Mixed convection and role of multiple solutions in lid-driven trapezoidal enclosures

Linear three-dimensional modal instability of steady laminar two-dimensional states developing in a lid-driven cavity of isosceles triangular cross-section is investigated theoretically and experimentally for the case in which the equal sides form a rectangular corner. An asymmetric steady two-dimensional motion is driven by the steady motion of one of the equal sides. If the side moves away from the rectangular corner, a stationary three-dimensional instability is found. If the motion is directed towards the corner, the instability is oscillatory. The respective critical Reynolds numbers are identified both theoretically and experimentally. The neutral curves pertinent to the two configurations and the properties of the respective leading eigenmodes are documented and analogies to instabilities in rectangular lid-driven cavities are discussed.

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Three-dimensional flow instability in a lid-driven isosceles triangular cavity

Cartesian grids used with the immersed boundary method (IBM) offer an attractive alternative for simulating fluid flows in complex geometries. We present a ghost fluid method for incompressible flows solved with staggered grids. The primary feature is the satisfaction of local mass continuity for ghost pressure cells, rather than extrapolating the pressures from within the flow domain. The method preserves local continuity in each cell and also global continuity. As a result, no explicit mass sources or sinks are needed. We have applied the method to study shear- and buoyancy-driven flows in a number of complex cavities.

http://absimage.aps.org/image/DFD08/MWS_DFD08-2008-020051.pdf

Immersed Boundary Simulations of Shear and Buoyancy Driven Flows in Complex Enclosures

A finite element formulation of the Navier—Stokes equations, written in terms of the stream function, ψ, and vorticity, ω, for a Newtonian fluid in the absence of body forces, is used to so...

A finite element analysis of steady viscous flow in triangular cavities

Three-dimensional instabilities for two circular lid-driven cavities are investigated by a linear stability analysis and direct numerical simulations using high order spectral techniques. Two circular geometries have been analysed and compared: a circular cavity with a horizontal top boundary and a circular cavity with a circular lid. Compared to more classic results for squared and rectangular lid driven cavities, the corners of these rounded geometries have been partially or totally removed. Critical Reynolds numbers, neutral curves, and three dimensional structures associated with the least stable modes have been identified by the linear stability analysis and then confirmed by spectral direct numerical simulations. We show that the geometries that present fewer sharp corners have enhanced stability: the circular cavity with a flat lid presents the first bifurcation at (Rec,kc)≈(1362,25) whilst the circular lid bifurcates at (Rec,kc)≈(1438,18), where Rec is the critical Reynolds number based on the cavity diameter and lid tangential velocity, and kc is the spanwise wavenumber. Neutral curves and properties of the leading three-dimensional flow structures are documented, and analogies to instabilities in other lid-driven cavities are discussed. Additionally, we include results for the adjoint problem and the structural sensitivity 3D iso-maps (i.e., wavemaker regions), to show that the cavity corners play a relevant role in the generation of 3D instabilities.

Onset of three-dimensional flow instabilities in lid-driven circular cavities

Numerical solutions of the Navier–Stokes equations governing shear flow in a trapezoidal enclosure have been obtained at fine grid resolutions using an efficient multigrid calculation procedure. The effects of flow Reynolds number and wall boundary conditions are studied in detail. Interesting flow patterns are observed to develop, especially at high Reynolds numbers. To the author’s knowledge, such a flow development has not been reported previously.

Joseph H. Darr

Papers

Separated flow in a driven trapezoidal cavity