Imposing a constraint on the discrete Reynolds–Orr equation demonstrated in shear flows

Abstract

The Reynolds–Orr equation predicts the unconditional stability limit of a flow. Although this seems to be a desirable aim in engineering applications, the predicted critical Reynolds numbers are one magnitude below the experimental observations. In this paper, an attempt is made to reduce this gap for incompressible shear flows. It is known that the Navier–Stokes equation has no regular solution at the initial time if the initial velocity field does not fulfill the compatibility condition. However, the original solution of the Reynolds–Orr equation, the critical perturbation, does not necessarily fulfill this condition. Therefore, the condition is added to the original problem as a non-linear constraint. This requires the use of a discrete functional, introduced in the paper. Two different formulations are implemented and discussed. The solution is assumed in a waveform. The augmented problem is solved in the cases of planar Poiseuille and the Couette flow. The result shows that adding the constraint increases the critical Reynolds number significantly in the case of a streamwise perturbation but only slightly in the case of a spanwise one. It was demonstrated using numerical simulations that the single waveform assumption was unreasonably strict. The usage of the compatibility condition without assuming the single waveform has a negligible effect on the critical Reynolds number. However, the presented methods can be used for adding other reasonable and complicated constraints to the variational problem.