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.     

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．     

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.     

A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier–Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier–Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier–Stokes equations are expressed in the convective form and approximated through the Lesaint–Raviart fluxes modified for DG methods.Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier–Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier–Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier–Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.     

Implementation of a transient adaptive sub-cell module for the parallel-DSMC code using unstructured grids

A method of transient adaptive sub-cells (TAS) suitable for unstructured grids that is modified from the existing one for the structured grids of DSMC is introduced. The TAS algorithm is implemented within the framework of a parallelized DSMC code (PDSC). Benchmarking tests are conducted for steady driven cavity flow, steady hypersonic flow over a two-dimensional cylinder, steady hypersonic flow over a cylinder/flare and the unsteady vortex shedding behind a two-dimensional cylinder. The use of TAS enables a reduction in the computational expense of the simulation since larger sampling cells and less simulation particles can be employed. Furthermore, the collision quality of the simulation is maintained or improved and the preservation of property gradients and vorticity at the scale of the sub-cells enables correct unsteady vortex shedding frequencies to be predicted. The use of TAS in a parallel-DSMC code allows simulations of unsteady processes at a level to be carried out efficiently, accurately and with acceptable computational time.     

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.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier–Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.     

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.     

LB-SGS方法和MRT-LBM方法对高雷诺数顶盖驱动流的数值模拟对比

针对格子Boltzmann方法(Lattice Boltzmann Method,LBM)广泛采用的LBGK模型虽然简单易行,但对高雷诺数流动模拟稳定性不佳的问题,分别采用结合亚格子模型的LBM(LBM with Sub Grid Scale(SGS),LB-SGS)方法和多松驰时间LBM(Multiple Relaxation Time(MRT)LBM,MRT-LBM)方法对高雷诺数顶盖驱动流进行数值模拟.取Re=7 500对比2种方法得到的涡位置与标准解之间的误差,结果表明LB-SGS方法更接近标准解;保持雷诺数和顶盖速度不变并减少格点数观察收敛情况,结果表明MRT-LBM方法更稳定.     

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.     

Numerical analysis of the unsteady flow in the near-tongue region in a volute-type centrifugal pump for different operating points

An investigation is presented on the unsteady flow behaviour near the tongue region of a single-suction volute-type centrifugal pump with a specific speed of 0.47. For this study, the flow through the test pump, which was available at laboratory, was simulated by means of a commercial CFD software that solved the Navier-Stokes equations for three-dimensional unsteady flow (3D-URANS). A sensitivity analysis of the numerical model was performed in order to impose appropriate parameters regarding grid size, time step size and turbulence model. The predictions of the numerical model were contrasted with experimental results of both global (flow-head curve and static pressure distribution at volute front side) and unsteady variables (unsteady pressure distribution at the volute front side filtered at the blade-passing frequency). Once validated, the model was used to study the flow pulsations associated to the interaction between the impeller blades and the volute tongue as a function of the flow rate, for several flow rates ranging from 20% to 160% of the nominal flow rate. The study allowed relating the blade passage with the pulsations of pressure and tangential and radial velocity at a number of reference locations in the near-tongue region. The numerical model was also used to evaluate the evolution of the leakage flow between the impeller-tongue gap and of the flow exiting the impeller through some specific angular intervals, during one single-blade passage.     

High-Reynolds number solutions of Navier-Stokes equations using incremental unknowns

The unsteady 2D Navier-Stokes equations on an unregularized driven cavity are solved in vorticity-streamfunction variables using Incremental Unknowns. Periodic asymptotic solutions have been found for Re = 10000 and Re = 12500.     

Numerical simulation of flow-induced cavity noise in self-sustained oscillations

The generation and near-field radiation of aerodynamic sound from a low-speed unsteady flow over a two-dimensional automobile door cavity is simulated by using a source-extraction-based coupling method. In the coupling procedure, the unsteady cavity flow field is first computed solving the Reynolds- averaged Navier–Stokes (RANS) equations. The radiated sound is then calculated by using a set of acoustic perturbation equations with acoustic source terms which are extracted from the time-dependent solutions of the unsteady flow. The aerodynamic and its resulting acoustic field are computed for the Reynolds number of 53,266 based on the base length of the cavity. The free stream flow velocity is taken to be 50.9 m/s. As first stage of the numerical investigation of flow-induced cavity noise, laminar flow is assumed. The CFD solver is based on a cell-centered finite volume method. A dispersion-relation-preserving (DRP), optimized, fourth-order finite difference scheme with fully staggered-grid implementation is used in the acoustic solver.     

