2.1 The Boussinesq model

The Boussinesq approximation is derived from the full NSE (2.3) by using asymptotic analysis (Schneider & Gauthier Reference Schneider and Gauthier2015a ). The main steps of the derivation are recalled here. The equations have first to be written in a dimensionless form and then dependent variables are expanded as powers of a small parameter, here the Atwood number, $A_{t}$ . In a third step, one takes the limit of vanishing stratification and Atwood numbers and infinite ratio of specific heats, $\unicode[STIX]{x1D6FE}$ . The system of equations (2.3) is adimensionalized with the following four reference quantities: (i) length $L_{y}$ , (ii) velocity $U_{r}=\sqrt{A_{t}gL_{y}}$ , (iii) density  $\unicode[STIX]{x1D70C}_{r}=(\unicode[STIX]{x1D70C}_{H}(0^{+})+\unicode[STIX]{x1D70C}_{L}(0^{-}))/2$ and (iv) temperature, the uniform temperature of the equilibrium state. The pressure reference is $p_{r}=\unicode[STIX]{x1D70C}_{r}(\mathscr{R}/\mathscr{M}_{r})T_{r}$ with $2/\mathscr{M}_{r}=1/\mathscr{M}_{H}+1/\mathscr{M}_{L}$ . As a result, system (2.3) reads

for a constant diffusion of species coefficient $\mathscr{D}$ . The expression of the dimensionless numbers (Atwood, stratification, Reynolds and Schmidt) are

In the following, we will use indifferently either $(x,y,z)$ or $x_{i}$ for the coordinates. The dependent variables $\unicode[STIX]{x1D711}=u_{i}$ , $\unicode[STIX]{x1D70C}$ , $c$ and $p$ are expanded as

This expansion (2.7) shows that the mean density $\unicode[STIX]{x1D70C}^{(0)}$ is steady and uniform for zero Atwood number $A_{t}=0$ , chosen such that $\unicode[STIX]{x1D70C}^{(0)}=1$ . For incompressible fluids, the speed of sound is infinite. This is equivalent to assuming that the adiabatic indexes go to the infinity, i.e. $\unicode[STIX]{x1D6FE}\rightarrow \infty$ , with $C_{v}\rightarrow 0$ and $C_{p}=O(1)$ (Zeytounian Reference Zeytounian1990). It also implies that the compressibility goes to zero $\unicode[STIX]{x1D70F}_{s}=\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x2202}p)_{s}=0$ , so that the density does not depend on pressure and it does not depend on the temperature either. By expanding the equation of state (2.4) in terms of the Atwood number, one has $\unicode[STIX]{x1D70C}^{(1)}=(2c^{(0)}-1)$ .

Let us now seek the limiting form of equations (2.5) when the stratification $\mathit{Sr}$ and the Atwood number $A_{t}$ simultaneously go to zero in such a way the similarity relation (Zeytounian Reference Zeytounian1990)

is satisfied. At zeroth order in the Atwood number, the velocity field is solenoidal, $\unicode[STIX]{x2202}_{i}u_{i}^{(0)}=0$ , and the temperature is uniform Schneider & Gauthier (Reference Schneider and Gauthier2015a ). At the order ${A_{t}}^{-2}$ , the momentum equation (2.5) reduces to ${A_{t}}^{-2}\widehat{B}^{-1}\unicode[STIX]{x2202}_{i}p^{(0)}=0$ , which implies that $p^{(0)}=p^{(0)}(t)$ . At the next order, ${A_{t}}^{-1}$ , we obtain the hydrostatic equilibrium

At zeroth order, the momentum equation (2.5) is

where the constant $\widehat{B}$ may be absorbed in the Lagrangian multiplier $p^{(2)}$ . Moreover, at zeroth order, the concentration equation is written

Using the density fluctuation expression given above, one obtains the Boussinesq equations for the velocity $u_{i}^{(0)}$ , pressure $p^{(2)}$ and concentration $c^{(0)}$ , in the case of RT flow

where we now omit the superscripts. For convenience, these equations have been rescaled by using the unit of velocity $U_{r}=\sqrt{gL_{y}}$ . The expression of the Reynolds number (2.6) is now $\mathit{Re}=g^{1/2}L_{y}^{3/2}/(\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}_{r})$ . Boundary conditions are periodic in the horizontal $(x,y)$ -directions for all physical quantities with stress-free conditions along the top and bottom boundaries for the velocity. With such periodic horizontal boundary conditions, a simulation is valid if the number of structures in the flow remains sufficiently high, so that interactions between the flow and its mirror images are negligible. This is the case for the simulation described in this paper. There is also no flow and therefore no mass flux through the top and bottom boundaries. Consequently the boundary conditions are

It is worth noting that the pressure $p$ involved in (2.12) is the second-order contribution in the expansion (2.7) while the velocity is of zeroth order. Moreover, as it is usual in Boussinesq approximations, the pressure $p$ is defined up to an additive constant, $p^{(0)}(t)$ .

2.2 The averaging procedure

For the data analysis in the turbulent regime, we will use the Reynolds averaging procedure where the ensemble averaging of a general variable $\unicode[STIX]{x1D711}$ is computed from a spatial average as

where $\overline{\unicode[STIX]{x1D711}}(z,t)=(L_{x}L_{y})^{-1}\int \unicode[STIX]{x1D711}(x,y,z,t)\,\text{d}x\,\text{d}y$ . As a result, one has $\overline{\unicode[STIX]{x1D711}^{\prime }}=0$ . The volume average is defined as $\langle \overline{\unicode[STIX]{x1D711}}(z,t)\rangle =L_{z}^{-1}\int \overline{\unicode[STIX]{x1D711}}(z,t)\,\text{d}z$ , to obtain global time-dependent quantities. We will also use the volume average over a vertical fraction of the mixing layer defined as

where $\unicode[STIX]{x1D6FD}$ is a positive constant. In this paper we will use the two values $\unicode[STIX]{x1D6FD}=0.2$ or $0.8$ . A first consequence of the horizontal periodicity is that the average value of the $z$ -velocity component is zero. Indeed, by averaging the continuity equation, $\unicode[STIX]{x2202}_{i}u_{i}=0$ , one obtains

Therefore $\overline{u}_{3}=\text{constant}$ . This constant is taken to be zero and the mean velocity field is

