Driven Cavity Problem Research Articles

A new compact scheme-based Lax–Wendroff method for high fidelity simulations

The classical Lax–Wendroff (LW) method employs explicit second order central difference schemes to solve partial differential equations. Recently, the authors in Sengupta et al. (2023) have identified numerical parameter ranges in performing large eddy simulation (LES) using this classical LW method. In a bid to improve these numerical parameter ranges for high fidelity computations, a new non-uniform grid based compact scheme for LW method (NUCLW) is developed here. This new method uses sixth order non-uniform compact scheme (NUC6) for the first and higher order derivatives. Reported global spectral analysis confirms substantial improvement in the resolution and accuracy of NUCLW scheme over the classical LW scheme. A significant advantage of the present scheme is its ability to compute solutions for non-uniform, structured grids, and that does not suffer degradation in accuracy. The potential of the NUCLW scheme for high accuracy computations is demonstrated by solving 2D incompressible Navier–Stokes equations for the benchmark unsteady flow inside a square lid driven cavity problem for a subcritical Reynolds number of 3200 and a post-critical Reynolds number of 10,000 and simulating the instability of the Taylor–Green vortex problem. These demonstrate the ability of the NUCLW scheme to solve the Navier–Stokes equations for high fidelity simulations such as LES/DNS with enhanced ranges of numerical parameters with respect to enhanced accuracy and faster computations.

Computational modelling of compressible nonisothermal viscoelastic fluids

The prediction of compressible nonisothermal flows of viscoelastic fluids is important in many industrial processes. Until relatively recently, there was a lack of tractable thermodynamically consistent mathematical models for this class of flows. In this paper a stabilised finite element scheme is presented for the models developed by the authors Mackay and Phillips (2019) which incorporates compressibility and nonisothermal effects. Numerical results are presented for variants of a couple of benchmark problems: the lid driven cavity and the natural convection problems. The temporal discretisation is based on the Taylor–Galerkin method. A compressible version of the discrete elastic viscous split stress (DEVSS) formulation is used to stabilise the numerical scheme. The combined and separate influence of compressible, viscoelastic and thermal effects on the characteristics of these benchmark flows is studied for the first time.

Extraction of Lagrangian Coherent Structures in the framework of the Lagrangian–Eulerian Stabilized Collocation Method (LESCM)

In recent years, the investigations on Lagrangian Coherent Structures (LCS) in complex flows have attracted increasing attention due to their ability to accurately describe flow field details. In this study, we propose a novel and accurate numerical technique based on the Lagrangian–Eulerian Stabilized Collocation Method (LESCM) for computing the Finite Time Lyapunov Exponents (FTLEs), which is essential for extracting LCSs in viscous incompressible flows. LESCM, previously employed for simulating fluid–structure interaction problems, was developed based on the Material Point Method (MPM). The hybrid Lagrangian–Eulerian​ description in LESCM enables explicit tracking of fluid flow trajectories throughout the simulation and direct evaluation of the FTLE field. Furthermore. The errors in FTLEs caused by the Particle Shifting Technique (PST) are completely avoided due to the Eulerian characteristic of LESCM as the deformation gradient is calculated based on fixed Eulerian nodes rather than fluid particles with unphysical shifting. Consequently, the novel technique based on LESCM surpasses the accuracy of pure Lagrangian particle methods and provides an accurate way of detecting complex LCSs in flow fields. Additionally, By harnessing the remarkable efficiency of LESCM, MATLAB can now handle up to 16 million particles with ease, eliminating the need for parallel computation techniques. Several numerical examples, such as 2D Taylor-Green​ vortices, flow passing a circular cylinder and a 3-dimensional shear driven cavity problem, have been tested to demonstrate the effectiveness of this novel approach under various conditions.

Variational multiscale stabilized finite element analysis of non-Newtonian Casson fluid flow model fully coupled with Transport equation with variable diffusion coefficients

This paper presents a new subgrid multiscale stabilized formulation for non-Newtonian Casson fluid flow model tightly coupled with variable diffusion coefficients Advection–Diffusion–Reaction equation (VADR). Both the subscales are chosen to be time dependent and the stabilized formulation has been reached through the complete elimination of the unresolvable scales in terms of the coarse scale solution. Hence the resultant formulation emerges as a set of equations involving only the coarse scale solution instead of multiple scales and makes the formulation simpler to handle. In this study the shear-rate dependent Casson viscosity coefficient is assumed to be a function of the solute mass concentration, resulting in a two way coupling. This paper investigates the stability and the convergence properties of the stabilized finite element solution, without imposing any additional regularity condition other than the admissible space requirement on it. The proposed expressions of the stabilization parameters play a significant role in obtaining optimal order of convergences. During various theoretical derivations the estimations of the non-linear apparent viscosity coefficient have been carefully carried out assuming sufficiently regular exact solution. The performance of the scheme has been numerically validated with lid driven cavity problem. The theoretically established rate of convergence results are appropriately verified through numerical studies. In addition the transient Casson fluid flow behavior in a flow past square cylinder has been analyzed thoroughly.

