5.1. Temporally growing Rayleigh–Taylor mixing layer

The results from TRT simulation T0 are presented here. In § 5.1.1, different growth regimes of T0 are identified. In § 5.1.2, the data from the self-similar phase are used to validate the scaling assumptions of the SRT equations.

5.1.1. Self-similar growth in TRT

Since TRT flow is non-stationary and homogeneous in the horizontal directions, it is customary to estimate ensemble averages with planar averages, i.e. $\langle {\cdot } \rangle (t,x_2) = \langle {\cdot } \rangle _{1,3}(t,x_2)$. However, this may result in profiles with significant statistical noise. To mitigate this issue, we leverage self-similar scaling and average results over a small time window, $t_1< t< t_2$. Specifically, dimensional data are averaged first in the horizontal directions, then normalized using self-similar scaling, and finally averaged over a small time window. For example, the ensemble-averaged normalized kinetic energy is computed as

(5.1) \begin{equation} \biggl[\frac{\langle \rho u_i^2/2 \rangle}{\rho_0Agh_p}\biggr]_{TRT}\left(t,\frac{x_2-\delta_I}{h_p}\right) = \left\langle \frac{\langle \rho u_i^2/2 \rangle_{1,3}(t,(x_2-\delta_I)/h_p(t))}{\rho_0Agh_p(t)} \right\rangle_{t_1\le t\le t_2}. \end{equation}

The assumed scaling is only valid if the flow is indeed self-similar. In the general case, a small time window is used simply to reduce statistical noise. However, in the self-similar growth regime, larger time windows can be used.

Case T0 is initialized as a perturbed planar interface, and its mathematical description is detailed in § 4.2. Large wavenumber perturbations ($\ell _I \ll h(t)$) are chosen to encourage late-time self-similar growth that is driven by mode-coupling processes. The time evolution of several quantities derived from the density field is shown in figure 1. Because there is no constant characteristic time scale in TRT, time is non-dimensionalized by $({\mathcal {L}}/Ag)^{1/2}$, an arbitrary choice of no physical significance. In figure 1(a), the mixing layer growth is represented by three different height definitions (defined in table 1): $h_p$ is the product height that will be used for comparisons with Cabot & Cook (Reference Cabot and Cook2006); $h_m$ is the mixing height that will be used as an input parameter to the SRT equations; and $H$ is a threshold-based height.

Figure 1. Temporal evolution for TRT-T0: (a) normalized heights and Reynolds number; (b) growth and mixedness parameters. Height definitions are summarized in table 1.

To minimize domain confinement effects, simulations were stopped shortly after $H(t)/{\mathcal {L}}>0.7$. The height $H(t)$ is also used to compute the Reynolds number, $Re=H\dot {H}/\nu$. Because $H$ is determined from averages over a plane, it is more susceptible to noise than integral height definitions like $h_p$ or $h_m$. Its time derivative $\dot {H}$ is computed using a second-order finite difference in time, which exacerbates statistical noise and contributes to significant fluctuations in $Re$. Nonetheless, a growing Reynolds number is clearly observed. The largest $Re$ considered in this simulation, attained at $t(Ag/{\mathcal {L}})^{1/2} = 3$, is approximately 8000.

The integral heights $h_p$ and $h_m$ from figure 1(a) are normalized using self-similar scaling (1.3) to obtain the growth parameters, $\alpha _i = \dot {h}_i^2/(4 A g h_i)$, shown in figure 1(b). The mixedness parameter $\varXi$ is defined identically to Cabot & Cook (Reference Cabot and Cook2006) as

(5.2) \begin{equation} \varXi = \frac{1}{h_p}\int^\infty_{-\infty} 2 \langle \min(X,1-X) \rangle \,{\rm d}\kern0.7pt x_2. \end{equation}

A diffusion-dominated regime is observed for $t(Ag/{\mathcal {L}})^{1/2} \lesssim 0.2$, where the flow is almost fully mixed at $\varXi \approx 1$ and the growth parameter decreases. Subsequently, RT instabilities cause large-amplitude changes in all parameters before the flow transitions to turbulence and reaches quadratic self-similar growth at $t(Ag/{\mathcal {L}})^{1/2} \ge 2$. While more sophisticated data analysis techniques can be used to quantify the extent of flow self-similarity (Morgan et al. Reference Morgan, Olson, White and McFarland2017), the self-similar phase in T0 is identified visually by the stationary behaviour of key statistical parameters. This onset of self-similar growth happens at $Re \approx 3000$. Although $Re \approx 2000$ was identified in § 4.2 as a lower bound for observed self-similarity in Cabot & Cook (Reference Cabot and Cook2006), figure 1(b) shows that this is true for $\alpha _p$, but not for $\alpha _m$. Indeed, it is well known that not all statistics in TRT flow achieve self-similar behaviour at the same time. Figure 2 shows one-dimensional scalar statistics and further illustrates this: in figure 2(a), the average mole fraction $\langle X \rangle$ appears to be converged by $t(Ag/{\mathcal {L}})^{1/2} = 1$, while the local mixed mass in figure 2(b) only converges around $t(Ag/{\mathcal {L}})^{1/2} = 1.5$.

Figure 2. Temporal evolution for TRT-T0: (a) heavy fluid mole fraction; (b) normalized local mixed mass. Lines represent $t(Ag/{\mathcal {L}})^{1/2} = 1.0$ (black dotted), 1.5 (green solid), 2.0 (magenta dashed), 2.5 (blue dash-dotted), 3.0 (red thick solid), corresponding to $Re \approx 1400$, 1900, 2800, 4400, and 8000, respectively. One-dimensional profiles are first normalized, then averaged over $t(Ag/{\mathcal {L}})^{1/2} \pm 0.1$ to reduce statistical variability.

For the purposes of comparison with Cabot & Cook (Reference Cabot and Cook2006) and subsequent SRT results, $2\le t(Ag/{\mathcal {L}})^{1/2} \le 3$ is taken as the time window where the scalar field exhibits self-similarity. Time averages are calculated over this window to yield $\alpha _m = 0.0190$, $\alpha _p = 0.0173$ and $\varXi = 0.814$. Values of $\alpha _p$ and $\varXi$ were extracted from figures 4 and 6 of Cabot & Cook (Reference Cabot and Cook2006) and averaged over time for $Re>3000$. These values, $\alpha _p \approx 0.019$ and $\varXi \approx 0.82$, compare well with those obtained from case T0.

