EQUADIFF 7. Terms of use:

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1 EQUADIFF 7 H. Berger; Gerald G. Warnecke; Wolfgang L. Wendland On the coupling of finite elements and boundary elements for transonic potential flow computations In: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conference on Differential Equations and Their Applications held in Prague, BSB B.G. Teubner Verlagsgesellschaft, Leipzig, Teubner-Texte zur Mathematik, Bd pp Persistent URL: Terms of use: BSB B.G. Teubner Verlagsgesellschaft, 1990 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 ON THE COUPLING OF FINITE ELEMENTS AND BOUNDARY ELEMENTS FOR TRANSONIC POTENTIAL FLOW COMPUTATIONS BERGER H.,WARNECKE G.,WENDLAND W., STUTTGART, FRG For the improved treatment of the far field boundary conditions in nonlinear compressible transonic flow computations around airfoil profiles we suggest the following method. Assuming that the far field is subsonic, the full potential equation may be linearized to the Prandtl-Glauert equation in the exterior of a bounded region Q containing the supersonic flow regions. The solution of the Unear equation in the exterior can easiuy be reduced to a boundary integral equation on the boundary Too- (see Figure 1). This leads to a coupled finite element/boundary element method for the flow problem. The numerical results show a considerable improvement in comparison with the commonly used method of taking the normal projection of the far field flow on Too [3], [5], [7].With our method we obtain results which show that the computational FEM domain can be chosen much smaller when the coupling is used. In the following, we describe briefly the method, a more detailed description including error analysis will be given in [4], A simple model for transonic flows with weak shocks is the full potential equation div (/>( Vu J ) Vu) = 0 (1) The density function is obtained under the assumption of isentropic flow from Bernoulli's law as p(s) = po 11 JTT~ 3 I where /c denotes the adiabatic exponent and ao the speed of sound in \ 2a o / the motionless gas. In order to have a flow potential in Q we have to introduce a sut E across which we assume Vw to be continuous, whereas the potential has a finite constant jump /?, which gives the circulation of the flow and which is determined by the Kutta- Joukowski condition. This implies a smooth flow with a stagnation point at TE (see [6]). For convenience we take the sut from the trailing edge point TE. On the profile, we take the non Figure 1 This research was supported by a grant from the Stiftung Volkswagenwerk, Germany 228

3 penetration condition d nu := Vu n = 0 on Vp where n denotes the normal vector. The simplest approximation of the exterior boundary condition is P(\Vu\ 2 )d nu - P(\voo\ 2 ) voo - a on r TO. (2) This condition was used in [3], [5] and [7]. Boundary condition (2) will be compared to the following coupling procedure. Prandtl-Glauert equation we introduce a perturbed potential For the linear (p := u Voo'X arctan (y]1 - M^\ x 2 / xy J in U c. (3) This leads to the following transmission problem. Find the functions u,(p and the constant /? satisfying Interior full potential problem, the div(p(\vu\ 2 )Vu) = 0 in ӣ, д n u = 0 on Vp, u + ~u~ = ß one, ð u+-ð u- = 0 F(ß) = Vu+ß в - Vu-ß E = 0, (4) Exterior Prandtl-Glauert problem, (1 - Ml) <p XlXl -r <p X2X2-0 in l c, Vy? = o(l) for \x\ > co ip+ _ (p- o on_ c, d n <p + -d n <p- = o (5) Coupling conditions on r^, u = ip + Voo-x + - arctan (y/1 - M^ x 2 / x x J, p(\vu\ 2 )0 nu = p(\voo\ 2 ) {((l-m^v?*.,^) >n + voo-n (6) + L v arc*an (^1 - M.r 2 /.n ) n}. The exterior Prandtl-Glauert solution u(x) is given by (3) where the perturbation potential <p is represented via Green's theorem as ^(x) = / ^ i Ч>Ш(x,y)ds y, J G(x ìу ) X(y)dsy y/l - Ml ^oo PooyJ\ - Ml ^oo (7) for x 6 Q c in terms of the Cauchy data ip^^ and Ky) = Poo{(l-M 2 00)(p yi n 1 (y) + ^y2 n 2 (y)} for y e T^. (8) Here G(x,y) and K(x,y) are given by ht \! / -/ \ *v \ (^i-yi) n i(y) + ( x 2-y2)n 2 (y) G ( x, y ) : = r(a; ~2~ ^ ' y) A ( ' X ' y ) 2*f(*-y) ' ( 9 ) where 229

