S. M. Deshpande

Kinetic flux vector splitting for Euler equations

This paper is mainly concerned with the development of a class of new upwind methods and a novel treatment of the boundary condition based on the concept of kinetic flux vector splitting (KFVS) for solving inviscid gasdynamic problems. KFVS utilizes the well-known connection that the Euler equations of motion are the moments of the Boltzmann equation whenever the velocity distribution function is a Maxwellian. After presentation of the analysis of the KFVS method in I-D in detail, it is described how KFVS can be performed in a different manner to construct various upwind methods for higher dimensions depending on the situations. Next, it is shown how the KFVS formulation together with the specular reflection model of the kinetic theory of gases at the solid boundary lead to the development of a new treatment of the flow tangency boundary condition which is robust, upwind and conservative and does not require any further assumptions or the use of fictitious grid points. Finally, the present method is tested on a wide variety of problems to demonstrate its capability in obtaining accurate and wiggle-free solutions.

Kinetic theory based new upwind methods for inviscid compressible flows

Two new upwind methods called the Kinetic Numerical Method (KNM) and the Kinetic Flux Vector Splitting (KFVS) method for the solution of the Euler equations have been presented. Both of these methods can be regarded as some suitable moments of an upwind scheme for the solution of the Boltzmann equation provided the distribution function is Maxwellian. This moment-method strategy leads to a unification of the Riemann approach and the pseudo-particle approach used earlier in the development of upwind methods for the Euler equations. A very important aspect of the moment-method strategy is that the new upwind methods satisfy the entropy condition because of the Boltzmann H-Theorem and suggest a possible way of extending the Total Variation Diminishing (TVD) principle within the framework of the H-Theorem. The ability of these methods in obtaining accurate wiggle-free solution is demonstrated by applying them to two test problems.

Minimum error solutions of the Boltzmann equation for shock structure

‘Best’ solutions for the shock-structure problem are obtained by solving the Boltzmann equation for a rigid sphere gas by applying minimum error criteria on the Mott-Smith ansatz. The use of two such criteria minimizing respectively the local and total errors, as well as independent computations of the remaining error, establish the high accuracy of the solutions, although it is shown that the Mott-Smith distribution is not an exact solution of the Boltzmann equation even at infinite Mach number. The minimum local error method is found to be particularly simple and efficient. Adopting the present solutions as the standard of comparison, it is found that the widely used v 2 x -moment solutions can be as much as a third in error, but that results based on Rosens method provide good approximations. Finally, it is shown that if the Maxwell mean free path on the hot side of the shock is chosen as the scaling length, the value of the density-slope shock thickness is relatively insensitive to the intermolecular potential. A comparison is made on this basis of present results with experiment, and very satisfactory quantitative agreement is obtained.

New developments in kinetic schemes

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.

The Boltzmann collision integrals for a combination of Maxwellians

The gain and loss integrals in the Boltzmann equation for a rigid sphere gas are evaluated in closed form for a distribution which can be expressed as a linear combination of Maxwellians. Application to the Mott-Smith bimodal distribution shows that the gain is also bimodal, but the two modes in the gain are less pronounced than in the distribution. Implications of these results for simple collision models in non-equilibrium flow are discussed.

Time Dependent Operator-split and Unsplit Schemes for One Dimensional Premixed Flames

This paper is concerned with a study of an operator split scheme and unsplit scheme for the computation of adiabatic freely propagating one-dimensional premixed flames. The study uses unsteady method for both split and unsplit schemes employing implicit chemistry and explicit diffusion, a combination which is stable and convergent. Solution scheme is not sensitive to the initial starting estimate and provides steady state even with straight line profiles (far from steady state) in small number of time steps. Two systems H2-Air and H2-NO (involving complex nitrogen chemistry) are considered in presentinvestigation. Careful comparison shows that the operator split approach is slightly superior than the unsplit when chemistry becomes complex. Comparison of computational times with those of existing steady and unsteady methods seems to suggest that the method employing implicit-explicit algorithm is very efficient and robust.

Least squares kinetic upwind method on moving grids for unsteady Euler computations

The present paper describes the extension of least squares kinetic upwind method for moving grids (LSKUM-MG). LSKUM is a kinetic theory based upwind Euler solver. LSKUM is a node based solver and can operate on any type of mesh or even on an arbitrary distribution of points. LSKUM-MG also has the capability to work on arbitrary meshes with arbitrary grid velocities. Results are presented for a moving piston problem and flow past an airfoil oscillating in pitch.

Higher Order Accurate Kinetic Flux Vector Splitting Method for Euler Equations

A new upwind method called Kinetic Flux Vector Splitting (KFVS) has been developed for the solution of the Euler equations of gas dynamics. This method is based on the fact that the Euler equations are the moments of the Boltzmann equation when the velocity distribution is a Maxwellian. It is shown that the KFVS is a suitable moment of the Courant-Isaacson-Rees (CIR) scheme applied to the Boltzmann equation and further that it is equivalent to the flux-difference splitting approach. It can also be regarded as a Kinetic Theory based Riemann solver. The KFVS has been combined with the TVD and UNO formalisms and its application to the test case of one-dimensional shock propagation has been shown to yield accurate wiggle-free solution with high resolution.

A New Grid-free Method for Conservation Laws

We present a grid-free or meshless approximation called the Kinetic Meshless Method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann-type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node

Euler Computations on Arbitrary Grids Using LSKUM

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