Unified Gas Kinetic Scheme Research Articles

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.

Onsager's cross coupling effects in gas flows confined to micro-channels

In rarefied gases, mass and heat transport processes interfere with each other, leading to the mechano-caloric effect and thermo-osmotic effect, which are of interest to both theoretical study and practical applications. We employ the unified gas-kinetic scheme to investigate these cross coupling effects in gas flows in micro-channels. Our numerical simulations cover channels of planar surfaces and also channels of ratchet surfaces, with Onsager's reciprocal relation verified for both cases. For channels of planar surfaces, simulations are performed in a wide range of Knudsen number and our numerical results show good agreement with the literature results. For channels of ratchet surfaces, simulations are performed for both the slip and transition regimes and our numerical results not only confirm the theoretical prediction [Phys. Rev. Lett. 107, 164502 (2011)] for Knudsen number in the slip regime but also show that the off-diagonal kinetic coefficients for cross coupling effects are maximized at a Knudsen number in the transition regime. Finally, a preliminary optimization study is carried out for the geometry of Knudsen pump based on channels of ratchet surfaces.

Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon Boltzmann transport equation

Numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved. In this work a discrete unified gas kinetic scheme (DUGKS) is developed for heat transfer in materials with different acoustic thickness based on the phonon Boltzmann equation. With discrete phonon direction, the Boltzmann equation is discretized with a second-order finite-volume formulation, in which the time-step is fully determined by the Courant–Friedrichs–Lewy (CFL) condition. The scheme has the asymptotic preserving (AP) properties for both diffusive and ballistic regimes, and can present accurate solutions in the whole transition regime as well. The DUGKS is a self-adaptive multiscale method for the capturing of local transport process. Numerical tests for both heat transfers with different Knudsen numbers are presented to validate the current method.

Discrete unified gas kinetic scheme with a force term for incompressible fluid flows

The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) and unified gas kinetic scheme (UGKS), including the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, similar to standard LBE, the DUGKS can also be considered as a compressible scheme, and the compressible effect may bring some undesirable errors when it is used to investigate incompressible fluid flows. To eliminate the compressible effect, in this work a new DUGKS with a force term is developed through modifying the equilibrium distribution function. And simultaneously, the non-equilibrium extrapolation (NEE) scheme is also introduced to treat the velocity and pressure boundary conditions. To illustrate the capacity of the present DUGKS, we first performed some numerical simulations of two-dimensional steady and unsteady flows, and conducted a comparison between the present DUGKS and the original one. The results indicate that the present DUGKS can reduce the compressible effect efficiently, and the NEE scheme is also consistent with the second-order accuracy of DUGKS. We then extended the present DUGKS to study the three-dimensional lid-driven flows (LDF) in cubic and deep cavities, and found that the present results are in good agreement with available benchmark results, which indicates the present DUGKS is also accurate and efficient in the study of three-dimensional problems. At last, the structures of vortex in the cubic and deep cavities are also considered, and the symmetric affiliated vortices aside the secondary vortex at Re≥600 can be observed in the deep LDF.

Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes

This paper presents an implicit unified gas-kinetic scheme (UGKS) for non-equilibrium steady state flow computation. The UGKS is a direct modeling method for flow simulation in all regimes with the updates of both macroscopic flow variables and microscopic gas distribution function. By solving the macroscopic equations implicitly, a predicted equilibrium state can be obtained first through iterations. With the newly predicted equilibrium state, the evolution equation of the gas distribution function and the corresponding collision term can be discretized in a fully implicit way for fast convergence through iterations as well. The lower-upper symmetric Gauss-Seidel (LU-SGS) factorization method is implemented to solve both macroscopic and microscopic equations, which improves the efficiency of the scheme. Since the UGKS is a direct modeling method and its physical solution depends on the mesh resolution and the local time step, a physical time step needs to be fixed before using an implicit iterative technique with a pseudo-time marching step. Therefore, the physical time step in the current implicit scheme is determined by the same way as that in the explicit UGKS for capturing the physical solution in all flow regimes, but the convergence to a steady state speeds up through the adoption of a numerical time step with large CFL number. Many numerical test cases in different flow regimes from low speed to hypersonic ones, such as the Couette flow, cavity flow, and the flow passing over a cylinder, are computed to validate the current implicit method. The overall efficiency of the implicit UGKS can be improved by one or two orders of magnitude in comparison with the explicit one.

