Stokes flow in closed,rectangular domains

Three practically relevant, Stokes flows in closed, rectangular cavities are considered. The first involves a solid-walled cavity where flow is driven by the motion of either one or both of its horizontal bounding walls; the other two have an upper free surface and are driven either by the motion of vertical side walls or by a horizontally-moving lower wall. Each problem is formulated as a biharmonic boundary value problem (bvp) for the streamfunction. The relative merits of two different coefficient determination methods for the corresponding analytical solutions are assessed and, in addition, each solution is compared with its numerical counterpart obtained using a finite element formulation of the governing equations. It is shown that, provided the number N of terms in each solution is sufficiently large, they are in extremely good agreement and, similarly, they compare well with work from other published theoretical and experimental studies. Streamlines are presented, over a wide range of operating parameters, for the geometries containing an upper free surface. For the flow generated by two moving vertical side walls two flow transformation mechanisms are identified. For cavities with small and decreasing width to depth (aspect) ratios, there is a sequence of critical aspect ratios at which flow bifurcations arise with a centre becoming a saddle point and vice versa, whereas for large aspect ratios increasing the ratio further leads to eddy growth from the lower wall, resulting in a regular sequence of separatrices along the cavity width. In the case of flow generated by a horizontally-moving lower wall the streamlines are simpler and exhibit the regular array of separatrices reported previously for flow in a solid-walled cavity with a single moving wall.     

Experiments on the transition to three-dimensional flow structures in two-sided lid-driven cavities

Flow instabilities in two-sided lid-driven cavities are studied experimentally. The transition of the nearly two-dimensional flow to steady or time-dependent three-dimensional flow structures is investigated for one-sided, for parallel, and for antiparallel motion of the driving walls and for two aspect ratios, Γ = 0.76 and Γ = 1. Stability diagrams are obtained by flow visualization. Six different three-dimensional flow patterns have been characterized, each corresponding to a particular critical mode of a linear-stability analysis. The structures of the near-critical flows and the critical curves are in good agreement with the corresponding numerical predictions. In a few cases, however, the critical Reynolds numbers deviate form the numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of 3D vortical cavity flow

The vortical flows of an incompressible fluid in a rectangular three-dimensional container with a large spanwise aspect ratio driven by a moving solid lid are studied using a combined compact finite difference (CCD) scheme with high accuracy and high resolution. The study focuses on the change of the steady flow structures in the cavity with Reynolds numbers ranging from 100 to 850. The results of the flow in the cavity with a spanwise aspect ratio 6.55 show that several stable closed streamlines localized near the symmetric plane are found at Re ≥500, while a closed stable streamline is found near the side wall at Re ≤300. The change of the flow pattern present in this system affects the diffusion properties in the flow but seems to have no qualitative effect on the global flow properties which include energy dissipation in the cavity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of magnetic field on the liquid-gas interfacial deformation in thermocapillary convection problem

Liquid contained in an open cavity flows owing to a temperature gradient applied along its free surface. The thermal variation of surface tension induces a steady viscous flow directed from hot end to cold end. For small aspect ratios, giving flow in thin, two dimensional slots, an asymptotic theory valid for (A ? 0)(A \rightarrow 0) is used to show that the interface undergoes an O(1) deformation from its flat position. Further, it is observed that for some values of the Hartmann number, the deformed interface can be made almost flat.     

Numerical study of the transition to chaos of a buoyant plume from a two-dimensional open cavity heated from below

The transition to a chaotic plume from a two-dimensional (2D) open cavity heated from below has been investigated using numerical simulation. A large range of Rayleigh numbers (Ra) pertaining to an aspect ratio of A = 1, and Prandtl number (Pr) of Pr = 0.71 (air) is numerically investigated. It is shown that there exists a complex transition of the plume from a steady reflection symmetry to a chaotic flow with a sequence of bifurcations. As the Rayleigh number increases, the plume from the open cavity undergoes a supercritical pitchfork bifurcation from a steady reflection symmetry to a steady reflection asymmetry flow. Once the Rayleigh number exceeds 7 × 10³, the plume appears as a distinct flapping namely, a Hopf bifurcation, and then as a distinct puffing. The chaotic plume has the possibility to exhibit an alternate appearance of flapping and puffing in the event the Rayleigh number exceeds 8 × 10⁴. Moreover, the dynamics of the plume from the open cavity is discussed, and the dependence on the Rayleigh number of heat and mass transfer of the plume from the open cavity is quantified.     

