Lid-driven Cavity Flow Problem Research Articles

Shannon Entropy Computations in Navier-Stokes Flow Problems Using the Stochastic Finite Volume Method.

The main aim of this study is to achieve the numerical solution for the Navier-Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation technique and the Monte Carlo scheme, are engaged for this purpose. It is implemented with the aid of the polynomial bases for the pressure-velocity-temperature (PVT) solutions, for which the weighted least squares method (WLSM) algorithm is applicable. The deterministic problem is solved using the freeware OpenFVM, the computer algebra software MAPLE 2019 is employed for the LSM local fittings, and the resulting probabilistic quantities are computed. The first two probabilistic moments, as well as the Shannon entropy spatial distributions, are determined with this apparatus and visualized in the FEPlot software. This approach is validated using the 2D heat conduction benchmark test and then applied for the probabilistic version of the 3D coupled lid-driven cavity flow analysis. Such an implementation of the SFVM is applied to model the 2D lid-driven cavity flow problem for statistically homogeneous fluid with limited uncertainty in its viscosity and heat conductivity. Further numerical extension of this technique is seen in an application of the artificial neural networks, where polynomial approximation may be replaced automatically by some optimal, and not necessarily polynomial, bases.

Entropically damped artificial compressibility for the discretization corrected particle strength exchange method in incompressible fluid mechanics

We present a consistent mesh-free numerical scheme for solving the incompressible Navier–Stokes equations. Our method is based on entropically damped artificial compressibility for imposing the incompressibility constraint explicitly, and the Discretization-Corrected Particle Strength Exchange (DC-PSE) method to consistently discretize the differential operators on mesh-free particles. We further couple our scheme with Brinkman penalization to solve the Navier–Stokes equations in complex geometries. The method is validated using the 3D Taylor–Green vortex flow and the lid-driven cavity flow problem in 2D and 3D, where we also compare our method with hr-SPH and report better accuracy for DC-PSE. In order to validate DC-PSE Brinkman penalization, we study flow past obstacles, such as a cylinder, and report excellent agreement with previous studies.

Approximating the Navier-Stokes Equations with the Explicit Finite Difference Method to Solve the Lid-driven Cavity Flow Problem

The Navier-Stokes equations, one of the seven Millennium Problems, are a set of 2nd-order partial differential equations that are near impossible to solve. Instead of solving it, this paper uses the finite difference method to convert the differential terms in the Navier-Stokes Equations into algebraic equations to generate approximate solutions. The procedure to use the explicit finite difference method is outlined and proved in the context of solving the lid-driven cavity flow problem. The solution discussed is then applied in Mat lab and its accuracy and speed is evaluated. The finite difference method is then evaluated and its applications are outlined.

Mesoscopic simulation of MHD mixed convection of non-newtonian ferrofluids with a non-uniformly heated plate in an enclosure

Mixed convective study has been popular in recent years because of its large applications, including the cooling of electronic devices, furnaces, lubrication technologies, high-performance building insulation, multi-shield structures used in nuclear reactors, food processing, glass manufacturing, solar power collectors, drying technologies, chemical processing equipment, and others involve mixed convection in a lid-driven cavity flow problems. Graphics process unit (GPU) based multiple-relaxation-time(MRT) lattice Boltzmann method (LBM) has been employed for investigating the numerical simulation of magnetohydrodynamic(MHD) mixed convection with a non-uniformly heated plate at the mid of an enclosure. The physical model consists of a two-dimensional square enclosure with the top wall moving at a constant speed. Thermally adiabatic conditions are imposed on the top and bottom walls, while the two vertical walls are cold. In the center of the enclosure, a plate has been placed that is non-uniformly heated. A magnetic field is applied with different angles of inclination. Numerical simulations were performed for various influential parameters such as Richardson number (Ri), Hartmann number (Ha), power-law index (n), ferroparticles volume fraction (ϕ), magnetic field angle (γ) to study the flow phenomena in terms of the velocity and temperature distributions as well as streamlines and isotherms, respectively. The present study also investigates entropy generation due to the convective heat transfer flow for industrial purposes. The results reveal that as the Richardson number rises, the average Nusselt number rises, and as the Hartmann number rises, the average Nusselt number reduces. Furthermore, it is found that the average Nusselt number is inversely proportional to the power-law index. Total entropy generation increases with the increase of the power-law index and Richardson number. Entropy due to fluid friction, heat transfer, and total entropy shows a maximum at γ = 90°. Previously, MHD natural convection of the ferrofluid has been studied in a square cavity with a non-uniformly heated plate for the Newtonian fluid using the finite volume method. The aim of the present investigation is to study the MHD mixed convection with non-Newtonian ferrofluid using the GPU-based lattice Boltzmann method, which provides quicker and more robust results.

