Influence of zero-modes on the inertial-range anisotropy of Rayleigh-Taylor and unstably stratified homogeneous turbulence

The purpose of this work is to study the anisotropic properties of the inertial range of Rayleigh-Taylor and unstably stratified homogeneous (USH) turbulence. More precisely, we aim to understand the role played by the so-called zero-modes, i.e., modes that nullify the anisotropic part of transfer terms. To this end, we determine several characteristic properties of zero-modes using an eddy-damped quasinormal Markovianized (EDQNM) model. Then we perform a high-Reynolds-number EDQNM simulation of a USH flow and check whether the predicted zero-mode properties are indeed observed in this idealized setting. Finally, we carry out a large-eddy simulation of a Rayleigh-Taylor flow and verify if zero-modes can also be identified in this configuration. Among the main findings of this work, we show that the small-scale anisotropy of the velocity and concentration spectra is dominated by the nonlocal contribution of zero-modes rather than by the local action of buoyancy forces. As a result, we predict inertial scaling exponents close to $-7/3$ (rather than $-3$) for the second-order harmonics of the velocity and concentration spectra. By contrast, the concentration flux spectrum remains controlled by buoyancy forces. Still, we show that the zero-mode contribution vanishes slowly as the Reynolds number increases. This translates into a slow convergence of the scaling exponent of the second-order harmonic of the concentration flux to $-7/3$.

Figure 1

Sketch of the idea followed in Ref. [8] to analyze the inertial-range anisotropy of Rayleigh-Taylor and USH turbulence.

Figure 2

Sketch of the idea followed in this work to analyze the inertial-range anisotropy of Rayleigh-Taylor and USH turbulence.

Figure 3

Value of the determinants ${\mathrm{\Delta }}_{\ell }$ as a function of the anisotropy exponent $\xi$ for $\ell =2$, 4, and 6.

Figure 4

Value of $L_{\ell }^F$ as a function of the anisotropy exponent $\xi$ for $\ell =2$, 4, and 6.

Figure 5

Value of $L_{\ell }^B$ as a function of the anisotropy exponent ${\xi }_{\ell }$ for $\ell =2$, 4, and 6.

Figure 6

Evolution of the maximum of the ratio $R_{33}^{\text{dir}}/R_{33}^{\text{pol}}$ in the inertial range (a) as a function of $N_{\theta }$ for $N_k=128$ and (b) as a function of $N_k$ for $N_{\theta }=60$.

Figure 7

Evolution of several one-point statistics for (a) Reynolds number $\text{Re}$ and (b) kinetic energy and concentration variance.

Figure 8

Normalized spectra $E^{*}\left(k\right)$ and $E_B^{*}\left(k\right)$ for $\text{Re}\approx {10}^7$.

Figure 9

Second-order harmonics of the velocity spectrum. (a) Comparison between $r_0^{\text{dir}}, r_2^{\text{dir}}$, and $r_2^{\text{pol}}$. (b) Compensated spectra $r_2^{\text{dir}}{k}^{7/3}$ and $r_2^{\text{pol}}{k}^{7/3}$.

Figure 10

Ratios of anisotropic components of the velocity spectrum: (a) ratio $r_2^{\text{dir}}/r_2^{\text{pol}}$ and (b) ratio $R_{33}^{\text{pol}}/R_{33}^{\text{dir}}$.

Figure 11

Nonlinear frequencies $|T_2^{\text{dir}}/r_2^{\text{dir}}|$ and $|T_2^{\text{pol}}/r_2^{\text{pol}}|$.

Figure 12

(a) Compensated spectra $r_2^{\text{dir}}\left(k\right)k^{7/3}/r_2^{\text{dir}}\left(k_I\right)k_I^{7/3}$ (solid lines) and $r_2^{\text{pol}}\left(k\right)k^{7/3}/r_2^{\text{pol}}\left(k_I\right)k_I^{7/3}$ (dashed lines) at different Reynolds numbers. (b) Maximum value of $r_2^{\text{dir}}\left(k\right)/r_2^{\text{pol}}\left(k\right)$ in the inertial range as a function of the Reynolds number.

Figure 13

Ratios (a) $|r_{\ell }^{\text{dir}}/r_2^{\text{dir}}|$ and (b) $|r_{\ell }^{\text{pol}}/r_2^{\text{pol}}|$ for $\ell =4$ and 6.

Figure 14

Second-order harmonic of the concentration flux $f_2\left(k\right)$ (a) without renormalization and (b) with the $k^{7/3}$ renormalization defined by Eq. (25): $f_2^{★}=\frac{f_2}{c_F{\overline{\varepsilon }}^{1/3}\left(c_K+c_O{\overline{\varepsilon }}_B/\overline{\varepsilon }\right)k^{-7/3}}$.

Figure 15

Difference between $f_2^{★}$ and its expected asymptotic value 1.

Figure 16

Difference between $7/3$ and the inertial slope of the concentration flux measured at $k=k_{ei}/2$ as a function of ${\text{Re}}_{\lambda }$.

Figure 17

Nonlinear frequency $|T_2^F/f_2|$.

Figure 18

Ratio $|f_{\ell }/f_2|$ for $\ell =4$ and 6.

Figure 19

Second-order harmonics of the concentration spectrum. (a) Comparison between $b_0$ and $b_2$. (b) Compensated spectra $b_2{k}^{7/3}$.

Figure 20

Ratio between $b_2\left(k\right)$ and $b_2^{\text{ZM}}\left(k\right)$.

Figure 21

Second-order harmonic of the nonlinear transfer term of $B, T_2^B$.

Figure 22

Ratio $|b_{\ell }/b_2|$ for $\ell =4$ and 6.

Figure 23

Evolution of $\alpha$ as a function of the mixing zone width $L$.

Figure 24

Compensated energy and concentration modulus spectra $E(k,t)$ and $E_B(k,t)$ at the final time of the simulation.

Figure 25

Second-order harmonics of the velocity spectrum for the Rayleigh-Taylor simulation. (a) Comparison between $E, r_2^{\text{dir}}$, and $r_2^{\text{pol}}$. (b) Compensated spectra $r_2^{\text{dir}}{k}^{7/3}$ and $r_2^{\text{pol}}{k}^{7/3}$.

Figure 26

(a) Ratio $r_2^{\text{dir}}/r_2^{\text{pol}}$. (b) Nonlinear frequencies $|T_2^{\text{dir}}/r_2^{\text{dir}}|$ and $|T_2^{\text{pol}}/r_2^{\text{pol}}|$.

Figure 27

Second-order harmonic of the concentration flux $f_2\left(k\right)$ for the Rayleigh-Taylor simulation (a) without renormalization and (b) with a $k^{7/3}$ renormalization: $f_2^{★}=\frac{f_2}{c_F{\langle \overline{\varepsilon }\rangle }^{1/3}(c_K+c_O\langle {\overline{\varepsilon }}_B\rangle /\langle \overline{\varepsilon }\rangle )k^{-7/3}}$.

Figure 28

Nonlinear frequency $|T_2^F/f_2|$.

Figure 29

Second-order harmonics of the concentration spectrum for the Rayleigh-Taylor simulation. (a) Comparison between $b_0$ and $b_2$. (b) Compensated spectra $b_2{k}^{7/3}$.