4.1. Local compressibility

For compressible flow, the normalised velocity divergence serves as an excellent indicator of the local compressibility. To obtain an impression of the changes in the local compressibility of the mixing layer induced by convective Mach number, instantaneous fields of the normalised velocity divergence $\varTheta \delta _{\theta }/\Delta U$ in the middle $x$–$y$ plane are shown in figure 5 at two different times, together with an isoline of vorticity magnitude $\omega =0.01\Delta U/\delta _{\theta }^0$ selected as the nominal threshold to identify the turbulent–non-turbulent interface (TNTI) (Jahanbakhshi & Madnia Reference Jahanbakhshi and Madnia2016; Watanabe, Zhang & Nagata Reference Watanabe, Zhang and Nagata2018). The vorticity magnitude is computed by $\omega = \sqrt {\omega ^2_1+\omega ^2_2+\omega ^2_3}$, where $\omega _i$ are vorticity components. For the subsonic flow with $M_c=0.2$, the results are omitted here because there is no obvious region of high compression. At $M_c=0.8$, several shocklets can be seen outside the mixing zone at $\tau =250$ in the transition region, whereas no shocklets are found at $\tau =1000$ in the self-similar region. At $M_c=1.8$, the shocklets can be observed during both transition and self-similar periods, and they tend to exhibit much smaller length scales inside the mixing layer, which is consistent with previous observations (Vaghefi et al. Reference Vaghefi, Nik, Pisciuneri and Madnia2013; Buchta & Freund Reference Buchta and Freund2017).

Figure 5. Visualisation of normalised velocity divergence $\varTheta \delta _{\theta }/\Delta U$ in middle $x$–$y$ plane at two different time instants: (a) $M_c=0.8$, $\tau =250$, (b) $M_c=0.8$, $\tau =1000$, (c) $M_c=1.8$, $\tau =500$ and (d) $M_c=1.8$, $\tau =1250$. The red solid lines correspond to contours of $\omega =0.01\Delta U/\delta _{\theta }^0$, showing the TNTI.

To the best of the authors’ knowledge, the appearance of shocklets in simulations of three-dimensional compressible mixing layers was first reported by Vreman, Kuerten & Geurts (Reference Vreman, Kuerten and Geurts1995). However, there has been no certain critical Mach number beyond which the shocklets appear. Wang, Gotoh & Watanabe (Reference Wang, Gotoh and Watanabe2017) found that the shocklets start to form when turbulent Mach number $M_t \ge 0.6$ in solenoidally forced compressible isotropic turbulence. Chen et al. (Reference Chen, Wang, Li, Wan and Chen2018) observed shocklets at relatively smaller turbulent Mach numbers $M_t \ge 0.4$ in compressible homogeneous shear turbulence owing to the higher level of velocity fluctuation. In compressible mixing layers, the occurrence of shocklets has been captured when the convective Mach number is higher than 0.7 in two-dimensional simulations (Lele Reference Lele1989). For most of the three-dimensional simulations, shocklets are observed at higher convective Mach number 1.2 (Vreman et al. Reference Vreman, Kuerten and Geurts1995; Kourta & Sauvage Reference Kourta and Sauvage2002; Vaghefi Reference Vaghefi2014; Buchta & Freund Reference Buchta and Freund2017). It is noteworthy that Zhou et al. (Reference Zhou, He and Shen2012) observed shocklets in the transition region of spatially developing compressible mixing layer at lower convective Mach number $M_c=0.7$. They inferred that larger initial disturbance, which leads to stronger vortical structures, might contribute to the shocklet formation at this low convective Mach number.

