This paper describes the theory and application of a grid-free kinetic upwind scheme known as LSKUM.The basic principle in LSKUM is the determination of derivatives occurring in the conservation laws using the least squares method. The grid-free nature of the scheme is obtained because the least squares method can be used on an arbitrary distribution of nodes, i.e. the nodes need not form a structured/unstructured grid.

Theory and application of 3‐D LSKUM based on entropy variables

Kinetic schemes for the numerical solution of PDE of fluid dynamics are a new class of upwind schemes. The moment method strategy used in constructing these schemes is based on the connection between the Boltzmann equation of kinetic theory of gases and governing equations of fluid dynamics. Any upwind scheme for the solution of the Boltzmann equation becomes an upwind scheme for the solution of PDE of fluid dynamics by taking suitable moments. This idea has been exploited in the present paper in developing Least Squares Kinetic Upwind Method (LSKUM) and Kinetic Flux vector splitting method on Moving Grid (KFMG). The robustness and versatility of LSKUM has been demonstrated by applying it to 2-D flow problems of inviscid gas dynamics. The LSKUM has been found to operate on any type of grid and can be called a grid fault tolerant scheme in view of the use of least squares approximation to the space derivatives. The KFMG is a promising new method showing how easily the kinetic schemes lend themselves easily to problems involving moving grids which are generally employed while solving problems in unsteady aerodynamics.

New Developments in Kinetic Schemes

A new upwind Boltzmann scheme termed Peculiar Velocity based Upwind (PVU) method is developed for solving Euler equations. Upwinding is done based on peculiar velocity of Kinetic Theory of Gases. This new method is more efficient and physically more meaningful than its predecessor, the Kinetic Flux Vector Splitting (KFVS) method. The PVU method is applied to some standard test problems. The results demonstrate the soundness of the new idea.

Peculiar velocity based upwind method for inviscid compressible flows

A gas-kinetic solver is developed for the ultrarelativistic Euler equations. The scheme is based on the direct splitting of the flux function of the Euler equations with inclusion of ``particle'' collisions in the transport process. Consequently, the artificial dissipation in the new scheme is much reduced in comparison with the usual kinetic flux-vector splitting (KFVS) schemes which are based on the free particle transport at the cell interfaces in the gas evolution stage. Although in a usual KFVS scheme the free particle transport gives robust solutions, it gives a smeared solution at the contact discontinuities. The new BGK-type KFVS scheme solves this problem and gives robust and reliable solutions as well as good resolution at the contact discontinuity. In contrast to the classical kinetic theory, we have only a finite domain of dependence due to the structure of the light cone. The scheme is naturally multidimensional and is extended to the two-dimensional case in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction. In the numerical case studies the results obtained from the BGK-type KFVS schemes are compared with the exact Riemann solution, KFVS schemes, upwind schemes, and central schemes.

A BGK-Type Flux-Vector Splitting Scheme for the Ultrarelativistic Euler Equations

https://cronfa.swan.ac.uk/Record/cronfa35652/Download/0035652-04102017103312.pdf

Nano-particle drag prediction at low Reynolds number using a direct Boltzmann-BGK solution approach

3-Dimensional upwind Euler solver using kinetic flux vector splitting method

The usefulness of the 3D Euler code BHEEMA (Boltzmann Hypersonic Euler Equation solver for Missile Aerodynamics) based on KFVS method has been demonstrated for a variety of design and analysis problems arising in vehicle aerodynamics.

S. Sekar

Papers

3-Dimensional upwind Euler solver using kinetic flux vector splitting method

3-D KFVS Euler code “BHEEMA” as aerodynamic design and analysis tool for complex configurations

A 3-D time marching upwind Euler code for handling complex flows