**3.4 The interface stability problem**

*g* = *tan*�<sup>1</sup>

*Aerodynamics*

of 10<sup>2</sup>

present.

**Figure 6.**

**112**

*The time sequence is from lower curve to upper.*

angle *ξ* = *tan*�<sup>1</sup>

the hot gas volume.

(*vy*/*vx*). The shock wave and the flow behind it will diverge at a relative

Numerical results for dimensionless vorticity *ω* ¼ *ωi=*ð Þ *V*1*=Rb* developing in the flow with the homogeneous plasma parameter distribution vs. the coordinate *Yi* are presented in **Figure 6a**, for times between *n* = 0.18 and 0.42 with *Δn* = 0.03 increments, and the last curve corresponds to *n* = 0.44 chosen for better resolution on the graph. The time sequence for the curves is in the up-direction. Only the upper half of the picture is shown; thus, the results on the graph correspond to a vortex sheet with rotations in both halves of the picture in opposite directions (in cylindrical geometry) or a toroidal vortex ring if considered in the spherical geometry.

Due to a small size of the structure, the *Y*-coordinate was scaled with the factor

no areas on the shock front where its shape would change sharply enough

rotation angles to move closer to the axis [45]. The total vorticity intensity

*Dimensionless vorticity ϖ vs. Yi, for the uniform (a), exponential (b), and power law (c) density distributions.*

(**Figures 2b** and **4b**), except a very narrow region near the symmetry axis, significant vorticity develops mostly in this region corresponding to the very tips of the fronts. The maximum vorticity intensity is about two orders of magnitude less compared to the levels found in the uniform density distribution case. Considerably weaker vorticity and a very small size of the structure found here can probably explain why sometimes the vortex system "does not develop" in experiments even though noticeable changes in the shock structure and pressure/drag reduction are

In the presence of the energy accumulation effect (power law density distribution), with more stretched fronts and softer distribution of the front inclination angle *φ*, the vorticity develops slower (**Figure 6c**), but still, its intensity steadily grows with time at all locations on the front. The curves on the graph correspond to times *n* = 0.025–0.250 with 0.025 increments and the distance *a* = 3.0*R*b. The vorticity originates further from the symmetry axis confirming a larger size of the vortex structure. As in other density distribution cases, the maxima of vorticity (centers) shift toward each other as the shock advances through the volume, following the same trend for the sharpest bending on the front and the shock and flow

, and thus the picture is greatly zoomed in on the narrow region next to the symmetry axis. The spike in the vorticity intensity observed in close proximity to the symmetry axis exactly corresponds to the location where abrupt change in the shock front structure occurs (**Figures 2a** and **4a**). This trend is pointing at the fact that the local rate of change of the front inclination angle *φ* is the key factor in the origin of the vorticity. The nonlinear growth of the vortex intensity (**Figure 6**) demonstrates the possibility of a strong positive dynamics in its development. The specific effect of the exponential density distribution has been studied for the interaction times between *n* = 0.05 and 0.40 through *Δn* = 0.05 increments, and the last two curves correspond to *n* = 0.43 and 0.45. (**Figure 6b**). Due to practically

(*vy*/*vx*)�*γ* (double refraction) giving rise to a non-zero circulation in

The existence of phenomenological connection between the shock's and the interface stability makes the chain of the transformations in the flow to continue. Since the flow instabilities take place upstream from the interface, the perturbations to the flow parameters propagating downstream, toward the interface, will disturb it. The overall pressure drop behind the refracted shock continuously mounting with time is responsible for the sucking effect resulting in the large-scale interface perturbation, moving it closer to the shock. The positive and essentially nonlinear dynamic in the pressure perturbation evolution [17] will support amplification of this global perturbation to the interface and thus determines the pattern in the interface instability structure. With increasing distortion of the interface, Kelvin-Helmholtz (KH) shearing instability may start to contribute resulting in the characteristic mushroom shapes of the interface perturbations [47]. The initial instability pattern associated with the global pressure drop behind the refracted shock will be of a larger scale, and the KH instability turning on at later stages would finally determine the smaller characteristic structure.
