Discrete Unified Gas Kinetic Scheme Research Articles

Free-Energy-Based Discrete Unified Gas Kinetic Scheme for van der Waals Fluid.

The multiphase model based on free-energy theory has been experiencing long-term prosperity for its solid foundation and succinct implementation. To identify the main hindrance to developing a free-energy-based discrete unified gas-kinetic scheme (DUGKS), we introduced the classical lattice Boltzmann free-energy model into the DUGKS implemented with different flux reconstruction schemes. It is found that the force imbalance amplified by the reconstruction errors prevents the direct application of the free-energy model to the DUGKS. By coupling the well-balanced free-energy model with the DUGKS, the influences of the amplified force imbalance are entirely removed. Comparative results demonstrated a consistent performance of the well-balanced DUGKS despite the reconstruction schemes utilized. The capability of the DUGKS coupled with the well-balanced free-energy model was quantitatively validated by the coexisting density curves and Laplace’s law. In the quiescent droplet test, the magnitude of spurious currents is reduced to a machine accuracy of . Aside from the excellent performance of the well-balanced DUGKS in predicting steady-state multiphase flows, the spinodal decomposition test and the droplet coalescence test revealed its stability problems in dealing with transient flows. Further improvements are required on this point.

Three-Dimensional Simulations of Anisotropic Slip Microflows Using the Discrete Unified Gas Kinetic Scheme.

The specific objective of the present work study is to propose an anisotropic slip boundary condition for three-dimensional (3D) simulations with adjustable streamwise and spanwise slip length by the discrete unified gas kinetic scheme (DUGKS). The present boundary condition is proposed based on the assumption of nonlinear velocity profiles near the wall instead of linear velocity profiles in a unidirectional steady flow. Moreover, a 3D corner boundary condition is introduced to the DUGKS to reduce the singularities. Numerical tests validate the effectiveness of the present method, which is more accurate than the bounce-back and specular reflection slip boundary condition in the lattice Boltzmann method. It is of significance to study the lid-driven cavity flow due to its applications and its capability in exhibiting important phenomena. Then, the present work explores, for the first time, the effects of anisotropic slip on the two-sided orthogonal oscillating micro-lid-driven cavity flow by adopting the present method. This work will generate fresh insight into the effects of anisotropic slip on the 3D flow in a two-sided orthogonal oscillating micro-lid-driven cavity. Some findings are obtained: The oscillating velocity of the wall has a weaker influence on the normal velocity component than on the tangential velocity component. In most cases, large slip length has a more significant influence on velocity profiles than small slip length. Compared with pure slip in both top and bottom walls, anisotropic slip on the top wall has a greater influence on flow, increasing the 3D mixing of flow. In short, the influence of slip on the flow field depends not only on slip length but also on the relative direction of the wall motion and the slip velocity. The findings can help in better understanding the anisotropic slip effect on the unsteady microflow and the design of microdevices.

Spatio-temporal error coupling and competition in meso-flux construction of discrete unified gas-kinetic scheme

This research systemically studies the numerical characteristics of the discrete unified gas-kinetic scheme (DUGKS) in terms of tracking interface motions driven by prespecified velocity fields. Multiple reconstruction methods have been utilized for the meso-flux construction of DUGKS and the performance of each method is evaluated accordingly. The spatio-temporal error coupling mechanism exhibited in the meso-flux evaluation is identified and analyzed theoretically for the first time. The spatio-temporal error competition mechanism (STECM) appeared during the construction of upwind meso-flux is specifically illustrated and discussed. On the basis of the newly discovered mechanism, we propose a high-resolution DUGKS by employing the fourth-order reconstruction scheme and the one-stage marching technique. Four benchmark tests have been conducted and numerical results have demonstrated the superiority of this high-resolution DUGKS over the lattice Boltzmann method (LBM) in capturing dynamic interfaces. Comparative results have also revealed that the STECM profoundly impacts the overall resolution of DUGKS implemented with different cell-based reconstruction methods. The STECM provides a theoretical understanding of the spatio-temporal characteristics of numerical errors of DUGKS and could serve as helpful guidance for the design of high-resolution DUGKS in future research.

