5.1 Numerical resolution and spectra at transition

A crucial aspect in numerical experiments is to ensure that the computational aspects are conducive to exploring the physics under investigation. To this purpose, we inspect the resolution and the spectra at the transitional time.

Turbulent flows are characterized by the activity of a wide range of scales, the smallest dynamic ones being related to the turbulent kinetic energy dissipation $\langle \unicode[STIX]{x1D700}\rangle \equiv \langle \unicode[STIX]{x1D61B}_{ij}^{\prime }\unicode[STIX]{x2202}u_{i}^{\prime }/\unicode[STIX]{x2202}x_{j}\rangle$ , where $\langle \,\rangle$ symbolizes averages over homogeneous $(x_{1},x_{3})$ planes and $(\cdot )^{\prime }$ denotes fluctuations from the average. Generally, it is possible to associate a characteristic length scale with the dynamic, the thermal-diffusion and the mass-diffusion phenomena. A measure of the smallest dynamic active scale is provided by the Kolmogorov scale, defined as (e.g. Jagannathan & Donzis Reference Jagannathan and Donzis2016)

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{K}=\left(\frac{\langle \unicode[STIX]{x1D707}\rangle ^{3}}{\langle \unicode[STIX]{x1D70C}\rangle ^{2}\langle \unicode[STIX]{x1D700}\rangle }\right)^{1/4}.\end{eqnarray}$$

Thermal processes are related to the Prandtl number $Pr$ , which measures the importance of momentum transfer with respect to heat transfer. An estimate of the smallest thermal length scale is provided by $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D702}_{K}Pr^{-0.5}$ (Goto & Kida Reference Goto and Kida1999), which suggests that, at locations where $Pr>1$ , thermal phenomena involve scales smaller than  $\unicode[STIX]{x1D702}_{K}$ . Similarly, mixing may also occur at scales smaller than  $\unicode[STIX]{x1D702}_{K}$ ; for reactive flows, resolving the mixing scales is of paramount importance since species can react only if they are brought together. The Schmidt number, which measures the importance of momentum transfer with respect to species-mass transfer, can provide an estimate of the mixing scales. For a binary species mixture the mixing scale is the Batchelor scale, $\unicode[STIX]{x1D702}_{B}$ , related to $\unicode[STIX]{x1D702}_{K}$ by $\unicode[STIX]{x1D702}_{B}=\unicode[STIX]{x1D702}_{K}\,Sc^{-0.5}$ , but it is noted that then $Sc$ is only defined for a single species mass-diffusion coefficient. Masi et al. (Reference Masi, Bellan, Harstad and Okong’o2013) circumvented this problem for a mixture of several species modelled utilizing a complete mass-diffusion matrix, by defining an effective species-specific Schmidt number, $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ , which accounts for all relevant species’ mass-diffusion coefficients, for $D_{T,\unicode[STIX]{x1D6FC}}$ and $D_{p,\unicode[STIX]{x1D6FC}}$ of (2.6) and for the state of the flow represented by $\unicode[STIX]{x1D735}T$ , $\unicode[STIX]{x1D735}p$ and $\unicode[STIX]{x1D735}Y_{\unicode[STIX]{x1D6FC}}$ . The corresponding effective Prandtl number, $Pr_{\mathit{eff}}$ , is defined in Masi et al. (Reference Masi, Bellan, Harstad and Okong’o2013) using the same strategy.

Figure 1 shows the cross-stream profile of $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K}$ and $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}$ at $t_{tr}^{\ast }$ for the $Re=2000$ cases, and for the two-species $Re=1000$ and $Re=3500$ cases; $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}$ was computed as $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D702}_{K}\langle Pr_{\mathit{eff}}\rangle ^{-0.5}$ using the model of Masi et al. (Reference Masi, Bellan, Harstad and Okong’o2013) to compute $Pr_{\mathit{eff}}$ . The resolution is assessed at $t_{tr}^{\ast }$ because the core of the analysis is performed at this time; moreover, $t_{tr}^{\ast }$ is the time of most severe grid requirements since $\unicode[STIX]{x1D702}_{K}$ increases monotonically past this station. The results show that $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K}\lesssim O(1)$ , indicating that all the relevant turbulent scales are well resolved. The maximum values are in the range $0.9<(\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K})_{max}<1.15$ for all p60 simulations; case s7R2p80X displays an even better resolution, with $(\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K})_{max}<0.8$ . The resolution is comparable for R1, R2 and R3 cases, highlighting the suitability of the scaling of the computational grids with the $Re_{0}$ value. The thermal resolution requirements are shown to be here slightly more demanding than the dynamic requirements ( $\langle Pr_{\mathit{eff}}\rangle$ being larger than unity), with $(\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}})_{max}\lesssim 1.3$ . The computational grids are thus slightly finer compared to the atmospheric- $p$ temporal mixing layer simulations of Almagro et al. (Reference Almagro, Garcia-Villalba and Flores2017) ( $(\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K})_{max}<1.8$ ) and Pantano & Sarkar (Reference Pantano and Sarkar2002) ( $(\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K})_{max}\approx 3{-}4$ ), so as to satisfy the more stringent thermal- and mass-diffusion requirements.

Figure 1. Averaged resolution in terms of (a) local Kolmogorov length scale $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{K}$ and (b) local thermal length scale $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}$ at the transition time $t_{tr}^{\ast }$ .

Figure 2. One-dimensional spectra at $t_{tr}^{\ast }$ for several simulations: streamwise spectra of (a)  $u_{1}$ , (b)  $u_{2}$ , (c)  $u_{3}$ , (g)  $T$ and (h)  $Y_{\text{C}_{7}\text{H}_{16}}$ ; and spanwise spectra of (d)  $u_{1}$ , (e)  $u_{2}$ , (f)  $u_{3}$ , (i)  $T$ and (j)  $Y_{\text{C}_{7}\text{H}_{16}}$ . Dashed, solid and dashed-dotted lines represent cases R1, R2 and R3, respectively. Same colour legend as in figure 1.

Further, the excellent resolution of all simulations is highlighted in figure 2 illustrating the spanwise and streamwise spectra, $E(k)$ , of the three velocity components  $u_{i}$ , and of the scalars $Y_{\text{C}_{7}\text{H}_{16}}$ and $T$ at $t_{tr}^{\ast }$ , as a function of the wavenumber  $k$ . Because, as explained above, the relationship $\unicode[STIX]{x1D702}_{B}=\unicode[STIX]{x1D702}_{K}\,Sc^{-0.5}$ is only valid for a binary-species mixture, examining the $Y_{\unicode[STIX]{x1D6FC}}$ -values resolution through the spectra is just as important as assessing the resolution of the dynamic scales; the $T$ spectra re-ascertain the resolution of the thermal scales. The results show the spectra smoothness associated with turbulent characteristics (except for the small peak at the perturbation frequency in the spanwise direction), and the appropriate filter behaviour manifested by the lack of energy accumulation at the smallest scales. As expected, the simulations at the larger $Re_{0}$ values have spectra which extend over a wider range of scales and have more energy in the smallest scales than at the smaller $Re_{0}$ values. Increasing $p_{0}$ from 60 atm to 80 atm has no visible influence on the spectra, as well as the number of species and the different initial density ratio.

As an additional test, the streamwise and spanwise correlation coefficients of the three velocity components and of the temperature were computed at  $t_{tr}^{\ast }$ . The coefficients decrease rapidly to zero in both homogeneous directions (not shown), thereby ensuring that the domain lengths $L_{1}$ and $L_{3}$ are sufficiently large to enable the unconstrained evolution of the largest turbulent scales in the computational box.

