Three-dimensional tracking of the long time trajectories of suspended particles in a lid-driven cavity flow

Stereo imaging methods are used to measure the positions of solid spherical particles suspended in a viscous liquid and enclosed in a transparent cubic cavity. The liquid and particle motions are driven at the top lid by a conveyor belt operated at constant speed. Based on sequences of stereo views of the full cavity, the particles are tracked continuously along their three-dimensional orbits. The corresponding position histories are treated as noisy stochastic data and processed using Kalman filters to fill data gaps and attenuate the effect of measurement errors. The lid-driven viscous flow is characterised by an intricate internal structure which is mirrored in the particle paths. The tracks of the solid particles align with long exposure images of laser-illuminated micro-particles in selected transverse planes. Nevertheless, their long time trajectories appear to cluster along preferential pathways of the internal circulation pattern.     

Aspect ratio effects on three-dimensional incompressible flow in a two-sided non-facing lid-driven parallelepiped cavity

A numerical study of the three-dimensional fluid flow has been carried out to determine the effects of the transverse aspect ratio, Ay, on the flow structure in two-sided non-facing lid-driven cavities. The flow is complex, unstable and can undergo bifurcation. The numerical method is based on the finite volume method and multigrid acceleration. Computations have been investigated for several Reynolds numbers and various aspect ratio values. At a fixed Reynolds number, Re=500, the three-dimensional flow characteristics are analyzed considering four transverse aspect ratios, Ay=1,0.75,0.5 and 0.25. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. An analysis of the flow evolution shows that, at Ay=0.75, the flow bifurcates to a periodic regime at (Re=600) with a frequency f=0.093 less than the predicted value in the cubical cavity. A correlation is established when Ay=0.5 and gives the critical Reynolds number value. At Ay=0.25, the periodic regime occurs at high Re value beyond 3500, after which the flow becomes chaotic. It is shown that, when increasing Ay over the unit, the flow in the cavity exhibits a complex behavior. The kinetic energy transmission from the driven walls to the cavity center is reduced at low Ay values.     

Lattice Boltzmann simulation of two-sided lid-driven flow in a staggered cavity

In this article, the lattice Boltzmann method is employed in order to explore incompressible fluid flow inside a two-sided lid-driven staggered cavity. Results of the lattice Boltzmann simulation for antiparallel motion of lids are compared with the data from existing literature. For parallel motion of lids, the characteristics of flow pattern for a variety of Re numbers (50–3200) are presented. An asymmetric steady-state flow pattern for parallel motion of lids is obtained.     

Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity

The swirling flow between a rotating lid and a stationary cylinder is studied experimentally. The flow is governed by two parameters: the ratio of container height to disk radius, h, and the Reynolds number, Re, based on the disk angular velocity, cylinder radius and kinematic viscosity of the working liquid. For the first time, the onset of three-dimensional flow behavior is measured by combining the high spatial resolution of particle image velocimetry and the temporal accuracy of laser Doppler anemometry. A detailed mapping of the transition scenario from steady and axisymmetric flow to unsteady and three-dimensional flow is investigated for 1 ≥ h ≥ 3.5. The flow is characterized by the development of azimuthal modes of different wave numbers. A range of different modes is detected and critical Reynolds numbers and associated frequencies are identified. The results are compared to the numerical stability analysis of Gelfgat et al. (J Fluid Mech 438:363–377, 2001). In most cases, the measured onset of three-dimensionality is in good agreement with the numerical results and disagreements can be explained by bifurcations not accounted for by the numerical stability analysis.     

Numerical approximations for flow of viscoplastic fluids in a lid-driven cavity

The effect of inertia and rheology parameters on the flow of viscoplastic fluids inside a lid-driven cavity is investigated using a stabilized finite element approximation. The viscoplastic material behavior is described by the model introduced by de Souza Mendes and Dutra [30] – herein called SMD fluid – which is essentially a regularized viscosity function that involves only rheological properties of the material. The incompressible balance equations are coupled with the non-linear SMD model and are approximated by a multi-field Galerkin least-squares method in terms of extra-stress, pressure and velocity. The results obtained confirm the stability features of the multi-field formulation and the appropriateness of the rheological stress regularization introduced by the SMD fluid. The influence of inertia and rheological parameters on the morphology of the material yield surfaces is analyzed and discussed.     