Next, the velocity statistics of T0 are examined. Figure 3(a) shows the time evolution of vertically integrated kinetic energy, $K = \int ^{+\infty }_{-\infty } \langle \rho k \rangle \,{\rm d}\kern0.7pt x_2$, where $k=u_iu_i/2$, while figure 3(b) shows the planar-averaged profiles. Figure 3(a) illustrates that the normalized total kinetic energy has a small positive slope for $t(Ag/{\mathcal {L}})^{1/2} \ge 2$, indicating that the velocity statistics of T0 may not be fully self-similar, despite stationarity in $\alpha _i$ values. This faster-than-self-similar growth is also observed in the normalized one-dimensional profiles shown in figure 3(b), where the magnitudes continue to increase (albeit slowly) for $t(Ag/{\mathcal {L}})^{1/2} \ge 2$.

Figure 3. Temporal evolution of normalized kinetic energy for TRT-T0: (a) vertically integrated; (b) planar averaged. Lines represent $t(Ag/{\mathcal {L}})^{1/2} =1.0$ (black dotted), 1.5 (green solid), 2.0 (magenta dashed), 2.5 (blue dash-dotted), 3.0 (red thick solid), corresponding to $Re \approx 1400$, 1900, 2800, 4400 and 8000, respectively. One-dimensional profiles are first normalized, then averaged over $t(Ag/{\mathcal {L}})^{1/2} \pm 0.1$ to reduce statistical variability.

Although T0 does not demonstrate true self-similarity in velocity, the onset of slow growth in normalized kinetic energy aligns with the time window for scalar self-similarity. Hence, the same time window, $2\le t(Ag/{\mathcal {L}})^{1/2} \le 3$, is used to compute approximately self-similar velocity statistics for subsequent comparisons. Convergence to self-similarity in TRT is a complex issue that depends on the initial conditions and the specific flow statistic considered. It is beyond the scope of this paper to question the existence of true velocity self-similarity in TRT flow. We simply note that T0 exhibits faster-than-self-similar growth in velocity statistics for $2\le t(Ag/{\mathcal {L}})^{1/2} \le 3$ and bear this result in mind through subsequent sections.

5.1.2. A priori verification of SRT equations using TRT data

An important objective of the SRT flow configuration is to reproduce TRT flow dynamics. Although the derivation of the SRT equations is informed fundamentally by observations of self-similarity in TRT flow, the statistical equivalence of SRT and TRT flow, stated mathematically as (2.49), has not been demonstrated. Contrary to TRT flow, the time derivative of any ensemble-averaged quantity in a converged SRT flow is zero and is instead replaced by contributions from $S_\phi$ and $T_i$. For (2.49) to hold, the additional SRT terms must, on average, be exactly the negative of the time derivative in TRT flow. We derive the analytical mixed mass and kinetic energy budgets for SRT, assume the validity of (2.49) by evaluating the budget terms from T0 data, and verify that the contributions from $S_\phi$ and $T_i$ can indeed mimic the time derivative.

In the stationary state, the balance of local mixed mass for SRT can be obtained from (2.46) by setting the time derivative to zero. This is written as

(5.3)\begin{equation} 0 ={-} \frac{\partial \langle \rho^* m^*u^*_2\rangle}{\partial \xi_2} + \frac{\partial}{\partial \xi_2}\left\langle \rho^* D \frac{\partial m^*}{\partial \xi_2}\right\rangle + \langle \rho^* \chi^* \rangle + \left(\frac{1}{\tau_0}\xi_2+\delta'_0\right)\frac{\partial \langle \rho^* m^*\rangle}{\partial \xi_2}. \end{equation}

We assume relationship (2.49) and express (5.3) using TRT quantities, leading to

(5.4)\begin{equation} 0 = \left[-\frac{\partial \langle \rho mu_2\rangle}{\partial x_2} + \frac{\partial}{\partial x_2}\left\langle \rho D \frac{\partial m}{\partial x_2}\right\rangle + \langle \rho \chi \rangle \right]_{t=t_0} + \left[\frac{1}{\tau_0}(x_2-\delta_I)\frac{\partial \langle \rho m\rangle}{\partial x_2}\right]_{t=t_0}. \end{equation}

The first three terms of (5.4) are precisely the right-hand side of the TRT mixed mass budget shown in (2.17). Their sum is exactly the time derivative of $\langle \rho m \rangle$. The final term in (5.4) is simplified using relations for $\tau _0$ and $\delta _0'$ from (2.50) and (2.51). Hence

(5.5)\begin{equation} \left[\frac{\partial \langle \rho m\rangle}{\partial t} \right]_{t=t_0} ={-}\left[\frac{\dot{h}_m}{h_m}(x_2-\delta_I)\frac{\partial \langle \rho m\rangle}{\partial x_2}\right]_{t=t_0}. \end{equation}

The SRT term $S_\phi$ contributes a mixed mass transport term that is exactly the negative of the TRT time derivative at $t=t_0$. Both terms of (5.5) are computed from T0 data over the self-similar region, normalized by $\rho _0 (Ag/h_p)^{1/2}$, and shown in figure 4(a). The comparison in figure 4(a) shows that the right-hand side of (5.5) recovers the time derivative of the mixed mass convincingly.

Figure 4. Verification of SRT scaling assumptions using TRT-T0 data, normalized and averaged over $2.0\le t(Ag/{\mathcal {L}})^{1/2} \le 3.0$. Left-hand side and right-hand side terms from (5.5) and (5.8) are compared. (a) Mixed mass: time derivative (black solid); right-hand side of (5.5) (blue dashed). (b) Kinetic energy: time derivative, ${\mathcal {T}}$ (black solid); ${\mathcal {T}}_S$ (blue dashed); ${\mathcal {T}}_T$ (green dotted); ${\mathcal {T}}_S + {\mathcal {T}}_T$ (red dash-dotted).