In figure 6, we present the temporal evolutions of the centreline turbulent Mach number $M_{t,y0}$ and the r.m.s. velocity divergence $\varTheta _{rms,y0}/(\Delta U/\delta _{\theta _0})$ for simulations with $M_c=0.2, 0.8$ and $1.8$. As can be seen, a significant overshoot in the centreline turbulent Mach numbers can be observed in the transitional region, as well as the centreline r.m.s. velocity divergence, indicating that the shocklet occurs more easily in this region. We find that the critical turbulent Mach number for shocklets to appear is close to $M_{t,y0} = 0.4$, which agrees well with that in compressible homogeneous shear turbulence (Chen et al. Reference Chen, Wang, Li, Wan and Chen2018), and the corresponding r.m.s. velocity divergence is $\varTheta _{rms,y0}/(\Delta U/\delta _{\theta _0}) \approx 0.2$ for the free shear flow.

Figure 6. Temporal evolution of the centreline (a) turbulent Mach number $M_{t,y0}$ and (b) r.m.s. velocity divergence $\varTheta _{rms,y0}/(\Delta U/\delta _{\theta _0})$ at the centreplane for simulations with $M_c=0.2$, 0.8 and 1.8. The dashed vertical lines mark the self-similar time periods.

Figures 7(a) and 7(b) show the probability density function (PDF) of the normalised velocity divergence inside and outside the turbulent region at several time instants for convective Mach number $M_c = 0.8$. It was shown in previous results that the PDF of velocity divergence became more skewed towards the negative value as the turbulent Mach number and Taylor Reynolds number increase, due to strong compression region associated with shocklets which occurs much more frequently than the strong expansion region (Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2012; Vaghefi & Madnia Reference Vaghefi and Madnia2015; Chen et al. Reference Chen, Wang, Li, Wan and Chen2019). For the moderately compressible case with $M_c = 0.8$, the PDF of velocity divergence in the turbulent region exhibits a skewness toward the negative side only at transition period $\tau =250$, and then it rapidly becomes nearly symmetric. The PDF of velocity divergence outside the turbulent region is skewed toward the negative side until $\tau =750$ in the self-similar period, owing to the shocklets radiated from the mixing zone.

Figure 7. PDF of the normalised velocity divergence $\varTheta /(\Delta U/\delta _{\theta _0})$ at five different time instants (a),(c) in the turbulent region and (b),(d) outside the turbulent region for cases at (a),(b) $M_c=0.8$ and (c),(d) $M_c=1.8$.

Figures 7(c) and 7(d) show the PDF of the normalised velocity divergence inside and outside the turbulent region at several time instants for convective Mach number $M_c = 1.8$. As we can see, the PDFs of velocity divergence both inside and outside the turbulent region are always strongly skewed towards the negative values. At $\tau =2000$, the PDF of velocity divergence inside the turbulent region is in excellent agreement with the result of Vaghefi & Madnia (Reference Vaghefi and Madnia2015) which is sampled at the beginning of the self-similar evolution. The two tails of PDF become shorter as time goes on, indicating that the local compressibility continuously decreases. We also find that the two tails of PDF inside the turbulent region are always longer than that outside the turbulent region, showing stronger local compressibility in the mixing zone.

4.2. Helmholtz decomposition of velocity fields

To reveal the underlying physics in the compressible turbulence and specifically the characteristic of local compressibility, we shall employ the well-known Helmholtz decomposition (Erlebacher & Sarkar Reference Erlebacher and Sarkar1993; Samtaney et al. Reference Samtaney, Pullin and Kosović2001; Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2012) to the velocity field

(4.1)\begin{equation} \textbf{u} = \textbf{u}_s + \textbf{u}_d, \end{equation}

where the solenoidal component $\textbf{u}_s$ satisfies $\boldsymbol {\nabla } \boldsymbol {\cdot } \textbf{u}_s = 0$ and the dilatational component $\textbf{u}_c$ is irrotational, i.e. $\boldsymbol {\nabla }\times \textbf{u}_d = 0$.