A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations

Fluid dynamic equations are valid in their respective modeling scales. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the flow study in this regime. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well.

Discrete unified gas kinetic scheme on unstructured meshes

The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for multiscale flow computations with asymptotic preserving property. The solution of the Boltzmann model equation is directly used for the construction of numerical flux and makes the scheme applicable in all flow regimes. In previous applications of the DUGKS, structured meshes have been mostly employed, which may have difficulties for problems with complex geometries. In this paper we will extend the DUGKS to unstructured meshes, with the implementation of computational fluid dynamics techniques to the DUGKS. Several test cases, i.e., the cavity flow ranging from continuum to free molecular regimes, a multiscale flow problem between two connected cavities with large pressure and density variations, high speed flows past multiple cylinders in slip and transitional regimes, and an impulsive start problem are performed. The results are compared with the well-defined Direct Simulation Monte Carlo (DSMC) or Navier–Stokes (NS) solutions in their applicable regimes. The numerical results demonstrate the effectiveness of the proposed DUGKS for the study of multiscale flow problems.

Numerical simulation of flows from free molecular regime to continuum regime by a DVM with streaming and collision processes

A discrete velocity method (DVM) with streaming and collision processes is presented in this work for simulation of flows from free molecular regime to continuum regime. The present scheme can be considered as a semi-Lagrangian like scheme. At first, we follow the conventional DVM to discretize the phase velocity space by a number of discrete velocities. Then, for each discrete velocity, the kinetic equation with BGK-Shakhov model is integrated in space and time within one time step. As a result, a simple algebraic formulation can be obtained, and its solution can be marched in time by the streaming and collision processes. However, differently from the conventional semi-Lagrangian scheme, the present scheme uses the MUSCL approach with van Albada limiter in the process of reconstructing the distribution function at the surrounding points of the cell center, and the transport distance is controlled in order to avoid extrapolation. This makes the present scheme be capable of simulating the hypersonic rarefied flows. In addition, as compared to the unified gas kinetic scheme (UGKS), the present scheme is simpler and easier for implementation. Thus, the computational efficiency can be improved accordingly. To validate the proposed numerical scheme, test examples from free molecular regime to continuum regime are simulated. Numerical results showed that the present scheme can predict the flow properties accurately even for hypersonic rarefied flows.

STUDY ON IMPLICIT IMPLEMENTATION OF THE UNIFIED GAS KINETIC SCHEME

The current explicit unified gas kinetic scheme (UGKS) is very time-consuming for high speed flows due to the massive phase space mesh demand, which is a bottleneck for complex engineering problems. In order to improve the efficiency, the motion and collision term in the model equation is implicitly treated and the evolving time averaged flux across the cell interface is introduced, then the implicit UGKS can be obtained applying the approximate LU decomposition on the matrix of the governing equations. The tests on the flows over a cylinder with different freestream Mach numbers show that the implicit method can give the same result as the original explicit method with a properly chosen evolving time step. Meanwhile, the computational efficiency can be improved by 1-2 orders.

An asymptotic preserving unified gas kinetic scheme for frequency-dependent radiative transfer equations

This paper presents an extension of previous work (Sun et al., 2015 [22]) of the unified gas kinetic scheme (UGKS) for the gray radiative transfer equations to the frequency-dependent (multi-group) radiative transfer system. Different from the gray radiative transfer equations, where the optical opacity is only a function of local material temperature, the simulation of frequency-dependent radiative transfer is associated with additional difficulties from the frequency-dependent opacity. For the multiple frequency radiation, the opacity depends on both the spatial location and the frequency. For example, the opacity is typically a decreasing function of frequency. At the same spatial region the transport physics can be optically thick for the low frequency photons, and optically thin for high frequency ones. Therefore, the optical thickness is not a simple function of space location. In this paper, the UGKS for frequency-dependent radiative system is developed. The UGKS is a finite volume method and the transport physics is modeled according to the ratio of the cell size to the photon's frequency-dependent mean free path. When the cell size is much larger than the photon's mean free path, a diffusion solution for such a frequency radiation will be obtained. On the other hand, when the cell size is much smaller than the photon's mean free path, a free transport mechanism will be recovered. In the regime between the above two limits, with the variation of the ratio between the local cell size and photon's mean free path, the UGKS provides a smooth transition in the physical and frequency space to capture the corresponding transport physics accurately. The seemingly straightforward extension of the UGKS from the gray to multiple frequency radiation system is due to its intrinsic consistent multiple scale transport modeling, but it still involves lots of work to properly discretize the multiple groups in order to design an asymptotic preserving (AP) scheme in all regimes. The current scheme is tested in a few frequency-dependent radiation problems, and the results are compared with the solutions from the well-defined implicit Monte Carlo (IMC) method. The UGKS is much more efficient than IMC, and the computational times of both schemes for all test cases are listed. The UGKS seems to be the first discrete ordinate method (DOM) for the accurate capturing of multiple frequency radiative transport physics from ballistic particle motion to the diffusive wave propagation.

