Compressible flows exhibit a diverse set of behaviors, where individual particle transports and their collective dynamics play different roles at different scales. At the same time, the atmosphere is composed of diffe...Compressible flows exhibit a diverse set of behaviors, where individual particle transports and their collective dynamics play different roles at different scales. At the same time, the atmosphere is composed of different components that require additional degrees of freedom for representation in computational fluid dynamics. It is challenging to construct an accurate and efficient numerical algorithm to faithfully represent multiscale flow physics across different regimes. In this paper, a unified gas-kinetic scheme(UGKS) is developed to study non-equilibrium multicomponent gaseous flows. Based on the Boltzmann kinetic equation, an analytical space-time evolving solution is used to construct the discretized equations of gas dynamics directly according to cell size and scales of time steps, i.e., the so-called direct modeling method. With the variation in the ratio of the numerical time step to the local particle collision time(or the cell size to the local particle mean free path), the UGKS automatically recovers all scale-dependent flows over the given domain and provides a continuous spectrum of the gas dynamics. The performance of the proposed unified scheme is fully validated through numerical experiments.The UGKS can be a valuable tool to study multiscale and multicomponent flow physics.展开更多
Sound wave propagation in rarefied monatomic gases is simulated using a newly developed unified gaskinetic scheme （UGKS）. The numerical calculations are carfled out for a wide range of wave oscillating frequencies. ...Sound wave propagation in rarefied monatomic gases is simulated using a newly developed unified gaskinetic scheme （UGKS）. The numerical calculations are carfled 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 os- cillating wall boundary condition and the methods for eval- uating 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 sim- ulation 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 cur- rent study clearly demonstrates the capability of the UGKS method in capturing the sound wave propagation and its usefulness for the rarefied flow study.展开更多
Aim.The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term.We propose using gas-kinetic BGK(Bhatnagar-Gross-Krook) method,which is based on the more fundamental Boltzmann equation,in ...Aim.The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term.We propose using gas-kinetic BGK(Bhatnagar-Gross-Krook) method,which is based on the more fundamental Boltzmann equation,in order to obviate the use of dissipation term and obtain,we believe,an improved solution.Section 1 deals essentially with three things:(1) as analytical solution of molecular probability density function at the cell interface has been obtained by the Boltzmann equation with BGK model,we can compute the flux term by integrating the density function in the phase space;eqs.(8) and(11) require careful attention;(2) the integrations can be expressed as the moments of Maxwellian distribution with different limits according to the analytical solution;eqs.(9) and(10) require careful attention;(3) the discrete equation by finite volume method can be solved using the time marching method.Computations are performed by the BGK method for the Sod′s shock tube problem and a two-dimensional shock reflection problem.The results are compared with those of the conventional JST scheme in Figs.1 and 2.The BGK method provides better resolution of shock waves and other features of the flow fields.展开更多
Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics ...Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.展开更多
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 val- idated by comparison with direct simulation Monte Carlo results. ...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 val- idated by comparison with direct simulation Monte Carlo results. After valida- tion, 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.展开更多
A coupled gas-kinetic scheme(GKS)and unified gas-kinetic wave-particle(UGKWP)method for the disperse dilute gas-particle multiphaseflow is proposed.In the two-phaseflow,the gas phase is always in the hydrodynamic regi...A coupled gas-kinetic scheme(GKS)and unified gas-kinetic wave-particle(UGKWP)method for the disperse dilute gas-particle multiphaseflow is proposed.In the two-phaseflow,the gas phase is always in the hydrodynamic regime and is fol-lowed by GKS for the Navier-Stokes solution.The particle phase is solved by UGKWP in all regimes from particle trajectory crossing to the hydrodynamic wave interac-tion with the variation of particle’s Knudsen number.In the intensive particle colli-sion regime,the UGKWP gives a hydrodynamic wave representation for the particle phase and the GKS-UGKWP for the two-phaseflow reduces to the two-fluid Eulerian-Eulerian(EE)model.In the rarefied regime,the UGKWP tracks individual particle and the GKS-UGKWP goes back to the Eulerian-Lagrangian(EL)formulation.In the tran-sition regime for the solid particle,the GKS-UGKWP takes an optimal choice for the wave and particle decomposition for the solid particle phase and connects the EE and EL methods seamlessly.The GKS-UGKWP method will be tested in allflow regimes with a large variation of Knudsen number for the solid particle transport and Stokes number for the two-phase interaction.It is confirmed that GKS-UGKWP is an efficient and accurate multiscale method for the gas-particle two-phaseflow.展开更多
This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity dis...This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation,which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function.Their convergence is derived through mathematical treatment.The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method.These two expressions agree with those used in the unified algorithm.Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.展开更多