Unified X-space parallelization algorithm for conserved discrete unified gas kinetic scheme

In this paper, the open source multiscale flow solver dugksFoam is optimized with a newly proposed parallelization strategy and conserved algorithm. A novel X-space parallel framework is introduced. In contrast to traditional discrete physical/velocity space decomposition parallelization methods, it can decompose each space in a hybrid manner and efficiently parallelize the computation. For better conservation of the macroscopic physical variables, flow properties in each cell are updated by moments of microscopic variable fluxes, which preserves particle multiscale information under sparse velocity space mesh. Several test cases, which include 2D lid-driven cavity flow in the transition flow regime, hypersonic rarefied flow past a 3D sphere, are carried out to verify the accuracy of the conserved discrete unified gas kinetic scheme to simulate the flows in all flow regimes. A large-scale parallel computational test case of 3D sphere with 101,600 physical mesh cells and 62,744 velocity mesh cells is conducted to measure the parallel efficiency of the conserved dugksFoam code. A significant superlinear speedup of 1631.87 has been achieved on 1120 cores.

On the nonlinear thermal stress, thermal creep, and thermal edge flows in triangular cavities

Here, rarefied thermally driven flow is investigated in two-dimensional equilateral triangular cavities with different uniform wall temperatures. We used three different solvers, i.e., the direct simulation Monte Carlo solver, discrete unified gas kinetic scheme solver, and continuum set of equations of a slow non-isothermal flow solver. Two main cases were considered; in the first case, the cavity's base is considered hot, and the other sides were set cold. In the second case, the right half of the bottom wall was regarded as a diffuse reflector with high temperature, while the left half of the bottom border was set as a specular reflector. The adjacent side walls were set cold with diffuse reflector boundary conditions. The imposed temperature difference/wall boundary condition induces various vortices in the geometry. In case 1, we observe that principal vortices appearing in the triangle are due to nonlinear thermal stress effects, and the thermal creep effects cause other smaller, confined ones. In case 2, a thermal edge flow is set up from the specular wall on the way to the diffusive hot wall, creating a large vortex in the geometry. As the Knudsen number decreases, another small vortex appears near the left cold border.

Well-balanced discrete unified gas-kinetic scheme for two-phase systems

In this paper, a well-balanced discrete unified gas-kinetic scheme (WB-DUGKS) is developed to capture the physical equilibrium state for two-phase fluid systems. Based on the strategies adopted in the well-balanced lattice Boltzmann equation (WB-LBE) [Z. Guo, “Well-balanced lattice Boltzmann model for two-phase systems,” Phys. Fluids 33, 031709 (2021)], a novel equilibrium distribution function and a modified force term are employed in the DUGKS framework. Unlike the LBE model, the time step in DUGKS is decoupled from the mesh size such that the numerical stability can be enhanced. First, the well-balanced properties of the method are validated by simulating a stationary droplet. The numerical results show that the WB-DUGKS can successfully reach an equilibrium state and exhibits superior numerical stability at low viscosity compared with the WB-LBE model. Then, the dynamic process of the coalescence of two droplets is simulated. The time scaling predicted by the present model is in good quantitatively agreement with the previous numerical results and experimental data. Overall, the proposed model provides a promising tool for simulating two-phase systems.

Discrete unified gas-kinetic scheme for the conservative Allen-Cahn equation.