A coupled discrete unified gas-kinetic scheme for Boussinesq flows

Recently, the discrete unified gas-kinetic scheme (DUGKS) [Guo et al., (2013) [6]] based on the Boltzmann equation is developed as a new multiscale kinetic method for isothermal flows. In this paper, a thermal and coupled discrete unified gas-kinetic scheme is derived for the Boussinesq flows, where the velocity and temperature fields are described independently. Kinetic boundary conditions for both velocity and temperature fields are also proposed. The proposed model is validated by simulating several canonical test cases, including the porous plate problem, the Rayleigh–Bénard convection, and the natural convection with Rayleigh number up to 1010 in a square cavity. The results show that the coupled DUGKS is of second order accuracy in space and can well describe the convection phenomena from laminar to turbulent flows. Particularly, it is found that this new scheme has better numerical stability in simulating high Rayleigh number flows compared with the previous kinetic models.

Unified Gas Kinetic Scheme and Direct Simulation Monte Carlo Computations of High-Speed Lid-Driven Microcavity Flows

AbstractAccurate simulations of high-speed rarefied flows present many physical and computational challenges. Toward this end, the present work extends the Unified Gas Kinetic Scheme (UGKS) to a wider range of Mach and Knudsen numbers by implementing WENO (Weighted Essentially Non-Oscillatory) interpolation. Then the UGKS is employed to simulate the canonical problem of lid-driven cavity flow at high speeds. Direct Simulation Monte Carlo (DSMC) computations are also performed when appropriate for comparison. The effect of aspect ratio, Knudsen number and Mach number on cavity flow physics is examined leading to important insight.

Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case

This paper is a continuation of our work on the development of multiscale numerical scheme from low-speed isothermal flow to compressible flows at high Mach numbers. In our earlier work [Z. L. Guo et al., Phys. Rev. E 88, 033305 (2013)], a discrete unified gas kinetic scheme (DUGKS) was developed for low-speed flows in which the Mach number is small so that the flow is nearly incompressible. In the current work, we extend the scheme to compressible flows with the inclusion of thermal effect and shock discontinuity based on the gas kinetic Shakhov model. This method is an explicit finite-volume scheme with the coupling of particle transport and collision in the flux evaluation at a cell interface. As a result, the time step of the method is not limited by the particle collision time. With the variation of the ratio between the time step and particle collision time, the scheme is an asymptotic preserving (AP) method, where both the Chapman-Enskog expansion for the Navier-Stokes solution in the continuum regime and the free transport mechanism in the rarefied limit can be precisely recovered with a second-order accuracy in both space and time. The DUGKS is an idealized multiscale method for all Knudsen number flow simulations. A number of numerical tests, including the shock structure problem, the Sod tube problem in a whole range of degree of rarefaction, and the two-dimensional Riemann problem in both continuum and rarefied regimes, are performed to validate the scheme. Comparisons with the results of direct simulation Monte Carlo (DSMC) and other benchmark data demonstrate that the DUGKS is a reliable and efficient method for multiscale flow problems.

A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

AbstractThe Ellipsoidal Statistical model (ES-model) and the Shakhov model (Smodel) 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 parameter 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 continuum flow regime. Through the adjustment of the new parameter, it provides abundant 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 Boltzmann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the Smodel 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.