This paper concerns the development of high-order multidimensional gas kinetic schemes for the Navier-Stokes solutions.In the current approach,the state-of-the-art WENO-type initial reconstruction and the gas-kinetic ...This paper concerns the development of high-order multidimensional gas kinetic schemes for the Navier-Stokes solutions.In the current approach,the state-of-the-art WENO-type initial reconstruction and the gas-kinetic evolution model are used in the construction of the scheme.In order to distinguish the physical and numerical requirements to recover a physical solution in a discretized space,two particle collision times will be used in the current high-order gas-kinetic scheme(GKS).Different from the low order gas dynamic model of the Riemann solution in the Godunov type schemes,the current method is based on a high-order multidimensional gas evolution model,where the space and time variation of a gas distribution function along a cell interface from an initial piecewise discontinuous polynomial is fully used in the flux evaluation.The high-order flux function becomes a unification of the upwind and central difference schemes.The current study demonstrates that both the high-order initial reconstruction and high-order gas evolution model are important in the design of a high-order numerical scheme.Especially,for a compact method,the use of a high-order local evolution solution in both space and time may become even more important,because a short stencil and local low order dynamic evolution model,i.e.,the Riemann solution,are contradictory,where valid mechanism for the update of additional degrees of freedom becomes limited.展开更多
How to solve the hypersonic aerothermodynamics around large-scale uncontrolled spacecraft during falling disintegrated process from outer space to earth,is the key to resolve the problems of the uncontrolled Tiangong-...How to solve the hypersonic aerothermodynamics around large-scale uncontrolled spacecraft during falling disintegrated process from outer space to earth,is the key to resolve the problems of the uncontrolled Tiangong-No.1 spacecraft reentry crash.To study aerodynamics of spacecraft reentry covering various flow regimes,a Gas-Kinetic Unified Algorithm(GKUA)has been presented by computable modeling of the collision integral of the Boltzmann equation over tens of years.On this basis,the rotational and vibrational energy modes are considered as the independent variables of the gas molecular velocity distribution function,a kind of Boltzmann model equation involving in internal energy excitation is presented by decomposing the collision term of the Boltzmann equation into elastic and inelastic collision terms.Then,the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions by developing the discrete velocity ordinate method and numerical quadrature technique.The unified algorithm of the Boltzmann model equation involving thermodynamics non-equilibrium effect is presented for the whole range of flow regimes.The gas-kinetic massive parallel computing strategy is developed to solve the hypersonic aerothermodynamics with the processor cores 500~45,000 at least 80%parallel efficiency.To validate the accuracy of the GKUA,the hypersonic flows are simulated including the reentry Tiangong-1 spacecraft shape with the wide range of Knudsen numbers of 220~0.00005 by the comparison of the related results from the DSMC and N-S coupled methods,and the low-density tunnel experiment etc.For uncontrolling spacecraft falling problem,the finite-element algorithm for dynamic thermalforce coupling response is presented,and the unified simulation of the thermal structural response and the hypersonic flow field is tested on the Tiangong-1 shape under reentry aerodynamic environment.Then,the forecasting analysis platform of end-of-life largescale spacecraft flying track is established on the basis of ballistic computation combined with reentry aerothermodynamics and deformation failure/disintegration.展开更多
In this paper,we extend the unified gas-kinetic wave-particle(UGKWP)methods to the multi-species gas mixture and multiscale plasma transport.The construction of the scheme is based on the direct modeling on the mesh s...In this paper,we extend the unified gas-kinetic wave-particle(UGKWP)methods to the multi-species gas mixture and multiscale plasma transport.The construction of the scheme is based on the direct modeling on the mesh size and time step scales,and the local cell’s Knudsen number determines the flow physics.The proposed scheme has the multiscale and asymptotic complexity diminishing properties.The multiscale property means that according to the cell’s Knudsen number the scheme can capture the non-equilibrium flow physics when the cell size is on the kinetic mean free path scale,and preserve the asymptotic Euler,Navier-Stokes,and magnetohydrodynamics(MHD)when the cell size is on the hydrodynamic scale and is much larger than the particle mean free path.The asymptotic complexity diminishing property means that the total degrees of freedom of the scheme reduce automatically with the decreasing of the cell’s Knudsen number.In the continuum regime,the scheme automatically degenerates from a kinetic solver to a hydrodynamic solver.In the UGKWP,the evolution of microscopic velocity distribution is coupled with the evolution of macroscopic variables,and the particle evolution as well as the macroscopic fluxes is modeled from a time accumulating solution of kinetic scale particle transport and collision up to a time step scale.For plasma transport,the current scheme provides a smooth transition from particle-in-cell(PIC)method in the rarefied regime to the magnetohydrodynamic solver in the continuum regime.In the continuum limit,the cell size and time step of the UGKWP method are not restricted by the particle mean free path and mean collision time.In the highly magnetized regime,the cell size and time step are not restricted by the Debye length and plasma cyclotron period.The multiscale and asymptotic complexity diminishing properties of the scheme are verified by numerical tests in multiple flow regimes.展开更多
The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as th...The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past a circular cylinder and a NACA 002 airfoil to verify the reliability of the present high-order algorithm and simulate gas transport phenomena covering various flow regimes. The computed results are found in good agreement both with the theoretical prediction from continuum to rarefied gas dynamics, the related DSMC solutions, and with the experimental results. The numerical effect of the schemes with the different precision and the different types of Boltzmann collision models on the computational efficiency and computed results is investigated and analyzed. The numerical experience indicates that an approach developing and applying the gas-kinetic high-order algorithm is feasible for directly solving the Boltzmann model equation.展开更多
In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike resolution will be presented.Based on the high-order gas evolution model,both the flux function and conservative flow variables ...In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike 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.展开更多
