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ABSTRACT: The transition from two-dimensional (2D) steady to three-dimensional (3D) flow is investigated experimentally in a two-sided lid-driven cavity for anti-parallel motion of two facing walls. Both wall speeds (Reynolds numbers) have the same absolute value. We present critical Reynolds numbers at the onset of steady three-dimensional flow as a function of the cross-sectional aspect ratio. The critical curve is composed of different neutral branches belonging to different instability modes. Comparison with numerical results enables insight into the corresponding mechanisms.
ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 03/2011; 81(S3):781 - 782. · 0.86 Impact Factor
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ABSTRACT: 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)
PAMM 03/2008; 7(1):3050001 - 3050002.
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ABSTRACT: A linear stability analysis of the buoyant-thermocapillary flow in open rectangular cavities with aspect ratios in the range Gamma=1.2 to 8 is carried out for Prandtl number Pr=10 and conditions of previous experiments. The results are in very good agreement with most available experimental data. The energy transfer between the basic and the perturbation flow reveals that buoyancy is not directly instrumental in the instabilities. For aspect ratios less than about three a stationary three-dimensional cellular flow arises. The instability relies on the lift-up mechanism operating in the shear layer below the free surface and it is aided by weak Marangoni forces. For larger aspect ratios Marangoni effects play a more significant role. While plane hydrothermal waves may appear a certain distance away from the hot wall for sufficiently large aspect ratios, the instability at intermediate aspect ratios is strongly influenced by the local nonparallel basic-flow structure.
Physical Review E 03/2008; 77(3 Pt 2):036303. · 2.26 Impact Factor
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ABSTRACT: The three-dimensional flow in a lid-driven cuboid is investigated numerically. The geometry is an extension to three dimensions of the lid-driven square cavity by translating the two-dimensional lid-driven cavity parallel to the third orthogonal direction. The incompressible Navier–Stokes equations are discretized by a pseudospectral Chebyshev-collocation method. The singularities caused by the discontinuous velocity boundary conditions are reduced by including asymptotic analytical solutions in the solution ansatz. The flow is computed for Reynolds numbers above the critical onset of Taylor–Görtler vortices. Nonlinear Taylor–Görtler cells are calculated for periodic and for realistic no-slip endwall conditions. For periodic boundary conditions the bifurcation is either sub- or supercritical, depending on the wavenumber. The limiting tricritical case arises near the critical wavenumber of the linear-stability problem. On the other hand, no-slip endwall conditions have a significant effect on the supercritical three-dimensional flow. In agreement with recent experimental results we find that Taylor–Görtler vortices are suppressed near no-slip endwalls.
Journal of Fluid Mechanics 12/2006; 569:465 - 480. · 2.46 Impact Factor
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ABSTRACT: Vortices are of fundamental importance in fluid mechanics. An interesting aspect of vortex dynamics is the transition from two- to three-dimensional flow. The transition to three-dimensional flow of a circular vortex can be caused by a weak two-dimensional non-axisymmetric strain field deforming the circular streamlines. After briefly reviewing the three-dimensional instability of strained vortices in unbounded domains, the effect on vortex stability of bounding walls of an enclosure will be investigated by exploring the 2D–3D transition in a driven cavity. This system, in which the flow is driven by the steady tangential motion of one or two facing walls, is a paradigm for closed flows. The rich multitude of instabilities and bifurcations is demonstrated and analogies to unstable unbounded vortices are established. Contrary to vortices in open flows, the amplitudes of three-dimensional perturbations in driven cavities typically saturate. Hence, stationary or time-periodic three-dimensional vortices can be realized with ease, facilitating detailed experimental investigations.
ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 04/2005; 85(6):387 - 399. · 0.86 Impact Factor
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ABSTRACT: The flow in an infinite slab of rectangular cross-section is investigated numerically
by a finite volume method. Two facing walls which move parallel to each other with
the same velocity, but in opposite directions, drive a plane flow in the cross-section
of the slab. A linear stability analysis shows that the two-dimensional flow becomes
unstable to different modes, depending on the cross-sectional aspect ratio, when the
Reynolds number is increased. The critical mode is found to be stationary for all
aspect ratios. When the separation of the moving walls is larger than approximately
twice the height of the cavity, the basic flow forms two vortices, each close to one
of the moving walls. The instability of this flow is of centrifugal type and similar to
that in the classical lid-driven cavity problem with a single moving wall. When the
moving walls are sufficiently close to each other (aspect ratio less than 2) the two
vortices merge and form an elliptically strained vortex. Owing to the dipolar strain
this flow becomes unstable through the elliptic instability. When both moving walls
are very close, the finite-length plane-Couette flow becomes unstable by a similar
elliptic mechanism near both turning zones. The critical mode produces wide streaks
reaching far into the cavity. For a small range of aspect ratios near unity the flow
consists of a single vortex. Here, the strain field is dominated by a four-fold symmetry.
As a result the instability process is analogous to the instability of a Rankine vortex
in an quadripolar strain field, resulting from vortex stretching into the four corners
of the cavity.
Journal of Fluid Mechanics 05/2002; 458:153 - 180. · 2.46 Impact Factor
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ABSTRACT: The two-dimensional steady incompressible flow in rectangular cavities is calculated numerically by a finite volume method.
The flow is driven by two opposing cavity side walls which move with constant velocities tangentially to themselves. Depending
on the cavity aspect ratio and the two side-wall Reynolds numbers different flow states exist. Their range of existence and
the bifurcations between different states are investigated by a continuation method accurately locating the bifurcation points.
When both side walls move in opposite directions up to seven solutions are found to exist for the same set of parameters.
Three of these are point-symmetric and four are asymmetric with respect to the center of the cavity, if the side-wall Reynolds
numbers have the same magnitude. When the walls move in the same direction, up to five different flow states are found. In
this case only a single mirror symmetric solution exists for equal Reynolds numbers.
Theoretical and Computational Fluid Dynamics 12/2000; 14(4):223-241. · 1.03 Impact Factor
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ABSTRACT: The hydrodynamic stability of steady axisymmetric thermocapillary flow in a cylindrical liquid bridge is investigated by linear stability theory. The basic state and the three-dimensional disturbance equations are solved by various spectral methods for aspect ratios close to unity. The critical modes have azimuthal wavenumber one and the most dangerous disturbance is either a pure hydrodynamic steady mode or an oscillatory hydrothermal wave, depending on the Prandtl number. The influence of heat transfer through the free surface, additional buoyancy forces, and variations of the aspect ratio on the stability boundaries and the neutral mode are discussed.
Journal of Fluid Mechanics 01/1993; 247:247 - 274. · 2.46 Impact Factor
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ABSTRACT: The linear stability of two counter-rotating vortices driven by the parallel motion of two facing walls in a rectangular cavity is investigated by a finite volume method. Critical Reynolds and wave numbers are calculated for aspect ratios ranging from 0.1 to 5. This range is sufficient to find the asymptotic behavior of the critical parameters when the aspect ratio tends to zero and infinity, respectively. The critical curve is smooth for all aspect ratios and, hence, the character of the instability changes continuously. When the moving walls are far apart the mechanism is centrifugal, as in the classical lid-driven cavity. For aspect ratios near unity a combined mechanism, also involving strain near the vortex cores, leads to the instability which tends to asymmetrically displace the vortex cores, very similar to the cooperative short-wave instability of a free counter-rotating vortex pair. In the limit when plane Poiseuille flow is approached in the bulk, the three-dimensional perturbations are strongly localized near both downstream ends of the moving walls.
European Journal of Mechanics - B/Fluids.
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ABSTRACT: A Chebyshev-collocation method in space is introduced, which allows an accurate calculation of three-dimensional lid-driven cavity flows. The time integration is carried out by an Adams–Bashforth backward–Euler scheme. The accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field. This way the most severe discontinuities in the boundary conditions, which arise along the lines where moving and stationary walls meet, are taken care of analytically and thus do not spoil the numerical part of the solution. Calculations are carried out for no-slip boundary conditions at the cavity end-walls as well as for periodic end-wall conditions. In general, the spatial accuracy is better than fifth order. For rigid end-wall conditions, the accuracy is reduced near the end-walls to O(N3/2), but recovers in the bulk. Tabulated data are provided for the most interesting flow properties.
Journal of Computational Physics.