The same analysis is applied to the kinetic energy budget, which includes contributions from both $S_\phi$ and $T_i$. The TRT kinetic energy budget is derived by multiplying (2.2) by $u_i$ and invoking continuity to yield

(5.6)\begin{equation} \frac{\partial \langle \rho k \rangle }{\partial t} ={-} \frac{\partial \langle \rho k u_2\rangle }{\partial x_2} - \left \langle u_i \frac{\partial p}{\partial x_i} \right\rangle + \frac{\partial \langle u_i \tau_{i2}\rangle}{\partial x_2} - \left\langle \tau_{ij} \frac{\partial u_i}{\partial x_j}\right\rangle - \langle \rho u_2 \rangle g. \end{equation}

The kinetic energy balance for converged SRT flow can be derived similarly by multiplying (2.34) with $u^*_i$, and setting the time derivative to zero. This results in

(5.7)\begin{align} 0 &={-} \frac{\partial \langle \rho^* k^* u^*_2\rangle }{\partial \xi_2} - \left \langle u_i^*\frac{\partial p^*}{\partial \xi_i} \right\rangle + \frac{\partial \langle u^*_i \tau^*_{i2}\rangle}{\partial \xi_2} - \left\langle \tau^*_{ij} \frac{\partial u^*_i}{\partial \xi_j}\right\rangle- \langle \rho^* {u^*_2} \rangle g \nonumber\\ &\quad + \left(\frac{1}{\tau_0}\xi_2 + \delta'_0\right)\frac{\partial \langle\rho^* k^*\rangle}{\partial \xi_2} - \frac{1}{\tau_0}\langle \rho^* k^*\rangle , \end{align}

where $k^* = u^*_iu^*_i/2$. After applying assumption (2.49) on (5.7) and substituting (5.6), we get

(5.8)\begin{equation} \underbrace{\left[ \frac{\partial \langle \rho k \rangle }{\partial t} \right]_{t=t_0}}_{{\mathcal{T}}(t_0)} = \underbrace{\left[ -\frac{\dot{h}_m}{h_m} (x_2-\delta_I) \frac{\partial \langle\rho k \rangle}{\partial x_2}\right]_{t=t_0}}_{{\mathcal{T}}_S(t_0)} + \underbrace{\left[ \frac{\dot{h}_m}{h_m} \langle\rho k\rangle \right]_{t=t_0}}_{{\mathcal{T}}_T(t_0)}, \end{equation}

where $\tau _0$ and $\delta _0'$ are expressed in terms of $h_m$ and $\delta _I$ using (2.50) and (2.51). The two terms on the right-hand side of (5.8) represent length and velocity scaling contributions, respectively.

The terms of (5.8) are computed from T0 data, normalized by $\rho _0(Ag)^{3/2}h_p^{1/2}$, and shown in figure 4(b). Both ${\mathcal {T}}_S$ and ${\mathcal {T}}_T$ contribute substantially to the estimate of the time derivative, with ${\mathcal {T}}_S$ being important near the edges of the mixing layer and ${\mathcal {T}}_T$ dominating near the core. The sum of both terms underestimates the time derivative, with a maximum difference of 17 % at $x_2\approx \delta _I$. Since the SRT equations are derived on the assumption of self-similarity in velocity, it is not surprising that the SRT terms underestimate the time derivative of T0, a flow that is experiencing faster-than-self-similar velocity growth (see figure 3).

5.2. Stationarity in the SRT configuration

The time evolution of SRT flow is studied using cases I1–I3. These cases have the same $\lambda =1.5$ and ${\textit {Gr}} = 7.4\times 10^6$, but start from different initial conditions, as detailed in § 4.2.

Figure 5(a,b) shows different height definitions that represent the large-scale growth of the mixing layer, while figure 5(c,d) shows the integrated scalar and viscous dissipation, respectively, as representations of the small-scale evolution. In these figures, ensemble averages are estimated with planar averages. After an initial transient, all cases exhibit statistical stationarity in all quantities. Case I1 is initialized with a TRT flow whose mixing height is approximately the input height, hence there are no noticeable changes in any of the quantities computed from I1. Case I2 is initialized as a perturbed flat interface. All quantities in I2 start from small values and grow to their statistically stationary values. Because $h^*_m$ is actively controlled via the update of $\tau _0$ according to closure equation (3.1), it approaches an approximately constant value in a manner consistent with (3.2). In contrast, $h^*_p$ responds via the flow dynamics of the SRT equations and oscillates to a greater extent. Case I3 is initialized with an SRT flow of the same height but a smaller ${\textit {Gr}}$. The dissipation integrals start from smaller values, but take less than one $\tau _0$ to grow to their stationary values. Beyond $5\tau _0$, all three signals are indistinguishable. The time-averaged results for $s>5\tau _0$ are summarized in table 4, with uncertainties computed using the method presented in Rah & Blanquart (Reference Rah and Blanquart2019). It is clear that the stationary solution does not depend on the initial conditions.

Figure 5. Evolution of normalized (a) mixing height, (b) product height, (c) scalar dissipation integral and (d) viscous dissipation integral for different initial conditions. Insets are zoomed in on early times to highlight differences in the initial transient. The initial conditions used in the SRT cases (I1–I3) correspond respectively to: TRT (red solid); perturbed flat interface (blue dashed); and lower ${\textit {Gr}}$ SRT (green dash-dotted).

Table 4. Effect of initial conditions on SRT for ${\textit {Gr}} = 7.4\times 10^{6}, \lambda = 1.5$.

These results have two practical implications. First, because the initial conditions of an SRT flow do not affect its stationary solution, initializations may be chosen to maximize efficiency without regard for their effect on the final results. Second, leveraging flow stationarity, all ensemble quantities that are extracted from SRT simulations are averaged in both homogeneous directions and time. Each SRT simulation is performed for at least $50\tau _0$ and statistics are collected after $5\tau _0$. Mathematically, $\langle {\cdot } \rangle _{SRT} = \langle {\cdot } \rangle _{1,3,s>5\tau _0}$.

5.3. Validity of the SRT equations

Three key assumptions are used in the derivation of the SRT equations. Assumption (III) was verified in § 5.1 through stationarity in $\alpha (t)$; assumption (II) was verified in § 5.2 through stationarity of all flow quantities in the SRT configuration. In this section, the validity of assumption (I) is assessed. To justify assumption (I), we first propose a candidate for how large this ‘small window of time’ needs to be, then quantify the errors incurred by the SRT simplifications over this time window.

5.3.1. Integral time scale

In § 5.2, it is shown that the flow loses memory of its initial conditions. Stated differently, the flow field loses its time correlation. To quantify how quickly this occurs, temporal autocorrelation functions and integral time scales of several global quantities are computed from I1. These are computed from SRT simulations, as the flow must be statistically stationary and sampled over long times. The temporal autocorrelation function for a general time-varying quantity $F^*(s)$ is

(5.9)\begin{equation} R(\tau) = \frac{\langle {F^*}(s+\tau){F^*}(s)\rangle_s}{\langle {F^*}^2\rangle_s}. \end{equation}

Autocorrelation functions are computed and shown in figure 6(a) for $F^* = h_p^*,K^*, \varPhi _\epsilon ^*$ and $\varPhi _\chi ^*$. All quantities become significantly uncorrelated by $\tau \approx \tau _0$. To quantify this more specifically, integral time scales are computed for each of these functions.

