Title 混合体モデルに基づく圧縮性流体と移動する固体の熱連成計算手法 Author(s) 鳥生, 大祐 ; 牛島, 省 Citation 土木学会論文集 A2( 応用力学 ) = Journal of Japan Civil Engineers, Ser. A2 (2017), 73 Issue Date 2017 URL http://hdl.handle.net/2433/229150 Right 2017 公益社団法人土木学会 Type Journal Article Textversion publisher Kyoto University
A2, Vol. 73, No. 2 Vol. 20, I_143-I_152, 2017. 1 2 1 ( ) 606-8501 E-mail: toriu.daisuke.8v@kyoto-u.ac.jp 2 606-8501 E-mail: ushijima.satoru.3c@kyoto-u.ac.jp Key Words : mixture model, compressible fluid, moving solid, thermal fluid-solid interaction 1. β T β T 1 Yamamoto 1, 2) Qi 3) 4) 5) MICS 6) TCUP 7) 4) CFL (Courant-Friedrichs-Lewy) 8) 4) CFL 4) 4) 9) I_143
10) 4) Lee & Ha 11) CFL Ra Ra 10 6 β T β T 0.3 2. (1) 1 ( 1) 4) 1 1 2 1 2 1 4) 1 1 T f1 T s T f1 T s 2 T f2 = T s 1 2 T f1 T f2 T T f1 T s 10) 7) 1 1 2 ( 1 2 1 ) 4) 4) (2) 1 ( ) ( ) 1 2 ( 1 ) 1 2 5) I_144
(ρu i ) (ρe) ρ + (ρu i) x i = 0 (1) + (ρu iu j ) + (ρeu j) = σ ij + ρf i (2) = σ ij u i (3) t x i f i x i ρ u i σ ij e (2) (3) 5) 1 (3) 1 2 (1) (2) (3) ρ σ ij u i e k w k,i 5) w k,i 0 w k,i k u k,i u i w k,i u k,i u i 5) ρ u i ρ = ϕ f1 ρ f1 + ϕ f2 ρ f2 (4) u i = ϕ f1ρ f1 u f1,i + ϕ f2 ρ f2 u f2,i ϕ f1 ρ f1 + ϕ f2 ρ f2 = 1 ρ (ϕ f1ρ f1 u f1,i + ϕ f2 ρ f2 u f2,i ) (5) f1 1 f2 2 ϕ f1 ϕ f2 1 2 0 ϕ f1, ϕ f2 1 ϕ f1 + ϕ f2 = 1 12) 12) 1 2 σ ij 4) σ ij = ϕ f1 σ f1,ij + ϕ f2 σ f2,ij pδ ij + τ ij (6) ( ui τ ij µ f + u ) j x i 2 3 µ u m f δ ij (7) x m δ ij p τ ij µ f 1 2 1 2 e T f1 T f2 (= T ) e = Σ kϕ k ρ k C V,k T k Σ k ϕ k ρ k T k = C V,f Σ k ϕ k ρ k Σ k ϕ k ρ k C V,f T (8) k k k = f1, f2 1 2 C V,f1 = C V,f2 = C V,f 1 2 e T p T f1 T f2 (= T ) p = ϕ f1 p f1 + ϕ f2 p f2 = ϕ f1 ρ f1 (γ f1 1)C V,f1 T f1 +ϕ f2 ρ f2 (γ f2 1)C V,f2 T f2 (ϕ f1 ρ f1 + ϕ f2 ρ f2 )(γ f 1)C V,f T = ρ(γ f 1)C V,f T (9) 1 2 γ f1 = γ f2 = γ f 1 2 2. (1) 1 1 (ρ f1 C V,f1 T f1 ) (ρ s C V,s T s ) = q f1,j (10) = q s,j (11) q f1,j q s,j 1 I_145
5) 1 (ρc V ) m T = q m,j (12) (ρc V ) m = ϕ f1 ρ f1 C V,f1 +ϕ s ρ s C V,s q m,j T f1 T f2 (= T ) q m,j T q m,j = ϕ f1 q f1,j + ϕ s q s,j = ϕ f1 λ f1 T f1 ϕ s λ s T s (ϕ f1 λ f1 + ϕ s λ s ) T = λ m T (13) λ m 1 λ m = ϕ f1 λ f1 + ϕ s λ s (3) 2. (1) 4) 4) 7) Q n Q Q Q n+1 a) 1 2 ρ + (ρu j) = 0 (14) Adv (ρu i ) + (ρu iu j ) = 0 (15) Adv (ρe) + (ρeu j) = 0 (16) Adv Euler MUSCL TVD 13) u i (15) (ρu i ) u i = 1 ρ m [ϕ f1 (ρu i ) + ϕ s (ρu i ) s ] (17) ρ m = ϕ f1 ρ + ϕ s ρ s 1 (9) p ( 2) ϕ s = ϕ f2 = 1 p = p n b) 1 2 (1) (2) (3) T 1 (12) T 2 1 2 (1) (2) (3) ρ = 0 (18) Diff (ρu i ) = τ ij (19) Diff (ρe) u i = τ ij (20) Diff1 2 I_146
