Songze Chen

A unified gas kinetic scheme with moving mesh and velocity space adaptation

There is great difficulty for direct Boltzmann solvers to simulate high Knudsen number flow due to the severe steep slope and high concentration of the gas distribution function in a local particle velocity space. Local mesh adaptation becomes necessary in order to make the Boltzmann solver to be a practical tool in aerospace applications. The present research improves the unified gas-kinetic scheme (UGKS) in the following two aspects. First, the UGKS is extended in a physical space with moving mesh. This technique is important to study a freely flying object in a rarefied environment. Second, the adaptive quadtree method in the particle velocity space is implemented in the UGKS. Due to the new improvements in the discretization of a gas distribution function in the six dimensional phase space, the adaptive unified gas kinetic scheme (AUGKS) is able to deal with a wide range of flow problems under extreme flying conditions, such as the whole unsteady flying process of an object from a highly rarefied to a continuum flow regime. After validating the scheme, the capability of AUGKS is demonstrated in the following two challenge test cases. The first case is about the free movement of an ellipse flying at initial Mach number 5 in a rarefied flow at different Knudsen numbers. The force on the ellipse and the unsteady trajectory of the ellipse movement are fully captured. The gas distribution function around the ellipse is analyzed. The second case is about the study of unsteady flight of a nozzle under a bursting process of the compressed gas expanding into a rarefied environment. Due to the strong expansion wave and the huge density difference between interior and exterior regions around the nozzle, the particle distribution function changes dramatically in the particle velocity space. The use of an adaptive velocity space in the AUGKS becomes necessary to simulate such a flow and to control the computational cost to a tolerable level. The second test is a challenge problem for any existing rarefied flow solver.

A hybrid numerical simulation of isotropic compressible turbulence

A novel hybrid numerical scheme with built-in hyperviscosity has been developed to address the accuracy and numerical instability in numerical simulation of isotropic compressible turbulence in a periodic domain at high turbulent Mach number. The hybrid scheme utilizes a 7th-order WENO (Weighted Essentially Non-Oscillatory) scheme for highly compressive regions (i.e., shocklet regions) and an 8th-order compact central finite difference scheme for smooth regions outside shocklets. A flux-based conservative and formally consistent formulation is developed to optimize the connection between the two schemes at the interface and to achieve a higher computational efficiency. In addition, a novel numerical hyperviscosity formulation is proposed within the context of compact finite difference scheme for the smooth regions to improve numerical stability of the hybrid method. A thorough and insightful analysis of the hyperviscosity formulation in both Fourier space and physical space is presented to show the effectiveness of the formulation in improving numerical stability, without compromising the accuracy of the hybrid method. A conservative implementation of the hyperviscosity formulation is also developed. Combining the analysis and test simulations, we have also developed a criterion to guide the specification of a numerical hyperviscosity coefficient (the only adjustable coefficient in the formulation). A series of test simulations are used to demonstrate the accuracy and numerical stability of the scheme for both decaying and forced compressible turbulence. Preliminary results for a high-resolution simulation at turbulent Mach number of 1.08 are shown. The sensitivity of the simulated flow to the detail of thermal forcing method is also briefly discussed.

A comparative study of an asymptotic preserving scheme and unified gas-kinetic scheme in continuum flow limit

Asymptotic preserving (AP) schemes target to simulate both continuum and rarefied flows. Many existing AP schemes are capable of recovering the Euler limit in the continuum regime. However, to get accurate Navier-Stokes solutions is still challenging. In order to distinguish physical mechanism underlining different AP schemes, an implicit-explicit (IMEX) AP method and the unified gas kinetic scheme (UGKS) are employed to solve the Bhatnagar-Gross-Krook (BGK) kinetic equation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for evaluating numerical performance of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles converge to the classical benchmark solutions in continuum regime with the mesh size being much larger than the local particle mean free path. However, the IMEX AP scheme seems to have difficulty to get these solutions in the corresponding limit. The analysis demonstrates that the dissipation of AP schemes has to be properly controlled in the continuum flow regime through a delicate numerical treatment of collision and convection of the kinetic equation. Physically, it becomes necessary to couple both the convection and collision terms in the flux evaluation in order to recover correct Navier-Stokes limit.

A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (S- model) were constructed to correct the Prandtl number of the original BGK model through the modification of stress and heat flux. With the introduction of a new pa- rameter to combine the ES-model and S-model, a generalized kinetic model can be developed. This new model can give the correct Navier-Stokes equations in the con- tinuum flow regime. Through the adjustment of the new parameter, it provides abun- dant dynamic effect beyond the ES-model and S-model. Changing the free parameter, the physical performance of the new model has been tested numerically. The unified gas kinetic scheme (UGKS) is employed for the study of the new model. In transition flow regime, many physical problems, i.e., the shock structure and micro-flows, have been studied using the generalized model. With a careful choice of the free parameter, good results can be achieved for most test cases. Due to the property of the Boltz- mann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the S- model predicts more accurate numerical solutions in most test cases presented in this paper than the ES-model, while ES-model performs better in the cases where the flow is mostly driven by temperature gradient, such as a channel flow with large boundary temperature variation at high Knudsen number. AMS subject classifications: 65M10, 78A48

