Gas Distribution Function Research Articles

The study of shallow water flow with bottom topography by high-order compact gas-kinetic scheme on unstructured mesh

A well-balanced compact high-order gas-kinetic scheme (GKS) on unstructured mesh is first developed for solving the shallow water equations with source terms. The distinguishable feature of the finite volume GKS is that based on the gas-kinetic formulation, a time-accurate gas distribution function can be constructed, from which both the fluxes and the flow variables can be explicitly evaluated at the cell interface. As a result, besides the update of cell-averaged conservative variables, the cell-averaged slopes of the flow variables can be updated as well. Equipped with both flow variables and their slopes, a fourth-order compact spatial reconstruction on unstructured mesh can be obtained as the initial condition at the beginning of each time step. For the shallow water flow, in order to preserve the well-balanced property, the advection and the source terms in the flux function have to be balanced properly. The current compact GKS achieves high-order accuracy, keeps the well-balanced property, and has super-robustness in the simulation of bore waves. The scheme is used in the shallow water flow studies, such as dam breaking and bore wave propagation. In addition, the pollution transport, morphodynamics, and bottom friction in the shallow water flow have been included in the scheme. In the end, the water discharge in the Pearl River estuary and the dam-break experiment with movable bed topography have been simulated.

Local non-equilibrium phase density reconstruction with Grad and Chapman-Enskog methods

The study of the non-equilibrium velocity gas distribution function reconstruction was carried out due to extended gas dynamics methods. The bimodal Mott-Smith solution for a plane shock wave was used as a reference distribution function. The shock-wave solutions were considered in the range of Mach numbers from Ma = 2.0 to Ma = 8.0. The reconstruction of the original function was carried out using the Navier-Stokes-Fourier, 13-moment, regularized 13-moment and 26-moment approximating functions.

The baryon distribution function in the expanding universe after the recombination era

In this paper, we derive a specific kinetic transport equation which as a partial differential equation describes the distribution function of cosmic H-gas (i.e. hydrogen atoms). We can infact solve this transport equation and find the gas distribution function as function f(v,t) of the particle velocity v and of the cosmic time t, with the surprising result that the effective temperature of these cosmic particles is not decreasing, but increasing with cosmic time. At the end of the cosmic recombination era, about 400000 years after the Big-Bang, electrons and protons are thought to recombine to neutral cosmic H-atoms. The question poses itself, what happens to this hydrogen (baryon) gas in thermodynamic terms, when it is exposed to the cosmic expansion dynamics of a Robertson-Walker Hubble universe. The result presented here is explained as due to the Hubble-induced velocity drift of the particles in velocity space.

Cosmic Baryon Kinetics Expressed by Its Velocity Moments over Times after the Matter Recombination

As we are going to show here it is not easily understandable how cosmic gases like H-atoms, after the recombination of cosmic matter, do thermodynamically behave under the ongoing Hubble-like expansion of the universe. The question namely is not easy to answer; how cosmic gas atoms do in fact recognize the expansion of cosmic 3- space. Contemporary mainstream cosmology takes for granted that gas atoms do react polytropically or even adiabatically to cosmic volume changes and thus do get more and more tenuous and colder in accordance with gas- and thermo- dynamics. However, one has to face the fact that cosmic gases at the recombination era are already nearly collisionless over scales of 10 AU, and how gases react to cosmic volume changes under such conditions is not a trivial problem. We derive in this article a kinetic transport equation which describes the evolution of the gas distribution function f(t, v) in cosmic time t and velocity space of v. This partial differential equation does not allow for a solution in form a separation of the two variables t and v, but instead we can find solutions for two moments of f(v, t), i.e. the density n(t) and the pressure P(t). Then we show that using kappa-like functions for the cosmic gas we can derive such functions as function of their velocity moments, i.e. as functions of cosmic time. It means we understand the kinetic evolution of the cosmic gas by understanding the evolution in cosmic time of their moments.