A less time-consuming upwind compact difference method with adjusted dissipation property for solving the unsteady incompressible Navier-Stokes equations

In this paper, a new strategy to establish less time-consuming upwind compact difference method with adjusted dissipation is introduced for solving the incompressible Navier-Stokes (N-S) equations in the streamfunction-velocity form efficiently. By weighted combination of the numerical solutions calculated using the upwind term and the downwind term of the general third-order upwind compact scheme (UCD3), a new fourth-order compact formulation and a third-order upwind compact formulation with adjusted dissipation nature are proposed for computing the first derivatives. Further, they are used to approximate the biharmonic term and the convective terms in the streamfunction-velocity formulation of the N-S equations, respectively. Meanwhile, the first derivatives of the streamfunction (velocities) in the coefficients in the convective terms are solved by the newly proposed fourth-order compact formulation. Temporal discretization for the streamfunction-velocity formulation is addressed with the help of the second-order Crank-Nicolson scheme. Moreover, the newly proposed scheme for the linear models is proved to be unconditionally stable by virtue of the discrete Fourier analysis. Finally, five numerical problems, viz. the analytic solution, Taylor-Green vortex problem, doubly periodic double shear layer flow problem, lid-driven square cavity flow problem and two-sided square cavity flow problem are solved numerically to verify the efficiency and accuracy of the present method. Results solved by the present method match well with the analytic solutions and the existing results proving the accuracy of it. What is more, it is less time-consuming and has lower dissipation than the existing method [20] .

A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains

In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using P1−P1 elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model.

Lagrangian differencing dynamics for incompressible flows

A Lagrangian meshless method is introduced for numerical simulation of flows of incompressible fluids with free surface. Poisson Pressure Equation (PPE) formulation of Navier-Stokes equations is discretised in Lagrangian context using recently introduced spatial operators based on finite differences. The advection adjusts to moving or deforming boundaries, and volume-conservation is enforced by solving a set of geometrical constraints. A technique for detecting the edge or boundary of a point cloud is introduced, which is based on the Laplacian properties for detecting jumps. Since accurate and fast discrete versions of the operator are not straightforward to implement in meshless context, the technique exploits favourable properties of a discrete Laplacian that handles highly irregular arrangements of points. A geometrical scan-sphere test is optionally performed to verify that points are on the point–cloud boundary. The boundary–detection technique is validated on significantly deformed point clouds by artificial numerical experiments, and by simulating unsteady incompressible flows. The method was validated by simulating lid-driven cavity flow, dam-break and water-entry problems, and comparing the kinematics of flow and hydrodynamic loading during the impact. The method reproduced complex free surface shapes, violent fluid-structure interaction, and pressure fields with second-order accuracy.

