Theoretical Foundation


M-Star CFD is a computational modeling tool used to predict heat, mass, momentum, and particle transport processes across three-dimensional space and time. The solver combines a lattice-Boltzmann-based fluid transport kernel with a high-resolution flux limiter scheme to solve the Navier-Stokes and advection-diffusion equations simultaneously. By writing algorithms that fully exploit modern computer architectures, simulations execute order-of-magnitude faster than conventional finite-difference schemes while providing superior accuracy and precision. The fundamentality of this modeling approach minimizes the need for user parameter tuning and shortens the learning curve associated with model building and post-processing.

Modeling Fluid Flow

Both the lattice-Boltzmann kernel and the high-resolution flux limiter schemes are adapted from approaches presented in the literature. Of foundational relevance are two papers developed by S. Chen et al., as presented in the 1996 Fields Institute for Research in the Mathematical Sciences [1] and the 1998 Annual Review of Fluid Mechanics. [2] The 1998 paper links the discrete Boltzmann transport equation to the low Mach number Navier-Stokes equations via a Chapman-Enskog expansion. The earlier 1996 paper outlines a link between the Smagorinsky large eddy simulation (LES) sub-grid turbulence model and lattice-Boltzmann. More specifically, the manuscript details how a filtered velocity and density can be incorporated into the collision/streaming process via the eddy viscosity to execute LES simulations in lattice-Boltzmann. This point was further addressed in work by Yu, Girimaji, and Luo, who validated the LBM-LES model against predictions from direct numerical simulations of turbulence. [3] Building upon this work, fluid-structure interactions are handled following the immersed boundary method of Z. Feng and E. Michaelides [4] and the interface between immiscible fluids is tracked following the approach of Bonger et al. [5]

Modeling Scalar Transport

The high-resolution flux limiter scheme used to solve for scalar advection and diffusion is adapted from the five-paper “Towards the Ultimate Conservative Difference Scheme” sequence published by Bram Van Leer in the Journal of Computational Physics between 1972 and 1979, then revisited in 1997. [6] In this series, Van Leer outlines the conditions necessary to guarantee both the stability and the consistency of high-resolution finite differences schemes. His findings are applied to solve the time-dependent advection-diffusion equation, using the velocity field predicted from the lattice-Boltzmann equations as input at each time-step.

Particle Dynamics

Particle trajectories, collisions, and interactions are all based on algorithms adapted from molecular dynamic simulation, as summarized in the textbook Computer Simulation of Liquids by Allen and Tildesley. [6] Key extractions from this work include the massively parallelizable approaches for performing trajectory integrations of large populations of interacting particles.

Implications and Benefits

The simulation parameter space that must be managed by users is very small. In fact, the only four user parameters available for tuning in M-Star CFD are (i) the spatial resolution, (ii) the Courant number, (iii) the Smagorinsky coefficient, and (iv) the flux limiter type. Among these four, only the first two are truly independent, as the Smagorinsky coefficient is informed by first-principals theory and the flux limiter type should rarely be changed to something other than the van Leer limiter. This small operating space is not a consequence of source code tuning, tweaking, or parameterization. Rather, as discussed above, it is a direct consequence of the primacy of the modeling approaches and the computational efficiency of the implementation.


  1. A Lattice Boltzmann Subgrid Model for High Reynolds Number Flows. S. Hou, J. Sterling, S. Chen and G. D. Doolen, arXiv:comp-gas/9401004 Jan. 1994.
  2. Lattice Boltzmann Method for Fluid Flows. S. Chen and G. Doolen, Annual Review of Fluid Mechanics 1998, 30:1, 329-364
  3. DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method.Y. Huidan, S.. Girimaji, L. Luo, Journal of Computational Physics 2005, 209:2, 599-616
  4. The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. Z. Feng, E. Michaelides, Journal of Computational Physics 2004,195:2,602-628,
  5. Curvature estimation from a volume-of-fluid indicator function for the simulation of surface tension and wetting with a free-surface lattice Boltzmann method. S. Bogner, U. Rude, and J. Harting, Phys. Rev. E 2016 93:043302
  6. Towards the Ultimate Conservative Difference Scheme. B. van Leer, Journal of Computational Physics 1997, 135:2,229-248
  7. Computer Simulation of Liquids, 2nd. Ed. M. Allen and D. Tildesley, Oxford University Press 2017, New York

Copyright M-Star Simulations, LLC 2018