Solution of the Navier-Stokes equations by the spectral-difference method

The Jacobi polynomial collocation method is extended to two and three dimensions by superimposing the individual one-dimensional expansions. Two innovative ideas are introduced in this article. The first is the cornerless/edgeless computational grid, and the second is the modified compressibility method, which is an iterative pressure solver. To evaluate the new method's applicability in solving the Navier-Stokes equation, the lid-driven cavity flow problem was solved. Two configurations were considered, the square cavity and the rectangular cavity with an aspect ratio of 2. A comparison of the center-line velocity profiles from two- and three-dimensional simulations at a Reynolds number of 1000 is provided for each of the configurations. The center-line velocity comparisons showed good quantitative agreement with previous studies. © 1994 John Wiley & Sons, Inc.     

Upper-Branch Instability of Flow in Pipes of Large Aspect Ratio with Three-Dimensional Nonlinear Viscous Critical Layers

We consider the upper-branch neutral stability of flow in pipesof large aspect ratio, basically extending the work of F. T.Smith to the nonlinear regime. The inclusion of weak nonlinearityleads to an eigenproblem whose solution depends on the propertiesof three-dimensional nonlinear critical layers. Two specialcases are considered. The first is for very small amplitude perturbations, where R is a Reynolds numberbased on the height of the tube and which is assumed large.Then a fully analytical solution of the three-dimensional criticallayers is possible, from which the linear results of Smith maybe deduced. The second case studied is that of flow in a rectangularpipe, where a solution of the nonlinear critical layer problemcan be obtained. Further analysis of neutral modes in this lattercase suggests the possible existence, inter alia, of neutralmodes for finite aspect ratio tubes. These modes depend on thescaled amplitude and have O(1) wavespeeds.     

Numerical study of flow asymmetry and self-sustained jet oscillations in geometrically symmetric cavities

In this paper we present the results of numerical investigation of self-sustained oscillations of a jet confined in a symmetric cavity. This work represents an attempt to reproduce empirical observations of asymmetric flows in geometrically symmetric systems and to extend the jet flow investigations to more complex possible scenarios. A well-known example of such two-dimensional flow has been reported experimentally and reproduced numerically for simple flow [E. Schreck, M. Schaefer, Numerical study or bifurcation in three-dimensional sudden channel expansions, Comput. Fluids 29 (2000) 583–593]. It has been found that for some particular control parameter, above its critical value (bifurcation point), the jet can be deflected to either of the two sides of the cavity. In this paper we report a similar behaviour which is, however, characterized by a more complicated flow pattern. While simple flow appears only within small cavity lengths the complex flows develops with increased cavity lengths. Unlike stationary asymmetric solutions accompanied by cavity jet oscillations, as experimentally reported in e.g., [A. Maurel, P. Ern, B.J.A. Zielinska, J.E. Wesfreid, Experimental study of self-sustained oscillations in a confined jet, Phys Rev. E 54 (1996) 3643–3651], in our investigations of both simple and complex asymmetric flow we observed the slow periodical drift of the jet from one to another side of the cavity. The essential control parameters were Reynolds number Re and the ratio length to inlet width L/d. According to experiments of Maurel et al. (1996), the jet is stable and symmetric, when both L/d and Re are below certain critical values, otherwise jet oscillations appear in both experiment and our simulation (cavity oscillations regime). However, further increase of either (or both) L/d and Re leads of so called free jet type oscillations regime. This paper describes complex jet behaviour within the later oscillations regime. We believe that both simple “classical” and “our” complex stationary asymmetric solutions (as well as superimposed cavity-type and free-jet oscillations) can be explained based on physical arguments as already done in previous works. However, the origin of slow drift motion remains still to be resolved. This might be of high importance for clear distinguishing between relevant physical and numerical features in future codes developments.     

Stability of two‐dimensional flow in the two‐sided lid‐driven cavity

The one‐sided lid‐driven cavity has been studied quite extensively as benchmark problem in CFD. In the present study we discus the stability of two‐dimensional flows with respect to three‐dimensional perturbations of a more general lid‐driven cavity with two moving walls facing each other. We show that, in addition to the known elliptic‐instability branch around aspect ratio 1.5, another branch of the elliptic instability exists for aspect ratios smaller unity. For high and around unit aspect ratio the instabilities are found to be of centrifugal and quadripolar type, respectively. The structure of critical modes as well as the instability mechanism are addressed. Additionally, the numerical results are compared to experimental results of [5].     

