Driven Cavity Flow Research Articles

Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier–Stokes equations

To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier--Stokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible Navier--Stokes equations, and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured.Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the truncation errors of the generalized approximate operators, an optimal weight function is derived by reducing the truncation errors over general particle distributions. The effectiveness of the generalized approximate operators with the optimal weight functions is confirmed using numerical results of truncation errors and driven cavity flow. As an application for flow problems with free surface effects, the explicit particle method is applied to a dam break flow.

A hybrid immersed boundary-lattice Boltzmann/finite difference method for coupled dynamics of fluid flow, advection, diffusion and adsorption in fractured and porous media

In this paper, a hybrid immersed boundary-lattice Boltzmann/finite difference method is extended to simulate the coupled dynamics of fluid flow, advection, diffusion and adsorption in fractured and porous media which can describe gas migration and adsorption in the cleat-matrix system of coal. The numerical method includes three important components: fluid solver, advection-diffusion solver, and immersed boundary method for fluid-solid interaction with coupled mass exchange. In the fluid solver, the single-relaxation time lattice Boltzmann method is adopted for the fluid dynamics, and immersed boundary method is employed to achieve the no-slip boundary conditions at the fluid-solid interface. The advection-diffusion equation is solved by using the finite difference method, with immersed boundary method for the Neumann boundary conditions. This integrated method is extremely efficient for flows involving complex geometries and large deformation problems, as usually encountered in geosciences. Benchmark studies including the lid driven cavity flow, heat transfer around a stationary cylinder, and gas migration with adsorption in a channel are conducted to validate the efficiency and accuracy of the current solver. It is found that results predicted by our solver are in good agreement with published data achieved by other numerical or analytical methods, showing that the current method is robust and capable to solve the fluid-solid interaction and mass exchange involving complex geometries in geosciences. Finally, this method is employed to model gas migration in the cleat-matrix system of coal, involving the process of kinetic adsorption. This contribution is the first development of its kind for solving problems in geosciences. It is held in a generic numerical formulation so that can be applied to many fields of geosciences involving fluid flow and reaction diffusion through fractured rocks, gas storage in geological formations, groundwater contaminant studies, geothermal and resource applications.

Transition to chaos in an acoustically driven cavity flow

We consider the unsteady regimes of an acoustically-driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific question being addressed is to know whether the system can sustain states of low-dimensional chaos when the acoustic intensity driving the jet is increased, and, if so, to characterise the pathway and underlying physical mechanisms. We adopt two complementary approaches, both based on data extracted from numerical simulations: (i) We first characterise successive bifurcations through the analysis of leading frequencies. Two successive phases in the evolution of the system are singled out in this way, both leading to potentially chaotic states. The two phases are separated by a drastic simplification of the dynamics that immediately follows the emergence of intermittency. The second phase also features a second intermediate state where the dynamics is simplified due to frequency-locking. (ii) Nonlinear time series analysis enables us to reconstruct the attractor of the underlying dynamical system, and to calculate its correlation dimension and leading Lyapunov exponent. Both these quantities bring confirmation that the state preceding the dynamic simplification that initiates the second phase is chaotic. Poincar\'e maps further reveal that this chaotic state in fact results from a dynamic instability of the system between two non-chaotic states respectively observed at slightly lower and slightly higher acoustic forcing.

A discontinuous Galerkin fast spectral method for the full Boltzmann equation with general collision kernels

The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical reactions. Despite its wide applicability, deterministic solution of the Boltzmann equation presents a huge computational challenge, and often the collision operator is simplified for practical reasons. In this work, we introduce a highly accurate deterministic method for the full Boltzmann equation which couples the Runge–Kutta discontinuous Galerkin (RKDG) discretization in time and physical space (Su et al. (2015) [1]) and the recently developed fast Fourier spectral method in velocity space (Gamba et al. (2017) [2]). The novelty of this approach encompasses three aspects: first, the fast spectral method for the collision operator applies to general collision kernels with little or no practical limitations, and in order to adapt to the spatial discretization, we propose here a singular-value-decomposition based algorithm to further reduce the cost in evaluating the collision term; second, the DG formulation employed has high order of accuracy at element-level, and has shown to be more efficient than the finite volume method; thirdly, the element-local compact nature of DG as well as our collision algorithm is amenable to effective parallelization on massively parallel architectures. The solver has been verified against analytical Bobylev–Krook–Wu solution. Further, the standard benchmark test cases of rarefied Fourier heat transfer, Couette flow, oscillatory Couette flow, normal shock wave, lid-driven cavity flow, and thermally driven cavity flow have been studied and their results are compared against direct simulation Monte Carlo (DSMC) solutions with equivalent molecular collision models or published deterministic solutions.

