arXiv subjects

Songze Chen

Publications and source records attributed to Songze Chen.

15 recordsLinked to original sources

arXiv2026

Semi-implicit Lax-Wendroff kinetic scheme for hydrodynamic phonon transport

A semi-implicit Lax-Wendroff kinetic scheme is developed for hydrodynamic phonon transport in solid materials based on the Boltzmann transport equation under the double relaxation time approximation, in which both the normal and resistive scattering processes are accounted. The trapezoidal and midpoint rules are adopted for the temporal integration of the scattering and migration terms under the framework of finite volume method, respectively. Instead of direct numerical interpolation, the kinetic equation is solved again when reconstructing the interfacial flux, in order to realize the coupling of phonon migration and scattering within a numerical time step. Specifically, the finite difference scheme is introduced and the second-order upwind or central schemes are used for the reconstruction of the interfacial distribution function and its spatial gradient. Consequently, the cell size and time step of the present method could be larger than the phonon mean free path and relaxation time in the limit of small Knudsen numbers. Numerical tests demonstrate that the present method can accurately capture multi-scale thermal conduction phenomena within different normal or resistive scattering rates.

physics.comp-ph↗

arXiv2024

Semi-implicit Lax-Wendroff kinetic scheme for multi-scale phonon transport

Fast and accurate predictions of the spatiotemporal distributions of temperature are crucial to the multi-scale thermal management and safe operation of microelectronic devices. To realize it, an efficient semi-implicit Lax-Wendroff kinetic scheme is developed for numerically solving the transient phonon Boltzmann transport equation (BTE) from the ballistic to diffusive regime. The phonon BTE at the cell center is discretized under the framework of finite volume method, where the trapezoidal and midpoint rules are used to deal with the temporal integration of phonon scattering and convection terms, respectively. For the reconstruction of the interfacial distribution function, the phonon BTE at the cell interface is discretized in the form of finite difference method and solved numerically, where second-order upwind and central scheme are used to deal with the spatial interpolation and gradient of interfacial distribution function, respectively. The macroscopic governing equations are invoked for the evolution of macroscopic fields at both the cell center and interface, where the macroscopic flux is obtained by taking the moment of the interfacial distribution function. Numerical results show that the present scheme could accurately predict the steady/unsteady heat conduction in solid materials from the ballistic to diffusive regime, and its time and cell size are not limited by the relaxation time and phonon mean free path. The present work could provide a useful tool for the efficient predictions of the macroscopic spatiotemporal distributions in the multi-scale thermal engineering.

physics.comp-ph↗

arXiv2022

Acceleration strategy of source iteration method for the stationary phonon Boltzmann transport equation

Mesoscopic numerical simulation has become an important tool in thermal management and energy harvesting at the micro/nano scale, where the Fourier's law failed. However, it is not easy to efficiently solve the phonon Boltzmann transport equation (BTE) from ballistic to diffusive limit. In order to accelerate convergence, an implicit synthetic iterative scheme is developed for the stationary phonon BTE, in which a macroscopic moment equation is invoked and solved iteratively coupled with the typical source iteration of the kinetic equation. Different from previous numerical interpolation, the phonon BTE is solved again at the cell interface along the group velocity direction within a certain length when reconstructing the interfacial phonon distribution function. Fourier stability analysis shows that the present method could converge faster than the source iteration method in the (near) diffusive regime. Numerical results prove that the present scheme can capture the ballistic-diffusive effects correctly and efficiently. The present acceleration framework could be a powerful tool for simulating practical thermal engineering problems in the future.

physics.comp-ph↗

arXiv2020

Heat vortexes of ballistic, diffusive and hydrodynamic phonon transport in two-dimensional materials

In this work, the heat vortexes in two-dimensional porous or ribbon structures are investigated based on the phonon Boltzmann transport equation (BTE) under the Callaway model. First, the separate thermal effects of normal (N) scattering and resistive (R) scattering are investigated with frequency-independent assumptions. And then the heat vortexes in graphene are studied as a specific example. It is found that the heat vortexes can appear in both ballistic (rare R/N scattering) and hydrodynamic (N scattering dominates) regimes but disappear in the diffusive (R scattering dominates) regime. As long as there is not sufficient R scattering, the heat vortexes can appear in present simulations.