5.2 Evolution of the global quantities

The parameter typically considered to understand and measure the growth of the mixing layer is the momentum thickness, computed as

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{m}={\displaystyle \frac{\displaystyle \int _{x_{2},min}^{x_{2},max}[\langle \unicode[STIX]{x1D70C}u_{1}\rangle _{x_{2},max}-\langle \unicode[STIX]{x1D70C}u_{1}\rangle ][\langle \unicode[STIX]{x1D70C}u_{1}\rangle -\langle \unicode[STIX]{x1D70C}u_{1}\rangle _{x_{2},min}]\,\text{d}x_{2}}{(\langle \unicode[STIX]{x1D70C}u_{1}\rangle _{x_{2},max}-\langle \unicode[STIX]{x1D70C}u_{1}\rangle _{x_{2},min})^{2}}},\end{eqnarray}$$

where $x_{2,min}=-0.4L_{2}$ and $x_{2,max}=0.6L_{2}$ , the values being consistent with the asymmetric extent of the mixing layer, as stated in § 4. Figure 3 displays $\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}$ as a function of $t^{\ast }$ for the runs listed in table 3. The analytical perturbation ensures a fast transition to a state close to self-similarity, in which the layer grows at a constant rate. The time history of $\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}$ exhibits a similar behaviour in each realization; namely, the initial constant growth rate is slightly reduced after the enstrophy peak (discussed below), but increases monotonically. The change of the slope is more evident for $Re=2000$ cases because of the stronger dampening of vortical structures. Simultaneously with layer growth, the vorticity content of the mixing layer rapidly grows as shown in figure 4, where the domain-averaged positive spanwise vorticity $\langle \langle \unicode[STIX]{x1D714}_{3}^{+}\rangle \rangle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}/\unicode[STIX]{x0394}U_{0}$ and the domain-averaged enstrophy $\langle \langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle \rangle (\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}/\unicode[STIX]{x0394}U_{0})^{2}$ are illustrated (where $\langle \langle \,\rangle \rangle$ denotes entire domain averaging). The amount of positive spanwise vorticity is representative of the small-scale formation (the initial value being negative), whereas the total enstrophy acts as an indicator of the turbulence activity, i.e. the balance between vorticity production by stretching and tilting mechanisms and its destruction by dissipation effects.

Figure 3. Time-wise evolution of the normalized momentum thickness $\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}$ .

Figure 4. Time-wise evolution of integral quantities: (a) normalized positive spanwise vorticity $\langle \langle \unicode[STIX]{x1D714}_{3}^{+}\rangle \rangle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}/\unicode[STIX]{x0394}U_{0}$ , and (b) normalized enstrophy $\langle \langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle \rangle (\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}/\unicode[STIX]{x0394}U_{0})^{2}$ . Legend as in figure 3.

The spanwise vorticity and enstrophy exhibit similar time-wise evolution for the same Reynolds number, regardless of the number of species. For the $Re=2000$ cases, a peak is observed at $t^{\ast }\approx 45$ , after which viscosity effects tend to emerge. The higher velocity gradients lead to a faster enstrophy decay due to the stronger dissipation with respect to $Re=1000$ cases that show less pronounced and delayed peaks (reflecting also in delayed transition as shown by $t_{tr}^{\ast }$ in table 3). Similarly, case s2R3p60Y exhibits higher vorticity content, peaks around $t^{\ast }\approx 40$ and experiences faster enstrophy decay with respect to R1 and R2 cases.

The difference among $\langle \langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle \rangle (\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}/\unicode[STIX]{x0394}U_{0})^{2}$ values for realizations with equal $Re_{0}$ and different number of species is related to $(\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U})$ : a larger $(\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U})$ value reduces the vorticity content, as already shown for both variable-density (Almagro et al. Reference Almagro, Garcia-Villalba and Flores2017) and compressible flows (Pantano & Sarkar Reference Pantano and Sarkar2002). Moreover, here enstrophy increases with the number of species because, if species other than n-heptane must be accommodated in the lower stream, the $Y_{\text{C}_{7}\text{H}_{16}}$ value must be reduced; since $m_{\text{C}_{7}\text{H}_{16}}$ has the largest value among all species by a substantial margin, through this $Y_{\text{C}_{7}\text{H}_{16}}$ reduction process $(\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U})$ decreases. This vortical behaviour is further confirmed by inspection of cases s2R2p60Ya and s7R2p60X, both of which have $(\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U})_{0}\approx 9$ : the time history of the global vortical quantities is essentially the same. Similar considerations hold for the $p_{0}$ dependence; the value of $(\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U})$ has more impact than the number of species as shown by the vortical evolution of case s7R2p80X being closer to s5R2p60Y than to s7R2p60X ( $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}=9.492$ , 9.8 and 9.015, respectively).

As an indication of turbulence level achieved, the value of the momentum-thickness-based Reynolds number, $Re_{m}\equiv Re_{0}\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}$ , corresponding to the $\unicode[STIX]{x1D6FF}_{m}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}$ value at transition, is provided in table 3. Clearly, similar growth rates of different mixing layers translate into approximately equal $Re_{m,tr}$ , which in turn explain the similarities in the vortical aspects of the mixing layers: for $Re_{0}=1000$ , $Re_{m,tr}\simeq 1800$ ; for $Re_{0}=2000$ , $Re_{m,tr}\simeq 3200$ . Table 3 also shows the Reynolds number based on the Taylor microscale, defined as $Re_{\unicode[STIX]{x1D706},tr}=q^{2}\sqrt{5\unicode[STIX]{x1D70C}/(\unicode[STIX]{x1D707}\unicode[STIX]{x1D700})}$ , $q^{2}$ being twice the turbulent kinetic energy. The $Re_{\unicode[STIX]{x1D706},tr}$ value displays slight variations for the $Re=2000$ cases and larger variations for the $Re=1000$ cases; this finding is consistent with the different $t_{tr}^{\ast }$ at which transition is achieved for the lower-Reynolds-number simulations. For case s2R3p60Y, values as high as $Re_{m,tr}\approx 5000$ and $Re_{\unicode[STIX]{x1D706},tr}\approx 450$ are achieved at transition.

Thus, the analysis of the global quantities indicates that $Re_{0}$ and $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ are the main parameters governing the dynamics of the layer evolution.

5.3 Uphill diffusion

Considering the large number of species and the wide range of their initial mass fractions in the free streams, it is useful to introduce a normalized mass fraction variable

(5.3) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}=\frac{Y_{\unicode[STIX]{x1D6FC}}-Y_{\unicode[STIX]{x1D6FC},ref}^{min}}{Y_{\unicode[STIX]{x1D6FC},ref}^{max}-Y_{\unicode[STIX]{x1D6FC},ref}^{min}},\end{eqnarray}$$

where $Y_{\unicode[STIX]{x1D6FC},ref}^{max}=\max (Y_{\unicode[STIX]{x1D6FC}}^{U},Y_{\unicode[STIX]{x1D6FC}}^{L})$ and $Y_{\unicode[STIX]{x1D6FC},ref}^{min}=\min (Y_{\unicode[STIX]{x1D6FC}}^{U},Y_{\unicode[STIX]{x1D6FC}}^{L})$ . In this manner, $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}=0$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}=1$ represent the initial species’ mass fraction of the upper and lower streams or vice versa, depending on the free-stream location of the species. For example, in the s7R2p60X case, a local value of $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}}=0$ means that $Y_{\text{C}_{7}\text{H}_{16}}=Y_{\text{C}_{7}\text{H}_{16},\mathit{ref}}^{min}=Y_{\text{C}_{7}\text{H}_{16}}^{U}$ ; similarly, $\unicode[STIX]{x1D6EF}_{\text{H}_{2}\text{O}}=1$ implies $Y_{\text{H}_{2}\text{O}}=Y_{\text{H}_{2}\text{O},\mathit{ref}}^{max}=Y_{\text{H}_{2}\text{O}}^{U}$ . This allows the evaluation of the degree of mixing for species inside the mixing layer compared to the relative reference stream values.

Figure 5. P.d.f. of $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ for all species of case s7R2p60X at $t_{tr}^{\ast }$ .

Since in the present situation each case of tables 2 and 3 has a different initial composition, the mixing layer region is identified by using a variable threshold. The bounds of the mixing layer are chosen as

(5.4) $$\begin{eqnarray}Y_{\text{C}_{7}\text{H}_{16}}^{U}+k\unicode[STIX]{x0394}Y_{\text{C}_{7}\text{H}_{16},ref}\leqslant Y_{\text{C}_{7}\text{H}_{16}}\leqslant Y_{\text{C}_{7}\text{H}_{16}}^{L}-k\unicode[STIX]{x0394}Y_{\text{C}_{7}\text{H}_{16},ref},\end{eqnarray}$$

where $k=0.05$ and $\unicode[STIX]{x0394}Y_{\text{C}_{7}\text{H}_{16},ref}=Y_{\text{C}_{7}\text{H}_{16}}^{L}-Y_{\text{C}_{7}\text{H}_{16}}^{U}$ . A sensitivity analysis to the value of $k$ is shown in appendix C and the conclusion is that the choice of the bounding value does not affect the physics derived from the analysis.