Least Square Finite Element Method for Viscous Splitting of Unsteady Incompressible Navier–Stokes Equations

In order to solve unsteady incompressible Navier–Stokes (N–S) equations, a new stabilized finite element method, called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation. After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.

A posteriori error estimation for incompressible viscous fluid with a new boundary condition

This paper describes numerical solutions of incompressible Navier-Stokes equations with a new boundary condition. To solve this problem, we use the discretization by mixed finite element method. We use a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equations. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. A numerical experiment on the driven cavity flow is given to demonstrate the effectiveness of the vector extrapolation method. We compare the result with the solution from commercial code like ADINA system as well as with values from other simulations.

Experimental study of vibration of metallic and composite plates inside channel driven cavity flow

An experimental study was conducted for the fluid–structure interaction (FSI) of the vibration of a flexible structure inside the channel driven cavity flow (CDCF). The structure was a thin flat plate whose one edge was clamped at the bottom of the three-dimensional CDCF system. Both aluminum and composite plates were examined. Initially the vibrational characteristics of the structures were measured without fluid flow. Subsequently, each structural vibration was measured under various flow rates through the CDCF system. The structural vibration was measured using strain gages attached to the structures as well as non-contact displacement sensors. Time histories of the strains and displacements were obtained, and their time averages and the vibrational magnitudes were computed as a function of the nominal flow rate through the channel cross-section. This set of test data is expected to be useful for validation of a computer program for analyzing a FSI problem.

Discrete unified gas kinetic scheme for all Knudsen number flows. III. Binary gas mixtures of Maxwell molecules.

Recently a discrete unified gas kinetic scheme (DUGKS) in a finite-volume formulation based on the Boltzmann model equation has been developed for gas flows in all flow regimes. The original DUGKS is designed for flows of single-species gases. In this work, we extend the DUGKS to flows of binary gas mixtures of Maxwell molecules based on the Andries-Aoki-Perthame kinetic model [P. Andries et al., J. Stat. Phys. 106, 993 (2002)JSTPBS0022-471510.1023/A:1014033703134. A particular feature of the method is that the flux at each cell interface is evaluated based on the characteristic solution of the kinetic equation itself; thus the numerical dissipation is low in comparison with that using direct reconstruction. Furthermore, the implicit treatment of the collision term enables the time step to be free from the restriction of the relaxation time. Unlike the DUGKS for single-species flows, a nonlinear system must be solved to determine the interaction parameters appearing in the equilibrium distribution function, which can be obtained analytically for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure problem under different Mach numbers and molar concentrations, the channel flow driven by a small gradient of pressure, temperature, or concentration, the plane Couette flow, and the shear driven cavity flow under different mass ratios and molar concentrations. The results are compared with those from other reliable numerical methods. The results show that the proposed scheme is an effective and reliable method for binary gas mixtures in all flow regimes.

A reconstruction-based cell-centered high-order finite volume method for incompressible viscous flow simulation on unstructured meshes

A new high-order (> 2nd order) cell-centered finite volume method is presented for incompressible flow simulation on unstructured meshes. Artificial compressibility is employed to couple the continuity and momentum equations in a manner that allows them to be solved simultaneously. A new numerical stencil, a so-called wrapping stencil, is utilized for linear and quadratic solution reconstruction in order to achieve more accurate and robust solution reconstruction not only for the interior cells, but also for the cells on the boundary, where fewer neighboring cells typically exist. The effectiveness of the current algorithm is demonstrated by various test cases, including an analytical solution reconstruction test, Kovasznay flow simulations with various Reynolds numbers, a driven cavity flow, and flow past a square cylinder. Based on the comparison with the standard low order scheme, the proposed second and third order schemes, based on linear and quadratic solution reconstruction, show superior accuracy, which sheds light on the method's applicability in solving more challenging incompressible flow problems.

Lattice Boltzmann Simulations of Cavity Flows on Graphic Processing Unit with Memory Management

AbstractLattice Boltzmann method (LBM) is adopted to compute two and three-dimensional lid driven cavity flows to examine the influence of memory management on the computational performance using Graphics Processing Unit (GPU). Both single and multi-relaxation time LBM are adopted. The computations are conducted on nVIDIA GeForce Titan, Tesla C2050 and GeForce GTX 560Ti. The performance using global memory deteriorates greatly when multi relaxation time (MRT) LBM is used, which is due to the scheme requesting more information from the global memory than its single relaxation time (SRT) LBM counterpart. On the other hand, adopting on chip memory the difference using MRT and SRT is not significant. Also, performance of LBM streaming procedure using offset reading surpasses offset writing ranging from 50% to 100% and this applies to both SRT and MRT LBM. Finally, comparisons using different GPU platforms indicate that Titan as expected outperforms other devices, and attains 227 and 193 speedup over its Intel Core i7-990 CPU counterpart and four times faster than GTX 560Ti and Tesla C2050 for three dimensional cavity flow simulations respectively with single and double precisions.