Contrast between steady and time-averaged unsteady combustion simulations

Time-averaged predictions from unsteady solutions of the two-dimensional Navier–Stokes equations are contrasted with Reynolds-averaged results for a reacting flow problem in a high pressure combustor. The goal is to determine whether the two-dimensional unsteady approximation can be useful as an engineering analysis in problems for which time-averaged quantities are of primary interest. The conditions are taken from an experiment in which non-premixed gaseous oxygen and hydrogen were injected into a combustion chamber through coaxial channels. The resulting flowfield is dominated by a large recirculation zone arising from the back-step created by the injector. The results of steady and time-averaged, unsteady solutions are strikingly different. The unsteady simulation produces strong unsteady structures whose time-averaged results lead to a much wider flame zone, a different recirculation zone structure, and a substantially different wall heat flux than those obtained with a steady RANS procedure. The time-averaged calculations yield the correct combustor chamber pressure and compare considerably more favorably with heat flux measurements than do the RANS results. The two-dimensional approximation, however, overstates the unsteady vortex roll up and precludes large scale mixing across the axis of symmetry, thereby giving deficient predictions near the centerline. Overall, the present results indicate that capturing large-scale unsteady characteristics can provide more accurate predictions of recirculation dominated reacting flows and suggest that two-dimensional, time-averaged solutions represent a potentially useful engineering tool for problems of this nature while also serving as a precursor for full three-dimensional simulations.     

A parallel boundary fitted dynamic mesh solver for applications to flapping flight

A parallel numerical solution procedure for unsteady incompressible flow is developed for simulating the dynamics of flapping flight. A collocated finite volume multiblock approach in a general curvilinear coordinate is used with Cartesian velocities and pressure as dependent variables. The Navier-Stokes equations are solved using a fractional-step algorithm. The dynamic grid algorithm is implemented by satisfying the space conservation law by computing the grid velocities in terms of the volume swept by the faces. The dynamic movement of grid in a multiblock approach is achieved by using a combination of spring analogy and Trans-Finite Interpolation. The spring analogy is used to compute the displacement of block corners, after which Trans-Finite Interpolation is applied independently on each computational block. The performance of the code is validated in forced transverse oscillations of a cylinder in cross-flow, a heaving airfoil, and hovering of a fruitfly. Finally, the unsteady aerodynamics of flapping flight at Re = 10,000 relevant to the development of Micro Air Vehicles is analyzed for forward flight. The results show the capability of the solver in predicting unsteady aerodynamics characterized by complex boundary movements.     

An improved immersed boundary method for curvilinear grids

In the present paper we propose an extension of the direct-forcing immersed boundary technique, recently developed and employed by Verzicco and co-authors [Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J Comput Phys 2000;161:35–60; Verzicco R, Fatica M, Iaccarino G, Moin P, Khalighi B. Large eddy simulation of a road vehicle with drag-reduction devices. AIAA J 2002;40(12):2447–55; Cristallo A, Verzicco R. Combined immersed boundary/large-eddy-simulations of incompressible three-dimensional complex flows. Flow Turbul Combust 2006;77(1–4):3–26.] and successively improved by Balaras and co-authors [Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. J Comput Phys 2003;191:660–9; Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput Fluids 2004;33:375–404]. We extend the aforementioned technique to curvilinear-coordinate, structured grid, Navier–Stokes solvers. This improved technique allows for more flexibility and efficiency when compared to standard methods in which the technique is coupled with orthogonal-grid solvers. Additional modifications are also proposed with respect to the state-of-art, which allow to deal with general shaped, multiple-body immersed surfaces and to make the interpolation of the velocity field off the body suitable for curvilinear grids. Several tests have been carried out to check the reliability of the proposed technique: first we have considered the three-dimensional Stokes flow around a sphere, and compared the numerical results with the analytical ones. Second we have considered the two-dimensional unsteady flow around a circular cylinder placed between two parallel solid walls and compared the results with those of the database of the Priority Research Program ‘Flow Simulation on High Performance Computers’ of the German Research Association (DFG). Third, we have considered the two-dimensional flow within a S-shaped duct containing an elliptical valve. Finally, we have applied the technique to the study of a practical high-Reynolds number industrial problem.The geometrical configuration of the first two test cases is suited for both Cartesian and curvilinear algorithms. The geometry of the third test case is suited for curvilinear meshes and makes the use of Cartesian grids very inefficient and less accurate than the curvilinear ones. In these cases Cartesian – as well as curvilinear – mesh simulations have been carried out. Finally, the geometry of the high-Reynolds industrial problem is suited for curvilinear grids.The proposed technique has shown to preserve at least the same level of accuracy of its Cartesian counterpart allowing to reduce in a considerable way the computational cost of the simulations. When the geometry is better suited for curvilinear meshes, the reduction of the computational cost is accompanied by an increased accuracy with respect to the Cartesian counterpart.We also propose a simplified direct-forcing, semi-implicit method, allowing reduced computational cost with respect to the literature techniques. We have checked the accuracy of the technique and shown that when the Reynolds number is large enough, the present simplified technique allows the use of time steps much larger than those allowed by the explicit time-advancement scheme, preserving the accuracy of the results.     

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both $\left(n\right)$th and $(n+1)$th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397-420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.