These resultsdemonstrate the effectiveness of the method for tran-
X1
Cp
0 0.25 0.5 0.75 1
-1.5
-1
-0.5
0
0.5
1
Initial Cp Distribution
Designed Cp Distribution
Target Cp Distribution
Fig. 12 Comparison of the Cp distributions for theNACA0015 cascade case.
sonic flows with shock waves.
B. The Hobson I Cascade
The second case is the Hobson I shock-free blade. Itconsists of a thick profile and is designed by the hodo-graph method to give a shock free supersonic pocket.3
However, a small perturbation in the flow field cancause the generation of shock waves and hence largechanges of the flow. The hodograph design has a spaceto chord ratio of 1.0121. The inflow angle is 43.544 de-grees with respect to the x1 axis. The design exit Machnumber is 0.476. The maximum Mach number in theblade passage is 1.520.
The Mach number distribution from the present Eu-ler flow solver is compared with the hodograph solutionin Figure 13. In order to better match the hodo-graph solution, the exit static pressure is adjusted to0.997 instead of 1 due to the finite distance of the exitboundary from the blades. Two design approachesare tested, both with the aim of reaching the originalHobson blade shape. The first design approach usesthe computed pressure distribution (corresponding tothe Mach number shown as dotted line in Figure 13) asthe target pressure, while the second approach uses theoriginal hodograph pressure distribution (correspond-ing to the Mach number shown as solid line in Figure13) as the target pressure. In both cases, the initialshape is obtained by deforming the original Hobsonblade shape. The size in the x2 direction is shrunkto 0.95 times of its original value. The size in the x1
direction remains the same. The distribution of theHicks-Henne shape functions is uniform.
Figure 14 shows the history of the cost function vs.the number of design cycles in the first approach. In
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X1
Mac
hN
umbe
r
0 0.25 0.5 0.75 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Hodograph Solution
Computational Solution
Fig. 13 Comparison between the hodograph so-lution and the computational solution for HobsonI.
Design Cycles
Cos
tFun
ctio
n
0 25 50 75 100
10-2
10-1
100
Fig. 14 History of cost function for the HobsonI cascade case with the computational target pres-sure.
this case, the cost function does not go down as muchas the NACA0015 design case in 100 design cycles.However, the cost function is still decreasing while inthe previous case the program reaches its limit at the98th cycle. The cost function history curve has someflat regions and some sudden drops. This is similar tothe behavior in the previous case. Figure 15 shows thefinal designed shape and compares it with the initialand target shapes. Figure 16 shows their correspond-ing pressure distributions. The designed shape andpressure distribution are very close to the target shapeand pressure distribution, respectively. No obvious
X1
X2
0 0.25 0.5 0.75 10
0.1
0.2
0.3
0.4
0.5
Initial Shape
Designed Shape
Target Shape
Fig. 15 Comparison of the blade shapes for theHobson I cascade case with the computational tar-get pressure.
X1
Cp
0 0.25 0.5 0.75 1
-4
-3
-2
-1
0
1
2
Initial Cp Distribution
Designed Cp Distribution
Target Cp Distribution
Fig. 16 Comparison of the Cp distributions forthe Hobson I cascade case with the computationaltarget pressure.
shock waves appear. A close examination of Figure 16shows that the designed pressure distribution has a no-ticeable deviation from the target pressure distributionnear the trailing edge. The reason is that the shapefunctions are not adequately distributed near the trail-ing edge, and hence the shape near the trailing edge isinsufficiently represented. Although the pressure dis-tribution is not completely determined by the localshape, adding more shape functions near the trail-ing edge improves the solution (see later discussions).In Figure 16, there is some unphysical fluctuation ofthe pressure distribution near the leading edge of the
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initial shape. For the target shape and the designedshape, there is no such obvious fluctuation.