In this paper, two discrete unified gas-kinetic scheme (DUGKS) methods with piecewise-parabolic flux reconstruction are presented for the conservative Allen-Cahn equation(CACE). One includes a temporal derivative of the order parameter in the force term while the other does not include temporal derivative in the force term but results in a modified CACE with additional terms. In the context of DUGKS, the continuum equationsrecovered from the piecewise-linear and piecewise-parabolic reconstructions for the fluxes at cell faces are subsequently derived. It is proved that the resulting equationwith the piecewise-linear reconstruction is a first-order approximation to the discrete velocity kinetic equationdue to the presence of the force term and the nonconservation property of the momentum of the collision model. To guarantee second-order accuracy of DUGKS, the piecewise-parabolic reconstruction for numerical flux is proposed. To validate the accuracy of the present DUGKS with the proposed flux evaluation, several benchmark problems, including the diagonal translation of a circular interface, the rotation of a Zalesak disk and the deformation of a circular interface, have been simulated. Numerical results show that the accuracy of both proposed DUGKS methods is almost comparable and improved compared with the DUGKS with linear flux reconstruction scheme.

Investigation of nonlinear squeeze-film damping involving rarefied gas effect in micro-electro-mechanical systems

In this paper, the nonlinear squeeze-film damping (SFD) involving rarefied gas effect in the micro-electro-mechanical systems (MEMS) is investigated. Considering the motion of structures (beam, cantilever, and membrane) in MEMS, the dynamic response of the structure is affected greatly by the SFD. In the traditional model, a viscous damping assumption that the damping force is linear with the moving velocity is used. As the nonlinear damping phenomenon is observed for a micro-structure oscillating at a high velocity, this assumption does not hold and will cause error results for predicting the response of the micro-structure. Meanwhile, due to the small size of the device and the low pressure of the encapsulation, the gas in MEMS is usually rarefied gas. Therefore, to correctly predict the damping force, the rarefied gas effect must be considered. To study the nonlinear SFD problem involving the rarefied gas effect, a kinetic method, i.e., discrete unified gas kinetic scheme (DUGKS), is introduced in this paper. Also, based on the DUGKS, two solving methods, i.e., a traditional decoupled method (Eulerian scheme) and a coupled framework (arbitrary Lagrangian-Eulerian scheme), are adopted. With these two methods, two basic motion forms, i.e., linear (perpendicular) and tilting motions of a rigid micro-beam, are studied under forced and free oscillations. For a forced oscillation, the nonlinear SFD phenomenon is investigated. For a free oscillation, in the resonance regime, some numerical results at different maximum oscillating velocities are presented and discussed. Besides, the influence of oscillation frequency on the damping force or torque is also studied, and the cause of the nonlinear damping phenomenon is investigated.

Conservative multilevel discrete unified gas kinetic scheme for modeling multiphase flows with large density ratios

A novel multilevel discrete unified gas kinetic scheme (MDUGKS) is proposed to efficiently model multiphase flows with large density ratios. By constructing the kinetic flux with a mutable time interval depending on the local mesh spacing, the MDUGKS overcomes the stability problems encountered by the standard DUGKS when operating with the multilevel mesh. With the interpolation of macroscopic variables and mesoscopic distributions handled separately, the moments of distribution functions are maintained consistent with the conservative flow variables. Two kinetic equations corresponding to the conservative Allen–Cahn equation and the hydrodynamic equation are individually solved by the MDUGKS, and six benchmark problems have been conducted to evaluate its performance. Numerical solutions in steady cases produced by the MDUGKS are in well accordance with the theoretical predictions. A limiting density ratio of 109 is achieved in the quiescent droplet. The dynamic processes in unsteady cases anticipated by the MDUGKS agree well with the reference predictions. Comparative results also demonstrate that the MDUGKS behaves consistently with different types of meshes. With the employment of the adaptive multilevel mesh, 80% improvement in computational efficiency could be achieved compared with the uniform mesh. Considering the kinetic nature and the high efficiency, the MDUGKS offers a powerful tool for presenting meaningful insight into understanding the realistic multiphase systems at the mesoscopic scale.

Particle distribution function discontinuity-based kinetic immersed boundary method for Boltzmann equation and its applications to incompressible viscous flows.

