空間多次元 Navier-Stokes 方程式に対する無反射境界条件

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1 81 Navier-Stokes Poinsot Lele Poinsot Lele Thompson Euler Navier-Stokes A Characteristic Nonreflecting Boundary Condition for the Multidimensional Navier-Stokes Equations Takaharu YAGUCHI, Kokichi SUGIHARA Graduate School of Information Science and Technology, University of Tokyo (Received 4 June, 004; in revised form 4 January, 005) Because the computational resources are finite, one needs to truncate the computational domain when he/she simulates a physical problem. This truncation gives rise to non-physical artificial boundaries and one cannot obtain proper solutions unless appropriate boundary conditions on such boundaries are imposed. Practically nonreflecting boundary conditions, which are boundary conditions that prevent the generation of reflections, are of great importance. By the reason of the practical robustness and the simplicity of implementation, the Poinsot Lele boundary condition is one of the most popular methods for the Navier-Stokes equations right now. Their method is based on Thompson s boundary condition for the Euler equations, which, however, is essentially one-dimensional. Therefore the Poinsot Lele boundary condition is valid only when the flow is perpendicular to the boundary theoretically. Here we propose a nonreflecting boundary condition for the Euler equations which does not have the assumption on the direction of flow. We also discuss its extension to the Navier-Stokes equations. Our basic idea is to estimate the direction of the flow from numerical data. KEY WORDS : Nonrefrecting Boundary Conditions, Absorbing Boundary Conditions, NSCBC, DNS, Computational Aeroacoustics, Poinsot Lele Boundary Conditions yaguchi@mist.i.u-tokyo.ac.jp

2 8 Navier-Stokes 3, 7, 17) DNS Poinsot Lele 1) Poinsot Lele ( Euler ) Thompson 15) Dutt 5) Thompson 1 Hedstrom 8) 1 3, 13, 17) 6) 11) 13) 14) PML (Perfectly Matched Layer) 1) Hedstrom Thompson 17) 4) 1 Euler Navier-Stokes Euler Euler ρ ρ u 1 + A 1 (ρ, u 1, u, s) u 1 t u u s s ρ u 1 +A (ρ, u 1, u, s) = 0 (1) u s

3 83 u 1 ρ 0 0 c /ρ u 1 0 p/ρs A 1 (ρ, u 1, u, s) =, 0 0 u u 1 u 0 ρ 0 0 u 0 0 A (ρ, u 1, u, s) = c /ρ 0 u p/ρs u ρ p u 1 u x y s γ s = pρ γ c c = γp/ρ Euler α 1 α α 1 A 1 + α A Navier-Stokes Poinsot Lele Thompson Hedstrom 1 Hedstrom 1(Hedstrom) t + A(u) = 0 x > 0 u A(u) l j t = 0 for j s.t. λ j > 0 x = 0 l j A(u) λ j u λ j Thompson (Thompson) Euler (1) x = 0 t + + A (u) = 0 u ( ) j = min{λ j, 0}l j r j r j, l j A 1 (u) λ j Hedstrom x Thompson Poinsot Lele A 1 (u) = 0x 3 Euler Hedstrom Thompson Hedstrom Jeffrey and Taniuti 9) John 18) 1.,

4 84 Navier-Stokes. du = 0 3 Euler (1) u(x, y, t) = φ(θ) () θ x, y, t φ θ = dφ dθ, = dφ dθ, t = (3) dφ t dθ () (1) (3) ( t I + A 1(u) + ) dφ A (u) dθ = 0 ( det t I + A 1(u) + ) A (u) = 0 ( t = A 1(u) + ) A (u) t + u 1 + u = 0 (4) t + u 1 + u ( ) ( ) ±c + = 0 (5) (4) (4) = u 1, dy = u (6) (5) = u 1 ± c sign ( ) ( ), 1 + dy = u sign ( ) ± c ) 1 + ( (7) (7) sign ( ) sign ( ) (3) j = ( ) j ( ) j (8) (v) j v j (8), θ (7) sign ( ) sign ( ) ± > 0 (7) A 1(u) + A (u) (9) 0 3 l j t = 0 j S

