基于预处理法非定常低速流动的数值模拟

本文基于预处理法,结合双时间步法,建立了应用高效隐式时间步进LU-SGS算法求解非定常低速流动问题的数值模拟方法．对典型的方腔顶盖瞬时启动驱动、周期振荡顶盖驱动等非定常低速流动问题进行了数值计算．结果表明,所建立的数值方法对非定常低速流动问题有较高的计算效率,并能有效的克服低速流动问题的系统刚性问题．     

Multiplicity of states for two-sided and four-sided lid driven cavity flows

Numerical simulations for incompressible flow in two-sided and four-sided lid driven cavities are reported in the present study. For the two-sided driven cavity, the upper wall is moved to the right and the left wall to the bottom with equal speeds. For the four-sided driven cavity, the upper wall is moved to the right, the lower wall to the left, while the left wall is moved downwards and the right wall upwards, with all four walls moving with equal speeds. At low Reynolds numbers, the resulting flow field is symmetric with respect to one of the cavity diagonals for the two-sided driven cavity, while it is symmetric with respect to both cavity diagonals for the four-sided driven cavity. At a critical Reynolds number of 1073 for the two-sided driven cavity and 129 for the four-sided driven cavity, the flow field bifurcates from a stable symmetric state to a stable asymmetric state. Three possible flow solutions exist above the critical Reynolds number, an unstable symmetric solution and two stable asymmetric solutions. All three possible solutions are recovered in the present study and flow bifurcation diagrams are constructed. Moreover, it is shown that the marching direction of the iterative solver determines which of the two asymmetric solutions is recovered.     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

Incompressible flow calculations with a consistent physical interpolation finite volume approach

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).     

The Estimation of Truncation Error by \tau -Estimation for Chebyshev Spectral Collocation Method

In this paper we show how to accurately estimate the local truncation error of the Chebyshev spectral collocation method using $\tau $ -estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error. Then, we show the validity of the analysis for the incompressible Navier–Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the truncation error if the precision of the approximations increases with the polynomial order.     

Numerical experiments with the lid driven cavity flow problem

The centerline velocity profiles obtained from the solution of the two- and three-dimensional representations of the lid driven cavity flow problem are compared for different Reynolds numbers. Two configurations were used in this study: a unit cavity and a cavity with an aspect ratio of 2. The Reynolds numbers ranged from 100 to 5000 for all of the configurations studied. A new method of extending the Jacobi collocation technique called spectral difference is developed in this paper together with a unique computational grid. In addition, an iterative method for solving the pressure problem is also developed. This new numerical method allowed the calculation of three-dimensional Navier-Stokes equations to be performed in computers with very modest computational capabilities such as workstations.     

A Conservative Isothermal Wall Boundary Condition for the Compressible Navier–Stokes Equations

We present a conservative isothermal wall boundary condition treatment for the compressible Navier-Stokes equations. The treatment is based on a manipulation of the Osher solver to predict the pressure and density at the wall, while specifying a zero boundary flux and a fixed temperature. With other solvers, a non-zero mass flux occurs through a wall boundary, which is significant at low resolutions in closed geometries. A simulation of a lid driven cavity flow with a multidomain spectral method illustrates the effect of the new boundary condition treatment.     

Hybrid atomistic�Ccontinuum simulations of fluid flows involving interfaces

In this work, we use an hybrid atomistic–continuum (HAC) simulation method to study transient and steady isothermal flows of Lennard-Jones fluids near interfaces. Our hybrid method is based on a domain decomposition algorithm. The flow domain is composed of two overlapping regions: an atomistic region described by molecular dynamics, and a continuum region described by a finite volume discretization of the incompressible Navier–Stokes equations. To show the interest of such an hybrid method to compute flows near fluid/solid interface, we first applied our hybrid scheme to the classical Couette flow, where the moving wall is modelled at the atomistic scale. In addition, we also studied an oscillatory shear flow. Then, to compute flows near fluid/fluid interface, we applied our method to a two-phase Couette flow (liquid/gas), where the interface is modelled at the molecular scale. We show that hybrid results can sometimes differ from those provided by analytical solutions deduced from continuum mechanics equations combined with usual boundary/interface relations. For the Couette and oscillatory shear flows, a good agreement is found between hybrid simulations and macroscopic analytical solutions, however, we noticed that the fluid in contact with the wall can be more entailed than what expected. For the liquid/gas Couette flow, the hybrid simulation exhibits an unexpected jump of the velocity in the interfacial region, corresponding to a partial slip between the two fluid phases. Those interesting results highlight the interest of using an HAC method to deal with systems for which surfaces/interfaces effects are important.     