The advantage of the $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ definition is immediately apparent when examining the probability density functions (p.d.f.s) of $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ illustrated in figure 5: the probability of the most likely value is the same for all species, although the most likely value is not the same for all species ( $\simeq 0.65$ for $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}},\simeq 0.35$ for $\unicode[STIX]{x1D6EF}_{\text{O}_{2}}$ and $\simeq 0.4$ for all minor species $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ ). However, although for n-heptane and oxygen the p.d.f.s only extend over the $[0,1]$ interval, meaning that all the corresponding $Y_{\unicode[STIX]{x1D6FC}}$ values are between those specified in the free streams, for all minor species the p.d.f.s extend outside of the $[0,1]$ interval, hence $Y_{\unicode[STIX]{x1D6FC}}$ values outside the range bounded by the free-stream values occur. This situation for the minor species is the first evidence that uphill diffusion occurs (Taylor & Krishna Reference Taylor and Krishna1993) wherein due to strong coupling among fluxes, species mass diffusion may occur against a species’ gradient. Comparing the p.d.f.s of all minor species, it appears that H₂ has the broadest p.d.f. (i.e. it exhibits the largest standard deviation corresponding to heaviest tails of the p.d.f.).

To examine the effect of uphill diffusion on the species’ spatial distribution and assess how it relates to $T$ and the compressibility factor $Z=p/(\unicode[STIX]{x1D70C}TR_{u}/m)$ which indicates the deviation from the perfect-gas behaviour, in figure 6 are displayed the spatial distributions of all $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ (except for N₂), $T$ and $Z$ in the braid plane for case s7R2p60X. Regions having $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}<0$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}>1$ are highlighted in white and black, respectively. The values of $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ for the first five species display similar values, except for the peripheral regions of the mixing layer: here, the minor species experience uphill diffusion (hydrogen being the most affected species) whereas oxygen is only subjected to regular diffusion, as shown by the p.d.f. As expected, $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}}$ shows the opposite distribution, with small fuel-rich pockets (red zones in figure 6 h) penetrating in the upper stream. As apparent in figure 6, uphill diffusion is not dependent on the value of either $T$ or  $Z$ , that is, it occurs over a wide range of $T$ values and, according to the $Z$ values, does not depend on whether the fluid behaves as a perfect gas ( $Z\simeq 1$ ) or a real gas (e.g. here it occurs at $Z<0.65$ ). The functional relationship between $T$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ is analysed in § 5.7.

Figure 6. Braid plane ( $x_{3}/L_{3}=1/16$ ) distribution of different quantities at $t_{tr}^{\ast }$ for s7R2p60X: (a) temperature $T$ (K), (b) local compressibility factor  $Z$ , and normalized mass fractions (c)  $\unicode[STIX]{x1D6EF}_{\text{H}_{2}\text{O}}$ , (d)  $\unicode[STIX]{x1D6EF}_{\text{H}_{2}}$ , (e)  $\unicode[STIX]{x1D6EF}_{\text{CO}}$ , (f)  $\unicode[STIX]{x1D6EF}_{\text{CO2}}$ , (g)  $\unicode[STIX]{x1D6EF}_{\text{O}_{2}}$ and (h)  $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}}$ . In panels (c–h), white and black denote regions having $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}<0$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}>1$ , respectively.

5.4 Density gradients

The Navier–Stokes equations rely on the hypothesis that all dependent variables are infinitely differentiable, and thus no discontinuities can be accommodated. The definition of a discontinuity depends on the scale of interest. For example, whereas material surfaces separating two phases were considered to be discontinuities prior to the studies of van der Waals, he showed that by magnifying the linear extent of the interface it is possible to study the density change from one phase to another as manifested by large density gradients. It has been experimentally shown that, in flows, large density gradients act as material surfaces do, and modify the vortical character of the flow (Hannoun et al. Reference Hannoun, Fernando and List1988). Korteweg (Reference Korteweg1901) proposed an elegant way to modify the momentum equation should there be large density gradients, and thus to account for the additional vorticity created by these density gradients. Since uphill diffusion results in the formation of density gradients, it is pertinent to examine the $\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}$ field, and correspondingly whether modifications are necessary to (2.2) due to possible significant influence of the Korteweg stresses, before focusing on the other effects of uphill diffusion in § 5.5. Effectively, the magnitude of the Korteweg stresses provides here the criterion enabling one to determine whether the magnitude of the density gradients is sufficiently large to warrant modification of the Navier–Stokes equations.

Figure 7 shows visualizations of $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ at $t_{tr}^{\ast }$ for several realizations; $|\cdot |$ denotes here the ${\mathcal{L}}2$ norm. Clearly, higher turbulence levels (i.e. larger values of $Re_{0}$ ) enhance the complexity of the morphological features of $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ because of an increase in stirring; this aspect is seen by comparing figure 7(a), 7(b) and 7(c) representing cases s2R1p60Y, s2R2p60Y and s2R3p60Y, respectively. At the same conditions otherwise, an increase in $p_{0}$ does not significantly change the morphological complexity of the HDGM regions, but it clearly results in an increase of $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ values. This increase of the $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ value occurs because both the molecular mass diffusion and effective mass diffusion decrease with larger  $p_{0}$ , these phenomena governing the composition at relatively low and relatively high turbulence levels, respectively. Changing the number of species does not seem to influence the distribution or complexity of the HDGM regions, except for the indirect effect due to change in $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ . Further, the p.d.f.s of $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ for some of the runs listed in table 3 are displayed in figure 8. For clarity, only the R2 category of cases, which contains the largest number of realizations, is completely shown; all R1 cases have p.d.f.s in the vicinity of the illustrated case s2R1p60Y and there is a single run in the R3 category. Inspection of these p.d.f.s in figure 8(a) shows that there is a quantifiable difference among the realizations: the right tails of the p.d.f.s clearly become heavier with increasing $Re_{0}$ and $p_{0}$ (the maximum values being obtained for case s2R3p60Y and, among R2 simulations, for s7R2p80X). The influence of $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ is highlighted by comparing the R2p60 cases; namely, the heavier tails are obtained for s2R2p60Y and s3R2p60Y (for which $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ is in the range [12.5–13]), and the lighter tails for s2R2p60Ya and s7R2p60X (having $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}\approx 9$ ). Of note, cases s2R2p60Ya and s7R2p60X show that the number of species is unimportant as far as the HDGM regions are concerned, provided that the initial composition does not change $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ . Normalization of $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ by $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ has similar variation to $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ as a function of the runs listed in table 3) in figure 8(b) shows that indeed $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ scales with $\unicode[STIX]{x1D70C}_{L}/\unicode[STIX]{x1D70C}_{U}$ to a great extent. Furthermore, it is also clear that the maximum values attained by $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|/\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ increase with increasing $Re_{0}$ . The explanation is that turbulence acts only on the continuum scales and thus cannot influence diffusion, which occurs at molecular scales. Since $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ represents the manifestation of diffusion at the continuum scales, the present results confirm the competition at the continuum scales of mass-diffusion manifestation with turbulence in creating the HDGM regions. Essentially, as turbulence is higher, its characteristic time is smaller, a fact which only permits limited diffusion, thus maintaining larger density gradients.

Figure 7. Braid plane ( $x_{3}/L_{3}=1/16$ ) distribution of the local $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ in logarithmic scale $\log _{10}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ at $t_{tr}^{\ast }$ for (a) s2R1p60Y, (b) s2R2p60Y, (c) s2R3p60Y, (d) s3R2p60Y, (e) s5R2p60Y, (f) s5R2p60X, (g) s7R2p60X and (h) s7R2p80X. Units are $\text{kg}~\text{m}^{-4}$ .

The composition of the HDGM regions is examined in figure 9, depicting the conditional averages of $Y_{\unicode[STIX]{x1D6FC}}$ on $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ for case s7R2p60X at $t_{tr}^{\ast }$ . For low $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$  values, $Y_{\text{C}_{7}\text{H}_{16}}$ increases rapidly with $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ , reaching $Y_{\text{C}_{7}\text{H}_{16}}\approx 0.8$ already at $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|\approx 5\times 10^{4}~\text{kg}~\text{m}^{-4}$ and asymptotically reaching 0.875. Thus, the composition of the regions of largest $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ values is dominated by n-heptane. The other species exhibit the opposite behaviour, with a decreased presence in higher- $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ regions. This is distinctly shown in figure 9(b) where all $Y_{\unicode[STIX]{x1D6FC}}$ values are normalized with respect to $Y_{\unicode[STIX]{x1D6FC},max}=\max (Y_{\unicode[STIX]{x1D6FC}}^{L},Y_{\unicode[STIX]{x1D6FC}}^{U})$ . This variation confirms the observation of Bellan (Reference Bellan2017a ) stating that the HDGM regions appear to initiate in regions of relatively small (with respect to the maximum) amount of fuel. Similarly, figure 10 displays the values of $\langle Y_{\text{C}_{7}\text{H}_{16}}\rangle$ and $\langle Y_{\text{C}_{7}\text{H}_{16}}/Y_{\text{C}_{7}\text{H}_{16}}^{max}\rangle$ conditioned on $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ for different realizations. The trend is similar for all simulations, and the normalization with respect to $Y_{\text{C}_{7}\text{H}_{16}}^{max}$ (different value for each case in order to accommodate the minor species) narrows the range of variation of the R2p60 profiles. For s7R2p80X, the n-heptane contribution to HDGM regions is slightly smaller and is here attributed to the larger difficulty of entraining a denser fluid.

