Prediction of flow instabilities and transition using high-resolution methods

We have considered the problem of flow through a rectangular channel with a suddenly-expanded and suddenly-contracted part and have conducted a computational investigation to examine numerical effects on the prediction of flow instabilities and bifurcation phenomena for both fine-resolved and under-resolved grid computations. The results revealed that the solution of the flow depends on the numerical method employed especially in the case for under-resolved grid computations. We have employed high-resolution (Godunov-type) methods in conjunction with first-, second- and third-order accurate interpolation schemes. It is shown that the order of accuracy of the interpolation used in the discretisation of the wave-speed dependent term (non-linear dissipation term) and averaged part of the intercell flux affects the prediction of the instability. Computations using first-order discretisation for the calculation of the flux components results in symmetric stable flow for all schemes except one (the characteristics-based scheme), whereas second- and third-order discretisations lead to a symmetry breaking bifurcation for all schemes within a critical range of Reynolds numbers. The results obtained for all numerical schemes confirm that the flow is steady and symmetric at low Reynolds numbers, becomes asymmetric at a critical Reynolds number then regains symmetry at another critical Reynolds number. In under-resolved grid computations the onset of symmetry breaking bifurcation is different for each numerical scheme and the degree of asymmetry is strongly dependent on the non-linear dissipation of the numerical scheme employed. Fine-resolved simulations showed little difference in the degree of asymmetry predicted by the three numerical schemes employed.