3.1 Characteristics of the mixing layer

Table 1 confirms that requirements were met at this location to obtain self-similar behaviour, i.e. geometric affinity of the statistical quantity profiles and linear growth of the mixing layer thickness. Indeed, Bradshaw (Reference Bradshaw1966) showed that transition typically comes to an end at around $x/\unicode[STIX]{x1D703}_{h}\sim 1000$ , where $\unicode[STIX]{x1D703}_{h}$ is the momentum thickness of the boundary layer on the high-speed side. Moreover, Huang & Ho (Reference Huang and Ho1990) showed that a high Reynolds number ( $Re>10\,000{-}20\,000$ ) and a high pairing parameter ( $\unicode[STIX]{x1D706}(x/L_{0})>8$ , where $\unicode[STIX]{x1D706}$ is the modified velocity ratio and $L_{0}$ the wavelength of the initial structure) are also required to support small-scale turbulence. Figure 5 shows the downstream evolution of the mixing layer thicknesses $\unicode[STIX]{x1D6FF}_{u}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ . The self-similar state was also confirmed by the linear growth of the mixing layer thicknesses shown in this figure and the geometric affinity of the turbulence profiles with an asymptotic behaviour achieved by turbulence intensities (see Ndoye et al. Reference Ndoye, Delville, Heitz and Arroyo2010; Sodjavi & Carlier Reference Sodjavi and Carlier2013). Note again that, in the presence of thermal stratification, that self-similar state would not be valid further downstream because of the growing influence of buoyancy (see § 1.3, Effects of stratification). The spreading rates were higher for the unstably stratified configuration than for the neutrally and stably stratified configurations. In parallel, comparing figure 5(a,b), a classical result can be seen for the ratios between the velocity and temperature spreading rates (0.82 and 0.74 for the stably and unstably stratified configurations, respectively), corresponding to faster spread values for temperature than for velocity.

Figure 5. Linear velocity and temperature spreads of the mixing layer for neutrally, stably and unstably stratified configurations (figure reproduced from Sodjavi & Carlier (Reference Sodjavi and Carlier2013) with permission of Springer). Mixing layer thicknesses $\unicode[STIX]{x1D6FF}_{u}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ and downstream location $x$ relatively to the virtual origin $x_{o}$ of the mixing layer are made dimensionless using the boundary layer momentum thickness on the high-speed side $\unicode[STIX]{x1D703}_{h}$ .

The mixing layer in this type of configuration is not a symmetrical flow. Symmetry with respect to the $y=0$ plane can be obtained for a temporally developing mixing layer, but not for a spatially developing mixing layer, for which there is a slight deviation of the central plane of the mixing layer (or the axis in two dimensional analysis) towards the low-velocity side (see Pope Reference Pope2000, for example). This deflection stems from the greater volumetric flux coming from the high-speed stream (asymmetric entrainment of fluid into the mixing layer) as shown by Dimotakis (Reference Dimotakis1986) and D’Ovidio & Coats (Reference D’Ovidio and Coats2013). Statistical quantities based only on velocity are symmetric if this deflection is taken into account using the similarity variable $\unicode[STIX]{x1D702}_{u}=(y-y_{o})/\unicode[STIX]{x1D6FF}_{u}$ for the ordinate axis, where $y_{o}$ is the half-profile location used to compute the deflection angle (see caption of table 1 for definitions). Even so, dissymmetry remains noticeable as soon as temperature is involved (see figure 6). For instance, the mean longitudinal velocity profile was symmetric with one single inflection point located on the axis of the mixing layer. In contrast, the mean temperature profile had three inflection points (see also Ndoye et al. Reference Ndoye, Delville, Heitz and Arroyo2010). The $\langle u^{\prime }u^{\prime }\rangle$ and $\langle v^{\prime }v^{\prime }\rangle$ were symmetric. However, the peak of $\langle v^{\prime }v^{\prime }\rangle$ often appeared slightly shifted toward the low-velocity side, as reported by Wygnanski & Fiedler (Reference Wygnanski and Fiedler1970). The two peaks of $\langle \unicode[STIX]{x1D703}^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ were not symmetric, the largest peak being on the low-velocity side. Heat fluxes $\langle u^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ and $\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ , rarely measured and often inferred, are discussed in relation to JPDFs and conditional analysis in the dedicated sections below.

Figure 6. Evolution of turbulence statistics through the mixing layer in the self-similar region for neutrally, stably and unstably stratified configurations. Note: for the unstably stratified configuration, the $y$  axis was turned upside down to make the comparison easier; dimensionless vertical position $\unicode[STIX]{x1D702}_{u}$ then stands for $-\unicode[STIX]{x1D702}_{u}$ in this figure and figures thereafter.

The effects of buoyancy on the mixing layer can be seen through the spreading rates (figure 5) and turbulence quantities (figure 6). In view of the Richardson numbers involved in this study ( $|Ri_{f}|_{max}\simeq 0.02$ ), the mixing layer can be considered to be dominated by shears associated with roller vortices from the Kelvin–Helmholtz instability, and the buoyancy force was very small compared to the main driving forces. In the stably stratified configuration, the curves were close to those of the neutrally stratified configuration and therefore temperature seemed to behave as a passive scalar. In the unstably stratified configuration, such a small buoyancy force appeared to be sufficient to significantly increase turbulence quantities (up to 60 % for the shear stress), spreading rates being affected to a lesser extent (10 % and 20 % for the velocity and temperature spreading rates, respectively, see figure 5). We pointed out in Sodjavi & Carlier (Reference Sodjavi and Carlier2013) that the buoyancy-driven increase in turbulence production involved changes in the equilibrium between the different terms of the transport equation for turbulent kinetic energy, whereas it seemed not to modify the balance in the temperature variance budget, thus leading to a greater direct effect on the temperature spreading rate than on the velocity spreading rate.

