Lid-driven Cavity Flow Problem Research Articles

Hemodynamic Analysis in an Idealized Artery Tree: Differences in Wall Shear Stress between Newtonian and Non-Newtonian Blood Models

Development of many conditions and disorders, such as atherosclerosis and stroke, are dependent upon hemodynamic forces. To accurately predict and prevent these conditions and disorders hemodynamic forces must be properly mapped. Here we compare a shear-rate dependent fluid (SDF) constitutive model, based on the works by Yasuda et al in 1981, against a Newtonian model of blood. We verify our stabilized finite element numerical method with the benchmark lid-driven cavity flow problem. Numerical simulations show that the Newtonian model gives similar velocity profiles in the 2-dimensional cavity given different height and width dimensions, given the same Reynolds number. Conversely, the SDF model gave dissimilar velocity profiles, differing from the Newtonian velocity profiles by up to 25% in velocity magnitudes. This difference can affect estimation in platelet distribution within blood vessels or magnetic nanoparticle delivery. Wall shear stress (WSS) is an important quantity involved in vascular remodeling through integrin and adhesion molecule mechanotransduction. The SDF model gave a 7.3-fold greater WSS than the Newtonian model at the top of the 3-dimensional cavity. The SDF model gave a 37.7-fold greater WSS than the Newtonian model at artery walls located immediately after bifurcations in the idealized femoral artery tree. The pressure drop across arteries reveals arterial sections highly resistive to flow which correlates with stenosis formation. Numerical simulations give the pressure drop across the idealized femoral artery tree with the SDF model which is approximately 2.3-fold higher than with the Newtonian model. In atherosclerotic lesion models, the SDF model gives over 1 Pa higher WSS than the Newtonian model, a difference correlated with over twice as many adherent monocytes to endothelial cells from the Newtonian model compared to the SDF model.

A Parallel Multigrid Finite-Volume Solver on a Collocated Grid for Incompressible Navier-Stokes Equations

Multigrid techniques are widely used to accelerate the convergence of iterative solvers. Serial multigrid solvers have been efficiently applied to a broad class of problems, including fluid flows governed by incompressible Navier-Stokes equations. With the recent advances in high-performance computing (HPC), there is an ever-increasing need for using multiple processors to solve computationally demanding problems. Thus, it is imperative that new algorithms be developed to run the multigrid solvers on parallel machines. In this work, we have developed a parallel finite-volume multigrid solver to simulate incompressible viscous flows in a collocated grid. The coarse-grid equations are derived from a pressure-based algorithm (SIMPLE). A domain decomposition technique is applied to parallelize the solver using a Message Passing Interface (MPI) library. The multigrid performance of the parallel solver has been tested on a lid-driven cavity flow. The scalability of the parallel code on both single- and multigrid solvers was tested and the characteristics were analyzed. A high-fidelity benchmark solution for lid-driven cavity flow problem in a 1,024 × 1,024 grid is presented for a range of Reynolds numbers. Parallel multigrid speedup as high as three orders of magnitude is achieved for low-Reynolds-number flows. The optimal multigrid efficiency is validated, i.e., the computational cost is shown to increase proportionally with the problem size.

Divergence-Free Wavelet Projection Method for Incompressible Viscous Flow on the Square

We present a wavelet numerical scheme for the discretization of two-dimensional Navier--Stokes equations with Dirichlet boundary condition on the square. This work is an extension to nonperiodic boundary conditions of the previous method of Deriaz and Perrier [E. Deriaz and V. Perrier, Multiscale Model. Simul., 7 (2008), pp. 1101--1129]. Here the temporal discretization is borrowed from the projection method. The projection operator is defined through a discrete Helmholtz--Hodge decomposition using divergence-free wavelet bases; this prevents the use of a Poisson solver as in usual methods, while improving the accuracy of the boundary condition. The stability and precision order of the new method are stated in the linear case of Stokes equations, confirmed by numerical experiments. Finally, the effectiveness, stability, and accuracy of the method are validated by simulations conducted on the benchmark problem of lid-driven cavity flow at Reynolds number $Re=1000$ and $Re=10000$.

