A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

Abstract

SUMMARYA high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.

Simulation of three‐dimensional incompressible flows in generalized curvilinear coordinates using a high‐order compact finite‐difference lattice Boltzmann method

“…For the higher kinematic viscosity ν = 10 −2 , the order of the accuracy of the solution is obtained 3.2 and it may be due to the compressibility errors because the characteristics velocity u 0 = 0.01 is not small enough here. It has been shown that the CFDLBM can lead to a more accurate solution compared with the analytical solution by considering a small characteristics velocity . Herein, the grid refinement study of the decaying double shear wave is also performed for ν = 10 −2 with the smaller characteristics velocity u 0 = 0.001 and the fourth‐order accuracy of the solution for this condition is indicated in Table (b).…”

“…It indicates that the boundary conditions imposed on the sphere wall based on the local distribution function with the extrapolation of the pressure (approximate BCs) do not affect the accuracy of the solution. It should be note that for the present numerical approach based on the compact finite‐difference LBM, the effects of two different procedures for implementing boundary conditions, namely, approximate and physical boundary conditions, are applied and assessed for 2D flow problems and it has been indicated that the accuracy of the solution is not affected by using these two procedures for the boundary conditions implementation and they give the same results.…”

“…The high-order one-sided boundary condition formulas are expressed as 23 Table 2 gives the values of the constants a, b, c, d, and for the third-and fourth-order one-sided compact finite-difference formulas. Here, the fourth-order compact formula (16) together with the one-sided third-order relations (17) and (18) are used to compute f / implicitly for i = 1, … , Imax each j and k. The calculation of the first derivative of the distribution function f in the -and -directions (denoted by ′ α = α ∕ and ′ α = α ∕ , respectively) in Equation (14) are performed for the grid points j = 1, … , Jmax for each i and k and for the grid points k = 1, … , Kmax for each i and j, respectively, in the same manner. Note that all the metrics that appeared in the formulation (x , y , … ) are also calculated by the fourth-order compact finite-difference formula described above.…”

A high‐order compact scheme for solving the 2D steady incompressible Navier‐Stokes equations in general curvilinear coordinates

“…In recent years, some high‐order compact finite difference schemes have been developed for incompressible Navier‐Stokes (N‐S) equation or lattice Boltzmann equation to obtain the numerical solutions with good resolution characteristics and smaller computational stencil size. For example, Hejranfar and Ezzatneshan applied a high‐order compact scheme to the lattice Boltzmann method to solve incompressible flows . Different from the primitive variable formulation and the vorticity formulation, the use of the pure stream function or stream function‐velocity formulation of incompressible N‐S equation, in which the stream function and its first‐order partial derivatives (velocities) are taken as the unknown variables, is attractive because the boundary conditions of stream function and velocities are generally known and are easy to implement computationally.…”

Implementation of a curved wall- and an absorbing open-boundary condition for the D3Q19 lattice Boltzmann method for simulation of incompressible fluid flows

“…The LBM studies the fluid flow with mesoscopic physics in which the particles 3 interpolation-or differential-type schemes [9][10][11]. These approaches eliminate the instability problem of the LBM with decoupling the time and space discretizations [12].…”