Driven flow motion by a dually moving lid of a square cavity

One of the important prototype problems of fluid mechanics, the lid-driven cavity flow is revisited such that the single lid is separated into two joint active/passive walls representing potential stirrers during a chemical mixing process. The right part of the wall is released freely to move right with a uniform velocity, while the left part attached to the adjacent wall at a point of dislocation is considered stationary or movable freely with a constant speed. The novel cavity wall mechanism introduced in this manner has the capability of driving the pressure of the fluid and resulting in formation of the primary/secondary vortex structures. This sort of cavity setup might be useful if mixing process of a fluid is demanded to be in different sections of the cavity or if a mixture is desired to be purified in different sections of the cavity. Special attention should be drawn here that this kind of wall action is dissimilar to the familiar lid-driven cavity flow problem due to moving one, two or more parallel/crossing sides of the cavity studied earlier in the literature. Having conveniently made the usual equations of motion non dimensional, three dimensionless parameters are arisen governing the physical phenomenon, namely the point linking the lid parts, the velocity of left part of the lid and the Reynolds number associated with the right part. A finite element approach is adopted to simulate the resulting pressure and flow fields affected by variation of these parameters. From the simulations, the mixing properties of flow as a result of the triggered pressure are elucidated at three distinct parts of the lid; (a) when the lid is separated close to the left portion of the wall, (b) when the lid separation is set at the middle of the wall and (c) when the parts of the lid are connected next to the right portion of the lid. Depending on the above scenarios, a quite rich phenomenon of pressure distribution and the resultant recirculating primary/secondary vortex flow in the form of streamline patterns is observed. Assisting and opposing influences of the dually moving lid and its control over the small and high Reynolds number driven flow regimes are explored through graphical illustrations as well as tabular forms.

Parallelization of lattice Boltzmann method for CFD using message passing interface

The lattice Boltzmann method has become a promising numerical technique and is now being considered as an alternative to the conventional CFD methods owing a possibility to simulate more complex geometries at relatively low computational costs. The simulations of complex and very fine resolution computational domains in CFD is challenging due to the lack of memory resources and long processing times, therefore parallel computing is being considered as a promising way to cope with this ever increasing mission of computational power. In this work parallelization of 2-D lattice Boltzmann model based CFD code using message passing interface was performed to increase speedup factor for parallel computing. Lattice Boltzmann approach based CFD code of a benchmark Lid driven cavity flow problem was parallelized using different message passing interface subroutines and made to run on a cluster system of processors. The time and speed up factor for execution of the benchmark problem was investigated. The results showed that these message passing communications have little influence on the performance of the parallel lattice Boltzmann method.

A New Approach to Solving Two-Dimensional Unsteady Incompressible Navier-Stokes Equations

This paper proposes a new approach that combines the reduced differential transform method (RDTM), a resummation method based on the Yang transform, and a Padé approximant to the kinetically reduced local Navier-Stokes equation to find approximate solutions to the problem of lid-driven square cavity flow. The new approach, called PYRDM, considerably improves the convergence rate of the truncated series solution of RDTM and also is based on a simple process that yields highly precise estimates. The numerical results achieved by this method are compared to earlier studies’ results. Our results indicate that this method is more efficient and precise in generating analytic solutions. Furthermore, it provides highly precise solutions with good convergence that is simple to apply for great Reynolds and low Mach numbers. Moreover, the new solution’ graphs demonstrate the new approach’s validity, usefulness, and necessity.

Numerical Simulation of Slip effect on Lid-Driven Cavity Flow Problem for High Reynolds Number: Vorticity – Stream Function Approach

Incompressible 2-D Navier-stokes equations for various values of Reynolds number with and without partial slip conditions are studied numerically. The Lid-Driven cavity (LDC) with uniform driven lid problem is employed with vorticity - Stream function (VSF) approach. The uniform mesh grid is used in finite difference approximation for solving the governing Navier-stokes equations and developed MATLAB code. The numerical method is validated with benchmark results. The present work is focused on the analysis of lid driven cavity flow of incompressible fluid with partial slip conditions (imposed on side walls of the cavity). The fluid flow patterns are studied with wide range of Reynolds number and slip parameters.

Numerical modelling of vertical axis turbines using the actuator surface model

This paper presents the development and validation of a lightweight fluid solver based on the Navier–Stokes equations coupled to a special blade element method (BEM) for vertical axis turbines. This method is called Actuator Surface Model (ASM) and leads to a medium fidelity tool designed to bridge the gap between expensive blade resolving computational fluid dynamics (CFD) calculations and simple, often inaccurate momentum models such as the double-multiple streamtube (DMST) model. Since the simulation runs on coarse meshes, it is possible to use an explicit solver with a reasonable time step to significantly reduce the computing time to almost real-time speed. The fluid model is validated for the classical lid-driven cavity flow problem, while the coupled ASM code is compared with three different experimental measurements from the literature. For a more accurate description of blade specific phenomena, several higher order corrections were implemented, such as dynamic stall, parasitic drag and finite aspect ratio. The model is able to calculate the performance of all tested turbines reasonably well and can predict the maximum efficiency up to a few percent. Therefore it is an excellent alternative for a fast and precise initial design of vertical axis turbines compared to traditional BEM methods.

