Xiaowen Shan

Efficient kinetic method for fluid simulation beyond the Navier-Stokes equation

We present a further theoretical extension to the kinetic-theory-based formulation of the lattice Boltzmann method of Shan [J. Fluid Mech. 550, 413 (2006)]. In addition to the higher-order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the nonequilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to third-order hydrodynamic moments. Numerical evidence demonstrates that the extended model overcomes some major defects existing in conventionally known lattice Boltzmann models, so that fluid flows at finite Knudsen number Kn can be more quantitatively simulated. Results from force-driven Poiseuille flow simulations predict the Knudsens minimum and the asymptotic behavior of flow flux at large Kn.

Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria

Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non-ideal equation of state.

Galilean invariance of lattice Boltzmann models

It is well known that the original lattice Boltzmann (LB) equation deviates from the Navier-Stokes equations due to an unphysical velocity-dependent viscosity. This unphysical dependence violates the Galilean invariance and limits the validation domain of the LB method to near incompressible flows. As previously shown, recovery of correct transport phenomena in kinetic equations depends on the higher hydrodynamic moments. In this letter, we give specific criteria for recovery of various transport coefficients. The Galilean invariance of a general class of LB models is demonstrated via numerical experiments.

Lattice ellipsoidal statistical BGK model for thermal non- equilibrium flows

A thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small Mach number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct Monte Carlo and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately 0:5. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations that have been chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

Multiscale lattice Boltzmann approach to modeling gas flows

For multiscale gas flows, the kinetic-continuum hybrid method is usually used to balance the computational accuracy and efficiency. However, the kinetic-continuum coupling is not straightforward since the coupled methods are based on different theoretical frameworks. In particular, it is not easy to recover the nonequilibrium information required by the kinetic method, which is lost by the continuum model at the coupling interface. Therefore, we present a multiscale lattice Boltzmann (LB) method that deploys high-order LB models in highly rarefied flow regions and low-order ones in less rarefied regions. Since this multiscale approach is based on the same theoretical framework, the coupling precess becomes simple. The nonequilibrium information will not be lost at the interface as low-order LB models can also retain this information. The simulation results confirm that the present method can achieve modeling accuracy with reduced computational cost.

Lattice Boltzmann models for nonideal fluids with arrested phase-separation.

The effects of midrange repulsion in lattice Boltzmann models on the coalescence and/or breakup behavior of single-component, nonideal fluids are investigated. It is found that midrange repulsive interactions allow the formation of spraylike, multidroplet configurations, with droplet size directly related to the strength of the repulsive interaction. The simulations show that just a tiny 10% of midrange repulsive pseudoenergy can boost the surface:volume ratio of the phase-separated fluid by nearly two orders of magnitude. Drawing upon a formal analogy with magnetic Ising systems, a pseudopotential energy is defined, which is found to behave similar to a quasiconserved quantity for most of the time evolution. This offers a useful quantitative indicator of the stability of the various configurations, thus helping the task of their interpretation and classification. The present approach appears to be a promising tool for the computational modeling of complex flow phenomena, such as atomization, spray formation, microemulsions, breakup phenomena, and possibly glassylike systems as well.

Continuum free-energy formulation for a class of lattice Boltzmann multiphase models

It is shown that the Shan-Chen (SC) model for non-ideal lattice fluids can be made compliant with a pseudo-free-energy principle by simple addition of a gradient force, whose expression is uniquely specified in terms of the fluid density. This additional term is numerically simulated and shown to provide fairly negligible effects on the system evolution during phase-separation. To the best of our knowledge, these important properties of the SC model were not noted before. The approach developed in the present work is based on a continuum analysis: Further extensions, more in line with a discrete lattice theory (Shan X., Phys. Rev. E, 77 (2008) 066702), can be envisaged for the future.

A Lattice-Boltzmann / Finite-Difference Hybrid Simulation of Transonic Flow

A lattice-Boltzmann / flnite-difierence hybrid method has been developed for transonic ∞ow. In the method, we introduce an interaction force to reduce the speed of sound so that high Mach number ∞ows can be simulated. In this model, the conservation of mass and momentum is represented by a lattice-Boltzmann model while the energy dynamics is described by a flnite difierence scheme. It has been a challenge problem for a long time to couple between a complete energy equation and a lattice-Boltzmann model. The proposed hybrid scheme has been used to simulate ∞ows past a wedge and ∞ows past an airfoil. The results are consistent with theoretic or experimental results.

Application of a Higher Order Lattice Boltzmann/ Hybrid Method for Simulation of Compressible Viscous Flows with Curved Boundary

Hermite polynomials and the resulting expansion coefficients are the hydrodynamics moments . Such high order LB model is able to recover the high order hydrodynamics moments precisely to conser ve the mass, moments and energy; however it is still facing instability problems when the local flow Mach number is high. In this study, a 39 state LB model is presented and coupled with a scalar energy equation for solving conservation of mass, momentum a nd energy in a fluid system. This hybrid approach is able to increase simulation Mach number substantially for compressible viscous flows. The validation results compare well with N -S (Navier -Stokes) equations based prediction for a two dimensional converg ence -divergence nozzle case, three dimensional simulation of a Mach 0.5 DLR -F4 aircraft also shows favorable agreements with experimental measurement. Nomenclature i f particle distribution function moving in the ith direction i ) , ( ) , ( ) , (