Predictor-corrector finite-difference lattice Boltzmann schemes

Predictor-corrector finite-difference lattice Boltzmann schemes are proposed. An approach with separate approximation of spatial derivatives in the advective term of kinetic equations and an approach when this term is replaced by a single finite difference are considered. Explicit finite-difference schemes are used at both the stages of the computation process. The lid-driven cavity flow problem and the Taylor — Green problem are solved numerically in a wide range of the Reynolds number. It is shown that the proposed schemes allow a larger time step compared to well-known upwind schemes. Mathematics Subject Classification: 76-04; 76M10; 65N30

The localized method of approximated particular solutions for solving two-dimensional incompressible viscous flow field

The purpose of this paper is to demonstrate that the localized method of approximated particular solutions (LMAPS) is a stable, accurate tool for simulating two-dimensional incompressible viscous flow fields with Chorin׳s projection method. Totally there are two numerical experiments conducted: the two-dimensional lid-driven cavity flow problem, and the two-dimensional backward facing step problem. Throughout this study, the LMAPS has been tested by non-uniform point distribution, extremely narrow rectangular domain, internal flow, velocity or pressure driven flow and high velocity or pressure gradient, etc. All results are similar to results of finite element method (FEM) or other literature, and it is concluded that the LMAPS has high potential to be applied to more complicated engineering applications.

2D lid-driven cavity flow simulation using GPU-CUDA with a high-order finite difference scheme

The high parallelism and low cost of the graphic processing units (GPUs) have attracted the interest of scientists and engineers requiring high computational power with a modest investment. This work explores the use of a GPU in the solution of the 2D lid-driven cavity flow problem using the pressure–velocity formulation for Reynolds numbers up to $$10{,}000$$ and turning to a 4th order finite difference scheme for spatial discretization. Results showed good agreement with those reported in the literature. The solver was implemented in both the CPU and the GPU in order to compare their performance, whereupon the latter was seventy times faster.

An Online Manifold Learning Approach for Model Reduction of Dynamical Systems

This article discusses a newly developed online manifold learning method, subspace iteration using reduced models (SIRM), for the dimensionality reduction of dynamical systems. This method may be viewed as subspace iteration combined with a model reduction procedure. Specifically, starting with a test solution, the method solves a reduced model to obtain a more precise solution, and it repeats this process until sufficient accuracy is achieved. The reduced model is obtained by projecting the full model onto a subspace that is spanned by the dominant modes of an extended data ensemble. The extended data ensemble in this article contains not only the state vectors of some snapshots of the approximate solution from the previous iteration but also the associated tangent vectors. Therefore, the proposed manifold learning method takes advantage of the information of the original dynamical system to reduce the dynamics. Moreover, the learning procedure is computed in the online stage, as opposed to being computed offline, which is used in many projection-based model reduction techniques that require prior calculations or experiments. After providing an error bound of the classical POD-Galerkin method in terms of the projection error and the initial condition error, we prove that the sequence of approximate solutions converge to the actual solution of the original system as long as the vector field of the full model is locally Lipschitz on an open set that contains the solution trajectory. Good accuracy of the proposed method has been demonstrated in two numerical examples, from a linear advection-diffusion equation to a nonlinear Burgers equation. In order to save computational cost, the SIRM method is extended to a local model reduction approach by partitioning the entire time domain into several subintervals and obtaining a series of local reduced models of much lower dimensionality. The accuracy and efficiency of the local SIRM are shown through the numerical simulation of the Navier--Stokes equation in a lid-driven cavity flow problem.

Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Strokes equations

Meshless methods for solving fluid flow problems have become a promising alternative to mesh-based methods. In this paper, a meshless method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier–Stokes (N–S) equations in primitive variable form. Pressure–velocity decoupling has been achieved using a fractional step method whereas time splitting has been done using both explicit and implicit schemes. The RBF-FD implicit scheme shows better accuracy and stability, and is able to accurately capture higher gradients of field variables even at coarser grids; unlike the RBF-FD explicit scheme where loss of accuracy was especially prominent at places with larger gradients. To overcome the ill-conditioning and accuracy problems arising from the use of non-uniform and random node distribution, a novel concept of adaptive shape parameter (ASP) for RBF functions is introduced. The use of ASP allows much finer nodal distribution at regions of interest enabling accurate capturing of gradients and leading to better results. The performance of the Implicit RBF-FD scheme with the ASP strategy is validated against a variety of benchmark problems, including lid driven cavity flow problems, and steady and unsteady laminar flow around circular cylinder at various Reynolds, and is found to be in good agreement with the existing results.

