Chebyshev pseudospectral method of viscous flows with corner singularities

Chebyshev pseudospectral solutions of the biharmonic equation governing two-dimensional Stokes flow within a driven cavity converge poorly in the presence of corner singularities. Subtracting the strongest corner singularity greatly improves the rate of convergence. Compared to the usual stream function/ vorticity formulation, the single fourth-order equation for stream function used here has half the number of coefficients for equivalent spatial resolution and uses a simpler treatment of the boundary conditions. We extend these techniques to small and moderate Reynolds numbers.     

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.     

A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows

In this paper, we extend a previous work on a compact scheme for the steady Navier–Stokes equations [Li, Tang, and Fornberg (1995), Int. J. Numer. Methods Fluids, 20, 1137–1151] to the unsteady case. By exploiting the coupling relation between the streamfunction and vorticity equations, the Navier–Stokes equations are discretized in space within a 3×3 stencil such that a fourth order accuracy is achieved. The time derivatives are discretized in such a way as to maintain the compactness of the stencil. We explore several known time-stepping approaches including second-order BDF method, fourth-order BDF method and the Crank–Nicolson method. Numerical solutions are obtained for the driven cavity problem and are compared with solutions available in the literature. For large values of the Reynolds number, it is found that high-order time discretizations outperform the low-order ones.     

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.     

Finite difference methods for viscous incompressible global stability analysis

In the present article some high-order finite-difference schemes and in particularly dispersion-relation-preserving (DRP) family schemes, initially developed by Tam and Webb [Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys. 107 (1993) 262-281.] for computational aeroacoustic problems, are used for global stability issue. (The term global is not used in weakly-non-parallel framework but rather for fully non-parallel flows. Some authors like Theofilis [Advances in global linear instability analysis of non-parallel and three-dimensional flows, Progress in Aerospace Sciences 39 (2003) 249-315] refer to this approach as “BiGlobal”.) These DRP schemes are compared with different classical schemes as second and fourth-order finite-difference schemes, seven-order compact schemes and spectral collocation scheme which is usually employed in such stability problems. A detailed comparative study of these schemes for incompressible flows over two academic configurations (square lid-driven cavity and separated boundary layer at different Reynolds numbers) is presented, and we intend to show that these schemes are sufficiently accurate to perform global stability analyses.     

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.     

Multiplicity of steady solutions in two-dimensional lid-driven cavity flows by Lattice Boltzmann Method

This work is concerned with the computation of two- and four-sided lid-driven square cavity flows and also two-sided rectangular cavity flows with parallel wall motion by the Lattice Boltzmann Method (LBM) to obtain multiple stable solutions. In the two-sided square cavity two of the adjacent walls move with equal velocity and in the four-sided square cavity all the four walls move in such a way that parallel walls move in opposite directions with the same velocity; in the two-sided rectangular lid-driven cavity flow the longer facing walls move in the same direction with equal velocity. Conventional numerical solutions show that the symmetric solutions exist for all Reynolds numbers for all the geometries, whereas multiplicity of stable states exist only above certain critical Reynolds numbers. Here we demonstrate that Lattice Boltzmann method can be effectively used to capture multiple steady solutions for all the aforesaid geometries. The strategy employed to obtain these solutions is also described.     

Numerical study of fourth-order linearized compact schemes for generalized NLS equations

The fourth-order compact approximation for the spatial second-derivative and several linearized approaches, including the time-lagging method of Zhang et al. (1995), the local-extrapolation technique of Chang et al. (1999) and the recent scheme of Dahlby et al. (2009), are considered in constructing fourth-order linearized compact difference (FLCD) schemes for generalized NLS equations. By applying a new time-lagging linearized approach, we propose a symmetric fourth-order linearized compact difference (SFLCD) scheme, which is shown to be more robust in long-time simulations of plane wave, breather, periodic traveling-wave and solitary wave solutions. Numerical experiments suggest that the SFLCD scheme is a little more accurate than some other FLCD schemes and the split-step compact difference scheme of Dehghan and Taleei (2010). Compared with the time-splitting pseudospectral method of Bao et al. (2003), our SFLCD method is more suitable for oscillating solutions or the problems with a rapidly varying potential.     

A comparison of the method of lines to finite difference techniques in solving time-dependent partial differential equations

Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (i) Burger's equation over a finite space domain by a forward time—central space explicit method, and (ii) the stream function—vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to “set up” time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.     

A family of fourth-order difference schemes on rotated grid for two-dimensional convection–diffusion equation

We derive a family of fourth-order finite difference schemes on the rotated grid for the two-dimensional convection–diffusion equation with variable coefficients. In the case of constant convection coefficients, we present an analytic bound on the spectral radius of the line Jacobi’s iteration matrix in terms of the cell Reynolds numbers. Our analysis and numerical experiments show that the proposed schemes are stable and produce highly accurate solutions. Classical iterative methods with these schemes are convergent with large values of the convection coefficients. We also compare the fourth-order schemes with the nine point scheme obtained from the second-order central difference scheme after one step of cyclic reduction.     