cond-mat.mes-hall↗

arXiv2020

Perturbation theory of thermal rectification

Thermal rectification which is a diode-like behavior of heat flux has been studied over a long time. However, a universal and systematic physical description is still lacking. In this letter, a perturbation theory of thermal rectification is developed, which provides an analytical formula of the thermal rectification ratio. It reveals the linear relationship between the thermal rectification ratio and temperature difference. Furthermore, the size-dependence of the thermal rectification relies on the specific form of the thermal conductivity. In addition, several experimental and numerical observations in previous literatures are well explained. This theory can be applicable to any system in which a differentiable effective thermal conductivity can be derived, and is helpful to unveil general principle for thermal rectification.

cond-mat.mes-hall↗

arXiv2020

Non-ideal gas dynamics under confinement: rarefaction effect, dense effect and molecular interaction

The effects of volume exclusion and long-range intermolecular attraction are investigated by the simplified kinetic model for surface-confined inhomogeneous fluids. Gas dynamics of the ideal gas, the hard-sphere fluid and the real gas are simulated by the Boltzmann equation, the Enskog equation and the simple kinetic equation, respectively. Only the Knudsen minimum appears for the ideal gas, while both the Knudsen minimum and the Knudsen maximum occur for the hard-sphere fluid and the real gas under certain confinements, beyond which the maximum and minimum may disappear. The Boltzmann equation and the Enskog equation overestimates and underestimates the mass flow rate of the real gas dynamics under confinement, respectively, where the volume exclusion and the long-range intermolecular attractive potential among molecules are not ignorable. With the increase of the channel width, gas dynamics of the hard-sphere fluid and the real gas tends to the Boltzmann prediction gradually. The density inhomogeneity, which hinders the flow under confinement, is more obvious when the solid fraction is larger. The anomalous slip occurs for real gas under constant confinement. The flow at a smaller Knudsen number (larger solid fraction or channel width) contributes more practical amount of mass transfer, although the rarefaction effects is more prominent at larger Knudsen numbers. The temperature has no effect on density and velocity profiles of the ideal gas and the hard-sphere fluid, but the energy parameter among the real gas molecules decreases with the increasing temperature and the real gas dynamics tends to the hard-sphere ones consequently.

physics.flu-dyn↗

arXiv2019

Radial thermal rectification in the concentric silicon ring from ballistic to diffusive regime

The radial thermal rectification in the concentric silicon ring from ballistic to diffusive regime is investigated based on the phonon Boltzmann transport equation. In the ballistic and diffusive limits, the analytical solutions prove that there is no thermal rectification. In the ballistic-diffusive regime, the heat flux prefers to flow from the inner boundary to the outer boundary. Furthermore, as the characteristic length (the distance between two circular boundaries) increases from tens of nanometers to tens of microns, the thermal rectification ratio enhances first and then fades away gradually. It attributes to that as the direction of the temperature gradient changes, the average phonon mean free path changes. The difference of the average phonon mean free path finally leads to the change of the heat flux or thermal conductivity. As the temperature decreases, the maximum thermal rectification ratio decreases. In addition, as the radius ratio between the inner and outer boundary increases, the thermal rectification ratio decreases for a given characteristic length.

cond-mat.mes-hall↗

arXiv2019

A well-balanced gas kinetic scheme for Navier-Stokes equations with gravitational potential

The hydrostatic equilibrium state is the consequence of the exact hydrostatic balance between hydrostatic pressure and external force. Standard finite volume or finite difference schemes cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibrium state and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations with a global reconstruction. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested numerically to demonstrate the accuracy and the stability of the new scheme, and the results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. The viscous effect is also illustrated through the propagation of small perturbation and the Rayleigh-Taylor instability. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibrium state from a highly non-balanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

physics.flu-dyn↗

arXiv2018

A fast synthetic iterative scheme for the stationary phonon Boltzmann transport equation

