3.1. SGS mixing on the jet centreline

The profiles of the mean filtered scalars, $\langle \langle \phi _1 \rangle _L \rangle$ and $\langle \langle \phi _2 \rangle _L \rangle$, on the jet centreline for Case I are shown in figure 3. The difference in the mean filtered scalars between different filter scales were negligible. The mean filtered scalars were very close to the mean scalar profiles for the filter scales considered. The general trends were similar for other cases (figures not shown).

Figure 3.

Centreline profiles of the filtered mean scalar for Case I.

The profiles of the mean SGS scalar variances, $\langle \langle \phi _1^{''2} \rangle _L \rangle$ and $\langle \langle \phi _2^{''2} \rangle _L \rangle$, on the jet centreline for Case I are shown in figure 4. The SGS scalar variances evolved similarly as the scalar variances. Their values, however, were significantly smaller than the scalar variances. The peak value of $\langle \langle \phi _1^{''2} \rangle _L \rangle$ was approximately 8 %, 20 % and 32 % of $\langle \phi _1^{'2} \rangle$ for the three filter widths ($\varDelta =0.25,0.53,0.8\ \textrm {mm}$), respectively, while $\langle \langle \phi _2^{''2} \rangle _L \rangle$ is 6.5 %, 17 % and 27 % of $\langle \phi _2^{'2} \rangle$. The general trends were similar for the other cases (figures not shown). The relative magnitudes of the mean SGS variances among the cases were similar to those of the scalar variances (Li et al. Reference Li, Yuan, Carter and Tong2017), with the peak values generally larger for cases with the higher velocity ratio and larger annulus width.

Figure 4.

Centreline profiles of the filtered mean SGS variance for Case I.

The results for the f.j.d.f. are given as a conditional mean, $\langle f|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$, referred to simply as f.j.d.f. hereafter for convenience. The f.j.d.f. conditional on the small SGS variance on the centreline for Case I are shown in figure 5. The values of the conditional variables, $\langle \phi _1\rangle _L$ and $\langle \phi _1''^2\rangle _L$, are given in each figure. The value of $\langle \phi _1\rangle _L$ was chosen to be its local mean, $\langle \langle \phi _1\rangle _L\rangle$, at the physical location. We used greyscales and isocontours to represent the f.j.d.f. The outermost contour represents the boundary within which the f.j.d.f. integrates to 99 %. The f.j.d.f. should be confined to a triangle in the $\phi _1$–$\phi _2$ space with the vertices at (1, 0), (0, 1) and (0, 0), where the coordinates denote the sample-space variables for $\phi _1$, $\phi _2$ and $\phi _3$, respectively. For small SGS variance, the f.j.d.f. was always unimodal and appeared to have a Gaussian-like shape. For $x/d<5$ (for convenience we use $d$ to denote the inner diameter of the inner tube $D_{ji}$ in table 1), it was centred on the $\phi _1$–$\phi _2$ mixing line connecting (1, 0) and (0, 1), which indicated that the SGS scalars contained little co-flow air (not shown). Further downstream the f.j.d.f. moved away from the mixing line towards (0, 0) due to mixing with the co-flow air. The evolution of the peak of the f.j.d.f. was generally consistent with filtered mean values. The general trends were similar for other cases (figures not shown).

Figure 5.

Evolution of the scalar f.j.d.f. conditional on the small SGS variance on the centreline for Case I. The last three contours correspond to boundaries within which the f.j.d.f. integrates to 90 %, 95 % and 99 %, respectively throughout the paper. The rest of the contours scale linearly over the remaining range. The filter width is 0.53 mm hereafter.

For large SGS variance (generally more than four times the mean SGS variance), the f.j.d.f. close to the jet exit ($x/d<8$) was unimodal (figure 6), with the peak near (1, 0) and a long tail. The area of the f.j.d.f. was much larger than for the small SGS variance, consistent with the relative magnitudes of the SGS variance. At $x/d=10.8$, the f.j.d.f. became bimodal for both Cases I and II, which indicated that the SGS mixing was to a large extent between two distinct and segregated SGS mixtures. The two SGS scalars ($\phi _1^{''}$ and $\phi _2^{''}$) were negatively correlated at this location. At $x/d=14.6$, the two peaks became closer and were further away from the mixing line due to the presence of more co-flow air. The ridgeline of the f.j.d.f. was horizontal for both cases with the f.j.d.f. of Case II having a more slender shape, consistent with better molecular mixing due to the existence of mean shear between the centre jet and the annular stream. Further downstream ($x/d=23.6$), the f.j.d.f. was still bimodal while moving closer to (0, 0). The two peaks were also much closer with the SGS scalars becoming positively correlated, which indicated that they were well mixed, and that they were mixing largely in unison with the co-flow air.

Figure 6.

Evolution of the scalar f.j.d.f. conditional on the large SGS variance on the centreline for Case I (a,c,e,g) and Case II (b,d,f,h).