The effects of inertia on the structure of viscous corner eddies

Viscous eddies in the region close to a sharp corner are examined. The asymmetry in their structure that is apparent in numerical solutions for moderate values of the Reynolds number is derived analytically. A comparison is given with previous numerical studies and the agreement is found to be good. Some numerical verification of the analytical results is obtained from a study of the driven cavity flow problem for Reynolds numbers in the range 0–1000.     

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.     

Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries

The article presents new developments in boundary domain integral method (BDIM) for computation of viscous fluid flows, governed by the Navier–Stokes equations. The BDIM algorithm uses velocity–vorticity formulation and is based on Poisson velocity equation for flow kinematics. This results in accurate determination of boundary vorticity values, a crucial step in constructing an accurate numerical algorithm for computation of flows in complex geometries, i.e. geometries with sharp corners. The domain velocity computations are done by the segmentation technique using large segments. After solving the kinematics equation the vorticity transport equation is solved using macro-element approach. This enables the use of macro-element based diffusion–convection fundamental solution, a key factor in assuring accuracy of computations for high Reynolds value laminar flows. The versatility and accuracy of the proposed numerical algorithm is shown for several test problems, including the standard driven cavity together with the driven cavity flow in an L shaped cavity and flow in a Z shaped channel. The values of Reynolds number reach 10,000 for driven cavity and 7500 for L shaped driven cavity, whereas the Z shaped channel flow is computed up to Re = 400. The comparison of computational results shows that the developed algorithm is capable of accurate resolution of flow fields in complex geometries.     

Numerical solution of the incompressible Navier–Stokes equations with exponential type schemes

A new exponential type finite-difference scheme of second-order accuracy for solving the unsteady incompressible Navier–Stokes equation is presented. The driven flow in a square cavity is used as the model problem. Numerical results for various Reynolds numbers are given, and are in good agreement with those presented by Ghia et al. (Ghia, U., Ghia, K.N. and Shin, C.T., 1982, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multi-grid method. Journal of Computational Physics, 48, 387–411.).     

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.     

Flow around the impulsively started elliptic cylinder at various angles of attack

The flow around an impulsively started elliptic cylinder at 0, 30, 45 and 90° incidence is investigated. The fluid is viscous, incompressible and its flow is governed by the Navier-Stokes equations. Semi-analytical solutions are calculated by solving numerically the system of coupled partial differential equations which are obtained by substituting the expanded finite Fourier Series of the stream and vorticity functions in the Navier-Stokes equations. The symmetrical solutions are presented for Reynolds number 200 and eccentricity 0.809 and 0.943 in terms of patterns of streamlines, lines of constant vorticity, pressure and vorticity distributions around the surface, drag coefficient and wake length at 0 and 90° and compared with the experimental results. A comparison of the calculations has been made for Reynolds number 100 and eccentricity 0.648 with different number of terms at 90°. A Kármán vortex street develops for Reynolds numbers 200 and 60 at 30 and 45° incidence and the solutions are presented in terms of various characteristics including Strouhal number. The vanishing of wall-shear does not denote separation in any meaningful sense in various cases.     

Viscous incompressible flow by a penalty function finite element method

A quantitative evaluation for the penalty function finite element method for two-dimensional viscous incompressible flow using primitive variables is made, using bilinear and biquadratic elements. We conclude that this procedure, more efficient than full velocity pressure formulation, can achieve excellent accuracy using rather coarse meshes, that biquadratic elements produce improved accuracy over the more commonly used bilinears, and that the penalty method effectively imposes the continuity constraint. We point out the dangers of modelling problems containing singularities, as is the case of the driven cavity flow, with finite element formulations using primitive variables and how to overcome these problems. The driven cavity flow for Reynolds number up to 400 is solved and the answers compared with the best available solutions, the Jeffery-Hamel flow in a convergent control channel is solved for Reynolds number up to 61 and the results compared with the analytical solution. Finally extensions of the method are indicated via an example of natural convection in a square cavity for Rayleigh number up to 10⁶, and the advantage of the method discussed.     

A Family of Fourth-Order and Sixth-Order Compact Difference Schemes for the Three-Dimensional Poisson Equation

In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.     

High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

Transient state analysis of separated flow around a sphere

Transient state solutions of the Navier-Stokes equations were obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry about the axis in the direction of motion, the Navier-Stokes equations and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity of the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations first occurred at a Reynolds number 20 for steady state flows and at Reynolds numbers 22·46 and 28·24 for transient state flows with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions, and drag coefficents were calculated for both steady and unsteady states. Good agreement was obtained with existing experimental data in the steady state.