Computational Analysis of Gas Kinetic Bhatnagar-Grosskrook Scheme for Inviscid Compressible Flow

Many numerical schemes have been developed in the field of computational fluid dynamics to simulate inviscid, compressible flows.Among those most notable and successful are the Godunov-type schemes and flux vector splitting schemes.Besides these numerical schemes, schemes based on the gas kinetic theory have been developed in the past few years.Stemming from this approach, the gas kinetic Bhatnagar-Gross-Krook (BGK) scheme is realized.In this thesis, the BGK scheme based on the BGK model of the approximate Boltzmann equation has been fully analyzed and developed accordingly.The numerical algorithms for the BGK scheme are first developed for simulating one-dimensional flow, and then follow by the-two dimensional flow realms. Higher-order spatial accuracy of the scheme is achieved through the reconstruction of the flow variables via the Monotone Upstream-Centered Schemes for Conservation Laws (MUSCL) approach. For time integration method, an explicit method is adopted for the first-order schemes in both one and two-dimensional flow problems.The classical Runge-kutta multistage method is employed only for schemes with higher-order of accuracy. In addition, an implicit time integration method known as the Approximate Factorization-Alternating Direction Implicit (AF-ADI) would be employed when dealing with two-dimensional flow problems in higher-order.In order to investigate the computational characteristics of the BGK scheme in detail, several cases of shock-shock interaction problem have been numerically analyzed.Developed code for the onedimensional flow is validated with three typical test cases, namely,quasi-onedimensional supersonic-subsonic nozzle flow, shock tube, and two interacting blast waves.Likewise,four typical two-dimensional test cases that are found in the literatures are used to validate the developed code for the two-dimensional flow.They are regular shock reflection,supersonic flow over a wedge, channel with a fifteen-degree ramp, and flow past a cylinder.From these validation cases, computed results are compared with the available exact solutions and with other computational results obtained by using some well known numerical discretization schemes.In comparison,the BGK scheme exhibits the most accurate shock resolution capabilities,least diffusiveness, least oscillatory,and great robustness

A gas-kinetic BGK solver for two-dimensional turbulent compressible flow

In this paper, a gas kinetic solver is developed for the Reynolds Average Navier-Stokes (RANS) equations in two-space dimensions. To our best knowledge, this is the first attempt to extend the application of the BGK (Bhatnagaar-Gross-Krook) scheme to solve RANS equations with a turbulence model using finite difference method. The convection flux terms which appear on the left hand side of the RANS equations are discretized by a semi-discrete finite difference method. Then, the resulting inviscid flux functions are approximated by gas-kinetic BGK scheme which is based on the BGK model of the approximate collisional Boltzmann equation. The cell interface values required by the inviscid flux functions are reconstructed to higher-order spatial accuracy via the MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) variable interpolation method coupled with a minmod limiter. As for the diffusion flux terms, they are discretized by a second-order central difference scheme. To account for the turbulence effect, a combined k-ε / k-ω SST (Shear-Stress Transport) two-equation turbulence model is used in the solver. An explicit-type time integration method known as the modified fourth-order Runge-Kutta method is used to compute steady-state solutions. The computed results for a supersonic flow past a flat plate where the transition is artificially triggered at 50% of plate length are presented in this paper. Validating the computed results against existing analytical solutions and also comparing them with results from other well-known numerical schemes show that a very good agreement is obtained

The Efficiency of Gas-Kinetic BGK scheme for solving 2-d compressible inviscid regular shock reflection problem

In this paper, the 1st and 2nd order gas-kinetic BGK scheme is developed and tested for its ability in solving the two-dimensional compressible inviscid flow fields. The BGK (Bhatnagar-Gross-Krook) scheme uses the collisional Boltzmann equation as the governing equation for flow evolutions. Second-order BGK scheme is also developed for flow simulation. This is achieved by means of reconstructing the initial data via MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) method. In addition, a multisage TVD (Total Variation Diminishing) Runge-Kutta method is employed for the time integration of the finite volume gas-kinetic scheme. A typical two-dimensional regular reflection of an oblique shock wave from a solid surface is chosen for testing the accuracy and robustness of the BGK scheme. The computational results are validated against the numerical results of Roe’s scheme

The accuracy of the gas-kinetic BGK finite difference method for solving 3-D compressible inviscid flows

In this paper, the descriptions on the development of a flow solver for the threedimensional compressible Euler equations are presented. The underlying numerical scheme for the solver was based on the collisional Boltzmann model that produces the gas-kinetic BGK (Bhatnagaar-Gross-Krook) scheme. In constructing the desired algorithm, the convection flux terms were discretized by a semi-discrete finite difference method. The resulting inviscid flux functions were approximated by the gas-kinetic BGK scheme. To achieve higher order spatial accuracy, the cell interface primitive flow variables were reconstructed via the MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) interpolation method coupled with a min-mod limiter. As for advancing the solutions to another time level, an explicit-type time integration method known as the modified fourth-order Runge-Kutta was employed in the current flow solver to compute steady-state solutions. Two numerical cases were used to validate the flow solver where the computed results obtained were compared with available analytical solutions and published results from literature to substantiate the accuracy and robustness of the developed gas-kinetic BGK flow solver

Application of Gas-Kinetic BGK Scheme for Solving 2-D Compressible Inviscid Flow around Linear Turbine Cascade

Fluid flows within turbomachinery tend to be extremely complex. Understanding such flows is crucial in the effort to improve current turbomachinery designs. Hence, computational approaches can be used to great advantage in this regard. In this paper, gas-kinetic BGK (Bhatnagar-Gross-Krook) scheme is developed for simulating compressible inviscid flow around a linear turbine cascade. BGK scheme is an approximate Riemann solver that uses the collisional Boltzmann equation as the governing equation for flow evolutions. For efficient computations, particle distribution functions in the general solution of the BGK model are simplified and used for the flow simulations. Second-order accuracy is achieved via the reconstruction of flow variables using the MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) interpolation technique together with a multistage Runge-Kutta method. A multi-zone H-type mesh for the linear turbine cascades is generated using a structured algebraic grid generation method. Computed results are compared with available experimental data and found to be in agreement with each other. In order to further substantiate the performance of the BGK scheme, another test case, namely a wedge cascade, is used. The numerical solutions obtained via this test are validated against analytical solutions, which showed to be in good agreement