Figure 6. (a) Autocorrelation functions from simulation I1 and (b) corresponding integral time scales computed as the integral of $R(\tau )$ up to the $n$th zero crossing. Legend: $h_p^*$ (black solid line, squares); $K^*$ (blue dash-dotted, triangles); $\varPhi _\epsilon ^*$ (green dotted, circles); and $\varPhi _\chi ^*$ (red dashed, crosses).

The integral time scale is defined as $\tau _{int} = \int ^\infty _0 R(\tau ) \,{\rm d}\tau$. Since these time signals (and their autocorrelation functions) have finite lengths, $\tau _{int}$ is estimated with an integral up to the $n$th zero crossing of the autocorrelation function, $\tau _{int}\approx \tau _n$. The integral time scale of each quantity is estimated for varying $n$ and shown in figure 6(b). In general, $\tau _n$ varies with the choice of quantity and decreases slowly with $n$. A conservative upper bound is taken as $\tau _{int}\approx 0.4\tau _0$. After $\tau _{int}$, flow states become uncorrelated, and a subsequent, independent realization of the flow begins. Hence, the SRT approximations need to be valid for a time window bounded by $t_0\pm \tau _{int}/2$.

5.3.2. Error in mixed mass transport and kinetic energy

Recall that the SRT equations are derived from the NSE in § 2.2 through two main steps. The original NSE, (2.1)–(2.3), are first transformed to the rescaled NSE, (2.23)–(2.25), then simplified to the final SRT equations, (2.33)–(2.35), by evaluating them at $q(t)=q_0$ based on assumption (I). Assessing the validity of assumption (I) is equivalent to quantifying the magnitude of the terms that are neglected even though $q\neq q_0$. The SRT errors in mixed mass and kinetic energy transport are estimated by comparing the budget terms evaluated at $q(t_0\pm \tau _{int}/2)$ (rescaled TRT) with those at $q(t_0)$ (SRT).

First, assumption (III) is used to relate $q(t_0\pm \tau _{int}/2)/q_0$ to $\tau _{int}/\tau _0$. Differentiating (2.32) in time yields $q'' = 2\alpha _q A g$, and $q^{(n)}=0$ for $n>2$. The function $q(t)$ can be written as an exact Taylor series, which leads to

(5.10)\begin{equation} \frac{q(t_0\pm \tau_{int}/2)}{q_0} = \frac{1}{q_0}\Bigg( q_0 \pm q_0'\frac{\tau_{int}}{2} + q_0''\frac{\tau_{int}^2}{8}\Bigg) = 1 \pm \frac{1}{2}\frac{\tau_{int}}{\tau_0} + \frac{1}{16}\left( \frac{\tau_{int}}{\tau_0}\right)^2. \end{equation}

Substituting $\tau _{int}/\tau _0 = 0.4$, we obtain $0.81\le q/q_0 \le 1.21$.

The SRT errors in mixed mass and kinetic energy budgets are estimated by comparing budgets for the RNSE ($0.81\le q(t)/q_0 \le 1.21$) and the SRT equations ($q=q_0$). The full RNSE budgets and their derivations are found in Appendix E. They are concisely stated as

(5.11)\begin{gather} \frac{\partial \langle \rho^* m^* \rangle}{\partial s} ={-} \frac{\partial \langle \rho^* m^*u^*_2\rangle}{\partial \xi_2} + \frac{\partial }{\partial \xi_2}\left\langle \rho^* D \frac{\partial m^*}{\partial \xi_2}\right\rangle + \langle \rho^* \chi^* \rangle + \left(\frac{\xi_2}{\tau_0}+\delta'_0\right)\frac{\partial \langle \rho^* m^*\rangle}{\partial \xi_2} + H_m(t), \end{gather}

(5.12) \begin{align} \frac{\partial \langle\rho^*k^*\rangle}{\partial s} &={-} \frac{\partial \langle \rho^* k^* u^*_2\rangle }{\partial \xi_2} - \left \langle u_i^*\frac{\partial {p^*}'}{\partial \xi_i} \right\rangle + \frac{\partial \langle u^*_i \tau^*_{i2}\rangle}{\partial \xi_2} - \langle \rho^*\epsilon^*\rangle - \langle (\rho^*-\langle \rho^* \rangle) {u^*_2} \rangle g \nonumber\\ &\quad + \left(\frac{\xi_2}{\tau_0} + \delta'_0\right)\frac{\partial \langle\rho^* k^*\rangle}{\partial \xi_2} - \frac{\langle \rho^* k^*\rangle}{\tau_0} + H_k(t). \end{align}

The first four terms on the right-hand side of (5.11) constitute the full SRT ($q=q_0$) mixed mass budget. They represent convective transport, diffusive transport, scalar dissipation and the SRT source terms contribution, respectively. The first seven terms on the right-hand side of (5.12) constitute the full SRT kinetic energy budget. They represent convective transport, pressure transport, viscous transport, viscous dissipation, production by buoyancy, and source term contributions from length and velocity scaling, respectively. The final term of each budget equation, $H_m$ and $H_k$, is the only term that depends on $q(t)$, and represents the error of the SRT equations. In figure 7(a,b), these error terms are compared with their corresponding SRT budget terms and shown to be much smaller for both budgets considered. Therefore, they can be neglected without significant impact on the overall flow dynamics.

Figure 7. Comparison of SRT budgets with their $q(t)$-dependent errors: (a) mixed mass, normalized by $\rho _0(Ag/h_p^*)^{1/2}$: convection (red circles); scalar dissipation (green triangles); SRT source term (blue squares); mixed mass diffusion is negligible (not shown). (b) Kinetic energy, normalized by $\rho _0(Ag)^{3/2}h_p^{*1/2}$: convection (red circles); viscous dissipation (green triangles); SRT source terms (blue squares); production (orange crosses); pressure transport (magenta solid line, no symbols); viscous transport is negligible (not shown). In both panels, dashed and dotted black lines represent the total error for $q/q_0 = 0.81$ and $1.21$, respectively.

6.1. Effect of mixing layer aspect ratio

The effect of mixing layer aspect ratio, $\lambda =\bar {h}_m/{\mathcal {L}}$, on SRT is shown qualitatively in figure 8. At all $\lambda$ considered, the two reservoirs of pure heavy and light fluid are separated by a highly mixed region that exhibits three-dimensional, random and multiscale features commonly observed in turbulent flows. Consistently, the largest flow structures appear to occupy the entire lateral domain width. For wider (small $\lambda$) configurations, sharp interfaces between pockets of heavy and light fluid are evident in many areas, a feature that disappears with enhanced mixing observed in the narrower (large $\lambda$) configurations.

