On the characteristics-based ACM for incompressible flows

In this paper, the revised characteristics-based (CB) method for incompressible flows recently derived by Neofytou [P. Neofytou, Revision of the characteristic-based scheme for incompressible flows, J. Comput. Phys. 222 (2007) 475–484] has been further investigated. We have derived all the formulas for pressure and velocities from this revised CB method, which is based on the artificial compressibility method (ACM) [A.J. Chorin, A numerical solution for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967) 12]. Then we analyze the formulations of the original CB method [D. Drikakis, P.A. Govatsos, D.E. Papatonis, A characteristic based method for incompressible flows, Int. J. Numer. Meth. Fluids 19 (1994) 667–685; E. Shapiro, D. Drikakis, Non-conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows, Int. J. Numer. Meth. Eng. 66 (2006) 1466–1482; D. Drikakis, P.K. Smolarkiewicz, On spurious vortical structures, J. Comput. Phys. 172 (2001) 309–325; F. Mallinger, D. Drikakis, Instability in three-dimensional, unsteady stenotic flows, Int. J. Heat Fluid Flow 23 (2002) 657–663; E. Shapiro, D. Drikakis, Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Parts I. Derivation of different formulations and constant density limit, J. Comput. Phys. 210 (2005) 584–607; Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756] to investigate their consistency with the governing flow equations after convergence has been achieved. Furthermore we have implemented both formulations in an unstructured-grid finite volume solver [Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756]. Detailed numerical experiments show that both methods give almost identical solutions and convergence rates. Both can generate solutions which agree well with published results and experimental measurements. We thus conclude that both methods, being upwind schemes designed for the ACM, have the same performances in terms of accuracy and convergence speed, even though the revised method is more complex with less stringent assumptions made, while the original CB method is simpler due to the use of extra simplifying assumptions.     

Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part I. Derivation of different formulations and constant density limit

The paper presents various formulations of characteristics-based schemes in the framework of the artificial-compressibility method for variable-density incompressible flows. In contrast to constant-density incompressible flows, where the characteristics-based variables reconstruction leads to a single formulation, in the case of variable density flows three different schemes can be obtained henceforth labeled as: transport, conservative and hybrid schemes. The conservative scheme results in pseudo-compressibility terms in the (multi-species) density reconstruction. It is shown that in the limit of constant density, the transport scheme becomes the (original) characteristics-based scheme for incompressible flows, but the conservative and hybrid schemes lead to a new characteristics-based variant for constant density flows. The characteristics-based schemes are combined with second and third-order interpolation for increasing the computational accuracy locally at the cell faces of the control volume. Numerical experiments for constant density flows reveal that all the characteristics-based schemes result in the same flow solution, but they exhibit different convergence behavior. The multigrid implementation and numerical studies for variable density flows are presented in Part II of this study.     

Lattice Boltzmann modeling of microchannel flow in slip flow regime

We present the lattice Boltzmann equation (LBE) with multiple relaxation times (MRT) to simulate pressure-driven gaseous flow in a long microchannel. We obtain analytic solutions of the MRT-LBE with various boundary conditions for the incompressible Poiseuille flow with its walls aligned with a lattice axis. The analytical solutions are used to realize the Dirichlet boundary conditions in the LBE. We use the first-order slip boundary conditions at the walls and consistent pressure boundary conditions at both ends of the long microchannel. We validate the LBE results using the compressible Navier–Stokes (NS) equations with a first-order slip velocity, the information-preservation direct simulation Monte Carlo (IP-DSMC) and DSMC methods. As expected, the LBE results agree very well with IP-DSMC and DSMC results in the slip velocity regime, but deviate significantly from IP-DSMC and DSMC results in the transition-flow regime in part due to the inadequacy of the slip velocity model, while still agreeing very well with the slip NS results. Possible extensions of the LBE for transition flows are discussed.     