Modeling and computation for non-equilibrium gas dynamics: Beyond single relaxation time kinetic models

Many kinetic relaxation models have been proposed for the study of rarefied flows. Based on the single relaxation time model, a discrete velocity method-based unified gas-kinetic scheme (UGKS) has been constructed. The UGKS models the gas dynamics on the discretized space directly on account of accumulating flow evolution from particle transport and collision within a time step. Under the UGKS framework, a unified gas-kinetic wave-particle (UGKWP) method has been further developed for non-equilibrium flow simulation, where the time evolution of the gas distribution function is composed of analytical wave and individual particles. In the highly rarefied regime, the flow evolution is mainly described by the particle transport and collision. Because of the use of single relaxation time for particle collision, there is a noticeable discrepancy between the UGKWP solution and the full Boltzmann or direct simulation Monte Carlo (DSMC) result, such as the temperature distribution inside a shock layer at high Mach numbers. In this Letter, a modification of the particle collision time according to the particle velocity will be implemented in the UGKWP. As a result, the new model greatly improves the performance of the UGKWP in the capturing of non-equilibrium flows. There is an excellent match between UGKWP and DSMC or Boltzmann solution in the highly rarefied regime. In the continuum flow regime, due to the absence of particles, the modification of the particle collision time will not take effect and the UGKWP will get back to the hydrodynamic Navier–Stokes flow solver with correct dissipative coefficients at small cell Knudsen numbers.

A simplified discrete unified gas kinetic scheme for incompressible flow

The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows, which combines the benefits of both the lattice Boltzmann method and UGKS. By the reconstruction of the gas distribution function using particle velocity characteristic lines, the flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with the reconstruction stage on a whole time step instead of a half time step in the original DUGKS. Using the temporal/spatial integral Boltzmann Bhatnagar–Gross–Krook equation, the auxiliary distribution function with the inclusion of the collision effect is adopted. The macroscopic and mesoscopic fluxes of the cell on the next time step are predicted by the reconstruction of the auxiliary distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on the next time step can be updated through its flux, which is a moment of the predicted mesoscopic flux at cell interfaces. The equilibrium distribution function on the next time step can also be updated. The gas distribution function is updated by the FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. The numerical time step is only limited by the Courant–Friedrichs–Lewy condition, and a relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, and microcavity flow cases, are conducted to validate the present scheme. The observed numerical simulation results reasonably agree with the reported results.

Reduced order modeling-based discrete unified gas kinetic scheme for rarefied gas flows

Boltzmann solvers face significant difficulty in simulating rarefied flows at high Knudsen numbers. In this flow regime, the gas distribution function is widely scattered and highly concentrated with a very steep slope in the particle velocity space. In order to capture the feature of such a flow, the Boltzmann solvers such as the Discrete Unified Gas Kinetic Scheme (DUGKS) discretize the particle velocity space with a very fine mesh (many discrete particle velocities) using the Discrete Velocity Method (DVM) due to which the load for computation becomes unendurable. In this paper, a Reduced Order Modeling (ROM) method is used to generate a reduced discrete velocity space for the DUGKS. More specifically, the discrete empirical interpolation method [S. Chaturantabut and D. C. Sorensen, SIAM J. Sci. Comput. 32, 2737–2764 (2010)] is used to select the dominant nodes in the original discrete velocity space to form a reduced discrete velocity space, which represents important dynamical characteristics. In this way, most grid points in the discrete velocity space, which are of negligible importance on the integration, are removed in practical computation, which yields a significant improvement in computational efficiency. The proposed ROM approach is not limited to a specific DVM-based solver. For illustration, in this paper, we developed the Reduced Order Modeling-based Discrete Unified Gas Kinetic Scheme (ROM-DUGKS) by applying the reduced velocity space to the conventional DUGKS. Validations are performed in both low-speed and hypersonic rarefied flows at various Knudsen numbers. The results show that the ROM-DUGKS is much more efficient than the original DUGKS while still maintaining high accuracy. This significant improvement in computational efficiency will unleash the potential of the DVM-based solvers such as the DUGKS for practical applications to rarefied flow problems.