Experiments on the transition to time‐dependence in highly strained vortex flow inside a rectangular cavity

The flows in a two‐sided lid‐driven cavity with a cross‐sectional aspect ratio Γ = 1.96 is experimentally investigated by laser‐Doppler anemometry and by hot‐film measurements. When the two facing walls move in opposite directions, a robust three‐dimensional flow consisting of four cells arises for Reynolds numbers Re ≥ 275 as a result of the elliptic instability. The flow within these steady cells is measured to high precision. For Re > 825 the four cell flow becomes oscillatory. The oscillations are due to a standing wave with the same wavelength as the underlying cellular flow.     

Numerical investigation of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls

A comprehensive numerical investigation on the natural convection in a rectangular enclosure is presented. The flow is induced due to the constant partial heating at lower half of the left vertical wall and partial cooling at upper half of the right vertical wall along with rest walls are adiabatic. In this investigation the Special attention is given to understand the effect of aspect ratio and heat source intensity i.e. Rayleigh number, Ra, on the fluid flow configuration as well as on the local and average heat transfer rates. The range of Rayleigh (Ra) and aspect ratio (A) is taken [10³, 10⁶] and [0.5, 4] respectively. The results are presented in terms of stream function (ψ), temperature (θ) and heat transfer rates (local Nusselt numbers Nu_L, and average Nusselt numbers Nu). The numerical experiments show that increasing of Ra implies the enhancement of thermal buoyancy force, which in turn increases the thermal convection in the cavity. As a result, the local as well as average heat transfer rate is expected to increase. The local transfer rate (Nu_L) is increases in the small region near the left vertical wall of the left wall of the cavity and after that it is decreases in the middle portion of heated region. And, it start to increase near to the middle point of left wall. It is also observed that the local heat transfer is increases as increases the aspect ratio. The average heat transfer rate (Nu) is increases as the aspect ratio A increases from 0.5 to 1 and beyond that it is decreases smoothly. It is also found that the heat transfer rate attains its maximum value at aspect ratio one.     

Free boundary fluid flow in a trough: A numerical approach

This paper examines the slow flow of a viscous liquid in an open rectangular container, one side (the base) of which moves steadily along its own plane, thereby providing the driving force the liquid needs. Unlike the two vertical sides that are rigid and stationary, the top side is left open so that the upper part of the liquid is in contact with air and is being controlled by surface tension and gravity. A numerical procedure for obtaining solutions for the cases when the capillary numbers are small is provided and the curves of the free boundaries obtained here are presented for some flow parameters. The deviation of the shape of the free boundary is observed to be strongly dependent on the aspect ratio of the boundary (i.e., the ratio of the vertical to horizontal spread of the liquid) with its curvature changing sign in the interval [1, 1.5].     

Flow separation behind ellipses at Reynolds numbers less than 10

Flow separation behind two-dimensional ellipses with aspect ratios ranging from 0, a flat plate, to 1, a circular cylinder, were investigated for Reynolds numbers less than 10 using both a cellular automata model and a commercial computational fluid dynamics software program. The relationship between the critical aspect ratio for flow separation and Reynolds number was determined to be linear for Reynolds numbers greater than one. At slower velocities, the critical aspect ratio decreases more quickly as the Reynolds number approaches zero. The critical Reynolds numbers estimated for flow separation behind a flat plate and circular cylinder agree with extrapolations from experimental observations. Fluctuations in the values of the stream function for laminar flow behind the ellipses were found at combinations of Reynolds number and aspect ratio near the critical values for separation.     

Towards a better understanding of wall-driven square cavity flows using the lattice Boltzmann method

Wall-driven flow in square cavities has been studied extensively, yet it appears some main flow characteristics have not been fully investigated. Previous research on the classic lid-driven cavity (S1) flow has produced the critical Reynolds numbers separating the laminar steady and unsteady flows. Wall-driven cavities with two opposing walls moving at the same speed and the same (S2p) or opposite (S2a) directions have seldom been studied in the literature and no critical Reynolds numbers characterizing transitional flows have ever been investigated. After validating the LBM code for the three configurations studied, extensive numerical simulations have been undertaken to provide approximate ranges for the critical Hopf and Neimark-Sacker bifurcations for the classic and two two-sided cavity configurations. The threshold for transition to chaotic motion is also reported. The symmetries of the solutions are monitored across the various bifurcations for the two-sided wall driven cavities. The mirror-symmetry of the base solution for case S2p is lost at the Hopf bifurcation. The exact same scenario occurs with the pi-rotational symmetry of the base state for case S2a.     

Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes

In this work, buckling and post-buckling analysis of fluid conveying multi-walled carbon nanotubes are investigated analytically. The nonlinear governing equations of motion and boundary conditions are derived based on Eringen nonlocal elasticity theory. The nanotube is modeled based on Euler–Bernoulli and Timoshenko beam theories. The Von Karman strain–displacement equation is used to model the structural nonlinearities. Furthermore, the Van der Waals interaction between adjacent layers is taken into account. An analytical approach is employed to determine the critical (buckling) fluid flow velocities and post-buckling deflection. The effects of the small-scale parameter, Van der Waals force, ends support, shear deformation and aspect ratio are carefully examined on the critical fluid velocities and post-buckling behavior.     

Slow viscous flow near a superhydrophobic surface with trapped gas bubbles

The Boundary Element Method (BEM) is used to solve the problem of Stokes flow of a viscous fluid over a periodic striped texture of a superhydrophobic surface (SHS), partially filled with frictionless gas bubbles. The shape of the bubble surfaces and the position of the meniscus pinning points relative to the cavity walls are taken into account in the study. Two kinds of flows important for practical applications are considered: a pressure-driven flow in a thin channel with a bottom superhydrophobic wall and a shear-driven flow over a periodic texture. We study the flow pattern in the fluid over a single cavity containing a bubble with a curved phase interface shifted into the cavity. A parametric numerical study of the averaged slip length of the SHS is performed as a function of the geometric parameters of the texture. It is shown that the curvature of the phase interface and/or its shift into the cavity both result in the decrease in the average slip length. It is demonstrated that the BEM can be an efficient tool for studying Stokes flows over textured superhydrophobic surfaces with different geometries of microcavities and phase interfaces. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experiments on the flow stability in a double-lid-driven cavity

The stability of the two-dimensional, steady, incompressible flow in a rectangular square cavity is investigated experimentally for the parallel motion of two facing walls. The critical Reynolds numbers for the onset of three-dimensional steady flow, its structure, and the bifurcation diagram of the velocity field, measured by LDV, agree with numerical predictions. It is observed that the wavelength of the selected pattern increases with the Reynolds number. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flow instabilities in thermocapillary liquid pools

The instability of steady axisymmetric thermocapillary flow in liquid pools heated from above is investigated numerically. The conditions for the onset of three-dimensional motion depend on the thermal conditions as well as the geometrical constraints. The critical Reynolds number is calculated as a function of the Prandtl number for a cylindrical pool of unit aspect ratio. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low-Reynolds boundary driven cavity flows in a thin liquid shell

Low-Reynolds recirculating cavity flows are traditionally generated from lid-driven boundary motion at a solid-fluid interface or result from shear flow over an opening. Such flows are typically described by the equations of creeping motion, where viscous forces are dominant. We illustrate using Particle Image Velocimetry (PIV) an original family of boundary-driven cavity flows occurring, in contrast to classic configurations, at a liquid-gas interface: thermally-induced Marangoni flows in a thin liquid shell generate forced, steady-state recirculating flows inside the cavity. Forcing relies on viscous mechanisms at the boundary but resulting flow patterns are, however, inviscid. Here, the inviscid equations of fluid motion are not used as an approximation, but rather come as a result from the solution of the creeping motion equations in the region inside the sphere. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Performance comparison of the NWF and DC methods for implementing High-Resolution schemes in a fully coupled incompressible flow solver

This paper reports on the use of the Normalized Weighting Factor (NWF) method and the Deferred Correction (DC) approach for the implementation of High Resolution (HR) convective schemes in an implicit, fully coupled, pressure-based flow solver. Four HR schemes are realized within the framework of the NWF and DC methods and employed to solve the following three laminar flow problems: (i) lid-driven flow in a square cavity, (ii) sudden expansion in a square cavity, and (iii) flow in a planar T-junction, over three grid systems with sizes of 10⁴, 5 × 10⁴, and 3 × 10⁵ control volumes. The merit of both approaches is demonstrated by comparing the computational costs required to solve these problems using the various HR schemes on the different grid systems. Whereas previous attempts to use the NWF method in a segregated flow solver failed to produce converged solutions, current results clearly demonstrate that both methods are suitable for utilization in a coupled flow solver. In terms of CPU efficiency, there is no global and consistent superiority of any method over another even though the DC method outperformed the NWF method in two of the three test problems solved.