Figure 8. Density fields for SRT cases L1–L5 at ${\textit {Gr}} = 7.4\times 10^{6}$, with increasing mixing layer aspect ratios from left to right: $\lambda = 0.5$, 1.0, 1.5, 2.0, 2.5. Images are rescaled to have the same $h_m^*$, and non-mixed regions of the flow in the vertical direction are intentionally removed. Blue and red regions correspond to $\rho _H$ and $\rho _L$, respectively.

The role of the lateral domain size ${\mathcal {L}}$ is different in TRT and SRT flows. In self-similar TRT flow, Zhou & Cabot (Reference Zhou and Cabot2019) showed that both vertical and horizontal length scales of the flow grow at approximately fixed ratios to the mixing layer height. The TRT flow configuration is solved over a finite time, and domains can be chosen to be sufficiently large (relative to the finite time window) so that they do not affect the flow. Unlike TRT, SRT flow is designed to be simulated over infinitely long times. Vertical growth is restricted by $\bar {h}_m$ but lateral growth is not; horizontal length scales can grow continuously until they are constrained by the lateral domain width if there are no other fluid dynamical constraining mechanisms. As lateral length scales approach the size of the domain, the flow physics is affected by the periodic lateral boundaries. In other stationary configurations of both isotropic (Rosales & Meneveau Reference Rosales and Meneveau2005; Dhandapani & Blanquart Reference Dhandapani and Blanquart2020) and buoyant (Chung & Pullin Reference Chung and Pullin2010; Carroll & Blanquart Reference Carroll and Blanquart2015) turbulent flows, integral turbulence length scales have similarly been shown to depend on the domain dimensions.

To quantify this behaviour, additional flow length scales, similar to Zhou & Cabot (Reference Zhou and Cabot2019), are computed. The correlation length $\varLambda ^*_i$ is defined as the integral of the longitudinal autocorrelation function, i.e.

(6.1a,b)\begin{equation} \varLambda^*_i = \int^\infty_0 C_i(r) \,{\rm d}r, \quad C_i(r) = \frac{\langle {u^*_i}' (s,\boldsymbol{\xi} + \boldsymbol{e}_i r) {u^*_i}'(s,\boldsymbol{\xi})\rangle}{\langle {u^*_i}'^2 \rangle}, \end{equation}

with no sum over index $i$ (where $i=1, 2$ or 3), and ${u_i^*}' = u_i^* - \langle u_i^*\rangle$. Since the flow is periodic in $\xi _1$ and $\xi _3$, the horizontal correlation lengths can be defined in terms of their energy spectra as

(6.2) \begin{equation} L^*_i = \left.{\displaystyle\int^\infty_0 \kappa^{{-}1} E^*_i(\kappa) \,{\rm d}\kappa}\right/{\displaystyle\int^\infty_0 E^*_i(\kappa)\,{\rm d}\kappa}, \end{equation}

where $E^*_i(\kappa )$ is the two-dimensional spectral energy density of $u^*_i$, and $\kappa$ is the two-dimensional wavenumber in the horizontal plane. To characterize the layer core, $L_1^*$ and $L_3^*$ are computed on the $\xi _2$-plane closest to $\delta ^*_I$. Zhou & Cabot (Reference Zhou and Cabot2019) verified that definitions (6.1) and (6.2) are approximately equivalent, hence, $\varLambda ^*_i$ and $L^*_i$ can be treated interchangeably in the horizontal directions.

The correlation lengths are shown in figure 9(a,b) as a function of $\lambda$. The normalized correlation lengths in both the vertical and horizontal directions decrease as the mixing layer aspect ratio is increased. Consistent with visual observations in figure 8, the lateral domain size physically restricts the growth of horizontal length scales in the SRT configuration. A less obvious consequence is that the vertical correlation length also decreases in a similar fashion, despite being normalized by the mixing layer height. Unlike TRT, SRT flow is affected independently by a second large length scale, ${\mathcal {L}}$. Hence, a mapping of SRT domain length scales to observed flow length scales is necessary for comparison with TRT.

Figure 9. Effect of mixing layer aspect ratio on normalized (a) vertical and (b) horizontal correlation lengths, (c) growth and (d) mixedness parameters. Red markers represent SRT values at ${\textit {Gr}}=7.4\times 10^6$; the maximum and minimum values from TRT simulations ($Re>3000$) are shown as: T0 (black dashed); reference dataset (Cabot & Cook Reference Cabot and Cook2006; Zhou & Cabot Reference Zhou and Cabot2019) (blue dotted).

Time-averaged TRT values (corresponding to $Re>3000$) for $\varLambda _2$ and $L_{1,3}$ are extracted from figure 13 of Zhou & Cabot (Reference Zhou and Cabot2019) and T0. Due to significant statistical variation, these are shown as maximum and minimum values in figure 9. Figure 9(a) shows that $\varLambda _2/h_p$ from both TRT sources are well matched by SRT at $\lambda =1.5$. Figure 9(b) shows a much larger spread of $L_{1,3}/h_p$ values from the TRT data. Arguably, all $L^*_{1,3}/h^*_p$ values from the SRT simulations of $0.5 \le \lambda \le 2.0$ are within the statistical variation of the two TRT simulations. There are two reasons for this larger statistical variation. First, $L_{1,3}$ is computed from spectra on a single plane, it is expected to be less converged than $\varLambda _2$, which is computed from an integral over the full domain. Second, in both Zhou & Cabot (Reference Zhou and Cabot2019) and T0, it is observed that $L_{1,3}/h_p$ continues to decrease slowly with time. Because the matching of $L_{1,3}/h_p$ values is inconclusive, the excellent agreement in $\varLambda _2/h_p$ is the main criterion for selecting $\lambda =1.5$ as the SRT configuration that corresponds best with TRT growth.

Additionally, supporting evidence of the $\lambda =1.5$ SRT configuration being an equivalent flow unit to self-similar TRT can be found in the spectra of Cabot & Cook (Reference Cabot and Cook2006). A wavenumber corresponding to the peaks of the vertical velocity and density spectra, $k_{peak}$, is extracted from figure 2 of Cabot & Cook (Reference Cabot and Cook2006) and used to estimate an aspect ratio, $h_p k_{peak}/2{\rm \pi} \approx 1.2$. This compares well with the value of $h_p^*/{\mathcal {L}}\approx 1.25$ for SRT at $\lambda = 1.5$. Hence, it appears that the lateral domain width of the SRT configuration may be dynamically equivalent to the most energetic lateral wavelength in TRT flow.