Euler u i T T T t 1 = ρ C V,f u i u i ρ t { (τ ij u i ) = τ ij (21) ρ (u 2 i u 2 2 t i ) } (22) 1 (12) T T T T 1 T = (λ ) t (ρc V ) m m (ρc V ) m (23) (ρc V ) m = ϕ f1 ρ f1c V,f1 + ϕ s ρ s C V,s (24) ρ f2 (24) ϕ f1 ρ f1 ϕ f1 ρ f1 = ρ ϕ f2 ρ f2 (25) (23) T T T = (1 ϕ sc )Θ + ϕ sc T sc (26) t Θ (23) ϕ sc T sc 1 2 7) k p k = γ k 1 ρk C P,k ρ k C (Tk Tk ) (27) P,kµ J,k + 1 p γ k k = f1 f2 C P,k k µ J,k k Joule-Thomson p = ϕ f1 p f1 + ϕ f2 p f2 1 2 µ J,k = 0 p p γ f 1 γ f (ϕ f1 ρ f1 + ϕ f2 ρ f2)c P,f (T T ) c) = γ f 1 γ f ρ C P,f (T T ) (28) 1 2 (1) (2) (3) (ρu i ) (ρe) ρ = 0 (29) Acous = p + ρf i (30) Acous x i = p u i (31) Acous x i 4) TCUP 7) 1 p n+1 p ρ a 2 = t ( 1 p n+1 ) x i ρ t + u i x i (32) a a = (γ f p )/ρ (32) CFL 4) 3. Lee & Ha 11) 11) 11) 11) 3 T h T c I_147
(a) t = 12 [s] (b) t = 60 [s] 3 (c) t = 72 [s] (d) 5 (Ra = 10 4 ) 4 11) T = T h T c = 5 [K] (β T = 0.017) 3 g x 2 ρ s /ρ f0 = 10 C s /C P,f = 10 λ s /λ f = 50 ρ f0 C P,f (T h +T c )/2 P r 0.70 L Ra 10 3 10 4 10 5 10 6 Ra Ra = gβ T L3 ν f0 α f0 (33) ν f0 α f0 Ra = 10 3 10 4 100 100 Ra = 10 5 10 6 152 152 Lee & Ha 11) (a) t = 10 [s] (b) t = 20 [s] (c) t = 24 [s] (d) 6 (Ra = 10 6 ) 4 5 6 Ra 10 4 10 6 T/10 5(a) 6(a) I_148
1 Nu h U 1 U 2 Ra = 10 4 Nu h U 1 U 2 100 100 1.53 0.103 0.110 152 152 1.52 0.103 0.109 (a) Ra = 10 3 (b) Ra = 10 4 Ra = 10 6 Nu h U 1 U 2 152 152 6.32 0.347 0.390 200 200 6.32 0.347 0.389 2 11) Nu h (a) Ra = 10 5 (b) Ra = 10 6 7 ( T/10, T = 5 [K]) T 7 T/10 7 2. (1) 2. (2) T f1 T s u k,i u i (w k,i = 0) Nu h x 1 = L/2 U 1 x 2 = L/2 U 2 Nu h 1 Nu h = 0 θ X 2 1 (34) X2=0dX θ X i θ = (T T c )/ T X i = x i /L gβl T 1 Ra = 10 4 100 100 152 152 Ra = 10 6 Nu h Ra 10 3 10 4 10 5 10 6 Present 1.29 1.53 4.04 6.32 Lee & Ha 11) 1.28 1.55 3.96 6.31 152 152 200 200 1 Nu h U 1 U 2 CFL C a 14) C a C a,max { u1 + a C a = max t, u } 2 + a t x 1 x 2 (35) t x i C a,max Ra = 10 3 10 4 10 5 10 6 3.50 10 2 3.90 10 2 4.58 10 2 4.25 10 2 CFL t Nu h Lee & Ha 11) 2 2 T = 100 [K] (β T = 0.286) Ra = 10 6 8 T/10 8 T = 100 [K] I_149
8 ( T = 100 [K]) ( T/10) 3 T ρ/ ρ 0 T [K] Max Min 5 1.008 0.992 100 1.157 0.870 9 T = 5 [K] T = 5 [K] Nu h 6.31 T = 5 [K] C a,max 2.86 10 2 ρ ρ 0 ρ/ ρ 0 3 ρ/ ρ 0 3 T = 5 [K] 1% T = 100 [K] 15.7% T 4. 9 9 ( r h ) T h T c T h = 400 [K] T c = 300 [K] L 5.0 10 2 [m] r t L/3 r h 3L/40 g 9.8 [m/s 2 ] ω ω ω = t t b π (0 t t b ) (36) ω = π (t b < t) (37) t b = 3.0 [s] t b 0 12) (Fe) ( ) 4 4 6.73 10 3 3.21 10 3 I_150