5 85 S S = { j v j } v j (9) l j l j (9) r M λ k 0 r km kλ (9) A 1 + A (10) 4( ) j S l j t = 0 (9) v j S S = { j v j } l j (10) Euler ) ) (5) Fourier Thompson Poinsot Lele

6 86 Navier-Stokes 4 1, A 1 (u) + A (u) 3 l j l j t = , (8) j = ( ) j ( ) j (v) j v j (8) j α α 4 ( min. ) j α ( ) α j α = α = 1 4 ( ) j ( ) j 4 ( ) j ( ) j α A 1(u) + A (u) (6) (7) α dy = u 1, = u 1 ± = u (11) 1 c, 1 + α dy = u ± sign (α ) c 1 + α (1) 3 l j l j t = 0 Thompson 16) t + A(u) = 0 (13)

7 87 l j t = 0 Thompson A(u) A = PΛP 1 Λ λ ˇΛ = 0 λ λ 3 t + Ǎ(u) = 0, Ǎ(u) = P ˇΛP 1 (14) t = A(u) (15) l j t = 0 l j A(u) = 0 λ j = 0 Ǎ(u) (14) l j t = 0 (15) t (15) u(x, y, t) = φ(θ) Euler t + A 1(u) + A (u) dφ dθ = 0 l j A 1(u) + A (u) A 1 (u) + A (u) (3) α t + (A 1(u) + α A (u)) = 0 (16) Thompson 1 1. α α ( ) j ( ) j. λ 1 u 1 + α u, λ u 1 + α u c, λ 3 u 1 + α u + c 3. A 1 (u) + α A (u) λ j 1 1 λ j 0 () λ 1 = u 1 + α u = u 1, dy = u 1 λ = u 1 + α u = u 1 + c, 1+α α c dy = u + sign(α ) c 1+α 1 λ 3 = u 1 + α u = u 1 c, 1+α 1 + α c dy = u sign(α ) c 1+α 4. u 4. t + Ǎ(u) = 0, λ λ Ǎ(u) = P P λ λ 3 1 α

8 88 Navier-Stokes (16) 1 α = ( ) j ( ) j ( ) j 0 ( ) j 0 x y x y α 1 = ( ) j ( ) j (17) α 1,α α 1, α sign( ) sign( ) (16) 1 x y x y (16) t + (α 1A 1 + A ) = 0 (18) α 1 1 (17) α 1 α α 1 α α 1 α = 1 (16) (18) t α 1 (α 1A 1 + A ) ( sign(α 1 ) + ) = 0 (19) t α (A 1 + α A ) ( + sign(α ) ) = 0 (0) (19) (0) 1 α 1 α () 1. α 1,α. α > α 1 α 1 α 1 1

9 89 3. u t + 1 α 1 + α (α 1A 1 + α A ) ( sign(α 1 ) + sign(α ) ) = 0 (1) α 1 A 1 (u)+α A (u) Navier-Stokes Navier-Stokes Poinsot Lele Thompson Dutt Dutt Navier-Stokes Dutt Navier-Stokes L x = ( ) 5, 1) τ 1 = 0, T = 0. T τ 1 Poinsot 1) T = 0 Dutt 5) Dutt Dutt ( ) k(γ 1) T T T Ω R T T dσ 6 Naviser-Stokes 6 ( 4 ) 10) 4 Runge-Kutta Lele 10) Poinsot Lele 1 1 Poinsot Lele Poinsot Lele

