The flat interface between two fluids in a vertically vibrating vessel may be parametrically excited, leading to the generation of standing waves. The equations constituting the stability problem for the interface of two viscous fluids subjected to sinusoidal forcing are derived and a Floquet analysis is presented. The hydrodynamic system in the presence of viscosity cannot be reduced to a system of Mathieu equations with linear damping. For a given driving frequency, the instability occurs only for certain combinations of the wavelength and driving amplitude, leading to tongue-like stability zones. The viscosity has a qualitative effect on the wavelength at onset: at small viscosities, the wavelength decreases with increasing viscosity, while it increases for higher viscosities. The stability threshold is in good agreement with experimental results. Based on the analysis, a method for the measurement of the interfacial tension, and the sum of densities and dynamic viscosities of two phases of a fluid near the liquid-vapour critical point is proposed.

A turbulent–laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x, y, z) ≈ U0(y) + Uc(y) cos(kz) + Us(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent–laminar patterns in other shear flows – Taylor–Couette, rotor–stator, and plane Poiseuille – are compared.

Turbulent-laminar patterns near transition are simulated in plane Couette flow using an extension of the minimal flow unit methodology. Computational domains are of minimal size in two directions but large in the third. The long direction can be tilted at any prescribed angle to the streamwise direction. Three types of patterned states are found and studied: periodic, localized, and intermittent. These correspond closely to observations in large aspect ratio experiments.Plane Couette flow -the flow between two infinite parallel plates moving in opposite directions -undergoes a subcritical (discontinuous) transition from laminar flow to turbulence as the Reynolds number is increased. Due to its simplicity, this flow has long served as one of the canonical examples for understanding shear turbulence and the subcritical transition process typical of channel and pipe flows [1,2,3,4,5,6,7,8,9,10,11,12]. Only recently was it discovered in very large aspect ratio experiments by Prigent et al. [13,14,15] that this flow also exhibits remarkable pattern formation near transition. Figure 1 shows such a pattern, not from experiment, but from numerical computations reported here. An essentially steady, spatially periodic pattern of distinct regions of turbulent and laminar flow emerges spontaneously from uniform turbulence as the Reynolds number is decreased. It now appears that turbulent-laminar patterns are inevitable intermediate states on the route from turbulent to laminar flow in large aspect ratio plane Couette flow.Related patterns have a long history in fluid dynamics. In Taylor-Couette flow between counter-rotating cylinders, Coles [16] first discovered a state known as spiral turbulence with coexisting turbulent and laminar regions. This state was famously commented on by Feynman [17] and has attracted attention as an example of a coherent structure comprising both turbulence and longrange order [18,19,20,21]. Until recently all experimental studies of this state showed only one turbulent and one laminar patch. Prigent et al. [13,14,15] found that in a very large-aspect-ratio Taylor-Couette system, the turbulent and laminar regions form a periodic pattern, of which the original observations of Coles comprised only one wavelength. Cros and Le Gal [22] discovered large-scale turbulent spirals as well, in experiments on the shear flow between a stationary and a rotating disk. The Reynolds-number thresholds, wavelengths, and angles are very similar for all of these turbulent patterned flows. Moreover, Prigent et al. suggest that the turbulent spots [2,3,4,6,8,10,12,22,23] long known to exist near transition are essentially a manifestation of the same mechanism. FIG. 1: Turbulent-laminar pattern at Reynolds number 350.The computational domain (outlined in white, aligned along x ′ , z ′ ) is repeated periodically to tile an extended region. The kinetic energy is visualized in a plane midway between and parallel to the plates moving in the streamwise (x) direction. Uniform gray or blue corresponds to laminar flow. The s...