2.3 The vorticity transport equation

The vorticity is a solenoidal vector field defined by $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D700}_{ijk}\unicode[STIX]{x2202}_{j}u_{k}$ . This is a key quantity in turbulence since it gives access to vortex motion (see for example, Schneider & Gauthier (Reference Schneider and Gauthier2015b )) and it is involved in enstrophy, helicity and also palintrosphy. Its evolution equation may be found either by taking the limit value of the vorticity equation of the full NSE or by applying the curl on the Boussinesq velocity equation (2.12). One obtains the same result but in the second case, the baroclinic term comes from the buoyancy term, while in the first case, it classically comes from the pressure term. Let us first take the curl of the momentum equation (2.3) (Thompson Reference Thompson1972, § 2.4). It can be written in the form

with the units defined above. The asymptotic analysis is carried out with the same methodology used to derive the Boussinesq model. Variables are first expanded as in (2.7) and terms of order ${A_{t}}^{-1}$ and ${A_{t}}^{-2}$ are zero since both $\unicode[STIX]{x1D70C}^{(0)}$ and $p^{(0)}$ do not depend on space variables. The order ${A_{t}}^{0}$ gives the transport equation of vorticity within the Boussinesq approximation

where we have used $\unicode[STIX]{x1D70C}^{(0)}\equiv 1$ and $\unicode[STIX]{x2202}_{i}u_{i}^{(0)}=0$ . The pressure $p^{(1)}$ is actually the hydrostatic pressure whose the vertical gradient is $\unicode[STIX]{x2202}_{3}p^{(1)}=-1$ , given by (2.12). The baroclinic production is therefore

With the expression for the density fluctuation, one obtains the vorticity transport equation for the RT-Boussinesq model

where the baroclinic production term acts only in the horizontal directions through the horizontal concentration gradients. As a result, the equation for the vertical vorticity component is identical to the vorticity equation for a homogeneous isotropic turbulence. In particular, the two source terms are homogeneous, i.e. no vertical vorticity can be generated from an initial irrotational vector field.

2.4 The turbulent kinetic energy equation

The turbulent kinetic energy is a key quantity in turbulence and it introduces naturally the dissipation rate, which is frequently used in turbulence and analysed in this paper. The equation for the mean turbulent kinetic energy $\overline{k}_{B}=1/2\overline{u_{i}^{\prime }u_{i}^{\prime }}$ is obtained by first writing down the evolution equation for the velocity fluctuation $u_{i}^{\prime }$ from (2.12). Multiplying this equation by $u_{i}^{\prime }$ and taking the mean value, one obtains

where the dissipation rate $\overline{\unicode[STIX]{x1D700}}$ is defined as

It has been checked that the same equation is obtained by starting from the full compressible kinetic energy equation and taking the limit of vanishing Atwood numbers.

2.5 Linear stability

The linear stability analysis has been studied with both the heuristic formula of Duff, Harlow & Hirt (Reference Duff, Harlow and Hirt1962) and the multidomain pseudo-spectral numerical stability code speclmd (Gauthier et al. Reference Gauthier, Le Creurer, Abéguilé, Boudesocque-Dubois and Clarisse2005; Lafay, Le Creurer & Gauthier Reference Lafay, Le Creurer and Gauthier2007). The Duff et al. formula is

where $\unicode[STIX]{x1D713}$ is a factor that takes into account the dynamic diffusion. Equation (2.24) also takes into account viscosity and diffusion effects. The maximum of the growth rate, the corresponding wavenumber and the cutoff wavenumber, beyond which the mode is stable, are

with $\unicode[STIX]{x1D713}=1$ and for $A_{t}=0.1$ , $\mathit{Re}=3\times 10^{4}$ and $\mathit{Sc}=0.7$ . On the other hand, the code speclmd solved the linear boundary value problem for the full Navier–Stokes equations. So we have used limit values for the stratification parameter, $\mathit{Sr}=0.1$ , the Prandtl number, $\mathit{Pr}=10^{-4}$ and the ratio of specific heats, $\unicode[STIX]{x1D6FE}=10$ , in addition to the parameter values of the simulation. The density, temperature and concentration profiles are regularized with an $\text{erf}(z/\unicode[STIX]{x1D6FF})$ function, with $\unicode[STIX]{x1D6FF}=5\times 10^{-3}$ . The same value will be used in the numerical simulation. The results obtained are close to, but slightly below, the previous ones

The differences between these two estimates (2.25) and (2.26) of the maximum of the growth rate, the corresponding wavenumber and the cutoff wavenumber are $18\,\%$ , $7\,\%$ and $1.5\,\%$ , respectively. Except for the growth rate, these differences are rather small.

4.1 Velocity

The self-similar character of the Boussinesq turbulence has been investigated in detail in Ristorcelli & Clark (Reference Ristorcelli and Clark2004). In particular, scalings of various quantities have emerged from this analysis. These scalings are used below to check the self-similarity of the flow. Figure 3 displays the superimposition of velocity moments $\overline{u_{x,z}^{\prime 2}}/2h$ (contributions to the kinetic energy) versus the self-similar variable $z/h$ . A trend to convergence is observed for the horizontal contribution, while it is clearly confirmed for the vertical contribution. Figure 4 confirms this trend. Note that the velocity profile supports are larger than the mixing layer thickness. These velocities correspond to large-scale bubbles escaped from the mixing layer and located beyond the front defined by $h(t)$ .

Figure 3.

Velocity fluctuation kinetic energy versus the self-similar coordinate $z/h$ at four different times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . (a) Horizontal velocity component $\overline{u_{x}^{\prime 2}}/2h$ . (b) Vertical velocity $\overline{u_{z}^{\prime 2}}/2h$ . The trend to convergence of the latter appears clearly.

Figure 4.

Velocity fluctuation kinetic energies, $\langle \overline{k}\rangle _{\unicode[STIX]{x1D6FD}}$ (black), $\langle \overline{u_{x}^{\prime 2}}\rangle _{\unicode[STIX]{x1D6FD}}/2$ (blue) and $\langle \overline{u_{z}^{\prime 2}}\rangle _{\unicode[STIX]{x1D6FD}}/2$ (red), versus time in log–log scale. Beginning of convergence is more pronounced for the vertical contribution. Two values of $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D6FD}=0.2$ (a) and 0.8 (b). The result does not depend significantly on this value.