Finally, the effect of the mixing layer aspect ratio is examined in terms of the growth parameter $\alpha ^*_p$ and mixedness $\varXi ^*$. Although SRT is statistically stationary, a growth parameter, $\alpha _m^*= \alpha _m (t_0)$, corresponding to a TRT mixing layer at $h_m(t_0)$ can be computed using the SRT parameter $\tau _0$. Evaluating the definition of $\alpha _m$ at $t=t_0$, and using (2.50), we can relate the TRT growth parameter to SRT flow quantities with

(6.3)\begin{equation} \alpha^*_m = \alpha_m(t_0) = \left. \frac{1}{4Ag} \frac{\dot{h}_m^2}{h_m} \right|_{t=t_0} = \frac{h_m^*}{4Ag} \left( \frac{1}{\tau_0} \right)^2. \end{equation}

It is then corrected for different height definitions using

(6.4)\begin{equation} \alpha^*_i = \alpha^*_m \frac{h^*_i}{h^*_m} = \frac{h_i^*}{4Ag} \left( \frac{1}{\tau_0} \right)^2. \end{equation}

The mixedness parameter for TRT is defined in (5.2) and $\varXi ^*$ has an identical definition, differing only in the coordinate system used. Figure 9(c,d) shows that the SRT values at $\lambda =1.5$ match self-similar TRT values from Cabot & Cook (Reference Cabot and Cook2006) and T0 well, verifying that SRT can reproduce TRT flow statistics as long as the correlation lengths are matched.

By matching the statistical flow length scales between SRT and TRT, we identify the SRT configuration with $\lambda =1.5$ as being representative of self-similar TRT flow (for $A=0.5$ and $\textit {Sc}=1$). This is verified by good agreement in $\alpha _p$ and $\varXi$ values. This configuration will be referred to as SRT-1.5 in the subsequent sections.

6.2. Effect of Grashof/Reynolds number

The effect of Grashof number on SRT-1.5 is studied. As discussed in § 3.2, ${\textit {Gr}}\propto Re^2$ for a self-similar flow, and either quantity can be used to characterize turbulence intensity. Since $Re$ is more commonly used in TRT studies, all results are presented in terms of $Re$ (defined in (1.4) and (3.7)) for ease of comparison. The * superscript may be omitted when comparing SRT and TRT quantities, as the specific definition should be clear from the context.

The effect of $Re$ (or ${\textit {Gr}}$) on SRT flow is shown qualitatively in figure 10. All cases have the same $\bar {h}_m$ and ${\mathcal {L}}$. Visually, the largest length scales observed are similar across all Reynolds numbers. Expectedly, the largest vertical scale is constrained by $\bar {h}_m$ and the largest horizontal scale is constrained by ${\mathcal {L}}$. In contrast, the smallest length scales in the flow decrease noticeably with Reynolds number. Additionally, three-dimensional mixing is observed across all $Re$. For small $Re$, this mixing primarily manifests through a smearing of the large-scale interfaces via laminar diffusion. For large $Re$, this mixing process is driven by the generation of small-scale turbulent structures.

Figure 10. Density fields for SRT cases G1–G6 at $\lambda = 1.5$ and increasing Reynolds (or Grashof) number from left to right: $Re \approx 500$, 1200, 1800, 2700, 6400 and 10 800. Non-mixed regions of the flow in the vertical direction are intentionally removed. Blue and red regions correspond to $\rho _H$ and $\rho _L$, respectively.

Figure 11 shows the variation of several key parameters with $Re$ for both SRT and TRT flow. Besides $\alpha _p$ and $\varXi$, we examine two other parameters, $K/\delta P$ and $\eta$. The ratio of kinetic-to-potential energy is defined using the vertically integrated kinetic energy $K(t)$ and the loss of potential energy relative to the initial state:

(6.5)\begin{align} \delta P(t) & = \int^{\infty}_{-\infty} \langle \rho(\boldsymbol{x}, t=0) - \rho(\boldsymbol{x}, t)\rangle g x_2\, {\rm d}\kern0.7pt x_2 \end{align}

(6.6)\begin{align} & \approx g\Bigg[\int^{\infty}_0 \langle \rho_H-\rho \rangle x_2 \,{\rm d}\kern0.7pt x_2 + \int^0_{-\infty} \langle \rho_L-\rho \rangle x_2 \,{\rm d}\kern0.7pt x_2 - \frac{\rho_H-\rho_L}{2} \delta^2_I\Bigg]. \end{align}

We compute $\delta P$ using (6.6), which is exactly equal to (6.5) if the initial interface is assumed to be infinitely thin. The corresponding definitions of $K^*$ and $\delta P^*$ for SRT are similar, with the appropriate variable substitution. The Kolmogorov length scale $\eta$ presented in figure 11(d) is the minimum value of $\eta$ found in the mixing layer. It is normalized by the mixing layer height and compared against the expected Kolmogorov scaling.

Figure 11. Comparison of SRT-1.5 with two different sources of TRT as a function of Reynolds number: SRT-1.5 (red squares); TRT-T0 (black points); data extracted from Cabot & Cook (Reference Cabot and Cook2006) (blue solid line). (a) Growth parameter; (b) mixedness; (c) ratio of kinetic to potential energy; (d) normalized Kolmogorov length scale ($Re^{-3/4}$ line represents Kolmogorov scaling).

Both SRT and TRT approach similar behaviour at high Reynolds numbers. The parameters $\alpha _p$ and $\varXi$ approach statistically stationary values, $K/\delta P$ exhibits a slow growth and $\eta$ scales in accordance with Kolmogorov scaling. We note that both sets of TRT data predict a lower $K/\delta P$ than SRT-1.5 and exhibit slightly faster growth than SRT-1.5 for $Re>2000$; it is likely that both TRT simulations have not achieved self-similarity in velocity. This will be addressed in greater detail in § 6.3. While the general high $Re$ behaviour between SRT and TRT flow is similar, the manner in which this self-similar state is approached is drastically different. A distinct change in flow regime is observed at $Re \approx 2000$ for TRT, corresponding to the onset of self-similar growth discussed in § 5.1.1. In contrast, the values in SRT approach high $Re$ behaviour smoothly. Because SRT is independent of initial conditions and enforces self-similar behaviour through timewise recycling, SRT flow decouples memory effects from $Re$ effects, leading to the possibility of low Reynolds number self-similar RT mixing.

6.3. Detailed comparisons between SRT and TRT turbulence

In this section, a more detailed comparison of SRT-1.5 and TRT is performed. We first present planar-averaged statistics for G1–G6 to examine $Re$ effects, then proceed to compare self-similar results from G5, T0 and the Cabot & Cook (Reference Cabot and Cook2006) dataset where available.

