Unified Gas Kinetic Scheme Research Articles

Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes

A unified gas-kinetic scheme (UGKS) is constructed for both continuum and rarefied flow computations. The underlying principle for the development of UGKS is the direct modeling for the gas evolution process from the kinetic to the hydrodynamic scale, which is used in the flux construction across a cell interface. More specifically, the physical process from the kinetic particle free transport to the hydrodynamic pressure wave propagation is recovered in the flux function. In the previous study, the UGKS has been developed mainly for monatomic gas with particle translational motion only. The construction of time evolution solution is based on the BGK, Shakhov, and ES–BGK models. The UGKS has been validated through extensive numerical tests. In this paper, a UGKS for diatomic gas will be constructed, where the gas-kinetic Rykov model with a Landau–Teller–Jeans-type rotational energy relaxation is used in the numerical scheme. The new scheme will be tested in many cases, such as homogeneous flow relaxation, shock structure calculations, hypersonic flow passing a flat plate, and the flow around a blunt circular cylinder. The analytic, DSMC, and experimental measurements will be used for validating the solutions of UGKS.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows III: Microflow Simulations

AbstractDue to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But, the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wall surface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.

Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case

Based on the Boltzmann-BGK (Bhatnagar-Gross-Krook) equation, in this paper a discrete unified gas kinetic scheme (DUGKS) is developed for low-speed isothermal flows. The DUGKS is a finite-volume scheme with the discretization of particle velocity space. After the introduction of two auxiliary distribution functions with the inclusion of collision effect, the DUGKS becomes a fully explicit scheme for the update of distribution function. Furthermore, the scheme is an asymptotic preserving method, where the time step is only determined by the Courant-Friedricks-Lewy condition in the continuum limit. Numerical results demonstrate that accurate solutions in both continuum and rarefied flow regimes can be obtained from the current DUGKS. The comparison between the DUGKS and the well-defined lattice Boltzmann equation method (D2Q9) is presented as well.

On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models

The unified gas kinetic scheme (UGKS) of K. Xu et al. (2010) [37], originally developed for multiscale gas dynamics problems, is applied in this paper to a linear kinetic model of radiative transfer theory. While such problems exhibit purely diffusive behavior in the optically thick (or small Knudsen) regime, we prove that UGKS is still asymptotic preserving (AP) in this regime, but for the free transport regime as well. Moreover, this scheme is modified to include a time implicit discretization of the limit diffusion equation, and to correctly capture the solution in case of boundary layers. Contrary to many AP schemes, this method is based on a standard finite volume approach, it does neither use any decomposition of the solution, nor staggered grids. Several numerical tests demonstrate the properties of the scheme.

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 rotational speed in the transitional flow regime.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases

AbstractWith discretized particle velocity space, a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case [J. Comput. Phys., vol. 229 (2010), pp. 7747-7764]. For the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is not so straightforward. The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional, rather than axially symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented. Instead of particle-based modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC) method, the philosophical principal underlying the current study is a partial-differential-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different, the gas evolution in a discretized space is modeled with the help of kinetic equation, instead of directly solving the partial differential equation. Due to the use of both hydrodynamic and kinetic scales flow physics in a gas evolution model at the cell interface, the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and Navier-Stokes flow solvers, the current method is much more efficient than DSMC in low speed transition and continuum flow regimes, and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes. As a result, the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single computation. This paper will extensively evaluate the performance of the unified scheme from free molecule to continuum NS solutions, and from low speed micro-flow to high speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations, and the scheme provides an important tool in the study of non-equilibrium flow.

The study of sound wave propagation in rarefied gases using unified gas-kinetic scheme

Sound wave propagation in rarefied monatomic gases is simulated using a newly developed unified gaskinetic scheme (UGKS). The numerical calculations are carried out for a wide range of wave oscillating frequencies. The corresponding rarefaction parameter is defined as the ratio of sound wave frequency to the intermolecular particle collision frequency. The simulation covers the flow regime from the continuum to free molecule one. The treatment of the oscillating wall boundary condition and the methods for evaluating the absorption coefficient and sound wave speed are presented in detail. The simulation results from the UGKS are compared to the Navier-Stokes solutions, the direct simulation Monte Carlo (DSMC) simulation, and experimental measurements. Good agreement with the experimental data has been obtained in the whole flow regimes for the corresponding Knudsen number from 0.08 to 32. The current study clearly demonstrates the capability of the UGKS method in capturing the sound wave propagation and its usefulness for the rarefied flow study.

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.

An improved unified gas-kinetic scheme and the study of shock structures

With discretized particle velocity space, a unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the Bhatnagar–Gross–Krook (BGK) model (2010. J. Comput. Phys., 229, 7747–7764). In comparison with many existing kinetic schemes for the Boltzmann equation, besides accurate capturing of non-equilibrium flows, the unified method has no difficulty to get accurate solution in the continuum flow regime as well, such as the solution of the Navier–Stokes equations. More importantly, in the continuum flow regime, the time step used by the unified scheme is determined by the Courant-Friedrichs-Lewy condition, which can be many orders larger than the particle collision time. In some sense, the unified method overcomes the time step barrier for many kinetic methods, such as direct simulation Monte Carlo (DSMC), direct Boltzmann solver and many other kinetic solvers. The unified scheme is a multiscale method, where the macroscopic flow variables and microscopic gas distribution function are updated simultaneously. In an early approach in the unified BGK (U-BGK) scheme, the heat flux in the BGK model is modified through the update of macroscopic flow variables, then this modification feeds back into the update of non-equilibrium gas distribution function. In this paper, we are going to develop a unified scheme for the Shakhov model, the so-called unified Shakhov (U-Shk), where the heat flux is corrected directly through the modification of gas distribution function. Theoretically, it will be shown that current U-Shk is more consistent than the U-BGK for the highly non-equilibrium flow computations. The study of shock structures from low to high Mach numbers will be presented and the simulation results will be compared with DSMC solutions as well as possible experimental measurements. The result improvement of U-Shk over U-BGK is clearly achieved. Based on the simulation results, now we fully believe that the unified scheme is an accurate and efficient flow solver in all Knudsen number flow regime.

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.

On the Construction and Comparison of Difference Schemes

On the Construction and Comparison of Difference Schemes