A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

An unconditionally stable fully conservative semi-Lagrangian method

Semi-Lagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFL-based time step restriction of the smallest cell can be overwhelming. Since these schemes are based on characteristic tracing and interpolation, they do not readily lend themselves to a fully conservative implementation. However, we propose a novel technique that applies a conservative limiter to the typical semi-Lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection. In addition, we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi-Lagrangian advection. We show that this new scheme can be used to conserve both mass and momentum for incompressible flows. For incompressible flows, we further explore properly conserving kinetic energy during the advection step, but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy (even for inviscid flows where it should be conserved). For compressible flows, we rely on a recently proposed splitting technique that eliminates the acoustic CFL time step restriction via an incompressible-style pressure solve. Then our new method can be applied to conservatively advect mass, momentum and total energy in order to exactly conserve these quantities, and remove the remaining time step restriction based on fluid velocity that the original scheme still had.     

A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a Multi-Implicit Spectral Deferred Correction (MISDC) strategy that iteratively couples advection, diffusion, and reactions evolving subject to a constraint. Our new approach overcomes a stability limitation of our previous second-order method encountered when trying to incorporate higher-order polynomial representations of the solution in time to increase accuracy. We have developed a new iterative scheme that naturally fits within our MISDC framework and allows us to conserve mass and energy while simultaneously satisfying the equation of state. We analyse the conditions for which the iterative schemes are guaranteed to converge to the fixed point solution. We present numerical examples illustrating the performance of the new method on premixed hydrogen, methane, and dimethyl ether flames.     

A novel coupled level set and volume of fluid method for sharp interface capturing on 3D tetrahedral grids

We present a new three-dimensional hybrid level set (LS) and volume of fluid (VOF) method for free surface flow simulations on tetrahedral grids. At each time step, we evolve both the level set function and the volume fraction. The level set function is evolved by solving the level set advection equation using a second-order characteristic based finite volume method. The volume fraction advection is performed using a bounded compressive normalized variable diagram (NVD) based scheme. The interface is reconstructed based on both the level set and the volume fraction information. The novelty of the method lies in that we use an analytic method for finding the intercepts on tetrahedral grids, which makes interface reconstruction efficient and conserves volume of fluid exactly. Furthermore, the advection of volume fraction makes use of the NVD concept and switches between different high resolution differencing schemes to yield a bounded scalar field, and to preserve both smoothness and sharp definition of the interface. The method is coupled to a well validated finite volume based Navier–Stokes incompressible flow solver. The code validation shows that our method can be employed to resolve complex interface changes efficiently and accurately. In addition, the centroid and intercept data available as a by-product of the proposed interface reconstruction scheme can be used directly in near-interface sub-grid models in large eddy simulation.     

The MAST FV/FE scheme for the simulation of two-dimensional thermohaline processes in variable-density saturated porous media

A novel methodology for the simulation of 2D thermohaline double diffusive processes, driven by heterogeneous temperature and concentration fields in variable-density saturated porous media, is presented. The stream function is used to describe the flow field and it is defined in terms of mass flux. The partial differential equations governing system is given by the mass conservation equation of the fluid phase written in terms of the mass-based stream function, as well as by the advection–diffusion transport equations of the contaminant concentration and of the heat. The unknown variables are the stream function, the contaminant concentration and the temperature. The governing equations system is solved using a fractional time step procedure, splitting the convective components from the diffusive ones. In the case of existing scalar potential of the flow field, the convective components are solved using a finite volume marching in space and time (MAST) procedure; this solves a sequence of small systems of ordinary differential equations, one for each computational cell, according to the decreasing value of the scalar potential. In the case of variable-density groundwater transport problem, where a scalar potential of the flow field does not exist, a second MAST procedure has to be applied to solve again the ODEs according to the increasing value of a new function, called approximated potential. The diffusive components are solved using a standard Galerkin finite element method. The numerical scheme is validated using literature tests.     