These results were in contrast to the j.p.d.f. (for the smaller annulus cases), which was always unimodal on the centreline (Li et al. Reference Li, Yuan, Carter and Tong2017). The different behaviours of the j.p.d.f. and f.j.d.f. were similar to those of the p.d.f. (unimodal) and f.d.f. (unimodal and bimodal) in binary mixing. Therefore, similar to the binary SGS mixing, there also existed two regimes for the three-scalar SGS mixing: for small SGS variance the SGS scalars were relatively well mixed, whereas for large SGS variance the SGS scalars were highly segregated.

The physics behind the two regimes is also likely to be similar to that of binary SGS mixing. Previous results (Tong Reference Tong2001; Wang & Tong Reference Wang and Tong2002) on binary SGS mixing have shown that for small SGS variance, the spectral transfer of the scalar variance is in equilibrium, i.e. the instantaneous spectral transfer rate of the scalar variance is comparable to or smaller than the instantaneous locally averaged scalar dissipation rate, whereas for large SGS variance, the SGS variance is in spectral non-equilibrium with the spectral transfer rate much larger. The bimodal f.j.d.f. observed here indicated that in three-scalar SGS mixing, spectral non-equilibrium also exists for large SGS variance, and is responsible for the bimodal f.j.d.f. Furthermore, there exists a ramp–cliff structure in the SGS scalar, $\phi _1^{''}$, which separates the two SGS mixtures. Bimodal f.j.m.d.f. of mixture fracture and temperature has also been observed in turbulent partially premixed flames (Wang et al. Reference Wang, Tong, Barlow and Karpetis2007a; Cai et al. Reference Cai, Wang, Tong, Barlow and Karpetis2009).

The loss of the initial three-scalar mixing configuration for small SGS variance suggested that for a SGS volume having the three-scalar mixing configuration, subsequent reduction of the SGS variance due to SGS mixing can destroy the configuration, rendering the SGS volume well mixed. The implication for a non-premixed turbulent flame is that the loss of the mixing configuration which generally corresponds to non-premixed flamelets, would imply destruction of such flamelets (Wang et al. Reference Wang, Tong, Barlow and Karpetis2007a). Our previous work using the data of the Sandia Flames (Wang et al. Reference Wang, Tong, Barlow and Karpetis2007a; Cai et al. Reference Cai, Wang, Tong, Barlow and Karpetis2009, Reference Cai, Barlow, Karpetis and Tong2011a) have shown that for small SGS variance, the SGS mixture fraction fluctuations are smaller than the reaction zone width in the mixture fraction space, based on 10 % of the peak CO oxidation reaction rate obtained in laminar flame calculations of Frank, Kaiser & Long (Reference Frank, Kaiser and Long2002), i.e. there are turbulent fluctuations inside the reaction zones. These results further showed that for small SGS variance, the SGS mixture fraction, the scalar dissipation (Wang et al. Reference Wang, Tong, Barlow and Karpetis2007a; Cai et al. Reference Cai, Wang, Tong, Barlow and Karpetis2009), and the scalar and temperature diffusion (Cai et al. Reference Cai, Barlow, Karpetis and Tong2011a) in burning SGS flames, are consistent with the distributed reaction zones (Cai et al. Reference Cai, Wang, Tong, Barlow and Karpetis2009). Therefore, the loss of the mixing configuration also implies generation of such reaction zones in burning SGS fields. In SGS fields having small SGS variance and containing extinguished mixtures, the results imply that premixed flamelets and stratified reaction layers can exist, as suggested by the temperature dissipation (Cai et al. Reference Cai, Barlow, Karpetis and Tong2011a).

The general trends for larger annulus cases (figure 7) are generally similar to those of the smaller annulus. The f.j.d.f. extends further along the $\phi _1$–$\phi _2$ mixing line before bending towards (0, 0), which indicates slower progression of SGS mixing for the larger annulus cases. The f.j.d.f. is unimodal for Case III at $x/d=10.9$ while it is bimodal for Case IV, in spite of the smaller value of the SGS variance chosen for the latter (because the mean SGS variance is smaller). This difference is in contrast to the j.p.d.f. on the centreline in that j.p.d.f. is bimodal at some locations for Case III but is always unimodal for Case IV (Li et al. Reference Li, Yuan, Carter and Tong2017). Moving downstream ($x/d=14.6$), the f.j.d.f. also becomes bimodal for Case III. Similar to the smaller annulus cases, the f.j.d.f. is bimodal for both cases further downstream ($x/d=23.6$) and the two SGS scalars are positively correlated. These results show that similar to binary SGS mixing (Tong Reference Tong2001; Wang & Tong Reference Wang and Tong2002), the f.j.d.f. can be bimodal even when the j.p.d.f. is unimodal everywhere (Cases II and IV). The earlier appearance of the bimodal f.j.d.f. for Case IV is likely due to the stronger SGS transport resulting from the SGS velocity and scalar fluctuation generated by the mean shear between the centre stream and the annular stream and between the annular stream and the co-flow.

Figure 7.

Evolution of the scalar f.j.d.f. conditional on the large SGS variance on the centreline for Case III (a,c,e,g) and Case IV (b,d,f,h).