DGIRM: Discontinuous Galerkin based isogeometric residual minimization for the Stokes problem

In this paper, we introduce a stable isogeometric analysis discretization of the Stokes system of equations. We use this standard constrained problem to demonstrate the flexibility and robustness of the residual minimization method on dual stable norms [16], which unlocks the extraordinary approximation power of isogeometric analysis [44]. That is, we introduce an isogeometric residual minimization method (IRM) for the Stokes equations, which minimizes the residual in a dual discontinuous Galerkin norm; thus we use the acronym DGiRM. Following Calo et al. [16], we start from an inf-sup stable discontinuous Galerkin (DG) formulation to approximate in a highly continuous trial space that minimizes the dual norm of the residual in a discontinuous test space. We demonstrate the performance and robustness of the methodology considering a manufactured solution and the well-known lid-driven cavity flow problem. First, we use a multi-frontal direct solver, and, using the Pareto front, compare the resulting numerical accuracy and the computational cost expressed by the number of floating-point operations performed by the direct solver algorithm. Second, we use an iterative solver. We measure the number of iterations required when increasing the mesh size and how the configuration of spaces affect the resulting accuracy. This paper is an extension of the paper A Stable Discontinuous Galerkin Based Isogeometric Residual Minimization for the Stokes Problem by M. Łoś, et al. (2020) published in Lecture Notes in Computer Science. In this extended version, we deepen in the mathematical aspects of the DGiRM, include the iterative solver algorithm and implementation, and discuss the influence of different discretization spaces on the iterative solver's convergence.

A new modified skew upwind differencing scheme for convective flows

In this article, a new modified skew upwind differencing scheme (MSUDS) for convective flow is presented. This scheme is tested by pure advection test cases such as Step, Sine-square, Semi-ellipse, and Smith-Hutton test profiles. The benchmark lid-driven cavity flow problem is also solved with this MSUDS scheme. All the results are well plotted and validated. All the data regarding overshoots/undershoots are presented in tabular form for each test case. The maximum value of overshoot/undershoot using this scheme is of the order of about which is quite insignificant and that may be smoothened during iteration. The presented MSUDS scheme offers a range of choices depending on the convection problem. It is observed that the results obtained with the application of the presented MSUDS scheme is as good as that obtained with the other schemes such as high resolution equi-gradient scheme and higher order schemes developed till so far for solving convection problems. Its implementation is simple with less computational effort.

Numerical verification of a non-residual orthogonal term-by-term stabilized finite element formulation for incompressible convective flow problems

The modeling of turbulent flows is relevant in many engineering applications and, is an active field of research on numerical methods. Convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows. In this work, a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual based formulations, the introduction of two velocity subscale components. The theoretical rate of convergence of the method is verified numerically using linear and quadratic equal-order finite element discretizations. To this end, a standard convergence test of L2-norm is presented where the computed solutions are compared when manufactured solutions are imposed at Gauss-point level. Moreover, the Hopf bifurcation is studied for two well-known benchmark problems: flow past a cylinder and the three-dimensional lid-driven cavity flow. For the flow past a cylinder case, the Hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution. In particular, a dominant convective problem (Re=4000) is solved using both the quasistatic and dynamic versions of the method, evaluating the performance of each one in the quality of the solution and in the CPU time needed to obtain a converged solution. For the 3D lid-driven cavity flow problem, the Hopf bifurcation is determined using two different boundary conditions, analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer. The final test case is the turbulent 3D lid-driven cavity problem (Re=12000), where velocity profiles are compared with experimental, LES and DNS reference solutions. Additionally, pressure and velocity spectra are shown at certain representative points of the domain, as well as phase diagrams, correlation function graphs, the Poincaré map, and the Lyapunov exponent, typical mathematical tools used in dynamical system analysis. From the results, the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones.

