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.     

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.     

Transient solutions for three-dimensional lid-driven cavity flows by a least-squares finite element method

A time-accurate least-squares finite element method is used to simulate three-dimensional flows in a cubic cavity with a uniform moving top. The time- accurate solutions are obtained by the Crank-Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix-free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least-squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time-accurate solutions to large-scale three-dimensional fluid flows, we present results for three-dimensional lid-driven flows in a cubic cavity for Reynolds numbers up to 3200.     

Three-dimensional flow in a lid-driven cavity with width-to-height ratio of 1.6

The flow in a lid-driven cavity with width-to-height ratio of 1.6 is investigated numerically and experimentally. Experimental investigation use an apparatus with a spanwise length-to-height ratio of $\Uplambda = 10.85.$ Λ = 10.85 . Increasing the Reynolds number, we experimentally find a gradual change from the quasi-two-dimensional basic flow to a three-dimensional flow pattern. The three-dimensional flow has a significant amplitude at considerably low Reynolds numbers. Streak-line photographs and PIV vector maps are presented to illustrate the structure of the finite-amplitude flow pattern. The smooth transition is in contrast to the linear instability predicted by a linear-stability analysis for a cavity with infinite span. LDV measurements confirm the absence of a distinct threshold Reynolds number and indicate an imperfect bifurcation. The deviations between experimental observations and numerical critical Reynolds number for infinite span are explained by conducting three-dimensional simulations for a finite-span geometry. A good agreement between experimental and numerical simulation is obtained. The numerical and experimental data lead to the conjecture of a premature onset of the three-dimensional flow caused by strong secondary flows which are induced by the cavity end walls. Nevertheless, the flow structure in the finite-span cavity carries the same characteristic signatures as the nonlinear flow in the corresponding infinite-length cavity. We conclude that the observed flow can be identified as the continuation of the normal mode C _e ⁴ earlier identified in a linear-stability analysis.     

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.     

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.     

The effect of real viscosity on the heat transfer of water based Al2O3 nanofluids in a two-sided lid-driven differentially heated rectangular cavity

The heat transfer and fluid flow behavior of water based Al₂O₃ nanofluids are numerically investigated inside a two-sided lid-driven differentially heated rectangular cavity. Physical properties which have major effects on the heat transfer of nanofluids such as viscosity and thermal conductivity are experimentally investigated and correlated and subsequently used as input data in the numerical simulation. Transport equations are numerically solved with finite volume approach using SIMPLEC algorithm. It was found that not only the thermal conductivity but also the viscosity of nanofluids has a key role in the heat transfer of nanofluids. The results show that at low Reynolds number, increasing the volume fraction of nanoparticles increases the viscosity and has a deteriorating effect on the heat transfer of nanofluids. At high Reynolds number, the increase in the viscosity is compensated by force convection and the increase in the volume fraction of nanoparticles which results in an increase in heat transfer is in coincidence with experimental results.     

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.     

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.     

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.     

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).     

A study on validity of using average fiber aspect ratio for mechanical properties of aligned short fiber composites with different fiber aspect ratios

A closed-form solution using the actual distribution of the fiber aspect ratio is proposed for predicting the stiffness of aligned short fiber composite. The present model is the simplified form of Takao and Taya’s model and the extended version of Taya and Chou’s model, where Eshelby’s equivalent inclusion method modified for finite fiber volume fraction is employed. The validity of using average fiber aspect ratio for predicting the composite stiffness is justified in terms of the scatter of fiber aspect ratio, fiber volume fraction, and constituents‘ Young’s modulus ratio, comparing with the results by the present model. The guideline for selection of either the actual distribution or the average fiber aspect ratio is presented for the better prediction of the composite stiffness.     

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.     

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.