A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations

For highly coupled nonlinear incompressible magnetohydrodynamic (MHD) system, a well-known numerical challenge is how to establish an unconditionally energy stable linearized numerical scheme which also has a fully decoupled structure and second-order time accuracy. This paper simultaneously reaches all of these requirements for the first time by developing an effective numerical scheme, which combines a novel decoupling technique based on the “zero-energy-contribution” feature satisfied by the coupled nonlinear terms, the second-order projection method for dealing with the fluid momentum equations, and a finite element method for spatial discretization. The implementation of the scheme is very efficient, because only a few independent linear elliptic equations with constant coefficients need to be solved by the finite element method at each time step. The unconditional energy stability and well-posedness of the scheme are proved. Various 2D and 3D numerical simulations are carried out to illustrate the developed scheme, including convergence/stability tests and some benchmark MHD problems, such as the hydromagnetic Kelvin-Helmholtz instability, and driven cavity problems.

Metaheuristic to Optimize Computational Convergence in Convection-Diffusion and Driven-Cavity Problems

This work presents an optimization proposal to better the computational convergence time in convection-diffusion and driven-cavity problems by applying a simulated annealing (SA) metaheuristic, obtaining optimal values in relaxation factors (RF) that optimize the problem convergence during its numerical execution. These relaxation factors are tested in numerical models to accelerate their computational convergence in a shorter time. The experimental results show that the relaxation factors obtained by the SA algorithm improve the computational time of the problem convergence regardless of user experience in the initial low-quality RF proposal.

Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number

We review a number of preconditioners for the advection-diffusion operator and for the Schur complement matrix, which, in turn, constitute the building blocks for Constraint and Triangular Preconditioners to accelerate the iterative solution of the discretized and linearized Navier-Stokes equations. An intensive numerical testing is performed onto the driven cavity problem with low values of the viscosity coefficient. We devise an efficient multigrid preconditioner for the advection-diffusion matrix, which, combined with the commuted BFBt Schur complement approximation, and inserted in a 2×2 block preconditioner, provides convergence of the Generalized Minimal Residual (GMRES) method in a number of iteration independent of the meshsize for the lowest values of the viscosity parameter. The low-rank acceleration of such preconditioner is also investigated, showing its great potential.

Convergence Studies on Popular Computational Schemes for Solution of Navier-Stokes Equation for Incompressible Flow

Most of the fluid flow situations may be described by non-linear partial differential Navier-Stokes equations. For incompressible flow, fluid density is assumed constant and hence equation of state cannot be utilized for coupling the continuity equation with the momentum equation. Researchers have developed and used the SIMPLE, SIMPLER and derived quantity based schemes for solution of incompressible Navier Stokes equations. Solution convergence is an important criteria for judging any numerical scheme and hence convergence studies are performed on a standard 2-D lid driven cavity problem for SIMPLE, SIMPLER and stream function-vorticity based method to solve N-S equations for incompressible flow. Convergence studies are performed and reported for two values of Reynolds number i.e. 100 and 400.

Preliminary verification of incompressible Navier-Stokes equations solved by The Newton method

The thermal-hydraulics module for reactor multi-physics coupling is studied. Multi-physics coupling for high temperature gas cooling reactor is very important due to the interaction between different physics, especially neutronics and thermal-hydraulics. This paper demonstrates an Incompressible Navier-Stokes (INS) module in the Multiphysics Object-Oriented Simulation Environment (MOOSE) to validate the accuracy and efficiency of the module, and the research on INS module is the significant basis of the future coupling work. MOOSE is an open-source finite element platform developed by Idaho National Lab (INL), which is for solving nonlinear equations from different physics simultaneously and can also couple different codes implicitly. The Jacobian-free Newton Krylov (JFNK) method adopted in MOOSE uses the finite difference method to avoid explicitly solving the Jacobian matrix so as to effectively save memory, and different preconditioning methods are also applied to accelerate the Krylov iteration. Besides, the Newton method is also supplied in MOOSE to solve nonlinear equations. Incompressible Navier-Stokes equations have been solved by the Newton method through the finite element method in MOOSE, and different numerical solution methods can be used to solve different cases. MOOSE regards problem equations as kernels and boundary conditions, and solves the discretized equations using the embedded Petsc or Trilinos solver. A convective heat transfer lid driven cavity problem is calculated using the INS module and the CFD tool Fluent to compare simulation results, and the temperature average error of 0.31% validates the module accuracy. In addition, JFNK and Newton methods are adopted to solve the same problem, and the results reveal that Newton method can save nearly 23% of the calculation time and has higher efficiency. The simulation results prove the module capability of applying to the reactor thermal-hydraulics, multi-physics coupling and safety analysis.