3.2. Cross-stream SGS profiles

The radial profiles of the mean SGS variances are shown in figure 8. They have similar shapes and peak locations to the scalar variances (Li et al. Reference Li, Yuan, Carter and Tong2017). Similar to the variance of $\phi _1$, the peak location of $\langle \langle \phi _1^{''2} \rangle _L \rangle$ moves towards the centreline as $x/d$ increases. The peak value of $\langle \langle \phi _1^{''2} \rangle _L \rangle$ decreases as $x/d$ increases for all cases, whereas the peak value of $\phi _1$ variance for Case III increases from $x/d=3.29$ to $x/d=6.99$. This difference is likely because the scalar integral length scale increases with $x/d$; for a fixed filter width, the fraction of the variance contained in the SGS decreases. The mean SGS variance of $\phi _2$ also have the same trend as the variance of $\phi _2$. The peak values decrease with increasing $x/d$. The mean SGS variance, however, decreases faster than the variance due to the increasing integral length scale. The relative magnitudes of the mean SGS variances among the cases are also similar to those of scalar variances. The peak values of both $\phi _1$ and $\phi _2$ mean SGS variances are smaller and decrease faster for Case II (IV) than for Case I (III). However, the profiles of $\langle \langle \phi _1^{''2} \rangle _L \rangle$ are wider for Case II (IV) than for Case I (III). The peak values are generally smaller and decrease faster for the smaller annulus cases than for the larger annulus cases, except that the peak value of $\langle \langle \phi _1^{''2} \rangle _L \rangle$ at $x/d=3.29$ is larger for Case I than for Case III. The general trends for the other filter width are similar but with different magnitudes.

Figure 8.

Cross-stream filtered mean SGS variance profiles.

The SGS correlation coefficient between $\phi _1$ and $\phi _2$,

(3.3)\begin{equation} \rho=\frac{\langle \langle\phi_1''\phi_2''\rangle_L \rangle}{\langle \langle \phi_1^{''2} \rangle_L\rangle^{{1}/{2}}\langle \langle \phi_1^{''2} \rangle_L \rangle^{{1}/{2}}}, \end{equation}

is shown in figure 9. The correlation coefficient generally has the value of negative one close to the centreline, increasing towards unity far away from the centreline. Close to the centreline, $\phi _1''$ and $\phi _2''$ are anticorrelated ($\rho \approx -1$) because there is virtually no $\phi _3$. It begins to increase when $\phi _1$ and $\phi _2$ begin to mix with $\phi _3$, and approaches unity far away from the centreline, which indicates that the two SGS scalars are well mixed and are in phase. The correlation at $x/d=6.99$ begins to dip at approximately $r/d=1.8$, due to residual measurement noise, i.e. after noise correction. At $x/d=3.29$, the results for both close to the centreline and towards the edge of the jet (at approximately $r/d=1$) are not shown, because the scalar fluctuations are small and the measured correlation coefficient is dominated by the residual measurement noise. The correlation coefficient begins to increase at smaller $r/d$ values at $x/d=6.99$ than at $x/d=3.29$, which results from the progression of scalar mixing. The differences between Cases I and II and between Cases III and IV are small. Comparisons between Cases I and III and between Cases II and IV show that the evolution of the correlation coefficient is much slower for the larger annulus than for the smaller annulus.

Figure 9.

Cross-stream SGS correlation coefficient between $\phi _1$ and $\phi _2$.

3.3. Cross-stream f.j.d.f. and conditionally filtered diffusion

In this section we discuss the f.j.d.f. and the conditionally filtered diffusion at several radial locations in the near field ($x/d=3.29$ and $6.99$). The conditionally filtered diffusion is given as conditional means, $\langle \langle D_1\nabla ^2\phi _1|\hat \phi _1,\hat \phi _2\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$ and $\langle \langle D_2\nabla ^2\phi _2|\hat \phi _1,\hat \phi _2\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$. The conditionally filtered scalar diffusion terms in the f.j.d.f. equation transport the f.j.d.f. in the scalar space; therefore, the conditionally filtered diffusion represents the two components of a diffusion (or transport) velocity in the scalar space. We present the conditionally filtered diffusion as the diffusion velocity, represented by streamlines and magnitude isocontours in the same way that conditional diffusion was represented (Cai et al. Reference Cai, Dinger, Li, Carter, Ryan and Tong2011b; Li et al. Reference Li, Yuan, Carter and Tong2017). Both conditionally filtered diffusion terms are non-dimensionalized by the square root of $\phi _1$ SGS variance and the conditionally filtered dissipation time scale for $\phi _1''$, ${\langle \phi _1''^2\rangle _L}/{\langle \langle \chi _1\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle }$.