The velocity statistics are now described. The skewness of one horizontal velocity component, $\overline{u_{x}^{\prime 3}}/\overline{u_{x}^{\prime 2}}^{3/2}$ and the vertical component, $\overline{u_{z}^{\prime 3}}/\overline{u_{z}^{\prime 2}}^{3/2}$ , are represented in figure 5. Skewness measures the asymmetry of the PDF of the corresponding quantity. For the $x$ component, the skewness has no particular structure. Very small values are obtained in the middle of the mixing layer and large values are obtained on the mixing layer boundaries. We may thus anticipate symmetric velocity component PDFs. Recall that the Reynolds number reaches its maximum at the centre of the mixing layer and decreases sharply at the boundaries. The skewness of the $z$ -velocity component is made of two peaks roughly centred around $z/h=\pm 1.4$ , corresponding to the mixing layer boundaries, matched by a region of small values. Peaks have the sign of the predominant events, i.e. positive velocities on the upper side and negative on the lower side. Flatness of the two velocity components are given in figure 6. Once again, flatnesses have peaks corresponding to the mixing layer boundaries. However inside the mixing layer flatnesses are close to 3, the value of a Gaussian random process.

Figure 5.

Skewness of the velocity fluctuations versus the self-similar coordinate $z/h$ at four different times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . (a) Horizontal velocity component $\overline{u_{x}^{\prime 3}}/\overline{u_{x}^{\prime 2}}^{3/2}$ . (b) Vertical velocity $\overline{u_{z}^{\prime 3}}/\overline{u_{z}^{\prime 2}}^{3/2}$ .

Figure 6.

Flatness of the velocity fluctuations versus the self-similar coordinate $z/h$ at four different times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . (a) Horizontal velocity component $\overline{u_{x}^{\prime 4}}/\overline{u_{x}^{\prime 2}}^{2}$ . (b) Vertical velocity $\overline{u_{z}^{\prime 4}}/\overline{u_{z}^{\prime 2}}^{2}$ . The flatness of a Gaussian process is represented by the horizontal dashed line.

Figure 7.

PDFs of the velocity components, $u_{x}^{\prime }$ (a) and $u_{z}^{\prime }$ (b). PDFs have been built with data located in the domain $[0:1;0:1;z_{1}:z_{2}]$ , where $z_{1}$ and $z_{2}$ are defined by $\overline{c}(z_{1})=0.4$ and $\overline{c}(z_{2})=0.6$ (see appendix B). On each plot, the dashed lines show Gaussian distribution with the same mean and variance.

These skewness and flatness plots have to be read with the velocity component PDFs represented in figure 7. It appears that the $x$ -velocity component PDF follows a Gaussian behaviour, even at the first selected time $t=12.4$ . Only small departures from Gaussianity is observed for large-amplitude events. This is mainly the case for the $z$ component. The Gaussian behaviour is associated with random interactions (She, Jackson & Orszag Reference She, Jackson and Orszag1990) and no particular structure is attached to this behaviour. However, velocity data located around the boundaries, which are not turbulent, have been excluded. We will see in § 5 that this Gaussian behaviour does not hold for vorticity, i.e. for the velocity gradients.

Figure 8.

Reynolds stress anisotropy tensor. (a) 2-D map in the plane $(t,z)$ of the component $\overline{\unicode[STIX]{x1D623}}_{zz}$ . (b) Evolution of the averaged value, $\langle \overline{\unicode[STIX]{x1D623}}_{zz}\rangle _{\unicode[STIX]{x1D6FD}}$ versus time. Two values of $\unicode[STIX]{x1D6FD}$ have been used, 0.2 (red line) and 0.8 (blue line).

Since anisotropy at various scales is an issue of RT-Boussinesq turbulence, we investigate this issue with three indicators. First, turbulence anisotropy is classically measured with the Reynolds stress anisotropy tensor

This quantity is mainly related to the large scales. The behaviour of the third diagonal component $\unicode[STIX]{x1D623}_{zz}$ is reported in figure 8. The 2-D map of $\overline{\unicode[STIX]{x1D623}}_{zz}$ in the plane $(t,z)$ shows the inhomogeneity of the anisotropy in the $z$ -direction. However the evolution of the mean value $\langle \overline{\unicode[STIX]{x1D623}}_{zz}\rangle _{\unicode[STIX]{x1D6FD}}$ shows a trend to convergence. The anisotropy decrease is faster at the very middle of the layer ( $\unicode[STIX]{x1D6FD}=0.2$ , red dashed line). This is in agreement with previous findings about large-scale anisotropy. Indeed Vladimirova & Chertkov (Reference Vladimirova and Chertkov2009) and Boffetta et al. (Reference Boffetta, Mazzino, Musacchio and Vozella2009) report an anisotropy between the two velocity components equal to $w_{rms}/u_{rms}\approx 1.8$ . Second velocity anisotropy at various scales may be investigated with the following indicator

with summation on the index $i$ , but not on $k=x,z$ . The superscript $(d)$ is the order of the derivative. Definition (4.2) generalizes the Reynolds stress tensor anisotropy (4.1) and gives some indications on the return to isotropy of the various scales, depending on the value of $d$ . For $d=0$ , one has $B_{\unicode[STIX]{x2202}_{k}^{(0)}u_{z}^{\prime }}\equiv b_{zz}$ , while for $d=1$ , this quantity may be seen as a measure of intermediate-scale anisotropy. For $d=2$ , it gives the small-scale anisotropy. Similar quantities have been introduced by Lin, Thiffeault & Doering (Reference Lin, Thiffeault and Doering2011) and called ‘multiscale variances’ in order to quantify the efficiency of mixing on different scales. The evolution of these indicators is given in figure 9. In figure 9(a), horizontal gradient evolution is shown. First- and second-order derivatives roughly behave in the same way. During the nonlinear regime, $t\leqslant 5.5$ , the indicator $B$ reaches large values, approximately 0.6, so that the horizontal gradients of the vertical velocity are much larger than the horizontal gradients of the horizontal velocities. This is related to the mainly vertical motions (updrafts and downdrafts) of the mushroom-like structures. Anisotropy decays slowly during the regime of transition to turbulence. At the end of the simulation, in the horizontal direction, intermediate and small scales have roughly the same ratio of anisotropy, $B_{\unicode[STIX]{x2202}_{x}^{(1)}u_{z}^{\prime }}\approx B_{\unicode[STIX]{x2202}_{x}^{(2)}u_{z}^{\prime }}\approx 0.20$ , so that the horizontal gradients of the vertical velocity are around twice those of the horizontal velocities. In other words, a positive anisotropy means that $u_{z}^{\prime }$ -intermediate and small scales are stronger than the horizontal components. The specific value of $\unicode[STIX]{x1D6FD}$ is not essential. The situation is different in the vertical direction (figure 9 b), where anisotropies depend significantly on the value of $\unicode[STIX]{x1D6FD}$ . The indicator $B_{\unicode[STIX]{x2202}_{z}^{(1)}u_{z}^{\prime }}$ starts at approximately $-1/3$ , so that the vertical gradient of the vertical velocity is close to zero. It thus grows to approximately $1/3$ in the nonlinear regime. This means that the vertical gradient of the vertical velocity is roughly twice the same gradient of the horizontal velocity. However, for $\unicode[STIX]{x1D6FD}=0.8$ , the averaging partially encompasses the boundaries and anisotropy is therefore much larger. In the transitional regime, this difference vanishes, the anisotropy does not depend on $\unicode[STIX]{x1D6FD}$ . At the end of the simulation, anisotropies are both slightly negative. At intermediate scales, one has $B_{\unicode[STIX]{x2202}_{z}^{(1)}u_{z}^{\prime }}\approx -0.06$ , while at small scales $B_{\unicode[STIX]{x2202}_{z}^{(2)}u_{z}^{\prime }}\approx -0.15$ . It turns out that, at small scales, $u_{z}^{\prime }$ is smaller than the fluctuations in the horizontal directions.