In this paper, a general immersed boundary force density is introduced for the Boltzmann equation and finally expressed as the desired particle distribution function discontinuity across the immersed boundary. Because of its independence of any specific boundary conditions and any specific solvers for the Boltzmann equation, it actually establishes a unified framework to incorporate various types of boundary conditions and several different kinds of solvers for the Boltzmann equation. Hence, a particle distribution function discontinuity-based kinetic immersed boundary method (KIBM) for the Boltzmann equation is proposed based on this general immersed boundary force density. Subsequently, this paper primarily focuses on the isothermal incompressible fluid-solid flows, and uses the discrete unified gas kinetic scheme to solve the Boltzmann Bhatnagar-Gross-Krook model equation. Meanwhile, the regularized delta function and the bounce-back rule combined with an iterative IBM correction procedure are employed in obtaining the general immersed boundary force density to enforce the no-penetration and no-slip boundary conditions on the solid wall. Finally, some numerical experiments for typical incompressible fluid-solid flows show that the present KIBM could provide good agreement with other numerical and experimental results.

Simulation of three-dimensional forced compressible isotropic turbulence by a redesigned discrete unified gas kinetic scheme

In this paper, we implemented the Boltzmann-equation-based mesoscopic model, developed recently by Chen et al. [“Inverse design of mesoscopic models for compressible flow using the Chapman–Enskog analysis,” Adv. Aerodyn. 3, 5 (2021)], to simulate three-dimensional (3D) forced compressible isotropic turbulence. In this model, both the Prandtl number and the ratio of bulk to shear viscosity can be arbitrary prescribed. The statistically stationary turbulent flow is driven by a large-scale momentum forcing in the Fourier space, with the internal heating due to the viscous dissipation at small scales being removed by a thermal cooling function. Under the framework of discrete unified gas kinetic scheme (DUGKS), a 3D direct numerical simulation code has been developed, incorporating a generalized Strang-splitting scheme. The weighted essentially non-oscillatory (WENO) scheme is used to increase local spatial accuracy in the reconstruction of particle distribution functions at the cell interface. A 3D discrete particle velocity model with a ninth-order Gauss–Hermite quadrature accuracy is used to ensure accurate evaluation of viscous stress and heat flux in the continuum regime. We simulate forced compressible isotropic turbulence at both low and high turbulent Mach numbers. A direct comparison is performed with the results obtained from a hybrid compact finite difference-WENO scheme solving directly the Navier–Stokes–Fourier system. The comparison validates our DUGKS code and indicates that DUGKS is a reliable and promising tool for simulating forced compressible isotropic turbulence. The work represents a first study to directly simulate forced compressible turbulence by a mesoscopic method based on the Boltzmann equation.

Central-Moment Discrete Unified Gas-Kinetic Scheme for Incompressible Two-Phase Flows with Large Density Ratio

Central-Moment Discrete Unified Gas-Kinetic Scheme for Incompressible Two-Phase Flows with Large Density Ratio

Numerical study on slip flow using the discrete unified gas-kinetic scheme

PurposeThis paper aims to explore the mechanism of the slip phenomenon at macro/micro scales, and analyze the effect of slip on fluid flow and heat transfer, to reduce drag and enhance heat transfer.Design/methodology/approachThe improved tangential momentum accommodation coefficient scheme incorporated with Navier’s slip model is introduced to the discrete unified gas kinetic scheme as a slip boundary condition. Numerical tests are simulated using the D2Q9 model with a code written in C++.FindingsVelocity contour with slip at high Re is similar to that without slip at low Re. For flow around a square cylinder, the drag is reduced effectively and the vortex shedding frequency is reduced. For flow around a delta wing, drag is reduced and lift is increased significantly. For Cu/water nanofluid in a channel with surface mounted blocks, drag can be reduced greatly by slip and the highest value of drag reduction (DR) (67.63%) can be obtained. The highest value of the increase in averaged Nu (11.78%) is obtained by slip at Re = 40 with volume fraction φ=0.01, which shows that super-hydrophobic surface can enhance heat transfer by slip.Originality/valueThe present study introduces and proposes an effective and superior method for the numerical simulation of fluid/nanofluid slip flow, which has active guidance meaning and applied value to the engineering practice of DR, heat transfer, flow control and performance improvement.