Figure 8. P.d.f.s at $t_{tr}^{\ast }$ of (a)  $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ (units are $\text{kg}~\text{m}^{-4}$ ), and (b)  $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|/\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ (units are $\text{m}^{-1}$ ). Legend as in figure 3.

Figure 9. Conditional average of species mass fraction conditioned on local density-gradient magnitude for case s7R2p60X at $t_{tr}^{\ast }$ : (a)  $\langle Y_{\unicode[STIX]{x1D6FC}}\mid |\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|\rangle$ , and (b)  $\langle Y_{\unicode[STIX]{x1D6FC}}/Y_{\unicode[STIX]{x1D6FC}}^{max}\mid |\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|\rangle$ , where $Y_{\unicode[STIX]{x1D6FC}}^{max}=\max (Y_{\unicode[STIX]{x1D6FC}}^{L},Y_{\unicode[STIX]{x1D6FC}}^{U})$ . Units for $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ are $\text{kg}~\text{m}^{-4}$ .

Figure 10. Average value of the n-heptane mass fraction conditioned on local density-gradient magnitude at $t_{tr}^{\ast }$ : (a)  $\langle Y_{\text{C}_{7}\text{H}_{16}}\mid |\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|\rangle$ , and (b)  $\langle Y_{\text{C}_{7}\text{H}_{16}}/Y_{\text{C}_{7}\text{H}_{16}}^{max}\mid |\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|\rangle$ , where $Y_{\text{C}_{7}\text{H}_{16}}^{max}=\max (Y_{\text{C}_{7}\text{H}_{16}}^{L},Y_{\text{C}_{7}\text{H}_{16}}^{U})$ . Units for $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ are $\text{kg}~\text{m}^{-4}$ . Legend as in figure 3.

The large $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ values encountered in the mixing region motivate an inspection of whether sufficiently large Korteweg stresses are created, thus potentially necessitating a revision of (2.2) to accommodate the effect of interfaces. The second-gradient theory (van der Waals Reference van der Waals1893; Korteweg Reference Korteweg1901) naturally allows one to account for the contribution of large density gradients in the Navier–Stokes equations. This theory represents the theoretical basis for the formulation of the well-known diffuse interface models (Anderson, McFadden & Wheeler Reference Anderson, McFadden and Wheeler1998; Jamet et al. Reference Jamet, Lebaigue, Coutris and Delhaye2001), widely employed to describe gas–liquid interfaces in multiphase flows. The changes to the momentum, species and energy equations originating from the second-gradient theory are derived in appendix D for a multi-species situation in concert with the mixing rules of Harstad & Bellan (Reference Harstad and Bellan2004a ). To summarize these derivations, including these second-gradient terms in the momentum equation yields (in compact notation)

(5.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u}\otimes \boldsymbol{u})=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D652}=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D64B}+\unicode[STIX]{x1D64F}),\end{eqnarray}$$

where

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D64B}=\left[p-\mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}=1}^{N}\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\left(\frac{\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FD}}}{2}+\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FD}}\right)\right]\unicode[STIX]{x1D644}+\mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}=1}^{N}\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FD}}\otimes \unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D64F}=(\hat{\unicode[STIX]{x1D707}}-{\textstyle \frac{2}{3}}\unicode[STIX]{x1D707})(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u})\unicode[STIX]{x1D644}+\unicode[STIX]{x1D707}[\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}] & \displaystyle\end{eqnarray}$$

are the reversible and irreversible components of the momentum flux, respectively, $\boldsymbol{u}$ is the velocity vector, $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ is the species-dependent influence parameter, $\unicode[STIX]{x1D644}$ is the identity matrix, and $\hat{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$ are the bulk and shear viscosities. The term multiplying the identity matrix in (5.6) can be interpreted as an equivalent pressure incorporating capillarity effects; whereas the last term is commonly known as the Korteweg tensor. Compared to the typical single-phase flow momentum equation, the additional terms in $\unicode[STIX]{x1D64B}$ do not produce entropy because they are associated with reversible processes. The irreversible component $\unicode[STIX]{x1D64F}$ of (5.7) is not modified by the gradient energy, and represents the classical irreversible processes related to viscous dissipation.

By comparing the magnitudes of the gradient of the capillarity terms and the pressure gradient, an order-of-magnitude analysis can be performed in order to estimate the influence of the Korteweg stresses in the momentum equation. Since at $t_{tr}^{\ast }$ the density-gradient p.d.f. scales with $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|/\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ practically independent of the number of species, and since additionally the highest- $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|$ regions primarily contain n-heptane, a representative magnitude of the Korteweg tensor can be computed for all cases by reducing the complexity of the Korteweg tensor to the situation of a single species: n-heptane. This procedure furthermore avoids confronting the issue of uncertainty in some values of $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ . Thus, the influence parameter is chosen to be $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}\approx 10^{-5}~\text{cm}^{7}~\text{g}^{-1}~\text{s}^{-2}=10^{-16}~\text{m}^{7}~\text{kg}^{-1}~\text{s}^{-2}$ , a value slightly larger than that reported in the literature (Cornelisse Reference Cornelisse1997) so as to provide an upper bound of the Korteweg effects. Figure 11 displays a comparison between the magnitude of the pressure gradient, $|\unicode[STIX]{x1D735}p|$ , and the term $|\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C})^{2}]|$ . The case considered is s2R3p60Y, for which density gradients reach values as high as $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}|\approx 8\times 10^{5}~\text{kg}~\text{m}^{-4}$ and for which there are only two species. The gradient term is shown to be almost nine orders of magnitude smaller than the pressure gradient, yielding thus a negligible contribution to  $\unicode[STIX]{x1D64B}$ . Even in the regions of the largest $|\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C})^{2}]|$ values, figure 11 shows that the value of $|\unicode[STIX]{x1D735}p|$ at those locations is orders of magnitude larger than the values of $|\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C})^{2}]|$ . A grid convergence analysis, presented in appendix E, shows that the criteria used in § 5.1 for the grid resolution are very stringent because, when the grid is refined by a factor of 50 % in each direction, the quantitative results obtained with the coarser grid are recovered. The conclusion is that the modification to the actual thermodynamic pressure is also negligible, and thus that the present DNS database is appropriate for the purpose of this study, enabling one to pursue the examination of the specific features bringing a deeper understanding of uphill diffusion.

Figure 11. Values of (a)  $|\unicode[STIX]{x1D735}p|$ and (b)  $|\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C})^{2}]|$ for case s2R3p60Y. Units are $\text{kg}~\text{m}^{-2}~\text{s}^{-2}$ .

5.5 Effective transport properties

In fluid mechanics, non-dimensional numbers are useful in providing a single length scale, or equivalently a single time scale, associated with a process; comparison of these length (or time) scales related to various phenomena yields crucial information on the physics governing a situation. However, unlike for a binary-species mixture, in a situation involving several species equation (2.6) shows that it is not possible to define a single diffusion length scale. To overcome this difficulty Masi et al. (Reference Masi, Bellan, Harstad and Okong’o2013) derived a methodology to express the species diffusive fluxes and heat flux as a function of single effective coefficients. Under the assumption of non-null gradients, valid for mixing layers and in turbulent flows, equation (2.4) can be rewritten as

(5.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}Y_{\unicode[STIX]{x1D6FC}})}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}u_{j}Y_{\unicode[STIX]{x1D6FC}})}{\unicode[STIX]{x2202}x_{j}}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(\unicode[STIX]{x1D70C}D_{\unicode[STIX]{x1D6FC},\mathit{eff}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{j}}\right),\end{eqnarray}$$

where

(5.9) $$\begin{eqnarray}D_{\unicode[STIX]{x1D6FC},\mathit{eff}}\equiv Y_{\unicode[STIX]{x1D6FC}}\frac{D_{T,\unicode[STIX]{x1D6FC}}}{T}\frac{\unicode[STIX]{x1D6FF}T}{\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}}+Y_{\unicode[STIX]{x1D6FC}}\frac{D_{p,\unicode[STIX]{x1D6FC}}}{p}\frac{\unicode[STIX]{x1D6FF}p}{\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}}+\mathop{\sum }_{\unicode[STIX]{x1D6FD}=1}^{N-1}\mathfrak{D}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\frac{\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}}\end{eqnarray}$$

represents a single species diffusion coefficient accounting for the entire flux matrix and $\unicode[STIX]{x1D6FF}$ signifies functional derivatives. Then $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ can be used to define a species-effective Schmidt number