Figure 9.

Velocity anisotropies calculated from $B_{\unicode[STIX]{x2202}_{k}^{(d)}u_{z}^{\prime }}$ (definition (4.2)). (a) Intermediate-scale anisotropy $B_{\unicode[STIX]{x2202}_{x}^{(1)}u_{z}^{\prime }}$ (blue) and small-scale anisotropy $B_{\unicode[STIX]{x2202}_{x}^{(2)}u_{z}^{\prime }}$ (red). (b) Intermediate-scale anisotropy $B_{\unicode[STIX]{x2202}_{z}^{(1)}u_{z}^{\prime }}$ (blue) and small-scale anisotropy $B_{\unicode[STIX]{x2202}_{z}^{(2)}u_{z}^{\prime }}$ (red). The return to isotropy is quicker for the intermediate vertical scales. Two values of $\unicode[STIX]{x1D6FD}$ have been used, 0.2 (full lines) and 0.8 (dashed lines).

Figure 10.

Velocity anisotropies calculated from $C_{\unicode[STIX]{x2202}_{z}^{(d)}u_{i}^{\prime }}$ (definition (4.3)). (a) Intermediate-scale anisotropy $C_{\unicode[STIX]{x2202}_{z}^{(1)}u_{x}^{\prime }}$ (blue) and small-scale anisotropy $C_{\unicode[STIX]{x2202}_{z}^{(1)}u_{z}^{\prime }}$ (red). (b) Intermediate-scale anisotropy $C_{\unicode[STIX]{x2202}_{z}^{(2)}u_{x}^{\prime }}$ (blue) and small-scale anisotropy $C_{\unicode[STIX]{x2202}_{z}^{(2)}u_{z}^{\prime }}$ (red). Two values of $\unicode[STIX]{x1D6FD}$ have been used, 0.2 (full lines) and 0.8 (dashed lines).

We use a third indicator to characterize the velocity anisotropy. We define

This definition applies to the three velocity components and it will allow us to compare with concentration anisotropies calculated in § 6 and displayed in figure 33. Criterion (4.3) is displayed in figure 10 for an horizontal and the vertical velocity component and for the intermediate ( $d=1$ ) and small ( $d=2$ ) scales. They all start with very large values and decay abruptly. At the beginning of the transitional regime they evolve slowly toward their final values. Within this transitional regime, the vertical gradient of the horizontal velocity reaches small values at the end of the transition for $\unicode[STIX]{x1D6FD}=0.8$ , while in the very interior of the mixing layer ( $\unicode[STIX]{x1D6FD}=0.2$ ) small anisotropy values are reached very early during the nonlinear regime. Vertical gradients of the vertical velocity are very small during the transitional regime and reach small negative values compatible with the quasi-isotropy of the intermediate scales. At the end of the simulation, it turns out that $C_{\unicode[STIX]{x2202}_{z}^{(1)}u_{x}^{\prime }}$ and $C_{\unicode[STIX]{x2202}_{z}^{(2)}u_{x}^{\prime }}$ goes to zero, while $C_{\unicode[STIX]{x2202}_{z}^{(1)}u_{z}^{\prime }}\approx -0.17$ and $C_{\unicode[STIX]{x2202}_{z}^{(2)}u_{z}^{\prime }}\approx -0.24$ . It means that with respect to the vertical gradient the horizontal velocity is almost isotropic but the vertical velocity component remains anisotropic both at intermediate and small scales. It is related to updraft and downdraft motions superimposed to the background turbulent flow. On the other hand, spectral velocity anisotropy have been investigated by Livescu et al. (Reference Livescu, Ristorcelli, Petersen and Gore2010) by taking the Fourier transform of the Reynolds stress anisotropy tensor, $\overline{\unicode[STIX]{x1D623}}_{zz}$ . We use here a fourth indicator, which will also be used for the concentration-gradient anisotropy

where $\widehat{u}$ denotes the Fourier transform of the variable $u$ . The Fourier spectra calculation is detailed below. Results for velocity anisotropy are shown in figure 11(a), where the integral scale and the horizontal and vertical Taylor length scales have also been indicated. Anisotropy reaches very small values around these two length scales. For smaller scales, the anisotropy increases with the wavenumber $k_{x}$ up to the Kolmogorov length scale. It confirms that the intermediate velocity scales are nearly isotropic while small scales are anisotropic.

Figure 11.

(a) Spectral anisotropy of the velocity field at the final time of the simulation. The vertical dashed black lines stand for the integral scale, the horizontal and vertical Taylor length scales and the Kolmogorov scale. (b) Spectra of the vertical velocity averaged over various heights in the vertical direction, from $0.27h$ to $11h$ . Spectra have been normalized by the first-mode amplitude. Final time of the simulation.