Application of Lattice Boltzmann Method for fluid flow modelling of FSLDR domain

Abstract In this work Lattice Boltzmann method is used to solve the Four Sided lid-driven rectangular cavity flow (FSLDR) problem. The fluid is considered as incompressible. In the present problem all the four walls moves with a constant velocity. The left wall moves in positive y-direction, the right wall moves in negative y-direction. The top wall moves in positive x-direction and the bottom wall moves in negative x-direction. The aspect ratio of the cavity taken is 0.50. The present code is validated for single lid-driven cavity flow problem. Next, the study is extended to FSLSR problem. The position of vortex obtained are studied at Reynolds number Re=50, 100, 500, 1000. In addition to the primary vortex, two secondary vortices are also obtained. Thus, the present study shows that Lattice Boltzmann Method can be used to capture the details of vortex dynamics

BOUNDARY ELEMENT METHOD FOR THE LID-DRIVEN PROBLEM RELATED TO THE DARCY-FORCHHEIMER-BRINKMAN SYSTEM

The Darcy-Forchheimer-Brinkman system is one mathematical model that describes the motion of a viscous fluid located in a porous domain. Recently, many papers have established the existence of a unique solution for this system depending on the boundary condition imposed. In this paper we investigate numerically a special Dirichlet-Robin boundary value problem associated with the nonlinear Darcy-Forchheimer-Brinkman system, i.e., the lid-driven cavity flow problem, by using the dual reciprocity boundary element method (DRBEM). The physical properties of such a flow are discussed by the geometry of the streamlines of the fluid flow for different Reynolds and Darcy numbers. Moreover, an additional sliding parameter is imposed on the moving wall, and its importance is highlighted by segmenting the upper lid into two opposite moving parts.

A Multilayer Nonlinear Elimination Preconditioned Inexact Newton Method for Steady-State Incompressible Flow Problems in Three Dimensions

We develop a multilayer nonlinear elimination preconditioned inexact Newton method for a nonlinear algebraic system of equations, and a target application is the three-dimensional steady-state incompressible Navier--Stokes equations at high Reynolds numbers. Nonlinear steady-state problems are often more difficult to solve than time-dependent problems because the Jacobian matrix is less diagonally dominant, and a good initial guess from the previous time step is not available. For such problems, Newton-like methods may suffer from slow convergence or stagnation even with globalization techniques such as line search. In this paper, we introduce a cascadic multilayer nonlinear elimination approach based on feedback from intermediate solutions to improve the convergence of Newton iteration. Numerical experiments show that the proposed algorithm is superior to the classical inexact Newton method and other single layer nonlinear elimination approaches in terms of the robustness and efficiency. Using the proposed nonlinear preconditioner with a highly parallel domain decomposition framework, we demonstrate that steady solutions of the Navier--Stokes equations with Reynolds numbers as large as 7,500 can be obtained for the lid-driven cavity flow problem in three dimensions without the use of any continuation methods.

Discretized pressure Poisson algorithm for steady incompressible flow on two-dimensional triangular unstructured grids

Abstract Many numerical algorithms have developed for Computational Fluid Dynamics (CFD). Previously, we provided a discretized pressure Poisson algorithm for steady incompressible flow on structured grids. Though this algorithm is very simple, there are still some problems for this algorithm to be implemented on unstructured grids. In order to solve incompressible flow problems on unstructured grids simply and easily, this paper developed some strategies to implement discretized pressure Poisson algorithm for steady incompressible flow on unstructured grids. Firstly, this paper discretized the convective term and diffusion term of momentum equations easily by the quantities of nodes and nearby points. Combined with the incenter-based Finite Volume Method (FVM), discretization of governing equations on triangular unstructured grids could be more easily implemented. Secondly, for the application of discretized pressure Poisson algorithm, this paper provided a discretization strategy for the continuity equation on unstructured grids. Thus, the mass conservation could be easily satisfied during the iteration of velocity calculation, which bring great advantages for the velocity calculation on unstructured grids. Then, discretized pressure Poisson algorithm was successfully used to calculate the lid-driven cavity flow problem on unstructured grids; the whole process was very simple and the results were also reliable.