A truly three-dimensional(3D)gas-kinetic flux solver for simulation of incompressible and compressible viscous flows is presented in this work.By local reconstruction of continuous Boltzmann equation,the inviscid and ...A truly three-dimensional(3D)gas-kinetic flux solver for simulation of incompressible and compressible viscous flows is presented in this work.By local reconstruction of continuous Boltzmann equation,the inviscid and viscous fluxes across the cell interface are evaluated simultaneously in the solver.Different from conventional gaskinetic scheme,in the present work,the distribution function at cell interface is computed in a straightforward way.As an extension of our previous work(Sun et al.,Journal of Computational Physics,300(2015)492–519),the non-equilibrium distribution function is calculated by the difference of equilibrium distribution functions between the cell interface and its surrounding points.As a result,the distribution function at cell interface can be simply calculated and the formulations for computing the conservative flow variables and fluxes can be given explicitly.To validate the proposed flux solver,several incompressible and compressible viscous flows are simulated.Numerical results show that the current scheme can provide accurate numerical results for three-dimensional incompressible and compressible viscous flows.展开更多
The development of high-order schemes has been mostly concentrated on the limiters and high-order reconstruction techniques.In this paper,the effect of the flux functions on the performance of high-order schemes will ...The development of high-order schemes has been mostly concentrated on the limiters and high-order reconstruction techniques.In this paper,the effect of the flux functions on the performance of high-order schemes will be studied.Based on the same WENO reconstruction,two schemes with different flux functions,i.e.,the fifthorderWENO method and the WENO-Gas-kinetic scheme(WENO-GKS),will be compared.The fifth-order finite difference WENO-SW scheme is a characteristic variable reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms,the sixth-order central difference for viscous terms,and three stages Runge-Kutta time stepping for the time integration.On the other hand,the finite volume WENO-GKS is a conservative variable reconstruction based method with the same WENO reconstruction.But,it evaluates a time dependent gas distribution function along a cell interface,and updates the flow variables inside each control volume by integrating the flux function along the boundary of the control volume in both space and time.In order to validate the robustness and accuracy of the schemes,both methods are tested under a wide range of flow conditions:vortex propagation,Mach 3 step problem,and the cavity flow at Reynolds number 3200.Our study shows that both WENO-SW and WENO-GKS yield quantitatively similar results and agree with each other very well provided a sufficient grid resolution is used.With the reduction of mesh points,the WENO-GKS behaves to have less numerical dissipation and present more accurate solutions than those from the WENO-SW in all test cases.For the Navier-Stokes equations,since theWENO-GKS couples inviscid and viscous terms in a single flux evaluation,and the WENO-SW uses an operator splitting technique,it appears that theWENO-SWismore sensitive to theWENO reconstruction and boundary treatment.In terms of efficiency,the finite volume WENO-GKS is about 4 times slower than the finite differenceWENO-SW in two dimensional simulations.The current study clearly shows that besides high-order reconstruction,an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions.In a physical flow,the transport,stress deformation,heat conduction,and viscous heating are all coupled in a single gas evolution process.Therefore,it is preferred to develop such a scheme with multi-dimensionality,and unified treatment of inviscid and dissipative terms.A high-order scheme does prefer a high-order gas evolution model.Even with the rapid advances of high-order reconstruction techniques,the first-order dynamics of the Riemann solution becomes the bottleneck for the further development of high-order schemes.In order to avoid the weakness of the low order flux function,the development of high-order schemes relies heavily on the weak solution of the original governing equations for the update of additional degree of freedom,such as the non-conservative gradients of flow variables,which cannot be physically valid in discontinuous regions.展开更多
The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstr...The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstruction method at a cell interface.The iteration method is used to determine the velocity of fluid interface,based on the force balance between both sides due to the incidence and bounce back of particles at the interface.An improved Aitken method is proposed with a simple hybrid of the modified Aitken method(Aitken-Chen)and the Steffensen method.Numerical tests validate its efficiency with significantly less calls to the function not only for the average number but also for the maximum.The new reconstruction is based on the tangent of hyperbola for interface capturing(THINC)but applied only to the volume fraction,which is very simple to be implemented under the stratified frame-work and capable of resolving fluid interface in mixture.Furthermore,the directional splitting is adopted rather than the previous quasi-one-dimensional method.Typical numerical tests,including several watergas shock tube flows,and the shock-water cylinder interaction flow show that the improved gas-kinetic scheme can capture fluid interfaces much sharper,while preserving the advantages of the original one.展开更多
Many configurations in plasma physics are axisymmetric,it will be more convenient to depict them in cylindrical coordinates compared with Cartesian coordinates.In this paper,a gas-kinetic scheme for collisional Vlasov...Many configurations in plasma physics are axisymmetric,it will be more convenient to depict them in cylindrical coordinates compared with Cartesian coordinates.In this paper,a gas-kinetic scheme for collisional Vlasov-Poisson equations in cylindrical coordinates is proposed,our algorithm is based on Strang splitting.The equation is divided into two parts,one is the kinetic transport-collision part solved by multiscale gas-kinetic scheme,and the other is the acceleration part solved by a Runge-Kutta solver.The asymptotic preserving property of whole algorithm is proved and it’s applied on the study of charge separation problem in plasma edge and 1D Z-pinch configuration.Numerical results show it can capture the process fromnon-equilibrium to equilibrium state by Coulomb collisions,and numerical accuracy is obtained.展开更多