In this section and below, we show Fourier spectra of various quantities. In that purpose, spectra are computed by performing a 1-D Fourier transform in an horizontal direction and averaged with respect to the second horizontal direction. A vertical averaging is also performed. As a result, spectra depend on the horizontal wavenumber $k_{x}$ and on time $t$ . The question arises of whether the result depends on the proportion of the mixing layer on which the average is performed. The answer is given in figure 11(b) where the spectra have been averaged on various fractions of the mixing layer. These spectra have also been normalized by the amplitude of the largest scale. It appears clearly that the spectral structure is invariant with respect to the vertical averaging. The 1-D energy spectra of the velocity fluctuations are presented in figure 12 at four different times. The two slopes, $k^{-5/3}$ and $k^{-2}$ , have also been displayed since the $k^{-2}$ energy spectrum has been suggested. However, LES and DNS seem to be in favour of the classical KO spectrum, at least within the Sandoval approximation (Cabot & Cook Reference Cabot and Cook2006; Livescu Reference Livescu2013). A short inertial range is visible between large and Taylor scales in figure 12, while a large dissipation zone is present. Since RT turbulence is continuously fed, spectra shift with time to higher energy. They also widen toward large scales on the one hand, and toward smaller scales on the other hand.

Figure 12.

One-dimensional kinetic energy spectra as function of the wavenumber $k_{x}$ in a log–log scale. The vertical lines stand for the integral scale, the vertical and horizontal Taylor length scales and the Kolmogorov scale. Selected times are $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ .

Figure 13.

(a) PDF of the pressure fluctuations $p^{\prime }$ at two different times $t=12.4$ and $t=15.7$ . The corresponding Gaussian distributions are plotted in dashed lines. Departures from Gaussianity and asymmetry appear clearly. The PDFs have quasi-exponential wings in the lows and in the highs. However, negative values are more probable than positive values. This is related to the fact that centre of vortices correspond to pressure minima. (b) Power spectra of the pressure fluctuations at the four selected times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . The $k^{-7/3}$ -scaling has also been plotted. The vertical dashed lines have the same meaning as in figure 12.

The pressure PDFs have been plotted in figure 13(a) at two times, $t=12.4$ and $t=15.7$ . Departures from Gaussianity and asymmetry appear clearly on these plots. The positive side of the PDF behaves as an exponential. On the negative side, there are two different exponential regimes, which can also be approximated by straight lines of different slopes. This behaviour is consistent with earlier observations on various turbulent flows (Lesieur Reference Lesieur1997, § 8). Indeed, it occurs in decaying isotropic turbulence with Taylor–Green initial conditions (Brachet Reference Brachet1991). It also occurs in statistically steady turbulence (Pumir Reference Pumir1994a ; Cao, Chen & Doolen Reference Cao, Chen and Doolen1999) and it has been recently confirmed for the decaying isotropic turbulence (Kalelkar Reference Kalelkar2006). This asymmetry is related to the fact that centres of vortices correspond to pressure minima. Therefore high vorticity fluctuations are correlated with low pressure fluctuations. Moreover, it has been demonstrated by Pumir (Reference Pumir1994a ) that the superexponential part of the pressure PDF comes from intermittent fluctuations of vorticity. The pressure power spectra are displayed in figure 13(b). The $k^{-7/3}$ -scaling obtained from a quasi-normal approximation within an isotropic turbulence (Batchelor Reference Batchelor1971; Leslie Reference Leslie1973, § 11.5) is also plotted. This scaling appears to be a good fit for scales above the vertical Taylor length scale.

4.2 Mass flux spectrum

Figure 14 displays the spectra of the vertical mass flux $\unicode[STIX]{x1D70C}^{\prime }u_{z}^{\prime }$ at the four selected times. We have also plotted the $k^{-7/3}$ -scaling given by Lumley (Reference Lumley1967) to predict the effect of buoyancy on the turbulent energy spectrum (see also Kaneda & Yoshida (Reference Kaneda and Yoshida2004)). It provides a good scaling around the Taylor length scales.

Figure 14.

One-dimensional spectra of the vertical component $\unicode[STIX]{x1D70C}^{\prime }u_{z}^{\prime }$ of the mass flux. The Lumley $k^{-7/3}$ -scaling has been plotted (dashed black line) and the vertical lines stand for the integral scale, the vertical and horizontal Taylor length scales, the horizontal and vertical Taylor mixing length scales and the Kolmogorov scale, respectively.

4.3 Dissipation

Dissipation is one of the key turbulent quantities that has to be investigated. It has been introduced in (2.22) and (2.23). In Ristorcelli & Clark (Reference Ristorcelli and Clark2004) it has been proved rigorously that the third dissipation component $\unicode[STIX]{x1D700}_{zz}$ scales as $h^{1/2}\propto t$ . Moreover, by dimensional analysis, it appears that the dissipation rate should grow linearly with time (Souffland et al. Reference Souffland, Grégoire, Gauthier and Schiestel2002; Boffetta et al. Reference Boffetta, Mazzino, Musacchio and Vozella2010). This scaling is used to check the self-similarity of the dissipation contributions. Figure 15 displays the $x$ (a) and $z$ (b)-contribution versus the self-similar variable, i.e. $\overline{\unicode[STIX]{x2202}_{j}u_{1}\unicode[STIX]{x2202}_{j}u_{1}}/h^{1/2}$ and $\overline{\unicode[STIX]{x2202}_{j}u_{3}\unicode[STIX]{x2202}_{j}u_{3}}/h^{1/2}$ versus $z/h$ . The $z$ -contribution is closer to convergence than the $x$ -contribution. On the other hand, if dissipation has reached a self-similar regime, one has $\overline{\unicode[STIX]{x1D700}}(z,t)=h^{1/2}(t)\unicode[STIX]{x1D6F6}(\unicode[STIX]{x1D709})$ , where $\unicode[STIX]{x1D6F6}$ is the structure function of the dissipation and $\unicode[STIX]{x1D709}=z/h(t)\sim z/t^{2}$ . In the self-similar regime, we must have $\int _{-\infty }^{+\infty }\unicode[STIX]{x1D6F6}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=C^{st}$ . Therefore

As a result, the $z$ -averaged value over the thickness $h(t)$ behaves as

This result is well known in IHT (Sreenivasan Reference Sreenivasan1998). The evolution of the $x$ - and $z$ -contributions and the total dissipation rate is represented in figure 16. Both contributions show a satisfactory agreement with the linear temporal scaling (4.6) while the total dissipation rate displays a very good agreement.

Figure 15.