4 For the coupled FEM-BEM procedure we introduce a family of quasi-regular triangulations of (2\S with maximum meshwidth h and associated piecewise linear continuous finite elements Vh C W 1,2 (Q\E) satisfying additionally v vj~ = /? on E. On r TO we consider a family of continuous piecewise linear periodic splines j/^ subject to particions of maximum meshwidth h of the arc length. The coupled boundary value problem (4), (5), (6) is then approximated by: Find Uh G Vh, ^h V?h> ^ ^ where Uh satisfies the speed condition V«/. 2 < SQ < -^- and a modified entropy condition - / Vu h V7Thi/>dx < K / TThHx + o(l)max \i/;(x)\ Jtt In r ft For h i 0 anfl? /or a// ^ CJ (ft) itf-i/i ^ > 0.swell t^at Me variational equations a(u h \uh,vh) - (A^,v^>= (/>oo(voo + -Varcfan ft/l - M 2, x 2 / xij) n,v/.y, (GAj, ^> = - /[I - K] (u h - t/oo x - ^arc/an (^/l - M^x 2 / xm, tf-\ ana* Me Kutta-Joukowski condition \Vut\ TE = IVt.jg.jj (11) ore satisfied for all «/, 6 V* and «// ^ 6,i > i(r oo ). Here we have used the notations (v,tj>) := f r<xy v(x)tl>(x) ds x, g(u u,i;) := / p( Vu ')Vu Vudx, GX(x) := 2J r X(y)G(x,y)ds y / (p^l-mj,), K V (x) := 2f v(y) K(x,y) ds, / Jl - Ml. «Toc 7T/j denotes the interpolation operator associated with Vh. The above coupled problem is an extension of the linear coupling method of Johnson/Nedelec [8] to nonlinear equations. For the solution of the nonlinear equations we use an improvement by Berger [2] of the conjugate gradient method by Glowinski/Pironneau [7]. Some error analysis for this coupled FEM-BEM method will be presented in [4]. For two standard test cases offlows around the NACA-0012 profile, we made some numerical computations. We compare the use of the boundary condition (2) in the coupling method described above. Two different sized C-grids were used, a large grid with 115 by 15 nodes and Too with 6 chord lengths distance from the profile and a smaller grid with 111 by 13 nodes and Too 3 chord lengths distance. The lift coefficients c a were calculated from the pressure distribution along the profile for a purely subsonic flow with Moo = 0.63, a = 2. We obtained for the large and for the small grid versus due to Kroll/Jain [9] with a potential and an Euler code. (Without coupling for the large and for the small grid.) For a transonic flow with Moo = 0.8,«= 1.25 we obtained for c a on the large and on the small grid via lift coefficients between 0.5 and 1.1 due to Rizzi/Viviant [10] and between 0.35 and 0.37 due to AGARD [1] obtained with Euler equations. 230

5 Figuгe2 Mach numbsr dlstributionв (laгgo grid) Mach number distributions (small grid) References [1] AGARD Advisoгy Repoгt No. 211, Test Cases for Inviscid Flow Field Methods, Repoгt of Fluid Dynamics Panel Woгking Group 07, [2] Beгgeг, H., "A conveгgent finite element foгmulation foг tгansonic flow", Numerische Mathematik, to appear. [3] Bergeг, H,, Warnecke, G. and Wendland, W., " On the convergence of a coupled finite and boundaгy element method foг transsonic flows". [4] Beгgeг, H., Waгnecke, G. and Wendland, W., "Finite Elements foг Tгansonic Flows", Numerìcal Methods foг Partial Differential Equaťюns, to appeaг. [5] Bгisteau, M.O., Glowinski, R., Peгiaux, J., Perrieг, P., Piгonneau, 0. and Poiгieг, G., "Applicatioa of Optimal Control and Finite Element Methods to the Calculation of Tгansonic Flows and Incompressible Flows," in Numerical Methods in Applied Fluid Dynamics, B. Hunt, Ed., Academic Pгess, New Yoгk, 1980, [6] Ciavaldini, J.F., Pogu, M. and Touгnemine, G., "Existence and Regularity of Stгeam Functions foг Subsonic Flows Past Pгofiles with a Shaгp Trailing Edge, n Aгch. Raťюnal Mech. Aпai., 93, 1-14 (1984). [7] Glowinski, R. and Pironneau, 0., "On the Computation of Tгansonic Flows," in Functional Analysis and Numerical Analysis, H. Fujita, Ed., Japan Society foг the Pгomotion of Science, [8] Johnson, C and Nedelec, J.C, " On the Coupling of Boundaгy Integгal and Finite Element Methods", Math. Comp., 35, 1980, [9] Kгoll, N. and Jain, R.K., Soluťюn of Two-Dimensional Euler Equations-Experience with a Finite Volume Code, DFVLR-Foгschungsbericht 87-41, DFVLR Institut fueг Entwuгfsaerodynamik, Bгaůnschweig, [10] Rizzi, A. and Viviand, H. Eds., Numerical Methods foг the Computaťюn of Inviscid Transonic Flows with Shock Waves - JVotes on Numerical Fluid Mechanics III, Vieweg, Bгaunschweig,