Behaviour of macroscopic rigid spheres in lid-driven cavity flow

Experiments are conducted to investigate the behaviour of macroscopic rigid particles suspended in a fully three-dimensional viscous flow. The flow considered takes place in a closed cubic cavity, steadily driven along its upper face by a translating lid. Navier–Stokes computations are first performed to characterize the fluid flow, and the resulting kinematic template is checked using laser-illuminated micro-particles. Nearly neutrally buoyant rigid spheres are then inserted in the cavity, and their three-dimensional motions are tracked using stereoscopic imaging. The measured macro-particle motions are compared with those of simulated passive tracers, and their responses to changes in experimental conditions are examined. Although steric effects are observed to hinder passage through narrow throats of the flow field, macro-particle trajectories are otherwise found to align closely with passive tracer paths. The macro-particle orbits, however, are not evenly distributed within the cavity, and cluster closer to the periphery as the Reynolds and Stokes numbers increase. With support from observations of particle rotations relative to the fluid, we interpret this behaviour as resulting from weak forces pulling the macroscopic spheres towards preferential paths, similar to the Segré–Silberberg effect in Poiseuille flow.     

Theoretical analysis of three-dimensional bifurcated flow inside a diagonally lid-driven cavity

The instability mechanism of fully three-dimensional, highly separated, shear-driven confined flow inside a diagonally lid-driven cavity was investigated. The analysis was conducted on 100³ and 200³ stretched grids by a series of direct numerical simulations utilizing a standard second-order accuracy finite volume code, openFoam. The observed oscillatory instability was found to set in via a subcritical symmetry breaking Hopf bifurcation. Critical values of the Reynolds number Re _cr = 2320 and the non-dimensional angular oscillating frequency \({\omega_{\rm cr}=0.249}\) for the transition from steady to oscillatory flow were accurately determined. An oscillatory regime of the bifurcated flow was analyzed in depth, revealing and characterizing the spontaneous symmetry breaking mechanism. Characteristic spatial patterns of the base flow and the main flow harmonic were determined for the velocity, vorticity and helicity fields. Lagrangian particle tracers were utilized to visualize the mixing phenomenon of the flow from both sides of the diagonal symmetry plane.     

Integral transform solution for the lid-driven cavity flow problem in streamfunction-only formulation

The basic ideas in the generalized integral transform technique are further advanced to allow for the hybrid numerical-analytical solution of the two-dimensional steady Navier-Stokes equations in streamfunction-only formulation. The classical lid-driven square cavity problem is selected for illustration of the approach. The corresponding biharmonic-type non-linear partial differential equation for the streamfunction is integral transformed in one of the co-ordinates and an infinite system of coupled non-linear ODEs for the transformed potential results in the other independent variable. Upon truncation to an appropriate finite order, the ODE system is numerically solved by well-established algorithms with automatic error control devices. The convergence behaviour of the eigenfunction expansion is demonstrated and reference results are provided for typical values of Reynolds number.     

Cessation of the lid-driven cavity flow of Newtonian and Bingham fluids

We provide benchmark results for a transient variant of the lid-driven cavity problem, where the lid motion is suddenly stopped and the flow is left to decay under the action of viscosity. Results include Newtonian as well as Bingham flows, the latter having finite cessation times, for Reynolds numbers Re ∈ [1, 1000] and Bingham numbers Bn ∈ [0, 10]. The finite-volume method and Papanastasiou regularisation were employed. A combination of Re and Bn, the effective Reynolds number, is shown to convey more information about the flow than either Re or Bn alone. A time scale which characterises the flow independently of the geometry and flow parameters is proposed.     

Experimental and numerical study of mixed convection and flow pattern in a lid-driven arc-shape cavity

Experimental and numerical study has been performed to investigate the combined effects of lid movement and buoyancy on flow and heat transfer characteristics for the mixed convective flow inside a lid-driven arc-shape cavity. The numerical methodology is based on a numerical grid generation scheme that maps the complex cross section onto a rectangular computation domain. The discretization procedure for the governing equations is based on the finite-volume method. In experiments, steady-state temperature data are measured by T-type thermocouples, and the flow field is visualized by using kerosene smoke. Reynolds number and Grashof number are two major independent parameters representing the effects of lid movement and buoyancy, respectively. Flow pattern, friction factor, and Nusselt numbers are investigated in wide ranges of these independent parameters. Close agreement in the comparison between the predicted and the visualized flow patterns shows the validity of the numerical methods.     

Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios

This study presents a continuation method to calculate flow bifurcation in a two-sided lid-driven cavity with different aspect ratios for anti-parallel motion. In anti-parallel motion, the top and bottom walls of the cavity move in opposite directions simultaneously, while the two walls both moving to the right give parallel motion at the same speed. Comprehensive bifurcation diagrams of the cavity flows with different aspect ratios of the cavities are derived via Keller’s continuation method, and linear- stability analysis is used to identify the nature of the various flow solutions. The Reynolds number (1 ≤ Re ≤ 1,200) is used as the continuation parameter to trace the solution curves. In anti-parallel motion, the evolution of the bifurcation diagrams in cases with different aspect ratios (1 ≤ AR ≤ 2.5) is illustrated. Two stable symmetric flows and one stable asymmetric flow are identified, and the existent regions of the stable flows in the aspect ratios and Reynolds numbers are distinguished. The newly found asymmetric flow state can be obtained at a high aspect ratio and a low Reynolds number.     

