Self-Similar Wave Produced by Local Perturbation of the Kelvin-Helmholtz Shear-Layer Instability

We show that the Kelvin-Helmholtz instability excited by a localized perturbation yields a self-similar wave. The instability of the mixing layer was first conceived by Helmholtz as the inevitable growth of any localized irregularity into a spiral, but the search and uncovering of the resulting self-similar evolution was hindered by the technical success of Kelvin’s wavelike perturbation theory. The identification of a self-similar solution is useful since its specific structure is witness of a subtle nonlinear equilibrium among the forces involved. By simulating numerically the Navier-Stokes equations, we analyze the properties of the wave: growth rate, propagation speed and the dependency of its shape upon the density ratio of the two phases of the mixing layer.

Figure 1

Kelvin-Helmholtz instability excited by two types of initial perturbation, shown by the deformation of the interface. $\mathrm{We}=1000$, $\mathrm{Re}=100$. The two fluids have the same density ($r=1$). For the wavelike initial condition, we observe the familiar roll-up of the interface in vortices whose size is fixed by the initially imposed wavelength. For the localized initial perturbation, we observe the creation of a system of two vortices growing in size without alteration in shape.

Figure 2

Measured wave size $L=\sqrt{A}$ growing in time for several density ratios. We observe an algebraic growth of the wave. The speed of growth depends on the density ratio $r$: the lighter the gas, the slower the growth. As inset, we scale the growth with the law in square root of $r$ according to the theoretical analysis $L\propto \sqrt{r}Ut$. The wave shapes for $r\ne 1$ are shown in Fig. 4.

Figure 3

Schematic representation of the wave as an obstacle to the gas stream. The acceleration of the gas stream above the wave induces a pressure drop. The pressure gradient $(P^{+}-P^{-})/L$ sucks liquid into the wave.

Figure 4

Evolution of the wave represented as a spatiotemporal diagram for three representative density ratios. The characteristic cone delimited from the virtual origin by the liquid and gas speed is represented as a frame of reference. The speed of advancement of the wave is compared to the Dimotakis speed, characteristic of propagation in this two-phase system. The origin of the self-similar shape ($x_0$, $t_0$) is visualized as the tip of the characteristic cone. We observe that it differs from (0,0). This effect is due to the transient from the initial forcing.

Figure 5

Wave interface overlapped in time, recast in the self-similar reference frame ($x^{\prime }$, $y^{\prime }$) for density ratio $r=0.1$. We can observe the self-similar shape of the wave, disturbed by instationarity at the wave tip and drop shedding. We see on Fig. 4 that the initial transient for $r=0.1$ lasts until about $t=70$.