10 90 Navier-Stokes ρ ρ ( ρu t ρu α 1 + α Ǎ sign(α 1 ) + sign(α ) ) u 1 = 0, () u ρ(e + u 1 +u ) p d 3 α 1 l 1 α l 1 m u 1 d 3 α 1 (u 1 l 1 + α 1 m 1 ) + m 4 α (u 1 l 1 + α 1 m 1 ) u 1 m + α 1 m 3 Ǎ =. u d 3 α 1 (u l 1 + α m 1 ) α (u l 1 + α m 1 ) + m 4 u m + α m 3 d 3 (ẽ c γ 1 ) α 1l + u 1 ρd 3 α l + u ρd 3 ẽm + r 0 m 3 + d 3 γ 1 d 1 = r+r1, d = r r1, d 3 = r 3, m 1 = ρ(d 1 d 3 ) κ, m = d 1 d 3 c, m 3 = d cκ, m 4 = ρd 3, l 1 = ρd cκ, l = r 0 m 1 + ρd ẽ cκ, ẽ = u 1 +u + c γ 1 7 Euler Navier-Stokes Poinsot Lele : 1 COE (S) 1. α ( ) j ( ) j α ( ) j ( ) j. α 1 = α = α > α 1 α 1 α κ α 1 + α r 0 α 1 u 1 + α u r 1 r 0 κc r r 0 + κc r 3 r 0 5. j = 1,, 3

11 91 r j 0 ( ) r 1 r r 3 = u 1 sign(α 1) c, α 1 +α dy = u sign(α ) α 1 +α = u 1 + sign(α 1) c α 1 +α c, dy = u + sign(α ) α 1 +α = u 1, dy = u 6. u () r 0 r 3 0 1) J. P. Berenger : A Perfectly Matched Layer for the Absorption of Electromagnetic Waves, Journal of Computational Physics, ) C. H. Bruneau and E. Creuse: Towards a Transparent Boundary Condition for Compressible Navier- Stokes Equations, International Journal for Numerical Methods in Fluids, ) T. Colonius: Modeling Artificial Boundary Conditions for Compressible Flow, Annual Review of Fluid Mechanics, ) R. Courant and D. Hilbert: Methods of Mathematical Physics, vol. John Wiley and Sons, , ) P. Dutt: Stable Boundary Conditions and Difference Schemas for Navier-Stokes Equations, SIAM Journal on Numerical Analysis, ) B. Engquist and A. Majda: Absorbing Boundary Conditions for the Numerical Simulation of c Waves, Mathematics of Computation, ) D. Givoli: Non-Reflecting Boundary Conditions, Journal of Computational Physics, ) G. W. Hedstrom: Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems, Journal of Computational Physics, ) A. Jeffrey and T. Taniuti: Non-Linear Wave Propagation Academic Press ) S. Lele: Compact Finite Difference Schemes with Spectral-Like Resolution, Journal of Computational Physics, ) P. Luchini and R. Tognaccini: Direction-Adaptive Nonreflecting Boundary Conditions, Journal of Computational Physics, ) T. J. Poinsot and S. K. Lele: Boundary Conditions for Direct Simulations of Compressible Viscous Flows Journal of Computational Physics, ) K. Mazaheri and P. Roe: Numerical Wave Propagation and Steady-State Solutions: Soft Wall and Outer Boundary Conditions, AIAA Journal, ) S. Ta asan and D. M. Nark: An Absorbing Buffer Zone Technique for Acoustic Wave Propagation, AIAA Paper, ) K. W. Thompson: Time Dependent Boundary Conditions for Hyperbolic Systems, Journal of Computational Physics, ) K. W. Thompson: Time Dependent Boundary Conditions for Hyperbolic Systems II, Journal of Computational Physics, ) S. V. Tsynkov: Numerical Solution of Problems on Unbounded Domains. A Review, Applied Numerical Mathematics, ) F., :,

JFE.dvi

JFE.dvi ,, Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112 8551, JAPAN E-mail : atsu1005@kc.chuo-u.ac.jp E-mail : kawa@civil.chuo-u.ac.jp SATO KOGYO CO., LTD. 12-20, Nihonbashi-Honcho