Dissipation rate contributions $\unicode[STIX]{x1D700}_{jj}$ . (a) Scaled $x$ -contribution of the dissipation, $\overline{\unicode[STIX]{x2202}_{j}u_{1}\unicode[STIX]{x2202}_{j}u_{1}}/h^{1/2}$ , as a function of the self-similar variable $z/h$ for four different times. (b) Scaled $z$ -contribution of the dissipation, $\overline{\unicode[STIX]{x2202}_{j}u_{3}\unicode[STIX]{x2202}_{j}u_{3}}/h^{1/2}$ .

Figure 16.

Dissipation rate $\unicode[STIX]{x1D700}$ . Evolution of the $x$ - and $z$ -contributions and the total dissipation, averaged over $xyz$ . The dissipation rate shows a very good agreement with the linear temporal scaling (4.6).

Finally, the PDF of the dissipation is displayed in figure 17 at two different times. Since dissipation is homogeneous to the square of a velocity gradient, we also display the PDF of the square-root of the dissipation $\unicode[STIX]{x1D700}$ . It clearly exhibits exponential wings. The slope increases with time – or the Reynolds number – and so does the intermittency.

Figure 17.

PDF of the dissipation $\unicode[STIX]{x1D700}$ , at two different times. (a) PDF ( $\unicode[STIX]{x1D700}$ ). (b) PDF ( $\unicode[STIX]{x1D700}^{1/2}$ ), to highlight the exponential wings. The slope of these wings increases with time, i.e. the Reynolds number.

5.1 Vorticity properties

In this section we study the properties of the vorticity whose transport equation has been derived in § 2.3 for the RT-Boussinesq flow. On the one hand, the mean vorticity components are given by

On the other hand, from the vorticity equation (2.21), one finds with the mean value definition (2.14)

In these evolution equations, there is neither transport nor baroclinic production. The baroclinic term vanishes due to the symmetry of the RT-Boussinesq flow. Turbulence production is limited to the stretching contribution, i.e. production through correlation between the third vorticity component and the gradients of the horizontal velocity components. There are also two diffusion terms, one turbulent, one molecular. Moreover these non-zero mean vorticity values come from a small large-scale drift in the velocity field. As such, the averaged values of the vorticity components do not appear to be relevant to the understanding of vorticity motion. Let us note that such simplification does not occur in the compressible case where baroclinic contributions of the form $\unicode[STIX]{x1D700}_{ijk}\overline{\unicode[STIX]{x2202}_{j}\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x2202}_{k}p^{\prime }}$ do not vanish. We thus turn our attention to the square of the mean vorticity. Owing to vorticity equation (2.21), we have

where we have introduced the dissipation of vorticity square $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D714}}=2\mathit{Re}^{-1}\unicode[STIX]{x2202}_{j}\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x2202}_{j}\unicode[STIX]{x1D714}_{i}$ . In this (5.3) summation over the index $i$ may be performed or not. If it is performed, we obtain an equation for the vorticity norm and thus for the enstrophy (see below equation (5.11)). If not, equation (5.3) leads to the three equations for the three vorticity component square

The baroclinic production is only present in the horizontal directions. On the other hand, it has been demonstrated in Ristorcelli & Clark (Reference Ristorcelli and Clark2004) that the third component of the velocity dissipation asymptotically scales as $t$ or $h(t)^{1/2}$ (see figure 15). For dimensional reasons, we will used the same scaling for the vorticity square. We thus may use the same argument given in § 4.3 for the dissipation. Therefore

If vorticity has reached a self-similar regime, one has $\int _{-\infty }^{+\infty }\unicode[STIX]{x1D6FA}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=C^{st}$ , so that

As a result, the value of the vorticity norm averaged over the thickness $h(t)$ behaves as

In the same way, the mean values of the vorticity components $\langle \overline{\unicode[STIX]{x1D714}_{i}^{2}}^{1/2}\rangle _{1}(t)$ , $i=1,2,3$ should also behave as $t^{1/2}$ . This is indeed the case for the horizontal contribution, the agreement with this temporal scaling is less good for the $z$ -contribution (not represented).

Figure 18.

2-D vorticity map in the plane $(t,z)$ . (a) Horizontal vorticity component $\overline{\unicode[STIX]{x1D714}_{1}^{2}}^{1/2}$ . (b) Vertical vorticity component $\overline{\unicode[STIX]{x1D714}_{3}^{2}}^{1/2}$ .

Figure 18 displays the 2-D maps in the $(t,z)$ plane of the horizontal and vertical vorticity component $\overline{\unicode[STIX]{x1D714}_{x}^{2}}^{1/2}$ and $\overline{\unicode[STIX]{x1D714}_{z}^{2}}^{1/2}$ . The horizontal component reaches higher values sooner than the vertical component. In other words, it reaches a turbulent behaviour and a self-similar behaviour sooner. This is obvious from the plots given in figure 19 where both contributions, normalized by $h^{1/4}$ , are represented versus the self-similar coordinate $z/h$ . The norm of the vorticity is displayed in figure 20(a) and shows the self-similar behaviour (5.7). The vorticity anisotropy computed from $\overline{\unicode[STIX]{x1D714}_{z}^{2}}/\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}-1/3$ is displayed in the same figure 20(b). It appears that the permanent regime, where the self-similar solution holds, is not fully reached. The vorticity anisotropy decreases with time, as the self-similar regime is more and more established. The final value is approximately $-0.05$ , i.e. a small value corresponding to a quasi-isotropy of the vorticity (Cabot & Cook Reference Cabot and Cook2006). This is in accord with the quasi-isotropy of both velocity and mixing at Taylor scales observed in §§ 4.1 and 6.

Figure 19.

Vorticity component scaling. (a)  $\overline{\unicode[STIX]{x1D714}_{x}^{2}}^{1/2}/h^{1/4}$ versus the vertical coordinate $z/h$ at times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . (b)  $\overline{\unicode[STIX]{x1D714}_{z}^{2}}^{1/2}/h^{1/4}$ versus $z/h$ , at the same times.

Figure 20.

Vorticity. (a) Vorticity norm, $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}^{1/2}/h^{1/4}$ , as a function of $z/h$ at times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . (b) Vorticity anisotropy $\overline{\unicode[STIX]{x1D714}_{z}^{2}}/\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}-1/3$ at the same times.