A Comparative Study of LBE and DUGKS Methods for Nearly Incompressible Flows

AbstractThe lattice Boltzmann equation (LBE) methods (both LBGK and MRT) and the discrete unified gas-kinetic scheme (DUGKS) are both derived from the Boltzmann equation, but with different consideration in their algorithm construction. With the same numerical discretization in the particle velocity space, the distinctive modeling of these methods in the update of gas distribution function may introduce differences in the computational results. In order to quantitatively evaluate the performance of these methods in terms of accuracy, stability, and efficiency, in this paper we test LBGK, MRT, and DUGKS in two-dimensional cavity flow and the flow over a square cylinder, respectively. The results for both cases are validated against benchmark solutions. The numerical comparison shows that, with sufficient mesh resolution, the LBE and DUGKS methods yield qualitatively similar results in both test cases. With identical mesh resolutions in both physical and particle velocity space, the LBE methods are more efficient than the DUGKS due to the additional particle collision modeling in DUGKS. But, the DUGKS is more robust and accurate than the LBE methods in most test conditions. Particularly, for the unsteady flow over a square cylinder at Reynolds number 300, with the same mesh resolution it is surprisingly observed that the DUGKS can capture the physical multi-frequency vortex shedding phenomena while the LBGK and MRT fail to get that. Furthermore, the DUGKS is a finite volume method and its computational efficiency can be much improved when a non-uniform mesh in the physical space is adopted. The comparison in this paper clearly demonstrates the progressive improvement of the lattice Boltzmann methods from LBGK, to MRT, up to the current DUGKS, along with the inclusion of more reliable physical process in their algorithm development. Besides presenting the Navier-Stokes solution, the DUGKS can capture the rarefied flow phenomena as well with the increasing of Knudsen number.

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.

An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations

The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon's mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction between radiation and material. In the optical thin limit, the photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the gray radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [14], and is an extension of a recent work [12] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional gray radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon's mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach.

Unified gas-kinetic scheme for microchannel and nanochannel flows

A Unified Gas-Kinetic Scheme (UGKS) is devised for micro and nano channel flow simulations. The principle underlying the scheme is a partial-differential-equation-based modeling in which the gas evolution is modeled with the help of kinetic equation, instead of solving the partial differential equation directly. Elastic collision and inelastic collision have both been taken into consideration in the presented scheme where the inelastic collision is modeled by a Landau–Teller–Jeans-type relaxation term. In this study, two dimensional channel flows are selected as our test cases. First, a thermal conduction case with a wide range of Knudsen number is carried out using the present scheme with different kinetic model equations in order to show the validity of the present scheme and to compare the accuracies of different model equations. Then, a series of pressure driven micro and nano channel flows whose working medium are nitrogen gas are used to validate the proposed scheme. In these cases, the results in both the stream direction and the radial direction agree well with the results predicted by the direct simulation Monte Carlo method, and UGKS results are smoother and without statistical fluctuation. When the density of working gas is high, the non-ideal effect should be taken into consideration. So, at the end of this paper, the presented scheme is also modified using the Enskog theory for non-ideal gas. Based on the gas evolution modeling process which automatically covers the whole scale mechanisms, the cell size and time step of UGKS are not limited by the molecular free path and collision time. Due to the advantages of cell size, time step, and accuracy (no statistical fluctuation), the UGKS is expected to be a practical method for the simulation of micro and nano channel flows.

Direct modeling-based computational fluid dynamics

A new principle for constructing CFD scheme for multiscale and multiphysics flow computation is proposed. The principle is about the direct modeling in the discrete space and the use of cross-scale time evolution solution in the construction of numerical method. In comparison with traditional CFD method, where a direct discretization is used for the partial differential equation (PDE), the new numerical methodology takes into account the mesh size effect and its associated evolution solution in the mesh size scale. As a result, with the variation of mesh size with respect to the local particle mean free path, a continuous spectrum of gas evolution model from the kinetic to the hydrodynamic scales can be recovered. For many multiscale flow problems, the governing equations may not be available, but the direct modeling can be still used to simulate different scale gas evolution process. The cross-scale time evolution solution from the model Boltzmann equation plays a central role in the direct modeling. Based on the direct modeling principle, the unified gas-kinetic scheme for the entire Knudsen number flow has been constructed. The excellent performance of the scheme validates the feasibility and advantage of the present CFD principle.

Unified gas-kinetic simulation of slider air bearing

The unified gas-kinetic scheme (UGKS) is presented and used in this letter to study the slider air bearing problem. The UGKS solutions are first validated by comparison with direct simulation Monte Carlo results. After validation, the UGKS is used to study the air-bearing problem under different non-equilibrium conditions. On the surface of the slider, the dependency of the gas pressure and normal force on the Mach and Knudsen numbers are fully evaluated. The non-equilibrium effect on the force loading in the whole transition regime up to the free molecular limit is also studied.