(5.10) $$\begin{eqnarray}Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}\equiv \frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}D_{\unicode[STIX]{x1D6FC},\mathit{eff}}}.\end{eqnarray}$$

Similarly, equation (2.3) can be recast in the form

(5.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}e_{t})}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}[(\unicode[STIX]{x1D70C}e_{t}+p)u_{j}-u_{i}\unicode[STIX]{x1D70E}_{ij}]}{\unicode[STIX]{x2202}x_{j}}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(\unicode[STIX]{x1D6EC}_{\mathit{eff}}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{j}}\right),\end{eqnarray}$$

where

(5.12) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathit{eff}}\equiv \unicode[STIX]{x1D706}+\unicode[STIX]{x1D70C}\mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{N-1}\left[\left(\frac{h_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}}-\frac{h_{N}}{m_{N}}\right)-R_{u}T\left(\frac{\bar{\unicode[STIX]{x1D6FC}}_{T,\unicode[STIX]{x1D6FC}}^{b}}{m_{\unicode[STIX]{x1D6FC}}}-\frac{\bar{\unicode[STIX]{x1D6FC}}_{T,N}^{b}}{m_{N}}\right)\right]D_{\unicode[STIX]{x1D6FC},\mathit{eff}}\frac{\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D6FF}T}.\end{eqnarray}$$

Thus $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ represents an effective thermal conductivity accounting for the entire heat flux pertaining to all species. Therefore, an effective Prandtl number can be defined as

(5.13) $$\begin{eqnarray}Pr_{\mathit{eff}}\equiv \frac{C_{p}\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6EC}_{\mathit{eff}}},\end{eqnarray}$$

along with a species-effective Lewis number,

(5.14) $$\begin{eqnarray}Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}\equiv \frac{\unicode[STIX]{x1D6EC}_{\mathit{eff}}}{\unicode[STIX]{x1D70C}D_{\unicode[STIX]{x1D6FC},\mathit{eff}}C_{p}}\equiv \frac{Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}}{Pr_{\mathit{eff}}}.\end{eqnarray}$$

The functional derivatives $\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FD}}/\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FD}}/\unicode[STIX]{x1D6FF}T$ , $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FF}p/\unicode[STIX]{x1D6FF}Y_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6FF}p/\unicode[STIX]{x1D6FF}T$ are modelled as in Masi et al. (Reference Masi, Bellan, Harstad and Okong’o2013). For example, defining coefficients of type $C_{pT}\equiv \unicode[STIX]{x1D6FF}p/\unicode[STIX]{x1D6FF}T$ which exactly satisfy the relation $C_{pT}\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}x_{i}=\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x_{i}$ , the model employing contracted products

(5.15a,b ) $$\begin{eqnarray}C_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}}=\left.\left(\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\right)\right/\!\left(\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\right),\quad C_{\unicode[STIX]{x1D6FD}T}=\left.\left(\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\right)\right/\!\left(\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\right),\end{eqnarray}$$

(5.16a,b ) $$\begin{eqnarray}C_{T\unicode[STIX]{x1D6FC}}=\left.\left(\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\right)\right/\!\left(\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\right),\quad C_{p\unicode[STIX]{x1D6FC}}=\left.\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\right)\right/\!\left(\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x_{i}}\right),\end{eqnarray}$$

(5.17) $$\begin{eqnarray}C_{pT}=\left.\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\right)\right/\!\left(\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{i}}\right)\end{eqnarray}$$

eliminates directional variations and has a finite value everywhere in the mixing layer. The model accuracy was validated by comparing the modelled fluxes with those computed from the DNS database, showing excellent agreement.

Figure 12. P.d.f.s for case s7R2p60X at $t_{tr}^{\ast }$ : (a)  $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ ( $10^{-7}~\text{m}^{2}~\text{s}^{-1}$ , not ${\mathcal{F}}$ -scaled), (b)  $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ , and (c)  $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ .

Figure 13. Braid plane ( $x_{3}/L_{3}=1/16$ ) distribution of the local effective Schmidt number $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ for different species at $t_{tr}^{\ast }$ for s7R2p60X: (a) O₂, (b) C₇H₁₆ (c) CO, (d) CO₂, (e) H₂O, and (f) H₂.

Figure 14. P.d.f.s computed over the three-dimensional mixing layer at $t_{tr}^{\ast }$ : (a)  $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ ( $10^{-7}~\text{m}^{2}~\text{s}^{-1}$ , not ${\mathcal{F}}$ -scaled), (b)  $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ , (c)  $D_{\text{H}_{2}\text{O},\mathit{eff}}$ ( $10^{-7}~\text{m}^{2}~\text{s}^{-1}$ , not ${\mathcal{F}}$ -scaled), and (d)  $Sc_{\text{H}_{2}\text{O},\mathit{eff}}$ .

We proceed by first investigating statistical aspects and further scrutinizing details not available through the statistical analysis. Figures 12(a) and 12(b) exhibit the p.d.f.s of the effective diffusion coefficients and of the species-effective Schmidt number. Oxygen and n-heptane have relatively narrow $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ distributions (figure 12 a) with a mean $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}\approx D_{\text{O}_{2},\mathit{eff}}\approx 4.5\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$ (a value which is approximately midway between the range of $O(10^{-5})~\text{m}^{2}~\text{s}^{-1}$ for gases and $O(10^{-9})~\text{m}^{2}~\text{s}^{-1}$ for liquids) and the p.d.f.s are located in the $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}>0$ range. Conversely, the $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ of all minor species exhibit much broader symmetric bell-shaped p.d.f.s in a range encompassing both positive and negative $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ with mean values $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}\approx 3\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$ ; the $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}<0$ region corresponds to the locations experiencing uphill diffusion. All minor species’ p.d.f.s are similar except for that of H₂, which is the widest, and correspondingly has the heaviest tails. We attribute the peculiar H₂ behaviour compared to all other minor species to H₂ being a much smaller (i.e. $0.74\times 10^{-10}$  m diameter) and lighter (i.e. $2.016~\text{kg}~\text{kmol}^{-1}$ molar mass) molecule which makes it very mobile, thus enhancing its diffusional interaction with other molecules, compared to species of larger and heavier molecules. This interaction manifests in the H₂ flux being more strongly coupled to those of the other species, thereby inducing enhanced uphill diffusion. For all minor species, their Gaussian-like p.d.f. shape is representative of tracer diffusion.

According to the p.d.f.s of figure 12(b), the most probable value of $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ and $Sc_{\text{O}_{2},\mathit{eff}}$ is approximately 1.2. In contrast to n-heptane and oxygen, the minor species exhibit a much broader distribution having symmetric bell-shaped p.d.f.s similar in concept to those of $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ and extending to $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}<0$ values. Correspondingly, the p.d.f.s of $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ for the minor species span a much wider range than those of the major species; the most probable values of $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ are between 1.5 and 2 for CO, CO₂ and H₂O, whereas it is approximately 0.7 for H₂, for which heavier tails are observed than those of all other minor species. These mean values, more representative for the major species than the minor species, indicate that the constant $Sc=0.7$ value, widely employed in modelling species mass transfer, is not appropriate for either the major or the minor species, but it would especially fail in describing the diffusion behaviour of minor species, except for H₂ for which however the mean is not very representative considering its wide $Sc_{\text{H}_{2},\mathit{eff}}$ p.d.f.

To assess the commonly used assumption $Le=1$ , the $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ p.d.f.s are displayed in figure 12(c) (the corresponding $Pr_{\mathit{eff}}$ is discussed in the following). For all species, the most likely values $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ are in the range $[0,1]$ , but they can locally be as large as 3 or as small as  $-1$ , the negative values being due to $\unicode[STIX]{x1D6EC}_{\mathit{eff}}<0$ , according to (5.13) and (5.14). The $Le=1$ assumption appears inaccurate, the most probable $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ values being ${\approx}0.45$ for n-C₇H₁₆ and O₂, ${\approx}0.75$ for CO, CO₂ and H₂O, and ${\approx}0.35$ for H₂. The narrower p.d.f.s for the major species indicate that the mean values for these species are more representative than for the minor species.