The instantaneous fields of decomposed velocity in the $x$–$y$ plane at $z=L_z/2$ are presented in figures 8 and 9 for the cases with $M_c=0.8$ at $\tau =1000$ and $M_c=1.8$ at $\tau =1700$, respectively. Note that all visualisations are selected at the middle of the self-similar region. For solenoidal velocity component, the visualisations in figures 8(a),(c),(e) and 9(a),(c),(e) show the typical features of a turbulent mixing layer, with patches of mixed fluid in the central region entraining patches of unmixed fluid from both external streams. It can be seen that the $u_s^{\prime \prime }$ field in the $x$–$y$ plane is characterised by elongated low- and high-speed regions, with characteristic sizes of the order of $\delta _\omega$. Comparing figures 8(a) and 8(c) or figures 9(a) and 9(c) reveals a strong anticorrelation between $u_s^{\prime \prime }$ and $v_s^{\prime \prime }$ events. The large-scale negative $u_s^{\prime \prime }$ events are characterised by regions of positive $v_s^{\prime \prime }$, upraising the low-speed fluid. The large-scale positive $u_s^{\prime \prime }$ events are accompanied by regions of negative $v_s^{\prime \prime }$, moving the high-speed fluid downwards. Moreover, the solenoidal velocity fields become visibly smoother at small scales for higher convective Mach number $M_c = 1.8$. Figures 10 and 11 show the instantaneous fields of decomposed velocity in $x$–$z$ plane at $y=0$ for cases with $M_c=0.8$ at $\tau =1000$ and $M_c=1.8$ at $\tau =1700$, respectively. The main characteristic of the solenoidal velocity field is consistent with that found previously in the $x$–$y$ plane. The presence of large-scale streamwise elongated streaks of $u_s^{\prime \prime }$ is presented more clearly in the $x$–$z$ plane visualisation, with spanwise sizes of about $L_z/4$ in the self-similar state. This indicates that the spanwise length of the domain is large enough for the present simulation.

Figure 8. Instantaneous fields of (a) $u_s^{\prime \prime }$, (b) $u_d^{\prime \prime }$, (c) $v_s^{\prime \prime }$, (d) $v_d^{\prime \prime }$, (e) $w_s^{\prime \prime }$ and ( f) $w_d^{\prime \prime }$ in the central $x$–$y$ plane for case M08 at $\tau =1000$.

Figure 9. Instantaneous fields of (a) $u_s^{\prime \prime }$, (b) $u_d^{\prime \prime }$, (c) $v_s^{\prime \prime }$, (d) $v_d^{\prime \prime }$, (e) $w_s^{\prime \prime }$ and ( f) $w_d^{\prime \prime }$ in the central $x$–$y$ plane for the case M18 at $\tau =1700$.

Figure 10. Instantaneous fields of (a) $u_s^{\prime \prime }$, (b) $u_d^{\prime \prime }$, (c) $v_s^{\prime \prime }$, (d) $v_d^{\prime \prime }$, (e) $w_s^{\prime \prime }$ and ( f) $w_d^{\prime \prime }$ in the central $x$–$z$ plane for case M08 at $\tau =1000$.

Figure 11. Instantaneous fields of (a) $u_s^{\prime \prime }$, (b) $u_d^{\prime \prime }$, (c) $v_s^{\prime \prime }$, (d) $v_d^{\prime \prime }$, (e) $w_s^{\prime \prime }$ and ( f) $w_d^{\prime \prime }$ in the central $x$–$z$ plane for case M18 at $\tau =1700$.

As can be seen in the right subfigures of figures 8 to 11, the increasing of convective Mach number induces different flow patterns of dilatational velocity component both in the $x$–$y$ and $x$–$z$ planes. For the case with $M_c=0.8$, all three dilatational velocity components exhibit large block-like spatial structures. The $u_d^{\prime \prime }$ and $v_d^{\prime \prime }$ structures are aligned at oblique angles to the streamwise direction in $x$–$y$ plane, whereas the $w_d^{\prime \prime }$ structure is almost perpendicular. The $v_d^{\prime \prime }$ and $w_d^{\prime \prime }$ structures extend over the entire spanwise length of the domain, as shown in figures 10(d) and 10( f). These results suggest the presence of the spanwise rollers at $M_c=0.8$, which also can be observed in the instantaneous fields of solenoidal velocity in the $x$–$y$ plane, as shown in figures 8(a), 8(c) and 8(e).