3.2 A priori illustration of the topology of mixing

With a view to analysing mixing and entrainment through PDF analysis, an illustration of the organization of the turbulent mixing layer in the self-similar region is proposed as an a priori in figure 7. This illustration is based on previous studies found in the literature representing streamlines of the flow in a moving or fixed reference frame (Coles Reference Coles1981; Broadwell & Breidenthal Reference Broadwell and Breidenthal1982; Lasheras & Choi Reference Lasheras and Choi1988). The spatio-temporal organization of the flow is described as a succession of well-identified events:

1. (i) Entrained fluid from free streams (potential flow with $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0$ and $\unicode[STIX]{x1D735}\times \boldsymbol{u}=0$ ) into the turbulence region bounded by the turbulent/non-turbulent interfaces. Entrained fluid by the entrainment process can be either nibbled at these boundaries or engulfed into the outermost boundaries of the mixing layer under the effect of individual growth or pairing of roller vortices.

2. (ii) Mixed fluid due to the mixing process. Mixed fluid and entrained fluid are mainly located in large roller vortices. In the context of mixing by turbulence, the region of mixed fluid can be seen as a turbulence region, but bearing in mind that heterogeneities exist between ‘mixed fluid’ and ‘unmixed fluid’ associated with entrainment and engulfment.

3. (iii) Fronts of high shear in the braid region, the shear being located between two roller vortices and aligned with the divergent axis of the saddle point (or stagnation point).

Mixed, entrained and engulfed fluids are entities described by Sandham et al. (Reference Sandham, Mungal, Broadwell and Reynolds1988). A measurement probe immersed in the mixing layer is likely to be submitted to a complex time sequence of events corresponding to entrained fluid from the slower stream, the front, entrained fluid from the faster stream and mixed fluid movement. The front is then seen as a temperature jump corresponding to the border crossing between entrained fluid from the slower and faster streams.

Figure 7. Illustration of the organization of the turbulent mixing layer in the self-similar region and in fixed reference frame. The underlying image came from flow visualization by laser tomography (Sobel operator was applied on the image to emphasize turbulence regions and turbulent/non-turbulent interfaces). The solid lines $-\unicode[STIX]{x1D6FF}/2$ and $+\unicode[STIX]{x1D6FF}/2$ correspond to the velocity-based expansion of the mixing layer and the dash-dotted line corresponds to the axis of the mixing layer, with a slight deflection angle toward the low-velocity free stream.

3.3 Skewness and kurtosis

Overall information on PDF shapes can be deduced from the skewness factor  $S=\unicode[STIX]{x1D707}_{3}/\unicode[STIX]{x1D70E}^{3}$ and the kurtosis factor $K=\unicode[STIX]{x1D707}_{4}/\unicode[STIX]{x1D70E}^{4}$ where $\unicode[STIX]{x1D70E}$ is the standard deviation and $\unicode[STIX]{x1D707}_{3}$ and $\unicode[STIX]{x1D707}_{4}$ are the third and fourth moments around the mean, respectively. The skewness factor measures the symmetry of the fluctuations around the mean and the kurtosis factor measures the peaked or flat shape characterizing a deviation from a Gaussian distribution ( $S=0$ and $K=3$ for a Gaussian distribution). Figure 8 gives distributions of these factors computed from $u$ , $v$ and $\unicode[STIX]{x1D703}$ signals across the mixing layer for neutrally, stably and unstably stratified configurations. In the free streams ( $|\unicode[STIX]{x1D702}_{u}|>1$ ), skewness and kurtosis factors were equal to  $0$ and  $3$ , respectively, suggesting a Gaussian distribution of the residual turbulence. The skewness and kurtosis factors values changed suddenly near the border of the mixing layer ( $|\unicode[STIX]{x1D702}_{u}|\simeq 1$ ), more significantly for $S_{\unicode[STIX]{x1D703}}$ and $K_{\unicode[STIX]{x1D703}}$ . These changes resulted from intermittencies between turbulent puffs (mixed fluid) into the free stream and fluid incursions into the mixing layer (entrained fluid). From the free stream, this can be roughly seen as follows:

Figure 8. Skewness and kurtosis for $u$ , $v$ and $\unicode[STIX]{x1D703}$ through the mixing layer in the self-similar region for neutrally, stably and unstably stratified configurations.

1. (i) On the high-velocity side ( $\unicode[STIX]{x1D702}_{u}<0$ ), events that disturbed the flow were fluid particles coming from the ‘turbulence region’. These fluid particles were then hotter ( $\unicode[STIX]{x1D703}^{\prime }>0$ ), slower ( $u^{\prime }<0$ ) and pointing downwards ( $v^{\prime }<0$ ) so that $S_{\unicode[STIX]{x1D703}}>0$ , $S_{u}<0$ and $S_{v}<0$ .

2. (ii) On the low-velocity side ( $\unicode[STIX]{x1D702}_{u}>0$ ), the corresponding fluid particles were colder ( $\unicode[STIX]{x1D703}^{\prime }<0$ ), faster ( $u^{\prime }>0$ ) and pointing upwards ( $v^{\prime }>0$ ) so that $S_{\unicode[STIX]{x1D703}}<0$ , $S_{u}>0$ and $S_{v}>0$ .