It was shown in § 6.2 that the global quantities of both TRT and SRT-1.5 show minimal Reynolds number sensitivity for $Re \gtrsim 3000$. We compare G5 ($Re^*\approx 6400$) and the self-similar region of T0 ($3000 \lesssim Re \lesssim 8000$). Statistics from T0 are first normalized, then averaged over this time window. Although the reference dataset has a much larger $Re$ range that reaches up to 32 000, we expect the results between all three simulations to compare relatively well after quantities are suitably normalized. One-dimensional profiles are extracted from Livescu et al. (Reference Livescu, Ristorcelli, Gore, Dean, Cabot and Cook2009) when available. Due to the limited number of snapshots presented, an approximate spread corresponding to all data in the range $Re>3000$ is shown. This spread represents the maximum and minimum values of the normalized profiles observed at each $x_2$ location.

A qualitative comparison of SRT-1.5 and TRT-T0 density fields at $Re \approx 6400$ is shown in figure 12. Both flow fields look similar, revealing a similar range of length scales and flow features. Movie 1 shows the time evolution for case G5. After the flow evolves from its lower ${\textit {Gr}}$ initial state to higher ${\textit {Gr}}$ behaviour by $s\approx \tau _0$, the flow appears stationary, ergodic and similar to TRT flow. Additionally, we note that the mixing layer aspect ratio for SRT-1.5 is approximately 4 times that of the TRT configuration. For the same $Re$, each in-plane direction for SRT flow is 4 times smaller, resulting in a sixteenfold reduction in the total number of grid points. This has obvious computational benefits that will be further discussed in § 7.2.

Figure 12. Comparison of TRT-T0 and SRT-1.5 (G5) density fields for $Re\approx 6400$. Images are rescaled to have the same $h_m$, and non-mixed regions of the flow in the vertical direction are intentionally removed.

The average heavy fluid mole fraction profiles are compared between SRT-1.5 and TRT. Figure 13(a) shows the variation of $\langle X \rangle$ with Reynolds number for SRT-1.5. These profiles show almost no sensitivity to $Re$, even at very low Reynolds numbers. Case G5 is compared with TRT results in figure 13(b). There are minor differences in the shape of these profiles, with both sources of TRT data having slightly extended tails towards the outer edges of the mixing layer. Next, the average local mixed mass is considered. The variation of mixed mass with Reynolds number for SRT-1.5 is shown in figure 14(a). Profile shapes are similar across different $Re$. Magnitudes are within 5 % across the entire range and converge smoothly towards high $Re$. Figure 14(b) shows that the average local mixed mass of TRT-T0 compares well with SRT-1.5, but with minor deviations at the tails similar to figure 13(b).

Figure 13. Heavy fluid mole fraction. (a) Reynolds number effects on SRT-1.5 at $Re^* \approx 500$ (red dotted), 1200 (blue dash-dotted), 1800 (magenta solid), 2700 (green dashed), 6400 (thick red solid) and 10 800 (thick black dashed). (b) Comparison of SRT-G5 (red), TRT-T0 (black dashed) and approximate spread from Livescu et al. (Reference Livescu, Ristorcelli, Gore, Dean, Cabot and Cook2009) (blue shaded).

Figure 14. Normalized mixed mass. (a) Reynolds number effects on SRT-1.5 at $Re^* \approx 500$ (red dotted), 1200 (blue dash-dotted), 1800 (magenta solid), 2700 (green dashed), 6400 (thick red solid) and 10 800 (thick black dashed). (b) Comparison of SRT-1.5 G5 (red) and TRT-T0 (black dashed).

The differences at the outer edges of the mixing layer may be attributed to a deviation from self-similarity in TRT flow. In TRT flow, the outer edges of the mixing layer are characterized by a small number of isolated mushroom-shaped structures with laminar fronts that extend primarily in a single direction (bubbles move up; spikes move down). Examples of these isolated structures are noticeable in the TRT snapshot towards the bottom of figure 12. These structures are most likely to retain some properties of the initial conditions. In contrast, SRT flow retains no memory of the initial perturbations and self-similar mixing is experienced throughout the entire domain.

Next, the normalized planar-averaged kinetic energy is considered. Figure 15(a,b) shows the $Re$ dependence of SRT-1.5 and a comparison with TRT, respectively. Shapes of kinetic energy profiles are largely similar across $Re$ but there appears to be a weak $Re$ dependence in the peak magnitudes. In figure 15(b), SRT-1.5 is compared with T0 at a similar Reynolds number. Similar to trends observed in the scalar statistics, SRT-1.5 has shorter tails than T0. Additionally, the peak magnitude is 14 % larger than the corresponding value from T0. Although figure 15(b) compares only G5 with T0, it can be easily compared with figure 15(a) to show that T0 has a lower peak than SRT-1.5 for the entire range of $Re$. This is consistent with the smaller values of $K/\delta P$ for TRT across all $Re >2000$ in figure 11(c). This discrepancy might be attributed to the fact that the SRT equations are designed to enforce velocity self-similarity, but T0 has not in fact reached a state of velocity self-similarity. It was shown in § 5.1 that normalized velocity statistics continue to grow for T0. It is likely that the SRT-1.5 kinetic energy profiles represent an upper limit for TRT and the profiles for TRT may eventually converge to this limit if T0 is simulated for a longer time window.

Figure 15. Normalized kinetic energy. (a) Reynolds number effects on SRT-1.5 at $Re^* \approx 500$ (red dotted), 1200 (blue dash-dotted), 1800 (magenta solid), 2700 (green dashed), 6400 (thick red solid) and 10 800 (thick black dashed). (b) Comparison of SRT-G5 (red) and TRT-T0 (black dashed).

Finally, we consider the anisotropy of the velocity field using the anisotropy tensor

(6.7)\begin{equation} {\mathsf{b}}_{ij} = \frac{\langle \rho u_i u_j \rangle}{\langle \rho u_k u_k \rangle} - \frac{1}{3}\delta_{ij}, \end{equation}

of which, the ${\mathsf{b}}_{22}$ component is shown in figure 16(a,b). A value of 0 describes isotropic behaviour and a value of 2/3 represents pure one-dimensional vertical flow. Equation (6.7) involves a ratio of velocities, and these approach zero in the reservoirs. Hence, values of ${\mathsf{b}}_{22}$ near and beyond the edges of the mixing layer should not be overly scrutinized. Figure 16(a) shows that anisotropy increases toward the outer edges of the mixing layer and decreases with $Re$. In the layer core, ${\mathsf{b}}_{22}$ converges to approximately 0.27 at high Reynolds number. Although magnitudes differ, profile shapes are similar even at low $Re$ and converge smoothly toward high $Re$. In figure 16(b), the anisotropy of SRT-1.5 is compared with TRT. Both sources of TRT data have slightly larger values of ${\mathsf{b}}_{22}$ compared with SRT-1.5. However, trends are similar and magnitudes are not significantly different. In addition, because the velocity statistics of the TRT simulations may not yet be fully self-similar, it is possible that the anisotropy of the TRT cases could converge to SRT-1.5 values with longer simulation times.