The norm of the baroclinic source term is obtained from (2.21). This term can be expressed as $2A_{t}(\overline{(\unicode[STIX]{x2202}_{1}c^{\prime })^{2}}^{1/2}+\overline{(\unicode[STIX]{x2202}_{2}c^{\prime })^{2}}^{1/2})$ and its evolution is displayed in figure 21(a). This plot shows that the baroclinic production is larger in the middle of the mixing layer and reaches a maximum in time between transition to turbulence and transition to self-similarity. Afterwards, it slightly decays and reaches roughly a constant and uniform value over a small region around $z=0$ . It means that concentration gradients slightly decrease after transition to self-similarity. This might be explained by a more complete mixing after transition. This is also a classic way to reach an equilibrium state by first exceeding this final state.

Figure 21.

(a) Baroclinic production of the vorticity equation (2.21), $2A_{t}(\overline{(\unicode[STIX]{x2202}_{1}c^{\prime })^{2}}^{1/2}+\overline{(\unicode[STIX]{x2202}_{2}c^{\prime })^{2}}^{1/2})$ : 2-D map in the $(t,z)$ -plane. (b) Enstrophy budget. Source terms of (5.14): stretching (blue), enstrophy–baroclinic production (green), dissipation (red) and full right-hand side (black).

On the other hand, the expression of the enstrophy is

With (5.3) and the symmetry properties (2.17) one finds

Through an integration by parts, the last term of (5.11) may also be written

The second source term in (5.11) is

As a result, the equation (5.11) is written

where the first contribution of the right-hand side comes from the stretching source term of the vorticity equation (2.18), while the second and the third are the baroclinic source term as detailed in § 2.3. The fourth contribution is the dissipation term. These contributions have been plotted as functions of time in figure 21. The enstrophy–baroclinic production term grows during the transition stage and stabilizes at late time, at around $t\approx 14$ . The stretching grows strongly as well as the dissipation (with the opposite sign), such that the full right-hand side grows with time. This production by stretching is by far the most important. As a result, there is an increasing enstrophy disequilibrium between production and dissipation. A fit of the right-hand side of (5.14) with the function $t^{b}$ gives $b=1.70$ . A fit of the enstrophy $\mathscr{E}(t)\sim t^{c}$ gives the value $c=2.71$ . According to (5.14), one should have indeed $b=c-1$ . This result may be compared with the exponent $5/2$ obtained for the time dependence of the integral of the vorticity norm square-root (5.8).

5.2 Intermittent vorticity structure

The intermittent vorticity structure in homogeneous turbulence has been revealed by a series of works (Siggia Reference Siggia1981; She et al. Reference She, Jackson and Orszag1990; Vincent & Meneguzzi Reference Vincent and Meneguzzi1991, Reference Vincent and Meneguzzi1994) and small-scale properties of a passive scalar in a homogeneous turbulence have been studied by Pumir (Reference Pumir1994b ,Reference Pumir c ). Similar analysis may be conducted in RT flows. However, the RT flow is bounded, i.e. the thickness of the mixing layer is finite. Visually figure 22(a) indicates clearly the energetic large-scale baroclincally generated structures that drive the turbulent Rayleigh–Taylor flow. Indeed vorticity appears in toroidal structures corresponding to the mushroom-like features in concentration isosurfaces (not included). These very large-scale structures are not turbulent. Indeed, the Reynolds number is fairly low on the boundaries. However, vortices and coherent structures in a turbulent flow are better detected with the $Q$ -criterion, since vorticity does not always correspond to vortices (Jeong & Hussain Reference Jeong and Hussain1995; Dubief & Delcayre Reference Dubief and Delcayre2000). This $Q$ -criterion refers to the second invariant of the velocity-gradient tensor $\unicode[STIX]{x1D735}_{i}u_{j}$ , $Q=1/2(\unicode[STIX]{x1D6FA}_{ij}\unicode[STIX]{x1D6FA}_{ij}-\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij})$ , where $\unicode[STIX]{x1D6FA}$ and $S$ are respectively the antisymmetric and symmetric parts of $\unicode[STIX]{x1D735}_{i}u_{j}$ . Positive values of $Q$ correspond to vortices, while negative values are associated with shear. The same figure 22(b) shows the $Q$ -criterion for $Q=50$ , i.e. a rather large value. This picture shows that intermediate scales have some coherence. Indeed vortices take the form of tubes of finite length (Vincent & Meneguzzi Reference Vincent and Meneguzzi1994). These vortices are localized inside the mixing layer. To be more specific, we have plotted the vorticity PDF in figure 23, at two different times, $t=12.4$ and $t=15.7$ . These PDFs reveal strong non-Gaussian character, as opposed to the velocity PDFs in figure 7. At the final time of the simulation, the horizontal component $\unicode[STIX]{x1D714}_{x}$ PDF has exponential wings. At the same time, the vertical component $\unicode[STIX]{x1D714}_{z}$ PDF has bended wings, i.e. more than exponential. This is the signature of some organized motion at intermediate scales. Symmetry of these PDFs with respect to zero must also be noted.

Figure 22.

(a) Vorticity isosurface coloured by the concentration (heavy fluid in red and light fluid in blue), top view, $|\unicode[STIX]{x1D714}|=3$ . (b) Isosurface $Q$ -criterion coloured by the concentration, $Q=50$ . The $Q$ -criterion reveals the small-scale vortex structure of the mixing layer.

Figure 23.

Vorticity PDFs at two different times $t=12.4$ and $t=15.7$ . (a) The horizontal component $\unicode[STIX]{x1D714}_{x}$ . (b) The vertical component $\unicode[STIX]{x1D714}_{z}$ . They all exhibit a non-Gaussian behaviour. The component $\unicode[STIX]{x1D714}_{x}$ has exponential wings and the component $\unicode[STIX]{x1D714}_{z}$ is beyond exponential. PDFs have been built with data located in the domain $[0:1;0:1;z_{1}:z_{2}]$ , where $z_{1}$ and $z_{2}$ are defined by $\overline{c}(z_{1})=0.4$ and $\overline{c}(z_{2})=0.6$ .

Figure 24.

Molecular mixing fraction $\unicode[STIX]{x1D703}(z,t)=\overline{c(1-c)}/\overline{c}(1-\overline{c})$ . (a) 2-D $(t,z)$ map. (b)  $\unicode[STIX]{x1D703}$ versus the self-similar coordinate $z/h$ at four different times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ .

Figure 25.

