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 1 113 8656 7 3 1 E-mail: yaguchi@mist.i.u-tokyo.ac.jp
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
83 u 1 ρ 0 0 c /ρ u 1 0 p/ρs A 1 (ρ, u 1, u, s) =, 0 0 u 1 0 0 0 0 u 1 u 0 ρ 0 0 u 0 0 A (ρ, u 1, u, s) = c /ρ 0 u p/ρs 0 0 0 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.,
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
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 3 0 0 0 1) ) (5) Fourier Thompson Poinsot Lele
86 Navier-Stokes 4 1, A 1 (u) + A (u) 3 l j l j t = 0 4.1 1, (8) j = ( ) j ( ) j (v) j v j (8) j α α 4 ( min. ) j α ( ) α j α = 1 4 4 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)
87 l j t = 0 Thompson A(u) A = PΛP 1 Λ λ 1 0 0 ˇΛ = 0 λ 0 0 0 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 1 4 4 ( ) 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 + 1 + 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 1 0 0 0 0 λ 1 0 0 Ǎ(u) = P P 1 0 0 λ 0 0 0 0 λ 3 1 α
88 Navier-Stokes (16) 1 α = 1 4 4 ( ) j ( ) j ( ) j 0 ( ) j 0 x y x y α 1 = 1 4 4 ( ) 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 + 1 1 + α 1 (α 1A 1 + A ) ( sign(α 1 ) + ) = 0 (19) t + 1 1 + 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
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) 1 0 5 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
90 Navier-Stokes ρ ρ ( ρu 1 1 + 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 1 1 4 4 ( ) j ( ) j α 1 4 4 ( ) j ( ) j. α 1 = α = 0 3 3. α > α 1 α 1 α 1 1 4. 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
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