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

As an extension of a fourth-order compact gas kinetic scheme (GKS) on structured mesh [24], this work is about the development of a third-order compact GKS on unstructured mesh for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution, the time-dependent gas distribution function at a cell interface in GKS provides not only the flux function and its time derivative, but also the time accurate flow variables there at the next time level. As a result, besides updating the conservative flow variables inside each control volume through the interface fluxes, the cell averaged first-order spatial derivatives of flow variables can be obtained as well using the updated flow variables at the closed cell interfaces around that cell through the divergence theorem. Therefore, with the cell-averaged flow variables and their first-order spatial derivatives inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of the next time step. Following the reconstruction technique in [64], a new HWENO reconstruction on triangular mesh is designed in the current scheme. Combined with the two-stage temporal discretization and second-order time accurate flux function, a third-order compact scheme on unstructured mesh has been constructed. Accurate solutions can be obtained for both inviscid and viscous flows without sensitive dependence on the quality of triangular mesh. The robustness and accuracy of the scheme have been validated through many cases, including strong shocks in the hypersonic viscous flow and smooth Navier-Stokes solution.

Fluid-thermal modeling of hypersonic vehicles via a gas-kinetic BGK scheme-based integrated algorithm

In our previous paper (doi: 10.1016/j.ijheatmasstransfer.2019.119016), the gas-kinetic BGK scheme was used to simulate both hypersonic flow and structural heat transfer, between which loose coupling was adopted for solving two-dimensional fluid-thermal problems. To model the unitive and continuous fluid-thermal processes more actually, this paper presents a BGK scheme-based integrated algorithm and the research object is extended to three-dimensional hypersonic vehicles. To start with, an appropriate BGK model is constructed for the unified macroscopic governing equations, the effectiveness of which is verified by the Chapman-Enskog expansion analysis. Then following similar procedures in the original scheme, the gas distribution function at each cell interface is derived, from which the numerical fluxes can be evaluated. The dual-time stepping approach, together with an implicit JFNK–BGK method, is applied to efficiently obtain time-accurate solutions. With an effective parallelization strategy that is in line with the integrated algorithm, the computational efficiency is improved. As another aspect which distinguishes from the loosely-coupled methods, the fluid-structure interface conditions are also addressed. For validation, several test cases are investigated, including an individual hypersonic flow, an individual transient heat transfer, and two fluid-thermal problems. By comparison with other methods and analysis of the computed results, features of the developed method are demonstrated.

Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow

The unified gas-kinetic scheme (UGKS) provides a framework for simulating multiscale transport with the updates of both gas distribution function and macroscopic flow variables on the cell size and time step scales. The multiscale dynamics in UGKS is achieved through the coupled particle transport and collision in the particle evolution process within a time step. In this paper, under the UGKS framework, we propose an efficient multiscale unified gas-kinetic wave-particle (UGKWP) method. The gas dynamics in UGKWP method is described by the individual particle movement coupled with the evolution of the probability density function (PDF). During a time step, the trajectories of simulation particles are tracked until collision happens, and the post-collision particles are evolved collectively through the evolution of the corresponding distribution function. The evolution of simulation particles and distribution function is guided by the evolution of macroscopic variables, and guarantees the conservation of the scheme in the final wave-particle formulation. A new concept of multiscale multi-efficiency preserving (MMP) method is introduced. Multiscale preserving means UGKWP method preserves the flow regime from collisionless regime to hydrodynamic regime without requiring the cell size and time step to be less than the mean free path and collision time. Multi-efficiency preserving means the computational cost of the scheme including the computational time and memory cost is on the same level as the most efficient method in the corresponding regime, such as the particle methods in the rarefied regime and hydrodynamic solvers in continuum regime. The UGKWP method is shown to satisfy the MMP requirement. The UGKWP method is specially efficient for hypersonic flow simulation in all regimes in comparison with the discrete ordinate methods, and presents a much lower stochastic noise in the continuum flow regime in comparison with the particle-based Monte Carlo methods. Numerical tests for flows over a wide range of Mach and Knudsen numbers are presented. The examples include the hypersonic flow passing a circular cylinder at Mach numbers 20 and 30 and Knudsen numbers 1 and 10−4, low speed lid-driven cavity flow, laminar boundary layer, shock structure, and shock tube problems. These results validate the accuracy, efficiency, and multiscale and multi-efficiency property of UGKWP method.