A fully conservative Eulerian–Lagrangian method for a convection–diffusion problem in a solenoidal field

Tracer transport is governed by a convection–diffusion problem modeling mass conservation of both tracer and ambient fluids. Numerical methods should be fully conservative, enforcing both conservation principles on the discrete level. Locally conservative characteristics methods conserve the mass of tracer, but may not conserve the mass of the ambient fluid. In a recent paper by the authors [T. Arbogast, C. Huang, A fully mass and volume conserving implementation of a characteristic method for transport problems, SIAM J. Sci. Comput. 28 (2006) 2001–2022], a fully conservative characteristic method, the Volume Corrected Characteristics Mixed Method (VCCMM), was introduced for potential flows. Here we extend and apply the method to problems with a solenoidal (i.e., divergence-free) flow field. The modification is a computationally inexpensive simplification of the original VCCMM, requiring a simple adjustment of trace-back regions in an element-by-element traversal of the domain. Our numerical results show that the method works well in practice, is less numerically diffuse than uncorrected characteristic methods, and can use up to at least about eight times the CFL limited time step.     

Wave-induced bed flows by a Lagrangian vortex scheme

Nominally 2-dimensional viscous flow induced by gravity waves over a spatially periodic bed is simulated by a Lagrangian vortex scheme. A vortex sheet is introduced on the surface at each time step to satisfy the zero velocity conditions. The sheet is discretised; the vortex-in-cell method is used to convect vorticity and random walks are added to effect viscous diffusion. Good agreement with analytical theory is obtained for velocity profiles in uniform sinusoidal flow and for mass transport due to linear waves. Mass transport for finite amplitude waves is also obtained. For separated flow over rippled beds, which is still liminar, a vortex decay factor is required to produce agreement with experiment and is thought to compensate for large scale 3-dimensional effects.     

High-performance high-resolution semi-Lagrangian tracer transport on a sphere

Current climate models have a limited ability to increase spatial resolution because numerical stability requires the time step to decrease. We describe a semi-Lagrangian method for tracer transport that is stable for arbitrary Courant numbers, and we test a parallel implementation discretized on the cubed sphere. The method includes a fixer that conserves mass and constrains tracers to a physical range of values. The method shows third-order convergence and maintains nonlinear tracer correlations to second order. It shows optimal accuracy at Courant numbers of 10–20, more than an order of magnitude higher than explicit methods. We present parallel performance in terms of strong scaling, weak scaling, and spatial scaling (where the time step stays constant while the resolution increases). For a 0.2° test with 100 tracers, the implementation scales efficiently to 10,000 MPI tasks.     

Transient chaotic mixing during a baroclinic life cycle

We discuss how atmospheric eddies affect transport and mixing of tracers at midlatitudes. To this purpose, we study baroclinic life cycles in a simple dynamical model of the atmosphere. We consider the trapping properties of the developing eddies and the characteristics of meridional transport, and we identify regions of increased mixing. Although the flow is in principle three-dimensional, we illustrate how some of the concepts developed in the study of two-dimensional chaotic advection provide useful information on tracer dynamics in more complicated flows. (c) 2000 American Institute of Physics.     

An improved particle correction procedure for the particle level set method

The particle level set method [D. Enright, R. Fedkiw, J. Ferziger, I. Mitchell, A hybrid particle level set method for improved interface capturing, J. Comput. Phys. 183 (2002) 83–116.] can substantially improve the mass conservation property of the level set method by using Lagrangian marker particles to correct the level set function in the under-resolved regions. In this study, the limitations of the particle level set method due to the errors introduced in the particle correction process are analyzed, and an improved particle correction procedure is developed based on a new interface reconstruction scheme. Moreover, the zero level set is “anchored” as the level set functions are reinitialized; hence the additional particle correction after the level set reinitialization is avoided. With this new scheme, a well-defined zero level set can be obtained and the disturbances to the interface are significantly reduced. Consequently, the particle reseeding operation will barely result in the loss of interface characteristics and can be applied as frequently as necessary. To demonstrate the accuracy and robustness of the proposed method, two extreme particle reseeding strategies, one without reseeding and the other with reseeding every time step, are applied in several benchmark advection tests and the results are compared with each other. Three interfacial flow cases, a 2D surface tension driven oscillating droplet, a 2D gas bubble rising in a quiescent liquid, and a 3D drop impact onto a liquid pool are simulated to illustrate the advantages of the current method over the level set and the original particle level set methods with regard to the smoothness of geometric properties and mass conservation in real physical applications.     

