Ordered and disordered dynamics in inertialess stratified three-layer shear flows (original) (raw)
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The interaction of double-layer density stratified interfaces with initial non-uniform velocity shear is investigated theoretically and numerically, taking the incompressible Richtmyer–Meshkov instability as an example. The linear analysis for providing the initial conditions in numerical calculations is performed, and some numerical examples of vortex double layers are presented using the vortex sheet model. We show that the density stratifications (Atwood numbers), the initial distance between two interfaces, and the distribution of the initial velocity shear determine the evolution of vortex double layers. When the Atwood numbers are large, the deformation of interfaces is small, and the distance between the two interfaces is almost unchanged. On the other hand, when the Atwood numbers are small and the initial distance between two interfaces is sufficiently close (less than or equal to the half of the wavelength of the initial disturbance), the two interfaces get closer to each ...
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Journal of Fluid Mechanics, 2008
The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the 'shear-Marangoni number'. From the weakly nonlinear PDEs, a single Stuart-Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart-Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart-Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < C s and subcritical for C > C s , where C s is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near C s , but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch ('Floquet') disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacialsurfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.
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