The particularly high extreme values of $S_{\unicode[STIX]{x1D703}}$ and $K_{\unicode[STIX]{x1D703}}$ expressed stronger intermittency for temperature than for velocity. Deeper inside the mixing layer ( $|\unicode[STIX]{x1D702}_{u}|<1$ ), progressive deviations from Gaussian distribution reflected the extension of this phenomenon from the corresponding boundaries into the mixing layer. Skewness distributions were not exactly antisymmetric, nor were kurtosis distributions, defined in terms of even power, strictly symmetric. This was particularly noticeable for temperatures near the centre of the mixing layer where the $S_{\unicode[STIX]{x1D703}}$ value was slightly positive, while the $K_{\unicode[STIX]{x1D703}}$ value was $3$ on the high-velocity side but below $3$ on the low-velocity side. All of this is in line with the hypothesis of asymmetry between entrainment of fluid from the high- and low-speed streams into the mixing layer.

Figure 9. Evolution through the mixing layer of the probability density function of temperature $P_{\unicode[STIX]{x1D703}}$ in the self-similar region: left panels (a) and (c) for the stably stratified configuration and right panels (b) and (d) for the unstably stratified configuration. Three-dimensional graphs $P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D702}_{u})$ are shown in top panels (a) and (b) as colour map views and two-dimensional graphs $P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D703})$ are shown in bottom panels (c) and (d) for various $\unicode[STIX]{x1D702}_{u}$ coordinates (red–blue colour palette). The mean temperature profiles are superimposed on the top views.

3.4 Probability density function of temperature

Figure 9 shows the evolution through the mixing layer of the PDF of temperature $P_{\unicode[STIX]{x1D703}}$ in the self-similar region for stably and unstably stratified configurations. For all the views, $(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6E9}_{l})/\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}$ was chosen as abscissa variable and $P_{\unicode[STIX]{x1D703}}$ has been normalized to integrate to one. The multimodal shape of $P_{\unicode[STIX]{x1D703}}$ is striking, with one large ellipsoidal central mode and two pointed lateral modes. The large ellipsoidal central mode corresponds to the temperature distribution in the mixed fluid region where the mixing takes place inside the roller vortices, and in the braid region between these vortices. The pointed lateral modes correspond to surrounding fluid (free stream) entrained into the mixing layer.

Typical PDF shapes involving one central mode and two lateral modes have been described, for different flow conditions, in several published results and a distinction has been made between ‘marching’ and ‘non-marching’ types. If the most probable value of the scalar in the mixed fluid core is independent of the transverse position, the PDF is called non-marching, otherwise the PDF is called marching (or tilted or hybrid, depending on how this value varies across the layer). A non-marching PDF has been attributed to homogeneous mixing by turbulence transport in the roller vortices. The stirring process leads to molecular diffusion at the Kolmogorov scale and produces a uniform scalar distribution across the layer. A marching PDF is due to molecular diffusion taking place directly in the high shear of the braid region, allowing lateral variation in PDF. These two processes are the foundation of the PDF-based mixing models proposed by Broadwell & Breidenthal (Reference Broadwell and Breidenthal1982), Dimotakis (Reference Dimotakis1989) and Broadwell & Mungal (Reference Broadwell and Mungal1991). Although in most studies a high Péclet number ( $Pe=Re\times Sc$ ) suggests that turbulent transport diffusion dominates over molecular diffusion, experimental and numerical results (see compilation in Cortesi et al. Reference Cortesi, Smith, Sigg and Banerjee2001) have shown the PDF shape to be dependent on the Reynolds and Schmidt numbers, and also on the initial conditions the effects of which on the structure of the flow persist far downstream. Non-marching PDFs (Konrad Reference Konrad1977; Mungal & Dimotakis Reference Mungal and Dimotakis1984; Koochesfahani & Dimotakis Reference Koochesfahani and Dimotakis1986; Masutani & Bowman Reference Masutani and Bowman1986; Pickett & Ghandhi Reference Pickett and Ghandhi2001; Wiltse & Glezer Reference Wiltse and Glezer2004) and marching PDFs (Batt Reference Batt1977; Karasso & Mungal Reference Karasso and Mungal1996; Pickett & Ghandhi Reference Pickett and Ghandhi2002; Meyer et al. Reference Meyer, Dutton and Lucht2006) have often been described in the literature, but a clear distinction between the shapes has still to be established. In figure 9, the most probable value did not follow the mean temperature profile and scrolled very slightly from the high-velocity side to the low-velocity side, giving a non-marching PDF for this experiment conducted at a high Reynolds number, low Schmidt number and beyond the mixing transition. This non-marching PDF suggests that, in this experiment, roller vortices were coherent and turbulent enough to promote the stirring process while preventing non-mixed fluid penetrating into the core of the mixing layer, at least from a statistical point of view.

There was no visible difference between the PDF distributions of the stably and unstably stratified configurations. The most probable value of the mixed fluid was approximately $(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6E9}_{l})/\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}\simeq 0.4$ . The temperature in the roller vortices was therefore colder than the average of the two free-stream temperatures. According to Koochesfahani & Dimotakis (Reference Koochesfahani and Dimotakis1986) and Pickett & Ghandhi (Reference Pickett and Ghandhi2002), asymmetry of the mixed fluid composition is not only due to preferential entrainment from the faster stream but also to the initial composition of the first rolling, with excess from the faster stream (as we chose mean vertical velocity and temperature gradients of opposite sign for both the stably and unstably stratified configurations, high speed was always associated with low temperature). However, we can assume that this initial contribution so far away from the trailing edge of the splitter plate would be small compared to the continuous entrainment process in the self-similar region, this process then being clearly asymmetric.

The asymmetry and the bimodal shape on both sides of the axis of the mixing layer were consistent with the previously shown turbulence statistics. First, the temperature difference between entrained and mixed fluids was higher on the low-speed side than on the high-speed side, and moreover temperature variations were preferentially a consequence of this temperature jump between entrained and mixed fluids, rather than of the temperature fluctuations in the roller vortices. This explains, in particular, the large peak on the low-velocity side of the temperature variance curve in figure 6(g).