Figure 16. Anisotropy. (a) Reynolds number effects on SRT-1.5 at $Re^* \approx 500$ (red dotted), 1200 (blue dash-dotted), 1800 (magenta solid), 2700 (green dashed), 6400 (thick red solid) and 10 800 (thick black dashed). (b) Comparison of SRT-G5 (red), TRT-T0 (black dashed) and approximate spread from Livescu et al. (Reference Livescu, Ristorcelli, Gore, Dean, Cabot and Cook2009) (blue shaded).

6.4. Higher-order statistics

To verify that the SRT framework preserves the spatial structure of RT turbulence, two-point statistics are considered. Two-dimensional spectra $E_i(\kappa )$ are computed using SRT-G5 data at $x_2 = \delta _I$ (or $\xi _2 = \delta ^*_I$), where $E_i(\kappa )$ is the two-dimensional spectral energy density of $u_i$, and $\kappa$ is the two-dimensional wavenumber in the horizontal plane. The spectrum for vertical velocity is $E_2$, and the spectrum for horizontal velocity is computed as $(E_1+E_3)/2$. Equivalent spectra are computed from TRT-T0 for a similar $Re\approx 6400$ and both sets of data are compared in figure 17(a). The SRT and TRT spectra show good agreement throughout the entire range of scales. Consistent with Cabot & Cook (Reference Cabot and Cook2006), an inertial range resembling Kolmogorov $\kappa ^{-5/3}$ scaling is observed in the $E_2$ spectrum, but not in the horizontal directions. The smallest wavenumber in the SRT spectra is approximately four times as large as that of TRT, due to the difference in domain sizes. The smallest wavenumber in SRT flow corresponds exactly to the most energetic wavenumber of TRT flow. This observation is not a coincidence; the SRT configuration acts as a minimum flow unit for RT turbulence, and this will be discussed in § 7.1.

Figure 17. Spectra comparison between TRT-T0 and SRT-G5 at $Re\approx 6400$ on horizontal plane $x_2\approx \delta _I$. (a) Two-dimensional spectra for vertical and horizontal velocities: TRT $E_2$(black solid); TRT $(E_1+E_3)/2$ (black dashed); SRT $E^*_2$ (blue circles); SRT $(E^*_1+E^*_3)/2$ (red squares). (b) Anisotropy spectra: TRT (black dashed); SRT (red solid); and approximate spread from Chung & Pullin (Reference Chung and Pullin2010) at $Re \approx 32\,000$ (blue shaded).

The structure of the small scales is further examined using the anisotropy spectrum

(6.8)\begin{equation} B_i(\kappa) = \frac{E_i}{E_1+E_2+E_3}-\frac{1}{3}, \end{equation}

and computed from SRT-G5 and TRT-T0 data. The two spectra are shown in figure 17(b), alongside an approximate spread of the equivalent spectrum for the Cabot & Cook (Reference Cabot and Cook2006) dataset that is found in figure 13 of Chung & Pullin (Reference Chung and Pullin2010). All three sets of data seem to suggest an increase in anisotropy in the dissipation range towards the smaller scales, albeit with some differences in magnitudes. The values from SRT-G5 are bounded by those of TRT-T0 and the reference data, verifying that SRT flow reproduces similar spectra to TRT flow. Differences in the two sets of TRT results may be related to initial conditions and/or Reynolds number, but it is difficult to conclude without further study.

Finally, we verify that SRT flow can also capture localized, irregular behaviour by comparing its intermittency characteristics with TRT flow. A procedure similar to Boffetta et al. (Reference Boffetta, Mazzino, Musacchio and Vozella2009) is used. In one dimension, the $p$th-order longitudinal velocity structure function is

(6.9)\begin{equation} S_p(r) = \langle (u_i(x_i+r) - u_i(x_i))^p \rangle. \end{equation}

These are computed at $x_2\approx \delta _I$ using $u_1(x_1)$ and $u_3(x_3)$, and averaged over the plane in both horizontal directions. The structure functions for $p=$ 2, 4 and 8 are shown in figure 18(a).

Figure 18. (a) Longitudinal velocity structure functions for $p=$ 2 (circles), 4 (triangles) and 8 (plus signs) for SRT-G5 at $\xi _2=\delta ^*_I$. Symbols represent SRT data and dashed lines correspond to the scaling exponents $3\zeta _p/p = 1.0$, 0.97 and 0.84, respectively. (b) Relative scaling exponents computed using the ESS method based on their deviation from $S_3(r)$: SRT-G5 (red squares); TRT-T0 (black circles); data from Boffetta et al. (Reference Boffetta, Mazzino, Musacchio and Vozella2009) (blue crosses); Kolmogorov's four-fifth law (dashed green line); assumed $\zeta _3=1$ (filled symbol).

Turbulence intermittency can be quantified by computing the scaling exponents, $S_p(r)\propto r^{\zeta _p}$, from the data. A deviation of $\zeta _p$ from $p/3$ is commonly considered evidence of turbulence intermittency (Anselmet et al. Reference Anselmet, Gagne, Hopfinger and Antonia1984; Meneveau & Sreenivasan Reference Meneveau and Sreenivasan1987). However, obtaining these scaling exponents requires power-law fitting over the limited inertial ranges that are found in the current simulations. The extended self-similarity (ESS) procedure from Benzi et al. (Reference Benzi, Ciliberto, Tripiccione, Baudet, Massaioli and Succi1993) is used to overcome this issue. The premise of this procedure is that the dissipation range can also be used in the power-law fitting after it is normalized by the third-order structure function. In doing so, $\zeta _3=1$ is assumed.

The computed relative scaling exponents are shown in figure 18(b) for both SRT-G5 and TRT-T0, and the respective fits are reflected in figure 18(a) for comparison with the original SRT structure functions. The scaling exponents from SRT-G5 and TRT-T0 show good agreement, suggesting that SRT flow is able to capture a similar level of intermittency that is present in TRT flow. Although Boffetta et al. (Reference Boffetta, Mazzino, Musacchio and Vozella2009) studied the RT configuration at a much lower Atwood number (in the Boussinesq approximation), the scaling exponents are extracted from figure 4 of their study and shown in figure 18(b) for comparison purposes. There is little difference between all the results compared, suggesting that Atwood number may not affect the scaling exponents of the structure functions significantly.