Higher-order lattice Boltzmann model for thermohydrodynamics

We present an energy conserving lattice Boltzmann model on a body-centered-cubic arrangement for thermohydrodynamics. It exhibits accurate thermohydrodynamic behavior with a high degree of accuracy and is therefore capable of simulating compressible and thermal hydrodynamics. The theoretical requirements and the methodology to construct this model have been described in detail and can be employed to construct even more accurate models. Simulations of canonical test cases related to compressible flows like shock tube and thermoacoustic convection and thermal flows like viscous heat dissipation and the Rayleigh-Bénard convection are performed to demonstrate the effectiveness of the model.

Figure 1

Finding the discrete equilibrium by evaluating the Maxwell-Boltzmann distribution at discrete points does not conserve the moments. In the figure, the area beneath the curve represents the density, which is different for the Maxwell-Boltzmann distribution and discrete equilibrium as Maxwell-Boltzmann distribution at discrete points.

Figure 2

Building block of a crystallographic lattice in two dimensions, simple cubic links (left) and body-centered links (right) are depicted here.

Figure 3

Building block of a crystallographic grid in three dimensions, simple cubic links (left), face-centered-cubic link (middle), and body-centered links (right). The dashed grid is offset by a distance $0.5\mathrm{\Delta }x$ in each direction.

Figure 4

Communication between two processors on a crystallographic grid.

Figure 5

The energy shells in $RD3Q67$ model: sc1, sc2, sc3 (left), $\mathrm{bcc}\frac{1}{2}$, bcc1, $\mathrm{bcc}\frac{3}{2}$ (center), fcc2, fcc3 (right).

Figure 6

Populations at each layer that see the top wall and need to be repopulated post streaming are listed in Table 5.

Figure 7

Nondimensionalized mean planar velocity profiles obtained from $RD3Q67$ at various diffusion times compared against the analytical solution.

Figure 8

Temperature plots without including the thermal conductivity correction for different magnitudes of the top wall temperature.

Figure 9

$L^2$ norms of temperature: deviation from linear temperature profile at ${\theta }_{\mathrm{top}}=1.2{\theta }_0$ and different grid resolutions (left) and comparison for corrected and uncorrected thermal conductivity at various magnitudes of the top wall temperature (right).

Figure 10

Sketch representing the geometry of the two-dimensional box and the imposed wall temperatures.

Figure 11

Steady-state conduction in a two-dimensional plate: temperature plots along constant $x$ lines (left) and along constant $y$ lines (right). The solid lines are the analytical solution of the conduction equation while the symbols are from simulations.

Figure 12

Mean planar temperature profiles obtained from $RD3Q67$ at steady state compared against the analytical solution.

Figure 13

The figure shows the variation of density ($\rho$), velocity ($\mathit{u}$), and pressure ($P$) along the tube for the Sod's shock test. The figure contrasts $RD3Q67$ simulation results from runs $A$ (left) and $B$ (right) with NSF equations at $t^{*}=0.2$.

Figure 14

Normalized temperature $\overline{\theta }=(\theta -{\theta }_{\mathrm{cold}})/({\theta }_{\mathrm{hot}}-{\theta }_{\mathrm{cold}})$, density $\overline{\rho }=\rho /{\rho }_0$, pressure $\overline{P}=\rho \theta /\left({\rho }_0{\theta }_{\mathrm{cold}}\right)$, and volume averaged pressure $〈\overline{P}〉$ profiles for thermoacoustic convection at various times. Here, $\overline{t}=t\nu /H^2$. The symbols are obtained from simulation while the lines are from the solution to the NSF equations.

Figure 15

Temperature field for the Rayleigh-Bénard convection at $\mathrm{Ra}=1.0\times {10}^4\left(\mathrm{top}\mathrm{left}\right),3.0\times {10}^4\left(\mathrm{top}\mathrm{right}\right),1.5\times {10}^5\left(\mathrm{bottom}\mathrm{left}\right),1.0\times {10}^7\left(\mathrm{bottom}\mathrm{right}\right)$. The lines represent isotemperature contours of temperature normalized from 0 to 1 in steps of 0.05.

Figure 16

Rayleigh Bénard convection at $\mathrm{Ra}={10}^9$: temperature field at the boundaries (left) and on a horizontal slice close to the top wall (right).

Figure 17

$\mathrm{Nu}$ vs $\mathrm{Ra}$, triangles are from current simulation, squares are from Ref. [93], line is empirical power law $\mathrm{Nu}=1.56{(\mathrm{Ra}/{\mathrm{Ra}}_{\mathrm{c}})}^{0.296}$.

Figure 18

Grid convergence study for $\mathrm{Ra}={10}^4$ reveals second-order convergence. The converged ${\mathrm{Nu}}_{\mathrm{R}}$ is 2.6311 with 360 points in the vertical direction. The line is the fitted curve.