4.1 Self-similar evolution

It is well known that temporal mixing layers reach a self-similar evolution after an initial transient, in which the initial perturbations evolve into the structure of the fully developed turbulent mixing layer (Rogers & Moser Reference Rogers and Moser1994; Pantano & Sarkar Reference Pantano and Sarkar2002). In the self-similar evolution, the mixing layer thickness grows linearly with time, and large-scale quantities scaled with the variation across the mixing layer (i.e., $\unicode[STIX]{x0394}U$ , $\unicode[STIX]{x1D70C}_{b}-\unicode[STIX]{x1D70C}_{t}$ , etc.) collapse into a single profile when plotted as a function of $y/\unicode[STIX]{x1D6FF}_{m}(t)$ or $y/\unicode[STIX]{x1D6FF}_{w}(t)$ .

In order to evaluate the self-similar evolution of the present DNS results, figure 2(a) shows the evolution of $\unicode[STIX]{x1D6FF}_{m}(t)$ for the four cases considered here. The variability in $\unicode[STIX]{x1D6FF}_{m}$ is estimated using the standard deviation of the momentum thicknesses over the four runs, and is indicated with error bars in the figure. Also, figure 2(b) shows the time evolution of the integrated dissipation rate of turbulent kinetic energy

The quantity $\unicode[STIX]{x1D701}$ scales with $\unicode[STIX]{x0394}U^{3}$ and, therefore, should be constant with time, once self-similarity has been achieved. The expression for the dissipation rate of turbulent kinetic energy for variable-density flows can be found in Chassaing et al. (Reference Chassaing, Antonia, Anselmet, Joly and Sarkar2002), and is reproduced here for completeness

where primed variables denote fluctuations with respect to the mean, $\unicode[STIX]{x1D703}=\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{i}$ is the divergence of the velocity, and $\unicode[STIX]{x1D714}_{i}$ are the components of the vorticity.

The results presented in figure 2 show that self-similarity is achieved after an initial transient, with $\unicode[STIX]{x1D6FF}_{m}(t)$ growing linearly with time and $\unicode[STIX]{x1D701}(t)$ becoming approximately constant (at least within the errors in $\unicode[STIX]{x1D701}$ ). However, comparing figure 2(a,b) it can be observed that the linear growth of $\unicode[STIX]{x1D6FF}_{m}$ starts at $\unicode[STIX]{x1D70F}=t\unicode[STIX]{x0394}U/\unicode[STIX]{x1D6FF}_{m}^{0}\approx 200$ , a time at which $\unicode[STIX]{x1D701}$ is still growing. This behaviour was also observed by Rogers & Moser (Reference Rogers and Moser1994), and it indicates that the determination of the time interval where self-similarity is achieved needs careful consideration, and should not be determined exclusively from a linear evolution of $\unicode[STIX]{x1D6FF}_{m}(t)$ .

Figure 2. Temporal evolution of (a) the momentum thickness $\unicode[STIX]{x1D6FF}_{m}$ divided by the initial momentum thickness $\unicode[STIX]{x1D6FF}_{m}^{0}$ , and (b) the non-dimensional integrated turbulent energy dissipation rate, $\unicode[STIX]{x1D701}/\unicode[STIX]{x0394}U^{3}$ . Line types are black for $s=1$ , blue for $s=2$ , green for $s=4$ and red for $s=8$ . The values correspond to the ensemble average of the four runs for each density ratio, and the error bars are the corresponding standard deviation. The thick line shows the ranges of self-similar evolution for each density ratio.

In the present study, and for the purpose of collecting statistics, we have defined the time interval $[\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D70F}_{f}]$ where the mixing layer is self-similar by analysing the collapse of the instantaneous (i.e., averaged in the horizontal directions only) profiles of the normal Reynolds stresses, $\unicode[STIX]{x1D619}_{11}(y/\unicode[STIX]{x1D6FF}_{m},\unicode[STIX]{x1D70F})$ , $\unicode[STIX]{x1D619}_{22}(y/\unicode[STIX]{x1D6FF}_{m},\unicode[STIX]{x1D70F})$ , and $\unicode[STIX]{x1D619}_{33}(y/\unicode[STIX]{x1D6FF}_{m},\unicode[STIX]{x1D70F})$ . We have computed the temporal mean and standard deviation of these Reynolds stresses for several time intervals, selecting for each run the longest time interval in which the standard deviation of the normal Reynolds stresses is smaller than 5 % of the maximum. The resulting time intervals (more explicitly, the maximal time interval over the four runs for each density ratio) are shown in figure 2 and reported in table 1, yielding a total self-similar range of at least 10 eddy-turnover times per density ratio. For illustration, figure 3 shows all the $\unicode[STIX]{x1D619}_{11}$ profiles within the self-similar range for the cases $s=1$ and $s=4$ , using different colour for each run. The agreement of the profiles is good, especially taking into account that there are 26 and 31 curves on each plot, respectively. The differences are more apparent near the maximum of the Reynolds stresses. It is interesting to note that the variability of the profiles within each run is small, similar to that reported by Pantano & Sarkar (Reference Pantano and Sarkar2002). On the other hand, the variability between different runs is a bit larger, and it is probably linked to differences between the largest structures developed in each run (i.e., by different realizations of the initial conditions), emphasizing the importance of running several realizations of each density ratio to accumulate statistics for the largest structures in the mixing layer.

Figure 3. Reynolds stress $\unicode[STIX]{x1D619}_{11}$ profiles within the self-similar range for (a) case $s=1$ , all runs with a total of 26 profiles, and (b) case $s=4$ , all runs, with a total of 31 profiles. Colours are used to differentiate between runs.

4.2 Effects of the density ratio on the growth rate

Once the self-similar time interval has been defined, we analyse the effect that the density ratio has on the growth rate of the temporal mixing layer, comparing the results of the present zero-Mach cases with those obtained by Pantano & Sarkar (Reference Pantano and Sarkar2002) for convective Mach number $M_{c}=0.7$ . First, consider the growth rate of the momentum thickness, $\dot{\unicode[STIX]{x1D6FF}}_{m}$ , which is evaluated here following the expression derived in Vreman et al. (Reference Vreman, Sandham and Luo1996),

This expression is obtained by differentiating (2.8) with respect to time, and neglecting viscous terms. An alternative method to compute $\dot{\unicode[STIX]{x1D6FF}}_{m}$ is to fit a linear law to the data shown in figure 2(a). The differences in the mean and standard deviation of the growth rate of the momentum thickness obtained from both methods are small: for $s=1$ , the first method yields $\dot{\unicode[STIX]{x1D6FF}}_{m}/\unicode[STIX]{x0394}U=0.0168\pm 0.0003$ , while the second method yields $\dot{\unicode[STIX]{x1D6FF}}_{m}/\unicode[STIX]{x0394}U=0.0170\pm 0.0002$ .