Multi-moment ADER-Taylor methods for systems of conservation laws with source terms in one dimension

A new integration method combining the ADER time discretization with a multi-moment finite-volume framework is introduced. ADER runtime is reduced by performing only one Cauchy–Kowalewski (C–K) procedure per cell per time step and by using the Differential Transform Method for high-order derivatives. Three methods are implemented: (1) single-moment WENO (WENO), (2) two-moment Hermite WENO (HWENO), and (3) entirely local multi-moment (MM-Loc). MM-Loc evolves all moments, sharing the locality of Galerkin methods yet with a constant time step during p-refinement.Five 1-D experiments validate the methods: (1) linear advection, (2) Burger’s equation shock, (3) transient shallow-water (SW), (4) steady-state SW simulation, and (5) SW shock. WENO and HWENO methods showed expected polynomial h-refinement convergence and successfully limited oscillations for shock experiments. MM-Loc showed expected polynomial h-refinement and exponential p-refinement convergence for linear advection and showed sub-exponential (yet super-polynomial) convergence with p-refinement in the SW case.HWENO accuracy was generally equal to or better than a five-moment MM-Loc scheme. MM-Loc was less accurate than RKDG at lower refinements, but with greater h- and p-convergence, RKDG accuracy is eventually surpassed. The ADER time integrator of MM-Loc also proved more accurate with p-refinement at a CFL of unity than a semi-discrete RK analog of MM-Loc. Being faster in serial and requiring less frequent inter-node communication than Galerkin methods, the ADER-based MM-Loc and HWENO schemes can be spatially refined and have the same runtime, making them a competitive option for further investigation.     

蒸汽-冷流体接触冷凝流动的数值模拟

介绍了关于蒸汽-冷流体直接接触冷凝流动与传热的数值计算模型与部分研究结果。用Level Set方法确定蒸汽-冷流体接触界面的位置和形状,建立了对蒸汽和冷流体普遍适用的动量、能量和质量守恒方程,在能量和质量寺恒方程中增加了部分项用于计算蒸汽冷凝所产生的影响。用有限差分法在交错网格上离散控制方程,用Runge-Kutta法-五阶WENO组合格式求解Level Set输运方程,用压力修正的迭代Projection方法求解动量方程,而用SIMPLE方法求解温度控制方程。对算例的计算结果表明,本文所建立的数值计算模型能反映物理现象的宏观特性。根据计算结果,分析了本文模型的优缺点,并指出了今后改进的方向。     

A Monte Carlo study of ionic transport in a simple cubic random alloy via the interstitialcy mechanism: effects of non-collinear and direct interstitial jumps

Self-diffusion and ionic conduction via the interstitialcy mechanism in a simple cubic, binary random alloy AB were investigated as a function of composition using Monte Carlo simulation. It was found that allowance for non-collinear jumps (partly) replacing concurrent collinear site exchanges leads to a reduction in diffusion correlation effects. This goes along with a shift of the diffusion percolation threshold to lower concentrations of the (more) mobile component B. Even stronger changes of mass and charge transport compared to an exclusively collinear interstitialcy scheme are observed for additional contributions of direct interstitial jumps. It is remarkable that for both extensions of interstitialcy mediated diffusion, the Haven ratio appears to be greater than unity in certain composition ranges poor in B. All results rely on the calculation of tracer and interstitialcy correlation factors in the simplest possible three-dimensional lattice structure. Yet they may have more general relevance to the interpretation of tracer self-diffusion data and ionic conductivity measurements on crystalline materials.     