At convective Mach number $M_c=1.8$, all three dilatational velocity components show severe discontinuity at the position of shocklets, particularly in the non-turbulent region, as presented in figures 9(b), 9(d) and 9( f). As can be seen, the fluids on two sides of shocklets move towards the shocklets and generate a sheet-like strong compressible region, causing a strong anticorrelation between $u_d^{\prime \prime }$ and $v_d^{\prime \prime }$ events. In the middle $x$–$z$ plane, the $u_d^{\prime \prime }$ and $v_d^{\prime \prime }$ events are also anticorrelated, which becomes less apparent masked by the trivial spanwise structures of $u_d^{\prime \prime }$.

To have a better picture of the effect of convective Mach number on the correlation between streamwise and vertical velocity components, we present the joint PDF of the velocity components, computed inside and outside the turbulent region at an instant in the self-similar region. As shown in figure 12(a), the joint PDF of $u_s^{\prime \prime }/u_{s,rms}^{\prime \prime }$ and $v_s^{\prime \prime }/v_{s,rms}^{\prime \prime }$ in the turbulent region reveals a statistical preference for points in the second and fourth quadrants, appearing as the oval contour lines. The long axis of the oval shape of PDF($u_s^{\prime \prime }$, $v_s^{\prime \prime }$) exhibits an inclined angle with the symmetry line of the $u_s^{\prime \prime }$–$v_s^{\prime \prime }$ plane, indicating the component anisotropy of solenoidal velocity components. This oval shape of the PDF($u_s^{\prime \prime }$, $v_s^{\prime \prime }$) is consistent well with that in incompressible homogeneous shear flow (Adrian & Moin Reference Adrian and Moin1988) and different from the results in the wall-bounded turbulent flow where axisymmetric shapes are usually observed (Wallace Reference Wallace2016; Yu, Xu & Pirozzoli Reference Yu, Xu and Pirozzoli2019; Xu et al. Reference Xu, Wang, Wan, Yu, Li and Chen2021a). PDF($u_s^{\prime \prime }$, $v_s^{\prime \prime }$) in the non-turbulent region decays rapidly, illustrating the solenoidal velocity components are insignificant here, as can be seen from figure 12(b).

Figure 12. Joint PDF of (a),(b) $u_s^{\prime \prime }/u_{s,rms}^{\prime \prime }$ and $v_s^{\prime \prime }/v_{s,rms}^{\prime \prime }$ and (c),(d) $u_d^{\prime \prime }/u_{d,rms}^{\prime \prime }$ and $v_d^{\prime \prime }/v_{d,rms}^{\prime \prime }$ for case with $M_c=0.8$ at $\tau =1000$ and $M_c=1.8$ at $\tau =1250$: (a),(c) in the turbulent region and (b),(d) in the non-turbulent region. The thick dashed auxiliary line indicates the long axis of the oval shape. The thin dashed-dotted line represents the symmetry line of the $u_s^{\prime \prime }$–$v_s^{\prime \prime }$ plane. The contour levels 0.001, 0.01 and 0.1 are shown.

