Simulation of Thermal Convection of Supercritical Fluid

Transcription

1 Simulation of Thermal Convection of Supercritical Fluid Satoko Komurasaki, Nihon University, Kanda-Surugadai, Chiyoda-ku, Tokyo Eruption of geothermally heated water from the hydrothermal vent in deep oceans of depth over 2,000 m is numerically simulated. The hydrostatic pressure of water is assumed to be over 200 atmospheres, and temperature of heated water occasionally more than 400 C. Under these conditions, a part of heated water can be in the supercritical state, where the physical properties can change significantly by the temperature. To investigate the mechanism of a supercritical hydrothermal-convection including transition to the subcritical state, the compressible Navier-Stokes equations are solved using a method for the incompressible equations under the assumption that the pressure is almost constant at the hydrostatic pressure and the density is a function of only the temperature. The equations are approximated by the multidirectional finite difference method. For the highly-unsteady-flow computation, KK scheme is used to stabilize the high-accuracy computation. To treat high temperature gradients in the computation, the energy equation is solved which is derived by transformation of thermodynamic variable φ into ϕ using logarithmic function. Solving the equation about ϕ instead of φ allows the sharp boundaries of φ to be properly preserved in the computation. As the result of the computation, highly complicated and unsteady flow is obtained m MPa 374 NS KK log NS 1 Dρ ρ Dt = u j (1) x j Du i Dt = 1 p 1 ( μ u ) i g i (2) ρ x i ρ x j x j (g 1,g 2,g 3 )=(0, 0,g) DH κ H. (3) ρ x j C p x j H :, κ:, C p : T = T (H) :, ρ = ρ(t ): p b Fig. 1 23MPa(> ) Fig. 1 23MPa T cp ( 651K) δp p b (p = p b δp) p b / x 3 = ρ 0 g (ρ 0 : ) (2) Du i Dt = 1 δp 1 ( μ u ) i ρ 0 ρ g i (4) ρ x i ρ x j x j ρ ρ D ( ) 1 = ũ j (5) D t ρ x j 1 Copyright c 2015 by JSFM

2 Dũ i D t = 1 ρ δp 1 ρ ( ) ( ) 1 ũ i 1 x i x j Re x j ρ 1 g i (6) D H ( D t = 1 ρ 1 H ) (7) x j Re Pr x j u i =Uũ i,h = H 0 H,δp=(U 2 ρ 0 ) δp, 1/ρ=(1/ρ 0 ) (1/ ρ), x i = L x i,t= L U t, g i = U 2 L g i, Re = ULρ 0 μ, Pr = μ κ/c p δp δ p μ, κ C p Re Pr projection (8) div 1 ρ gradp = F 1 ( )) (div u ρ D 1 (8) Δt Dt ρ [ u i F =div u j 1 ( ) ( ) ] 1 u i 1 x j ρ x j Re x j ρ 1 g i (8) ( )] 1/Δt (5) [div u ρ D 1 Dt ρ NS (6) 3 KK 2 - Pseudocritical temperature Tcp Fig. 1 : Enthalpy-temperature diagram for water at P =23 MPa KK - (7) H Eq.(9) h = h(h) h Yabe (3) tan Eq.(9) d2 H h = 1 (log(α H) log(α H)) (9) 2 (α 1, α <H<α) d 2 / H dh dh d 2 H = 2H α dh dh = 2tanhh 2sgnh (10) 1 (h>0) sgnh = 0 (h =0) 1 (h<0) (7) h Dh 1 h ρ x j Re Pr x j ( d 2 / ) [ ( ) 2 ( ) 2 ( ) ] 2 H dh 1 h h h dh ρ Re Pr x 1 x 2 x 3 (10) (11) Dh 1 h ρ x j [ Re Pr x j ( 2sgnh ) 2 ( ) 2 ( ) ] 2 h h h (11) ρ Re Pr x 1 x 2 x 3 α = 1 H H Fig.2 1 KK (11) KK Fig.2(a) =0 Fig.2(b) Fig.2(c) Fig.2 (11) (11) KK H ±α ±α 0 2 Copyright c 2015 by JSFM

3 2 Q (a) : Initial rectangular wave. (b) : Computation of an advection equation. Fig. 3 : Experiment of thermal convection using supercritical carbon dioxide (2). (c) : Computation of an advection diffusion equation. Fig. 2 : Computation of an advection diffusion equation using the translation Eq.(9). 2.3 (2) Fig. 3 Tamba (2) 0.1mm T w T b p T c (=304.21K) p c (=7.38MPa) Tw = T w/t c,tb = T b/t c,p ast = p/p c Fig. 3 p =1.016 Tb = T w=2.307 Fig. 4 1mm Fig. 4 : Computation using the present scheme for thermal convection in supercritical carbon dioxide. 2.4 Fig. 5 5cm 3 Copyright c 2015 by JSFM

4 m/s 2000m p 0 = 23MPa p c = 22.1MPa T 0 = 275K T pc =651K T M = 675K Fig K 5h Heated water pool Sea floor h 15h Fig. 5 : Computational domain for hydrothermal convection. high 23MPa Pressure Solid Liquid Temperature Pseudocritical Point Critical Point (647.1K, 22.1MPa) 675K 651K Gas SCS Pseudocritical line (T, P) in the pres. comp. high Fig. 6 : Pressure-temperature phase diagram of water Fig. 7 23MPa m 3 Fig. 8 Fig. 8 Ps e udocr i etalae mplot sot lcto Fig. 7 : Physical properties depending on temperature for water at P = 23MPa log 5. ( ) (1) Komurasaki, S., A Hydrothermal Convective Flow at Extremely High Temperature, Proc. 7th Int. Conf. on Computational Fluid Dynamics, (2012), ICCFD (2) Tamba, J., Takahashi, T., Ohara, T. and Aihara, T., Transition from boiling to free convection in supercritical fuild, Experimental Thermal and Fluid Science 17 (1998), pp (3) Yabe, T. and Xiao, F., Description of complex and sharp interface during shock wave interaction with liquid drop, J. Phys. Soc. Jpn. Letters, 62 (1993), pp (4) Kawamura, T. and Kuwahara, K., Computation of high Reynolds number flow around a circular cylinder with surface roughness, AIAA paper (1984), (5) Suito, H., Ishii, K. and Kuwahara, K., Simulation of Dynamic Stall by Multi-Directional Finite Difference Method, AIAA paper (1995), (6) Hahne, E. and Neumann, R. T., Boiling-like phenomena in free-convection heattransferatsupercritical pressures, Heat and Mass Transfer, 15 (1981), pp Copyright c 2015 by JSFM

5 (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 8 : Time development of temperature field visualized by isothermal surface of 290K. 5 Copyright c 2015 by JSFM