Explicit algebraic Reynolds stress model for compressible flow turbulence

Algebraic Reynolds stress model (ARSM) is often employed in practical turbulent flow simulations. Most of previous works on ARSM have been carried out for incompressible flows. In the present paper, a new ARSM model is suggested for compressible flows. The model adopts a compressibility factor function involving the turbulent Mach number and the gradient Mach number. Compared to incompressible flow, explicit solution for ARSM for compressible flow can hardly be obtained due to dilatation terms. We propose approximate representations for these dilatation-related terms to obtain an explicit procedure for compressible flow turbulence. The model is applied to compressible mixing layer, supersonic flat-plate boundary and planar supersonic wake flow. It is found that the model works very well yielding results that are in good agreement with the DNS and the experimental data.     

Lattice Boltzmann simulations of 2D laminar flows past two tandem cylinders

We apply the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model to simulate laminar flows in two-dimensions (2D). In order to simulate flows in an unbounded domain with the LBE method, we need to address two issues: stretched non-uniform mesh and inflow and outflow boundary conditions. We use the interpolated grid stretching method to address the need of non-uniform mesh. We demonstrate that various inflow and outflow boundary conditions can be easily and consistently realized with the MRT-LBE. The MRT-LBE with non-uniform stretched grids is first validated with a number of test cases: the Poiseuille flow, the flow past a cylinder asymmetrically placed in a channel, and the flow past a cylinder in an unbounded domain. We use the LBE method to simulate the flow past two tandem cylinders in an unbounded domain with Re = 100. Our results agree well with existing ones. Through this work we demonstrate the effectiveness of the MRT-LBE method with grid stretching.     

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows

A momentum exchange-based immersed boundary-lattice Boltzmann method is presented in this Letter for simulating incompressible viscous flows. This method combines the good features of the lattice Boltzmann method (LBM) and the immersed boundary method (IBM) by using two unrelated computational meshes, an Eulerian mesh for the flow domain and a Lagrangian mesh for the solid boundaries in the flow. In this method, the non-slip boundary condition is enforced by introducing a forcing term into the lattice Boltzmann equation (LBE). Unlike the conventional IBM using the penalty method with a user-defined parameter or the direct forcing scheme based on the Navier–Stokes (NS) equations, the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. Numerical examples show that the present method can provide very accurate numerical results.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.     

Cumulative compressibility effects on slow reactive dynamics in turbulent flows

Reactions in turbulent flows, chemical reactions or combustion, are common. Typically reaction time scales are much shorter than turbulence timescales. In biological applications, as it is the case for bacterial and plankton populations living under the influence of currents in oceans and lakes, the typical lifetime can be long and thus can fall well within the inertial range of turbulence time scales. Under these conditions, turbulent transport interacts in a very complex way with the dynamics of growth and death of the individuals in the population. In the present paper, we quantitatively investigate the effect of the flow compressibility on the dynamics of populations. Small effective compressibility can be induced by several physical mechanisms, such as, e.g., by the density mismatch, by a small but finite size of microorganisms, and by gyrotaxis (an interaction between swimming and shear). We report, for the first time, how even a tiny effective compressibility can produce a dramatically large effect on global quantities like the carrying capacity of the ecosystem. We interpret our findings by means of a cumulative effect made possible by the long replication times of the organisms with respect to turbulence time scales. A statistical quantification of the fluctuations of population concentration is presented.     

Kinetic theory based lattice Boltzmann equation with viscous dissipation and pressure work for axisymmetric thermal flows

A lattice Boltzmann equation (LBE) for axisymmetric thermal flows is proposed. The model is derived from the kinetic theory which exhibits several features that distinguish it from other previous LBE models. First, the present thermal LBE model is derived from the continuous Boltzmann equation, which has a solid foundation and clear physical significance; Second, the model can recover the energy equation with the viscous dissipation term and work of pressure which are usually ignored by traditional methods and the existing thermal LBE models; Finally, unlike the existing thermal LBE models, no velocity and temperature gradients appear in the force terms which are easy to realize in the present model. The model is validated by thermal flow in a pipe, thermal buoyancy-driven flow, and swirling flow in vertical cylinder by rotating the top and bottom walls. It is found that the numerical results agreed excellently with analytical solution or other numerical results.     

