Bifurcation of Low Reynolds Number Flows in Symmetric Channels

Abstract

The ow elds in two-dimensional channels with discontinuous expansions are studied numerically to understand how the channel expansion ratio in uences the symmetric and non-symmetric solutions that are known to occur. For improved con dence and understanding, two distinct numerical techniques are used. The general ow eld characteristics in both symmetric and asymmetric regimes are ascertained by a time-marching nite volume procedure. The ow elds and the bifurcation structure of the steady solutions of the Navier-Stokes equations are determined independently using the nite-element technique. The two procedures are then compared both as to their predicted critical Reynolds numbers and the resulting ow eld characteristics. Following this, both numerical procedures are compared with experiments. The results show that the critical Reynolds number decreases with increasing channel expansion ratio. At a xed supercritical Reynolds number, the location at which the jet rst impinges on the channel wall grows linearly with the expansion ratio.

Stability analysis and control of the flow in a symmetric channel with a sudden expansion

“…II, ER = 3, which is similar to what is done in the literature for the same flow (see, for instance, Hawa and Rusak 7 and Battaglia et al 5 ). A mesh composed of triangular elements is generated though the Delaunay-Voronoi algorithm.…”

“…3 of the flow configuration, the two reattachment points differ as reported in the figure. Data from Alleborn et al 8 and Battaglia et al 5 are also shown, in order to validate the present results. An overall very good agreement is observed.…”

“…It was found that there exists a critical value of the Reynolds number, Re c , such that, when Re < Re c the flow in the channel is steady, two-dimensional and symmetric, and when Re > Re c the flow is still two-dimensional and steady, but it becomes asymmetric. The value of Re c is shown to decrease as the expansion ratio increases in Battaglia et al 5 and Drikakis. 6 The physical mechanism originating the bifurcation has been numerically investigated in Hawa and Rusak 7 and Alleborn et al 8 by asymptotic techniques.…”

“…Therefore, if we consider an arbitrary base-flow modification δU b that, anywhere in the flow, is oriented in the same direction as the growth rate sensitivity, the variation of the growth rate is positive (δλ > 0) which indicates a destabilization of the flow. We now consider the sensitivity with respect to a steady force in the momentum equation (5). As mentioned before M + is real, thus the adjoint base flow, solution of problem (16) is real too.…”

“…The asymmetric case is also considered in Alleborn et al 8 Three-dimensional effects on the considered flow have been investigated in Chiang et al 10 and Battaglia and Papadopoulos. 11 In this study, we consider a value of the expansion ratio equal to ER = 3 as, for instance, in Hawa and Rusak 7 and Battaglia et al 5 A 2D linear stability analysis of the symmetric steady solution in the considered configuration has shown that for Re > Re c , one real unstable mode exists which a) s.camarri@ing.unipi.it. Phys.…”