The dynamic mechanism of a moving Crookes radiometer

The dynamics of a 2D rotating Crookes radiometer is studied using a moving mesh unified gas kinetic scheme. The whole evolution process of a fan from an initial unsteady start-up to a final steady state rotational movement in a rarefied gas environment is simulated numerically. Through the numerical study, the unsteady force distribution along a vane which dynamically drives the fan movement is captured. And a quantitative connection between total torque and rotational speed of the fan in the Knudsen number regime of 10−3 < Kn < 102 is obtained. Based on the dimensional analysis, the total radiometric torque can be decomposed into a net radiometric driving torque and a rotational resistance. Based on the numerical data, the analytical functions of the torque and angular velocity of a rotating fan in terms of Knudsen number are quantitatively constructed. This relationship is used to explain the experimental observation of the Knudsen number shift for the appearance of the maximum torque and the maximum ro...

Gas-kinetic scheme with discontinuous derivative for low speed flow computation

The present paper concerns the improvement of the gas-kinetic scheme (GKS) for low speed flow computation. In the modified GKS scheme, the flow distributions with discontinuous derivatives are used as an initial condition at the cell interface for the flux evaluation. This discontinuity is determined by considering both the flow characteristic and grids resolution. Compared with GKS method with a continuous slope for the flow variables at a cell interface, the new scheme is more robust and accurate. In the under resolved flow computation, the new scheme presents much less numerical oscillation. The extension of the current scheme to unstructured mesh is straightforward. To validate the method, both computations of 2D lid-driven cavity flow and 3D flow past a sphere are performed. The numerical results validate the current method.

Cartesian grid method for gas kinetic scheme on irregular geometries

A Cartesian grid method combined with a simplified gas kinetic scheme is presented for subsonic and supersonic viscous flow simulation on complex geometries. Under the Cartesian mesh, the boundaries are represented by a set of direction-oriented boundary points, and the computational grid points are classified into four different categories, the fluid point, the solid point, the drop point, and the interpolation point. A constrained weighted least square method is employed to evaluate the physical quantities at the interpolation points. Different boundary conditions, including isothermal boundary, adiabatic boundary, and Euler slip boundary, are presented by different interpolation strategies. We adopt a simplified gas kinetic scheme as the flux solver for both subsonic and supersonic flow computations. The methodology of constructing a simplified kinetic flux function can be extended to other flow systems. A few numerical examples are used to validate the Cartesian grid method and the simplified flux solver. The reconstruction scheme for recovering the boundary conditions of compressible viscous and heat conducting flow with a Cartesian mesh can provide a smooth distribution of physical quantities at solid boundary, and present an accurate solution for the flow study with complex geometry.

A Cartesian Grid-Based Unified Gas Kinetic Scheme

A Cartesian grid-based unified gas kinetic scheme is developed. In this approach, any oriented boundary in a Cartesian grid is represented by many directional boundary points. The numerical flux is evaluated on each boundary point. Then, a boundary flux interpolation method (BFIM) is constructed to distribute the boundary effect to the flow evolution on regular Cartesian grid points. The BFIM provides a general strategy to implement any kind of boundary condition on Cartesian grid. The newly developed technique is implemented in the unified gas kinetic scheme, where the scheme is reformulated into a finite difference format. Several typical test cases are simulated with different geometries. For example, the thermophoresis phenomenon for a plate with infinitesimal thickness immersed in a rarefied flow environment is calculated under different orientations on the same Cartesian grid. These computational results validate the BFIM in the unified scheme for the capturing of different thermal boundary conditio...

Simplification of the unified gas kinetic scheme.

The unified gas kinetic scheme (UGKS) is an asymptotic preserving (AP) scheme for kinetic equations. It is superior for transition flow simulation and has been validated in the past years. However, compared to the well-known discrete ordinate method (DOM), which is a classical numerical method solving the kinetic equations, the UGKS needs more computational resources. In this study, we propose a simplification of the unified gas kinetic scheme. It allows almost identical numerical cost as the DOM, but predicts numerical results as accurate as the UGKS. In the simplified scheme, the numerical flux for the velocity distribution function and the numerical flux for the macroscopic conservative quantities are evaluated separately. The equilibrium part of the UGKS flux is calculated by analytical solution instead of the numerical quadrature in velocity space. The simplification is equivalent to a flux hybridization of the gas kinetic scheme for the Navier-Stokes (NS) equations and the conventional discrete ordinate method. Several simplification strategies are tested, through which we can identify the key ingredient of the Navier-Stokes asymptotic preserving property. Numerical tests show that, as long as the collision effect is built into the macroscopic numerical flux, the numerical scheme is Navier-Stokes asymptotic preserving, regardless the accuracy of the microscopic numerical flux for the velocity distribution function.