A contractive operator solution of an airfoil design inverse problem

Jan Simak∗ and Jaroslav Pelant

Aeronautical Research and Test Institute (VZLU), Department of High Speed Aerodynamics, Beranovych 130,199 05 Prague 9, Czech Republic.

This paper deals with a numerical method for an airfoil design which was presented in [3, 5]. This method is intended fordesign of an airfoil from a given velocity distribution along a mean camber line. The method is based on searching for afixed point of a contractive operator. This operator combines an inexact inverse operator and equations describing the flow. Asubsonic flow is assumed, the flow is described by a system of the Euler equations which is solved by an implicit finite volumemethod. The Newton method is applied to the solution of the nonlinear system. The resulting system of linear algebraicequations is solved by GMRES method, the Jacobian-free version is described. The inexact inverse operator consists of amiddle curve function and a thickness function, both depending on the given velocity distribution. In addition to the velocitydistribution the velocity in infinity is given. The angle of attack is determined so that the stagnation point is in a specificposition. Successful numerical results are presented.

1 Introduction

A method how to get an airfoil from a given velocity distribution is presented. This method can be used in the case of asubsonic inviscid compressible flow described by the Euler equations. The method was described in [3,5]. The main idea is touse a couple of a direct and inverse operator. The direct operator represents an assignment of a velocity distribution to a shapeof an airfoil and the second operator is its inversion. Since we are not able to construct the exact inversion, some iterationmethod has to be used. The direct and the approximate inverse operator are put together to create a contractive operator andits fixed point is found. The method can be used to modify existing airfoils. In the next sections these two operators aredescribed. The theory of a numerical solution of flow dynamics can be found in [1]. The method can also be used to deal witha prescribed pressure distribution.

2 Inverse problem

Denote by P the direct operator representing the solution of the flow problem. In this case it is represented by the Eulerequations, but it is possible to modify the method to use another description of the flow. By L is denoted the approximateinversion. These operators are put together and an equation

PL(u) = f (1)

for an unknown function u : 〈−b, b〉 → R is obtained, b is the length of the chord. The function f : 〈−b, b〉 → R is aprescribed velocity distribution, which is given on the upper and lower side of an airfoil along its mean camber line. Thatmeans f(x) = fupper(−x) for x < 0 and f(x) = flower(x) for x ≥ 0, x is an x-coordinate of the mean camber line. Thestagnation point on the leading edge has to be at the origin of the coordinate system which is also the beginning of the meancamber line. The result of the Richardson’s iteration method is the following sequence

{uk}∞

k=0 , uk+1 = uk + α (f − PLuk) . (2)

The parameter α ∈ (0, 1〉 is suitably chosen in order that the sequence tends to a limit u∗. The initial condition u0 can bechosen for example u0 = f . The desired airfoil represented by a curve ψ : 〈−b, b〉 → R

2 is obtained by application of L tothe limit u∗. The condition for the stagnation point is satisfied by the choice of the angle of attack in every iteration.

The inverse operator L is based on the thin airfoil theory. It consists of a function su(x) describing the mean camber lineand a function tu(x) describing the thickness. The coordinates of the curve ψ = L(u) can be expressed as the following(see [5]):

ψ1(x) = |x| + sign(x)tu(|x|)s′u(|x|)√

1 + s′u2(|x|)

, (3)

ψ2(x) = su(|x|) − sign(x)tu(|x|)1√

1 + s′u2(|x|)

, x ∈ 〈−b, b〉 . (4)

∗ Corresponding author E-mail: simak@vzlu.cz, Phone: +420 266 310 578, Fax: +420 284 825 347

PAMM · Proc. Appl. Math. Mech. 7, 2100023–2100024 (2007) / DOI 10.1002/pamm.200700136

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

X

V

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Y0

0.2

0.4

0.6

0.8

Fig. 1 Prescribed velocity f/v∞ and result-ing airfoil.

Xf-

p0 0.2 0.4 0.6 0.8 1

-0.005

0

0.005

X

f-v

0 0.2 0.4 0.6 0.8 1-0.005

0

0.005

Fig. 2 Distribution of error (f − v)/v∞ onthe chord (for both examples)

X

P

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

Y0

0.2

0.4

0.6

0.8

Fig. 3 Prescribed pressure p/p∞ and result-ing airfoil.

3 Solution of the Euler equations using the Newton-Krylov method

It is needed to express the operator Pψ = u. Thus our goal is to find a steady-state solution of the Euler equations describinginviscid compressible flow. The problem is solved by the implicit finite volume method:

wk+1i = wk

i − τk 1

|Di|

∑j∈S(i)

H(wk+1

i , wk+1j , nij

)|Γij |

(= wk

i − τkΦi

(wk+1

)). (5)

The Osher-Solomon numerical flux is used and the Van Leer κ-scheme or the Van Albada limiter is applied in order to have ahigher order method.

Linearization of this problem by the Newton method leads to a sparse system of linear algebraic equations, which issolved by GMRES method. In order to accelerate the convergence of the method the ILU preconditioning is applied. In theGMRES algorithm (see [4]) the Jacobian matrix A(w) is required only to form a matrix-vector product. This product can beapproximated with the use of the Taylor series expansion, A(w)v ≈

(Φ(w + εv) − Φ(w)

)/ε. The parameter ε is computed

in every GMRES iteration as a function of a machine precision and the norm of v. Using this technique construction of theJacobian matrix can be avoided (see [2]). For the purpose of preconditioning it is sufficient to have a less precise Jacobianmatrix and update it only when it is needed.

4 Numerical examples

Some numerical results were obtained by this method. In order to prove the correctness of this method, it was applied to avelocity distribution belonging to a known airfoil and the results were compared. Prescribed velocity distribution f togetherwith the resulting airfoil are in Fig. 1. The distribution of error f − v on the chord (v - resulting velocity) is in Fig. 2. TheFig. 3 demonstrates the possibility to modify the operators to solve a viscous problem.

5 Conclusion

A numerical method for a design of an airfoil based on a solution of an inverse problem is described. It was shown bynumerical results that this approach can be used. Since the relations between the direct and inverse operator are not so strong,the method can be modified, for example, to obtain solution of a viscous problem.

Acknowledgements Results have been achieved with the support of the Ministry of Education, Youth and Sports of the Czech Republic,the project MSM 0001066902 of Aeronautical Research and Test Institute.

References

[1] M. Feistauer, J. Felcman and I. Straskraba, Mathematical and Computational Methods for Compressible Flow (Clarendon Press,Oxford, 2003).

[2] D. A. Knoll and D. E. Keyes, Jacobian-free Newton-Krylov Methods: A Survey of Approaches and Applications, Journal ofComputational Physics 193, 357–397 (2004).

[3] J. Pelant, Inverse Problem for Two-dimensional Flow around a Profile, VZLU Report Z-69 (Prague, 1998).[4] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).[5] J. Simak, Solution of 2D Navier-Stokes Equations by Implicit Finite Volume Method and Application in Inverse Problem, VZLU

Report R-4003 (Prague, 2006), (in Czech).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2100024