Entropy Generation Rates in Two-Dimensional Rayleigh–Taylor Turbulence Mixing

Entropy generation rates in two-dimensional Rayleigh–Taylor (RT) turbulence mixing are investigated by numerical calculation. We mainly focus on the behavior of thermal entropy generation and viscous entropy generation of global quantities with time evolution in Rayleigh–Taylor turbulence mixing. Our results mainly indicate that, with time evolution, the intense viscous entropy generation rate su and the intense thermal entropy generation rate Sθ occur in the large gradient of velocity and interfaces between hot and cold fluids in the RT mixing process. Furthermore, it is also noted that the mixed changing gradient of two quantities from the center of the region to both sides decrease as time evolves, and that the viscous entropy generation rate 〈Su〉V and thermal entropy generation rate 〈Sθ〉V constantly increase with time evolution; the thermal entropy generation rate 〈Sθ〉V with time evolution always dominates in the entropy generation of the RT mixing region. It is further found that a “smooth” function 〈Su〉V∼t1/2 and a linear function 〈Sθ〉V∼t are achieved in the spatial averaging entropy generation of RT mixing process, respectively.


Introduction
Rayleigh-Taylor (RT) instability occurs in a large number of engineering applications. In general, it mainly originates at the interface between a heavy fluid and a light fluid due to a gravitational field [1][2][3][4][5][6]: the colder layer is placed above the hotter layer due to a gravitational field causing the accumulation of two layers in a single-phase fluid [3]. Zhou [4] studied the statistical properties of the kinetic energy dissipation rates and thermal energy dissipation rates in 2D RT turbulence. One of the critical issues is a deeper comprehension of the transport mechanisms of both the viscosity and thermal entropy generation rate inside the mixing zone during RT turbulence. The local entropy generation rates that play a substantial role in energy-loss are the viscosity and thermal entropy generation rate fields [7,8], which are given in two dimensions. And

Macroscopic Dynamics Equation of Thermal Fluid
The classical Oberbeck-Boussinesq equations are given by the following equations to study the thermal fluid dynamics equations [6,8].
where ν represents the kinematic viscosity coefficient, κ represents the coefficient of diffusivity, ρ represents the density of fluid, u represents the macroscopic velocity, p represents the fluid pressure, S is the shear stress, θ is the macroscopic temperature, and ∆θ is the temperature difference, respectively.

Double Distribution Lattice Boltzmann Method
The lattice Boltzmann equation to simulate fluid flow is given as [29][30][31]: Here f i (x, t) denotes the distribution function of density, c i is the discrete velocity. F i represents the discrete force term in Equation (6), τ ν is the relaxation time for density evolution equation in lattice Boltzmann equation, and f eq i is the density distribution equilibrium function. The lattice Boltzmann equation for the temperature field: where g i (x, t) denotes the temperature distribution function, τ θ denotes the relaxation times for temperature evolution equation in the above equation and g eq i is the temperature distribution equilibrium function. The density distribution equilibrium function and the temperature distribution equilibrium function are represented by Equations (8) and (9) [30], respectively.
where w i represents the weight coefficient [25]. The relations among kinematic viscosity ν and the coefficient of thermal diffusivity κ, and the relaxation time are given as: where ∆t is the unit time and ∆x is the unit space. The Macroscopic density, velocity, and temperature are represented by Equation (9).
The Mesoscopic equation for density, momentum (Equation (6)), and temperature (Equation (7)) are spread by a Chapman-Enskog expansion. The Macroscopic Oberbeck-Boussinesq equations (Equations (3)-(5)) are obtained by a macroscopic length scale (x 1 = εx) and two macroscopic time scales (t 1 = εt, t 2 = εt). As one spatial scale ∂ x = ε∂ α , and two time scales ∂ t = ε∂ t 1 + ε 2 ∂ t 2 are implemented. The macroscopic classical Oberbeck-Boussinesq equations (Equations (3) and (4)) can be reproduced by the Chapman-Enskog expansion of executing the Equations (6) and (7). The Rayleigh number (Ra) is an important dimensionless parameter in the turbulent RT mixing flow. The expression of Ra in the simulation of LBM is given by: where β is the coefficient of thermal conductivity, g represents the acceleration of gravity. The Nusselt number (Nu) is also one of most important dimensionless parameters in the turbulent RT mixing flow.
The expression of Nu in the simulation of LBM is given by: where ∆θ represents the difference of temperature between the bottom boundary and the top boundary, H denotes the height of channel, u y is the vertical macroscopic velocity, and . is the average value of the entire computational domain. In this paper, the nonequilibrium extrapolation method and the periodic condition are used. The expressions of nonequilibrium extrapolation method are given by [20]: where the nonequilibrium contribution is derived from the fluid node x f next to x b along the boundary normal vector [27]. The expressions of the periodic condition method are given by [27]: where the vector L is the periodicity direction and the length of the flow pattern.