The value of the growth rate of the momentum thickness for $s=1$ is in good agreement with previous works, especially taking into account the scatter of the available data. For instance, in the ‘unforced’ experiments quoted by Dimotakis (Reference Dimotakis1991) the growth rate of the momentum thickness varies from 0.014 to 0.022. Also, Rogers & Moser (Reference Rogers and Moser1994) report $\dot{\unicode[STIX]{x1D6FF}}_{m}/\unicode[STIX]{x0394}U=0.014$ in simulations of incompressible temporal mixing layers, and the experimental data of Bell & Mehta (Reference Bell and Mehta1990) yield a value of 0.016. For $M_{c}=0.3$ and $s=1$ , Pantano & Sarkar (Reference Pantano and Sarkar2002) report $\dot{\unicode[STIX]{x1D6FF}}_{m}/\unicode[STIX]{x0394}U\approx 0.0184$ , a value that decreases to $0.0108$ when the Mach number is increased to $M_{c}=0.7$ .

As the density ratio increases, the values of $\dot{\unicode[STIX]{x1D6FF}}_{m}$ decrease. This reduction of the growth rate is quantified in figure 4(a), in terms of the ratio of growth rates,  $\dot{\unicode[STIX]{x1D6FF}}_{m}(s)/\dot{\unicode[STIX]{x1D6FF}}_{m}(1)$ . For $s=8$ , our results show that the growth rate of $\unicode[STIX]{x1D6FF}_{m}$ has been reduced by 60 % with respect to the growth rate of the case with $s=1$ . A similar behaviour is observed for the subsonic cases of Pantano & Sarkar (Reference Pantano and Sarkar2002) at $M_{c}=0.7$ , also included in the figure. The ratio $\dot{\unicode[STIX]{x1D6FF}}_{m}(s)/\dot{\unicode[STIX]{x1D6FF}}_{m}(1)$ is very similar for the $M_{c}=0$ and $M_{c}=0.7$ cases for large density ratios, with significant differences for the smaller density ratio, $s=2$ . Careful inspection shows that the density ratio $s=2$ is indeed somewhat anomalous in Pantano & Sarkar (Reference Pantano and Sarkar2002), presenting a non-monotonic behaviour for some quantities (see for instance the growth rates and the profiles of Reynolds stress, as shown in table 6 and figure 18 respectively in their paper).

Figure 4. Mixing-layer growth rate as a function of the density ratio. Growth rate based on (a) momentum thickness $\dot{\unicode[STIX]{x1D6FF}}_{m}$ , and (b) vorticity thickness $\dot{\unicode[STIX]{x1D6FF}}_{w}$ , normalized by the growth rate for $s=1$ . In both panels the horizontal axis is in logarithmic scale. Coloured dots with error bars stand for the present results, squares represent results for $M_{c}=0.7$ (Pantano & Sarkar Reference Pantano and Sarkar2002). The dashed curve in (b) corresponds to equation (4.4), from Ramshaw (Reference Ramshaw2000).

The results shown in figure 4(a) for $\dot{\unicode[STIX]{x1D6FF}}_{m}$ are very similar to those obtained for $\dot{\unicode[STIX]{x1D6FF}}_{w}$ , which are plotted in figure 4(b). Again, our results are compared to the $M_{c}=0.7$ cases of Pantano & Sarkar (Reference Pantano and Sarkar2002), and the theoretical prediction by Ramshaw (Reference Ramshaw2000). The latter is based on a model for the growth of the visual thickness of a variable-density mixing layer at $M_{c}=0$ , directly comparable to the present results. The model is obtained by extending a linear stability analysis to the nonlinear regime through scaling hypothesis, leading after proper manipulation to

Figure 4(b) shows a very good agreement between Ramshaw’s model and our data. The agreement is also fairly good with the subsonic data of Pantano & Sarkar (Reference Pantano and Sarkar2002) at $M_{c}=0.7$ , except for the case $s=2$ as it happened also for $\dot{\unicode[STIX]{x1D6FF}}_{m}$ . It should be noted that, to the best of our knowledge, this is the first direct validation of the Ramshaw model with a variable-density DNS at $M_{c}=0$ .

Overall, the results presented in this subsection show that the growth rates of the $M_{c}=0$ cases are significantly higher than those reported by Pantano & Sarkar (Reference Pantano and Sarkar2002) for $M_{c}=0.7$ , in agreement with previous works. However, the effect of $s$ on the reduction of the growth rate seems to be very similar at both Mach numbers, except for maybe the low-density-ratio case, $s=2$ . Also, the effect of $s$ seems to be stronger on $\unicode[STIX]{x1D6FF}_{m}$ than on $\unicode[STIX]{x1D6FF}_{w}$ , with $\dot{\unicode[STIX]{x1D6FF}}_{m}(s=8)/\dot{\unicode[STIX]{x1D6FF}}_{m}(s=1)\approx 0.4$ and $\dot{\unicode[STIX]{x1D6FF}}_{w}(s=8)/\dot{\unicode[STIX]{x1D6FF}}_{w}(s=1)\approx 0.6$ . As a consequence, the ratio between the two thicknesses, $D_{w}=\unicode[STIX]{x1D6FF}_{w}/\unicode[STIX]{x1D6FF}_{m}$ , increases with $s$ , as can be observed in table 1. Note that since $\unicode[STIX]{x1D6FF}_{w}$ and $\unicode[STIX]{x1D6FF}_{m}$ grow linearly with time, $D_{w}\approx \dot{\unicode[STIX]{x1D6FF}}_{w}/\dot{\unicode[STIX]{x1D6FF}}_{m}$ for sufficiently long times. For reference, Pantano & Sarkar (Reference Pantano and Sarkar2002) report a value of $D_{w}=5.0$ for a compressible mixing layer with $M_{c}=0.3$ and $s=1$ , in good agreement with $D_{w}=4.83$ for our $s=1$ case.