3.5 Joint probability density functions

Variable-temperature hot-wire thermo-anemometry was developed to measure both velocity and temperature of the flow instantaneously. The joint nature of the measurement was used in a previous study (Sodjavi & Carlier Reference Sodjavi and Carlier2013) to evaluate the heat fluxes involved in the production of temperature fluctuation and buoyancy, and then to calculate and compare the profiles of the separate terms of the transport equations for turbulent kinetic energy and temperature variance. In the present study, we benefited from the joint measurements to obtain JPDFs of velocity and temperature, with a view to obtaining information on the respective influences of the movements in the mixing process corresponding to entrained and mixed fluids. It is of note that quantitative measurements of heat or scalar fluxes in turbulent mixing layers are very rare. We only found them in Koochesfahani, Cohn & MacKinnon (Reference Koochesfahani, Cohn and MacKinnon2000), Li, Balaras & Wallace (Reference Li, Balaras and Wallace2010b ), Ndoye et al. (Reference Ndoye, Delville, Heitz and Arroyo2010) and Sodjavi & Carlier (Reference Sodjavi and Carlier2013), and only Li et al. (Reference Li, Balaras and Wallace2010b ) focused their study on JPDFs.

Figure 10. Normally distributed JPDFs associated with the distributions of $(u^{\prime },v^{\prime })$ through the mixing layer in the self-similar region for stably and unstably stratified configurations.

Figure 11. Normally distributed JPDFs associated with the distributions of $(u^{\prime },\unicode[STIX]{x1D703}^{\prime })$ through the mixing layer in the self-similar region for stably and unstably stratified configurations.

Figure 12. Normally distributed JPDFs associated with the distributions of $(v^{\prime },\unicode[STIX]{x1D703}^{\prime })$ through the mixing layer in the self-similar region for stably and unstably stratified configurations.

Figures 10, 11 and 12 present the JPDFs $P_{u^{\prime },v^{\prime }}$ , $P_{u^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ at five locations across the mixing layer in the self-similar region for stably and unstably stratified configurations. Each small illustration gives a two-dimensional view of the different JPDFs for a given transverse  $y$ location measured in proportion to the velocity-based thickness of the mixing layer $\unicode[STIX]{x1D6FF}_{u}$ . The JPDFs are normally distributed with mean values $\langle u\rangle$ , $\langle v\rangle$ or $\langle \unicode[STIX]{x1D703}\rangle$ and standard deviations $\unicode[STIX]{x1D70E}_{u}$ , $\unicode[STIX]{x1D70E}_{v}$ or $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D703}}$ . To facilitate a step by step analysis, we chose to divide each of them into only quadrants identified by numbers (i for northeast, ii for northwest, iii for southwest and iv for southeast) as proposed by Wallace, Eckelmann & Brodkey (Reference Wallace, Eckelmann and Brodkey1972) and Willmarth & Lu (Reference Willmarth and Lu1972), while aware that the octant decomposition proposed by Vinçont et al. (Reference Vinçont, Simoëns, Ayrault and Wallace2000) could have been used to consider the three-dimensional JPDFs $P_{u^{\prime },v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ .

It can be seen that near the velocity-based border of the mixing layer ( $|\unicode[STIX]{x1D702}_{u}|\simeq 1$ ), $P_{u^{\prime },v^{\prime }}$ had a circular shape centred on the origin, while $P_{u^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ had an elliptical shape with the minor axis aligned with the $\unicode[STIX]{x1D703}^{\prime }$ axis and the major axis aligned with the $u^{\prime }$ or $v^{\prime }$ axis. Note that, as these JPDFs were normally distributed, their shapes only translated the correlations and distributions of $u^{\prime }$ , $v^{\prime }$ and $\unicode[STIX]{x1D703}^{\prime }$ and not the intensity of their fluctuations. The symmetry of these JPDFs about the $\unicode[STIX]{x1D703}^{\prime }$  axis and the $u^{\prime }$ or $v^{\prime }$  axis corresponds to patterns associated with low correlation levels at this location (see figure 6). The circular shape of $P_{u^{\prime },v^{\prime }}$ corresponds with residual turbulence in the free stream where PDFs of $u^{\prime }$ and $v^{\prime }$ were Gaussian without significant Reynolds stress. The elliptical shape of $P_{u^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ was derived from the temperature distribution at this location situated within the temperature-based border of the mixing layer (the expansion rate was greater for the temperature than for velocity). At this location, the PDFs of $\unicode[STIX]{x1D703}^{\prime }$ were non-Gaussian with a flatness factor for temperature much higher than the flatness factor for velocity (see figure 8).

Near the axis of the mixing layer ( $|\unicode[STIX]{x1D702}_{u}|\leqslant 0.2$ ), $P_{u^{\prime },v^{\prime }}$ had an elliptical shape with a major axis oriented toward quadrants i and iii. This result agrees very well with $P_{u^{\prime },v^{\prime }}$ patterns presented in Li, Chang & Wang (Reference Li, Chang and Wang2010a ). $P_{u^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ had a completely different shape, with a centred mode corresponding to the velocity–temperature distribution in the mixed fluid and two lateral modes, located in quadrants ii and iv, corresponding to entrained fluid from the low- and high-speed streams.

A break-down analysis can be proposed as follows:

1. (i) Fluid entrained from the low-speed stream was hotter ( $\unicode[STIX]{x1D703}^{\prime }>0$ ), slower ( $u^{\prime }<0$ ) and pointing downward ( $v^{\prime }<0$ ), corresponding to the lateral mode in quadrant ii for $P_{u^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ .

2. (ii) Fluid entrained from the high-speed stream was colder ( $\unicode[STIX]{x1D703}^{\prime }<0$ ), faster ( $u^{\prime }>0$ ) and pointing upward ( $v^{\prime }>0$ ), corresponding to the lateral mode in quadrant iv for $P_{u^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime },\unicode[STIX]{x1D703}^{\prime }}$ .

The privileged orientations of $P_{u^{\prime },v^{\prime }}$ , $P_{u^{\prime }\unicode[STIX]{x1D703}^{\prime }}$ and $P_{v^{\prime }\unicode[STIX]{x1D703}^{\prime }}$ were consistent with positive production terms in transport equations for turbulent kinetic energy and temperature variance:

1. (i) Production of turbulence ( $-\langle u^{\prime }v^{\prime }\rangle (\unicode[STIX]{x2202}\langle u\rangle /\unicode[STIX]{x2202}y)>0$ ) with a negative velocity gradient ( $(\unicode[STIX]{x2202}\langle u\rangle /\unicode[STIX]{x2202}y)<0$ ) gave a positive transverse mass flux ( $\langle u^{\prime }v^{\prime }\rangle >0$ ).

2. (ii) Production of temperature variance ( $-\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle (\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D703}\rangle /\unicode[STIX]{x2202}y)>0$ ) with a positive temperature gradient ( $(\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D703}\rangle /\unicode[STIX]{x2202}y)>0$ ) gave a negative transverse heat flux ( $\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle <0$ ).

3. (iii) Positive mass flux and negative heat flux ( $\langle u^{\prime }v^{\prime }\rangle >0$ and $\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle <0$ ) gave a negative longitudinal heat flux ( $\langle u^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle <0$ ).

JPDF analysis based on quadrant decomposition thus indicated that movements of entrainment into the mixing layer contributed significantly to heat and mass fluxes. In the following section, we propose to go further in discriminating between entrained and mixed fluid within these fluxes by using a conditional analysis.

3.6 Conditional analysis

One way to separate the contributions of mixing and entrainment in the heat and mass fluxes is to perform conditional analysis. The approach we chose was to use the instantaneous value of temperature as a tag value to differentiate particles according to their origin (hot, cold or mixed region) and to apply a conditional analysis to the measurements affected by such a tag, or indicator function. The multimodal shape of the PDFs of temperature (see figure 9), with clear valleys and peaks, allowed simple thresholding to construct an indicator function (Antonia Reference Antonia1981; Prasad & Sreenivasan Reference Prasad and Sreenivasan1989), where two thresholds could be set to distinguish between mixed fluid, entrained fluid from the high-velocity side and entrained fluid from the low-velocity side:

1. (i) The first threshold $(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6E9}_{l})/\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}=0.1$ selected was the valley point between the centred mode (high probability of mixed fluid) and the left lateral mode (high probability of entrained fluid from the high-velocity side).

2. (ii) The second threshold $(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6E9}_{l})/\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}=0.9$ selected was the valley point between the centred mode (high probability of mixed fluid) and the right lateral mode (high probability of entrained fluid from the low-velocity side).

The indicator functions were then:

with $I_{m}$ for fluid parcels tagged as mixed fluid, $I_{h}$ for fluid parcels tagged as entrained fluid from the high-velocity side and $I_{l}$ for fluid parcels tagged as entrained fluid from the low-velocity side. Morphological filtering (opening and closing) was applied on the indicator functions to remove single detections due to the overlap of the modes while preserving $I_{h}+I_{m}+I_{l}=1$ . Due to the selection of temperature for the indication function, no differentiation was made between entrained fluid and engulfed fluid (engulfed fluid is the fluid entrained intermittently into the two outermost turbulent/non-turbulent boundaries of the mixing layer). The region of fluid mixed by turbulence can be considered or not as a turbulence region, depending on whether or not engulfed fluid is considered as turbulent. Such an indicator function can therefore be an alternative to a velocity-based indicator for marking the position of the turbulent/non-turbulent interfaces, giving a slightly different light on the separation.

The conditional expectation of $X$ given the event $I$ is defined by:

and in this paper is simply noted $\langle X\rangle _{\mid I}$ . The conditional covariance of $X$ and $Y$ , given $I$ , is defined by:

and simply noted $\langle X^{\prime }Y^{\prime }\rangle _{\mid I}$ .

As the indicator functions $I$ are binary, the conditional expectation and covariance of $X$ and $Y$ to $I$ defined by (3.2) and (3.3) can be regarded as expectation and covariance of $X$ and $Y$ in fluid parcels tagged by $I$ . The expectation of $X$ can be broken down into three terms:

and can be considered as the weighted sum of the conditional expectations using their respective contributions. Similarly the covariance of $X$ and $Y$ is given by:

where the weighted sum of the conditional covariances is supplemented by cross-products between conditional expectations.

Figures 13 and 14 present the conditional expectations and covariances of velocity and temperature across the mixing layer, given the indicator functions for the mixed fluid $I_{m}$ and for the entrained fluid from both sides $I_{h}$ and $I_{l}$ . Expectations of these indicator functions can be viewed as probabilities of presence of these events and are also given in figure 13(a).