In this paper, a fast synthetic iterative scheme is developed to accelerate convergence for the implicit DOM based on the stationary phonon BTE. The key innovative point of the present scheme is the introduction of the macroscopic synthetic diffusion equation for the temperature, which is obtained from the zero- and first-order moment equations of the phonon BTE. The synthetic diffusion equation, which is asymptomatically preserving to the Fourier's heat conduction equation in the diffusive regime, contains a term related to the Fourier's law and a term determined by the second-order moment of the distribution function that reflects the non-Fourier heat transfer. The mesoscopic kinetic equation and macroscopic diffusion equations are tightly coupled together, because the diffusion equation provides the temperature for the BTE, while the BTE provides the high-order moment to the diffusion equation to describe the non-Fourier heat transfer. This synthetic iterative scheme strengthens the coupling of all phonons in the phase space to facilitate the fast convergence from the diffusive to ballistic regimes. Typical numerical tests in one-, two-, and three-dimensional problems demonstrate that our scheme can describe the multiscale heat transfer problems accurately and efficiently. For all test cases convergence is reached within one hundred iteration steps, which is one to three orders of magnitude faster than the traditional implicit DOM in the near-diffusive regime.

physics.comp-ph↗

arXiv2018

GPU acceleration of an iterative scheme for gas-kinetic model equations with memory reduction techniques

This paper presents a Graphics Processing Units (GPUs) acceleration method of an iterative scheme for gas-kinetic model equations. Unlike the previous GPU parallelization of explicit kinetic schemes, this work features a fast converging iterative scheme. The memory reduction techniques in this method enable full three-dimensional (3D) solution of kinetic model equations in contemporary GPUs usually with a limited memory capacity that otherwise would need terabytes of memory. The GPU algorithm is validated against the DSMC simulation of the 3D lid-driven cavity flow and the supersonic rarefied gas flow past a cube with grids size up to 0.7 trillion points in the phase space. The performance of the GPU algorithm is assessed by comparing with the corresponding parallel CPU program using Message Passing Interface (MPI). The profiling on several models of GPUs shows that the algorithm has a medium to high level of utilization of the GPUs' computing and memory resources. A $190\times$ speedup can be achieved on the Tesla K40 GPUs against a single core of Intel Xeon-E5-2680v3 CPU for the 3D lid-driven cavity flow.

physics.comp-ph↗

arXiv2018

An implicit kinetic scheme for multiscale heat transfer problem accounting for phonon dispersion and polarization

An efficient implicit kinetic scheme is developed to solve the stationary phonon Boltzmann transport equation (BTE) based on the non-gray model including the phonon dispersion and polarization. Due to the wide range of the dispersed phonon mean free paths, the phonon transport under the non-gray model is essentially multiscale, and has to be solved differently and appropriately for varied phonon frequencies and branches. The proposed implicit kinetic scheme is composed of a microscopic iteration and a macroscopic iteration. The microscopic iteration is capable of automatically adapting with varied phonon mean free path of each phonon frequency and branch through solving the phonon BTE. The energy transfer of all phonons is gathered together by the microscopic iteration to evaluate the heat flux. The temperature field is predicted through a macroscopic heat transfer equation according to the heat flux, and the equilibrium state in the phonon BTE is also updated. The combination of the phonon BTE solver and the macroscopic equation makes the present method very efficient in a wide length scale. Three numerical tests, including the cross-plane, in-plane and nano-porous heat transfer in silicon, validate that the present scheme can handle with the phonon dispersion and polarization correctly and predict the multiscale heat transfer phenomena efficiently in a wide range. The present method could be tens of times faster than the typical implicit DOM and keeps the same amount of the memory requirements as the Fourier solver for multiscale heat transfer problem.

physics.comp-ph↗

arXiv2016

A simplification of the unified gas kinetic scheme

Unified gas kinetic scheme (UGKS) is an asymptotic preserving scheme for the kinetic equations. It is superior for transition flow simulations, and has been validated in the past years. However, compared to the well known discrete ordinate method (DOM) which is a classical numerical method solving the kinetic equations, the UGKS needs more computational resources. In this study, we propose a simplification of the unified gas kinetic scheme. It allows almost identical numerical cost as the DOM, but predicts numerical results as accurate as the UGKS. Based on the observation that the equilibrium part of the UGKS fluxes can be evaluated analytically, the equilibrium part in the UGKS flux is not necessary to be discretized in velocity space. In the simplified scheme, the numerical flux for the velocity distribution function and the numerical flux for the macroscopic conservative quantities are evaluated separately. The simplification is equivalent to a flux hybridization of the gas kinetic scheme for the Navier-Stokes (NS) equations and conventional discrete ordinate method. Several simplification strategies are tested, through which we can identify the key ingredient of the Navier-Stokes asymptotic preserving property. Numerical tests show that, as long as the collision effect is built into the macroscopic numerical flux, the numerical scheme is Navier-Stokes asymptotic preserving, regardless the accuracy of the microscopic numerical flux for the velocity distribution function.