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral-like resolution will be presented. Based on the high-order gas evolution model, both the flux function and conservative flow variables in GKS can be evaluated explicitly from the time-accurate gas distribution function at a cell interface. As a result, inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step. The flow variable update and slope update are coming from the same physical solution at the cell interface. This strategy needs time accurate solution at a cell interface, which cannot be achieved by the Riemann problem based flow solvers, even though they can also provide the interface flux functions and interface flow variables. Instead, in order to update the slopes in the Riemann-solver based schemes, such as HWENO, there are additional governing equations for slopes or equivalent degrees of freedom inside each cell. In GKS, only a single time accurate gas evolution model is needed at the cell interface for updating cell averaged flow variables through interface fluxes and updating the cell averaged slopes through the interface flow variables. Based on both cell averaged values and their slopes, compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed. As analyzed in this paper, the local linear compact reconstruction without limiter can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives. For nonlinear gas dynamic evolution, in order to avoid spurious oscillation in discontinuous region, the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction. Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction. In GKS, the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation, which covers a physical process from an initial non-equilibrium state to a final equilibrium one. Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction, and the equilibrium state is basically constructed from the linear symmetric reconstruction, the GKS evolution models unifies the nonlinear and linear reconstructions in a gas relaxation process in the determination of a time-dependent gas distribution function. This property gives GKS great advantages in capturing both discontinuous shock waves and the linear aero-acoustic waves in a single computation due to its dynamical adaptation of non-equilibrium and equilibrium states in different flow regions. This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions. Compared with discontinuous Galerkin (DG) scheme, the current compact GKS uses the same local and compact stencil, achieves the 6th-order and 8th-order accuracy, uses a much larger time step with CFL number ≥ 0.3, has the robustness as a 2nd-order scheme, and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process. The nonlinear reconstruction in the compact GKS is solely based on the WENO-Z technique. At the same time, the current scheme solves the Navier-Stokes equations automatically due to combined inviscid and viscous flux terms from a single time evolution gas distribution function at a cell interface. Due to the use of multi-stage multi-derivative (MSMD) time-stepping technique, for achieving a 4th-order time accuracy, the current scheme uses only two stages instead of four in the traditional Runge-Kutta method. As a result, the current GKS becomes much more efficient than the corresponding same order DG method. A variety of numerical tests are presented to validate the compact 6th and 8th-order GKS. The current scheme presents a state-of-art numerical solutions under a wide range of flow conditions, i.e., strong shock discontinuity, shear instability, aero-acoustic wave propagation, and NS solutions. It promotes the development of high-order scheme to a new level of maturity. The success of the current scheme crucially depends on the high-order gas evolution model, which cannot be achieved by any other approach once the 1st-order Riemann flux function is still used in the development of high-order numerical algorithms.

An extended gas-kinetic scheme for shock structure calculations

In super/hyper-sonic flows in near space and low speed flows around MEMS, the gas distribution function deviates from the equilibrium, because the molecular mean free path is comparable with the characteristic length of the flow in certain part of the flow flied. The non-equilibrium distribution function makes the constitutive relations of stress and heat flux deviate from the linear form used in NS equation. Methods for rarefied flow (DSMC and DVM) and unified methods (UGKS, GKUA and DUGKS) are able to simulate non-equilibrium flows, while, for flows in near continuum regimes, their computational costs are much larger than that of the continuum NS method. For the flows in near space and around MEMS, the rarefaction and non-equilibrium effects are not too strong as these in free molecular flows. Therefore, an Extend Gas-Kinetic Scheme (EGKS) is proposed in the present article for near continuum flows and weak non-equilibrium flows where certain amount of non-equilibrium exists. As an extension of the Gas-Kinetic Scheme (GKS), by using a perturbation of the distribution function that has the maximum entropy under the constrains of conservation variables, stress and heat flux, the nonlinear constitutive relations can be directly introduced into the EGKS. There are many nonlinear constitutive relations so far, whose numerical predictions are conducted in their own frameworks. There is a huge demand for a unified and physical framework in order to investigate their validity and accuracy. Since all these constitutive relations are derived from the gas-kinetic theory, a GKS framework is chosen in consideration of physical accommodation. Burnett, SCB, RCE and NCCR nonlinear constitutive relations are used in the EGKS along with linear Stokes-Fourier relations. Using the shock structure cases in a wide range of Mach number from weak non-equilibrium to high non-equilibrium (Mach number from 1.2 to 20), the EGKS scheme and the nonlinear constitutive relations are verified. The density, temperature, stress and heat flux profiles along with shock thickness and density-temperature separation, are systematically examined. Since the accuracy of the EGKS depends on the model of stress and heat flux used, the EGKS with nonlinear constitutive relations gives much better predictions than that with linear constitutive relations in high Mach number cases. Benefit from the present framework, the RCE and NCCR constitutive relations are also modified using the concept and property of the entropy generation of collision term, the modified constitutive relations provide good predictions in high Mach number shock structure calculations.