Figure 17 shows the history of the cost function forthe second design approach, where the original hodo-graph solution is used as the target pressure distribu-tion in the design. Obviously, the cost function ceasesto decrease after less than one order of magnitude re-duction. In the previous case where the computedpressure distribution of the Hobson I cascade is usedas the target pressure, we expect to recover both theoriginal Hobson I blade shape and the target pressuredistribution since the same Euler flow solver is used toobtain the flow solutions. This is confirmed by the overtwo orders of magnitude reduction in the cost functionshown in Figure 14 and also the close agreement be-tween the target pressure distribution and that of thedesigned blade although they differ noticeably fromthe original hodograph pressure distribution. Thehodograph solution is after all a solution from a dif-ferent numerical method. One cannot expect that thecurrent design method based on a particular numeri-cal method for the Euler equations to fully recover thehodograph design. Consequently, the reduction in thecost function is limited when the hodograph pressuredistribution is used as the target function.
Figure 18 shows that the agreement between the de-signed blade shape and the target shape is not as goodas that found in Figure 15. The agreement between thedesigned pressure distribution and the target hodo-graph pressure distribution shown in Figure 19 is alsonot as good as that found in Figure 16. These resultsare consistent with the above argument. However, thelatter design does give the best approximation to thespecified hodograph pressure distribution, given theconstraint on the number and location of the shapefunctions (the design space).
C. The Hobson II Cascade
The third case is the Hobson II shock-free cascade.3
It is thinner than the Hobson I blade. This cascadehas a space to chord ratio of 0.5259. The design exitMach number is 0.575. The maximum Mach num-ber in the blade passage is 1.132. The inflow angleis 46.123 degrees. The Mach number distributions ofthe hodograph solution and the computational solu-tion are shown in Figure 20. Similar to the HobsonI case, there are visible differences between the com-putational solution and the hodograph solution. Inthis case, only the computational target pressure isadopted. The initial shape is a deformed Hobson IIblade. The size in the x2 direction is shrunk to 0.95times of its original value. The size in the x1 directionremains the same. The distribution of the Hicks-Henneshape functions is uniform. Figure 21 shows the costfunction history. Figure 22 shows the comparison of
Design Cycles
Cos
tFun
ctio
n
0 25 50 75 1000.1
0.2
0.3
0.4
0.5
0.60.70.80.9
1
Fig. 17 History of cost function for the Hobson Icascade case with the hodograph target pressure.
X1
X2
0 0.25 0.5 0.75 10
0.1
0.2
0.3
0.4
0.5
Initial Shape
Designed Shape
Target Shape
Fig. 18 Comparison of the blade shapes for theHobson I cascade case with the hodograph targetpressure.
the blade shapes. Figure 23 shows the comparison ofpressure distributions. The cost function does notgo down as much as it does in the Hobson I case withthe same type of the target pressure. Nevertheless, thedesigned shape and pressure distribution are still veryclose to the target shape and pressure distribution.For the pressure distributions, the deviation from thetarget pressure near the trailing edge is more obviousin this case. The unphysical fluctuation of pressure atthe leading edge of the initial shape is also present.
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X1
Cp
0 0.25 0.5 0.75 1
-4
-3
-2
-1
0
1
2
Initial Cp Distribution
Designed Cp Distribution
Target Cp Distribution
Fig. 19 Comparison of the Cp distributions for theHobson I cascade case with the hodograph targetpressure.
X1
Mac
hN
umbe
r
0 0.25 0.5 0.75 10
0.2
0.4
0.6
0.8
1
1.2
Hodograph Solution
Computational Solution
Fig. 20 Comparison between the hodograph so-lution and the computational solution for HobsonII.
D. Effect of Shape Functions
The above test cases demonstrate the capability ofpresent design method under different flow conditionsand different geometries. The performance of the op-timization program depends on the path it can take inthe design space. For a given design target, how wellone can achieve the target pressure depends on thecompleteness of the design space. Proper distributionof the shape functions and the number of the shapefunctions are important parameters. We explore theireffect on the performance of our design program forthe NACA0015 case.
Design Cycles
Cos
tFun
ctio
n
0 25 50 75 100
10-2
10-1
100
Fig. 21 History of cost function for the Hobson IIcascade case.
X1
X2
0 0.25 0.5 0.75 1
0
0.1
0.2
0.3
Initial Shape
Designed Shape
Target Shape
Fig. 22 Comparison of the blade shapes for theHobson II cascade case.
