Large wave number vortices in time–periodic flows

Abstract

The linear stability properties of short wavelength vortices within time–periodic flows are examined using a WKB–based asymptotic analysis. Such vortices typically have an instantaneous growth rate considerably larger than the time–scale of the underlying flow and thus have often been studied using quasi–steady techniques. A major drawback of these quasi–steady approaches is that although they are able to determine the instantaneous flow structure they are of no use in describing the evolution of the modes. It is shown how rational asymptotic analysis based on WKB ideas overcomes this deficiency and enables the form of the neutral stability curve to be obtained for short wavelengths. The technique is applied to three problems. First, the key features of the analysis are exemplified by careful consideration of a model problem which is motivated by the physical situation of the sinusoidal heating of a horizontal flat plate bounding a semi–infinite layer of fluid. The governing linearized stability equations take the form of a system which is sixth order in the spatial variable and second order in time; however, the important features of this system are captured by our model which is only second order in the spatial coordinate and first order in time. The second physically important flow examined is the motion induced in a viscous fluid surrounding a long torsionally oscillating cylinder (a curved Stokes flow). Detailed asymptotic and numerical studies of both the model and the two physically motivated flows are undertaken and it is shown how the analytical results provide good qualitative agreement with the numerical findings at quite modest wavenumbers. The methodology adopted here may be used as a basis for the rational study of the stability properties of other time–periodic flows. A key result forthcoming from this study is the demonstration that the first few terms in the relevant high wavenumber form of the neutral curve for modes in an oscillatory flow may be derived relatively quickly with minimal computational effort.