At $x/d=3.29$, close to the jet centreline (not shown) the f.j.d.f. is largely limited to the $\phi _1$–$\phi _2$ mixing line. The spread of the measured f.j.d.f. is largely due to the measurement uncertainties. For small SGS variance at $r/d=0.372$ (figure 10), the f.j.d.f. is unimodal with the peak near the $\phi _1$–$\phi _2$ mixing line. The diffusion streamlines converge to the peak of the f.j.d.f. At $r/d=0.496$, the f.j.d.f. still has a Gaussian-like shape but the peak has already moved away from the $\phi _1$–$\phi _2$ mixing line, which indicates that $\phi _1''$ and $\phi _2''$ are well mixed and there is some co-flow air present. The diffusion streamlines again converge to the stagnation point near ($\langle \phi _1\rangle _L$, $\langle \langle \phi _2\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$), which is also the peak of f.j.d.f. Further away from the centreline (at $r/d=0.703$), the peak of the f.j.d.f. moves closer to $(0,0)$, consistent with the evolution of the filtered mean values. The general trends for other cases are also similar (figures not shown).

Figure 10.

The f.j.d.f. (a,c,e) and conditionally filtered diffusion streamlines (b,d,f) conditional on the small SGS variance at $x/d=3.29$ for Case I. The filtered scalar values ($\langle \phi _1\rangle _L$, $\langle \langle \phi _2\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$) are denoted by a bullet in the streamline figures.

For large SGS variance, the f.j.d.f. close to the centreline (not shown) is concentrated at (1, 0) with a tail extending toward (0, 1), which indicates that the SGS mixing is largely limited to between $\phi _1$ and $\phi _2$ but with only a small amount of $\phi _3$. At $r/d=0.372$ (figure 11), the f.j.d.f. begins to extend toward (0, 0) for both Cases I and II. A diffusion manifold begins to emerge, and the diffusion streamlines converge to a stagnation point that is different from both the local filtered mean scalars and the peak of f.j.d.f. For Case II, it appears that a second peak begins to emerge on the left-hand side of the f.j.d.f. At $r/d=0.496$ (figure 12), the f.j.d.f. has become bimodal for both cases with stronger bimodality for Case I, consistent with the larger SGS variance. The right peak is close to the $\phi _1$–$\phi _2$ mixing line without much $\phi _3$, while the left peak contains little $\phi _1$, which indicates that within the SGS field the two mixtures coming from the two mixing layers are segregated with a sharp interface (ramp–cliff) between them. The two SGS scalars are negatively correlated. The diffusion streamlines first move towards a well-defined and bell-shaped manifold, then continue along it to a stagnation point, which is again different from the filtered mean composition. Thus, the SGS scalars for large SGS variance have a structure due to the mixing configuration of the coaxial jet, whereas for small SGS variance the three scalars are quite well mixed and the configuration is lost. The curvature of the manifold appears to be larger for Case I than for Case II.

Figure 11.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=3.29$ and $r/d=0.372$ for Case I (a,b) and Case II (c,d).

Figure 12.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=3.29$ and $r/d=0.496$ for Case I (a,b) and Case II (c,d).

Further away from the centreline, the right peak of the f.j.d.f. becomes weaker. At $r/d=0.703$ (figure 13), the right peak completely disappears for Case I whereas a weak right peak still exists for Case II, which indicates that the bimodal f.j.d.f. exists over a wider range of physical locations for Case II. This trend is different from the cross-stream evolution of j.p.d.f. (Li et al. Reference Li, Yuan, Carter and Tong2017), which is bimodal at some locations for Case I whereas it is always unimodal for Case II. (This difference will be further discussed in detail along with the results at $x/d=6.99$.) The left peak of the f.j.d.f. has already moved very close to $(0,0)$. The streamlines converge directly to a stagnation point from larger $\phi _1$ values (from the right), but appear to move to a manifold first from smaller $\phi _1$ values (from the left) and then approach the stagnation point. The f.j.d.f. also becomes unimodal for Case II further away from the centreline.

Figure 13.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=3.29$ and $r/d=0.703$ for Case I (a,b) and Case II (c,d).

Moving downstream to $x/d=6.99$, some co-flow air has reached the centreline (Li et al. Reference Li, Yuan, Carter and Tong2017). For small SGS variance, the conditional f.j.d.f. (not shown) again has a Gaussian-like shape and is concentrated near the filtered mean scalar values. The conditionally filtered diffusion streamlines (also not shown) mostly converge to a stagnation point. The trends of the f.j.d.f. and the diffusion streamlines with increasing $r/d$ values are similar to those at $x/d=3.29$. The f.j.d.f. and the conditionally filtered diffusion again indicate that the SGS scalars are relatively well mixed.