4 ( ) ( ) Fluid (air) Solid (Fe) ρ [kg/m 3 ] 1.17 7.87 10 3 λ [W/(m K)] 2.50 10 2 80.30 C V [J/(kg K)] 4.20 10 3 4.42 10 3 150 150 t 1.50 10 4 [s] t = 12.00 [s] Ra Ra = 2.51 10 5 V ( u 2 1 + u2 2 ) Re t = 12.00 [s] Re 3.22 10 2 10 10 (T h T c )/10 10 10 (b) (c) 3.21 10 3 t = 12.00 [s] ( 10 (c)) t = 12.00 [s] C a 1.81 10 2 3,000 7,000 x2 x1 x2 x1 x2 x1 (a) t = 0.12 [s] (b) t = 3.00 [s] (c) t = 12.00 [s] 10 5. 4) 11) I_151
β T 0.3 16% 7,000 3,000 10 6 JSPS JP16K17552 1) Yamamoto, S., Niiyama, D. and Shin, B. R.: A numerical method for natural convection and heat conduction around and in a horizontal circular pipe, Int. J. Heat Mass Transfer, Vol. 47, pp. 5781 5792, 2004. 2) Yamamoto, S.: Preconditioning method for condensate fluid and solid coupling problems in general curvilinear coordinates, J. Compt. Phys., Vol. 207, pp. 240 260, 2005. 3) Qi, S., Furusawa, T. and Yamamoto, S.: A numerical method applied to forced and natural convection flows over arbitrary geometry, Int. J. Heat Mass Transfer, Vol. 85, pp. 375 389, 2015. 4), CFL, A2( ), Vol. 71, No.2, pp. I 213 I 222, 2015. 5), pp. 72 74,, 1991. 6),, 3, B, Vol. 64, No.2, pp. 128 138, 2008. 7), ( CCUP -TCUP - ), (B ), Vol. 69, No.678, pp. 266 273, 2003. 8) P. J. Roache(, ), pp. 28 32,, 1978. 9),,,,, Vol. 19, No. 6, pp. 1074 1080, 1993. 10),, A2( ), Vol. 72, No.2, pp. I 179 I 186, 2016. 11) Lee, J. R. and Ha, M. Y.: Numerical simulation of natural convection in a horizontal enclosure with a heat-generating conducting body, Int. J. Heat Mass Transfer, Vol. 49, pp. 2648 2702, 2006. 12),, 3,, Vol. 51, pp. 787 792, 2007. 13) Yamamoto, S. and Daiguji, H.: Higher-order-accurate upwind schemes for solving the compressible Euler and Navier-Stokes equations, Computers & Fluids, Vol. 22, pp. 259 270, 1993. 14),, (B ), Vol. 69, No. 682, pp. 1386 1393, 2003. (2017.6.23 ) COMPUTATIONAL METHOD FOR THERMAL INTERACTIONS BETWEEN COMPRESSIBLE FLUID AND MOVING SOLID BASED ON MIXTURE MODEL Daisuke TORIU and Satoru USHIJIMA In this paper, a computational method was proposed for thermal interactions between compressible fluids and moving solids with different physical properties based on the mixture model. In the present method, physical properties of solids are considered in computational stages for heat conduction and velocities of multiphase fields. The present method was applied to natural convection in the cavity containing a square solid and obtained averaged Nusselt numbers on the heated wall were in good agreement with reference results. In addition, applicability of the present method was confirmed for compressible high buoyancy flows and heat transfer around a rotating triangular solid. I_152