Anti-diffusion method for interface steepening in two-phase incompressible flow

In this paper, we present a method for obtaining sharp interfaces in two-phase incompressible flows by an anti-diffusion correction, that is applicable in a straight-forward fashion for the improvement of two-phase flow solution schemes typically employed in practical applications. The underlying discretization is based on the volume-of-fluid (VOF) interface-capturing method on unstructured meshes. The key idea is to steepen the interface, independently of the underlying volume-fraction transport equation, by solving a diffusion equation with reverse time, i.e. an anti-diffusion equation, after each advection time step of the volume fraction. As the solution of the anti-diffusion equation requires regularization, a limiter based on the directional derivative is developed for calculating the gradient of the volume fraction. This limiter ensures the boundedness of the volume fraction. In order to control the amount of anti-diffusion introduced by the correction algorithm we propose a suitable stopping criterion for interface steepening. The formulation of the limiter and the algorithm for solving the anti-diffusion equation are applicable to 3-dimensional unstructured meshes. Validation computations are performed for passive advection of an interface, for 2-dimensional and 3-dimensional rising-bubbles, and for a rising drop in a periodically constricted channel. The results demonstrate that sharp interfaces can be recovered reliably. They show that the accuracy is similar to or even better than that of level-set methods using comparable discretizations for the flow and the level-set evolution. Also, we observe a good agreement with experimental results for the rising drop where proper interface evolution requires accurate mass conservation.     

A 6th order staggered compact finite difference method for the incompressible Navier–Stokes and scalar transport equations

In a previous paper we have developed a staggered compact finite difference method for the compressible Navier–Stokes equations. In this paper we will extend this method to the case of incompressible Navier–Stokes equations. In an incompressible flow conservation of mass is ensured by the well known pressure correction method  and . The advection and diffusion terms are discretized with 6th order spatial accuracy. The discrete Poisson equation, which has to be solved in the pressure correction step, has the same spatial accuracy as the advection and diffusion operators. The equations are integrated in time with a third order Adams–Bashforth method. Results are presented for a 1D advection–diffusion equation, a 2D lid driven cavity at a Reynolds number of 1000 and 10,000 and finally a 3D fully developed turbulent duct flow at a bulk Reynolds number of 5400. In all cases the methods show excellent agreement with analytical and other numerical and experimental work.     

Influence of interactive stratospheric chemistry on large-scale air mass exchange in a global circulation model

A new globally uniform Lagrangian transport scheme for large ensembles of passive tracer particles is presented and applied to wind data from a coupled atmosphere-ocean climate model that includes interactive dynamical feedback with stratospheric chemistry. This feedback from the chemistry is found to enhance large-scale meridional air mass exchange in the northern winter stratosphere as well as intrusion of stratospheric air into the troposphere, where both effects are due to a weakened polar vortex.     

Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions

This paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems. It is shown that the Newton–Raphson (NR) method converges for any time step size, if the flux function is convex, concave, or linear, which is, in general, the case for CFD problems. In some problems, e.g., multiphase flow in porous media, the nonlinear flux function is S-shaped (not uniformly convex or concave); as a result, a standard NR iteration can diverge for large time steps, even if an implicit discretization scheme is used to solve the nonlinear system of equations. In practice, when such convergence difficulties are encountered, the current time step is cut, previous iterations are discarded, a smaller time step size is tried, and the NR process is repeated. The criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and/or NR iteration. Here, we propose a simple modification to the NR iteration scheme for conservation laws with S-shaped flux functions that converges for any time step size. The new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver. The proposed method can be implemented in an existing simulator, e.g., for CO₂ sequestration or reservoir flow modeling, quite easily. The numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media. The analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems.     