Concentration. (a) Concentration fluctuation variance, $\overline{c^{\prime 2}}^{1/2}$ , versus the self-similar coordinate $z/h$ . Profiles at four different times have been superimposed ( $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ ) to show the self-similar regime. (b) Molecular mixing fraction $\langle \unicode[STIX]{x1D703}\rangle _{\unicode[STIX]{x1D6FD}}=\langle \overline{c(1-c)}/\overline{c}(1-\overline{c})\rangle _{\unicode[STIX]{x1D6FD}}$ versus time. Two values of $\unicode[STIX]{x1D6FD}$ have been used, 0.2 (red dashed line) and 0.8 (blue line).

Figure 26.

Mixing statistics. (a) Skewness of the concentration fluctuations, $\overline{c^{\prime 3}}/\overline{c^{\prime 2}}^{3/2}$ . (b) Flatness of the concentration fluctuations, $\overline{c^{\prime 4}}/\overline{c^{\prime 2}}^{2}$ . Both are plotted versus the self-similar coordinate $z/h$ at four different times $t=11.5$ , $t=12.9$ , $t=14.3$ and $t=15.7$ . The flatness of the Gaussian process is represented by the horizontal dashed line.

6.1 Mixing anisotropy

In order to have a deeper insight of the mixing structure, we use several indicators. The multiscale variances, $\langle \overline{|\unicode[STIX]{x1D735}^{(d)}c^{\prime }|^{2}}\rangle _{\unicode[STIX]{x1D6FD}}$ (Lin et al. Reference Lin, Thiffeault and Doering2011), the anisotropy indicator, $C_{\unicode[STIX]{x2202}_{z}^{(d)}c^{\prime }}$ , defined in (4.3), with $d=1$ for the intermediate scales and $d=2$ for the small scales and the spectral indicator defined in (4.4). The anisotropy indicator evolution, $C_{\unicode[STIX]{x2202}_{z}^{(d)}c^{\prime }}(z,t)$ , is represented in figure 31 for $d=1$ . The 2-D map shows spatial oscillations along the $z$ -direction, which are confirmed by the plot of the anisotropy indicator as a function of the self-similar variable $z/h$ . For $d=2$ , the $z$ -variations are similar to the intermediate-scale mixing anisotropy, as shown in figure 32. The average of this small-scale mixing anisotropy over a part of the mixing layer shows good convergence (see figure 33 b). As for the intermediate scales, the vertical concentration gradients are close to zero in the nonlinear regime. However, in the transitional regime, they quickly reach their final values. This final value is approximately $0.10$ , i.e. some mixing anisotropy remains at small scales. Figure 33(a) displays the evolution of the anisotropy indicator (4.3) averaged over a part of the mixing layer in the direction $z$ , $\langle C_{\unicode[STIX]{x2202}_{z}^{(d)}c^{\prime }}\rangle _{\unicode[STIX]{x1D6FD}}(t)$ . Intermediate- and small-scale anisotropies start at very large values. In the nonlinear regime, where mushrooms develop, they drop to smaller negative values. The vertical concentration gradients are thus smaller than the horizontal gradients. Within the core of the mixing layer ( $\unicode[STIX]{x1D6FD}=0.2$ ), the vertical concentration gradients are even vanishing. In the transitional regime, the mixing anisotropy grows monotonically and reaches $0.03$ at the final time of the simulation. It means that the intermediate scales of the mixing are close to isotropy. Spectral velocity anisotropy have been investigated with the indicator whose expression is given in (4.4). We use here the same indicator written for the concentration-fluctuation gradient. It reads

The variations with respect to the horizontal wavenumber $k_{x}$ are shown in figure 34. As expected, large scales are strongly anisotropic. This anisotropy decreases for smaller scales, the isotropy being reached for the vertical velocity Taylor wavenumber. At these scales, buoyancy is negligible and the scalar behaves passively. For larger wavenumbers, anisotropy grows again, buoyancy is no longer negligible and the scalar can be considered as active. At the Kolmogorov scale the anisotropy is close to $-1/3$ , which means that the vertical concentration gradients vanish faster than the horizontal gradients. In other words, full mixing is achieved at this scale.

Figure 34.

Spectral anisotropy of the concentration gradient. The vertical dashed black lines stand for the integral scale, the horizontal and vertical Taylor length scales, the horizontal and vertical Taylor mixing length scales and the Kolmogorov scale, respectively.

Figure 35.

(a) Spatial average of multiscale concentration variances for large, intermediate and small scales, $\langle \overline{c^{\prime 2}}\rangle _{\unicode[STIX]{x1D6FD}}$ (blue), $\langle \overline{|\unicode[STIX]{x1D735}c^{\prime }|^{2}}\rangle _{\unicode[STIX]{x1D6FD}}$ (red) and $\langle \overline{|\unicode[STIX]{x1D6FB}^{2}c^{\prime }|^{2}}\rangle _{\unicode[STIX]{x1D6FD}}$ (black). Each quantity is normalized by its maximum over time. The order of temporal convergence, from the slowest to the fastest, is large, small and intermediate scales. (b) One-dimensional power spectra of the concentration fluctuation, $c^{\prime }$ , at the four selected times. The $k^{-17/3}$ -scaling have also been plotted. The vertical lines stand for the integral scale, the horizontal and vertical Taylor mixing length scales and the Kolmogorov scale, respectively.

The multiscale variances at orders 0, 1 and 2 are represented in figure 35(a). They describe the efficiency of the mixing at large, intermediate and small scales, respectively. These variances are 0 in pure fluid or in perfectly mixed fluid. The maximum of these quantities happens around the transition to turbulence of the flow. After that time, as turbulence develops to smaller and smaller scales, the two fluids mix more and more efficiently. Since the large scales are responsible for the engulfment of resting fluids into the continuously growing mixing layer, the large-scale variance remains high. Comparing the intermediate and the small scales, the mixing seems to be more efficient at the former. The anisotropy at the small scales may slow down the mixing at these scales. Finally the 1-D power spectrum of the concentration fluctuations, $c^{\prime }$ , is represented in figure 35(b). The region around the Taylor scales is not incompatible with the $k^{-17/3}$ -scaling (Batchelor, Howells & Townsend Reference Batchelor, Howells and Townsend1959; Ristorcelli Reference Ristorcelli2006). However this scaling is uncertain since the scalar may be active at small scales. Moreover, this scaling holds for small Schmidt numbers in the inertial diffusive range. In our case, the Schmidt number is $\mathit{Sc}=0.7$ , which is not so small.