Some Numerical Results and Discussions
The uniform grid is implemented for all of the following numerical simulations. The convergence criterion is set for all cases. A clear scaling can be seen for Nu(Ra) for nearly four decades from Ra ≈ 10 6 to 10 10 . The compensated plots in the insets give [4], The grid-dependence study of the results is implemented. One example of the Rayleigh number of Ra = 9.8 × 10 9 is presented in Table 1. In this study, the number of grid points is taken as the same in both the x and y directions. That is, the grid size is taken as M × N, where M is the grid number in the transverse coordinates direction and N is the grid number in the longitudinal coordinates direction. The calculated Nusselt number changing with M × N, is presented in Table 1. From this table, it is clearly seen that when M × N increases, the calculated Nusselt number quickly approaches the benchmark result. When the grid size further increases from 2056 × 4112 to 2200 × 4400, there is not much improvement in the result. So one can say that for Ra = 9.8 × 10 9 , the grid size of 2056 × 4112 can give very accurate results. As shown in Table 1, one can see that result of LBM for the relation of Nu(Ra) is well consistent with theoretical value of Nu(Ra) [4]. To ensure adequate resolution for S u and S θ , 2056 × 4112 lattices were implemented using the double distribution LBM of the present work. In the initial stage, the system is at rest. In the upper half of the calculation area, the fluid is cold. The fluid is hot in the lower half of the calculation area. If y is greater than H/2, the temperature equals −0.5 in Figure 1 and the temperature equals to 0.5 when y is less than H/2. An initial temperature, θ 0 , is executed in the colder uniform fluid layer and placed on top of the hotter one. To achieve the repeatability of whole flow field, a total of eight independent realizations in RT evolution were performed by giving some perturbed interfaces. In all the simulations, Ag = 0.25, Ra = 9.8 × 10 9 , and the corresponding Prandtl number is Pr = v/κ = 7. For the vertical boundaries, periodic boundary conditions are executed. The no-slip boundary conditions are adopted in the top and bottom boundary conditions. It is noted that in the previous studies, Zhou et al. [4] Entropy 2018, 20, 738 5 of 11 investigated the statistical properties of kinetic and thermal energy dissipation rates in RT turbulence mixing. Here, some new analysis investigating the viscous and thermal entropy generation are performed in RT turbulence mixing.
when y is less than H/2. An initial temperature, 0 θ , is executed in the colder uniform fluid layer and placed on top of the hotter one. To achieve the repeatability of whole flow field, a total of eight independent realizations in RT evolution were performed by giving some perturbed interfaces. In all the simulations, Ag = 0.25, Ra = 9.8 × 10 9 , and the corresponding Prandtl number is For the vertical boundaries, periodic boundary conditions are executed. The no-slip boundary conditions are adopted in the top and bottom boundary conditions. It is noted that in the previous studies, Zhou et al. [4] investigated the statistical properties of kinetic and thermal energy dissipation rates in RT turbulence mixing. Here, some new analysis investigating the viscous and thermal entropy generation are performed in RT turbulence mixing.    and / τ 4 t = . Figure 4 displays the snapshots of the viscous entropy generation with time evolution obtained at times / τ 1.2 t = and / τ 4 t = . As shown in Figures 2 and 3, one clearly sees that the viscous entropy generation rate ( u S ) and the velocity with time evolution always occur 3.2. Analysis of S u and S θ in RT Turbulence Mixing Figure 3 represents the distribution of constant velocity contours with time evolution obtained at times (a) t/τ = 1.2 and t/τ = 4. Figure 4 displays the snapshots of the viscous entropy generation with time evolution obtained at times t/τ = 1.2 and t/τ = 4. As shown in Figures 2 and 3, one clearly sees that the viscous entropy generation rate (S u ) and the velocity with time evolution always occur in the RT mixing region, which indicates that the loss of flow is also mainly concentrated in this mixing area. It is also further seen that the intense S u usually concentrates on the steepest velocity gradient in the mixing process, which is consistent with the viscous entropy generation in Rayleigh-Bénard convection [19].  and / τ 4 t = . Figure 4 displays the snapshots of the viscous entropy generation with time evolution obtained at times / τ 1.2 t = and / τ 4 t = . As shown in Figures 2 and 3, one clearly sees that the viscous entropy generation rate ( u S ) and the velocity with time evolution always occur in the RT mixing region, which indicates that the loss of flow is also mainly concentrated in this mixing area. It is also further seen that the intense u S usually concentrates on the steepest velocity gradient in the mixing process, which is consistent with the viscous entropy generation in Rayleigh-Bénard convection [19].     As shown in Figure 4, one clearly sees that with time evolution the θ S always occurs in the RT mixing region. Further, it was found that as the time evolution progressed, the intense θ S focuses on the interfaces between the hot and cold fluids in the RT mixing process, which is also consistent with the thermal entropy generation in Rayleigh-Bénard convection [19].   