A third order gas-kinetic scheme for unstructured grid

In order to study the high order gas-kinetic scheme on unstructured grid, we combine the third order gas-kinetic flux solver with the compact least-square reconstruction (CLS) method together. The CLS method can achieve arbitrary high order compact reconstruction using the stencil from the whole computational domain implicitly. A large sparse linear system resulted from the CLS reconstruction method is solved by applying the generalized minimal residual algorithm (GMRES). To accelerate the convergence, the Reverse-Cuthill–McKee (RCM) algorithm and the incomplete lower–upper (ILU) factorization method are implemented. Different from the traditional flux solver, the gas-kinetic scheme developed by Xu is of high spatial and temporal accuracy. Applying the second order Taylor expansion of the initial gas distribution function and equilibrium state at cell interface, the flux solver can be extended to third order accuracy directly. The accuracy of present method is validated by several numerical cases such as the advection of density perturbation problem, isotropic vortex propagation problem, Sod shock–wave problem, Lax shock tube test case, Shu–Osher problem, shock–vortex interaction, lid-driven cavity flow, and flat plate boundary layer. The advantages of this high order gas-kinetic scheme are exhibited in some benchmarks including incompressible flow and supersonic compressible flow, inviscid flow and viscous flow.

Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions

ABSTRACTFor the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.

High Rate, Selective, and Stable Electroreduction of CO2 to CO in Basic and Neutral Media

The electroreduction of carbon dioxide (CO2) to chemicals such as carbon monoxide (CO) shows great potential for renewable fuel and chemical production. Significant progress in individual performance metrics such as reaction rate, selectivity, and stability has been achieved, yet the simultaneous achievement of each of these key metrics within a single system, and in a wide range of operating conditions, has yet to be demonstrated. Here we report a composite multilayered porous electrode consisting of a polytetrafluoroethylene gas distribution layer, a conformal Ag catalyst, and a carbon current distributor. Separating the gas and current distribution functions provides endurance, and further reconstructing the catalyst to carbonate-derived Ag provides flexibility in terms of electrolyte. The resulting electrodes reduce CO2 to CO with a Faradaic efficiency over 90% at current densities above 150 mA/cm2, in both neutral and alkaline media for over 100 h of operation. This represents an important step towar...

A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier–Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate flow variables at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the cell averaged slopes inside each control volume through the differences of flow variables at the cell interfaces can be updated as well in GKS. So, with the updated flow variables and their slopes inside each cell, the HWENO techniques can be naturally implemented for the compact high-order reconstruction at the beginning of each time step. Therefore, a compact higher-order GKS, such as the two-stage fourth-order compact scheme can be constructed. This fourth-order compact GKS has the same stencil and as robust as the second-order scheme. In comparison with the fourth-order DG method, they have the same stencil. In order to get a fourth-order temporal accuracy, the GKS uses two stages and DG needs four stages in the time stepping methods. The CFL number used in GKS is on the order of 0.5 instead of 0.11 in the DG. This research concludes that beyond the first-order Riemann solver the use of high-order time evolution model at a cell interface is extremely helpful in the design of robust, accurate, and efficient higher-order compact schemes for the compressible flow simulations.

A unified gas-kinetic scheme for axisymmetric flow in all Knudsen number regimes

Based on kinetic model equation, a unified gas-kinetic scheme for axisymmetric flow (UGKS-AS) is proposed in the whole flow regimes. The flux transport of UGKS-AS is based on a reduced gas distribution function of a time integral solution of the full 3D BGK model. In the cylindrical coordinate system, the local time evolution of convection and source terms for the macroscopic and microscopic flow variables is fully coupled with particle free transport and collisions. As a result, a multiple scale UGKS-AS is developed and used for the flow simulation from the free molecular transport to the Navier–Stokes solutions. In comparison with the full 3D UGKS method for axisymmetric flow, the present scheme is much more efficient and gives accurate solution in all flow regimes. Different from operator splitting methods with decoupled treatment of particle transport and collision, for the Navier–Stokes solution in the continuum regime the UGKS-AS doesn't have the constraint on the time step being less than the particle collision time. To further improve the efficiency of the scheme for steady state solution, an implicit UGKS-AS method is also developed. A large number of numerical tests for axisymmetric flow are conducted by the proposed explicit and implicit UGKS-AS, which include inner and outer, steady and unsteady, low and high speed flows. Some tests are very challenging due to the delicate capturing of very small flow variations. The good agreement among the solutions from the present scheme and other analytical, numerical, and experimental studies, validates the high accuracy and efficiency of the UGKS-AS for non-equilibrium flow simulation in all regimes.

