Volumetric formulation for a class of kinetic models with energy conservation

We analyze a volumetric formulation of lattice Boltzmann for compressible thermal fluid flows. The velocity set is chosen with the desired accuracy, based on the Gauss-Hermite quadrature procedure, and tested against controlled problems in bounded and unbounded fluids. The method allows the simulation of thermohydrodyamical problems without the need to preserve the exact space-filling nature of the velocity set, but still ensuring the exact conservation laws for density, momentum, and energy. Issues related to boundary condition problems and improvements based on grid refinement are also investigated.

Figure 1

Left: diagram for finite volume implementation as reported in the papers of Peng and co-workers [32]. The points $P$, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, $P_8$ stand for mesh points while $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ constitute the edges of the control volume where the integration of the lattice Boltzmann model is performed. The gray area is where the algorithm is sketched in Eqs. (11, 12, 13). Right: a regular (rectangular) mesh with variable spacings $\Delta x_i$ $\left(i=1\dots N_x\right)$ and $\Delta y_j$ $\left(j=1\dots N_y\right)$.

Figure 2

(Color online) Left: time history for the transverse velocity in the shear wave decay experiment in a given location $y=\frac{L_y}{4}$ and for different relaxation parameters (different Prandtl numbers). Right: time history for the density in the thermal diffusion mode experiment. In both cases, the results of numerical simulations with TVLBM model given in Eqs. (7, 14) are obtained with different relaxation times as reported in the figure. The corresponding analytical prediction extracted from the linearized hydrodynamic Eqs. (18, 19, 20) is also reported (solid line). The details for the initial conditions and other simulation parameters are reported in the text.

Figure 3

(Color online) Simulation of the Sod-Riemann shock tube. For two given times [$t=0.1$ (left) and $t=0.2$ (right)] the numerical results obtained with TVLBM are compared with the solution of the inviscid Euler equations for a compressible gas (solid line). The numerical simulation with TVLBM is done with 21 off-lattice velocities whose properties are reported in [47]. All the other simulation details are reported in the text.

Figure 4

(Color online) Velocity slip and temperature jump from TVLBM model [Eqs. (7, 14)] with single time relaxation $\tau$ and diffuse boundary conditions (25). Two numerical experiments are designed to compute the slip length and temperature jump emerging at the walls from the extrapolation of the profiles away from the boundary layers. Details of the numerical simulations are described in the text. The numerical results are then compared with the result expected from a perturbative analysis of the BGK equation with such diffusive boundary condition [44, 51] and reported in Eq. (29).

Figure 5

(Color online) Temperature profile for the thermal Couette flow. We have defined the normalized temperature $\frac{T-T_{cold}}{T_{hot}-T_{cold}}$, and plotted it as a function of the normalized distance from the wall $x/L_y$. The Eckert number is kept fixed to $\text{Ec}=2.0$ while the Prandtl number is varied between $\text{Pr}=0.3333$ and $\text{Pr}=1.0$. The corresponding analytical profiles are also shown (solid line). All the numerical results have been obtained with Eq. (14) and Dirichlet boundary conditions imposed as explained in Sec. 8. In the inset we report the shear flow in the velocity field normalized with the wall velocity. As we can see, in both temperature and velocity profiles, slip is prevented to emerge at the boundaries.

Figure 6

(Color online) Temperature profile for the thermal Couette flow with refined grid. The temperature is plotted as a function of the normalized distance from the lower wall. The numerical results have been obtained with Eq. (14) and Dirichlet boundary conditions for the hydrodynamic fields, imposed as explained in Sec. 8: a fixed wall velocity $U=\pm 0.1$ on both walls (located at 0 and $y/L_y=1$) and a unit Prandtl number have been used. The corresponding analytical profile is also shown (solid line), as predicted from stationary hydrodynamics. Also, we have used a non uniform grid normal to the walls, whose details are reported in Eqs. (31, 32). In the inset, we highlight the effect of grid refinement close to the lower boundary layer, and compare it with the corresponding numerical simulation with the same number of grid points arranged in a uniform way (details are reported in the text). To make it a clear distinction between the two profiles, we have shifted the profile with uniform grid $\left(T_u\right)$ with respect to the one with refined grid $\left(T_r\right)$ by a constant $\delta T=0.001$.

Figure 7

(Color online) Left: stationary velocity vector profiles for Rayleigh-Bénard thermal convection. The results from TVLBM are shown. The simulated system has a size $L_x\times L_y=80\times 40$, unit Prandtl number $\text{Pr}=1$, and Rayleigh number $\text{Ra}=8424$. Other computational details are given in the text. Right: temperature and streamwise velocity profiles extracted from the stationary snapshots. We have chosen a fixed $x=x_0=21.25$ and we have plotted the temperature (main figure) and streamwise velocity profile (inset) from TVLBM and finite difference (FD) simulations.