Lattice Boltzmann simulations of the capillary pressure bump phenomenon in heterogeneous porous media

In this work, the Lattice Boltzmann method (LBM) is used to simulate the fluid displacement processes of primary drainage and imbibition in a randomly generated heterogeneous porous medium. The present procedure is based on the single-relaxation-time Bhatnagar–Gross–Krook (BGK) approximation and the original Shan-Chen multi-component multiphase (MCMP) model. The developed model is validated against classical single and multiphase flow problems, which include the single-phase Poiseulle flow and lid driven cavity flow, and the multiphase miscibility test, bubble test, and contact angle test. Simulation results show that the Lattice Boltzmann method successfully captures the capillary pressure bump phenomenon, which is a frequently encountered phenomenon in petroleum reservoirs. The bump in capillary pressure is shown to correspond to the instant when the fluid invades the narrow pore throats, which require a higher capillary pressure threshold.

Acceleration of a BEM based solution of the velocity–vorticity formulation of the Navier–Stokes equations by the cross approximation method

In this paper, we present a method to decrease the computational demand of the boundary-domain integral equation based 3D flow solver. We focus on the solution of the velocity–vorticity formulation of the Navier–Stokes equation, which governs incompressible viscous fluid flow. We introduce the cross approximation into the solution of the boundary vorticity values. This problem is governed by the Poisson type kinematics equation and presents a computational bottleneck of the algorithm. In order to accelerate the flow solver, we approximate the domain contribution of the kinematics integral equation by the cross approximation algorithm. The cross approximation method is used in combination with the hierarchical decomposition of the domain boundary combined by the hierarchical decomposition of domain interior. We propose to specify the approximation extent by controlling the depth of the hierarchical decomposition and the rank of the approximated integral matrix parts.The developed algorithm is tested using the Arnold–Beltrami–Childress and lid driven cavity flows. We study the accuracy of boundary vorticity estimation and of the flow solution for different flow complexities (Reynolds number values), computational mesh densities and cross approximation settings. We found that that by using the cross approximation technique in the flow solver, we were able to reduce the computational demands of storing matrices to approximately 30% of the storage space of the original matrices. Furthermore, we showed that achieved approximation extent depends on the complexity of the simulated flow problem.

Accelerated lattice Boltzmann simulation using GPU and OpenACC with data management

We assess the performance of the combined Open Accelerator (OpenACC) programming standard and graphics processing unit (GPU) accelerator for lattice Boltzmann (LB) simulations of fluid flow, heat and mass transfer. By optimizing the data layout, minimizing the memory access frequency, and adjusting the number of gangs and vector length, we show that the enhanced parallel computations can result in orders of magnitudes of speedup relative to the serial implementation of the LB algorithm. Based on such implementations, benchmark quality results are obtained with fine grid of 20492 for both two-dimensional lid driven cavity flow with Reynolds number up to 7500, and double diffusive cavity flow with solute Rayleigh number up to 108.

Lattice Boltzmann simulations of three-dimensional incompressible flows in a four-sided lid driven cavity

Numerical study on three-dimensional (3D), incompressible, four-sided lid (FSL) driven cavity flows has been conducted to show the effects of the transverse aspect ratio, K, on the flow field by using a multiple relaxation time lattice Boltzmann equation. The top wall is driven from left to right, the left wall is moved downward, whereas the right wall is driven upward, and the bottom wall is moved from right to left, all the four moving walls have the same speed and the others boundaries are fixed. Numerical computations are performed for several Reynolds numbers for laminar flows, up to 1000, with various transverse aspect ratios. The flow can reach a steady state and the flow pattern is symmetric with respect to the two cavity diagonals (i.e., the center of the cavity). At Reynolds number = 300, the flow structures of the 3D FSL cavity flow at steady state with various transverse aspect ratio, i.e., 3, 2, 1, 0.75, 0.5 and 0.25 only show the unstable symmetrical flow pattern. The stable asymmetrical flow pattern could be reproduced only by increasing the Reynolds number that is above a critical value which is dependent on the aspect ratio. It is found that an aspect ratio of more than 5 is needed to reproduce flow patterns, both symmetric and asymmetric flows, simulated by using 2D numerical models.