A third-order gas-kinetic CPR method for the Euler and Navier–Stokes equations on triangular meshes

A third-order accurate gas-kinetic scheme based on the correction procedure via reconstruction (CPR) framework is developed for the Euler and Navier–Stokes equations on triangular meshes. The scheme combines the accuracy and efficiency of the CPR formulation with the multidimensional characteristics and robustness of the gas-kinetic flux solver. Comparing with high-order finite volume gas-kinetic methods, the current scheme is more compact and efficient by avoiding wide stencils on unstructured meshes. Unlike the traditional CPR method where the inviscid and viscous terms are treated differently, the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly through the kinetic evolution model. In addition, the present scheme adopts a fully coupled spatial and temporal gas distribution function for the flux evaluation, achieving high-order accuracy in both space and time within a single step. Numerical tests with a wide range of flow problems, from nearly incompressible to supersonic flows with strong shocks, for both inviscid and viscous problems, demonstrate the high accuracy and efficiency of the present scheme.

An efficient high-order finite difference gas-kinetic scheme for the Euler and Navier–Stokes equations

Based on the temporal evolution of the Navier-Stokes gas distribution function and Weighted Essential Non-Oscillatory (WENO) interpolation, a high-order finite difference gas-kinetic scheme (FDGKS) is constructed. Different from the previous high-order finite volume gas-kinetic method [Li, Xu, and Fu, J. Comput. Phys. vol. 229, pp. 6715 (2010)], which uses a discontinuous initial reconstruction at the cell interface, the present scheme is a finite-difference one with a continuous flow distribution at the grid point.The time-accurate solution of the gas distribution function permits the FDGKS to be a one-step high-order scheme without multi-step Runge-Kutta temporal matching, which significantly reduces the computational time. Many numerical tests in solving one and two-dimensional Euler and Navier-Stokes equations demonstrate that FDGKS is a highly stable, accurate, and efficient scheme, which captures discontinuities without oscillations.

Unified gas-kinetic scheme with multigrid convergence for rarefied flow study

The unified gas kinetic scheme (UGKS) is based on direct modeling of gas dynamics on the mesh size and time step scales. With the modeling of particle transport and collision in a time-dependent flux function in a finite volume framework, the UGKS can connect the flow physics smoothly from the kinetic particle transport to the hydrodynamic wave propagation. In comparison with the direct simulation Monte Carlo (DSMC) method, the current equation-based UGKS can implement implicit techniques in the updates of macroscopic conservative variables and microscopic distribution functions. The implicit UGKS significantly increases the convergence speed for steady flow computations, especially in the highly rarefied and near continuum regimes. In order to further improve the computational efficiency, for the first time, a geometric multigrid technique is introduced into the implicit UGKS, where the prediction step for the equilibrium state and the evolution step for the distribution function are both treated with multigrid acceleration. More specifically, a full approximate nonlinear system is employed in the prediction step for fast evaluation of the equilibrium state, and a correction linear equation is solved in the evolution step for the update of the gas distribution function. As a result, convergent speed has been greatly improved in all flow regimes from rarefied to the continuum ones. The multigrid implicit UGKS (MIUGKS) is used in the non-equilibrium flow study, which includes microflow, such as lid-driven cavity flow and the flow passing through a finite-length flat plate, and high speed one, such as supersonic flow over a square cylinder. The MIUGKS shows 5–9 times efficiency increase over the previous implicit scheme. For the low speed microflow, the efficiency of MIUGKS is several orders of magnitude higher than the DSMC. Even for the hypersonic flow at Mach number 5 and Knudsen number 0.1, the MIUGKS is still more than 100 times faster than the DSMC method for obtaining a convergent steady state solution.