The above analysis highlights the range of values of effective non-dimensional numbers but does not provide an indication of their local distribution in the mixing layer. Such local distribution of $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ is illustrated in figure 13 for species involved in case s7R2p60X. The major species, namely O₂ and n-C₇H₁₆, display small variations of $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ , with the larger values occurring at the upper bound of the mixing layer. This small increase from the lower to the upper mixing layer domain results from the opposite variation of, for example, $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ in $(x_{1},x_{3})$ planes and the local value of  $\unicode[STIX]{x1D70C}$ : the former increases as a consequence of the larger  $T$ , whereas the latter decreases. The minor species CO, CO₂ and H₂O display a common behaviour; they exhibit higher variations compared to the major species, and negative local values indicate that these species undergo uphill diffusion primarily, but not exclusively, in the lower stream. Lastly, a specific behaviour is observed for H₂ for which $Sc_{\text{H}_{2},\mathit{eff}}>0$ and $Sc_{\text{H}_{2},\mathit{eff}}<0$ regions occur over the entire mixing layer. We attribute this H₂ prevalence of uphill diffusion over the entire mixing layer to a combination of the aforementioned properties of H₂, conferring to it greater mobility, and to the fact that $Y_{\text{H}_{2},ref}^{max}<0.01$ in both free streams, making it strongly influenced by turbulent stirring.

Having analysed the different behaviour of the species in a single realization, the focus is now on two species – n-heptane and water – and their behaviour in realizations obtained with different number of species and $Re_{0}=2000$ is assessed. Case s7R1p60X is also considered in order to address the influence of the $Re$ value. Figures 14(a) and 14(b) show the p.d.f.s of $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ and $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ for different realizations. Compared to the effect of  $p_{0}$ , the initial composition only has a relatively small influence on $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ . Indeed, as the number of species increases, the n-heptane molecules encounter a greater variety of molecules, both size-wise and mass-wise, a fact which increases the irregularity of the possible molecular pathways and slightly widens the p.d.f., resulting in a modest reduction of the mean value of $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ . In contrast, pressure plays a greater role since an increase in $p_{0}$ causes a decrease of the molecular diffusivity and of the $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ values (Masi et al. Reference Masi, Bellan, Harstad and Okong’o2013). As a consequence, the most likely value of $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ is similar for all the $p_{0}=60$  atm cases ( $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}\approx 1.15$ ), whereas it is shifted to $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}\approx 1.45$ for $p_{0}=80$  atm due the smaller $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ values. Although $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ is smaller for $p_{0}=80$  atm, the maximum $Sc_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ values are ${\approx}1.85$ for all cases, this behaviour being attributed to $\unicode[STIX]{x1D70C}$ increasing with  $p_{0}$ . The R1 case is mostly indistinguishable from the equivalent R2 case except for the most probable value of $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ which slightly decreases, indicating that the most probable value of $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ is susceptible to stirring. Figures 14(c) and 14(d) show the p.d.f.s of $D_{\text{H}_{2}\text{O},\mathit{eff}}$ and $Sc_{\text{H}_{2}\text{O},\mathit{eff}}$ for realizations in which water is present in the initial composition. Case s5R2p60Y has a uniform initial distribution of H₂O, meaning small $Y_{\text{H}_{2}\text{O}}$ gradients, leading to uphill diffusion; on the contrary, s5R2p60X has substantial initial gradients in $Y_{\text{H}_{2}\text{O}}$ , explaining why uphill diffusion is small despite this being a minor species. The value of $Re$ is seen to make a smaller impact on $D_{\text{H}_{2}\text{O},\mathit{eff}}$ than on the equivalent for C₇H₁₆, this being attributed to the smaller initial gradients than those of $Y_{\text{C}_{7}\text{H}_{16}},$ leading to more subdued diffusion. Pressure has a definite effect in that it makes the most probable value more statistically significant, just as for C₇H₁₆, but unlike for C₇H₁₆, the most probable value only decreases very slightly. The result of this behaviour is seen in the $Sc_{\text{H}_{2}\text{O},\mathit{eff}}$ : all simulations having the same initial composition exhibit similar p.d.f.s, independent of the number of species, and only slightly dependent on pressure (the most probable value increasing slightly with pressure). However, the initial composition has a major impact, as indeed it should, by substantially increasing the values of $Sc_{\text{H}_{2}\text{O},\mathit{eff}}$ in the negative range.

Figure 15. P.d.f.s computed over the three-dimensional mixing layer at $t_{tr}^{\ast }$ : (a)  $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ ( $\text{W}~\text{m}^{-1}~\text{K}^{-1}$ ), and (b)  $Pr_{\mathit{eff}}$ . J.p.d.f.s of $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ and $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ ( $\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$ ) for cases (c,d) s5R2p60Y and (e,f) s7R2p60X. All the coefficients are not ${\mathcal{F}}$ -scaled.

The p.d.f.s of $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ and $Pr_{\mathit{eff}}$ are displayed in figures 15(a) and 15(b), respectively. That for $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ exhibits small variations from the most likely value ( $\unicode[STIX]{x1D6EC}_{\mathit{eff}}\approx 0.02{-}0.03$ W  $\text{m}^{-1}~\text{K}^{-1}$ ). Although $\unicode[STIX]{x1D706}>0$ as molecular transport theory states, $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ is primarily positive, but negative values also occur in the mixing layer due to transport of enthalpy caused by the species-mass flux, as clearly shown by (5.12). The significance of $\unicode[STIX]{x1D6EC}_{\mathit{eff}}<0$ regions is that heat is transported against $\unicode[STIX]{x1D735}T$ . However, negative $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ values have small probabilities, leading to statistically insignificant $Pr_{\mathit{eff}}<0$ regions. To verify the fact that uphill conduction is statistically insignificant, p.d.f.s of a normalized  $T$ ,

(5.18) $$\begin{eqnarray}\unicode[STIX]{x1D703}=\frac{T-T_{L}}{T_{U}-T_{L}},\end{eqnarray}$$

were computed (not shown) for all cases shown in figure 15(a). All p.d.f.s have most probable values in the $\unicode[STIX]{x1D703}\in [0.1,0.2]$ range, the most probable value becomes less statistically significant with increased number of species, and uphill conduction is confined to regions where $\unicode[STIX]{x1D703}<0$ and $|\unicode[STIX]{x1D703}|<0.05$ , thus being negligible; also, the region $\unicode[STIX]{x1D703}<0$ becomes reduced with increasing number of species. Of note, $\unicode[STIX]{x1D703}<0$ values are mainly found along the lower limit of the mixing layer region, whereas negative values for the effective thermal diffusivity are registered across the entire mixing layer: this fact highlights that uphill thermal conduction is not limited to the peripheral zones, but its occurrence is widespread across the layer. Independent of the case, the p.d.f. assumes null values past $\unicode[STIX]{x1D703}>0.9$ ; this fact is a manifestation of the definition of the mixing layer that is based on the $(x_{1},x_{3})$ -averaged mass fraction of heptane (initially present only in the lower colder stream), thus resulting in the temperature of the mixture at the upper limit of the mixing layer being slightly lower than the reference temperature in the upper, unperturbed stream.

Globally, there is no direct clear influence of the initial composition on the $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ values, whereas an increase of $p_{0}$ seems to slightly shift the distribution towards positive values. As a consequence of both positive and negative values of $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ , the same variation holds for $Pr_{\mathit{eff}}$ having generally values in the range  $[0,7]$ , with most probable values as small as ${\approx}2.5$ and as large as ${\approx}3$ . These values are representative of dense gases; typically $Pr_{\mathit{eff}}\approx 4$ –5 for liquid refrigerants and ${\approx}7$ for liquid H₂O. As the number of species increases, the most probable value of the p.d.f. shifts towards lower values, being ${\approx}3$ for case s3R2p60Y to ${\approx}2$ for case s7R2p60X. The p.d.f.s for the seven-species cases exhibit lower dispersion, and higher $Pr_{\mathit{eff}}$ values are less probable. An increase of the free-stream pressure $p_{0}$ results in a slightly lower dispersion, whereas the most likely value remains unchanged. To examine the influence of the $\unicode[STIX]{x1D6EC}_{\mathit{eff}}<0$ values on the corresponding $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ for C₇H₁₆ and H₂O, figures 15(c–f) display joint probability density functions (j.p.d.f.s) of $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ and $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ for cases s5R2p60Y and s7R2p60X. According to (5.14), $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}<0$ only if $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ and $D_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ have different signs. The j.p.d.f.s show that $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ has much more extensive domain correlation with $D_{\text{C}_{7}\text{H}_{16},\mathit{eff}}$ (which is always positive) than with $D_{\text{H}_{2}\text{O},\mathit{eff}}$ (which can also become negative). Moreover, that larger correlation is increasingly in the $\unicode[STIX]{x1D6EC}_{\mathit{eff}}>0$ region as the number of species increases. Additionally, figures 15(c) and 15(f) also show the presence (even if statistically small) of $Le_{\text{H}_{2}\text{O}}>0$ regions given by simultaneous negative values of $\unicode[STIX]{x1D6EC}_{\mathit{eff}}$ and $D_{\text{H}_{2}\text{O},\mathit{eff}}$ . Thus, uphill diffusion may occur for species  $\unicode[STIX]{x1D6FC}$ , yet $Le_{\unicode[STIX]{x1D6FC},\mathit{eff}}>0$ .