First the distribution of the shape functions is ad-justed. The number of the shape functions remains thesame. It is 32. The locations of xMi for i = 1, 2, ..., 16are chosen to be 0.5× [1− cos(θi)]. θi = π× i/17. Fori = 17, 18, ..., 32, xMi=xMj , and j = i−16. The distri-bution is denser near the leading edge and the trailingedge. The cost function history is shown in Figure 24.At the 59th design cycle, the program reaches its limitand stops.
Secondly, the number of the shape functions are dou-bled, but the distribution of xMi remains uniform. Thecost function history is also shown in Figure 24. Atthe 67th design cycle, the program reaches its limitand stops. The original cost function history shown in
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X1
Cp
0 0.25 0.5 0.75 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Initial Cp Distribution
Designed Cp Distribution
Target Cp Distribution
Fig. 23 Comparison of the Cp distributions for theHobson II cascade case.
Figure 10 is also plotted in Figure 24 as a reference.The computational time needed for solving the adjointequation is independent of the number of design pa-rameters. However, in order to obtain the gradientinformation, extra time is needed to perturb the mesh.The time needed for perturbing the mesh is propor-tional to the number of design parameters. Thereforethe number of design parameters needs to be kept ina reasonable range although increasing the number ofparameters can improve the precision of the solution.In this study, the number of parameters is 32 or 64. Asshown in Figure 24, the computation with non-uniformdistribution shape functions gives the best result. Ityields the fastest reduction and also the lowest valueof the cost function. The second best one is the com-putation with double the number of shape functions.In this case, choosing better locations to put the shapefunctions has more benefit than doubling the numberof shape functions.
For the Hobson I and Hobson II design cases, if theshape functions are located too close to the leadingedge or trailing edge, problems may occur at the initialstage of the design process. For example, sometimesa negative-thickness leading edge or trailing edge mayappear during the design process. That causes prob-lems for the flow and adjoint solvers. After 100 designcycles with uniformly distributed shape functions, thedesign is very close to the target. The step size ofshape modification is smaller than that at the begin-ning of the design process. At this stage, the problemmentioned above is less likely to occur so that we mayshift to use non-uniformly distributed shape functionsin order to obtain a better design. For the Hobson IIdesign case, the number of shape functions is increased
Fig. 24 Performances for different arrangementsof design parameters.
Design Cycles
Cos
tFun
ctio
n
100 110 120 130 140 150
1.0x10-02
2.0x10-02
3.0x10-02
4.0x10-02
Uniform Distribution,32 Shape Functions
Non-uniform Distribution,64 Shape Functions
Fig. 25 Effect of changing the distribution of shapefunctions after 100 design cycles.
to 64 and distributed nonuniformly after the first 100design cycles to see if it can give a better result. Thecomputation is continued for another 50 design cy-cles. The history of the cost function for the additionaldesign cycles is shown in Figure 25. The correspond-ing pressure distributions near the trailing edge areshown in Figure 26. The solution near the trailing-edge of the blade is significantly improved when thenumber of shape functions is increased and the non-uniform distribution is adopted. During the designcycles, the distribution of shape functions can be mod-ified if different distributions are needed for differentdesign stages. More study is needed to determine the
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X1
Cp
0.9 0.95 1
0
0.2
0.4
0.6
0.8Non-uniform Distribution, 64 Shape Functions
Uniform Distribution, 32 Shape Functions
Target Cp Distribution
Fig. 26 Cp distributions near the trailing edge af-ter 150 design cycles.
optimum number and distribution of the shape func-tions for a given design case.
V. ConclusionsAn adjoint equation method based on the Euler
equations is developed for the aerodynamic design ofcascade blades. Three test cases involving a transoniccascade using the NACA0015 airfoil and the Hobson Iand Hobson II supercritical but shock-free cascades arestudied. The adjoint formulation and boundary condi-tions are derived by using cost functions based on thesquare integral of the deviation from a specified targetpressure distribution over the blade surface. Gradientinformation can be obtained with little dependence onthe number of design parameters. A 1-D search algo-rithm is used to accelerate convergence to the designpoint. Results for the three test cases demonstratethat this method is effective and efficient for the designof transonic and supercritical turbomachinery cascadeblades. The choice of the number and locations ofshape functions, however, may significantly influencethe efficiency and accuracy of the design. Better per-formance can be obtained by dynamically adjustingthe number and distribution of the shape functionsduring the design process.
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