The profiles of these probabilities of presence ( $\langle I_{m}\rangle$ , $\langle I_{h}\rangle$ and $\langle I_{l}\rangle$ in figure 13 a) behaved similarly to those computed from concentration data in Konrad (Reference Konrad1977) and Koochesfahani & Dimotakis (Reference Koochesfahani and Dimotakis1986). The mixed fluid probability $\langle I_{m}\rangle$ was bell shaped and symmetrical with regard to the mixing layer axis. It is equivalent to the intermittency factor (flatness factor of the velocity time derivative) computed by Wygnanski & Fiedler (Reference Wygnanski and Fiedler1970) and Spencer & Jones (Reference Spencer and Jones1971), i.e. the ratio between the observed turbulence time and the total observation time. The entrained fluid probabilities $\langle I_{h}\rangle$ and $\langle I_{l}\rangle$ decreased inside the mixing layer so that there was low probability of finding unmixed fluid at the centre of the mixing layer (approximately 8 %). This suggests very active nibbling at the border of the entrained region which allowed only a small proportion of entrained fluid to reach the centre of the mixing layer. Neither imbalance between the entrained fluid from the high- and low-velocity sides nor differences between the stably and unstably stratified configurations could be observed from these curves of probability of presence.

Figure 13. Expectation of the indicator functions for the mixed fluid $I_{m}$ and for the entrained fluid from both sides $I_{h}$ and $I_{l}$ and conditional expectations for the velocity and temperature (profiles across the mixing layer in the self-similar region for stably and unstably stratified configurations).

Figure 14. Conditional covariances for the velocity and temperature (profiles across the mixing layer in the self-similar region for stably and unstably stratified configurations).

The conditional expectations of temperature $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{h}}$ and $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{l}}$ shown in figure 13(b) represent the expectations of temperature in the entrained fluid. As expected, the normalized values were constant across the layer, $0$ for the entrained cold fluid from the high-speed stream and $1$ for the entrained hot fluid from the low-speed stream. The conditional expectation of temperature $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{m}}$ represents the expectation of temperature in the mixed fluid. The profiles across the layer were fairly flat and homogeneous, as was the conditional variance of temperature $\langle \unicode[STIX]{x1D703}^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle _{\mid I_{m}}$ in figure 14(e). These behaviours have already been described by Fiedler (Reference Fiedler1974), Koochesfahani & Dimotakis (Reference Koochesfahani and Dimotakis1986) and Karasso & Mungal (Reference Karasso and Mungal1996), indicating an active stirring process in the turbulent roller vortices before molecular diffusion, as expected with a high Péclet number. As observed in figure 9, these results were consistent with non-marching PDFs.

The conditional expectation of temperature $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{m}}$ averaged over the layer can provide an estimate of the entrainment ratio  $E=Q_{h}/Q_{l}$ , which is the ratio between the fractional flow rates $Q_{h}$ and $Q_{l}$ originating from the high- and low-velocity streams, respectively, and entrained into the mixing layer. Indeed, the entrainment balance equation yields:

(n.b. cold air $\unicode[STIX]{x1D6E9}_{l}$ blown on the high-velocity side $U_{h}$ and hot air $\unicode[STIX]{x1D6E9}_{h}$ blown on the low-velocity side $U_{l}$ ), leading to:

The entrainment balance is simply depicted in Dimotakis (Reference Dimotakis1991) as a bucket filled by hot and cold water faucets running with unequal flow rates, the average temperature in the bucket being only a function of the ratio of the two flow rates. The average value of $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{m}}$ integrated across the layer was approximately $0.43$ in the stably stratified configuration and 0.47 in the unstably stratified configuration, giving entrainment ratios of approximately 1.38 and 1.17, respectively. This greater entrainment from the high-speed stream was also found in the literature, namely 1.58 by Fiedler (Reference Fiedler1974), $1.32$ by Koochesfahani & Dimotakis (Reference Koochesfahani and Dimotakis1986) and 1.37 by Karasso & Mungal (Reference Karasso and Mungal1996).

The conditional expectations of the streamwise velocity are shown in figure 13(c). The expectations of the streamwise velocity in the entrained fluid $\langle u\rangle _{\mid I_{h}}$ and $\langle u\rangle _{\mid I_{l}}$ tended to retain the free-stream velocity value, as in Spencer & Jones (Reference Spencer and Jones1971). In contrast to the conditional expectation of the temperature $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{m}}$ , the conditional expectation of the streamwise velocity $\langle u\rangle _{\mid I_{m}}$ followed the mean streamwise velocity in the mixing layer. Indeed, the turbulent roller vortices trap the engulfed fluid and a significant part of the mixed fluid. The thresholding we made isolates the mixed fluid and associates it with $I_{m}$ while engulfed fluid remains with particles contributing to $I_{l}$ and $I_{h}$ . The conditional expectations in the mixed fluid account for the active stirring process in the turbulent roller vortices, leading to homogeneous distribution of the conditional expectation of the temperature in the mixed fluid $\langle \unicode[STIX]{x1D703}\rangle _{\mid I_{m}}$ and, with the rotation of the roller vortices, leading to the evolution of the conditional expectation of the streamwise velocity in the mixed fluid $\langle u\rangle _{\mid I_{m}}$ from $U_{l}$ at the low-speed side to $U_{h}$ at the high-speed side. Let us note that this describes only the mixing and movement of mixed fluid but engulfed fluid may also be present in the turbulent roller vortices, adding some heterogeneity or intermittency in the covered region.

The conditional expectations of transverse velocity are shown in figure 13(d). Entrained fluid dived into the mixing layer faster and faster because of entrainment by roller vortices and the values were higher for entrained fluid from the high-speed stream due to greater momentum. The expectation of the transverse velocity in the mixed fluid was negative for $\unicode[STIX]{x1D702}_{u}<0$ and positive for $\unicode[STIX]{x1D702}_{u}>0$ , in line with the expansion of the mixing layer.