Figure 16. Braid plane ( $x_{3}/L_{3}=1/16$ ) distribution of the local effective Prandtl number $Pr_{\mathit{eff}}$ at $t_{tr}^{\ast }$ : (a) 2pR260Y, (b) s3R2p60Y, (c) s5R2p60Y, (d) s5R2p60X, (e) s7R2p60X and (f) s7R2p80X.

To inquire about the variation of the local distribution of $Pr_{\mathit{eff}}$ with the number of species, portrayed in figure 16 are visualizations of these fields at $t_{tr}^{\ast }$ for realizations initiated with $Re_{0}=2000$ . The general observation is that high $Pr_{\mathit{eff}}$ values are intermingled over the entire mixing layer with low and negative values for all initial compositions. However, with increasing number of species, large values as well as negative values (i.e. uphill heat conduction) occur at a smaller number of locations, rendering the $Pr_{\mathit{eff}}$ field more uniform. That is, the transport of enthalpy by the species has a thermal-transport homogenizing effect on the mixture with increasing mixture complexity, whereas it has the opposite effect for a smaller number of species in the mixture.

5.6 Dissipation

The above analysis addressed both the generation of density gradients (and therefore of $\unicode[STIX]{x1D735}T$ ,  $\unicode[STIX]{x1D735}p$ and $\unicode[STIX]{x1D735}Y_{\unicode[STIX]{x1D6FC}}$ ) and the difference in transport properties when the complexity of a mixture was increased by adding species to a baseline mixture. Since the dissipation is the combined result of gradients and molecular transport properties, it is natural to inquire about the effect of species complexity on the dissipation.

Considering the entropy conservation equation (e.g. de Groot & Mazur Reference de Groot and Mazur1984) and using the transport coefficient calculation through the mixing rules (Masi et al. Reference Masi, Bellan, Harstad and Okong’o2013), a generalized formulation for the dissipation  $g$ , i.e. the irreversible entropy production, which is the source term in the entropy equation, has been derived (Masi et al. Reference Masi, Bellan, Harstad and Okong’o2013). The dissipation is written as the sum of three terms accounting for viscous, thermal and species-mass contributions, $g_{visc},g_{temp}$ and  $g_{mass}$ ,

(5.19) $$\begin{eqnarray}g=\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D707}}{T}}\left(2\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij}-{\textstyle \frac{2}{3}}\unicode[STIX]{x1D61A}_{kk}\unicode[STIX]{x1D61A}_{ll}\right)}_{g_{visc}}+\underbrace{\unicode[STIX]{x1D706}{\displaystyle \frac{1}{T^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{j}}}{\displaystyle \frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{j}}}}_{g_{temp}}+\underbrace{{\displaystyle \frac{1}{2}}\mathop{\sum }_{\unicode[STIX]{x1D6FD}=1}^{N}\mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{N}{\displaystyle \frac{R_{u}}{(-\mathbb{D}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}})}}{\displaystyle \frac{\unicode[STIX]{x1D70C}}{m_{\unicode[STIX]{x1D6FC}}Y_{\unicode[STIX]{x1D6FD}}}}\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}j}\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}j}}_{g_{mass}},\end{eqnarray}$$

where $(-\mathbb{D}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}m)/(Y_{\unicode[STIX]{x1D6FC}}m_{\unicode[STIX]{x1D6FD}})$ is a symmetric positive semidefinite matrix (Keizer Reference Keizer1987; Giovangigli, Matuszewsky & Dupoirieux Reference Giovangigli, Matuszewsky and Dupoirieux2011) and

(5.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}j} & = & \displaystyle -X_{\unicode[STIX]{x1D6FD}}\mathbb{D}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\left[\left(\frac{m_{\unicode[STIX]{x1D6FC}}}{m}\bar{\unicode[STIX]{x1D6FC}}_{T,\unicode[STIX]{x1D6FD}}^{b}-{\displaystyle \frac{m_{\unicode[STIX]{x1D6FD}}}{m}}\bar{\unicode[STIX]{x1D6FC}}_{T,\unicode[STIX]{x1D6FC}}^{b}\right){\displaystyle \frac{1}{T}}{\displaystyle \frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{j}}}+{\displaystyle \frac{m_{\unicode[STIX]{x1D6FC}}m_{\unicode[STIX]{x1D6FD}}}{mR_{u}T}}\left(\frac{v_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}}-\frac{v_{\unicode[STIX]{x1D6FD}}}{m_{\unicode[STIX]{x1D6FD}}}\right){\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{j}}}\right]\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{\unicode[STIX]{x1D6FE}=1}^{N-1}\left({\displaystyle \frac{m_{\unicode[STIX]{x1D6FD}}}{m_{\unicode[STIX]{x1D6FE}}}}\mathbb{D}_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FC}_{D\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}}-{\displaystyle \frac{m_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FE}}}}\mathbb{D}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FC}_{D\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\right)\left[{\displaystyle \frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x2202}x_{j}}}-Y_{\unicode[STIX]{x1D6FE}}\mathop{\sum }_{\unicode[STIX]{x1D6FF}=1}^{N-1}\left({\displaystyle \frac{m}{m_{\unicode[STIX]{x1D6FF}}}}-{\displaystyle \frac{m}{m_{N}}}\right){\displaystyle \frac{\unicode[STIX]{x2202}Y_{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x2202}x_{j}}}\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}j}=0$ and $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}j}=-\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}j}$ . The $g$ form of (5.19) has been previously used by Masi et al. (Reference Masi, Bellan, Harstad and Okong’o2013).

Figure 17. Planar averages in $(x_{1},x_{3})$ planes of the three modes, (a)  $\langle g_{\mathit{visc}}\rangle$ , (b)  $\langle g_{\mathit{temp}}\rangle$ and (c)  $\langle g_{\mathit{mass}}\rangle$ , and of (d)  $g$ at the respective $t_{tr}^{\ast }$ . Units are $\text{W}~\text{m}^{-3}~\text{K}^{-1}$ . Same colour legend as in figure 1.

The $g_{visc}$ ,  $g_{temp}$ and $g_{mass}$ ( $x_{1},x_{3}$ ) planar averages are here computed for the realizations presented in figure 1, and are separately illustrated as a function of $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}$ in figure 17(a), (b) and (c), respectively. The values of the transport coefficients are not ${\mathcal{F}}$ -scaled and units are $\text{W}~\text{m}^{-3}~\text{K}^{-1}$ . Comparing the magnitude of these terms, $\langle g_{visc}\rangle$ is the smallest by approximately a factor of 5 with respect to $\langle g_{temp}\rangle$ and $\langle g_{mass}\rangle$ ; this ordering is attributed to the relatively modest values of $Re_{m,tr}$ compared to fully turbulent flows. The $(x_{1},x_{3})$ planar average $\langle g_{visc}\rangle$ profiles are not symmetric with respect to $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}=0$ but are rather heavily skewed to the upper free-stream side where the fluid is lighter and is thus more amenable to experiencing larger velocities and perhaps larger velocity gradients. While the values of all p60 simulations nearly coincide, a larger $p_{0}$ value (case s7R2p80X) induces a slightly larger $\langle g_{visc}\rangle$ with respect to p60 cases on the lighter free-stream side, primarily caused by the more elevated values of the dynamic viscosity in the core of the mixing layer (not shown). Even though the single-species viscosity of the minor species can be largely different from that of the major species, the mixture viscosity (computed using the Wilke method, equation (A 1)) is strongly dependent on the molar fractions, explaining the small influence of the minor species for the p60 cases.

The cross-stream profiles of $\langle g_{temp}\rangle$ exhibit similar values for all realizations, a fact which is conjectured to reflect the dominance of $X_{\unicode[STIX]{x1D6FC}}$ , and thus of the major species’ contribution in the computation of $\unicode[STIX]{x1D706}$ (see (A 4)). The $\langle g_{temp}\rangle$ profiles are very heavily skewed towards the hotter upper free stream where according to figure 6 is the region of the larger $\unicode[STIX]{x1D735}T$ , and unlike $\langle g_{visc}\rangle$ , $\langle g_{temp}\rangle$ is unaffected by  $p_{0}$ .