Rectangular Lattice Boltzmann Equation for Gaseous Microscale Flow

The lattice Boltzmann equation (LBE) is considered as a promising approach for simulating flows of liquid and gas. Most of LBE studies have been devoted to regular square LBE and few works have focused on the rectangular LBE in the simulation of gaseous microscale flows. In fact, the rectangular LBE, as an alternative and efficient method, has some advantages over the square LBE in simulating flows with certain computational domains of large aspect ratio (e.g., long micro channels). Therefore, in this paper we expand the application scopes of the rectangular LBE to gaseous microscale flow. The kinetic boundary conditions for the rectangular LBE with a multiple-relaxation-time (MRT) collision operator, i.e., the combined bounce-back/specular-reflection (CBBSR) boundary condition and the discrete Maxwell's diffuse-reflection (DMDR) boundary condition, are studied in detail. We observe some discrete effects in both the CBBSR and DMDR boundary conditions for the rectangular LBE and present a reasonable approach to overcome these discrete effects in the two boundary conditions. It is found that the DMDR boundary condition for the square MRT-LBE can not realize the real fully diffusive boundary condition, while the DMDR boundary condition for the rectangular MRT-LBE with the grid aspect ratio a≠1 can do it well. Some numerical tests are implemented to validate the presented theoretical analysis. In addition, the computational efficiency and relative difference between the rectangular LBE and the square LBE are analyzed in detail. The rectangular LBE is found to be an efficient method for simulating the gaseous microscale flows in domains with large aspect ratios.     

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.     

用格子Boltzmann模型模拟非等温流场

根据微观和宏观之间的质量、动量、能量守恒准则和在原格子Boltzmann模型基础上,建立了几个新的格子Boltzmann模型,使得在外力场中的格子Boltzmann模型得到进一步完善.通过还原宏观流体力学方程,捕捉到了浮力强迫系数与Grashof数之间的关系.所得动量方程和Navier Stokes方程相比,在黏性输运项上有明显的改进,说明黏性应力不但与流体的速度梯度和流体的压缩性有关,而且还与非定常的内能梯度和动量通量有关.该模型对非等温流场的数值结果证明了其具有很好的数值稳定性和适用性. 关键词： Boltzmann模型 平衡分布函数 流体力学方程     

A study on the inclusion of body forces in the lattice Boltzmann BGK equation to recover steady-state hydrodynamics

When the lattice Boltzmann (LB) method is used to solve hydrodynamic problems containing a body force term varying in space and/or time, its modelling at the mesoscopic scale must be verified in terms of consistency in order to avoid the appearance of non-hydrodynamic error terms at the macroscopic scale. In the present work it is shown that the modelling of spatially varying steady body force terms in the LB equation must be different from the time-dependent case, when a steady-state flow solution is sought. For that, the Chapman-Enskog analysis is used to derive the LB body force model for the LB BGK equations in a steady-state flow problem. The theoretical findings are supported by numerical tests performed on two different 2D steady-state laminar flows driven by spatially varying body forces with known analytical solutions.     

一个新的用于不可压磁流体动力学的格子波尔兹曼模型

绝大多数现有的格子波尔兹曼磁流体动力学模型其实是用可压缩方法来模拟不可压磁流体。而这些可压缩效应在数值模拟中往往会带来意想不到的误差。在这篇文章中，我们提出了一个全新的可用于的不可压格子波尔兹曼磁流体动力学模型，并且进行了哈特曼流的数值模拟。模拟结果与哈特曼流的解析解非常吻合。这个方法需要一个假设条件来消除误差。我们做了大量的数值试验，并且与Dellar教授的模型进行了详细的分析与比较。     

The mixing efficiency of open flows

Mixing in open incompressible flows is studied in a model problem with inhomogeneous passive scalar injection on an inlet boundary. As a measure of the efficiency of stirring, the bulk scalar concentration variance is bounded, and the bound is shown to be sharp at low Péclet number. Although no specific flow saturating the bound at high Péclet number is produced here, the estimate is conjectured to be approached for flows possessing sufficiently sustained chaotic regions.     