4.3 Mean density, velocity and temperature

We now proceed to analyse the one-point statistics of the present DNS (mean values in this subsection, higher-order moments in § 4.4), averaging the data in the horizontal directions and in time, binning in $y/\unicode[STIX]{x1D6FF}_{m}(t)$ . In all the vertical profiles presented in this section, a shadowing has been applied around plus/minus one standard deviation of the horizontally averaged data with respect to the mean, in order to show the uncertainty of the statistics.

Figure 5. (a) Reynolds-average density profiles. (b) Favre-averaged streamwise velocity profiles. Different colours correspond to different density ratios: black, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; and red, $s=8$ . Solid lines are the present turbulent temporal mixing layers. Dashed lines are the laminar temporal mixing layers (see appendix B). Symbols: Rogers & Moser (Reference Rogers and Moser1994) for $s=1$ , Pantano & Sarkar (Reference Pantano and Sarkar2002) for $s=2,4$ and 8.

Figure 5(a) shows mean density profiles, comparing the present zero-Mach results with the results of the subsonic mixing layer of Pantano & Sarkar (Reference Pantano and Sarkar2002) at $M_{c}=0.7$ . The figure also includes for comparison the results from laminar temporal mixing layers, obtained as discussed in appendix B. As the density ratio increases, the density mixing layer extends further into the low-density stream, with small variations in the position where $\overline{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}_{0}$ . The profiles of the $M_{c}=0.7$ and $M_{c}=0$ cases are qualitatively similar at any given density ratio, although there are some differences in the profiles in the central part of the mixing layer ( $|y|\lesssim 3\unicode[STIX]{x1D6FF}_{m}$ ). The agreement between the $M_{c}=0$ and $M_{c}=0.7$ cases is better for the Favre-averaged velocity, shown in figure 5(b). The only exception is maybe the region closer to the high-density free stream, where the edge of the mixing layer seems sharper for the present simulations ( $M_{c}=0$ ). The figure also includes the incompressible data of Rogers & Moser (Reference Rogers and Moser1994) for $s=1$ , showing a very good agreement with our incompressible case.

Besides some small changes in the shape of the profiles (which will be discussed later), the most apparent effect of the density ratio in $\overline{\unicode[STIX]{x1D70C}}$ and $\tilde{u}$ is the shifting of the $\tilde{u}$ profile towards the low-density side. Note that this effect is apparent in both turbulent cases ( $M_{c}=0$ and $M_{c}=0.7$ ), as well as in the laminar self-similar profiles (dashed lines in figure 5). This shifting of the mean density and velocity profiles with the density ratio has already been reported in previous studies, both experimental and numerical, and it has been explained qualitatively in terms of the asymmetry in the momentum exchange of the large scales with the free streams (Brown & Roshko Reference Brown and Roshko1974) and their linear stability properties (Soteriou & Ghoniem Reference Soteriou and Ghoniem1995). Note that the mechanism proposed by Brown & Roshko (Reference Brown and Roshko1974) acts in both turbulent diffusion (as originally proposed by the authors) and mean velocity entrainment (i.e, $\langle \unicode[STIX]{x1D70C}v\rangle$ ). While in turbulent mixing layers the turbulent diffusion is dominant over the mean velocity entrainment, the latter is important in laminar mixing layers. This could explain why figure 5 also shows a clear shifting between $\tilde{u}$ and $\overline{\unicode[STIX]{x1D70C}}$ for the laminar cases, as well as the results reported by Bretonnet et al. (Reference Bretonnet, Cazalbou, Chassaing and Braza2007) in laminar mixing layers with density variations due to various effects: different velocities, temperatures or species.

Besides the numerous qualitative observations of the shift between $\tilde{u}$ and $\overline{\unicode[STIX]{x1D70C}}$ in the literature, few authors have tried to quantify it. In turbulent mixing layers, Pantano & Sarkar (Reference Pantano and Sarkar2002) proposed to quantify this shift using two semi-empirical relationships, $\overline{\unicode[STIX]{x1D70C}}(\tilde{u} )$ and $\unicode[STIX]{x1D619}_{12}(\tilde{u} )$ . They later used these relationships to estimate the reduction of the momentum thickness growth rate. In laminar mixing layers, Bretonnet et al. (Reference Bretonnet, Cazalbou, Chassaing and Braza2007) proposed to characterize the drift as the distance between the inflection points of the velocity and density mean profiles.

Here we propose to quantify the shift using $\unicode[STIX]{x1D6E5}$ , which is defined as the distance between the $y$ locations where $\tilde{u} =0$ and $\overline{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}_{0}$ , positive when $\tilde{u}$ is displaced towards $y$ -positive (low-density side in our simulations). The main advantage of the present definition with respect to those used by Pantano & Sarkar (Reference Pantano and Sarkar2002) and Bretonnet et al. (Reference Bretonnet, Cazalbou, Chassaing and Braza2007) is that it can be easily computed from the mean profiles of velocity and density, without having to compute higher-order derivatives. This distance is plotted in figure 6 as a function of the density ratio, for turbulent mixing layers with $M_{c}=0$ and $M_{c}=0.7$ , and for the laminar self-similar solutions. The figure shows two possible scalings for $\unicode[STIX]{x1D6E5}$ , with $\unicode[STIX]{x1D6FF}_{m}$ (figure 6 a) and with $\unicode[STIX]{x1D6FF}_{w}$ (figure 6 b). The different datasets collapse better with the second scaling, especially for $s=8$ cases, suggesting an empirical relation

with $C_{\unicode[STIX]{x1D6E5}}=0.25$ . This empirical approximation yields correlation coefficients of $R^{2}=0.998$ for the present DNS results at $M_{c}=0$ . Similar values of $C_{\unicode[STIX]{x1D6E5}}$ are obtained for the other datasets in the figure. The results of Pantano & Sarkar (Reference Pantano and Sarkar2002) at $M_{c}=0.7$ yield $C_{\unicode[STIX]{x1D6E5}}=0.23$ and $R^{2}=0.956$ , and the laminar self-similar solutions yield $C_{\unicode[STIX]{x1D6E5}}=0.23$ and $R^{2}=0.994$ . Finally, the good agreement between the laminar and turbulent data (and compressible and low-Mach-number data) is consistent with the discussion in the previous paragraphs: the mixing layer is able to erode more easily the lighter free stream, either by turbulent diffusion (in turbulent mixing layers) or by the mean entrainment (in laminar mixing layers).