Moving on to the conditional variance profiles in figure 14, for $\langle u^{\prime }u^{\prime }\rangle$ , $\langle u^{\prime }v^{\prime }\rangle$ and $\langle v^{\prime }v^{\prime }\rangle$ in figure 14(a–c), the values of the conditional variances to $I_{h}$ and $I_{l}$ (i.e. relating to the entrained fluid) were much lower than the respective total variances (in black) and values decreased from the mixing layer axis to the borders. Indeed, the velocity variances in the entrained fluid were both produced by velocity fluctuations inside these regions and by variability of the velocity of successive entrainment events from one event to another. In the same figure, the values of the conditional variances to $I_{m}$ (i.e. related to the mixed fluid) and the total variances were similar near the mixing layer axis, suggesting negligible effects of intermittency between entrained and mixed fluids on the fully velocity-based variances. These results can be clarified by applying (3.5) near the mixing layer axis in which the first term was governed by the conditional variances to $I_{m}$ ( $\langle I_{m}\rangle =0.92$ and $\langle I_{h}\rangle +\langle I_{l}\rangle =0.08$ ) and the second term was negligible due to the small difference between the expected velocities in the central regions (see figure 13 c).

For $\langle u^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ , $\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ and $\langle \unicode[STIX]{x1D703}^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ in figure 14(d–f), the conditional variances to $I_{h}$ and $I_{l}$ were very low. This result was consistent as it was related to entrained fluid which carried unmixed fluid with a temperature remaining near that of the free stream from which it came. In the same figure, the profiles of the conditional variances to $I_{m}$ were more homogeneous than for the total variances. In contrast to the fully velocity-based variances, there was a small gap between the conditional variances to $I_{m}$ for $\langle u^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ , $\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ and $\langle \unicode[STIX]{x1D703}^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ and the total variances near the mixing layer axis. The second term in (3.5) contributed significantly to the total variances here due to the high levels of difference between the expected temperatures in the entrained fluids (see figure 13 b). As there was very little temperature fluctuation in the entrained fluid, the temperature variance resulted mainly from the homogeneous turbulence in the mixed fluid and the level of temperature difference between mixed and entrained fluids. As already described in § 3.4, the double peak of $\langle \unicode[STIX]{x1D703}^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ in figure 14(e) (also displayed on figure 6 g) was due to the temperature jumps that occurred at the fronts, and thus the larger peak on the low-velocity side can be explained by the greater temperature difference between entrained and mixed fluids on the low-speed side (around $0.45\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}$ ) than on the high-speed (low-temperature) side (around $0.35\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}$ ).

In conclusion, this kind of conditional analysis gives a more complete view of the separate contributions of the entrainment and mixing to the velocity and temperature fluctuations and their expectations and covariances.

3.7 Effects of buoyancy on entrainment velocity

Entrainment velocity was defined by Townsend (Reference Townsend1976, p. 231) as the mean rate of formation of turbulent fluid in a mixing layer:

By using the Leibniz integral rule and the similarity variable $\unicode[STIX]{x1D702}_{u}$ , this definition becomes:

It should be noted that the velocity-based spreading rate ${\unicode[STIX]{x1D6FF}^{\prime }}_{u}=(\text{d}\unicode[STIX]{x1D6FF}_{u}/\text{d}x)$ is then proportional to the entrainment velocity. According to Brown (Reference Brown1974) and Konrad (Reference Konrad1977), entrainment velocity can also be broken down into the sum of the entrainment velocities across the two boundaries between non-turbulent and turbulent regions:

with

where $Ve_{l}$ is the entrainment velocity from the low-velocity side and $Ve_{h}$ the entrainment velocity from the high-velocity side. The scalar quantity $\unicode[STIX]{x1D709}$ transported by the flow makes it possible to discriminate between entrainments from the low- and high-velocity sides. It can be $(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6E9}_{l})/\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}$ for heat flux or $(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{l})/\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ for mass flux.

In many studies (e.g. Brown Reference Brown1974; Masutani & Bowman Reference Masutani and Bowman1986; Zhou & Pereira Reference Zhou and Pereira2000; D’Ovidio & Coats Reference D’Ovidio and Coats2013), simplifications were used to cope with the lack of turbulence detection and joint measurements $\unicode[STIX]{x1D709}\cdot u$ . Integrations were limited to the boundaries (based on mean quantities) and the mean of the product was estimated by the product of the means. In the present study, no simplification was necessary but entrainment velocity was related to the mean rate of formation of mixed fluid rather than turbulent fluid, and hence $I_{t}$ was replaced by $I_{m}$ in (3.11) and (3.12).

Table 3 gives entrainment velocities and their sums and ratios in the self-similar region for stably and unstably stratified configurations. Buoyancy amplified entrainment from both sides in the unstably stratified configuration, the increase being higher on the low-velocity side, while the total entrainment velocity $Ve$ was increased in the same proportion as the velocity-based spreading rate ${\unicode[STIX]{x1D6FF}^{\prime }}_{u}$ was (see table 1). The entrainment ratios (1.4 and 1.2) appeared to confirm the estimate obtained by (3.7) (1.38 and 1.17, respectively) and its dependence on buoyancy. The lower asymmetry for the unstably stratified configuration is consistent with the lower deflection angle (see table 1).

Dimotakis (Reference Dimotakis1986) proposed a model based on the work of Brown (Reference Brown1974) to predict the volumetric flux entrainment ratio $E_{Ve}$ , in addition to the vortex convection velocity $U_{v}$ and the velocity-based spreading rate ${\unicode[STIX]{x1D6FF}^{\prime }}_{u}$ :

where $r=(U_{l}/U_{h})$ is the velocity ratio, ${\unicode[STIX]{x1D6FF}^{\prime }}_{\unicode[STIX]{x1D714}}$ the vorticity-based spreading rate (with $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}=0.91\unicode[STIX]{x1D6FF}_{u}$ ) and $s=(\unicode[STIX]{x1D70C}_{h}/\unicode[STIX]{x1D70C}_{l})$ the density ratio.

