Symmetrized operator split schemes for force and source modeling in cascaded lattice Boltzmann methods for flow and scalar transport

Operator split forcing schemes exploiting a symmetrization principle, i.e., Strang splitting, for cascaded lattice Boltzmann (LB) methods in two- and three-dimensions for fluid flows with impressed local forces are presented. Analogous scheme for the passive scalar transport represented by a convection-diffusion equation with a source term in a novel cascaded LB formulation is also derived. They are based on symmetric applications of the split solutions of the changes on the scalar field or fluid momentum due to the sources or forces over half time steps before and after the collision step. The latter step is effectively represented in terms of the post-collision change of moments at zeroth and first orders, respectively, to represent the effect of the sources on the scalar transport and forces on the fluid flow. Such symmetrized operator split cascaded LB schemes are consistent with the second-order Strang splitting and naturally avoid any discrete effects due to forces or sources by appropriately projecting their effects for higher-order moments. All the force or source implementation steps are performed only in the moment space and they do not require formulations as extra terms and their additional transformations to the velocity space. These result in particularly simpler and efficient schemes to incorporate forces or sources in the cascaded LB methods unlike those considered previously. Numerical study for various benchmark problems in 2D and 3D for fluid flow problems with body forces and scalar transport with sources demonstrate the validity and accuracy, as well as the second-order convergence rate of the symmetrized operator split forcing or source schemes for the cascaded LB methods.

Figure 1

Comparison of the computed velocity profiles using the 2D symmetrized operator split cascaded LB forcing scheme with the analytical solution for Poiseuille flow for body force magnitudes of ${10}^{-7}$ and ${10}^{-8}$. The lines indicate the analytical results, and the symbols are the solutions obtained by our present numerical scheme.

Figure 2

Grid convergence for 2D Poiseuille flow with a constant Reynolds number $\text{Re}=100$ and relaxation time $\tau =0.55$ computed using the 2D symmetrized operator cascaded LB forcing scheme.

Figure 3

Comparison of the computed velocity profiles using the 2D symmetrized operator split cascaded LB forcing scheme with the analytical solution for Hartmann flow for Hartmann numbers $\mathrm{Ha}$ of 3 and 10. The lines indicate the analytical results, and the symbols are the solutions obtained by our present numerical scheme.

Figure 4

Comparison of computed and analytical velocity profiles at different instants within a time period of pulsatile flow at two different Womersley numbers of Wo = 4 Wo = 10.7. Here, lines represent the analytical solution and symbols refer to the numerical results obtained using the 2D symmetrized operator split cascaded LB forcing scheme.

Figure 5

Comparison of the computed velocity profiles using the 3D symmetrized operator split cascaded LB forcing scheme and the analytical solution, for flow through a square duct in presence of a body force magnitude of $F_x={10}^{-7}$ for different values of $y$. Here, lines represent the analytical solution and symbols refer to the results obtained using the present numerical scheme.

Figure 6

Comparison of the computed and analytical vertical velocity profiles $u_y\left(x\right)$ at $y=\pi$ for the four-rolls mill flow problem at $u_0=0.01, \nu =0.0011,$ and $N=96$. Here, line represents the analytical solution and the symbol refers to the numerical results obtained using the 2D symmetrized operator split cascaded LB forcing scheme.

Figure 7

Streamlines (a) computed using the 2D symmetrized operator split cascaded LB forcing scheme and (b) obtained using the analytical solution for the four-rolls mill flow problem at $u_0=0.01, \nu =0.0011,$ and $N=96$.

Figure 8

Grid convergence for the four-rolls mill flow problem at $u_0=0.01, \nu =0.0011$ computed using the 2D symmetrized operator split cascaded LB forcing scheme under the convective scaling.

Figure 9

Comparison between numerical results of the temperature profile computed using the 2D symmetrized operator split cascaded LB source scheme for a passive scalar transport and the analytical solution for the thermal Couette flow for various values of the Eckert number $\mathrm{Ec}$. Here, lines represent the analytical solution and symbols refer to the results obtained using the present numerical scheme.