A Pressure-Correction Method for Incompressible Flows Using Unstructured Meshes

Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin–reduced integration penalty method

The numerical simulation of incompressible turbulent flow at Re = 11 800 (equivalent Reynolds number Rem = 10 030) inside lid-driven cavities with varying spanwise aspect ratio and depthwise aspect ratio (DAR) has been carried out using the dynamic Smagorinsky model. A better comparison of central line averaged velocities and second order turbulent properties is obtained with the existing experimental result of Prasad and Koseff [“Reynolds number and end-wall effects on a lid-driven cavity flow,” Phys. Fluids A 1, 208–218 (1989)] than with the one provided by Leriche and Gavrilakis [“Direct numerical simulation of the flow in a lid-driven cubical cavity,” Phys. Fluids 12, 1363–1376 (2000)]. It is noticed that the size of the downstream secondary vortex increases with an increase in the aspect ratio. Lower DAR cavities are found to be having the maximum turbulent kinetic energy (TKE) and turbulent production, which is further confirmed by the TKE spectra shown at the point of maximum turbulent production on the symmetrical mid-plane. The power spectra maintain a slope of −53 in the inertial subrange as proposed by Kolmogorov and show a steeper slope in the dissipating range. The time integral scale shows that in low DAR cavities, eddies are formed and undergo complete distortion more frequently. The joint probability density function confirms the anisotropy in turbulence, and the anisotropy invariant map (Lumley triangle) shows the type of turbulent anisotropy which shows whether the turbulence is either one-component or two-component near the walls.

Buoyancy-induced periodic flow and heat transfer in lid-driven cavities with different cross-sectional shapes

The aim of the numerical study is to investigate the mixed convection flow and heat transfer in a square lid-driven cavity with corner heating and internal heat generation or absorption. The top wall of the cavity is moving at a constant speed in its own plane. A portion of the left and bottom walls of the cavity are kept at a constant temperature; that is, the heaters are fixed at the left bottom corner of the cavity. The right wall is maintained at lower temperature and the remaining portions of the cavity are insulated. The finite volume method is used to discretize the governing equations that are then solved iteratively. The results are obtained for different lengths of the heaters, Richardson numbers, and internal heat generation or absorption parameters. It is observed that the heat transfer rate is enhanced at forced convection dominated regime when the vertical length of the heater is higher than that of the horizontal length.

This paper presents the large-eddy simulation of the lid-driven cubic cavity flow by the spectral element method (SEM) using the dynamic model. Two spectral filtering techniques suitable for these simulations have been implemented. Numerical results for Reynolds number $\text{Re}=12'000$ are showing very good agreement with other experimental and DNS results found in the literature.

Numerical study of flow in a lid-driven cavity with internal square obstacle using FEM

ABSTRACTMixed convection flow in a 2D rectangular cavity is simulated by a novel finite element method, namely the projection- and characteristic-based operator-splitting algorithm. In each time step, the Navier–Stokes equations are split as follows: the diffusion part, the convection part by applying operator-splitting method, and the Poisson’s equation by adopting projection method. The implicit diffusion part is solved by the preconditioned conjugate gradient (PCG) method, whereas characteristic method is applied for the convection part in a multistep explicit scheme. The characteristic Galerkin approach is used to solve the energy equation. To validate the model, lid-driven cavity flow and natural convection flow are simulated.

In this study, a new filter-matrix lattice Boltzmann (FMLB) model is proposed and extended to include incompressible thermal flows. A new equilibrium solution is found in the improved FMLB model, which is derived from the Hermite expansion. As a result, the velocity-dependent pressure is removed, which is an inherent defect of Somers's FMLB model. In addition, the improved model is extended to include incompressible thermal flows by introducing a class of temperature-distribution function for evaluating the temperature field. Two different temperature-distribution functions are discussed. The improved FMLB model and the temperature-evaluation equation are combined into one coupled model. Numerical simulations are performed on the two-dimensional (2D) lid-driven square cavity flow and the 2D natural convection flow in a square cavity using the improved FMLB model and the two coupled models, respectively. The numerical results of the 2D lid-driven square cavity flow show that the improved FMLB model is superior to the lattice Bhatnagar-Gross-Krook (LBGK) model in terms of both accuracy and stability. When compared with the multi-relaxation-time (MRT) model, the similar accuracy and slightly enhanced stability can be obtained by the improved model. The advantage of the improved model is that it no longer relies on difficult selection of the free parameters requested by the MRT model; in addition, the force term is already included in the collision operator of the improved model. In the case of 2D natural convection flow, the numerical results of the two present models are almost the same, and both exhibit good agreement with the benchmark solution.

This paper reports on the SIMPLE-like solution algorithm that significantly improves velocity–pressure coupling, leading to the accelerated convergence. The algorithm is applicable to both incompressible and compressible flows. It is implemented within the framework of a cell centered finite volume method using colocated storage of flow variables on unstructured grids. The convergence acceleration is an outcome of the novel treatment of the pressure correction equation that accounts for two commonly neglected terms appearing in a general pressure correction equation. These terms, namely, the neighbor and nonorthogonal corrections, represent velocity corrections from neighboring cells and the cell-face pressure correction gradient associated with the grid nonorthogonality, respectively. Reminiscent of the PISO method, both neighbor and nonorthogonal corrections are taken into account by performing two or more correction steps. However, the full inclusion of neighbor velocity corrections can prevent the solution convergence. Following an analogy between time marching and a steady-state iterative approach, this problem has been resolved by introducing a pseudounsteady term into discretized velocity correction equations. Using this term, the contribution of neighbor corrections is underrelaxed, enabling a satisfactory convergence rate. The algorithm is applied to several benchmark test cases, covering laminar and turbulent, as well as incompressible and compressible, flows. For all test cases, the convergence rate can be significantly improved by performing two or more pressure correction steps. An optimal number of pressure corrections exists for which a meaningful reduction of computing time is possible.

