Lattice Boltzmann model for weakly compressible flows

We present an energy conserving lattice Boltzmann model based on a crystallographic lattice for simulation of weakly compressible flows. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics. Several test cases for acoustics and thermal and thermoacoustic flows are simulated to show the accuracy of the proposed model.

Figure 1

Building block of a crystallographic lattice in two dimensions: simple cubic links (blue) and body centered links (red) are depicted here.

Figure 2

A representation of a sphere and an ellipsoid with equal number of total lattice points $2N^3$ points on sc (left) and $N^3$ points on bcc (right) lattices.

Figure 3

Energy shells of the $RD3Q41$ model: sc-1(red), fcc-1(blue), bcc-1(light green) shown on a regular lattice and sc-2(orange), bcc-$\frac{1}{2}$(dark green).

Figure 4

Density fluctuations along the center line in isothermal case (solid line) at different time steps compared with analytical solution (points).

Figure 5

Comparison of pressure fluctuations along the center line at time $t^{*}$ from a thermal simulation (solid line) and isothermal simulation (points) at time $\sqrt{\gamma }\times t^{*}$.

Figure 6

Density fluctuations due to a 3D spherical pulse source along the $y$ axis at $(x,z)=(0.5,0.5)$ from LB simulation (line) and exact solution (points).

Figure 7

Pressure fluctuations normalized with $A$ at $y=24L$ after 50 convection times (LB: line, analytical: points).

Figure 8

Contours of pressure fluctuations normalized with $A$ at $t=0$ (up) and $t=50$ convection times (bottom).

Figure 9

Time evolution of pressure fluctuations normalized by $p_{0}A$ at points $r=5D_0$ and $\theta ={90}^{\circ },{135}^{\circ }$, and ${180}^{\circ }$ from top to bottom (LBM: solid line, analytical: points).

Figure 10

Isocontours of pressure fluctuations normalized by $p_{0}A$ at $t=1.6,4.0,6.0$, and 10.0 convection times where $D$ is the direct wave and $R$ is the wave reflected off the surface of the cylinder.

Figure 11

Mean planar temperature profiles obtained from $RD3Q41$ at steady state (symbols) compared to the analytical solution (lines).

Figure 12

Setup for the 2D cavity heated at the top.

Figure 13

Steady-state temperature profiles at sections along $x$ axis (left) and $y$ axis (right). The symbols are the solution form the $RD3Q41$ model while the lines represent the analytical solution.

Figure 14

A schematic of the setup for thermoacoustic convection.

Figure 15

Nondimensional temperature, density, and pressure at a few intermediate times scaled by diffusion time. (LBM: points; solution of NSF equations: lines).

Figure 16

Liquid vapor densities of the van der Waals fluid for various formulations at different reduced temperature ${\theta }^{*}=\theta /{\theta }_c$. The same plot is represented on the linear scale (top) and the logarithmic scale (bottom) to emphasize the error in density of the gas phase.

Figure 17

Spurious currents of the van der Waals fluid for various formulations at different reduced temperature ${\theta }^{*}=\theta /{\theta }_c$.

Figure 18

Speed of sound in a nonideal gas for liquid and vapor phases: theoretical prediction (solid line) and simulation (points).

Figure 19

Equilibrium liquid and gas density $\rho /{\rho }_c={\rho }^{*}$ obtained from $RD3Q41$ model for Carnahan-Starling EOS (top) and Peng-Robinson type EOS (bottom) compared against their respective Maxwell equal area construction at various $\theta /{\theta }_c={\theta }^{*}$.

Figure 20

Head-on collision between two droplets at Re $=297.03$ and $\mathrm{We}=19.47$.

Figure 21

Comparison of evolution of mean enstrophy with time (line: LBM, points: PS).

Figure 22

Mean velocity profiles from our simulation (points) and DNS results (line) from [94].

Figure 23

RMS velocity profiles from our simulation (points) and DNS results (line) from [94].

Figure 24

Isovorticity contours of flow over sphere at Reynolds number $=3700$.

Figure 25

$C_p$ distribution on the surface of the sphere compared with with the experimental [100] and DNS [98] studies.

Figure 26

Averaged profile of the normalized velocity at three different locations ($x/D=0.2,1.6,3.0$) in the wake of the sphere from this study (solid line) compared with DNS [98], LES [99], and experimental [100] studies.