Discrete unified gas kinetic scheme for incompressible Navier-Stokes equations

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the Mach number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

Oscillatory square cavity flows of binary gas mixtures

The linear oscillation of rarefied binary gas mixtures inside a two-dimensional square cavity is studied based on the McCormack model. The discrete unified gas kinetic scheme is applied to solving this problem over wide ranges of the Knudsen number and the oscillation frequency. Two typical groups of binary mixtures, i.e., the Neon–Argon (Ne–Ar) mixture with comparable molecular masses and the Helium–Xenon (He–Xe) with significant different masses are considered. The dependence of the flow characteristics and damping force on the mass ratio and the molar concentration is also investigated in detail. It is found that results of the Ne–Ar mixture with a small mass ratio have similar tendencies to those of a single gas as expected. However, the damping force of the He–Xe mixture is qualitatively different from that of a single gas. In the free-molecular and transition regimes, the molar concentration of the He–Xe mixture shows significant influence on the anti-resonant frequency. Furthermore, the anti-resonance appearing in a single gas can no longer be observed in the He–Xe mixture as the molar concentration of the light species is large in the near-continuum regime, which is helpful to avoid the damping damage for the micro-electro-mechanical devices.

Discrete unified gas kinetic scheme simulation of conjugate heat transfer problems in complex geometries by a ghost-cell interface method

In this paper, we extend the discrete unified gas kinetic scheme (DUGKS) to simulate conjugate heat transfer flows. The conjugate interface that separates two media with different thermophysical properties is handled by a ghost-cell (GC) immersed boundary method. Specifically, fictitious cells are set in both domains to implement the continuity conditions of both temperature and heat flux at the conjugate interface. Information at the ghost cells is unavailable and should be reconstructed from the interface and neighboring field. With interface constraints involved in the reconstruction, the conjugate condition can be incorporated smoothly into the process of solving the full flow field. Those features make the present GC-DUGKS capture the conjugate interface, without introducing an additional term or modifying the equilibrium distribution function as in previous studies. Furthermore, the Strang-splitting scheme is applied for convenient handling of the thermal force term in the governing equation. Simulations of several well-established natural convection flows containing complex conjugate interfaces are performed to validate the GC-DUGKS. The results demonstrate the accuracy and feasibility of the present method for conjugate heat transfer problems.

Discrete unified gas kinetic scheme simulation of microflows with complex geometries in Cartesian grid

Microscale gas flow attracts significant research interest in recent years since it is concerned with a wide range of engineering applications. It is noted that the Navier–Stokes equations-based scheme and the standard lattice Boltzmann method both encounter a great challenge in the simulation of such flows. The newly developed discrete unified gas kinetic scheme (DUGKS) has been demonstrated to be capable of modeling microflows, but presently it is mainly limited to the problems with straight boundaries. In this study, the ghost-cell (GC) immersed boundary method is introduced to the DUGKS for handling curved boundaries. The most attractive feature of the GC method is to set a ghost point inside the solid domain, at which the information is unknown and will be extrapolated linearly from the corresponding wall and image nodes. As for the two latter points, the distribution functions are first evaluated by the inverse distance weighted (IDW) method and then should be corrected according to the impenetrability condition and Maxwellian diffuse-scattering rule. Three typical test cases, including the plane Poiseuille flow, cylindrical Couette flow and flow through porous media are simulated to validate the present IDW-GC-DUGKS. The results demonstrate the accuracy and feasibility of the method for the gaseous microflows.