For large SGS variance, as shown in figure 6, the f.j.d.f. on the jet centreline is still concentrated near (1, 0), but extends further away from it, which indicates the penetration of both $\phi _2$ and $\phi _3$. Moving away from the centreline, the f.j.d.f. extends further towards lower $\phi _1$ values and bends further towards (0, 0). A second peak begins to emerge and the f.j.d.f. becomes bimodal at $r/d=0.248$ and $r/d=0.207$ for Case I and Case II (figures not shown), respectively. At $r/d=0.376$ (figure 14), similar to $x/d=3.29$, the f.j.d.f. is strongly bimodal for both Cases I and II with the left peak further away from the mixing line, which indicates that the SGS field contains predominately the $\phi _1$–$\phi _2$ mixture and the $\phi _1$–$\phi _2$–$\phi _3$ mixture coming from the two mixing layers. Again the mixtures are segregated with a cliff between them. Similar to $x/d=3.29$, the diffusion streamlines first converge to a manifold, and then continue on the manifold at a lower rate towards a stagnation point. At $r/d=0.538$ (figure 15), the f.j.d.f. becomes unimodal for Case I whereas it is still bimodal for Case II, although the SGS variance is again smaller for Case II. There is also a well-defined curved manifold for the conditional diffusion for each case. Towards the edge of the jet ($r/d=0.827$), the f.j.d.f. is still bimodal for Case II but with $\phi _1^{''}$ and $\phi _2^{''}$ positively correlated (figure 16). For Case I, the diffusion streamlines to the left of the stagnation point converge to a manifold and then to the stagnation point, whereas those to the right converge to the stagnation point. For Case II, the curved manifold is better defined. Moving further away from the centreline, the f.j.d.f. is also unimodal for Case II (not shown).

Figure 14.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.376$ for Case I (a,b) and Case II (c,d).

Figure 15.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.538$ for Case I (a,b) and Case II (c,d).

Figure 16.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.827$ for Case I (a,b) and Case II (c,d).

For the larger annulus at $x/d=3.29$ (figure 17), the general trends are similar to the smaller annulus cases. The main difference is that the peak of the f.j.d.f. evolves along the $\phi _1$–$\phi _2$ mixing line and reaches $(0,1)$ before bending towards $(0,0)$. At $r/d=0.331$, the f.j.d.f. peaks near $(1,0)$ while the ridgeline stays on the $\phi _1$–$\phi _2$ mixing line. A second peak begins to emerge on the left for Case IV. It is strongly bimodal for both Cases III and IV at $r/d=0.488$. The f.j.d.f. is nearly symmetric with respect to the $\phi _1$–$\phi _2$ mixing line for Case III whereas it extends towards $(0,0)$ on the left for Case IV. The right peak disappears for Case III at $r/d=0.62$ whereas a weak right peak still exists for Case IV. The peak near $(0,1)$ indicates that $\phi _1$ and $\phi _3$ are separated by pure $\phi _2$, and there are two separate mostly binary mixing processes. A diffusion manifold begins to emerge for Case IV at $r/d=0.62$ (figure 18), whereas there is no sign of a curved manifold for Case III. The f.j.d.f. would also become unimodal for Case IV moving further away from the centreline.

Figure 17.

The f.j.d.f. conditional on the large SGS variance at $x/d=3.29$ for Case III (a,c,e) and Case IV (b,d,f).

Figure 18.

Conditionally filtered diffusion streamlines conditional on the large SGS variance at $x/d=3.29$ and $r/d=0.62$ for Case III (a) and Case IV (b).

For the larger annulus at $x/d=6.99$ (figures 19–21), the general trends are again similar to the smaller annulus cases. The bimodal f.j.d.f. exists over a wider range of physical locations for Case IV than for Case III, again a trend different from that of the j.p.d.f. The curvature of the diffusion manifold is also larger for Case III than for Case IV, consistent with better mixing for Case IV.

Figure 19.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.331$ for Case III (a,b) and Case IV (c,d).

Figure 20.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.496$ for Case III (a,b) and Case IV (c,d).

Figure 21.

The f.j.d.f. (a,c) and conditionally filtered diffusion streamlines (b,d) conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.703$ for Case III (a,b) and Case IV (c,d).

The above results show that the strongest bimodal f.j.d.f. occurs in Cases I and III at physical locations near the peaks of the mean SGS variance of $\phi _1$. These cases have higher peak mean SGS variance values than Cases II and IV. The strongest bimodal j.p.d.f. also occurs (in Cases I and III, which have higher peak scalar variance values) near the peaks of the variance of $\phi _1$ (Li et al. Reference Li, Yuan, Carter and Tong2017). Therefore, a higher variance (as well as mean SGS variance) is conducive to a bimodal j.p.d.f. (f.j.d.f.). On the other hand, the f.j.d.f. is bimodal over a wider range of physical locations for Cases II and IV than for Cases I and III, in spite of the weaker bimodality at the location of the peak mean SGS variance. Furthermore, while Cases II and IV have wider SGS variance profiles with higher values towards the edge of the jets than Cases I and III, thereby favouring bimodal f.j.d.f., there are also instances (e.g. at $x/d=3.29$ and $r/d=0.372$ for the smaller annulus shown in figure 11, and at $x/d=3.29$ and $r/d=0.331$ for the larger annulus shown in figure 17) where the f.j.d.f. is unimodal for Case I (larger mean SGS variance) and is bimodal for Case II (relatively smaller SGS variance). Therefore, a large SGS variance and the non-equilibrium spectral transfer is only one important factor determining the bimodality of the f.j.d.f. The other important factor is the length scales of the turbulent fluctuations, which influence the SGS scalar structure. Cases II and IV have two shear layers; therefore, the length scales of the turbulent (both velocity and scalar) fluctuations are smaller than those of Cases I and III, which have a single shear layer. Therefore, for a fixed filter width and SGS variance value, which correspond to a fixed ratio of the filter width to the centre jet length scale, the largest SGS scales for Cases II and IV, which have a larger length scale ratio between the annulus and the centre jet, have a stronger influence on the SGS structure, and are more likely to result in a bimodal f.j.d.f.