Figure 6. Shifting of the mixing layer, normalized with (a) the momentum thickness, (b) the vorticity thickness. Circles for present DNS at $M_{c}=0$ . Squares for Pantano & Sarkar (Reference Pantano and Sarkar2002) at $M_{c}=0.7$ . Triangles for laminar self-similar solutions. The dashed line in (b) corresponds to $\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D6FF}_{w}=0.25\log (s)$ .

Although the present definition of shifting is not directly comparable to the one used by Pantano & Sarkar (Reference Pantano and Sarkar2002), it is also possible to relate the present $\unicode[STIX]{x1D6E5}$ to the ratio $\unicode[STIX]{x1D6FF}_{m}(s)/\unicode[STIX]{x1D6FF}_{w}(s)$ . Let us assume that the mean density and velocity profiles are

where $F_{u}(\unicode[STIX]{x1D709})$ and $F_{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D709})$ tend to $\pm 1$ when $\unicode[STIX]{x1D709}\rightarrow \pm \infty$ , and $\unicode[STIX]{x1D6E5}$ is assumed to be a function of the density ratio, $s$ . Note that this is equivalent to limiting the effect of $s$ to a shift between the profiles of $\overline{\unicode[STIX]{x1D70C}}$ and $\tilde{u}$ , with no explicit change in their shape. Introducing (4.6a,b ) into (2.8), it is possible to show that

where $\unicode[STIX]{x1D706}(s)=(s-1)/(s+1)$ . Note that by construction $G(0)=0$ and $G^{\prime }(0)<0$ . Hence, it is possible to simplify (4.7) to

Interestingly, piecewise linear expressions for $F_{\unicode[STIX]{x1D70C}}$ and $F_{u}$ yield $C=1/3$ and a cubic leading-order error in (4.8).

Figure 7. Effects of $s$ and $\unicode[STIX]{x1D6E5}$ on the reduction of the momentum thickness. (a)  $\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{w}$ versus $\unicode[STIX]{x1D706}(s)\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D6FF}_{w}$ . (b)  $\dot{\unicode[STIX]{x1D6FF}}_{m}/\dot{\unicode[STIX]{x1D6FF}}_{w}$ versus $s$ . In both panels, circles are the present DNS at $M_{c}=0$ , squares are Pantano & Sarkar (Reference Pantano and Sarkar2002) at $M_{c}=0.7$ , and triangles are the self-similar solution for the laminar temporal mixing layer. The solid lines in (a) correspond to equation (4.8) with: black, $C=0.188$ ; green, $C=0.190$ ; yellow, $C=0.32$ . The dashed lines in (b) correspond to (4.9) with: black, $C^{\prime }=0.047$ and $\dot{\unicode[STIX]{x1D6FF}}_{w}(1)/\dot{\unicode[STIX]{x1D6FF}}_{m}(1)=4.8$ , green, $C^{\prime }=0.047$ and $\dot{\unicode[STIX]{x1D6FF}}_{w}(1)/\dot{\unicode[STIX]{x1D6FF}}_{m}(1)=5.4$ .

In order to estimate $C$ from the DNS data, figure 7(a) shows the ratio $1/D_{w}=\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{w}$ as a function of $\unicode[STIX]{x1D706}(s)\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D6FF}_{w}$ . The figure shows that $C=0.188$ for the present $M_{c}=0$ data, yielding a correlation coefficient between the data and the linear approximation equal to $R^{2}=0.998$ . For the $M_{c}=0.7$ case, the ratio of the growth rates at $s=1$ is smaller, but the slope of the curve seems to be approximately the same ( $C=0.190$ , $R^{2}=0.920$ ), supporting the assumption that $F_{\unicode[STIX]{x1D70C}}$ , $F_{u}$ (and hence $C$ ) do not vary much with the density ratio. Note that for the laminar case, with notable differences in the shape of $\tilde{u}$ and $\overline{\unicode[STIX]{x1D70C}}$ (and hence in $F_{u}$ and $F_{\unicode[STIX]{x1D70C}}$ ), the value of the constant is $C=0.32$ and the linear approximation is exact ( $R^{2}=1$ ).

Finally, it is possible to combine (4.4), (4.5) and (4.8) to obtain a semi-empirical prediction of the reduction of the momentum thickness growth rate with the density ratio,

To obtain (4.9) we have also taken advantage of $1/D_{w}=\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{w}\approx \dot{\unicode[STIX]{x1D6FF}}_{m}/\dot{\unicode[STIX]{x1D6FF}}_{w}$ , which is a reasonable approximation for sufficiently long times. The performance of this simple model for the reduction of the momentum thickness growth rate is evaluated in figure 7(b), where the dashed lines correspond to equation (4.9) with $C^{\prime }=CC_{\unicode[STIX]{x1D6E5}}=0.047$ and the appropriate value for $\dot{\unicode[STIX]{x1D6FF}}_{w}(1)/\dot{\unicode[STIX]{x1D6FF}}_{m}(1)$ – black for $M_{c}=0$ and green for $M_{c}=0.7$ . The figure also includes the DNS data for both Mach numbers. The agreement between the DNS data and the model is very good, except for the lower density ratios of the $M_{c}=0.7$ cases, which already showed differences when compared to the present $M_{c}=0$ cases in figure 4.

Figure 8. (a,b) Profiles of the vertical gradients of the Reynolds-averaged density. (c,d) Profiles of the vertical gradients of the Favre-averaged streamwise velocity. (a,c) Normalized with the momentum thickness. (b,d) Normalized with the vorticity thickness. Different colours correspond to different density ratios: black, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; red, $s=8$ .

