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Multiplicity of steady state solutions in 2-D incompressible viscous wall driven arc-shaped cavity flow
Author(s): Ercan Erturk
Keywords: Arc-shaped cavity flow, Multiple steady state solutions, Bifurcation Reynolds number, 2-D incompressible viscous flow, Arc length ratio
Numerical simulations of the steady 2-D incompressible viscous flow in an arc-shaped cavity are presented. The Navier–Stokes equations in streamfunction and vorticity formulation are solved numerically using a body fitted mesh obtained by a conformal mapping. Our numerical results reveal that the arc-shaped cavity flow has multiple steady solutions above a bifurcation Reynolds number when the arc length ratio is less than 1/2 ($r$$<$1/2). Multiple steady state solutions of the arc-shaped cavity flow with different arc length ratios ($r$=2/5, 1/3, 1/4, 1/5 and 1/6) are presented at a variety of Reynolds numbers. Our results show that the bifurcation Reynolds number at which a second solution starts to exist changes as the arc length ratio of the arc-shaped cavity changes. Among the considered different arc length ratios ($r$=2/5, 1/3, 1/4, 1/5 and 1/6), the minimum bifurcation Reynolds number occurs at 1/3 arc length ratio with $Re$=5164. Detailed results are presented.