In the past century, the Boltzmann equation has been demonstrated as the most accurate model based on statistical mechanics to describe the gas flows. Due to the complexity of the time-dependent equation in six-dimensions, only very limited solutions are available for use in engineering applications. During the last decades, a numerical scheme called the lattice Boltzmann method (LBM) has been devised as a very power tool to solve the Boltzmann equation. However, up to dates it has been illustrated that the LBM scheme is only efficient for incompressible and isothermal flows. These flow restrictions apparently defy the original physics of the Boltzmann equation. The main objective of this study is to develop a new method, called dynamic lattice Boltzmann method (DLBM), with a novel feature to circumvent these incompressible and isothermal restrictions for numerical simulations of the gas flows.

Firstly, the conventional LBM scheme has been reviewed in details to reveal the root cause of the incompressible and isothermal difficulties encountered by conventional LBM. It is found that the lack of symmetry and uniformity in the Maxwellian equilibrium distribution f

^{eq} when the physical velocity c is used conventionally as the discrete velocities c

_{i} for LBM is the root cause of these difficulties. To resolve these asymmetric and in-homogeneous problems, f

^{eq} is conventionally expanded into power-series under small Mach number assumption and truncated the series at certain orders. These procedures render the conventional LBM schemes to lose accuracy and stability, and restrict the conventional LBM schemes to be efficient only for incompressible and isothermal flows.

To circumvent this root cause of the incompressible and isothermal restrictions encountered in the conventional LBM, a dynamic quadrature scheme (DQS), which transforms the c-space into a C*-space with the coordinate transformation defined by C* = (c − u) / a where a = √2RT is the sound speed of gas, was proposed in this study. The above velocity-space transformation basically consists of a local Galilean translation and a local thermal normalization. The equilibrium distribution f

^{eq} with respective to C* is isotropic and homogeneous, which matches perfectly to the symmetric behavior of C

_{i}* if C

_{i}* is used as discrete quadrature velocities for LBM procedure. Since c

_{i} = aC

_{i}* + u from the velocity-space transformation, the physical velocities c

_{i} will change from time to time and from location to location because a and u are functions of geometric space and time. By this dynamic nature in c

_{i} the new quadrature scheme is called as the dynamic quadrature scheme (DQS). A lattice Boltzmann method with DQS is then termed as the dynamic lattice Boltzmann method (DLBM).

For the formulation of the DLBM, a coordinate transformation from (r, c ,t) to (r* ,C* ,t*) , incorporated with DQS, was proposed to transform the Boltzmann equation into a new equation in (r* ,C* ,t*) , that contains additional terms due to acceleration and advection by a and u. The equilibrium distribution with respective to C* is isotropic and homogeneous and is independent of a and u. Therefore, there is no need to expand the equilibrium distribution based on small Mach number. This implies that the DLBM is free of the incompressible and isothermal restrictions. Through the coordinate transformation the spirit of LBM, i.e., an exact streaming of the distribution function from nodes to nodes at each time step, is maintained.

A simplified DLBM scheme is also proposed for the simulation of gas flows with moderate temperature variations. The scheme involves only local Galilean translation but not local thermal normalization. The velocity transformation of the simplified DLBM is given by C* = (c − u) / a

_{0} where a

_{0} = √2RT

_{0} and T

_{0} is a constant reference temperature. This simplified DLBM has greatly reduced the complexity of the general DLBM code, and has been demonstrated to retain most advantages of DLBM for simulating thermal gas flows.

The transformed Boltzmann equation in (r* ,C* , t* ) , with the moment constraints, is then discretized. Numerical codes in two-dimensions and three-dimensions were developed respectively to implement the general DLBM and the simplified DLBM. The quadrature in the transformed coordinate system is fixed while the corresponding nodes in the physical space change according to macroscopic variables. As a whole, the numerical scheme is similar to the conventional LBM, with a simple, additional step of acceleration and advection due to the variations of a and u. It is found that the DLBM retains many of the outstanding properties of original LBM, including the ease of parallel implementation.

The validations of the DLBM codes have been carried out by numerically simulating various bench mark problems, including Riemann problems, Poiseuille flows, lid driven cavity flows, Rayleigh-Benard convection. Excellent agreements are achieved when compared with the results provided in classic literatures. This DLBM scheme has been illustrated to have removed the incompressible and isothermal restrictions encountered by the conventional LBM.

In summary, a novel dynamic lattice Boltzmann method (DLBM) has been created and successfully demonstrated to be a more superior scheme for the numerical simulations of viscous gas flows. This is achieved by adopting the dynamic quadrature scheme (DQS) into a coordinate transformation, which transforms the Boltzmann equation system into a new system that is free of the incompressible and isothermal restrictions encountered by the conventional lattice Boltzmann method (LBM). As this the first attempt to the development of this novel DLBM scheme, further refinement of the scheme as well as the code may be needed. It is hoped that this study will shed lights on the development of new schemes for further improvement of the present DLBM.

## Post a Comment

Your email address will not be published.