The investigation on grid sensitivity for the bifurcation problem of the canonical lid-driven cavity (LDC) flow results is reported here with very fine grids. This is motivated by different researchers presenting different first bifurcation critical Reynolds number (Re_{cr1}), which appears to depend on the formulation, numerical method, and choice of grid. By using a very-high-accuracy parallel algorithm, and the same method with which sequential results were presented by Lestandi etal. [Comput. Fluids 166, 86 (2018)CPFLBI0045-793010.1016/j.compfluid.2018.01.038] [for (257 × 257) and (513 × 513) uniformly spaced grid], we present results using (1025×1025) and (2049×2049) grid points. Detailed results presented using these grids help us understand the computational physics of the numerical receptivity of the LDC flow, with and without explicit excitation. The mathematical physics of the investigated problem will become apparent when we identify the roles of numerical errors with the ambient omnipresent disturbances in real physical flows as interchangeable. In physical or in numerical setups, presence of disturbances cannot be ignored. In this context, the need for explicit excitation for the used compact scheme arises for a definitive threshold amplitude, below which the flow relaxes back to quiescent state after the excitation is removed in computations. We also implement the present parallel method to show the physical aspects of primary and secondary instabilities to be maintained for other numerical schemes, and we show the results to reflect the complex physics during multiple subcritical Hopf bifurcation. Also, we relate the various sources of errors during computations that is typical of such shear-driven flow. These results, with near spectral accuracy, constitute universal benchmark results for the solution of Navier-Stokes equation for LDC.

The evolution to fully developed laminar flow in low aspect ratio, two-dimensional, lid-driven cavities has been studied experimentally and numerically. Velocity measurements were made in water in a moving-lid apparatus using a laser Doppler velocimeter (LDV). Numerical solutions for the cavity flow were obtained by solving the two-dimensional mass-momentum equation set in a finite-volume framework. The measured and predicted results were in excellent agreement. Fully developed cavity flow is said to exist when the main regions of the flow field become independent of the aspect ratio. When fully developed conditions prevail, a region of countercurrent flow (CCF) separates the end structures, which are decoupled. The extent of the end regions is shown to grow linearly with increasing Reynolds number Re, based on the lid speed and the cavity height. Consequently, the critical aspect ratio for the onset of fully developed flow is also linearly dependent on Re. Above a critical Reynolds number, Re [Formula: see text] 300, the flow becomes unsteady, and a lower-wall, tertiary vortex appears, which is thought to be associated with the onset of hydrodynamic instability.Key words: lid-driven cavity, laminar flow, shallow water cover, countercurrent flow.

In this work, a mixed Eulerian-Lagrangian method is employed to explore fluid flows in two-dimensional, lid-driven, irregular-bottom cavities. The methodology can handle fluid flows in the presence of irregularly shaped solid boundaries. A fixed Cartesian grid is used in the discretization of the momentum and mass equations, precluding the need to generate a grid to accommodate the non-Cartesian walls. Special treatment is required to deal with the control volumes located at the fluid–solid interface, an approach that is described herein. The discretization of the governing equations uses the finite-volume method with a collocated, i.e., cell-centered arrangement of velocity and pressure. The procedure is verified by solving two-dimensional lid-driven cavity flows for different Reynolds numbers in standard cavity shapes such as semicircular, rectangular, and square, with or without an irregular bottom wall. Results for the velocity components along the geometric centerline, stream function patterns, and pressure contours are presented and discussed. Excellent congruence with benchmark solutions is obtained.

In this paper, two-dimensional lid-driven cavity flow phenomena at steady state were simulated using two different scales of numerical method: the finite difference solution to the Navier–Stokes equation and the cubic interpolated pseudo-particle lattice Boltzmann method. The aspect ratio of cavity was set at 1, 2/3, 1/2 and 1/3 and the Reynolds number of 100, 400 and 1,000 for every simulation condition. The results were presented in terms of the location of the center of main vortex, the streamline plots and the velocity profiles at vertical and horizontal midsections. In this study, it is found that at the simulation of Reynolds numbers 100 and 400, both methods demonstrate a good agreement with each other; however, small discrepancies appeared for the simulation at the Reynolds number of 1,000. We also found that the number, size and formation of vortices strongly depend on the Reynolds number. The effect of the aspect ratio on the fluid flow behavior is also presented.

A synthesis of observations of flow in a three-dimensional lid-driven cavity is presented through the use of flow visualization pictures and velocity and heat flux measurements. The ratio of the cavity depth to width used was 1:1 and the span to width ratio was 3:1. Flow visualization was accomplished using the thymol blue technique and by rheoscopic liquid illuminated by laser-light sheets. Velocity measurements were made using a two-component laser-Doppler-anemometer and the heat flux on the lower boundary of the cavity was measured using flush mounted sensors. The flow is three-dimensional and is weaker at the symmetry plane than that predicted by accurate two-dimensional numerical simulations. Local three-dimensional features, such as corner vortices in the end-wall regions and longitudinal Taylor-Go¨rtler-like vortices, are significant influences on the flow. The flow is unsteady in the region of the downstream secondary eddy at higher Reynolds numbers (Re) and exhibits turbulent characteristics in this region at Re = 10,000.

Heat transfer and entropy generation of mixed convection flow in Cu-water nanofluid-filled lid-driven cavity with wavy surface