A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh

The consistent and conservative scheme developed on a rectangular collocated mesh [M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: on a rectangular collocated grid system, Journal of Computational Physics 227 (2007) 174–204] and on an arbitrary collocated mesh [M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh, Journal of Computational Physics 227 (2007) 205–228] has been extended and specially designed for calculation of the Lorentz force on a staggered grid system (Part III) by solving the electrical potential equation for magnetohydrodynamics (MHD) at a low magnetic Reynolds number. In a staggered mesh, pressure (p) and electrical potential (φ) are located in the cell center, while velocities and current fluxes are located on the cell faces of a main control volume. The scheme numerically meets the physical conservation laws, charge conservation law and momentum conservation law. Physically, the Lorentz force conserves the momentum when the magnetic field is constant or spatial coordinate independent. The calculation of current density fluxes on cell faces is conducted using a scheme consistent with the discretization for solution of the electrical potential Poisson equation, which can ensure the calculated current density conserves the charge. A divergence formula of the Lorentz force is used to calculate the Lorentz force at the cell center of a main control volume, which can numerically conserve the momentum at constant or spatial coordinate independent magnetic field. The calculated cell-center Lorentz forces are then interpolated to the cell faces, which are used to obtain the corresponding velocity fluxes by solving the momentum equations. The “conservative” is an important property of the scheme, which can guarantee computational accuracy of MHD flows at high Hartmann number with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers. 2D fully developed MHD flows with analytical solutions available have been conducted to validate the scheme at a staggered mesh. 3D MHD flows, with the experimental data available, at a constant magnetic field in a rectangular duct with sudden expansion and at a varying magnetic field in a rectangular duct are conducted on a staggered mesh to verify the computational accuracy of the scheme. It is expected that the scheme for the Lorentz force can be employed together with a fully conservative scheme for the convective term and the pressure term [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, Journal of Computational Physics 143 (1998) 90–124] for direct simulation of MHD turbulence and MHD instability with good accuracy at a staggered mesh.     

一个三维多松弛时间全速域格子Boltzmann模型

构建一个既适用于低速不可压流体又适用于高速可压缩流体的三维自由参数多松弛时间格子Boltzmann模型.模型中,根据SO（3）群的不可约表述基函数构造转化矩阵,根据恢复可压Navier-Stokes方程的需要选取非守恒矩平衡值.通过von Neumann稳定性分析模型参数对数值稳定性的影响,并给出建议选择范围.模型经过基准问题的验证,模拟结果与解析解及其它数值结果符合较好.     

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

An asymptotic outer solution applied to the Keller box method

The Keller box method (“Numerical Solutions of Partial Differential Equations, Vol. 2” (B. Hubbard Ed.), pp. 327–350, Academic Press, New York, 1970) was applied to incompressible flow past a flat plate to demonstrate that the basic computation region must extend outward from the wall until the outer boundary conditions are effectively obtained. The Keller box method was modified to include an asymptotic outer solution for the case of the self-similar solution for compressible flow in a boundary layer. Initial application of the basic and modified Keller box methods to incompressible flow past a flat plate showed similar rates of convergence but smaller RMS error for the same basic range of the independent variable when the asymptotic outer solution is applied. Furthermore, extension of the solution beyond the range of the independent variable for the numerical solution using the resulting asymptotic solution produced RMS error at least as small as the RMS error on the range of the numerical solution. Also, when the asymptotic solution was applied, a smaller range of independent variables could be used in the numerical solution to obtain the same RMS error. Numerical results for compressible flow were qualitatively the same as for the case with the incompressible velocity profile except the rate of iterative convergence was slightly slower. Application of asymptotic outer solution for incompressible flow at a two dimensional stagnation point produced similar results with smaller relative improvements. For compressible flow with smaller favorable pressure gradients than the stagnation point and with adverse pressure gradients, significant improvements were again obtained. Examination of the errors associated with the asymptotic solution reveals that greatest success is obtained for flows with thicker boundary layers and shows that the boundary layer at a two dimensional stagnation point is too thin for small error in the asymptotic solution. Despite relatively large errors in the asymptotic solutions for boundary layer in strong favorable pressure gradients where the boundary layer is thin, the boundary layer solutions generally showed improvement in error and reduction in computation times.