The joint PDFs of $u_d^{\prime \prime }/u_{d,rms}^{\prime \prime }$ and $v_d^{\prime \prime }/v_{d,rms}^{\prime \prime }$ are plotted in figures 12(c) and 12(d). In the turbulent region, we find that there is no significant correlation between $u_d^{\prime \prime }$ and $v_d^{\prime \prime }$ for $M_c=0.8$, whereas PDF($u_d^{\prime \prime }$, $v_d^{\prime \prime }$) also has a statistical preference in the second and fourth quadrants for $M_c=1.8$, showing an oval shape similar to that of PDF($u_s^{\prime \prime }$, $v_s^{\prime \prime }$). In the non-turbulent region, both for $M_c=0.8$ and $M_c=1.8$, PDF($u_d^{\prime \prime }$, $v_d^{\prime \prime }$) has the inclined oval pattern, which is elongated in the $u_d^{\prime \prime }$ direction for $M_c=1.8$ compared with that in the turbulent region. Actually, we find that, as the development of turbulent flow, the inclination angle of the oval shape of the PDF($u_d^{\prime \prime }$, $v_d^{\prime \prime }$) for $M_c=0.8$ in non-turbulent region increases monotonically from $10^\circ$ at $\tau =200$ to nearly $30^\circ$ at $\tau =1000$. Only the results at $\tau =1000$ is illustrated here in figure 12(d) for brevity.

It is worth noting that the statistical preference in the second and fourth quadrants of PDF($u_s^{\prime \prime }$, $v_s^{\prime \prime }$) and PDF($u_d^{\prime \prime }$, $v_d^{\prime \prime }$) corresponds to totally different physical processes. The former is related to the mixing between the two streams, demonstrated in the left subfigures of figures 8–11, resembling the sweep and ejection events in wall-bounded turbulent shear flow, and the latter is due to the induction of strong compression or expansion motions, seen from the right subfigures of figures 8–11.

On the basis of Helmholtz decomposition of velocity, the Reynolds stress ${\mathsf{R}}_{ij}$ can be decomposed into

(4.2)\begin{equation} {\mathsf{R}}_{ij} = \overline{\rho u_i^{\prime\prime}u_j^{\prime\prime}} = \overline{\rho u_{s,i}^{\prime\prime}u_{s,j}^{\prime\prime}}+ \overline{\rho u_{d,i}^{\prime\prime}u_{d,j}^{\prime\prime}}+ \overline{\rho u_{s,i}^{\prime\prime}u_{d,j}^{\prime\prime}}+ \overline{\rho u_{d,i}^{\prime\prime}u_{s,j}^{\prime\prime}}. \end{equation}

However, it is found that in the turbulent region the magnitudes of the last three terms on the right-hand side of (4.2) are several orders of magnitude smaller than the first term, even for the highest convective Mach number $M_c=1.8$. Then we take Reynolds shear stress ${\mathsf{R}}_{12}=\overline {\rho u^{\prime \prime }v^{\prime \prime }}$ for example to show the effect of compressibility, and the Helmholtz decomposition of Reynolds normal stresses ${\mathsf{R}}_{11}$, ${\mathsf{R}}_{22}$ and ${\mathsf{R}}_{33}$ can be found in Appendix B. The statistical preference in the second and fourth quadrants of PDF($u_s^{\prime \prime }$, $v_s^{\prime \prime }$) and PDF($u_d^{\prime \prime }$, $v_d^{\prime \prime }$) above shows that both solenoidal and dilatational velocity have a contribution to the negative values of the Reynolds shear stress. Figure 13 provides the Reynolds shear stress and its decomposed components for cases with $M_c=0.8$ and $M_c=1.8$. We can see that ${\mathsf{R}}_{ij}$ and $\overline {\rho u_{s}^{\prime \prime }v_{s}^{\prime \prime }}$ overlap almost perfectly with each other for both $M_c=0.8$ and $M_c=1.8$, indicating the dominant role of the solenoidal velocity field. For $M_c=0.8$, $\overline {\rho u_{s}^{\prime \prime }v_{d}^{\prime \prime }}$ is positive with a peak value at $y=0$ and is significantly greater than the negligible two terms $\overline {\rho u_{d}^{\prime \prime }v_{d}^{\prime \prime }}$ and $\overline {\rho u_{d}^{\prime \prime }v_{s}^{\prime \prime }}$. For $M_c=1.8$, the values of three dilatation-related terms are on the same order. As the convective Mach number increases, the magnitudes of three dilatation-related terms become larger, demonstrating the effect of compressibility. It is interesting to note that for $M_c=1.8$, $\overline {\rho u_{d}^{\prime \prime }v_{d}^{\prime \prime }}$ has two peak values at $y/\delta _\omega \approx \pm 0.5$ and becomes larger than $\overline {\rho u_{s}^{\prime \prime }v_{s}^{\prime \prime }}$ in the non-turbulent region, which is probably due to the induction of the shocklets frequently formed near the TNTI, consistent with the analysis of figure 9.