Molecular dynamics simulation of compressible hot/cold moving lid-driven microcavity flow

Hard-sphere molecular dynamics simulations of lid-driven microcavity gas flow with various subsonic speeds and lid temperatures are conducted. Simulations with faster and colder lids show streamlines of stronger primary vortices. Variations of mass and energy centers with respect to lid speed and temperature are examined. Center of energy is less sensitive to employed lid conditions than center of gravity is. Although moving lid imparts energy into fluid, due to change of impingement rates on the walls of fixed temperature, average energy within the cavity seems quite insensitive to the subsonic lid speed. Behavior of compressibility at both top corners is observed even at low Mach numbers widely considered within incompressible flow region. While high Knudsen number causes considerable property slips near the lid, two-dimensional pressure, density, and temperature plots of excellent quality are generated. Results are promising in use of molecular dynamics simulations for compressible vortex flow analyses while providing insights for understanding microfluidics and nanofluidics in context of molecular mass, momentum and heat transfer in microscale and nanoscale systems.     

Implementation of density-based solver for all speeds in the framework of OpenFOAM

In the framework of open source CFD code OpenFOAM, a density-based solver for all speeds flow field is developed. In this solver the preconditioned all speeds AUSM+(P) scheme is adopted and the dual time scheme is implemented to complete the unsteady process. Parallel computation could be implemented to accelerate the solving process. Different interface reconstruction algorithms are implemented, and their accuracy with respect to convection is compared. Three benchmark tests of lid-driven cavity flow, flow crossing over a bump, and flow over a forward-facing step are presented to show the accuracy of the AUSM+(P) solver for low-speed incompressible flow, transonic flow, and supersonic/hypersonic flow. Firstly, for the lid driven cavity flow, the computational results obtained by different interface reconstruction algorithms are compared. It is indicated that the one dimensional reconstruction scheme adopted in this solver possesses high accuracy and the solver developed in this paper can effectively catch the features of low incompressible flow. Then via the test cases regarding the flow crossing over bump and over forward step, the ability to capture characteristics of the transonic and supersonic/hypersonic flows are confirmed. The forward-facing step proves to be the most challenging for the preconditioned solvers with and without the dual time scheme. Nonetheless, the solvers described in this paper reproduce the main features of this flow, including the evolution of the initial transient.     

Evaluation of multigrid acceleration for preconditioned time-accurate Navier-Stokes algorithms

The development of a two-dimensional time-accurate dual time step Navier-Stokes flow solver with time-derivative preconditioning and multigrid acceleration is described. The governing equations are integrated in time with both an explicit Runge-Kutta scheme and an implicit lower-upper symmetric-Gauss-Seidel scheme in a finite volume framework, yielding second-order accuracy in space and time. Issues concerning the implementation of multigrid for preconditioned, dual time step algorithms are discussed. Steady and unsteady computations were made of lid driven cavity flow, thermally driven cavity flow and pulsatile channel flow for a variety of conditions to validate the schemes and evaluate the effectiveness of multigrid for time-accurate simulations. Significant speedups were observed for steady and unsteady simulations. The speedups for unsteady simulations were problem dependent, a function of how rapidly the flow varied in time and the size of the allowable time step.     

Flow topology in a steady three-dimensional lid-driven cavity

We present in this paper a thorough investigation of three-dimensional flow in a cubical cavity, subject to a constant velocity lid on its roof. In this steady-state analysis, we adopt the mixed formulation on tri-quadratic elements to preserve mass conservation. To resolve difficulties in the asymmetric and indefinite large-size matrix equations, we apply the BiCGSTAB solution solver. To achieve stability, weighting functions are designed in favor of variables on the upstream side. To achieve accuracy, the weighting functions are properly chosen so that false diffusion errors can be largely suppressed by the equipped streamline operator. Our aim is to gain some physical insight into the vortical flow using a theoretically rigorous topological theory. To broaden our understanding of the vortex dynamics in the cavity, we also study in detail the longitudinal spiralling motion in the flow interior.     

Comparison of finite-volume numerical methods with staggered and colocated grids

The paper presents a detailed comparison of two finite-volume solution methods for two-dimensional incompressible fluid flows, one with staggered and the other with colocated numerical grids. The staggered method is well-known and well-established, and it is used here as a standard against which the relatively new colocated approach is compared. Three test cases were considered, employing orthogonal rectilinear grids: lid driven cavity flow, backward facing step flow and flow through a pipe with sudden contraction. The results of the computations demonstrate that the convergence rate, dependency on under-relaxation parameters, computational effort and accuracy are almost identical for both solution methods. The colocated method converges faster in some cases, and has advantages when extensions such as multigrid techniques and non-orthogonal grids are considered.     

