Flow Instability: Richtmyer-Meshkov and Pulsating Flows

We examine the linear stability of modulated Taylor--Couette flow in a system of independently rotating concentric cylinders, where the azimuthal velocities are periodically modulated in time. 

With Ω the pulsation rate of the cylinder velocities, the base flow is analytically obtained as U_θ(r,t) = Σ¹_k=-1 U_θ^(k)(r)e^ikΩt. The geometry is described by the inner to outer radius ratio η = R_i / R_o, which is varied across three representative cases: η =0.9 (narrow gap), η =0.75 (moderate gap), and η =0.5 (wide gap). The time-periodic base flow is then uniquely determined by 5 non-dimensional control parameters: two steady Reynolds numbers Re_i⁰ = (U_θ-i⁰D)/ν, Re_o⁰ = (U_θ-o⁰D)/ν, two pulsating Reynolds numbers: Re_i¹ = (U_θ-i¹D)/ν, Re_o¹ = (U_θ-o¹D)/ν, and the Womersley number W_o=(D/2)√(Ω/ν), where U_θ-i⁰, U_θ-o⁰, U_θ-i¹, U_θ-o¹ represent the steady and oscillating components of the inner and outer cylinder velocities respectively, and D=R_o-R_i . Linear instability features such as complex growth rates and Floquet multipliers depend on axial and azimuthal modenumbers of the perturbation and are computed with two different methods: generalized eigenvalue problems using Floquet theory and direct numerical simulations of the linearized Navier--Stokes equations. Neutral stability boundaries in the multi-parameter space are obtained with continuation methods.

Our study systematically explores the effects of varying modulation frequency W_o, and oscillation amplitudes of both inner and outer cylinders, Re_i¹ and Re_o¹. For narrow-gap configurations (η = 0.9), low W_o and increasing oscillation amplitudes initially show destabilizing effects, and eventually lead to the emergence of multiple neutral curves, demarcated by harmonic and subharmonic instability transitions. At higher W_o, these transitions shift to significantly larger oscillation amplitudes. In contrast, wider-gap systems display harmonic–subharmonic bifurcations at even lower values of oscillation amplitude and a wider range of W_o, indicating enhanced sensitivity of the flow to temporal modulation and large curvature effects.

Our findings provide a comprehensive map of the stability boundaries in modulated Taylor--Couette systems and reveal the critical role of geometry and modulation parameters in shaping the onset of instabilities.