Scaling law of mixing layer in cylindrical Rayleigh-Taylor turbulence

The nonlinear evolution of mixing layer in cylindrical Rayleigh-Taylor (RT) turbulence is studied theoretically and numerically. The scaling laws including the hyperbolic cosine growth for outward mixing layer and the cosine growth for inward mixing layer of the cylindrical RT turbulence are proposed for the first time and verified reliably by direct numerical simulation of the Navier-Stokes equations. It is identified that the scaling laws for the cylindrical RT turbulence transcend the classical power law for the planar RT turbulence and can be recovered to the quadratic growth as cylindrical geometry effect vanishes. Further, characteristic time- and length scales are reasonably obtained based on the scaling laws to reveal the self-similar evolution features for the cylindrical RT turbulence.

Figure 1

The diagram of cylindrical RT instability for (a) convergent case and (b) divergent case.

Figure 2

Snapshots of instantaneous density fields for 2D convergent (a)–(c) and divergent (e)–(g) cases at the rescaled time $\tau =3$. Isosurfaces of the density (orange for $\rho =0.41$, gray for $\rho =1$, and purple for $\rho =1.59$) for 3D convergent (d) and divergent (h) cases at the rescaled time $\tau =3$.

Figure 3

The density energy spectra in the circumferential direction at $r_0=1$ for (a) convergent and (b) divergent cases at the rescaled time $\tau =3$. The energy spectra of cases 4 and 8 are obtained by an average in the axial direction.

Figure 4

Mean density distributions in the radial direction for the typical cases at different moments: (a) case 2 and (b) case 6.

Figure 5

Mean density distributions versus similarity variable $\xi$ for typical cases at different moments: (a) case 2 and (b) case 6.

Figure 6

Temporal evolution of (a) the width of mixing layer $h$ and (b) the ratio of width of spike region to that of bubble region, $h_s/h_b$, for the cases listed in Table 1.

Figure 7

Temporal evolution of (a) the mixed mass $M$ and (b) the normalized mixed mass $\mathrm{\Psi }$ for the cases listed in Table 1.

Figure 8

Ratios of the square values of the circumferential velocity and growth velocity versus rescaled time for (a) the inward mixing layer and (b) the outward mixing layer.

Figure 9

The scaling law of outward mixing layer for (a) convergent cases and (b) divergent cases, and the scaling law of inward mixing layer for (c) convergent cases and (d) divergent cases.