High Level Languages Implementation and Analysis of 3D Navier-Stokes Solvers

In this work we introduce PRIN-3D (PRoto-code for Internal flows modeled by Navier-Stokes equations in 3-Dimensions), a new high level algebraic language (Matlab$^®$) based code, by discussing some fundamental aspects regarding its basic solving kernel and by describing the design of an innovative advection scheme. The main focus was on designing a memory and computationally efficient code that, due to the typical conciseness of the Matlab coding language, could allow for fast and effective implementation of new models or algorithms. Innovative numerical methods are discussed in the paper. The pressure equation is derived with a quasi-segregation technique leading to an iterative scheme obtained within the framework of a global preconditioning procedure. Different levels of parallelization are obtainable by exploiting special pressure variable ordering patterns that lead to a block-structured Poisson-like matrix. Moreover, the new advection scheme has the potential of a controllable artificial diffusivity. Preliminary results are shown including a fully three-dimensional internal laminar flow evolving in a relatively complex geometry and a 3D methane-air flame simulated with the aid of libraries based on the Flamelet model.     

Multiphysics modelling of the separation of suspended particles via frequency ramping of ultrasonic standing waves

A model was developed to determine the local changes of concentration of particles and the formations of bands induced by a standing acoustic wave field subjected to a sawtooth frequency ramping pattern. The mass transport equation was modified to incorporate the effect of acoustic forces on the concentration of particles. This was achieved by balancing the forces acting on particles. The frequency ramping was implemented as a parametric sweep for the time harmonic frequency response in time steps of 0.1 s. The physics phenomena of piezoelectricity, acoustic fields and diffusion of particles were coupled and solved in COMSOL Multiphysics? (COMSOL AB, Stockholm, Sweden) following a three step approach. The first step solves the governing partial differential equations describing the acoustic field by assuming that the pressure field achieves a pseudo steady state. In the second step, the acoustic radiation force is calculated from the pressure field. The final step allows calculating the locally changing concentration of particles as a function of time by solving the modified equation of particle transport. The diffusivity was calculated as function of concentration following the Garg and Ruthven [1] equation which describes the steep increase of diffusivity when the concentration approaches saturation. However, it was found that this steep increase creates numerical instabilities at high voltages (in the piezoelectricity equations) and high initial particle concentration. The model was simplified to a pseudo one-dimensional case due to computation power limitations. The predicted particle distribution calculated with the model is in good agreement with the experimental data as it follows accurately the movement of the bands in the centre of the chamber.     

Hybrid Flux-Splitting Schemes for a Two-Phase Flow Model

In this paper we deal with the construction of hybrid flux-vector-splitting (FVS) schemes and flux-difference-splitting (FDS) schemes for a two-phase model for one-dimensional flow. The model consists of two mass conservation equations (one for each phase) and a common momentum equation. The complexity of this model, as far as numerical computation is concerned, is related to the fact that the flux cannot be expressed in terms of its conservative variables. This is the motivation for studying numerical schemes which are not based on (approximate) Riemann solvers and/or calculations of Jacobian matrix. This work concerns the extension of an FVS type scheme, a Van Leer type scheme, and an advection upstream splitting method (AUSM) type scheme to the current two-phase model. Our schemes are obtained through natural extensions of corresponding schemes studied by Y. Wada and M.-S. Liou (1997, SIAM J. Sci. Comput.18, 633–657) for Euler equations. We explore the various schemes for flow cases which involve both fast and slow transients. In particular, we demonstrate that the FVS scheme is able to capture fast-propagating acoustic waves in a monotone way, while it introduces an excessive numerical dissipation at volume fraction contact (steady and moving) discontinuities. On the other hand, the AUSM scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves. This motivates us to propose other hybrid FVS/FDS schemes obtained by removing numerical dissipation at contact discontinuities in the FVS and Van Leer schemes.