In the previous discussion, the effect of $s$ on the shape of the profiles of $\overline{\unicode[STIX]{x1D70C}}$ and $\tilde{u}$ has been neglected, resulting in a reasonable approximation for the reduction in the growth rate of the mixing layer with $s$ . However, the density ratio has some effects in the shapes of $\overline{\unicode[STIX]{x1D70C}}$ and $\tilde{u}$ , which are responsible for changes in the structure of the turbulence in the mixing layer. These effects, which are difficult to evaluate in figure 5, are better observed in figure 8, which shows the vertical gradients of the mean profiles with different normalizations. In particular, the gradients of the mean density normalized with $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{b}-\unicode[STIX]{x1D70C}_{t}$ and $\unicode[STIX]{x1D6FF}_{w}$ seem to collapse reasonably well (see figure 8 b), especially in the high-density side (lower stream). More differences are visible near the low-density side, where it is apparent that the gradients tend to become smoother with increasing $s$ . Indeed, for $s=8$ , figure 8(a,b) show that the gradient of $\overline{\unicode[STIX]{x1D70C}}$ is roughly linear, so that $\overline{\unicode[STIX]{x1D70C}}$ becomes roughly parabolic for $y\gtrsim -2\unicode[STIX]{x1D6FF}_{m}\approx -0.25\unicode[STIX]{x1D6FF}_{w}$ . Although outside of the scope of the present paper, it would be interesting to check whether the same linear region in $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x2202}y$ is obtained for higher density ratios. The shifting of the velocity profiles discussed above is clearly visible when looking at their corresponding gradients, figure 8(c,d). For $\tilde{u}$ , the change of shape of the profile results in the maximum gradients appearing nearer to the lower-density side, with smoother gradients in the higher-density side. Indeed, opposite to what is observed for $\unicode[STIX]{x1D70C}$ , case $s=8$ seems to develop a nearly parabolic profile for $\tilde{u}$ towards the higher-density side of the mixing layer ( $y\lesssim 4\unicode[STIX]{x1D6FF}_{m}\approx y\lesssim 0.5\unicode[STIX]{x1D6FF}_{w}$ ).

To finalize this subsection, we turn our attention to the mean temperature distribution, more especifically to the non-dimensional temperature jump $\overline{\unicode[STIX]{x1D703}}=(\overline{T}-T_{b})/(T_{t}-T_{b})$ . It is interesting to study the temperature since it follows an advection–diffusion equation, equation (2.3). This allows the comparison of the variable-density cases ( $s=2$ , 4 and 8) with the passive scalar simulated for the uniform density case ( $s=1$ ). Note that although the temperature is inversely proportional to the density (equation of state), the same is not true for the mean temperature and mean density. Figure 9(a) shows the mean temperature profiles for all cases and figure 9(b) the corresponding profiles of the vertical gradients of the mean temperature. The passive scalar shows a roughly symmetric distribution, with $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}}/\unicode[STIX]{x2202}y$ peaking near the edges of the mixing layer ( $|y/\unicode[STIX]{x1D6FF}_{w}|\approx 0.5$ ). The small deviation with respect to a symmetric profile provides an impression about the convergence of the statistics.

With increasing $s$ , the mean temperature profiles shifts towards the upper stream (low-density stream) in a similar way as the Favre-averaged streamwise velocity. The profiles also become more asymmetric, which is more clearly visible in the mean temperature gradients shown in figure 9(b). As the density ratio increases, the gradients at the high-density edge of the mixing layer are strongly damped, while the gradients at the low-density edge are enhanced.

Figure 9. (a) Reynolds-averaged temperature profiles. (b) Profiles of the vertical gradients of the Reynolds-averaged temperature. Different colours correspond to different density ratios: black, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; red, $s=8$ .

4.4 Higher-order statistics

Figure 10. Vertical profiles of (a) $\unicode[STIX]{x1D619}_{11}/\unicode[STIX]{x0394}U^{2}$ , (b) $\unicode[STIX]{x1D619}_{22}/\unicode[STIX]{x0394}U^{2}$ , (c) $\unicode[STIX]{x1D619}_{12}/\unicode[STIX]{x0394}U^{2}$ , and (d) $\unicode[STIX]{x1D619}_{33}/\unicode[STIX]{x0394}U^{2}$ . Different colours correspond to different density ratios: black, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; and red, $s=8$ . Solid lines are the present turbulent temporal mixing layers. Symbols are data from incompressible mixing layers: dots from simulations of Rogers & Moser (Reference Rogers and Moser1994), triangles from experiments of Bell & Mehta (Reference Bell and Mehta1990) and diamonds from experiments of Spencer & Jones (Reference Spencer and Jones1971). Dashed lines in (c) represent results from $M_{c}=0.7$  Pantano & Sarkar (Reference Pantano and Sarkar2002).

Figure 11. (a) Profiles of $T_{rms}^{2}/\unicode[STIX]{x0394}T^{2}$ . (b) Profiles of $\unicode[STIX]{x1D70C}_{rms}^{2}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}^{2}$ . Different colours correspond to different density ratios: black, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; red, $s=8$ . Both magnitudes calculated using Reynolds average.

The shifts in the mean velocity and temperature, as well as the changes in their gradients, are also accompanied by changes in the root mean square of velocity and temperature fluctuations, which are analysed in figures 10 and 11. In particular, figure 10 displays the vertical profiles of the turbulent stress tensor, $\unicode[STIX]{x1D619}_{ij}$ . The plots include the data for the incompressible mixing layer of Rogers & Moser (Reference Rogers and Moser1994), and the experimental results of Bell & Mehta (Reference Bell and Mehta1990) and Spencer & Jones (Reference Spencer and Jones1971). Both datasets show profiles that are consistent with the shape of the present $s=1$ case, although there is considerable scatter between the three datasets. The scatter in $\unicode[STIX]{x1D619}_{12}$ (figure 10 c) is consistent with the scatter in the growth rates of the mixing layers, since these two quantities are related through equation (4.3). This could also explain the scatter in $\unicode[STIX]{x1D619}_{11}$ , $\unicode[STIX]{x1D619}_{22}$ and $\unicode[STIX]{x1D619}_{33}$ for the cases with $s=1$ . As the density ratio increases, $\unicode[STIX]{x1D619}_{ij}$ tend to shift towards the low-density region, following the maximum gradient of $\tilde{u}$ . Interestingly, while the peak values of $\unicode[STIX]{x1D619}_{22},\unicode[STIX]{x1D619}_{12}$ and $\unicode[STIX]{x1D619}_{33}$ decrease with increasing $s$ , the peak values of $\unicode[STIX]{x1D619}_{11}$ seem to remain roughly constant (at least within the uncertainty in the statistics, shown in the figure by the shaded areas around each curve). The high-speed data of Pantano & Sarkar (Reference Pantano and Sarkar2002) are also included in figure 10(c), and they also show a decrease of the peak values of $\unicode[STIX]{x1D619}_{12}$ with increasing $s$ , although for the $M_{c}=0.7$ data the decrease is not monotonic as it is for the present $M_{c}=0$ results. Note also that, as expected, the $M_{c}=0.7$ profiles have lower maximum values, consistent with the lower growth rate of the subsonic mixing layers (as discussed in § 4.2 and in Pantano & Sarkar Reference Pantano and Sarkar2002).