Figure 13. The Reynolds shear stress and its decomposed components for cases with (a) $M_c=0.8$ and (b) $M_c=1.8$. The legend is the same for both plots.

4.3. Integrated TKE transport

The physical mechanism of the suppression of layer growth rate due to compressibility is further examined by analysing the time evolution of integrated TKE transport. The terms in conservative form disappear after integration, hence the equation of the $y$-integrated TKE becomes (Vreman et al. Reference Vreman, Sandham and Luo1996; Attili & Bisetti Reference Attili and Bisetti2012)

(4.3)\begin{equation} \frac{\partial \hat{K}}{\partial t} = \hat{P} + \hat{\epsilon} + \hat{\varPhi} + \hat{\varSigma}, \end{equation}

where

(4.4)\begin{equation} \hat{K} = \int_{{-}L_y/2}^{L_y/2}\frac{1}{2}\overline{\rho u_i^{\prime\prime}u_i^{\prime\prime}}\,{{\rm d} y}, \end{equation}

is total TKE, and the integrated transport terms $\hat {P}$, $\hat {\epsilon }$, $\hat {\varPhi }$ and $\hat {\varSigma }$ are

(4.5a)$$\begin{gather} \hat{P}={-} \int_{{-}L_y/2}^{L_y/2}R_{ik}\frac{\partial \tilde{u}_i}{\partial x_k}\,{{\rm d} y}, \end{gather}$$

(4.5b)$$\begin{gather}\hat{\epsilon}={-}\int_{{-}L_y/2}^{L_y/2}\frac{1}{Re}\overline{\sigma^\prime_{ik}\frac{\partial u^{\prime\prime}_i}{\partial x_k}}\,{{\rm d} y}, \end{gather}$$

(4.5c)$$\begin{gather}\hat{\varPhi}= \int_{{-}L_y/2}^{L_y/2}\overline{p^\prime\frac{\partial u^{\prime\prime}_i}{\partial x_i}}\,{{\rm d} y}, \end{gather}$$

(4.5d)$$\begin{gather}\hat{\varSigma}= \int_{{-}L_y/2}^{L_y/2}\overline{u_i^{\prime\prime}}\left(\frac{1}{Re}\frac{\partial\bar{\sigma}_{ik}}{\partial x_k} - \frac{\partial \bar{p}}{\partial x_i}\right){{\rm d} y}. \end{gather}$$

The magnitude of the integrated mass flux coupling term is still negligible. The transport equations of the integrated TKE components $\hat {K}_i$ and the Reynolds shear stress $\hat {R}_{12}$ can be obtained similarly, which are omitted here for brevity.