Similar to the conditional diffusion for the j.p.d.f., for large SGS variance the diffusion streamlines first converge to a manifold and then continue along it towards a stagnation point. Thus, there are also two mixing processes in the SGS mixing, one slow and one fast. This phenomenon is related to the structure of the SGS scalars, in which $\phi _1$ is dominated by a ramp–cliff structure (Tong Reference Tong2001; Wang & Tong Reference Wang and Tong2002) and $\phi _2$ by a Gaussian-like scalar profile, both large-scale structures. The ramp–cliff structure is generated by a large-scale convergent–divergent separatrix (Holzer & Siggia Reference Holzer and Siggia1994; Tong & Warhaft Reference Tong and Warhaft1994) acting on a mean (or large-scale) scalar gradient. The large-scale scalar structures in $\phi _2$ are also likely due to the same reason. Smaller scalar fluctuations may be viewed as being superimposed on these structures. These fluctuations are likely due to mixing of small-scale (somewhat homogeneous) scalar fields by velocity fluctuations of smaller scales. Consequently, the SGS scalars diffuse (relax) towards the largest SGS structures first before the diffusion of these structures move the streamlines towards the stagnation point. Thus, the slow and fast processes in SGS mixing are likely the results of large- and small-scale velocity fluctuations acting on large- and small-scale scalar gradients, respectively. The large-scale scalar structures also form a mixing path in the scalar space along which mixing of $\phi _1$ and $\phi _3$ occurs.

3.4. Conditionally filtered dissipation and cross-dissipation

In this section we discuss the conditionally filtered dissipation, which are also given as conditional means, $\langle \langle \chi _i|\hat \phi _1,\hat \phi _2\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$ and $\langle \langle \chi _{12}|\hat \phi _1,\hat \phi _2\rangle _L|\langle \phi _1\rangle _L, \langle \phi _1''^2\rangle _L\rangle$. For convenience, the conditionally filtered dissipation and cross-dissipation are referred to as $\langle \chi _i|\phi _1,\phi _2\rangle _L$ and $\langle \chi _{12}|\phi _1,\phi _2\rangle _L$ hereafter. The conditionally filtered dissipation rates are non-dimensionalized by the maximum $\phi _1$ mean dissipation rate at the same $x/d$ location.

For small SGS variance, the conditionally filtered conditional dissipation rates for $\phi _1$ and $\phi _2$ share a similar pattern close to the centreline for Cases I and II (not shown). The dissipation rates are small close to (1, 0) and increase towards (0, 1). These similarities are because there is essentially no co-flow air at this location and the SGS mixing is only between $\phi _1$ and $\phi _2$. Thus, their fluctuations have equal magnitudes but are anticorrelated, which results in similar dissipation rates. The cross-dissipation is also similar but has negative values due to the anticorrelation. For Case I at $r/d=0.372$ (figure 22), both $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$ as well as $\langle \chi _{12} | \phi _1, \phi _2 \rangle _L$ are relatively uniform, consistent with the Gaussian-like f.j.d.f. since the SGS scalars are well mixed for small SGS variance. The cross-dissipation still has the same trend as $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$, but with negative values due to the mixing being primarily between $\phi _1$ and $\phi _2$. The magnitudes are between those of $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$. Moving towards the edge of the jet (figures not shown), the general trends are opposite to those close to the centreline, with the dissipation rates increasing with $\phi _1$. The cross-dissipation also has the same general trend but with positive values.

Figure 22.

Conditionally filtered dissipation conditional on small SGS variance at $x/d=3.29$ and $r/d=0.372$ for Case I (a,c,e) and Case II (b,d,f). (a,b), (c,d), (e,f) are for $\langle \chi _1 | \phi _1, \phi _2 \rangle _L$, $\langle \chi _2 | \phi _1, \phi _2 \rangle _L$, and $\langle \chi _{12} | \phi _1, \phi _2 \rangle _L$, respectively.

For large SGS variance, $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$ are generally higher than those for small SGS variances. Close to the centreline (not shown), they are higher on the mixing line towards (0, 1). They peak at the location in scalar space where the f.j.d.f. values are low, which indicates that the large dissipation rates are rare events, likely a result of strong SGS motions transporting $\phi _2$ to this physical location generating a cliff. The cross-dissipation also has the same trend.