Double-diffusive convection in a cubical lid-driven cavity with opposing temperature and concentration gradients

A numerical study of three-dimensional incompressible viscous flow inside a cubical lid-driven cavity is presented. The flow is governed by two mechanisms: (1) the sliding of the upper surface of the cavity at a constant velocity and (2) the creation of an external gradient for temperature and solutal fields. Extensive numerical results of the three-dimensional flow field governed by the Navier-Stokes equations are obtained over a wide range of physical parameters, namely Reynolds number, Grashof number and the ratio of buoyancy forces. The preceding numerical results obtained have a good agreement with the available numerical results and the experimental observations. The deviation of the flow characteristics from its two-dimensional form is emphasized. The changes in main characteristics of the flow due to variation of Reynolds number are elaborated. The effective difference between the two-dimensional and three-dimensional results for average Nusselt number and Sherwood number at high Reynolds numbers along the heated wall is analyzed. It has been observed that the substantial transverse velocity that occurs at a higher range of Reynolds number disturbs the two-dimensional nature of the flow.     

Novel wavelet-homotopy Galerkin technique for analysis of lid-driven cavity flow and heat transfer with non-uniform boundary conditions

In this paper, a brand-new wavelet-homotopy Galerkin technique is developed to solve nonlinear ordinary or partial differential equations. Before this investigation,few studies have been done for handling nonlinear problems with non-uniform boundary conditions by means of the wavelet Galerkin technique, especially in the field of fluid mechanics and heat transfer. The lid-driven cavity flow and heat transfer are illustrated as a typical example to verify the validity and correctness of this proposed technique. The cavity is subject to the upper and lower walls' motions in the same or opposite directions.The inclined angle of the square cavity is from 0 to π/2. Four different modes including uniform, linear, exponential, and sinusoidal heating are considered on the top and bottom walls, respectively, while the left and right walls are thermally isolated and stationary.A parametric analysis of heating distribution between upper and lower walls including the amplitude ratio from 0 to 1 and the phase deviation from 0 to 2π is conducted. The governing equations are non-dimensionalized in terms of the stream function-vorticity formulation and the temperature distribution function and then solved analytically subject to various boundary conditions. Comparisons with previous publications are given,showing high efficiency and great feasibility of the proposed technique.     

Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid

This work is focused on the numerical modeling of steady laminar mixed convection flow in a lid-driven inclined square enclosure filled with water–Al₂O₃ nanofluid. The left and right walls of the enclosure are kept insulated while the bottom and top walls are maintained at constant temperatures with the top surface being the hot wall and moving at a constant speed. The developed equations are given in terms of the stream function–vorticity formulation and are non-dimensionalized and then solved numerically subject to appropriate boundary conditions by a second-order accurate finite-volume method. Comparisons with previously published work are performed and found to be in good agreement. A parametric study is conducted and a set of graphical results is presented and discussed to illustrate the effects of the presence of nanoparticles and enclosure inclination angle on the flow and heat transfer characteristics. It is found that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by inclination of the enclosure at moderate and large Richardson numbers.     

Mixed convection in an inclined lid-driven cavity with non-uniform heating on both sidewalls

The present work reports a numerical simulation of mixed convection in an inclined square cavity. The vertical sidewalls are assumed to have a nonuniform temperature distribution. The finite volume method is used to solve dimensionless governing equations. Simulations are performed for different Richardson numbers, amplitude ratios, phase deviations, and cavity inclination angles. The results are presented graphically. The mean heat transfer significantly increases in the buoyancy-dominated mode on increasing cavity inclination angle if both walls have identical heating and cooling zones.     

Direct numerical simulation and global stability analysis of three-dimensional instabilities in a lid-driven cavity

The first bifurcation in a lid-driven cavity characterized by three-dimensional Taylor–Görtler-Like instabilities is investigated for a cubical cavity with spanwise periodic boundary conditions at Re=1000. The modes predicted by a global linear stability analysis are compared to the results of a direct numerical simulation. The amplification rate, and the shape of the three-dimensional perturbation fields from the direct numerical simulation are in very good agreement with the characteristics of the steady S1 mode from the stability analysis, showing that this mode dominates the other unstable unsteady modes. To cite this article: J. Chicheportiche et al., C. R. Mecanique 336 (2008).