A comprehensive finite element examination of Carreau Yasuda fluid model in a lid driven cavity and channel with obstacle by way of kinetic energy and drag and lift coefficient measurements

In current pagination the flow problems involving shear rate dependent nonlinear viscosity have been treated successfully in the sense of space discretization and solvers. The stable finite element pair Q2/P1disc is employed to approximate the velocity and pressure spaces independently. Discretized form of non-linear expressions is linearized by implementing Newton’s procedure and the resulting systems are solved using a geometric multigrid approach. The flow generated by way of driven cavity and by an obstacle are very important benchmarks of computational fluid dynamics. In current pagination shear rate reliant viscosity model renowned as Carreau Yasuda fluid is capitalized. The obtained results are demonstrated and analyzed with the help of velocity and viscosity plots. In addition, we have produced new reference data for kinetic energy (K.E) for driven cavity problem and drag and lift coefficients for circular obstacle problem. The obtained results for driven cavity problem reveal the fact that K.E is an increasing function of the relaxation parameter (λ) and power law exponent (n) whereas a decreasing function of the model parameter (a). For the case of flow around obstacle the drag coefficient (CD) and the lift coefficient (CL) show strong dependence on (λ) and (n), however a weak dependence on the parameter (a).

Pressure - velocity projection method with mixed type approximation for Oseen discrete operator

We are considering the problem of divergence - free projection in rectangular domains for the Oseen (stationary Navier-Stokes) operator. It uses nodal Finite Element method to reconstruct pressure and Finite Difference method for scalar functions and vector functions for other parts of the equations. This allows us to use simple finite difference schemes for linear part and WENO-type schemes for the the nonlinear part in the space of discrete velocities. The boundary conditions are formulated in Finite Volume conservative approach for velocity and in Finite Element approach for pressure. We illustrate that the convergence rate depends on the smoothness of the projected function and that the projection method allows one to obtain divergence-free solutions with desired tolerance. We numerically analize the projection operator that acts in the space of divergence operator image. It is shown that all eigenvalues of the operator are less than unity for Oseledets-type wall boundary conditions for velocity and zero Neumann conditions for the Poisson equation. We show that for smooth functions we obtain forth order convergence in divergence. Finally we demonstrate the method by solving the stationary lid driven cavity problem for the 2D Oseen operator.

Modeling and Simulation of Top and Bottom Lid Driven Cavity using Lattice Boltzmann Method

Simulation of computational fluid dynamic (CFD) in isothermal fluid flow is very important in engineering and modeling problem. The simulation can explain the closed-flow behavior as complement with experimental field. Lid Driven Cavity as the tool for CFD benchmark is important to investigate. Lattice Boltzmann Method (LBM) as non-conventional CFD is used as present method investigation. The keys parameters of lid driven cavity problem are Reynolds number and the parts of moving boundaries that driven the cavity. In the present investigation was used the variations of Reynolds numbers: 500, 1000 and 2000. The lid that driven the cavity are top and bottom part. The results presented in visual flow pattern and in two dimensional velocity plot.

Development of General Finite Differences for complex geometries using a sharp interface formulation

We apply sharp interface boundary conditions to collocated meshfree approximations. Using the sharp interface formulation with the General Finite Difference (GFD) shape functions, we solve several test cases on meshfree grids with uniform and variable resolutions. Numerical results for the 2D Poisson equation indicate that the sharp interface formulation performs as well as the traditional collocated boundary approach with respect to both the convergence rate and accuracy. The analytic lid driven cavity results indicate that our discretization of a semi-implicit approximate projection scheme is indeed consistent and spatially converges at a second order rate under diffusive time stepping and at a first order rate under convective time stepping due to the first order in time splitting error. To resolve sharp gradients, variable resolution meshfree grids are considered for the classic lid driven cavity problem and for the uniform flow over a cylinder problem. Comparison to appropriate data sets indicate that our approach accurately estimates the various flow parameters using small cloud sizes of 13 neighbors while using high clustering ratios.

