Suppression mechanism of Kelvin-Helmholtz instability in compressible fluid flows

The transformative influence of compressibility on the Kelvin-Helmholtz instability (KHI) at the interface between two fluid streams of different velocities is explicated. When the velocity difference is small (subsonic), shear effects dominate the interface flow dynamics causing monotonic roll-up of vorticity and mixing between the two streams leading to the KHI. We find that at supersonic speed differentials, compressibility forces the dominance of dilatational (acoustic) rather than shear dynamics at the interface. Within this dilatational interface layer, traveling pressure waves cause the velocity perturbations to become oscillatory. We demonstrate that the oscillatory fluid motion reverses vortex roll-up and segregates the two streams leading to KHI suppression. Analysis and illustrations of the compressibility-induced suppression mechanism are presented.

Figure 1

A schematic of a mixing layer with the initial perturbation of $u_2^{\prime }(x_1,t=0)$; pivot point $P$; stagnation points $S_1,S_2$; and quadrants marked by $Q_1\text{–}{Q}_4$.

Figure 2

The temporal evolution of the mixing metrics: momentum thickness (a), turbulent kinetic energy (b), and circulation around the inflection point of the base velocity at low and high $M_c$, (c) and (d), respectively. Horizontal axis is the normalized time: $\tau =t\mathrm{\Delta }U/{\delta }_m^0$.

Figure 3

KHI development in the incompressible $\left(M_c=0.3\right)$ case: Stage 1—initial development [(a), (b)]; Stage 2—merger and roll-up [(c) and (d)]; and Stage 3—asymptotic [(e)–(f)].

Figure 4

Schematic of a compressible mixing layer. The initial perturbation profile—$u_2^{\prime }(x_1,0)$—and the DIL are shown. Stagnation points ($P,S_1$, and $S_2$) and quadrants $\left(Q_1\text{–}{Q}_4\right)$ are also identified.

Figure 5

KHI development in the compressible $\left(M_c=1.2\right)$ case: Stage 1—initial development [(a), (b)]; Stage 2—vortex reversal [(c)–(f)]; dashed lines demarcate the dilatational interface layer.

Figure 6

Schematic representation of contrasting the stages of the KHI development in the incompressible (left) against the compressible (right) flows.