A discrete unified gas kinetic scheme(DUGKS)is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame(AAP)kinetic model.Although the species collision operator in the AAP model conserv...A discrete unified gas kinetic scheme(DUGKS)is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame(AAP)kinetic model.Although the species collision operator in the AAP model conserves fully the mass,momentum,and energy for the mixture,it does not conserve the momentum and energy for each species due to the inter-species collisions.In this work,the species collision operator is decomposed into two parts:one part is fully conservative for the species and the other represents the excess part.With this decomposition,the kinetic equation is solved using the Strang-splitting method,in which the excess part of the collision operator is treated as a source,while the kinetic equation with the species conservative part is solved by the standard DUGKS.Particularly,the time integration of the source term is realized by either explicit or implicit Euler scheme.By this means,it is easy to extend the scheme to gas mixtures composed of Maxwell or hard-sphere molecules,while the previous DUGKS[Zhang Y,Zhu L,Wang R et al,Phys Rev E 97(5):053306,2018]of binary gases was only designed for Maxwell molecules.Several tests are performed to validate the scheme,including the shock structure under different Mach numbers and molar concentrations,the Couette flow under different mass ratios,and the pressure-driven Poiseuille flow in different flow regimes.The results are compared with those from other reliable numerical methods based on different models.And the influence of molecular model on the flow characteristics is studied.The results also show that the present DUGKS with implicit source discretization is more stable and preferable for gas mixture problems involving different flow regimes.展开更多
This paper focuses on the development and application of a threedimensional gas-kinetic Bhatnagar-Gross-Krook(BGK)method for the viscous flows in rotating machinery.For such flows,a rotating frame of reference is usua...This paper focuses on the development and application of a threedimensional gas-kinetic Bhatnagar-Gross-Krook(BGK)method for the viscous flows in rotating machinery.For such flows,a rotating frame of reference is usually used in formulating the Navier-Stokes(N-S)equations,and there are two major concerns in constructing the corresponding BGK model.One is the change of the convective velocities in the N-S equations,which can be reflected through modification of the gas streaming velocity.The other one is the necessity to account for the effect of the additional Coriolis and centrifugal forces.Here,a specifically-designed acceleration term is added into the modified Boltzmann equation so that the source effects can be naturally included into the gas evolution process and the resulted fluxes.Under the finitevolume framework,the constructed BGK model is locally solved at each cell interface and then the numerical fluxes can be evaluated.When employing the BGK scheme,it is sometimes found that the calculated spatial derivatives of the initial and equilibrium distribution functions are sensitive to the mesh quality especially in complex rotating flow applications,which may significantly influence flux evaluation.Therefore,an improved approach for computing these slopes is adopted,through which the modeling capability for viscous flows is enhanced.For validation,several numerical examples are presented.The computed results show that the present method can be well applied to a wide range of flows in rotating machinery with favorable accuracy.展开更多
The paper introduces the gas-kinetic scheme for three-dimensional(3D)flow simulation.First,under a unified coordinate transformation,the 3D gaskinetic BGK equation is transformed into a computational space with arbitr...The paper introduces the gas-kinetic scheme for three-dimensional(3D)flow simulation.First,under a unified coordinate transformation,the 3D gaskinetic BGK equation is transformed into a computational space with arbitrary mesh moving velocity.Second,based on the Chapman-Enskog expansion of the kinetic equation,a local solution of gas distribution function is constructed and used in a finite volume scheme.As a result,a Navier-Stokes flow solver is developed for the low speed flow computation with dynamical mesh movement.Several test cases are used to validate the 3D gas-kinetic method.The first example is a 3D cavity flow with up-moving boundary at Reynolds number 3200,where the periodic solutions are compared with the experimental measurements.Then,the flow evolution inside a rotating 3D cavity is simulated with the moving mesh method,where the solution differences between 2D and 3D simulation are explicitly presented.Finally,the scheme is applied to the falling plate study,where the unsteady plate tumbling motion inside water tank has been studied and compared with the experimental measurements.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11772281,91530319,and 11521091)the Hong Kong Research Grant Council(Nos.16207715 and 16206617)
文摘Compressible flows exhibit a diverse set of behaviors, where individual particle transports and their collective dynamics play different roles at different scales. At the same time, the atmosphere is composed of different components that require additional degrees of freedom for representation in computational fluid dynamics. It is challenging to construct an accurate and efficient numerical algorithm to faithfully represent multiscale flow physics across different regimes. In this paper, a unified gas-kinetic scheme(UGKS) is developed to study non-equilibrium multicomponent gaseous flows. Based on the Boltzmann kinetic equation, an analytical space-time evolving solution is used to construct the discretized equations of gas dynamics directly according to cell size and scales of time steps, i.e., the so-called direct modeling method. With the variation in the ratio of the numerical time step to the local particle collision time(or the cell size to the local particle mean free path), the UGKS automatically recovers all scale-dependent flows over the given domain and provides a continuous spectrum of the gas dynamics. The performance of the proposed unified scheme is fully validated through numerical experiments.The UGKS can be a valuable tool to study multiscale and multicomponent flow physics.
基金supported by Hong Kong Research Grant Council(621709,621011)HKUST grants SRFI11SC05 and RPC10SC11the Nanoscience and Nanotechnology Program at HKUST
文摘Sound wave propagation in rarefied monatomic gases is simulated using a newly developed unified gaskinetic scheme （UGKS）. The numerical calculations are carfled 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 os- cillating wall boundary condition and the methods for eval- uating 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 sim- ulation 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 cur- rent study clearly demonstrates the capability of the UGKS method in capturing the sound wave propagation and its usefulness for the rarefied flow study.
文摘Aim.The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term.We propose using gas-kinetic BGK(Bhatnagar-Gross-Krook) method,which is based on the more fundamental Boltzmann equation,in order to obviate the use of dissipation term and obtain,we believe,an improved solution.Section 1 deals essentially with three things:(1) as analytical solution of molecular probability density function at the cell interface has been obtained by the Boltzmann equation with BGK model,we can compute the flux term by integrating the density function in the phase space;eqs.(8) and(11) require careful attention;(2) the integrations can be expressed as the moments of Maxwellian distribution with different limits according to the analytical solution;eqs.(9) and(10) require careful attention;(3) the discrete equation by finite volume method can be solved using the time marching method.Computations are performed by the BGK method for the Sod′s shock tube problem and a two-dimensional shock reflection problem.The results are compared with those of the conventional JST scheme in Figs.1 and 2.The BGK method provides better resolution of shock waves and other features of the flow fields.
基金supported by the National Natural Science Foundation of China (No.10621062)the Research Fund for Next Generation of General Armament Department (No.9140A13050207KG29)
文摘Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.
基金supported by Hong Kong Research Grant Council(621011 and 620813)HKUST(SRFI11SC05 and FSGRF13SC21)
文摘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 val- idated by comparison with direct simulation Monte Carlo results. After valida- tion, 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.
基金supported by National Numerical Windtunnel project,National Science Foundation of China(11772281,91852114,12172316)Hong Kong research grant council 16208021Department of Science and Technology of Guangdong Province(Grant No.2020B1212030001).
文摘A coupled gas-kinetic scheme(GKS)and unified gas-kinetic wave-particle(UGKWP)method for the disperse dilute gas-particle multiphaseflow is proposed.In the two-phaseflow,the gas phase is always in the hydrodynamic regime and is fol-lowed by GKS for the Navier-Stokes solution.The particle phase is solved by UGKWP in all regimes from particle trajectory crossing to the hydrodynamic wave interac-tion with the variation of particle’s Knudsen number.In the intensive particle colli-sion regime,the UGKWP gives a hydrodynamic wave representation for the particle phase and the GKS-UGKWP for the two-phaseflow reduces to the two-fluid Eulerian-Eulerian(EE)model.In the rarefied regime,the UGKWP tracks individual particle and the GKS-UGKWP goes back to the Eulerian-Lagrangian(EL)formulation.In the tran-sition regime for the solid particle,the GKS-UGKWP takes an optimal choice for the wave and particle decomposition for the solid particle phase and connects the EE and EL methods seamlessly.The GKS-UGKWP method will be tested in allflow regimes with a large variation of Knudsen number for the solid particle transport and Stokes number for the two-phase interaction.It is confirmed that GKS-UGKWP is an efficient and accurate multiscale method for the gas-particle two-phaseflow.
基金supported by the National Natural Science Foundation of China (Grant Nos. 91016027 and 91130018)
文摘This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation,which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function.Their convergence is derived through mathematical treatment.The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method.These two expressions agree with those used in the unified algorithm.Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.
基金supported by Hong Kong Research Grant Council(Grant No.621011)HKUST research fund(Grant No.SRFI11SC05)
文摘This paper concerns the development of high-order multidimensional gas kinetic schemes for the Navier-Stokes solutions.In the current approach,the state-of-the-art WENO-type initial reconstruction and the gas-kinetic evolution model are used in the construction of the scheme.In order to distinguish the physical and numerical requirements to recover a physical solution in a discretized space,two particle collision times will be used in the current high-order gas-kinetic scheme(GKS).Different from the low order gas dynamic model of the Riemann solution in the Godunov type schemes,the current method is based on a high-order multidimensional gas evolution model,where the space and time variation of a gas distribution function along a cell interface from an initial piecewise discontinuous polynomial is fully used in the flux evaluation.The high-order flux function becomes a unification of the upwind and central difference schemes.The current study demonstrates that both the high-order initial reconstruction and high-order gas evolution model are important in the design of a high-order numerical scheme.Especially,for a compact method,the use of a high-order local evolution solution in both space and time may become even more important,because a short stencil and local low order dynamic evolution model,i.e.,the Riemann solution,are contradictory,where valid mechanism for the update of additional degrees of freedom becomes limited.
基金The National Key Basic Research and Development Program(2014CB744100)and the National Natural Science Foundation of China(91530319 and 11325212)support the present researches in the design of the study and collection,analysis,and interpretation of data and in writing the manuscript.
文摘How to solve the hypersonic aerothermodynamics around large-scale uncontrolled spacecraft during falling disintegrated process from outer space to earth,is the key to resolve the problems of the uncontrolled Tiangong-No.1 spacecraft reentry crash.To study aerodynamics of spacecraft reentry covering various flow regimes,a Gas-Kinetic Unified Algorithm(GKUA)has been presented by computable modeling of the collision integral of the Boltzmann equation over tens of years.On this basis,the rotational and vibrational energy modes are considered as the independent variables of the gas molecular velocity distribution function,a kind of Boltzmann model equation involving in internal energy excitation is presented by decomposing the collision term of the Boltzmann equation into elastic and inelastic collision terms.Then,the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions by developing the discrete velocity ordinate method and numerical quadrature technique.The unified algorithm of the Boltzmann model equation involving thermodynamics non-equilibrium effect is presented for the whole range of flow regimes.The gas-kinetic massive parallel computing strategy is developed to solve the hypersonic aerothermodynamics with the processor cores 500~45,000 at least 80%parallel efficiency.To validate the accuracy of the GKUA,the hypersonic flows are simulated including the reentry Tiangong-1 spacecraft shape with the wide range of Knudsen numbers of 220~0.00005 by the comparison of the related results from the DSMC and N-S coupled methods,and the low-density tunnel experiment etc.For uncontrolling spacecraft falling problem,the finite-element algorithm for dynamic thermalforce coupling response is presented,and the unified simulation of the thermal structural response and the hypersonic flow field is tested on the Tiangong-1 shape under reentry aerodynamic environment.Then,the forecasting analysis platform of end-of-life largescale spacecraft flying track is established on the basis of ballistic computation combined with reentry aerothermodynamics and deformation failure/disintegration.
基金supported by National Numerical Windtunnel project and National Science Foundation of China 11772281,91852114.
文摘In this paper,we extend the unified gas-kinetic wave-particle(UGKWP)methods to the multi-species gas mixture and multiscale plasma transport.The construction of the scheme is based on the direct modeling on the mesh size and time step scales,and the local cell’s Knudsen number determines the flow physics.The proposed scheme has the multiscale and asymptotic complexity diminishing properties.The multiscale property means that according to the cell’s Knudsen number the scheme can capture the non-equilibrium flow physics when the cell size is on the kinetic mean free path scale,and preserve the asymptotic Euler,Navier-Stokes,and magnetohydrodynamics(MHD)when the cell size is on the hydrodynamic scale and is much larger than the particle mean free path.The asymptotic complexity diminishing property means that the total degrees of freedom of the scheme reduce automatically with the decreasing of the cell’s Knudsen number.In the continuum regime,the scheme automatically degenerates from a kinetic solver to a hydrodynamic solver.In the UGKWP,the evolution of microscopic velocity distribution is coupled with the evolution of macroscopic variables,and the particle evolution as well as the macroscopic fluxes is modeled from a time accumulating solution of kinetic scale particle transport and collision up to a time step scale.For plasma transport,the current scheme provides a smooth transition from particle-in-cell(PIC)method in the rarefied regime to the magnetohydrodynamic solver in the continuum regime.In the continuum limit,the cell size and time step of the UGKWP method are not restricted by the particle mean free path and mean collision time.In the highly magnetized regime,the cell size and time step are not restricted by the Debye length and plasma cyclotron period.The multiscale and asymptotic complexity diminishing properties of the scheme are verified by numerical tests in multiple flow regimes.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10621062 and 91016027)
文摘The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past a circular cylinder and a NACA 002 airfoil to verify the reliability of the present high-order algorithm and simulate gas transport phenomena covering various flow regimes. The computed results are found in good agreement both with the theoretical prediction from continuum to rarefied gas dynamics, the related DSMC solutions, and with the experimental results. The numerical effect of the schemes with the different precision and the different types of Boltzmann collision models on the computational efficiency and computed results is investigated and analyzed. The numerical experience indicates that an approach developing and applying the gas-kinetic high-order algorithm is feasible for directly solving the Boltzmann model equation.
基金The current research is supported by Hong Kong research grant council(16206617)and National Science Foundation of China(11772281,91852114,11701038).
文摘In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike 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.
基金National Natural Science Foundation of China(Grant Nos.11772157 and 11832012).
文摘A truly three-dimensional(3D)gas-kinetic flux solver for simulation of incompressible and compressible viscous flows is presented in this work.By local reconstruction of continuous Boltzmann equation,the inviscid and viscous fluxes across the cell interface are evaluated simultaneously in the solver.Different from conventional gaskinetic scheme,in the present work,the distribution function at cell interface is computed in a straightforward way.As an extension of our previous work(Sun et al.,Journal of Computational Physics,300(2015)492–519),the non-equilibrium distribution function is calculated by the difference of equilibrium distribution functions between the cell interface and its surrounding points.As a result,the distribution function at cell interface can be simply calculated and the formulations for computing the conservative flow variables and fluxes can be given explicitly.To validate the proposed flux solver,several incompressible and compressible viscous flows are simulated.Numerical results show that the current scheme can provide accurate numerical results for three-dimensional incompressible and compressible viscous flows.
基金supported by Hong Kong Research Grant Council(621709,621011)National Natural Science Foundation of China(Project No.10928205).
文摘The development of high-order schemes has been mostly concentrated on the limiters and high-order reconstruction techniques.In this paper,the effect of the flux functions on the performance of high-order schemes will be studied.Based on the same WENO reconstruction,two schemes with different flux functions,i.e.,the fifthorderWENO method and the WENO-Gas-kinetic scheme(WENO-GKS),will be compared.The fifth-order finite difference WENO-SW scheme is a characteristic variable reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms,the sixth-order central difference for viscous terms,and three stages Runge-Kutta time stepping for the time integration.On the other hand,the finite volume WENO-GKS is a conservative variable reconstruction based method with the same WENO reconstruction.But,it evaluates a time dependent gas distribution function along a cell interface,and updates the flow variables inside each control volume by integrating the flux function along the boundary of the control volume in both space and time.In order to validate the robustness and accuracy of the schemes,both methods are tested under a wide range of flow conditions:vortex propagation,Mach 3 step problem,and the cavity flow at Reynolds number 3200.Our study shows that both WENO-SW and WENO-GKS yield quantitatively similar results and agree with each other very well provided a sufficient grid resolution is used.With the reduction of mesh points,the WENO-GKS behaves to have less numerical dissipation and present more accurate solutions than those from the WENO-SW in all test cases.For the Navier-Stokes equations,since theWENO-GKS couples inviscid and viscous terms in a single flux evaluation,and the WENO-SW uses an operator splitting technique,it appears that theWENO-SWismore sensitive to theWENO reconstruction and boundary treatment.In terms of efficiency,the finite volume WENO-GKS is about 4 times slower than the finite differenceWENO-SW in two dimensional simulations.The current study clearly shows that besides high-order reconstruction,an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions.In a physical flow,the transport,stress deformation,heat conduction,and viscous heating are all coupled in a single gas evolution process.Therefore,it is preferred to develop such a scheme with multi-dimensionality,and unified treatment of inviscid and dissipative terms.A high-order scheme does prefer a high-order gas evolution model.Even with the rapid advances of high-order reconstruction techniques,the first-order dynamics of the Riemann solution becomes the bottleneck for the further development of high-order schemes.In order to avoid the weakness of the low order flux function,the development of high-order schemes relies heavily on the weak solution of the original governing equations for the update of additional degree of freedom,such as the non-conservative gradients of flow variables,which cannot be physically valid in discontinuous regions.
基金supported by Science Challenge Project(TZ2016001)National Natural Science Foundation of China(U1430235)Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund(the second phase).
文摘The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstruction method at a cell interface.The iteration method is used to determine the velocity of fluid interface,based on the force balance between both sides due to the incidence and bounce back of particles at the interface.An improved Aitken method is proposed with a simple hybrid of the modified Aitken method(Aitken-Chen)and the Steffensen method.Numerical tests validate its efficiency with significantly less calls to the function not only for the average number but also for the maximum.The new reconstruction is based on the tangent of hyperbola for interface capturing(THINC)but applied only to the volume fraction,which is very simple to be implemented under the stratified frame-work and capable of resolving fluid interface in mixture.Furthermore,the directional splitting is adopted rather than the previous quasi-one-dimensional method.Typical numerical tests,including several watergas shock tube flows,and the shock-water cylinder interaction flow show that the improved gas-kinetic scheme can capture fluid interfaces much sharper,while preserving the advantages of the original one.