Figure 11 displays the profiles of the variance of the temperature, $T_{rms}^{2}$ , and density, $\unicode[STIX]{x1D70C}_{rms}^{2}$ , normalized with the corresponding jumps across the mixing layer, $\unicode[STIX]{x0394}T=T_{t}-T_{b}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{b}-\unicode[STIX]{x1D70C}_{t}$ , respectively.

For $s=1$ the temperature corresponds to the passive scalar, which exhibits in figure 11(a) the double-peak r.m.s. observed in high-Reynolds-number mixing layers by others (e.g., see Pickett & Ghandhi Reference Pickett and Ghandhi2001). When the density ratio is increased, the peak on the high-density side gradually decreases, while the peak of $T_{rms}^{2}$ on the low-density side shifts with the mean temperature gradients (see figure 9 b). This suggests that $T_{rms}^{2}$ is governed by the mean temperature gradient, in a similar way as $\overline{\unicode[STIX]{x1D619}_{ij}}$ are governed by $\unicode[STIX]{x2202}\tilde{u} /\unicode[STIX]{x2202}y$ . Indeed, consistent with the mean temperature gradients, the peaks of $T_{rms}^{2}/\unicode[STIX]{x0394}T^{2}$ increase with $s$ , except for maybe case $s=8$ . At the present moment, the reason for the non-monotonous behaviour of $s=8$ is unclear. It could be related to a decrease in the value of $Re_{\unicode[STIX]{x1D706}}$ for this case. Another possible explanation could be the onset of interferences of the finite size of the computational domain with the evolution of the mixing layer.

Not surprisingly, the behaviour of $\unicode[STIX]{x1D70C}_{rms}^{2}$ shown in figure 11(b) suggests that $\unicode[STIX]{x1D70C}_{rms}^{2}$ is governed by the mean density gradients (figure 8 b), analogous to the behaviour of temperature and velocity fluctuations. As $s$ increases, the rms around $y/\unicode[STIX]{x1D6FF}_{w}\approx -0.5$ (high-density side) increases, while the fluctuations around $y/\unicode[STIX]{x1D6FF}_{w}\approx 0.5$ (low-density side) decrease. The behaviour is opposite to $\overline{\unicode[STIX]{x1D619}_{ij}}$ (which are more intense near the low-density side), which is consistent with the arguments of Brown & Roshko (Reference Brown and Roshko1974) for the shift and the asymmetry of the growth of the variable-density mixing layers.

Figure 12. (a) Skewness distribution and (b) kurtosis distribution of temperature $\unicode[STIX]{x1D703}$ . (c) Skewness distribution and (d) kurtosis distribution of streamwise velocity $u$ . (e) Skewness distribution and (f) kurtosis distribution of vertical velocity $v$ . Different colours correspond to different density ratios: black, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; and red, $s=8$ .

Finally, figure 12 shows the profiles of the skewness, $S$ , and kurtosis, $K$ , of the temperature and the velocity field. Since these profiles are more noisy than the second-order moments beyond the edge of the mixing layer, figure 12 only shows them in the region limited by 98 % of the free-stream velocity, indicated with vertical dotted lines. For reference, the horizontal dashed lines represent the expected value for a Gaussian distribution, i.e. $S=0$ and $K=3$ . Due to the symmetry of the configuration for the passive scalar case, $s=1$ , we expect an antisymmetric distribution for the skewness and a symmetric distribution for the kurtosis. Deviations from this symmetry in figure 12 are small and provide an impression of the convergence of the statistics. Note also that the almost linear profile of $\overline{\unicode[STIX]{x1D703}}$ in the centre of the mixing layer results in $S_{\unicode[STIX]{x1D703}}\approx 0$ for the case with $s=1$ (recall the broad maximum of the vertical gradient of $\overline{\unicode[STIX]{x1D703}}$ in figure 9).

Carlier & Sodjavi (Reference Carlier and Sodjavi2016) measured the skewness and kurtosis in a spatially developing mixing layer. Their neutral case is comparable to the present passive scalar case. They distinguish between two zones. First, a mixed region in the central part, characterized by a moderate slope of the temperature skewness profile and an almost constant value of all kurtosis profiles. The value of $K$ in this region is somewhat smaller than the Gaussian value. Second, the entrained region in the outer part that presents higher slopes of the temperature skewness profile than the mixed region, and also steep gradients of all kurtosis profiles. All these features are clearly observed in the present profiles for the passive scalar case.

Overall, increasing $s$ results in a shift of the profiles of $S$ and $K$ to the low-density side, for both temperature and velocity. This is especially clear in $S_{u}$ , $S_{v}$ and $K_{v}$ , which show small variations on the shape of the profiles (see figure 12 c,e,f). For the skewness of the temperature (see figure 12 $a$ ) we can observe the same shift, and a gradual increase of $S_{\unicode[STIX]{x1D703}}$ on the high-density half of the central region of the mixing layer. This is probably a consequence of the narrowing of the maximum of $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D703}}/\unicode[STIX]{x2202}y$ with $s$ , and its displacement towards the high-temperature (low-density) side: a sharper edge on the high-temperature side makes it more likely for a pocket of high-temperature fluid to be entrained into the mixing layer, biasing $S_{\unicode[STIX]{x1D703}}$ towards positive values. It is also interesting to observe that, on top of the shifting, $K_{\unicode[STIX]{x1D703}}$ and $K_{u}$ show some changes in their shape with $s$ . In particular, both kurtosis values become larger in the high-density half of the mixing layer ( $y\lesssim 0$ ). This can be interpreted as an increase in the intermittency of $u$ and $T$ , and it suggests that mixing becomes more difficult near the high-density region as $s$ increases, in agreement with the qualitative arguments of Brown & Roshko (Reference Brown and Roshko1974) regarding the reduced velocity fluctuations near the denser stream. As a result, the size of the well-mixed region (i.e., with values of $K$ below the Gaussian threshold) is reduced.

