Growth rate of Rayleigh-Taylor turbulent mixing layers with the foliation approach

For years, astrophysicists, plasma fusion, and fluid physicists have puzzled over Rayleigh-Taylor turbulent mixing layers. In particular, strong discrepancies in the growth rates have been observed between experiments and numerical simulations. Although two phenomenological mechanisms (mode-coupling and mode-competition) have brought some insight on these differences, convincing theoretical arguments are missing to explain the observed values. In this paper, we provide an analytical expression of the growth rate compatible with both mechanisms and is valid for a self-similar, low Atwood Rayleigh-Taylor turbulent mixing subjected to a constant or time-varying acceleration. The key step in this work is the presentation of foliated averages and foliated turbulent spectra highlighted in our three-dimensional numerical simulations. We show that the exact value of the Rayleigh-Taylor growth rate not only depends upon the acceleration history but is also bound to the power-law exponent of the foliated spectra at large scales.

Figure 1

(Color online) Image of a turbulent Rayleigh-Taylor mixing zone for a constant acceleration $\left(p=0\right)$ when $L\left(t\right)=H$ from our numerical simulations. The axes are defined in such a way that the mixing zone grows along $z$ and is spatially periodic along $x$ and $y$. The initial interface between pure light fluid and pure heavy fluid is located at $z=0$. Thus, along the $z$ axis at time $t$, the flow goes from a turbulent mixing of two fluids when $\left|z\right|<L\left(t\right)/2$ to an intermittent border at $\left|z\right|\approx L\left(t\right)/2$ and finally to a laminar pure fluid (light or heavy depending upon the direction) when $\left|z\right|>L\left(t\right)/2$. The vertical velocity $u_z^{\prime }$ of the flow is plotted in false color on the vertical sections. It puts to the fore large structures going up (light gray/red) and down (dark/blue), made up of mixed fluids. The 3D bubblelike shapes on top (at $z\approx H/2$) correspond to isosurfaces of $u_z^{\prime }$ at 80% of its maximum positive value.

Figure 2

(Color online) Image of the foliated concentration and of the foliated vertical velocity. The top slice displays $⟨c^{\prime }⟩\left(x,y\right)$ in false color at the instant when $L\left(t\right)=H$ for $p=0$. The bottom slice displays $-⟨u_z^{\prime }⟩\left(x,y\right)$ (the minus sign is introduced so that colors match between top and bottom). On the color chart, the value 1 corresponds to the normalized maximum. The striking similarity in location and shape of large scales on both planes (peculiar to foliated average) reveals that heavy mixed fluid tends to concentrate where the flow goes down (dark/blue) and light mixed fluid where the flow goes up (light gray/red).

Figure 3

(Color online) Normalized foliated spectra versus transverse spectra. The left figure represents the ratio of velocity and concentration spectra, ${\mathcal{E}}_z\left(k\right)/\left[{\mathcal{E}}_c\left(k\right)g\left(t\right)L\left(t\right)\right]$ in solid lines and $E_z\left(k\right)/\left[E_c\left(k\right)g\left(t\right)L\left(t\right)\right]$ in dashed lines. The right figure represents the ratio of production and concentration spectra, ${\mathcal{E}}_{cz}\left(k\right)/\left({\mathcal{E}}_c\left(k\right)\sqrt{g\left(t\right)L\left(t\right)}\right)$ in solid lines and $E_{cz}\left(k\right)/\left(E_c\left(k\right)\sqrt{g\left(t\right)L\left(t\right)}\right)$ in dashed lines. Displayed results come from the simulation at constant acceleration $p=0$. The colors correspond to different times in the evolution of the mixing zone: $t_1$ (dark/blue), $<t_2$ (medium gray/red), and $<t_3$ (light gray/green) defined by $L\left(t_1\right)=0.5H$, $L\left(t_2\right)=0.75H$, and $L\left(t_3\right)=H$. In the whole domain of wave numbers, ratios of foliated spectra are distinctively smoother than transverse spectra. In the domain where $n<6$ (large scales), the ratio of foliated spectra is constant and does not vary with time whereas, in the same domain, the ratio of transverse spectra can vary up to one decade.

Figure 4

(Color online) Normalized correlation spectra. The evolution of $\mathcal{R}\left(k,t\right)={\mathcal{E}}_{cz}^2\left(k,t\right)/2{\mathcal{E}}_z\left(k,t\right){\mathcal{E}}_c\left(k,t\right)$ (solid lines) and of $R\left(k,t\right)=E_{cz}^2\left(k,t\right)/2E_z\left(k,t\right)E_c\left(k,t\right)$ (dashed lines) at constant acceleration $p=0$ is compared at different times, $t_1$ (dark/blue), $<t_2$ (medium gray/red), and $<t_3$ (light gray/green). The simulation, at $p=0$ (but at $p=1$, 2 and 3 as well), shows that when $n\le 6$, $\mathcal{R}=1$ to within 6% and with the same characteristic smoothness as already depicted on Fig. 2 (vertical axis is linear). On the contrary, $R$ varies significantly as time goes by on the same range of wave numbers.

Figure 5

(Color online) Observed to predicted ratio of growth rate ${\alpha }_p$. Green squares correspond to results obtained with SURFER and performed by the authors and red circles to results performed with TURMOIL3D by D. Youngs and A. Llor. A least square best fit has been applied on the numerical results using the formula (41) and it was found, assuming $s$ does not depend on $p$, that $s=4.0\pm 0.1$. Error bars correspond to the size of the biggest interval containing all three different measures of ${\alpha }_p$: (1) ${\alpha }_p=L\left(t\right)/\left(\mathcal{A}g\left(t\right)t^2\right)$, (2) ${\alpha }_p=\dot{L}\left(t\right)/\left[\left(p+2\right)\mathcal{A}g\left(t\right)t\right]$, and (3) the same method described in Ref. [4]. Error bars were not provided in Ref. [28].