At $r/d=0.372$ (figure 23), $\langle \chi _1|\phi _1,\phi _2\rangle _L$ peaks near the lower edge of the f.j.d.f. at intermediate $\phi _1$ values, due to the SGS mixing of the $\phi _2$–$\phi _3$ mixture with $\phi _1$. On the $\phi _1$–$\phi _2$ mixing line, both $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$ are higher for intermediate $\phi _1$ and $\phi _2$ values because this location is near the mean $\phi _1$–$\phi _2$ interface. For $\langle \chi _2|\phi _1,\phi _2\rangle _L$ the peak on the mixing line is higher than that of $\langle \chi _1|\phi _1,\phi _2\rangle _L$ due to the higher ethylene diffusivity (the $\phi _1$ and $\phi _2$ gradients have the same magnitude). The lower edge value of $\langle \chi _2|\phi _1,\phi _2\rangle _L$ is lower than the $\langle \chi _1|\phi _1,\phi _2\rangle _L$ peak value because the $\phi _2$ values are approximately one half of the $\phi _1$ value, hence the smaller $\phi _2$ gradient and dissipation. The cross-dissipation has a similar trend with negative values. The strengths of the (negative) peaks are between those of $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$. The conditionally filtered dissipation rates and cross-dissipation rate for Case II have similar trends (Li et al. Reference Li, Yuan, Carter and Tong2017). However, the peak locations of both $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$ for Case II shift to higher $\phi _1$ and $\phi _2$ values compared with those of Case I.

Figure 23.

Conditionally filtered dissipation conditional on large SGS variance at $x/d=3.29$ and $r/d=0.372$ for Case I (a,c,e) and Case II (b,d,f). (a,b), (c,d), (e,f) are for $\langle \chi _1 | \phi _1, \phi _2 \rangle _L$, $\langle \chi _2 | \phi _1, \phi _2 \rangle _L$, and $\langle \chi _{12} | \phi _1, \phi _2 \rangle _L$, respectively.

At $r/d=0.496$ for Case I (figure 24), $\langle \chi _1|\phi _1,\phi _2\rangle _L$ still peak at the lower edge, which indicates that the most intense SGS mixing occurs when large SGS velocity fluctuations bringing together mixtures near the centreline ($\phi _1=1$) and far from the centreline (both $\phi _1$ and $\phi _2$ are low), which produces a ramp–cliff structure. The conditional dissipation, $\langle \chi _1 | \phi _1, \phi _2 \rangle$ (unfiltered), also has a peak near this location (Li et al. Reference Li, Yuan, Carter and Tong2017). The peak of $\langle \chi _1|\phi _1,\phi _2\rangle _L$, however is stronger and exists for a wider range of $r/d$. Our previous studies (Wang & Tong Reference Wang and Tong2002) have found that in the far field of turbulent round jets the scalar f.d.f. is bimodal and there is a ramp–cliff structure when the SGS variance is large, even when the scalar p.d.f. is unimodal. Thus, the bimodal f.j.d.f. and the peak in the conditionally filtered dissipation is primarily due to the ramp–cliff structure, whereas the bimodal j.p.d.f. is partly due to the flapping of $\phi _1$ and the $\phi _2$–$\phi _3$ mixtures. At this location, $\langle \chi _2|\phi _1,\phi _2\rangle _L$ is still large on the mixing line, but with two peaks (the left peak is fairly weak) at the lower edge of the f.j.d.f. These peaks are located on either side of the peak of $\langle \chi _1|\phi _1,\phi _2\rangle _L$ in the scalar space. In physical space the peak $\phi _2$ is located approximately in the centre part of the ramp–cliff structure, where the $\phi _2$ dissipation is small, but on either side of the peak the $\phi _2$ gradient is large, which results in two dissipation peaks. The peaks are located in regions of low-$\phi _2$ values because for these intense SGS mixing events, the $\phi _2$ values are reduced by the co-flow air. The right peak of $\langle \chi _2|\phi _1,\phi _2\rangle _L$ is close to the $\langle \chi _1|\phi _1,\phi _2\rangle _L$ peak as they likely come from the same mixing events. Their locations (the maximum gradient) do not coincide due to the presence of the co-flow air. For Case II the peaks shift to higher $\phi _2$ values, due to the shear layer between the $\phi _1$–$\phi _2$ streams enhancing mixing without transporting large amounts of $\phi _3$.

Figure 24.

Conditionally filtered dissipation conditional on the large SGS variance at $x/d=3.29$ and $r/d=0.496$ for Case I (a,c,e) and Case II (b,d,f). (a,b), (c,d), (e,f) are for $\langle \chi _1 | \phi _1, \phi _2 \rangle _L$, $\langle \chi _2 | \phi _1, \phi _2 \rangle _L$, and $\langle \chi _{12} | \phi _1, \phi _2 \rangle _L$, respectively.

The conditionally filtered cross-dissipation rate at this location has some of the characteristics of both $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$. It has a negative peak close to that of $\langle \chi _1|\phi _1,\phi _2\rangle _L$, with magnitudes between those of $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and the right peak of $\langle \chi _2|\phi _1,\phi _2\rangle _L$. It has a positive peak close to that of the left peak of $\langle \chi _2|\phi _1,\phi _2\rangle _L$. Here $\phi _1$ and $\phi _2$ are being mixed with $\phi _3$, hence the positive cross-dissipation. The value, however, is much lower because $\phi _1$ and $\langle \chi _1|\phi _1,\phi _2\rangle _L$ are low. The left peaks of both $\langle \chi _2|\phi _1,\phi _2\rangle _L$ and $\langle \chi _{12}|\phi _1,\phi _2\rangle _L$ are stronger for Case I than Case II.