Exploiting Kondo spin chains for generating long range distance independent entanglement

We investigate the entanglement properties of the Kondo spin chain when it is prepared in its ground state as well as its dynamics following a single bond quench. We show that a true measure of entanglement such as negativity enables to characterize the unique features of the gapless Kondo regime. We determine the spatial extent of the Kondo screening cloud and propose an ansatz for the ground state in the Kondo regime accessible to this spin chain; we also demonstrate that the impurity spin is indeed maximally entangled with the Kondo cloud. We exploit these features of the entanglement in the gapless Kondo regime to show that a single local quench at one end of a Kondo spin chain may always induce a fast and long lived oscillatory dynamics, which establishes a high quality entanglement between the individual spins at the opposite ends of the chain. This entanglement is a footprint of the presence of the Kondo cloud and may be engineered so as to attain—even for very large chains—a constant high value independent of the length; in addition, it is thermally robust. Moreover, we show that high entanglement between very distant boundary spins is generated by suddenly connecting two long Kondo spin chains. We show that this procedure provides an efficient way to route entanglement between multiple distant sites. To better evidence the remarkable peculiarities of the Kondo regime, we carry a parallel analysis of the entanglement properties of the Kondo spin chain model in the gapped dimerised regime where these remarkable features are absent.     

Multi relaxation time lattice Boltzmann simulations of deep lid driven cavity flows at different aspect ratios

In this paper, the multi relaxation time (MRT) lattice Boltzmann equation (LBE) was used to compute lid driven cavity flows at different Reynolds numbers (100–7500) and cavity aspect ratios (1–4 cavity width depth). Steady solutions were obtained for square cavity flows, however for deep cavity flows at 1.5 and 4 cavity width depth, unsteady solutions prevail at Re = 7500, where periodic flow exists manifested by the rapid changes of the shapes and locations of the corner vortices in strong contrast of the stationary primary vortex. The merger of the bottom corner vortices into a primary vortex and the reemergence of the corner vortices as the Reynolds number increases are more evident for the deep cavity flows. For the four cavity width depth cavity, four primary vortices were predicted by MRT model for Reynolds number beyond 1000, which were not predicted by previous single relaxation time (SRT) BGK LBE model, and this was verified by complementary Navier–Stokes simulations. Also, MRT model is more suitable for parallel computations than its BGK counterpart, due to the more intense local computations of the multi relaxation time procedure.     

求解Navier-Stokes方程的一类有限元有限体积迎风方法及其实现

Navier-Stokes方程是一类非线性的鞍点问题,在高Reynolds数流的情形下,标准Galerkin有限元方法会导致数值伪震荡.迎风有限元方法在算法结构上表征了流体＂上游＂决定＂下游＂的流动性态,它能够有效地消除高Reynolds数流的对流占优扩散所产生的非物理震荡.基于此,将Navier-Stokes方程的对流项采用有限体积框架下的迎风离散,对其它项仍使用Galerkin有限元离散,研究了二维定常Navier-Stokes方程的数值求解,编程藉助于有限元程序自动生成软件FEPG.通过对方腔流动和圆柱绕流问题与基准测试已有数值结果的比较,验证了所构造方法的可行性和有效性.     

A new method for handling flow problems with body forces

The solution of the Navier-Stokes equations by the numerical solution procedure of Patankar and Spalding has been found to be numerically unstable when applied to flows driven by body forces. A method (Pressure Reduction by Force Decomposition, PRFD) is outlined whereby such flows can be calculated without encountering numerical stability problems.The method is tested by calculating the laminar flow in a rotating square cavity with a moving lid. A stability limit is shown to exist when the numerical procedure in its usual form is applied to this problem. No limit is encountered when the PRFD technique is employed.     

Controllability, observability, and parameter identification of two coupled spin 1's

In this note, we study the control theoretic properties of a couple of interacting spin 1's driven by an electromagnetic field. In particular, we assume that it is possible to observe the expectation value of the total magnetization and we study controllability, observability, and parameter identification of these systems. We give conditions for controllability and observability and characterize the classes of equivalent models which have the same input-output behavior. The analysis is motivated by the recent interest in three level systems in quantum information theory and quantum cryptography as well as by the problem of modeling molecular magnets as spin networks.     

From multi- to single-grid CFD on massively parallel computers: Numerical experiments on lid-driven flow in a cube using pressure–velocity coupled formulation

A parallel implementation of a fully pressure–velocity coupled multigrid solver based on analytical solution accelerated coupled line Gauss Seidel (ASA-CLGS) smoother with grid partitioning is carried out. The parallelized algorithm is characterized by an enhanced scalability that results from a formulation enabling an intermediate analytical solution for the entire row (column) of control volumes. General strategies of applying single- or multigrid approach depending on flow characteristics are discussed. Performance of the parallelized algorithm is studied for up to 2048 processors. The developed approach is applied to analysis of a time-dependent three-dimensional incompressible lid-driven cavity flow. The steady state results of benchmark quality are reported for Re = 10³, 1.5 × 10³ and 1.9 × 10³. A new benchmark case of a fully 3D flow in a cubic cavity driven by the lid moving at 45° relatively to its lateral boundaries is proposed and the corresponding data is reported.     

Transitional cylindrical swirling flow in presence of a flat free surface

This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type—no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier-Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.