A meshless scheme for incompressible fluid flow using a velocity–pressure correction method

A meshless point collocation method is proposed for the numerical solution of the steady state, incompressible Navier–Stokes (NS) equations in their primitive u–v–p formulation. The flow equations are solved in their strong form using either a collocated or a semi-staggered “grid” configuration. The developed numerical scheme approximates the unknown field functions using the Moving Least Squares approximation. A velocity, along with a pressure correction scheme is applied in the context of the meshless point collocation method. The proposed meshless point collocation (MPC) scheme has the following characteristics: (i) it is a truly meshless method, (ii) there is no need for pressure boundary conditions since no pressure constitutive equation is solved, (iii) it incorporates simplicity and accuracy, (iv) results can be obtained using collocated or semi-staggered “grids”, (v) there is no need for the usage of a curvilinear system of coordinates and (vi) it can solve steady and unsteady flows. The lid-driven cavity flow problem, for Reynolds numbers up to 5000, has been considered, by using both staggered and collocated grid configurations. Following, the Backward-Facing Step (BFS) flow problem was considered for Reynolds numbers up to 800 using a staggered grid. As a final example, the case of a laminar flow in a two-dimensional tube with an obstacle was examined.

An unstructured grids-based discretization method for convection–diffusion equations in the two-dimensional cylindrical coordinate systems

The study on the application of unstructured grids in the solution of problems in two-dimensional cylindrical coordinate systems (r–z) is scarce, since one of the challenges facing this application is the accurate calculation of the control volumes. In this article, an unstructured grids-based discretization method, in the framework of a finite volume approach, is presented for the solution of the convection–diffusion equations in cylindrical coordinate systems. Numerical simulations are presented for the natural convection and lid-driven cavity flow problems. The numerical results calculated on unstructured grids are found to be in good agreement with those calculated on fine structured meshes. The employment of unstructured grids leads to flexibility of the discretization method for irregular domains of any shape.

A compact streamfunction–velocity scheme on nonuniform grids for the 2D steady incompressible Navier–Stokes equations

In this paper, an effective compact finite difference approximation which carries streamfunction and its first derivatives (velocities) as the unknown variables for the streamfunction–velocity formulation of the steady two dimensional incompressible Navier–Stokes equation is developed on non-uniform orthogonal Cartesian grids. To solve the resulting system of equations, a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques. Numerical experiments, involving two test problems with analytical solutions and the lid-driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid. Numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids, viz., second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized. The backward-facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points. The solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems. All of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes.

A global meshless collocation particular solution method for solving the two-dimensional Navier–Stokes system of equations

The two-dimensional Navier–Stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions (MAPS) in its global formulation. The fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a Stokes non-homogeneous system of equations with multiquadric (MQ) radial basis function as the source term. The nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field, and the resulting linear system of equations is solved by a simple direct iterative scheme (Picard iteration), with the velocity guess given by the solution at the previous iteration. Although the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation. The proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the Kovasznay flow problem at different Reynolds numbers, Re. From this analysis, it is observed that the MAPS results are stable and accurate for a wide range of shape parameter values. In addition, lid-driven cavity flow problems in rectangular and triangular domains up to Re=3200 and Re=1000, respectively, and the backward-facing step at Re=800 are solved, and the results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement.

Longitudinal and transverse flow over a cavity containing a second immiscible fluid

AbstractAn analytical solution for the low-Reynolds-number flow field of a shear flow over a rectangular cavity containing a second immiscible fluid is derived. While flow of a single-phase fluid over a cavity is a standard case investigated in fluid dynamics, flow over a cavity that is filled with a second immiscible fluid has received little attention. The flow field inside the cavity is considered to define a boundary condition for the outer flow, which takes the form of a Navier slip condition with locally varying slip length. The slip-length function is determined heuristically from the related problem of lid-driven cavity flow. Based on the Stokes equations and complex analysis, it is then possible to derive a closed analytical expression for the flow field over the cavity for both the transverse and the longitudinal case. The result is a comparatively simple function, which displays the dependence of the flow field on the cavity geometry and the medium filling the cavity. The analytically computed expression agrees well with results obtained from a numerical solution of the Navier–Stokes equations.