4.5 Turbulence structure

Figure 13. Visualization of $\unicode[STIX]{x1D703}$ on an $xy$ -plane, at the beginning of the self-similar evolution. The corresponding density ratios and times are (a)  $s=1$ , $t\unicode[STIX]{x0394}U/\unicode[STIX]{x1D6FF}_{m}^{0}=400$ ; (b)  $s=2$ , $t\unicode[STIX]{x0394}U/\unicode[STIX]{x1D6FF}_{m}^{0}=418$ ; (c)  $s=4$ , $t\unicode[STIX]{x0394}U/\unicode[STIX]{x1D6FF}_{m}^{0}=455$ ; and (d)  $s=8$ , $t\unicode[STIX]{x0394}U/\unicode[STIX]{x1D6FF}_{m}^{0}=570$ .

Figure 14. Visualization of $\unicode[STIX]{x1D703}$ on an $xz$ -plane at $y=0$ , at the beginning of the self-similar evolution. The corresponding density ratios are (a)  $s=1$ , (b)  $s=2$ , (c)  $s=4$ , (d)  $s=8$ . Times as in figure 13.

Figure 15. Visualization of streamwise velocity on an $xy$ -plane, at the beginning of the self-similar evolution. The corresponding density ratios are (a)  $s=1$ , (b)  $s=2$ , (c)  $s=4$ , (d)  $s=8$ . Times as in figure 13. The black lines show contours of $u=\pm \unicode[STIX]{x0394}U/2$ .

Figure 16. Visualization of streamwise velocity on an $xz$ -plane at $y=0$ , at the beginning of the self-similar evolution. The corresponding density ratios are (a)  $s=1$ , (b)  $s=2$ , (c)  $s=4$ , (d)  $s=8$ . Times as in figure 13.

We provide now visualizations to obtain an impression of the changes in the turbulent structures of the mixing layer induced by the density ratio. Instantaneous fields of the temperature and velocity field are shown using vertical planes (figures 13 and 15 for $\unicode[STIX]{x1D703}$ and $u$ , respectively) and horizontal planes (figures 14 and 16 for $\unicode[STIX]{x1D703}$ and $u$ at the plane $y=0$ , respectively). Note that all visualizations correspond to the same time snapshot and same plane locations. For case $s=1$ , in which the temperature is a passive scalar, the visualization in figure 13(a) shows the typical features of a turbulent mixing layer, with patches of mixed fluid in the central region alternating with patches of unmixed fluid that are entrained from both streams. The presence of quasi-2D rollers is visible in both temperature (figure 13 a) and velocity (figure 15 a) visualizations, but maybe more clearly so in the midplane visualization of the temperature shown in figure 14(a).

On the other hand, the presence of the quasi-2D rollers in the $u$ velocity (figure 15 a) is masked by the formation of more elongated structures, similar to the streaky structures observed in other free and wall-bounded turbulent shear flows (Lee, Kim & Moin Reference Lee, Kim and Moin1990; Flores & Jiménez Reference Flores and Jiménez2010; Sekimoto, Dong & Jiménez Reference Sekimoto, Dong and Jiménez2016).

Increasing the density ratio produces small changes in the flow visualizations. The quasi-2D rollers are also observed for $s=2$ , 4 and 8 in both temperature (figure 13 b–d) and velocity (figure 15 b–d). Also, in agreement with the results discussed in § 4.3, the mixing layer shifts upwards (towards the low-density side) with increasing $s$ , as can be observed in figures 13 and 15. In addition, the temperature field becomes somewhat smoother at small scales. This fact is reflected in the lower value of $Re_{\unicode[STIX]{x1D706}}$ obtained in the cases with large $s$ , as shown in table 1.

The shift of the mixing layer is also apparent in the visualization of the $y=0$ plane shown in figures 14 and 16. With increasing $s$ , the temperature field at this height is increasingly dominated by patches of fluid entrained from the lower stream, while the mean value of the $u$ field drifts to positive values. The footprint of the quasi-2D rollers is also clear in the temperature field (figure 14) for all density ratios, while this footprint becomes less apparent in the $u$ velocity as $s$ increases (figure 16).

Finally, it is interesting to observe in figure 15 that the turbulence within the mixing layer produces irrotational perturbations into the free stream, with characteristic sizes of the order of $\unicode[STIX]{x1D6FF}_{w}$ . This potential perturbations are relatively weak, and are highlighted in figure 15 by contours of $u=\pm \unicode[STIX]{x0394}U$ (in black).

In order to quantify the changes in the structure of the turbulent motions in the mixing layer due to the density ratio, we proceed to analyse the one-dimensional spectra of velocity and temperature fluctuations: $E_{ii}(k_{x},y)$ and $E_{ii}(k_{z},y)$ for $i=u,v$ and $T$ (no summation). These spectra are computed during runtime, as functions of $k_{x}\unicode[STIX]{x1D6FF}_{m}^{0}$ , $k_{z}\unicode[STIX]{x1D6FF}_{m}^{0}$ , $y/\unicode[STIX]{x1D6FF}_{m}^{0}$ and $t$ . Then, during post-processing, these spectra are interpolated into wavenumbers and vertical distances normalized with $\unicode[STIX]{x1D6FF}_{w}(t)$ , and averaged (ensemble and in time) for the self-similar evolution of the mixing layer. The smallest wavenumbers considered in the interpolation are $k_{x}^{0}\unicode[STIX]{x1D6FF}_{w}\approx 0.4{-}0.5$ and $k_{z}^{0}\unicode[STIX]{x1D6FF}_{w}\approx 1.1{-}1.3$ , depending on the density ratio.

Figure 17 shows the premultiplied spectra ( $k_{x}E_{ii}$ and $k_{z}E_{ii}$ ), as a function of the vertical position in the mixing layer and the streamwise or spanwise wavelength, $\unicode[STIX]{x1D706}_{x}=2\unicode[STIX]{x03C0}/k_{x}$ and $\unicode[STIX]{x1D706}_{z}=2\unicode[STIX]{x03C0}/k_{z}$ . The spectra is premultiplied by the wavenumber so that, when plotted in log-scale for the wavelength, the area under the surface corresponds to the actual energy content of a given range of wavelengths. The contours plotted in the figure correspond to 20 % and 40 % of the maxima among all cases, so that they represent equal levels of energy density for all cases. The small inset to the right of each panel shows the energy in wavenumbers smaller than $k_{x}^{0}$ and $k_{z}^{0}$ ,

From a physical point of view, these two quantities roughly correspond to the energy in structures that are infinitely long ( $E_{ii}^{L}$ ) or wide ( $E_{ii}^{W}$ ).