physics.flu-dyn↗

arXiv2015

Cartesian Grid Method for Gas Kinetic Scheme

A Cartesian grid method combined with a simplified gas kinetic scheme is presented for subsonic and supersonic viscous flow simulation on complex geometries. Under the Cartesian mesh, the computational grid points are classified into four different categories, the fluid point, the solid point, the drop point, and the interpolation point. The boundaries are represented by a set of direction-oriented boundary points. A constrained weighted least square method is employed to evaluate the physical quantities at the interpolation points. Different boundary conditions, including isothermal boundary, adiabatic boundary, and Euler slip boundary, are presented by different interpolation strategies. We also propose a simplified gas kinetic scheme as the flux solver for both subsonic and supersonic flow computations. The methodology of constructing a simplified kinetic flux function can be extended to other flow systems. A few numerical examples are used to validate the Cartesian grid method and the simplified flux function. The reconstruction scheme for recovering the boundary conditions of compressible viscous and heat conducting flow with a Cartesian mesh can provide a smooth distribution of physical quantities at solid boundary, and present an accurate solution for the flow study with complex geometry.

physics.flu-dyn↗

arXiv2013

A Comparative Study of an Asymptotic Preserving Scheme and Unified Gas-kinetic Scheme in Continuum Flow Limit

Asymptotic preserving (AP) schemes are targeting to simulate both continuum and rarefied flows. Many AP schemes have been developed and are capable of capturing the Euler limit in the continuum regime. However, to get accurate Navier-Stokes solutions is still challenging for many AP schemes. In order to distinguish the numerical effects of different AP schemes on the simulation results in the continuum flow limit, an implicit-explicit (IMEX) AP scheme and the unified gas kinetic scheme (UGKS) based on Bhatnagar-Gross-Krook (BGk) kinetic equation will be applied in the flow simulation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for the comparison of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles are very close to the classical benchmark solutions. However, the IMEX AP scheme seems have difficulty to get these solutions. Based on the analysis and the numerical experiments, it is realized that the dissipation of AP schemes in continuum limit is closely related to the numerical treatment of collision and transport of the kinetic equation. Numerically it becomes necessary to couple the convection and collision terms in both flux evaluation at a cell interface and the collision source term treatment inside each control volume.

physics.flu-dyn↗

arXiv2013

A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (S-model) are constructed for the correction of Prandtl number of the original BGK model through the modification of stress and heat flux. Even though in the continuum flow regime, both models can give the same Navier-Stokes equations with correct Prandtl number, their modification of the collision term may have different dynamic effect in the non-equilibrium transition flow regimes. With the introduction of one free parameter, a generalized kinetic model with the combination of the ES-model and S-model can be developed, and this new model can get the correct Navier-Stokes equations in the continuum flow regime as well, but with abundant dynamic effect through the adjustment of the new degree of freedom. In order to validate the generalized model, a numerical method based on the unified gas kinetic scheme (UGKS) has been developed for the new model. The physical performance of the new model with the variation of the free parameter has been tested, where the ES-model and S-model become the limiting cases. In transition flow regime, many physical problems, i.e., the shock structure and micro-flows, have been studied using the generalized model. With a careful choice of the free parameter, good results can be achieved for most test cases. The overall conclusion is that the S-model predicts more accurate numerical solutions in most tough test cases presented in this paper than the ES-model, while ES-model performs better in the cases when the flow is mostly driven by heat, such as a channel flow with large boundary temperature variations at high Knudsen number. The numerical study demonstrates the necessity of developing such a generalized model. With the inclusion of one more freedom, in the transition regime the new kinetic model may provide more accurate solution than the ES and Shakhov models.

physics.flu-dyn↗