Parallel numerical algorithm for simulation of a three-dimensional lid-driven cavity flow problem with the use of compact schemes

The parallel algorithm for simulation of the three-dimensional lid driven cavity flow problem with Reynolds numbers up to 10000 on a grid 128x128x128 was developed and implemented in the research. The problem is solved by numerical method where momentum equation is solved by a modified fractional step method with the use of compact scheme and pressure equation is solved by Fourier method in combination with the matrix sweep. The technique of the domain decomposition for parallelization of the proposed numerical method is described. In the process of computing and comparative experiment the dependency of the resulted acceleration and the coefficient of the calculation scalability on number of processors was found. Mathematics Subject Classication: 76F65

求解不可压缩Navier-Stokes方程的GLS平面八节点单元

给出了求解不可压缩Navier-Stokes方程的GLS平面八节点单元的“单元长度”的定义和相应的逆估计常数，由此可计算出该单元的GLS稳定因子。给出的“单元长度”只依赖单元形状且易于计算；各种形状的直边四边形单元的逆估计常数的最大值由遗传算法得出。对Reynolds数为20,000的方腔上盖板流的数值模拟表明，本文所提出的单元长度和逆估计常数的正确性。 The definition of the element length and the inverse estimate constant of straight-edged quadrilateral element with eight nodes for solving incompressible Navier-Stokes equations was proposed, based on this, the stabilization factor was readily calculated. The element length was only dependent on the quadrilateral geometry and easy to compute, and the largest inverse estimate constant among various quadrilateral was obtained by means of the optimization method of genetic algorithm (GA). The numerical solutions of the classical lid-driven cavity flow problem were ob- tained for Reynolds number of 20,000 which demonstrated the validity of the element length and the inverse estimate constant proposed.

Collocated discrete least squares (CDLS) meshless method for the stream function-vorticity formulation of 2D incompressible Navier–Stokes equations

The Least squares approach is a robust and simple method for function approximation. Collocated Discrete Least Squares (CDLS) is a meshless method based on least squares technique enjoying symmetric and positive-definite properties. In this paper, the CDLS method is extended for the stream function-vorticity formulation of 2D incompressible Navier–Stokes equations. Shape functions are constructed using Radial Point Interpolation Method (RPIM) because of its robustness and simplicity. The accuracy of the proposed scheme is investigated through solving lid-driven cavity flow and backward facing step problems for the different Reynolds numbers.

A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems

A global version of the Method of Approximate Particular Solutions (MAPS) is developed to solve two-dimensional Stokes flow problems in bounded domains. The velocity components and the pressure are approximated by a linear superposition of particular solutions of the non-homogeneous Stokes system of equations with a Multiquadric Radial Basis Function as forcing term. Although, the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative since the particular solutions exactly satisfy the mass conservation equation. The present scheme is validated by comparing the obtained numerical result with the analytical solution of two boundary value problems constructed from the Stokeson exterior fundamental solution, i.e. regular everywhere except at infinity. For these two cases, convergence of the method and the influence of the value of the Multiquadric’s shape parameter on the numerical results are studied by computing the relative Root Mean Square (RMS) error for several homogeneous distributions of collocation points and values of the shape parameter. From this analysis is observed that the proposed MAPS results are stable and accurate for a wide range of shape parameter values. In addition, the lid-driven cavity and backward-facing step flow problems are solved and the obtained results compared with the solutions found with more conventional numerical schemes, showing good agreement between them.

