Solution of 2D Euler Equations and Application to Airfoil Design

Transcription

1 WDS'6 Proceedings of Contributed Papers, Part I, 47 52, 26. ISBN MATFYZPRESS Solution of 2D Euler Equations and Application to Airfoil Design J. Šimák Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. This paper deals with a numerical method for an airfoil design. It is shown how to create an airfoil from a given velocity distribution along a mean camber line. The method is based on searching a fixed point of a contractive operator. We need to have a fast solver of the Euler equations. The Newton method for solving implicit finite volume scheme is described. The resulting system of linear algebraic equations is solved by GMRES, the Jacobian-free version is described. Numerical results are presented. Introduction We present here a method how to get an airfoil from a given velocity distribution. This method can be used in the case of a subsonic inviscid compressible flow described by the Euler equations. The method was presented by [Pelant, 1998]. The main idea is to use a couple of a direct and an inverse operator. The direct operator represents an assignment of a velocity distribution to a shape of an airfoil and the second operator is its inversion. Since we are not able to construct an exact inversion we have to use some iteration method. We put the direct and the approximate inverse operator together to create a contractive operator and find its fixed point. The method can be used to modify existing airfoils. We get a velocity distribution of some airfoil, then we change it and we get a new airfoil with required quality. In the next sections these two operators are described. The theory of a numerical solution of flow dynamics can be found in [Feistauer et al., 23]. Solution of Euler equations using Newton-Krylov method Our goal is to find a steady-state solution of the Euler equations describing inviscid compressible flow. The nonstationary equations can be written in the form where w t + f(w) x + g(w) =, (1) y w = (ρ,ρu,ρv,e) T, f(w) = (ρu,ρu 2 + p,ρuv,(e + p)u) T, g(w) = (ρv,ρuv,ρv 2 + p,(e + p)v) T. The symbol ρ denotes the density, u,v denote the velocity components, E denotes the energy and p denotes the pressure. The Pressure p can be expressed by ( p = (γ 1) E 1 2 ρ( u 2 + v 2)), (2) where γ is the Poisson adiabatic constant. The problem is solved by the implicit finite volume method. We assume a standard finite volume mesh D h (in experiments a quadrilateral structured grid was used). By w h we shall denote a vector of 4-dimensional blocks w i of the values of approximate solution on finite 47

2 volumes D i D h. For w h R n the vector Φ(w h ) consists of 4-dimensional blocks Φ i (w h ) given by Φ i (w h ) = 1 H (w i,w j,n ij ) Γ ij, (3) D i j S(i) where D i denotes the cell area, Γ ij denotes the length of the edge between D i and D j, n ij denotes the outer normal to D i and H(u,v,n) denotes the numerical flux. In this case the Osher- Solomon numerical flux is used. In order to have a higher order method we apply the Van Leer κ-scheme or the Van Albada limiter. By S(i) we denote a set of indices of neighbouring elements and by symbol τ we shall denote the time step. So we get an implicit FV scheme w k+1 i = w k i τ k Φ i (w k+1 h ). (4) Since we are interested in a steady-state solution, we can deal with the nonlinear system Φ(w h ) = and apply the Newton method to this system: DΦ(w k h ) Dw ( ) w k+1 h wh k = Φ(wh k ), k =,1,..., (5) where wh is some suitable initial approximation. Since the employed numerical flux is differentiable, it is possible to compute the Jacobian matrix DΦ(w)/Dw. We will refer to these Newton iterations as outer iterations. We get a large system of linear algebraic equations Ax = b. The matrix A is sparse and consists of four-by-four blocks. Although the structure is symmetric, the matrix itself is nonsymmetric. This leads us to the use of iterative Krylov subspace methods, namely GMRES. As the GMRES solver the program SPARSKIT2 was used. The algorithm can be found in [Saad, 23]. Since we use an iterative solver, it is necessary to stop the process after a sufficient precision is reached and avoid oversolving of the system. The stopping criterion can be residuum < 1 2 Φ(w k h ). To these iterations we will refer as inner iterations. Since the Jacobian matrix is very ill-conditioned, it is necessary to use some preconditioning. ILU(p) preconditioning is used. The choice of p influences the speed of GMRES and also the amount of memory necessary to store the preconditioning matrix. The optimum value appears to be p = 2. In the GMRES algorithm the Jacobian matrix A(w h ) is required only to form a matrixvector product. This product can be approximated with the use of the Taylor series expansion, Av Φ(w h + ǫv) Φ(w h ). (6) ǫ The parameter ǫ is computed in every inner iteration as a function of a machine precision and the norm of v: ǫmach ǫ = v. (7) Thus, we can avoid the generation of the Jacobian matrix and save some memory. In the matrix version the Jacobian matrix is generated at the beginning of iterations and in each iteration there is one matrix-vector multiplication. On the other hand, in the matrix-free version one numerical flux is computed in each iteration. The computing time depends on the number of inner iterations. Since the average number of iterations is small (about 8) the matrix-free version is slightly faster. However, we need some Jacobian matrix in order to generate preconditioning matrix. But this matrix is not generated in each outer iteration, it is update only when it is needed. This Jacobian matrix can be simpler than the true Jacobian matrix. Some results of the Jacobian-free method are presented by [Blanco, Zingg, 1997] and [Knoll, Keyes, 24]. 48