Solution of lid-driven cavity problems with an improved SIMPLE algorithm at high Reynolds numbers

The SIMPLE algorithm and its variants have been successfully employed to solve the driven-cavity problem at Re<10,000. However, converged steady solutions cannot be achieved at higher Re (15,000–30,000) by SIMPLE-family algorithms owing to its inherent drawbacks. In this work, a new segregated solver, ISTEC-N (incompressible flow solver with two explicit N correction loops), for determining the solution of incompressible flow on structured collocated grid systems is proposed. A pressure equation is deduced from the discretized continuity equation to overcome the “Semi-Implicit” characteristic of the SIMPLE algorithm. An explicit velocity update loop is introduced to improve the consistency between the pressure field and the velocity field in each outer iteration. The performance of the ISTEC-N algorithm is evaluated by solving the driven-cavity problem at high Re number of (15,000<Re<30,000). It is shown that the ISTEC-N algorithm manifests a strong robustness feature under these high convection conditions. The computational results given by the ISTEC-N algorithm are in excellent agreement with the benchmarking data.

A rational high-order compact difference method for the steady-state stream function–vorticity formulation of the Navier–Stokes equations

A rational high-order compact (RHOC) finite difference (FD) method on the nine-point stencil is proposed for solving the steady-state two-dimensional Navier–Stokes equations in the stream function–vorticity form. The resulting system of algebra equations can be solved by using the point-successive over- or under-relaxation (SOR) iteration. Numerical experiments, involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward-facing step flows, are carried out to validate the performance of the newly proposed method. Numerical solutions of the driven cavity problem with different grid mesh sizes (maximum being 513×513) for Reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature.

Localized method of approximate particular solutions for solving unsteady Navier–Stokes problem

The localized method of approximate particular solutions (LMAPS) is proposed to solve two-dimensional transient incompressible Navier–Stokes systems of equations in primitive variables. The equations contain the Laplacian operator. In avoiding ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear system within local supporting domain in which five nearest neighboring points and multiquadrics are used for interpolation. Then local matrices are reformulated in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The method is assessed on driven cavity problem and flow around cylinder. The numerical experiments show that the newly developed LMAPS is suitable for solving incompressible Navier–Stokes equations with high accuracy and efficiency.

Modified fully discretized projection method for the incompressible Navier–Stokes equations

The stability and convergence of a second-order fully discretized projection method for the incompressible Navier–Stokes equations is studied. In order to update the pressure field faster, modified fully discretized projection methods are proposed. It results in a nearly second-order method. This method sacrifices a little of accuracy, but it requires much less computations at each time step. It is very appropriate for actual computations. The comparison with other methods for the driven-cavity problem is presented.

Parallel control-volume method based on compact local integrated RBFs for the solution of fluid flow problems

In this paper, a high performance computing method based on the Integrated Radial Basis Function (IRBF), Control Volume (CV) and Domain Decomposition technique for solving Partial Differential Equations is presented. The goal is to develop an efficient parallel algorithm based on the Compact Local IRBF method using the CV approach, especially for problems with non-rectangular domain. The results showed that the goal is achieved as the computational efficiency is quite significant. For the case of square lid driven cavity problem with Renoylds number 100, super-linear speed-up is also achieved. The parallel algorithm is implemented in the Matlab environment using Parallel Computing Toolbox based on Distributed Computing Engine.

The Taylor Vortex and the Driven Cavity Problems in the Stream Function-Vorticity Formulation

In this work, two problems will be presented: The Taylor Vortex problem and the Driven Cavity problem. Both problems are solved using the Stream function-Vorticity formulation of the Navier-Stokes equations in 2D. Results are obtained using two methods: A fixed point iterative method and another one working with matrixes A and B resulting from the discretization of the Laplacian and the advective term, respectively. This second method resulted faster than the fixed point iterative one.

Moving least squares simulation of free surface flows

In this paper a Moving Least Squares method (MLS) for the simulation of 2D free surface flows is presented. The emphasis is on the governing equations, the boundary conditions, and the numerical implementation. The compressible viscous isothermal Navier–Stokes equations are taken as the starting point. Then a boundary condition for pressure (or density) is developed. This condition is applicable at interfaces between different media such as fluid–solid or fluid–void. The effect of surface tension is included. The equations are discretized by a moving least squares method for the spatial derivatives and a Runge–Kutta method for the time derivatives. The computational frame is Lagrangian, which means that the computational nodes are convected with the flow. The method proposed here is benchmarked using the standard lid driven cavity problem, a rotating free surface problem, and the simulation of drop oscillations. A new exact solution to the unsteady incompressible Navier–Stokes equations is introduced for the rotating free surface problem.