Numerical Simulations of the Square Lid Driven Cavity Flow of Bingham Fluids Using Nonconforming Finite Elements Coupled with a Direct Solver

In this paper, numerical simulations are performed in a single and double lid driven square cavity to study the flow of a Bingham viscoplastic fluid. The governing equations are discretized with the help of finite element method in space and the nonconforming Stokes elementQ~1/Q0is utilized which gives 2nd-order accuracy for velocity and 1st-order accuracy for pressure. The discretized systems of nonlinear equations are treated by using the Newton method and the inner linear subproablems are solved by the direct solver UMFPACK. A qualitative comparison is done with the results reported in the literature. In addition to these comparisons, some new reference data for the kinetic energy is generated. All these implementations are done in the open source software package FEATFLOW which is a general purpose finite element based solver package for solving partial differential equations.

Mixed Convection and Heat Transfer Studies in Non-Uniformly Heated Buoyancy Driven Cavity Flow

We analyse the mixed convection flow in a cavity flow which is driven by buoyancy generated due to a non-uniformly heated top wall which is moving uniformly. A fourth order accurate finite difference scheme is used in this study and our code is first validated against available data in the literature. The results are obtained for different sets of Reynolds number Re, Prandtl number Pr and Grashof number Gr which are in the ranges 100 - 3000, 0.0152 - 10 and 102 - 106 respectively. Here Gr is related to the Richardson number according to Ri=Gr/Re2. While increasing the Richardson number, the growth of upstream secondary eddy (USE) is observed together with a degradation of downstream secondary eddy (DSE). When mixed convection is dominant, the upstream secondary eddy and the downstream secondary eddy merge to form a large recirculation region. When the effect of Pr is studied in the forced convection regime, Ri<<1, the temperature in the central region of the cavity remains nearly a constant. However, in the mixed convection regime, the temperature in cavity undergoes non-monotonic changes. Finally, using the method of divided differences, it is shown that numerical accuracy of the derived numerical scheme used in this work is four.

Experimental Study of Channel Driven Cavity Flow for Fluid–Structure Interaction

An experimental setup was designed and fabricated for the channel driven cavity flow in order to provide benchmark data for validation of any numerical analysis program for solving fluid–structure interaction (FSI) problems. The channel driven cavity flow is a modification from the lid-driven cavity flow. To provide the fluid–structure interaction, the bottom face of the cavity is a deformable flat plate. All other boundaries are rigid. The fluid motion inside the cavity is driven by the flow through a narrow channel topside of the cavity. To establish suitable boundary conditions for numerical analyses of the experiment, the inlet of the channel has a given fluid velocity, while its outlet has a known pressure. Water is used as the fluid in this study. Multiple strain gages and laser displacement sensors were used to measure dynamic responses of the plate attached at the bottom of the cavity.

A local numerical solution of a fluid-flow problem on an irregular domain

This paper deals with a numerical solution of an incompressible Navier-Stokes flow on non-uniform domains. The numerical solution procedure comprises the Meshless Local Strong Form Method for spatial discretization, explicit time stepping, local pressure-velocity coupling and an algorithm for positioning of computational nodes inspired by Smoothed Particles Hydrodynamics method. The presented numerical approach is demonstrated by solving a lid driven cavity flow and backward facing step problems, first on regular nodal distributions up to 315,844 (562×562) nodes and then on domain filled with randomly generated obstacles. It is demonstrated that the presented solution procedure is accurate, stable, convergent, and it can effectively solve the fluid flow problem on complex geometries. The results are presented in terms of velocity profiles, convergence plots, and stability analyses.

High-order fluid solver based on a combined compact integrated RBF approximation and its fluid structure interaction applications

In this study, we present a high-order numerical method based on a combined compact integrated RBF (IRBF) approximation for viscous flow and fluid structure interaction (FSI) problems. In the method, the fluid variables are locally approximated by using the combined compact IRBF, and the incompressible Navier-Stokes equations are solved by using the velocity-pressure formulation in a direct fully coupled approach. The fluid solver is verified through various problems including heat, Burgers, convection-diffusion equations, Taylor-Green vortex and lid driven cavity flows. It is then applied to simulate some FSI problems in which an elastic structure is immersed in a viscous incompressible fluid. For FSI simulations, we employ the immersed boundary framework using a regular Eulerian computational grid for the fluid mechanics together with a Lagrangian representation of the immersed boundary. For the immersed fibre/membrane FSI problems, although the order of accuracy of the present scheme is generally similar to FDM approaches reported in the literature, the present approach is nonetheless more accurate than FDM approaches at comparable grid spacings. The numerical results obtained by the present scheme are highly accurate or in good agreement with those reported in earlier studies of the same problems.

Numerical Analysis of the High Speed Driven Cavity Flow in 2-D Curved Channel

Numerical Analysis of the High Speed Driven Cavity Flow in 2-D Curved Channel