3 Since the Newton method is not globally convergent, we cannot start the method from an arbitrary initial condition. From this reason we solve the time dependent problem w h t + Φ(w h ) = (8) until residuum is smaller than a given tolerance. For subsonic cases the tolerance could be 1 1, for transonic 1 3. After that we switch to the Newton iterations. This problem leads to a system of linear algebraic equations in the form (I + τa)x = τb, where A and b are the same as in the time independent problem. The matrix (I +τa) is better conditioned than the matrix A and for small time steps only few inner iterations are needed. Inverse problem The solution of the inverse problem is based on the application of a direct operator and an approximate inverse operator. The direct operator P is the solution of the Euler equations described above. The approximate inverse operator L represents an assignment of an airfoil shape to a velocity distribution on the surface of the airfoil. If we put these operators together we get an equation PL(u) = f, (9) where f denotes the required velocity distribution and u is the unknown fictitious velocity (it has no physical meaning). We transform this equation into a contractive operator equation and search for its fixed point. So we get a sequence {u k } k=,u k+1 = u k + α(f PLu k ), (1) where α (,1 is such that the sequence tends to a limit u. The first term u of the sequence is suitably chosen (for example u = f). The desired airfoil ψ is obtained by the application of L to the limit u. The velocity distribution is given on the upper and lower side of the airfoil along its mean camber line. This curve can be different from the standard mean camber line, because it connects the stagnation points on the leading and trailing edge. The position of the stagnation point on the leading edge depends on the angle of attack. If the chord line lies on the x-axis and the stagnation point on the leading edge is at the origin and the point on the trailing edge has the x-coordinate b, then the velocity distribution f = {f upper,f lower } must satisfy f upper () = f lower () = and f upper (b) = f lower (b). (11) We consider the approximate inverse operator L : u (ψ 1,ψ 2 ), where u is a fictitious velocity distribution and the (ψ 1,ψ 2 ) coordinates of the airfoil are given by the relations s (x) ψ 1 (x) = x t(x), (12) 1 + s 2 (x) ψ 2 (x) = s(x) ± t(x) 1, x,b. (13) 1 + s 2 (x) The function s(x) describes the mean camber line and the function t(x) describes the thickness of the airfoil. The derivation of these functions can be found in [Pelant, 1998]. 49

4 Since there is some restriction on the velocity distribution, it is necessary to find a suitable angle of attack to ensure that the conditions (11) are satisfied. Thus, we look for a root α of a function describing the position of the stagnation point. Each evaluation requires a steadystate solution of the Euler equations. If we remember the angle from previous inverse problem iteration, we are close enough and only a few evaluations are needed. Moreover, the change of the angle is so small that the flow remains nearly the same. Numerical experiments The described method for the solution of the inverse problem was applied to the velocity distribution along a known airfoil in order to show that our method yields this airfoil. In the first case we start from the airfoil NACA321 and get a velocity distribution satisfying the conditions for the stagnation points (the angle of attack 1.94 ). The Mach number at infinity is M =.52. In the second case we start from the airfoil Eppler E337. The Mach number at infinity is M =.5 and the angle of attack α = In both cases the parameter α in (1) is set to 1. By the symbol v we shall denote the computed velocity distribution on Mach.5 f-v Iterations Figure 1: Given velocity distribution, Figure 2: Convergence history of the L 2 -norm NACA321, M =.52, α = 1.94 f v 2 in a log-scale Y f-v Figure 3: Resulting airfoil shape (NACA321) Figure 4: Distribution of the error f v 5

5 the airfoil and by f the given velocity distribution, both along the mean camber line. Case 1 The given velocity distribution corresponding to NACA321 is shown in Figure 1. The convergence history of the error f v 2 is shown in Figure 2. After 2 iterations of the inverse problem the L 2 -norm of error f v 2 = Resulting shape shows Figure 3 and the error distribution shows Figure 4. Case 2 The given velocity distribution corresponding to E337 is shown in Figure 5. The convergence history of the error is shown in Figure 6. After 2 iterations of the inverse problem the L 2 -norm of error f v 2 = Resulting shape is shown in Figure 7 and the error distribution is shown in Figure 8..7 Mach.5 f-v Iterations Figure 5: Given velocity distribution, E337, Figure 6: Convergence history of the L 2 -norm M =.5, α = 2.48 f v 2 in a log-scale Y f-v Figure 7: Resulting airfoil shape (E337) Figure 8: Distribution of the error f v 51

6 Conclusion We have described a method, which allows us to get an airfoil shape for a given velocity distribution. It turns out that the ability of fast solving the Euler equations is essential. We need to compute at least one steady-state solution in each iteration of the inverse problem. From this reason explicit methods are unsuitable and we use an implicit method which needs the solution of a large ill-conditioned system of linear algebraic equations. GMRES with ILU preconditioning is used. Apparently, it follows from Figures 4 and 8 that the error distribution is not uniform. This is due to the behaviour of the velocity function and also the inaccuracy of the inverse operator. But the influence of the error near the leading edge is small. This method could be modified to the use of the Navier-Stokes equations or to compute an airfoil from the pressure distribution. Acknowledgments. Results have been achieved with the support of the Ministry of Education, Youth and Sports of the Czech Republic, the project MSM of Aeronautical Research and Test Institute and the project MSM of the Charles University. References Blanco M., Zingg D.W., A Fast Solver for the Euler Equations on Unstructured Grids Using a Newton- GMRES Method, AIAA Paper , Feistauer M., Felcman J., Straškraba I., Mathematical and Computational Methods for Compressible Flow, Clarendon Press, Oxford, 23. Knoll D.A., Keyes D.E., Jacobian-free Newton-Krylov Methods: A Survey of Approaches and Applications, Journal of Computational Physics 193, , 24. Pelant J., Inverse Problem for Two-dimensional Flow around a Profile, Report VZLU Z-69, Prague, Saad Y., Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM,