Discrete Unified Gas Kinetic Scheme Research Articles

Electron–phonon coupling and non-equilibrium thermal conduction in ultra-fast heating systems

A discrete unified gas kinetic scheme is developed to solve the Boltzmann transport equation (BTE) with coupled electron and phonon thermal transport, whose time step is not limited by the relaxation time. The key of the present scheme is that the electron/phonon advection, scattering and electron–phonon interactions are coupled together within one time step by solving the BTE again at the cell interface. Numerical results show that the present scheme can correctly predict the electron–phonon coupling effects, and is in agreement with typical two-temperature model and experimental results in existing literatures and our performed time-domain thermoreflectance technique. It can also capture the ballistic effects when the characteristic length is comparable to or smaller than the mean free path where the two-temperature model fails. In addition, two anomalous heat conduction phenomena are predicted by the present scheme, namely, heat flows from phonon to electron in transient thermal grating geometry and re-rise reflected signal appears in Au/Pt bilayer metal structure. The present work can provide theoretical guidance for the study of electron–phonon coupling and offer a useful tool for multi-scale thermal management.

DUGKS-GPU: An efficient parallel GPU code for 3D turbulent flow simulations using Discrete Unified Gas Kinetic Scheme

This paper presents a parallel implementation of the Discrete Unified Gas Kinetic Scheme (DUGKS) on the GPU system using the CUDA Fortran and CUDA C++ programming languages. Firstly, we conducted an extensive revision of our original CPU-based code, resulting in a threefold decrease in memory usage. This new implementation is also paired with a novel approach to compute cell face flux using trilinear interpolation. It is shown analytically that the interpolation-based approach to flux calculation is more accurate compared to the one used in the original DUGKS. The initial simulation results using this new approach suggest that trilinear interpolation can reduce numerical errors on a coarse mesh. For example, in the case of the decaying Taylor-Green vortex flow at a 1283 mesh resolution, the relative numerical error in the energy dissipation rate at t⁎=2, using the spectral simulation result as the benchmark, is approximately 30% lower than that of the original implementation. The improved GPU DUGKS method is applied to laminar and turbulent flows in periodic and wall-bounded boundary configurations. A performance comparison of the GPU implementation is also presented and compared to the previous CPU implementation. A maximum speedup of 7.64x was achieved on a desktop-level GPU compared to a 32-core CPU. The strong scaling test, conducted on an eight-GPU node, demonstrated the efficient utilization of available multiple GPU resources by the code. Program summaryProgram Title: DUGKS-GPUCPC Library link to program files:https://doi.org/10.17632/yykv5s9g2n.1Developer's repository link:https://github.com/kairzhan/DUGKS-GPUCode Ocean capsule:https://codeocean.com/capsule/3b1e4f74-cdd3-4781-8923-8514d5923dfb/Licensing provisions: GNU General Public License 3Programming language: CUDA C++, CUDA FortranSupplementary material:Nature of problem: DUGKS-GPU is an advanced numerical code designed to accelerate the simulation of turbulent fluid flows through the application of the kinetic method known as the Discrete Unified Gas Kinetic Scheme (DUGKS). This computational tool comprises a CUDA Fortran version, tailored for a single GPU, as well as a CUDA C++ version developed for multi-GPU platforms.Solution method: The code employs a second-order finite-volume discretization of the discrete velocity Boltzmann equation with the BGK collision model. It can be used to simulate small to medium-scale problems on modern multi-GPU platforms with CUDA architecture.

Thermal rarefied gas flow simulations with moving boundaries based on discrete unified gas kinetic scheme and immersed boundary method

Rarefied gas flows in complex geometries with heat transfer and moving boundaries have received remarkable attention due to the prevailing of micro/nano science and engineering applications. This study presents an immersed boundary (IB)-based discrete unified gas kinetic scheme (DUGKS) for the simulation of complex moving boundary thermal rarefied flows. The fluid and temperature fields are solved according to the DUGKS with Shakhov collision model (SDUGKS), which is capable of modeling non-unit Prandtl number problems in all Knudsen regimes. The IB approach, featured by its superior performance to incorporate complicated boundary constraints on Cartesian grids, is applied here to handling the interaction between the micro gas and curved boundaries. Specifically, for the IB adopted in this study, we use the least square method for the reconstruction of the distribution functions near the boundary. The accuracy of the IB-SDUGKS is explored comprehensively in six benchmark problems, including the micro lid-driven cavity flow, thermal equilibrium gas in a circular domain, thermal transpiration in an enclosure, moving shuttle in a rarefied gas, movement of a piston with pressure differences and particle free motion in a lid-driven cavity. The present results agree well with the corresponding analytical and direct simulation Monte Carlo (DSMC) solutions, illustrating the versatility of the present IB-SDUGKS complex thermal rarefied flows.

An efficient discrete unified gas-kinetic scheme for compressible thermal flows

In this paper, an efficient discrete unified gas-kinetic scheme (DUGKS) is developed for compressible thermal flows based on the total energy kinetic model for natural convection with a large relative temperature difference. A double distribution function model is designed with the second distribution representing the total energy. This efficient DUGKS enables the simulation of compressible thermal flows, governed by the compressible Navier–Stokes–Fourier system, using only a seventh-order, off-lattice Gauss–Hermite quadrature (GHQ) D3V27A7 combined with a fifth-order GHQ D3V13A5. The external force is included by truncated Hermite expansions. Based on the Chapman–Enskog approximation and Hermite projection, we propose a systematic approach to derive the discrete kinetic boundary conditions for the density and total energy distribution functions. The discrete kinetic boundary treatments are provided for the no-slip boundary condition, Dirichlet boundary condition and Neumann boundary condition. To validate our scheme, we perform simulations of steady natural convection (Ra=103−106) in two- and three-dimensional cavities with differentially heated sidewalls and a large temperature difference (ε=0.6), where the Oberbeck–Boussinesq approximation is invalid. The results demonstrate that the current efficient DUGKS is robust and accurate for thermal compressible flow simulations. With the D3V27A7 and D3V13A5 off-lattice discrete particle velocity model, the computational efficiency of the DUGKS is improved by a factor of 3.09 when compared to the previous partial energy kinetic model requiring the ninth-order Gauss–Hermite quadrature.

An immersed boundary method‐discrete unified gas kinetic scheme simulation of particle‐laden turbulent channel flow on a nonuniform orthogonal mesh

AbstractParticle‐resolved simulations of turbulent particle‐laden flows provide a powerful research tool to explore detailed flow physics at all scales. However, efficient particle‐resolved simulations for wall‐bounded turbulent particle‐laden flows remain a challenging task. In this article, we develop a simulation approach for a turbulent channel flow laden with finite‐size particles on a nonuniform mesh by combining the discrete unified gas kinetic scheme (DUGKS) and the immersed boundary method (IBM). The standard discrete delta function was modified according to reproducible kernel particle method to take into account mesh non‐uniformity and correctly conserve force moments. Simulation results based on uniform and nonuniform meshes are compared to validate and examine the accuracy of the nonuniform mesh DUGKS‐IBM. Finally, the computational performance of the nonuniform mesh DUGKS‐IBM is discussed.

Well-balanced kinetic schemes for two-phase flows

At equilibrium state of a two-phase fluid system, the chemical potential is constant and the velocity vanishes. However, such equilibrium state usually cannot be captured by numerical methods due to discretization errors. Consequently, inconsistent thermodynamic interfacial properties due to non-constant chemical potential, and spurious velocities due to discrete force imbalance, are frequently encountered in numerical simulations. Therefore, numerical schemes with well-balanced properties, which can capture the equilibrium state at discrete level, are preferred for simulating two-phase flows. In this paper, we report the progress of developing well-balanced mesoscopic numerical methods based on kinetic theory, focusing on the lattice Boltzmann equation (LBE) method and the discrete unified gas-kinetic scheme (DUGKS). First, the discrete balance property of a free-energy LBE for single-component two-phase flows is analyzed to identify the structure of force imbalance. Then, a well-balanced LBE (WB-LBE) model which has the same algorithm structure as the standard one is reviewed. The WB-LBE is theoretically shown to be able to achieve the discrete equilibrium state, and the well-balance properties are confirmed numerically. Some extensions to other two-phase LBE models are discussed with enhanced numerical stability. The well-balanced DUGKS, which can use non-uniform meshes and exhibits better numerical stability for large density ratio systems is also reported.

GPU implementation of the discrete unified gas kinetic scheme for low-speed isothermal flows

In this paper, two GPU parallel algorithms are proposed for the discrete unified gas kinetic scheme (DUGKS) for simulating low-speed isothermal flows. Algorithm-I uses a two-level fine-grain technique for the parallelization of physical spatial space, while Algorithm-II adopts this technique for both physical spatial and particle velocity spaces. To evaluate the performance of the proposed algorithms, several typical benchmark problems are simulated, including the two-dimensional (2D) and three-dimensional (3D) lid-driven cavity flows, the micro channel and cavity flows. Numerical results show that our GPU algorithms can achieve satisfactory computational efficiency. For Algorithm-I, the speedup can reach 250 and 338 on a Tesla V100 GPU card for the 2D and 3D continuum cavity flows, respectively, and a hundredfold acceleration can be obtained for the rarefied cases. While for Algorithm-II, a speedup of about 70 can be attained for rarefied cases. However, it is not applied to continuum problems that only require a small number of velocity points. Moreover, comparisons between the two GPU algorithms are also conducted for the rarefied flows with various grid meshes and velocity directions. The results show that Algorithm-I performs better when physical mesh size is large, while Algorithm-II can provide higher efficiency for a coarser mesh with medium number of discrete velocities. Special attention is also paid to comparisons between Algorithm-I and MPI parallelization with 128 CPU cores based on physical space discretization approach, and it is found that Algorithm-I has a clear advantage on V100 GPU when dealing with sparse physical grids in both continuum and rarefied cases.

Numerical simulation of lateral jet interaction with rarefied hypersonic flow over a two-dimensional blunt body

Reaction Control System (RCS) is a direct force control system that successfully adjusts a craft's attitude or orbit using the reaction force created by jet flow. RCS is frequently employed in the management of near-space vehicles due to its properties of fast response time and effective control efficiency. When the near-space vehicle is navigating at high altitude in a low density atmosphere, the Navier–Stokes equation is no longer applicable. The numerical approach utilized in this study is known as the Conserved Discrete Unified Gas Kinetic Scheme, and the governing equation is the Boltzmann equation, which is not constrained by the continuum hypothesis. In velocity space, an unstructured mesh is utilized, which minimizes the amount of discrete velocity points and considerably increases computation efficiency. The numerical results are in good agreement with the direct simulation Monte Carlo code DS2V when modeling large Knudsen number lateral jet flow. The interaction flow field between hypersonic free stream and lateral jet is then simulated at altitudes of 60–90 km using argon as the working gas and a two-dimensional blunt cone with lateral jet as the study object. Under a fixed jet pressure ratio, preliminary research was conducted on the variation of the lateral jet interference flow field characteristics with the freestream Knudsen number and angle of attack. The differences in surface pressure and heat flux caused by jet opening and shutting are compared. Under rarefied atmospheric conditions, the variation of the force/moment amplification coefficient is given. The numerical results show that when the angle of attack is 0°, the separation area in front of the nozzle and a pair of opposite vortices, which are common in the jet interference flow field, gradually disappear with increasing altitude, but the separation vortex reappears when the angle of attack of the freestream is increased. The high-pressure region generated upstream of the nozzle is the primary cause of the extra force/moment. The density of the main flow decreases as altitude increases, various shock wave patterns of the interference flow field gradually dissipate and the force/moment amplification factor changes considerably. The rarefied gas effect has a significant effect on the lateral jet interference flow.

Combined immersed boundary and discrete unified gas kinetic scheme for the motion of an autonomous underwater vehicle model with slip over a solid-liquid interface

To improve the endurance and speed of an autonomous underwater vehicle (AUV), we consider adopting the slip over a solid-liquid interface, which has the effect of drag reduction. However, the simulation of motion of an AUV with the slip remains a challenge due to the fluid-structure interaction and multiscale problems involving the moving boundary. To solve these problems, the present work proposes a novel and competitive mesoscopic method. Firstly, the Discrete Unified Gas Kinetic Scheme (DUGKS) coupled with the immersed boundary method is improved by correcting the boundary force, avoiding any additional computational time or implementation effort. Its effectiveness and accuracy are validated by two numerical tests involving the moving boundary. Furthermore, a fluid-solid interaction force is introduced to this method to model the slip condition due to its easy and effective implementation. We perform the experiment of motion of an AUV model in water to further validate the proposed method, which shows that it can predict acceptable results. Then, the method is applied to study the effect of the slip on the motion of the AUV model. The greater slip induced by the fluid-solid interaction force has a greater effect on reducing the drag and improving the velocity. It is indicated that increasing the fluid-solid interaction force can increase the amplitude and decrease the oscillation frequency of the drag curve. Overall, the slip can improve the endurance and speed of the AUV model.

A multiscale discrete velocity method for diatomic molecular gas

In the previous study, the multiscale discrete velocity method (MDVM) has been developed for monatomic gas with particle translational motion only. Unlike the unified gas-kinetic scheme (UGKS) and discrete unified gas-kinetic scheme, which are the typical representative of multiscale kinetic methods, MDVM achieves multiscale property by mixing the solution of macroscopic control equations and the Boltzmann equation, without the need to calculate complex interface flux. Therefore, MDVM has a higher computational efficiency. To broaden the application scope of MDVM, the Rykov model, which elucidates the exchange of energy between molecular translational and rotational energies, is introduced into MDVM in this paper. Numerical simulations are conducted for various cases, including one-dimensional shock tube, one-dimensional nitrogen shock structure, two-dimensional lid-driven cavity flow, and two-dimensional hypersonic flows around a flat plate and a blunt circular cylinder. The present results agree well with those from the diatomic UGKS method, demonstrating the developed diatomic MDVM can simulate multi-scale, strongly non-equilibrium, diatomic molecular gas flow while exhibiting certain efficiency improvements compared to the diatomic UGKS.

Novel Schemes of No-Slip Boundary Conditions for the Discrete Unified Gas Kinetic Scheme Based on the Moment Constraints.

The boundary conditions are crucial for numerical methods. This study aims to contribute to this growing area of research by exploring boundary conditions for the discrete unified gas kinetic scheme (DUGKS). The importance and originality of this study are that it assesses and validates the novel schemes of the bounce back (BB), non-equilibrium bounce back (NEBB), and Moment-based boundary conditions for the DUGKS, which translate boundary conditions into constraints on the transformed distribution functions at a half time step based on the moment constraints. A theoretical assessment shows that both present NEBB and Moment-based schemes for the DUGKS can implement a no-slip condition at the wall boundary without slip error. The present schemes are validated by numerical simulations of Couette flow, Poiseuille flow, Lid-driven cavity flow, dipole-wall collision, and Rayleigh-Taylor instability. The present schemes of second-order accuracy are more accurate than the original schemes. Both present NEBB and Moment-based schemes are more accurate than the present BB scheme in most cases and have higher computational efficiency than the present BB scheme in the simulation of Couette flow at high Re. The present Moment-based scheme is more accurate than the present BB, NEBB schemes, and reference schemes in the simulation of Poiseuille flow and dipole-wall collision, compared to the analytical solution and reference data. Good agreement with reference data in the numerical simulation of Rayleigh-Taylor instability shows that they are also of use to the multiphase flow. The present Moment-based scheme is more competitive in boundary conditions for the DUGKS.

A discrete unified gas kinetic scheme for simulating transient hydrodynamics in porous media with fractures

Tight gas reservoirs have typical characteristics of well-developed natural fractures, extremely low permeability, and low porosity. The flow in such media can be described by a dual-porosity dual-permeability (DPDP) model with the consideration of gas slippage and real gas effects. Based on this model, in this study, a discrete unified gas kinetic scheme is developed to describe the transient hydrodynamic behaviors in porous media with fractures. With an interporosity term, two kinetic equations are employed to capture the pressure propagation in matrix and fracture sub-systems, respectively. The proposed DUGKS can correctly recover the DPDP model through the Chapman–Enskog analysis. Unlike the traditional numerical methods, the velocity in DUGKS can be calculated locally by the non-equilibrium distribution functions. The presented method is verified by three typical problems, including the flow in a fractured porous medium, the flow around a vertical production well, and the flow around a fractured horizontal production well. The numerical results are in good agreement with the reference data. In the test, the global relative errors in pressures are on the order of 10−3. Moreover, the transient hydrodynamics in porous media with fractures are evaluated. The simulation results show that the slippage mainly occurs in the matrix, and this effect may result in a 20% increase in permeability. With the consideration of slippage and real gas effects, the predicted production is higher than that predicted by Darcy’s law. In addition, the results indicate that hydraulic fractures have a great impact on production.

Computational study of rarefied gas flow and heat transfer in lid-driven cylindrical cavities

The gas flow characteristics in lid-driven cavities are influenced by several factors, such as the cavity geometry, gas properties, and boundary conditions. In this study, the physics of heat and gas flow in cylindrical lid-driven cavities with various cross sections, including fully or partially rounded edges, is investigated through numerical simulations using the direct simulation Monte Carlo (DSMC) and the discrete unified gas kinetic scheme (DUGKS) methods. The thermal and fluid flow fields are systematically studied for both constant and oscillatory lid velocities, for various degrees of gas rarefaction ranging from the slip to the free-molecular regimes. The impact of expansion cooling and viscous dissipation on the thermal and flow fields, as well as the occurrence of counter-gradient heat transfer (also known as anti-Fourier heat transfer) under non-equilibrium conditions, is explained based on the results obtained from numerical simulations. Furthermore, the influence of the incomplete tangential accommodation coefficient on the thermal and fluid flow fields is discussed. A comparison is made between the thermal and fluid flow fields predicted in cylindrical cavities and those in square-shaped cavities. The present work contributes to the advancement of micro-/nano-electromechanical systems by providing valuable insight into rarefied gas flow and heat transfer in lid-driven cavities.

Multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration using preconditioned Krylov subspace method for multigroup neutron Boltzmann transport equation.

A multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration [accelerated steady discrete unified gas kinetic scheme (SDUGKS)] is proposed to improve the convergence of the original SDUGKS for an optically thick system in solving the multigroup neutron Boltzmann transport equation (NBTE) to analyze the distribution of fission energy in the reactor core. In the accelerated SDUGKS, by solving the coarse mesh macroscopic governing equations (MGEs) derived from the moment equations of the NBTE, the numerical solutions of the NBTE on fine meshes at the mesoscopic level can be rapidly obtained from the prolongation of the coarse mesh solutions of the MGE. Furthermore, the use of the coarse mesh can greatly reduce the computational variables and improve the computational efficiency of the MGE. The biconjugate gradient stabilized Krylov subspace method with the modified incomplete LU preconditioner and the lower-upper symmetric-Gauss-Seidel sweeping method are implemented to solve the discrete systems of the macroscopic coarse mesh acceleration model and mesoscopic SDUGKS to further improve the numerical efficiency. Numerical solutions validate good numerical accuracy and high acceleration efficiency of the proposed accelerated SDUGKS for the complicated multiscale neutron transport problems.

Adaptive partitioning-based discrete unified gas kinetic scheme for flows in all flow regimes

To improve the efficiency of the discrete unified gas kinetic scheme (DUGKS) in capturing cross-scale flow physics, an adaptive partitioning-based discrete unified gas kinetic scheme (ADUGKS) is developed in this work. The ADUGKS is designed from the discrete characteristic solution to the Boltzmann-BGK equation, which contains the initial distribution function and the local equilibrium state. The initial distribution function contributes to the calculation of free streaming fluxes and the local equilibrium state contributes to the calculation of equilibrium fluxes. When the contribution of the initial distribution function is negative, the local flow field can be regarded as the continuous flow and the Navier–Stokes (N-S) equations can be used to obtain the solution directly. Otherwise, the discrete distribution functions should be updated by the Boltzmann equation to capture the rarefaction effect. Given this, in the ADUGKS, the computational domain is divided into the DUGKS cell and the N-S cell based on the contribution of the initial distribution function to the calculation of free streaming fluxes. In the N-S cell, the local flow field is evolved by solving the N-S equations, while in the DUGKS cell, both the discrete velocity Boltzmann equation and the corresponding macroscopic governing equations are solved by a modified DUGKS. Since more and more cells turn into the N-S cell with the decrease of the Knudsen number, a significant acceleration can be achieved for the ADUGKS in the continuum flow regime as compared with the DUGKS.

Computational fluid dynamics with the coupled discrete unified gas kinetic scheme (CDUGKS)

ABSTRACTIn this paper, we introduce our open source implementation of the Coupled Discrete Unified Gas Kinetic Scheme (CDUGKS), a phase space scheme capable of handling a wide range of flow regimes. We demonstrate its performance on several well known test problems from the astrophysical fluid dynamics literature such as the 1D Sod shock tube and Einfeldt rarefaction, 2D Kelvin-Helmholtz instability, 1D thermoacoustic wave, a triangular Gresho vortex, a sine wave velocity perturbation. For these problems, we show that the code can simulate flows ranging from the inviscid/Eulerian regime to the free-streaming regime, capturing shocks and emergent diffusive processes in the appropriate regimes. We also use a variety of Prandtl numbers to demonstrate the scheme’s ability to simulate different thermal conductivities at fixed viscosity. The scheme is second-order accurate in space and time and, unlike many solvers, uses a time-step that is independent of the mean free path of the gas. Our code (mp-cdugks) is public under a CC0 1.0 Universal license and is available on GitHub.

A diffusion synthetic acceleration method for steady discrete unified gas kinetic scheme in radiative heat transfer

An efficient synthetic-type iterative method is proposed to improve the convergence rate of the steady discrete unified gas kinetic scheme (SDUGKS) for steady radiative heat transfer problems. The key ingredient of the present method is the tight coupling of the mesoscopic radiative transfer equation (RTE) and the macroscopic diffusion equation. The RTE is calculated by SDUGKS, and the diffusion equation is solved to predict the incident radiation energy and angular radiation intensity used in SDUGKS to accelerate the evolution, unlike the original diffusion synthetic acceleration (DSA) method which only predicts the incident radiation energy. Furthermore, a damping factor is adopted to remedy the inconsistency of spatial discretization between the mesoscopic SDUGKS and macroscopic diffusion equation. The theoretical convergence rate of the present method is studied via the discrete Fourier analysis. Several numerical tests under a wide range of optical thicknesses are performed to show the convergence speed. The damping factor and the correction of the angular radiation intensity added in solving the macroscopic diffusion equation greatly increase the stability of the present method and also reduce its computational costs. The convergence rate of the present method becomes faster than that of previous SDUGKS when the optical thickness is larger than 10. With the increase of extinction coefficient, the speedup ratio of the present method and SDUGKS in one-dimensional problems becomes extremely large and also reaches one or two orders of magnitude in two- and three-dimensional problems.

Central-moment discrete unified gas-kinetic scheme for incompressible two-phase flows with large density ratio

In this paper, we proposed a central moment discrete unified gas-kinetic scheme (DUGKS) for multiphase flows with large density ratio and high Reynolds number. Two sets of kinetic equations with central-moment-based multiple relaxation time collision operator are employed to approximate the incompressible Navier-Stokes equations and a conservative phase field equation for interface-capturing, respectively. In the framework of DUGKS, the first moment of the distribution function for the hydrodynamic equations is defined as velocity instead of momentum. Meanwhile, the zeroth moments of the distribution function and external force are carefully defined such that a artificial pressure evolution equation can be recovered. Moreover, the Strang splitting technique for time integration is employed to avoid the calculation of spatial derivatives in the force term at cell faces. For the interface-capturing equation, two equivalent DUGKS methods that deal with the diffusion term differently using a source term as well as a modified moments of the equilibrium distribution function are presented. Several benchmark tests that cover a wide a range of density ratios (up to 1000) and Reynolds numbers (up to 105) are subsequently carried out to demonstrate the capabilities of the proposed scheme. Numerical results are in good agreement with the theoretical and experimental data.

Diffuse interface model for a single-component liquid-vapor system.

We elucidate the theoretical relationships among fundamental physical concepts that are involved in the diffuse interface modeling for an isothermal single-component liquid-vapor system, which cover both the equationof state (EOS) and the surface tension force. As an example, a flat surface at equilibrium is discussed both theoretically and numerically by using two different approaches. Particularly, the force structure in the transition region is clearly presented, which demonstrates that the capillary contributions due to the density gradients can suppress the mechanical instability of the thermodynamic pressure and lead to constant hydrodynamic pressure (and chemical potential). Then, by comparing with the van der Waals (vdW) EOS for a flat interface at equilibrium, it is shown that applying the double-well approximation can give qualitative predictions for relatively high density ratio (ρ_{l}/ρ_{g}=7.784) and satisfactory results for relatively low density ratio (ρ_{l}/ρ_{g}=1.774). The main cause for this observation is attributed to the nonlinear variation of the generalized coefficient function in the double-well formulation at different density ratios. In addition, for the latter case, we simulate a droplet impact on a hydrophilic wall by using a recently proposed well-balanced discrete unified gas kinetic scheme (WB-DUGKS), which justifies the applicability of the double-well approximation to complex interfacial dynamics in the low-density-ratio limit. Furthermore, the reason for the inconsistency between the coefficients of the mean-field force expressions in the existing literature is explained.

Discrete unified gas kinetic scheme for continuum compressible flows.

In this paper, a discrete unified gas kinetic scheme (DUGKS) is proposed for continuum compressible gas flows based on the total energy kinetic model [Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704]. The proposed DUGKS can be viewed as a special finite-volume lattice Boltzmann method for the compressible Navier-Stokes equationsin the double distribution function formulation, in which the mass and momentum transport are described by the kinetic equationfor a density distribution function (g), and the energy transport is described by the other one for an energy distribution function (h). To recover the full compressible Navier-Stokes equationsexactly, the corresponding equilibrium distribution functions g^{eq} and h^{eq} are expanded as Hermite polynomials up to third and second orders, respectively. The velocity spaces for the kinetic equationsare discretized according to the seventh and fifth Gauss-Hermite quadratures. Consequently, the computational efficiency of the present DUGKS can be much improved in comparison with previous versions using more discrete velocities required by the ninth Gauss-Hermite quadrature.