Figure 14 displays the time evolution of the normalised $y$-integrated TKE $\hat {K}/\Delta U^2$, its three components $\hat {K}_i/\Delta U^2$ and the Reynolds shear stress $\hat {R}_{12}/\Delta U^2$ at $M_c=0.2$, $0.8$ and $1.8$. We can see that the integrated TKE and Reynolds shear stress grow linearly after a rapid growing transition region with a slight overshoot between two regions only at $M_c=0.8$. Because of the anisotropic nature of free shear flows, it is generally observed that the components of integrated TKE satisfy the relationship $\hat {K}_1 > \hat {K}_3 > \hat {K}_2$ in the self-similar region. It is interesting for $M_c=0.2$ that the vertical component $\hat {K}_2$ of the TKE is larger than the spanwise component $\hat {K}_3$ in transition region $\tau <150$, and undergoes an accelerating growth resulting in $\hat {K}_2 > \hat {K}_3$ again beyond the self-similar region $\tau >750$, as shown in figure 14(a). This behaviour is consistent with a previous numerical study of a spatially developing incompressible mixing layer by Attili & Bisetti (Reference Attili and Bisetti2012). We observe that the rapid growth of $\hat {K}_2$ in the late stages of development at $M_c=0.2$ is the characteristic of the larger spanwise rollers generated by the Kelvin–Helmholtz instability, which is in agreement with previous results by Rogers & Moser (Reference Rogers and Moser1994).

Figure 14. Time evolution of $\hat {K}/\Delta U^2$, $\hat {K}_i/\Delta U^2$, $0.5\hat {R}_{12}/\Delta U^2$ at (a) $M_c=0.2$, (b) $M_c=0.8$ and (c) $M_c=1.8$. The legend is the same for all plots. The dashed vertical lines mark the self-similar time periods.

Figure 15 displays the time evolution of the normalised $y$-integrated production $\hat {P}/\Delta U^3$, dissipation $\hat {\epsilon }/\Delta U^3$ and pressure–strain $\hat {\varPhi }/\Delta U^3$ in the equation of $\hat {K}$ at $M_c=0.2$, $M_c=0.8$ and $M_c=1.8$. Here, the integrated mass flux coupling and transport terms have been neglected because their magnitude is very small, even in the large $M_c$ case. At $M_c=0.2$, the asymptotic behaviour of $\hat {P}/\Delta U^3$ and $\hat {\epsilon }/\Delta U^3$ is in good agreement with the previous numerical results by Attili & Bisetti (Reference Attili and Bisetti2012) in a spatially evolving incompressible mixing layer, as compared in figure 15(b) where the abscissa is a local Reynolds number $Re_\omega = \rho _\infty \Delta U \delta _\omega /\mu _\infty$, which allows a direct comparison between the spatially and temporally evolving mixing layers. As convective Mach number increases, both magnitudes of the integrated production and dissipation become smaller, whereas the integrated pressure–strain term slightly increases only in the transition region. We can see that the integrated pressure–strain term is always much smaller than the other two terms, indicating that the net contribution of pressure dilatation to the kinetic energy transfer is negligibly small. This is consistent with previous studies on compressible turbulence (Wang et al. Reference Wang, Wan, Chen and Chen2018, Reference Wang, Wang, Li and Chen2021). In the transition region, $\hat {P}/\Delta U^3$ grows linearly for all three convective Mach numbers. In contrast, $\hat {\epsilon }/\Delta U^3$ grows linearly only at $M_c=1.8$ and it satisfies $\hat {\epsilon }/\Delta U^3 \propto -\tau ^2$ at $M_c=0.2$ and $M_c=0.8$. An overshoot in the integrated production and dissipation occurs near the end of the transition region, and then the overshoot decays and becomes nearly a constant in the self-similar region. We also observe that the dissipation overshoot clearly lags behind the production overshoot. The ratio between the corresponding time of the dissipation overshoot $\tau _\epsilon$ and the production overshoot $\tau _P$ is defined to measure the lag level between them. The values for $\tau _\epsilon /\tau _P$ are 1.8, 1.5 and 1.2 at $M_c=0.2$, $M_c=0.8$ and $M_c=1.8$, respectively. This result indicates that the dissipation starts to effectively balance the production earlier for larger convective Mach number, and partly reveals some reasons for reduction of growth rate with increased convective Mach number in the transition region.