Figure 4, one clearly sees that with time evolution the S θ always occurs in the RT mixing region. Further, it was found that as the time evolution progressed, the intense S θ focuses on the interfaces between the hot and cold fluids in the RT mixing process, which is also consistent with the thermal entropy generation in Rayleigh-Bénard convection [19]. / τ 1.2 t = , and (b) / τ 4 t = . Figure 5 displays the snapshots of thermal entropy generation with time evolution obtained at times (a) / τ 1.2 t = and (b) / τ 4 t = . As shown in Figure 4, one clearly sees that with time evolution the θ S always occurs in the RT mixing region. Further, it was found that as the time evolution progressed, the intense θ S focuses on the interfaces between the hot and cold fluids in the RT mixing process, which is also consistent with the thermal entropy generation in Rayleigh-Bénard convection [19]. In the above section, the various instantaneous viscous entropy generation rates and thermal entropy generation rates are presented in the field of space. In the following section, the mean values of the viscous entropy generation rate and thermal entropy generation rates are analyzed in space. Figures 6 and 7 display the temporal evolution of the mean vertical profiles of the horizontal and In the above section, the various instantaneous viscous entropy generation rates and thermal entropy generation rates are presented in the field of space. In the following section, the mean values of the viscous entropy generation rate and thermal entropy generation rates are analyzed in space. Figures 6 and 7 display the temporal evolution of the mean vertical profiles of the horizontal and vertical root-mean-square (rms) viscous entropy generation S u X and the thermal entropy generation S θ X at times t/τ = 1.5, t/τ = 2.4 and t/τ = 3.5, respectively, where i rms = (i − (i) j ) 2 j is the RMS value of i with i = S u , S θ and with j = x for a horizontal average. As shown in Figures 5 and 6, one can see that all profiles of S u X and S θ X display a similar shape, not far from a parabola at the temporal evolution. However, the behaviors of the amplitudes of two quantities vary as time evolves. It is also found that the mixed changing gradient of two quantities from the center of the region to both sides decrease as time evolves.  Figures 5 and 6, one can see that all profiles of u X S and θ X S display a similar shape, not far from a parabola at the temporal evolution. However, the behaviors of the amplitudes of two quantities vary as time evolves. It is also found that the mixed changing gradient of two quantities from the center of the region to both sides decrease as time evolves.     Figures 5 and 6, one can see that all profiles of u X S and θ X S display a similar shape, not far from a parabola at the temporal evolution. However, the behaviors of the amplitudes of two quantities vary as time evolves. It is also found that the mixed changing gradient of two quantities from the center of the region to both sides decrease as time evolves.     Figure 8 shows the time behaviors of the viscous entropy generation rate S u V normalized by the computational grid spacing. From Figure 8, it is clearly seen that the viscous entropy generation rate S u V always increases with time evolution. The solid line represents the theoretical prediction fitted approximately by the least square method according to the numerical results of LBM in Figure 8. A "smooth" function S u V ∼ t 1/2 is approximately achieved.  The time behaviors of thermal entropy generation rate θ V S normalized by the computational grid spacing in RT mixing are plotted in Figure 9. As shown in Figure 9, one can clearly see that the thermal entropy generation rate θ V S increases with time evolution. The solid line in Figure 9 represents the theoretical prediction fitted approximately by the least square method according to the numerical results of LBM. The linear function   The time behaviors of thermal entropy generation rate S θ V normalized by the computational grid spacing in RT mixing are plotted in Figure 9. As shown in Figure 9, one can clearly see that the thermal entropy generation rate S θ V increases with time evolution. The solid line in Figure 9 represents the theoretical prediction fitted approximately by the least square method according to the numerical results of LBM. The linear function S θ V ∼ t was approximately obtained. Comparing Figure 7 with Figure 8, it is seen that S θ V is almost four orders of magnitude greater than S u V in turbulent RT mixing. It is further indicated that the thermal entropy generation rate with time evolution plays a main role in the entropy generation of RT mixing. The time behaviors of thermal entropy generation rate θ V S normalized by the computational grid spacing in RT mixing are plotted in Figure 9. As shown in Figure 9, one can clearly see that the thermal entropy generation rate θ V S increases with time evolution. The solid line in Figure 9 represents the theoretical prediction fitted approximately by the least square method according to the numerical results of LBM. The linear function

Conclusions
In this paper, entropy generation rates in two-dimensional Rayleigh-Taylor turbulence mixing with time evolution are investigated. The various instantaneous viscous entropy generation rates and thermal entropy generation rates were studied in the field of space. Mean values of the viscous entropy generation rate and thermal entropy generation rate were also discussed in space. Several major findings are summarized.
First of all, it is shown that the intense viscous entropy generation rate S u with time evolution always focuses on the large gradient of velocity in the RT mixing region. With progressive time evolution, the intense thermal entropy generation rate S θ also focuses on the interfaces between the hot and cold fluids in the RT mixing process. In addition, all profiles of S u X (the mean vertical profiles of the horizontal and vertical root-mean-square) and S θ X possess a similar shape, not far from a parabola at the temporal evolution. The mixed changing gradient of two quantities from the center of the region to both sides decrease as time evolves. One can also obtain that the viscous entropy generation rate S u V and the thermal entropy generation rate S θ V constantly increase with time evolution. A "smooth" function S u V ∼ t 1/2 and a linear function S θ V ∼ t are achieved, respectively. Furthermore, it is found that the thermal entropy generation rate S θ V with time evolution always plays a main role in the entropy generation of RT mixing region.