[Paper Review] A simplified discrete unified gas kinetic scheme for incompressible flow

Mingliang Zhong, Sen Zou+3 more|arXiv (Cornell University)|Jul 12, 2020

Lattice Boltzmann Simulation Studies44 references4 citations

TL;DR

This paper proposes a simplified discrete unified gas kinetic scheme (SDUGKS) for incompressible flows by eliminating the half-time-step flux calculation of the original DUGKS, enabling more efficient and stable simulations. By reconstructing the transformed distribution function along particle velocity characteristic lines over a full time step, the method preserves multi-scale accuracy and stability while allowing larger time steps under the CFL condition, validated across laminar, complex, and rarefied flow cases with excellent agreement to reference solutions.

ABSTRACT

The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows which combines the benefits of both Lattice Boltzmann Method (LBM) and unified gas kinetic scheme (UGKS). By reconstruction of gas distribution function using particle velocity characteristic line, flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with reconstruction stage on a whole time step instead of half time step in original DUGKS. Using temporal/spatial integral Boltzmann Bhatnagar-Gross-Krook (BGK) equation, the transformed distribution function with inclusion of collision effect is constructed. The macro and mesoscopic fluxes of the cell on next time step is predicted by reconstruction of transformed distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on next time step can be updated through its macroscopic flux. Equilibrium distribution function on next time step can also be updated. Gas distribution function is updated by FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. Numerical time step is only limited by the Courant-Friedrichs-Lewy (CFL) condition and relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, as well as micro cavity flow cases are conducted to validate present scheme. The numerical simulation results agree well with the references' results.

Motivation & Objective

• To develop a more computationally efficient variant of the discrete unified gas kinetic scheme (DUGKS) for incompressible flows.
• To eliminate the half-time-step flux calculation in original DUGKS to streamline the algorithm and reduce computational overhead.
• To maintain the multi-scale accuracy and stability of DUGKS while enabling larger time steps under the CFL condition.
• To validate the proposed scheme on complex geometries and flow regimes, including laminar, viscous, and rarefied flows.
• To demonstrate the method's robustness on unstructured meshes for practical fluid dynamics applications.

Proposed method

• The method reconstructs the transformed distribution function at cell interfaces using particle velocity characteristic lines over a full time step, replacing the original half-time-step flux evaluation.
• The mesoscopic flux for the next time step is predicted via reconstruction of the transformed distribution function along particle trajectories.
• Macroscopic variables at the next time step are updated using conservation laws based on the reconstructed fluxes.
• The equilibrium distribution function at the next time step is updated using the predicted macroscopic variables.
• The gas distribution function is updated via a finite volume scheme using the predicted mesoscopic flux over the full time step.
• Spatial interpolation on unstructured meshes is performed using least-squares reconstruction (LLSR) to ensure accuracy and robustness.

Experimental results

Research questions

• RQ1 Can the half-time-step flux calculation in DUGKS be eliminated without sacrificing accuracy or stability in incompressible flow simulations?
• RQ2 How does the proposed full-time-step reconstruction strategy affect the time step size and computational efficiency compared to original DUGKS?
• RQ3 To what extent does the simplified scheme maintain multi-scale capability across continuum and rarefied flow regimes?
• RQ4 Can the method achieve second-order spatial accuracy on unstructured meshes for complex geometries?
• RQ5 How well does the scheme perform in simulating laminar flows over bodies like airfoils and cylinders compared to established solvers?

Key findings

• The SDUGKS achieves second-order spatial accuracy in the Couette flow test on unstructured meshes, confirming the effectiveness of the LLSR-based reconstruction.
• Lid-driven cavity flow results show good agreement with benchmark solutions, validating the method’s capability for complex vortical flows.
• Laminar flows over flat plates, circular cylinders, and NACA 0012 airfoils demonstrate excellent agreement with reference data from Fluent and other solvers.
• At $AOA=0^\circ$, the drag coefficient $C_d$ is 0.1757 and lift coefficient $C_l$ is 0.3813, closely matching Fluent’s $C_d = 0.1741$ and $C_l = 0.3962$.
• For micro cavity flow at $Kn=0.1$ to $Kn=8$, the SDUGKS results match DSMC simulations across all Knudsen numbers, confirming accuracy in rarefied regimes.
• The method maintains stability and accuracy even with larger time steps, limited only by the CFL condition, and avoids non-physical oscillations seen in standard LBM.