基金partially supported by Science Challenge project TZ2016002,NSFC(Nos.11871113,11171154,11671050,11771055,11771053)3D numerical simulation platform TB14-1 of China academy of engineering physics.
文摘Many configurations in plasma physics are axisymmetric,it will be more convenient to depict them in cylindrical coordinates compared with Cartesian coordinates.In this paper,a gas-kinetic scheme for collisional Vlasov-Poisson equations in cylindrical coordinates is proposed,our algorithm is based on Strang splitting.The equation is divided into two parts,one is the kinetic transport-collision part solved by multiscale gas-kinetic scheme,and the other is the acceleration part solved by a Runge-Kutta solver.The asymptotic preserving property of whole algorithm is proved and it’s applied on the study of charge separation problem in plasma edge and 1D Z-pinch configuration.Numerical results show it can capture the process fromnon-equilibrium to equilibrium state by Coulomb collisions,and numerical accuracy is obtained.
基金the National Natural Science Foundation of China(Grant Nos.11872024,and 12002131)the China Postdoctoral Science Foundation(Grant No.2020M672347).
文摘A discrete unified gas kinetic scheme(DUGKS)is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame(AAP)kinetic model.Although the species collision operator in the AAP model conserves fully the mass,momentum,and energy for the mixture,it does not conserve the momentum and energy for each species due to the inter-species collisions.In this work,the species collision operator is decomposed into two parts:one part is fully conservative for the species and the other represents the excess part.With this decomposition,the kinetic equation is solved using the Strang-splitting method,in which the excess part of the collision operator is treated as a source,while the kinetic equation with the species conservative part is solved by the standard DUGKS.Particularly,the time integration of the source term is realized by either explicit or implicit Euler scheme.By this means,it is easy to extend the scheme to gas mixtures composed of Maxwell or hard-sphere molecules,while the previous DUGKS[Zhang Y,Zhu L,Wang R et al,Phys Rev E 97(5):053306,2018]of binary gases was only designed for Maxwell molecules.Several tests are performed to validate the scheme,including the shock structure under different Mach numbers and molar concentrations,the Couette flow under different mass ratios,and the pressure-driven Poiseuille flow in different flow regimes.The results are compared with those from other reliable numerical methods based on different models.And the influence of molecular model on the flow characteristics is studied.The results also show that the present DUGKS with implicit source discretization is more stable and preferable for gas mixture problems involving different flow regimes.
基金This work has been supported by the National Natural Science Foundation of China(Grant No.11372135)the National Basic Research Program of China(“973”Project)(Grant No.2014CB046200).
文摘This paper focuses on the development and application of a threedimensional gas-kinetic Bhatnagar-Gross-Krook(BGK)method for the viscous flows in rotating machinery.For such flows,a rotating frame of reference is usually used in formulating the Navier-Stokes(N-S)equations,and there are two major concerns in constructing the corresponding BGK model.One is the change of the convective velocities in the N-S equations,which can be reflected through modification of the gas streaming velocity.The other one is the necessity to account for the effect of the additional Coriolis and centrifugal forces.Here,a specifically-designed acceleration term is added into the modified Boltzmann equation so that the source effects can be naturally included into the gas evolution process and the resulted fluxes.Under the finitevolume framework,the constructed BGK model is locally solved at each cell interface and then the numerical fluxes can be evaluated.When employing the BGK scheme,it is sometimes found that the calculated spatial derivatives of the initial and equilibrium distribution functions are sensitive to the mesh quality especially in complex rotating flow applications,which may significantly influence flux evaluation.Therefore,an improved approach for computing these slopes is adopted,through which the modeling capability for viscous flows is enhanced.For validation,several numerical examples are presented.The computed results show that the present method can be well applied to a wide range of flows in rotating machinery with favorable accuracy.
基金supported by grants from the National Natural Science Foundation of China(Project No.10772033)K.Xu was supported by Hong Kong Research Grant Council 621709.
文摘The paper introduces the gas-kinetic scheme for three-dimensional(3D)flow simulation.First,under a unified coordinate transformation,the 3D gaskinetic BGK equation is transformed into a computational space with arbitrary mesh moving velocity.Second,based on the Chapman-Enskog expansion of the kinetic equation,a local solution of gas distribution function is constructed and used in a finite volume scheme.As a result,a Navier-Stokes flow solver is developed for the low speed flow computation with dynamical mesh movement.Several test cases are used to validate the 3D gas-kinetic method.The first example is a 3D cavity flow with up-moving boundary at Reynolds number 3200,where the periodic solutions are compared with the experimental measurements.Then,the flow evolution inside a rotating 3D cavity is simulated with the moving mesh method,where the solution differences between 2D and 3D simulation are explicitly presented.Finally,the scheme is applied to the falling plate study,where the unsteady plate tumbling motion inside water tank has been studied and compared with the experimental measurements.