Designing a consistent implementation of the discrete unified gas-kinetic scheme for the simulation of three-dimensional compressible natural convection

Discrete unified gas-kinetic scheme (DUGKS) has been developed as a robust and accurate approach for thermal compressible flow simulations; however, designing an efficient and accurate lattice velocity model to take full advantage of DUGKS remains a challenge. In this study, we apply DUGKS to simulate three-dimensional compressible natural convection in an enclosure with a large temperature difference, without making the Boussinesq approximation. The Chapman–Enskog analysis indicates that the fourth-order moments of equilibrium is needed for the heat flux evaluation in the energy equation, implying that the fourth-order Hermite expansion of equilibrium and thus at least an eighth-order Gauss–Hermite quadrature are needed for accurate simulation of the Navier–Stokes–Fourier system. For this purpose, a highly efficient lattice velocity model, D3Q77A9, is derived, which provides a Gauss–Hermite quadrature of ninth-order accuracy in three dimensions. The accuracy of this D3Q77A9 model is demonstrated by simulating compressible natural convection flows in both two-dimensional and three-dimensional cavities. An error analysis is performed to emphasize the importance of combining a quadrature with an adequate degree of precision and a proper order of Hermite expansion of the equilibrium distribution.

Computational methods for pore-scale simulation of rarefied gas flow

Direct simulation at the pore-scale is crucial to unravel rarefaction effect on gas transport in tight porous media. To satisfy the dual demands on modeling accuracy and computation effort, an appropriate method must be chosen. This work, therefore, evaluates four numerical methods for pore-scale rarefied gas flows in a two-dimensional (2D) model porous media over a wide range of rarefaction. These methods are the incompressible Navier-Stokes equation with the first-order velocity-slip boundary condition, and three gas-kinetic solvers i.e. a finite-difference (FD) iterative solver for the linearized Bhatnagar, Gross and Crook (BGK) model kinetic equation, a finite-volume (FV) solver for the non-linear Shakhov model, and an open-source direct simulation Monte Carlos (DSMC) solver. The benchmark cases cover the characteristic Knudsen number ranging from 0.0231 to 4.62 while the characteristic Reynolds numbers are kept to be less than 1.0. All the solvers are developed in OpenFOAM, except the FD solver, allowing us to investigate the effect of local grid refinements using the automatic Cartesian grid generator in OpenFOAM. The flow fields and the apparent permeabilities predicted by all the solvers have been compared in detail. Besides, the computational time of these solvers is measured and analyzed to demonstrate the relative cost of the three kinetic solvers. It is found that the FD solver is the most efficient one and gives accurate results over the whole range of Knudsen number. Finally, this study also evaluates the feasibility of the recently developed discrete unified gas-kinetic scheme (DUGKS), the algorithm in the aforementioned non-linear Shakhov model equation solver, for pore-scale rarefied gas flow. It is found the results predicted by this solver agree well with the other two kinetic solvers.

An improved discrete unified gas kinetic scheme for simulating compressible natural convection flows

Discrete unified gas-kinetic scheme (DUGKS) has been developed recently as a general method for simulating flows at all Knudsen numbers. In this study, we extend DUGKS to simulate fully compressible thermal flows. We introduce a source term to the Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision model [1] to adjust heat flux and thus the Prandtl number. The fully compressible Navier-Stokes equations can be recovered by the current model. As a mesoscopic CFD approach, it requires an accurate mesoscopic implementation of the boundary conditions. Using the Chapman-Enskog approximation, we derive the “bounce-back” expressions for both temperature and velocity distribution functions, which reveal the need to consider coupling terms between the velocity and thermal fields. To validate our scheme, we first reproduce the Boussinesq flow results by simulating natural convection in a square cavity with a small temperature difference (ϵ=0.01) and a low Mach number. Then we perform simulations of steady natural convection (Ra=1.0×106) in a square cavity with differentially heated side walls and a large temperature difference (ϵ=0.6), where the Boussinesq approximation becomes invalid. Temperature, velocity profiles, and Nusselt number distribution are obtained and compared with the benchmark results from the literature. Finally, the unsteady compressible natural convection with Ra=5.0×109,ϵ=0.6 is studied and the turbulent fluctuation statistics are computed and analyzed.