Development of a Moving and Stationary Mixed Particle Method for Solving the Incompressible Navier-Stokes Equations at High Reynolds Numbers

In the current particle method, we propose a new semi-implicit particle method for more effectively solving the incompressible Navier-Stokes equations at a high Reynolds number. Within the Lagrangian framework, the convective terms in the equations of motion are eliminated, without the problem of convective numerical instability. Also, the crosswind diffusion error generated normally in the case of a large angle difference between the velocity vector and the coordinate line disappears. Only the Laplacian operator for the velocity components and the gradient operator for the pressure need to be approximated on the basis of particle interaction through the currently proposed kernel function. As the key to getting better predicted accuracy, the kernel function is derived subject to theoretical constraint conditions. In the conventional moving-particle method, it is almost impossible to get convergent solution at a high Reynolds number. To overcome this simulation difficulty so that the moving-particle method is applicable to a wider range of flow simulations, a new solution algorithm is proposed for solving the elliptic-parabolic set of partial differential equations. In the momentum equations, calculation of the velocity components is carried out in the particle-moving sense. Unlike the traditional moving-particle semi-implicit method, the pressure values are not calculated at the particle locations being advected along the flowfield. After updating the fluid particle locations within the Lagrangian framework, we interpolate the velocities at uniformly distributed pressure locations. In the current semi-implicit solution algorithm, pressure is governed by the elliptic differential equation with the source term being contributed entirely to the velocity gradient terms. The distribution of particle locations can become highly nonuniform in cases involving a high Reynolds number and under conditions having an apparently vortical flow. As a result, the elliptic nature of the pressure can be considerably destroyed in the course of Lagrangian motion. To retain the embedded ellipticity in the incompressible viscous flow equations, the Poisson equation adopted for the calculation of pressure is solved in a mathematically more plausible fixed uniform mesh so as to get not only fourth-order accuracy for the pressure but also to enhance ellipticity in the pressure Poisson equation. Moreover, the velocity–pressure coupling can be more enhanced in the semi-implicit solution algorithm. The proposed moving and stationary mixed particle semi-implicit solution algorithm and the particle kernel will be demonstrated to be suitable to simulate high-Reynolds number fluid flows by investigating the lid-driven cavity flow problem at Re = 100 and Re = 1,000. Besides the validation of the proposed semi-implicit particle method in the fixed domain, the broken-dam problem is also solved to demonstrate the ability of accurately capturing the time-evolving free surface using the proposed semi-implicit particle method.

A high-order compact local integrated-RBF scheme for steady-state incompressible viscous flows in the primitive variables

This study is concerned with the development of integrated radialbasis- function (IRBF) method for the simulation of two-dimensional steady-state incompressible viscous flows governed by the pressure-velocity formulation on Cartesian grids. Instead of using low-order polynomial interpolants, a high-order compact local IRBF scheme is employed to represent the convection and diffusion terms. Furthermore, an effective boundary treatment for the pressure variable, where Neumann boundary conditions are transformed into Dirichlet ones, is proposed. This transformation is based on global 1D-IRBF approximators using values of the pressure at interior nodes along a grid line and first-order derivative values of the pressure at the two extreme nodes of that grid line. The performance of the proposed scheme is investigated numerically through the solution of several linear (analytic tests including Stokes flows) and non-linear (recirculating cavity flow driven by combined shear & body forces and lid-driven cavity flow) problems. Unlike the global 1D-IRBF scheme, the proposed method leads to a sparse system matrix. Numerical results indicate that (i) the present solutions are more accurate and converge faster with grid refinement in comparison with standard finite-difference results; and (ii) the proposed boundary treatment for the pressure is more effective than conventional direct application of the Neumann boundary condition.

Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems

In this paper, a multigrid method based on the high order compact (HOC) difference scheme on nonuniform grids, which has been proposed by Kalita et al. [J.C. Kalita, A.K. Dass, D.C. Dalal, A transformation-free HOC scheme for steady convection–diffusion on non-uniform grids, Int. J. Numer. Methods Fluids 44 (2004) 33–53], is proposed to solve the two-dimensional (2D) convection diffusion equation. The HOC scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids, consequently, the multigrid method is brand-new for solving the discrete system arising from the difference equation on nonuniform grids. The corresponding multigrid projection and interpolation operators are constructed by the area ratio. Some boundary layer and local singularity problems are used to demonstrate the superiority of the present method. Numerical results show that the multigrid method with the HOC scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems. The present method is also applied to solve the 2D incompressible Navier–Stokes equations using the stream function–vorticity formulation and the numerical solutions of the lid-driven cavity flow problem are obtained and compared with solutions available in the literature.