Through this model, Dimotakis found a way to introduce the asymmetry between the entrainments in the case of a spatially developing mixing layer, by considering a differential in fluid entrainment between the upstream and downstream sides of a roller vortex, the differential being related to the expansion of the mixing layer. The Dimotakis models also introduced density gradient effects through dependence on mean roller vortex spacing to position ratio $l/x$ and the density ratio function $\sqrt{s}$ . In case of density gradient effects due to unsteadiness-induced acceleration, he assumed the ratio between the mean roller vortex spacing and the position to be $l/x=3.9{\unicode[STIX]{x1D6FF}^{\prime }}_{\unicode[STIX]{x1D714}}$ (Koochesfahani et al. Reference Koochesfahani, Dimotakis, Catherasoo, Gharib and Lang1979) with ${\unicode[STIX]{x1D6FF}^{\prime }}_{\unicode[STIX]{x1D714}}=0.17(1-r)/(1+r)$ or $0.17\unicode[STIX]{x1D706}$ (Brown & Roshko Reference Brown and Roshko1974), providing the $0.68$ coefficient in the expression of $E_{Ve}$ (3.13). Soteriou & Ghoniem (Reference Soteriou and Ghoniem1995) stated that the dependence on $\sqrt{s}$ , initially quasi-empirically assumed by Brown (Reference Brown1974) and later based on Bernoulli’s equation applied along a line through the saddle point by Dimotakis (Reference Dimotakis1986), provides coarse modelling of the effects of the baroclinic torque on entrainment (see § 1.3, Effects of stratification). These effects mainly take place through the generation, amplification and damping of eddy structures, on intermediate scales, similar to those of rib vortices. Together with gravity-induced acceleration, by its construction (3.13) the model integrates buoyancy effects, but only through the large-scale topological length values (spacing and expansion of roller vortices).

Figure 15. Volumetric flux entrainment ratio $E_{Ve}$ against density ratio  $s$ for three velocity ratios  $r$ . The solid lines are the predictions of the Dimotakis model, while the symbols are results from numerical simulations or experiments.

Figure 15 shows that the volumetric flux entrainment ratio for a mixing layer with or without an unsteadiness-induced acceleration was fairly well predicted by the Dimotakis model for a wide range of density ratios, including experimental results from Brown (Reference Brown1974) and Konrad (Reference Konrad1977) and numerical results from Soteriou & Ghoniem (Reference Soteriou and Ghoniem1995). It should be noted however that the excellent agreement between the Dimotakis model and the results of Konrad (Reference Konrad1977) and Soteriou & Ghoniem (Reference Soteriou and Ghoniem1995) was probably biased by adjustment artefacts, bearing in mind that Dimotakis constructed his model using results from Konrad (Reference Konrad1977), and that Soteriou & Ghoniem (Reference Soteriou and Ghoniem1995) corrected their results to take into account confinement in their numerical simulations. In the study presented here, the mixing layer configurations were all characterized by $r=0.5$ and $s=0.93$ but led to different values for the mean roller vortex spacing and spreading rate, due to differences in buoyancy magnitude. For the neutrally stratified configuration, the Dimotakis model provided $E_{Ve}=1.21$ . For the stably and unstably stratified configurations with both unsteadiness- and gravity-induced acceleration, the Dimotakis model provided $E_{Ve}=1.19$ and 1.23, respectively, which were still different from the measured values 1.4 and 1.2 (see table 3).

As the neutrally and stably stratified configurations of the mixing layer in this study provided similar characteristics (see table 1) and turbulence profiles (see figures 5, 6 and 8), we conclude that the density gradient effects were too weak to be observed in that particular case of weak stably stratified condition. We do not consider that the gap between our measured values and the Dimotakis model prediction for the neutrally and stably stratified configurations invalidates the model per se, bearing in mind that the gap may have been at least partially due to differences in the initial conditions and in the development length of the mixing layer (Dimotakis himself suggested such dependence), but an inadequacy of the model emerges in that it could not predict the buoyancy effect on the unstably stratified configuration. The inability to predict the buoyancy effect mainly shows that the modifications in the large-scale topological lengths (spacing and expansion of roller vortices) are not sufficient to capture and model the difference in asymmetry that buoyancy induces. The modifications in smaller-scale eddy structures are relatively significant, as shown by Piccirillo & Van Atta (Reference Piccirillo and Van Atta1997) in the case of uniform shear and stratification and Nagata & Komori (Reference Nagata and Komori2000) in the case of localized shear and stratification. To improve the model prediction, the modifications in the smaller-scale eddy structures should be better taken into account, in addition to the unsteadiness- and gravity-induced effects on roller vortices.

In the present configuration of a stratified mixing layer dominated by shear, the entrainment process results from both engulfing by large roller vortices (individual growth and pairing process) and nibbling by intermediate-scale eddy structures. In unstably stratified conditions with ${Ri_{f}}_{min}\simeq -0.02$ , we can assume that the lower entrainment ratio measured can be attributed to enhanced nibbling as compared with engulfing. In stably stratified conditions with ${Ri_{f}}_{max}\simeq 0.02$ , it is of note that, in our experiment, almost no difference appeared between the neutrally and stably stratified configurations, but higher Richardson numbers would probably provide measurable effects of buoyancy on the entrainment process. This could then be observed further downstream, as the Richardson number grows with  $x$ , in a longer wind tunnel. We therefore conclude that, for flow configurations where buoyancy significantly acts on the entrainment and turbulent mixing processes, at least some complement to the Dimotakis model (or even a new model) should be devised from more complete experiments, to account for the separate effects of buoyancy on engulfing and nibbling.