Figure 17. Vertical distribution of the premultiplied spectral energy distribution of velocity and temperature. (a)  $k_{x}E_{TT}(\unicode[STIX]{x1D706}_{x},y)$ . (b)  $k_{z}E_{TT}(\unicode[STIX]{x1D706}_{z},y)$ . (c)  $k_{x}E_{uu}(\unicode[STIX]{x1D706}_{x},y)$ . (d)  $k_{z}E_{uu}(\unicode[STIX]{x1D706}_{z},y)$ . (e)  $k_{x}E_{vv}(\unicode[STIX]{x1D706}_{x},y)$ . (f)  $k_{z}E_{vv}(\unicode[STIX]{x1D706}_{z},y)$ . The inset to the right of each panel shows the energy in wavenumbers not included in the corresponding panel (see text for discussion). The contours plotted correspond to 20 % (solid) and 40 % (dashed) of the maxima of all the spectra shown in each panel. Different colours correspond to different density ratios: black with shading, $s=1$ ; blue, $s=2$ ; green, $s=4$ ; and red, $s=8$ .

For the incompressible case, figure 17 shows that the spectra of $u$ tend to be longer than wide, while the spectra of $v$ and $T$ tend to be wider than long, consistently with the visualizations shown in figures 14 and 16. Indeed, both $T$ and $v$ show considerably more energy on structures that are wide ( $\unicode[STIX]{x1D706}_{z}>2\unicode[STIX]{x03C0}/k_{z}^{0}\approx 5\unicode[STIX]{x1D6FF}_{w}$ ) than in structures that are long ( $\unicode[STIX]{x1D706}_{x}>2\unicode[STIX]{x03C0}/k_{x}^{0}\approx 12\unicode[STIX]{x1D6FF}_{w}$ ), which is shown by $E_{vv}^{W}>E_{vv}^{L}$ and $E_{TT}^{W}>E_{TT}^{L}$ . It is also apparent in figure 17 that the spectra of $v$ is shifted towards smaller scales with respect to the spectra of $u$ and $T$ , both in $\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{z}$ . In terms of the vertical extension of the spectra, figure 17(a,b) show that the temperature spreads over $|y|\lesssim 0.8\unicode[STIX]{x1D6FF}_{w}$ , while $u$ and $v$ are limited to a narrower region ( $|y|\lesssim 0.5\unicode[STIX]{x1D6FF}_{w}$ ), in agreement with the results shown in figure 10. Interestingly, figure 17(e) shows that $E_{vv}$ has a larger spread in the vertical direction, at about $\unicode[STIX]{x1D706}_{x}\approx 4\unicode[STIX]{x1D6FF}_{w}$ . Careful inspection of figure 17(e,f) shows that those peaks correspond to infinitely wide structures ( $k_{z}=0$ ): note that $E_{vv}(\unicode[STIX]{x1D706}_{z},y)$ at $y=0.7\unicode[STIX]{x1D6FF}_{w}$ has little energy (i.e., below the 20 % contour), while $E_{vv}^{W}$ at the same height is still important (i.e., around 50 % of the maximum of $E_{vv}^{W}$ ).

Although not shown here, instantaneous visualizations of $v$ show that these wavelengths ( $\unicode[STIX]{x1D706}_{x}\approx 4\unicode[STIX]{x1D6FF}_{w}$ , $\unicode[STIX]{x1D706}_{z}\rightarrow \infty$ ) roughly correspond to potential perturbations of $v$ into the free stream, analogous to the potential perturbations of $u$ highlighted in figure 15.

As the density ratio increases, figure 17(a,b) show that the spectra of the temperature gradually shifts towards the low-density side (see the contours of 20 % in the figures). The shift occurs first on the high-density edge of the spectrum ( $y<0$ ), and a bit later on the low-density side ( $y>0$ ). Note that for $y/\unicode[STIX]{x1D6FF}_{w}\gtrsim 0.25$ , there are little differences between the spectra of the $s=1$ and $s=2$ cases, consistent with the agreement of $T_{rms}^{2}$ in figure 10(d) in these same vertical locations. In terms of the effect of $s$ in the streamwise and spanwise wavelengths, figure 17(a) shows that the longest scales in the high-density side are gradually inhibited ( $\unicode[STIX]{x1D706}_{x}/\unicode[STIX]{x1D6FF}_{w}\approx 5{-}10,y\approx 0$ ). The same effect, although weaker, is also present in the spanwise wavelengths (figure 17 b). In terms of the small scales, figure 17(a,b) suggest that the effect of $s$ is stronger on $\unicode[STIX]{x1D706}_{z}$ than on $\unicode[STIX]{x1D706}_{x}$ . This could be related to the fact that the small scales in $x$ are not only due to turbulent fluctuations (i.e., vortices), but to the formation of sharp gradients $\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}x$ , due to the roll-up of the shear layer (see blue lines in figures 13 and 14).

The behaviour of the spectra of $u$ and $v$ in figure 17(c–f) is qualitatively similar to that discussed for $T$ , with all spectra shifting towards the low-density side, with a gradual reduction of the energy in small scales (both $\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{z}$ ). There is also a clear reduction of the energy of large scales near the high-density edge of the mixing layer, more apparent for wide ( $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}_{w}\gtrsim 3{-}5$ ) structures than for long structures ( $\unicode[STIX]{x1D706}_{x}/\unicode[STIX]{x1D6FF}_{w}\gtrsim 5{-}10$ ). The $u$ and $v$ spectra of cases $s=1$ and $s=2$ also agree reasonably well near the low-density edge of the mixing layer ( $y\gtrsim 0.25\unicode[STIX]{x1D6FF}_{w}$ ), except for the $v$ spectrum at about $\unicode[STIX]{x1D706}_{x}\approx 4\unicode[STIX]{x1D6FF}_{w}$ , suggesting that even a small change on the density ratio has an important effect on the potential perturbations of the mixing layer into the free stream.