Viscous flow in a slightly asymmetric channel with a sudden expansion

Abstract

The effect of a slight asymmetry in the channel geometry on the flow transition from symmetric to nonsymmetric states in a long channel with a sudden expansion is presented. The asymptotic analysis is based on a study of the unsteady Navier–Stokes equations around the critical Reynolds number, Rec, where a bifurcation of asymmetric states in a symmetric channel occurs. It results in an ordinary, nonlinear, nonhomogeneous, first-order differential equation (similar to the Landau equation) which describes the evolution of the perturbation’s amplitude as function of Re near Rec and of the asymmetry of the channel, τ. It is found that channel asymmetry changes the pitchfork bifurcation diagram of a flow in a symmetric channel into two separate branches of equilibrium states. The primary branch describes a gradual and stable change of the flow states from symmetric to non-symmetric as Re is increased across Rec. The secondary branch appears at a certain modified critical Reynolds number, Recτ>Rec, and describes two additional non-symmetric flow states for each Re>Recτ which are disconnected from the primary branch. The large-amplitude asymmetric states along the secondary branch are also stable whereas the small-amplitude states are unstable. In the surroundings of Rec, the asymptotic results show an agreement with the results from numerical simulations using the Navier–Stokes equations. The asymptotic results also demonstrate an agreement with the available experimental data for Re near Rec. When Re increases above Rec, the second-order effects which are neglected in the asymptotic analysis become more dominant and are responsible for the deviation of the asymptotic results from the numerical and experimental results. The analysis clarifies the sensitive nature of the flow around Rec to geometry or flow asymmetries in both the experiments and the numerical computations and sheds more light on the nonlinear transition of a viscous flow in an expanding channel from symmetric to asymmetric states.