Three-dimensional Lid-driven Cavity Flow Research Articles

A new class of massively parallel direction splitting for the incompressible Navier–Stokes equations

We introduce in this paper a new direction splitting algorithm for solving the incompressible Navier–Stokes equations. The main originality of the method consists of using the operator ( I − ∂ xx )( I − ∂ yy )( I − ∂ zz ) for approximating the pressure correction instead of the Poisson operator as done in all the contemporary projection methods. The complexity of the proposed algorithm is significantly lower than that of projection methods, and it is shown the have the same stability properties as the Poisson-based pressure-correction techniques, either in standard or rotational form. The first-order (in time) version of the method is proved to have the same convergence properties as the classical first-order projection techniques. Numerical tests reveal that the second-order version of the method has the same convergence rate as its second-order projection counterpart as well. The method is suitable for parallel implementation and preliminary tests show excellent parallel performance on a distributed memory cluster of up to 1024 processors. The method has been validated on the three-dimensional lid-driven cavity flow using grids composed of up to 2 × 10 9 points.

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 = 10 3, 1.5 × 10 3 and 1.9 × 10 3. 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.

Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method

In this paper, the local multiquadric differential quadrature (LMQDQ) method is applied on three-dimensional incompressible flow problems. The LMQDQ method is among the newly proposed mesh-free methods. Unlike the traditional differential quadrature (DQ) method, the weighting coefficients of LMQDQ method are determined by using the radial basis functions (RBFs) as the trial functions instead of high-order polynomials. The main concern of this paper is to discuss the effectiveness of using LMQDQ method to solve 3-D incompressible Navier–Stokes (N–S) equations in the primitive-variable form. Three-dimensional lid-driven cavity flow problem with Reynolds numbers of 100, 400 and 1000 was chosen as a test case to validate the LMQDQ method. The computed velocity profiles along the vertical and horizontal centerlines are given and compared with available data in the literature.

Accurate three-dimensional lid-driven cavity flow

A Chebyshev-collocation method in space is introduced, which allows an accurate calculation of three-dimensional lid-driven cavity flows. The time integration is carried out by an Adams–Bashforth backward–Euler scheme. The accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field. This way the most severe discontinuities in the boundary conditions, which arise along the lines where moving and stationary walls meet, are taken care of analytically and thus do not spoil the numerical part of the solution. Calculations are carried out for no-slip boundary conditions at the cavity end-walls as well as for periodic end-wall conditions. In general, the spatial accuracy is better than fifth order. For rigid end-wall conditions, the accuracy is reduced near the end-walls to O ( N 3 / 2 ) , but recovers in the bulk. Tabulated data are provided for the most interesting flow properties.

A unified incompressible lattice BGK model and its application to three-dimensional lid-driven cavity flow

A unified lattice Bhatnagar–Gross–Krook (ILBGK) model iDdQq for the incompressible Navier–Stokes equation is presented. To test its efficiency, the lid-driven cavity flow in three-dimensional space for Reynolds number Re=3200 and span aspect ratio SAR=1, 2 and 3 is simulated in detail on a 48×48×(48×SAR) uniform lattice using the model. The test results agree well with those in previous experiments and numerical works and show the efficiency and strong numerical stability of the proposed ILBGK model.

TAYLOR SERIES EXPANSION AND LEAST SQUARES-BASED LATTICE BOLTZMANN METHOD: THREE-DIMENSIONAL FORMULATION AND ITS APPLICATIONS

The two-dimensional form of the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) was recently presented by Shu et al.8 TLLBM is based on the standard lattice Boltzmann method (LBM), Taylor series expansion and the least square optimization. The final formulation is an algebraic form and essentially has no limitation on the mesh structure and lattice model. In this paper, TLLBM is extended to the three-dimensional case. The resultant form keeps the same features as the two-dimensional one. The present form is validated by its application to simulate the three-dimensional lid-driven cavity flow at Re=100, 400 and 1000. Very good agreement was achieved between the present results and those of Navier–Stokes solvers.

Spectral/ hp least-squares finite element formulation for the Navier–Stokes equations

We consider the application of least-squares finite element models combined with spectral/ hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/ hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L 2 least-squares functional and L 2 error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.

TRANSIENT SOLUTIONS BY A LEAST-SQUARES FINITE-ELEMENT METHOD AND JACOBI CONJUGATE GRADIENT TECHNIQUE

Abstract We present a least-squares finite-element method that can provide implicit, fully coupled transient solutions for time-dependent incompressible fluid flows and thermal convection. The algorithm consists of the Crank-Nicolson scheme for time discretization, Newton's method for linearization, and a matrix-free Jacobi conjugate gradient method as an iterative solver for the symmetric, positive-definite linear system of equations. The combined algorithm is first validated by two-dimensional flows: flows in a square cavity with a periodically oscillating lid and mixed convection in a driven cavity. Then the algorithm is used to obtain transient solutions of a three-dimensional lid-driven cavity flow for Re = 400

On End Wall Effects in a Lid-Driven Cavity Flow

Experiments were conducted in a three-dimensional lid-driven cavity flow to study the effects of the end walls on the size of the downstream secondary eddy. The ratio of cavity depth to cavity width is 1:1. The span of the cavity was varied such that span-to-width ratios of 3:1, 2:1, and 1:1 were obtained. Flow visualization was accomplished by the thymol blue technique, and by rheoscopic liquid illuminated by laser-light sheets, for Reynolds numbers (based on lid speed and cavity width) between 1000 and 10,000. The results indicate that the corner vortices present at the end walls, in the region of the downstream secondary eddy, are a major influence on the size of this eddy. In addition, as the span of the cavity is reduced the size of the downstream secondary eddy at the symmetry plane becomes smaller with increasing Reynolds numbers, for Reynolds numbers greater than 2000.

Flow visualization of a recirculating flow by rheoscopic liquid and liquid crystal techniques

Experiments were conducted in a three-dimensional lid-driven cavity flow to study the behavior of longitudinal Taylor-Gortler-like vortices. Flow visualization was accomplished by use of a rheoscopic liquid and of liquid crystals, together with laser-light and white-light sheets, respectively. Photographs of the lighted planes in the flow confirmed the existence of the vortices for a wide range of Reynolds numbers and for stable, neutrally-buoyant and buoyant global flow conditions. As usual the flow visualization revealed flow patterns not deducible by in situ measurements; the liquid crystal photographs give both flow pathlines and temperature distribution on any lighted plane.