Among all dissipation modes, $\langle g_{mass}\rangle$ has the largest values; it also exhibits a slight increase with the number of species as a result of the larger interdiffusion in the seven- and five-species cases with respect to the two- and three-species realizations. The $\langle g_{mass}\rangle$ profile is less skewed towards the upper stream than those of $\langle g_{visc}\rangle$ and $\langle g_{temp}\rangle$ , and unlike the latter shows a substantial contribution from the lower-stream side. While all $\langle g_{mass}\rangle$ profiles nearly coincide on the upper-stream side of the $\langle g_{mass}\rangle$ peak, there is substantial variability among realizations on the lower-stream side that is indicative of the changes in mass-diffusion characteristics as the number of species varies. Compared to s7R2p60X, case s7R2p80X exhibits relatively lower $\langle g_{mass}\rangle$ values because of the reduced values of the diffusion coefficients in higher- $p$ environments.

Of note, $\langle g_{visc}\rangle$ ,  $\langle g\rangle$ and $\langle g_{mass}\rangle$ all culminate at $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}\approx 5$ , suggesting that most mixing layer activity occurs at this location. As a result, $\langle g\rangle$ , portrayed in figure 17(d), also culminates at the $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}\approx 5$ location and, while it retains the near-coincidence of all profiles on the upper-stream side of the layer, it also shows the number-of-species variability effect on the lower-stream side of $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}\approx 5$ . Thus, while the effect of the number of species on $g$ is modest, it is not negligible.

5.7 Relationship between species and temperature

In simulations of reacting flows using reduced kinetic mechanisms, one of the important considerations is maintaining the relationship between $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ (Bellan Reference Bellan2017b ). The DNS database generated here provides a unique opportunity to inquire about the possible deviations in this relationship as the mixture composition is considered in coarser detail. For example, possible causes of deviation would be the observed uphill diffusion of neglected minor species as the composition is simplified. To determine the relationship between $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ and  $\unicode[STIX]{x1D703}$ , we compute j.p.d.f.s which, being pointwise functions, provide very detailed information. A less stringent test of this relationship is obtained by computing conditional averages of $\unicode[STIX]{x1D703}$ given  $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ . These investigations are performed at $t_{tr}^{\ast }$ and they address the relationship for both major and minor species.

Figure 18. J.p.d.f. of $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ for heptane at $t_{tr}^{\ast }$ , in logarithmic scale: (a) s2R2p60Y, (b) s5R2p60X and (c) s7R2p60X. The p.d.f. is computed inside the mixing layer, defined according to (5.4).

Figure 19. J.p.d.f. in logarithmic scale of $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ for cases s7R2p60X (top line) and s5R2p60X (bottom line): (a,d) O₂, (b,e) CO₂ and (c,f) H₂O. The p.d.f. is computed inside the mixing layer, according to (5.4), at $t_{tr}^{\ast }$ .

Figure 18 illustrates the j.p.d.f.s for n-heptane in cases s2R2p60Y, s5R2p60X and s7R2p60X; discussing the j.p.d.f. for n-heptane does not imply that n-heptane will preserve its chemical structure at the higher $T$ values in the domain rather than pyrolysing. Upon careful scrutiny, the j.p.d.f. for s2R2p60Y extends perceptibly to $\unicode[STIX]{x1D703}<0$ regions; however, these $\unicode[STIX]{x1D703}<0$ regions are negligible for s5R2p60X and s7R2p60X, consistent with the discussion of § 5.5. Examination of figure 18 shows that, as expected given the initial conditions wherein n-heptane resides in the lower $T$ stream, the j.p.d.f. exhibits a decreasing relationship between $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}}$ and  $\unicode[STIX]{x1D703}$ ; the width of the j.p.d.f. becomes larger with decreasing $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}}$ and increasing  $\unicode[STIX]{x1D703}$ , this being a manifestation of the mixing and heat transfer occurring between the two streams. The j.p.d.f. values corresponding to the smallest $\unicode[STIX]{x1D6EF}_{\text{C}_{7}\text{H}_{16}}$ values and largest  $\unicode[STIX]{x1D703}$ values correspond to the largest values of $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}>0$ at which n-heptane is present in the upper stream (see figure 6 a,h). The j.p.d.f.s do not show much visual variation from s2R2p60Y to s5R2p60X, but it is visually apparent that the width of the j.p.d.f. increases from s5R2p60X to s7R2p60X. However, comparison of figures 18(b) and 18(c) indicates that regions with large j.p.d.f. values are less susceptible to change by the addition of species if these species are minor, and the width increase seems to affect only the tails of the j.p.d.f.; that is, the presence of the additional species decreases the preponderance of n-heptane as $\unicode[STIX]{x1D703}$ increases, this situation representing regions at larger $x_{2}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714},0}>0$ locations within the mixing layer (see figure 6 a,h). To show that the findings for n-heptane are not an isolated conclusion, the j.p.d.f.s for O₂, CO₂ and H₂O are displayed in figure 19. Since their corresponding $Y_{\unicode[STIX]{x1D6FC}}$ is initially larger in the upper stream compared to the lower stream, the j.p.d.f.s exhibit an increasing relationship between $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ and  $\unicode[STIX]{x1D703}$ . These j.p.d.f.s visually confirm the findings of figure 18: minor species addition widens the tails of the j.p.d.f.s but does not substantially affect the j.p.d.f.s large value region. Finally, figure 20 presents for information the j.p.d.f.s of CO and H₂ computed for s7R2p60X, the single simulation in which these species are present. Despite their initial $Y_{\unicode[STIX]{x1D6FC}}$ value being smaller than those of CO₂ and H₂O, the j.p.d.f.s of CO and H₂ are the widest, this being due to uphill diffusion, which for CO and H₂ enlarged the range of $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ values more than for CO₂ and H₂O (see figure 5).

Figure 20. J.p.d.f. in logarithmic scale of $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ for case s7R2p60X: (a) CO and (b) H₂. The p.d.f. is computed inside the mixing layer, according to (5.4), at  $t_{tr}^{\ast }$ .

To quantify the differences between the j.p.d.f.s obtained as the composition is simplified, relative errors were calculated. The relative deviations of the j.p.d.f.s, $dev_{\unicode[STIX]{x1D6FC}}$ , for the four common species to s5R2p60X and s7R2p60X, is computed using

(5.21) $$\begin{eqnarray}\mathit{dev}_{\unicode[STIX]{x1D6FC}}=\frac{\sqrt{\left[(\text{j.p.d.f.})_{\unicode[STIX]{x1D6FC}}^{\text{s7R2p60X}}-(\text{j.p.d.f.})_{\unicode[STIX]{x1D6FC}}^{\text{s5R2p60X}}\right]^{2}}}{(\text{j.p.d.f.})_{\unicode[STIX]{x1D6FC}}^{\text{s7R2p60X}}+\unicode[STIX]{x1D716}},\end{eqnarray}$$

where the small parameter $\unicode[STIX]{x1D716}=10^{-16}$ is added to the denominator in order to avoid division by zero. The addition of $\unicode[STIX]{x1D716}$ does not affect the $\mathit{dev}_{\unicode[STIX]{x1D6FC}}$ evaluation in regions of large j.p.d.f. values representing the significant combustion localities, but does affect the regions of small j.p.d.f. values in which the relative deviation thus computed may be inaccurate; however, this range of values has negligible contribution to combustion. Figure 21 shows that for all species under consideration – n-heptane, O₂, CO₂ and H₂O – $\mathit{dev}_{\unicode[STIX]{x1D6FC}}$ is very small in the regions of maximum and large j.p.d.f. values and is only large on the periphery of the j.p.d.f.s. These peripheral values occur primarily where the temperature is too low for combustion to occur. The fact that the deviation is small in the region of larger probability is here interpreted as a confirmation that conditional averages may be considered as a sufficient test of the change in j.p.d.f.s as the composition of a mixture of species is refined. The $\langle \unicode[STIX]{x1D703}\mid \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}\rangle$ conditional averages are depicted in figure 22 and show that, with the exception of the uphill diffusion regime of CO₂ and H₂O, where $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}>1$ , the conditional averages are primarily similar. Thus, we conclude that the simplification of the mixture composition by eliminating minor species does not affect the relationship between the remaining mass fractions and temperature, with the exception of the regions of uphill diffusion of those species that experience it, and in those regions the conditional averages of those species are different.

Figure 21. Estimation of the relative deviation (in logarithmic scale) between the j.p.d.f.s of $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ for cases s5R2p60X and s7R2p60X (defined by (5.21)), for the same species: (a) O₂, (b) C₇H₁₆, (c) CO₂ and (d) H₂O.

Figure 22. Conditional average of the normalized temperature $\unicode[STIX]{x1D703}$ given $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FC}}$ at $t_{tr}^{\ast }$ for cases (a) s7R2p60X and (b) s5R2p60X.