Moving further towards the edge of the jet (not shown), $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$ have similar trends, each having a peak caused by the $\phi _1$–$\phi _2$ mixture mixing with $\phi _3$. The cross-dissipation has the same trend as $\langle \chi _1|\phi _1,\phi _2\rangle _L$ and $\langle \chi _2|\phi _1,\phi _2\rangle _L$, and has positive values because $\phi _1$ and $\phi _2$ are well mixed and well correlated at this location.

The general trends for the larger annulus are similar to the smaller annulus. However, the peaks are located at higher $\phi _2$ (closer to the $\phi _1$–$\phi _2$ mixing line) for Case III than for Case IV (e.g. at $x/d=6.99$ and $r/d=0.496$ shown in figure 25), which is opposite to the relative locations between Case I and Case II.

Figure 25.

Conditionally filtered dissipation conditional on the large SGS variance at $x/d=6.99$ and $r/d=0.496$ for Case III (a,c,e) and Case IV (b,d,f). (a,b), (c,d) and (e,f) are for $\langle \chi _1 | \phi _1, \phi _2 \rangle _L$, $\langle \chi _2 | \phi _1, \phi _2 \rangle _L$ and $\langle \chi _{12} | \phi _1, \phi _2 \rangle _L$, respectively.

The results on the conditionally filtered dissipation suggest that there exist several SGS mixing scenarios in the near field of the coaxial jets studied. The first involves mixing of $\phi _1$ and a $\phi _2$–$\phi _3$ mixture, which is probably caused by large SGS velocity fluctuations bringing $\phi _1$ and $\phi _3$ together, which produces high dissipation rates. The second scenario involves primarily $\phi _1$–$\phi _2$ mixing, which generally does not require SGS velocity fluctuations as large as in the first scenario. The dissipation rates, therefore, are lower than those in the first scenario. These two scenarios generally occur in most regions of the jet, but with very low probabilities towards the edge. The third scenario involves mixing of the $\phi _1$–$\phi _2$–$\phi _3$ mixture with pure $\phi _3$, and occurs primarily towards the edge of the jet.

While these mixing scenarios occur under general conditions, they manifest themselves more clearly when the SGS variance is large. For small SGS variance, the SGS scalars are relatively well mixed. The conditionally filtered dissipation rates and their variations in the scalar space are moderate. For large SGS variance, the SGS fields contain the ramp–cliff structure for $\phi _1$. The conditionally filtered dissipation rates for both $\phi _1$ and $\phi _2$ are higher. In the first SGS mixing scenario, $\langle \chi _1|\phi _1,\phi _2\rangle _L$ has a peak near the centre of the cliff and $\langle \chi _2|\phi _1,\phi _2\rangle _L$ has two peaks, one on each side of the $\langle \chi _1|\phi _1,\phi _2\rangle _L$ peak. These peaks are located in the part of the scalar space with relatively low-$\phi _2$ values, since a significant amount of $\phi _3$ is brought in by the large SGS velocity fluctuations. In the second scenario the cliff for $\phi _1$ is not as sharp as in the first scenario. Thus $\langle \chi _1|\phi _1,\phi _2\rangle _L$ might not have a peak in the scalar space. The overall pattern of the conditionally filtered dissipation rates are largely determined by the relative probability and the dissipation magnitudes of these SGS mixing scenarios.

The results also show that $\langle \chi _2|\phi _1,\phi _2\rangle _L$ has some similarities to the conditionally filtered temperature dissipation in a turbulent non-premixed flame. There are two $\langle \chi _2|\phi _1,\phi _2\rangle _L$ peaks, one on each side of the peak of $\langle \chi _1|\phi _1,\phi _2\rangle _L$. These peaks are near the lower edge of the f.j.d.f., due to the large dissipation lowering the $\phi _2$ values. In flames, high temperatures are generated between mixture fraction values of one (fuel stream, similar to $\phi _1=1$) and zero (air stream), thus having a similar mixing configuration as the three-scalar mixing in the present turbulent coaxial jet. The conditionally filtered temperature dissipation has peaks on both sides of the peak temperature in the scalar space (Cai et al. Reference Cai, Wang, Tong, Barlow and Karpetis2009). Due to the heat release generating high temperatures, the locations of peaks relative to the f.j.d.f. peaks in the mixture fraction–temperature space are much higher than those of the $\langle \chi _2|\phi _1,\phi _2\rangle _L$ in the $\phi _1$–$\phi _2$ f.j.d.f. domain. The temperature dissipation for the locally extinguished samples are more similar to $\langle \chi _2|\phi _1,\phi _2\rangle _L$ in the present study due to the lack of a temperature source.