Figure 15. Time evolution of $\hat {P}/\Delta U^3$, $\hat {\epsilon }/\Delta U^3$, $\hat {\varPhi }/\Delta U^3$ at (a),(b) $M_c=0.2$, (c) $M_c=0.8$ and (d) $M_c=1.8$. The legend is the same for (a), (c) and (d). The black solid lines correspond to $\hat {\epsilon }/\Delta U^3 \propto -\tau ^2$. The dashed vertical lines mark the self-similar time periods.

In order to better understand the effect of compressibility, we plot the time evolution of transport terms for $\hat {K}_i$ in figure 16. In all cases, we can see that the production terms associated with $\hat {K}_2$ and $\hat {K}_3$, namely $\hat {P}_2$ and $\hat {P}_3$, are negligible concerning the $\hat {K}_1$ production $\hat {P}_1$. As is well known, on average, the pressure–strain correlation components $\hat {\varPhi }_i$ redistribute energy from $\hat {K}_1$ to the other components $\hat {K}_2$ and $\hat {K}_3$. Figure 16(a) shows that $\hat {\varPhi }_2 > \hat {\varPhi }_3$ for $\tau <100$ and $\tau >700$, leading to the preferential transfer of energy into the vertical velocity fluctuation, as shown in figure 14(a). It is interesting to note that all pressure–strain correlation components $\hat {\varPhi }_i$ and production almost reach their peak values at the same time for all three convective Mach numbers.

Figure 16. Time evolution of $\hat {P}_i/\Delta U^3$, $\hat {\epsilon }_i/\Delta U^3$, $\hat {\varPhi }_i/\Delta U^3$ at (a) $M_c=0.2$, (b) $M_c=0.8$ and (c) $M_c=1.8$. The legend is the same for all plots. The black solid lines correspond to $\hat {\epsilon }_i/\Delta U^3 \propto -\tau ^2$.

For the case with $M_c=0.2$, the three components of viscous dissipation $\hat {\epsilon }_i$ always overlap perfectly with each other, as shown in figure 16(a), suggesting local isotropy of the flow at small scales. As the convective Mach number increases, the viscous dissipation occurs more anisotropically and is dominated by the streamwise component $\hat {\epsilon }_1$ in the transition region. As is clearly presented in figure 16(c) for $M_c=1.8$, the magnitude of streamwise component $\hat {\epsilon }_1$ is found to be even larger than $\hat {\varPhi }_1$, whereas the $\hat {\epsilon }_2$ and $\hat {\epsilon }_3$ components are still proportional to $-\tau ^2$, resulting in linear growth of the total viscous dissipation $\hat {\epsilon }$ in figure 15(d). We depict the PDF of three instantaneous viscous dissipation components ${\epsilon }_i/\Delta U^3$ at $\tau =250$ in figure 17 for the convective Mach number $M_c = 0.2$ and $1.8$, where ${\epsilon }_i$ is defined as (avoiding summation convention)

(4.6)\begin{equation} \epsilon_i={-}\frac{1}{Re}{\sigma^\prime_{ik}\frac{\partial u^{\prime\prime}_i}{\partial x_k}}. \end{equation}

The PDFs of ${\epsilon }_i/\Delta U^3$ always collapse into the same curve for $M_c = 0.2$. At convective Mach number $M_c = 1.8$, we observe that the left tail of PDF of ${\epsilon }_1/\Delta U^3$ is much longer than that of ${\epsilon }_2/\Delta U^3$ and ${\epsilon }_3/\Delta U^3$, suggesting that stronger events of viscous dissipation occur in the transport of $\hat {K}_1$ during the transition region. This is further studied from aspect of vortical structures in § 4.4.

Figure 17. PDF of viscous dissipation components ${\epsilon }_i/\Delta U^3$ at $\tau =250$ for cases with (a) $M_c=0.2$ and (b $M_c=1.8$.

We also plot the time evolution of transport terms for $\hat {R}_{12}$ in figure 18. For all cases, we can see that the transfer of the Reynolds shear stress is dominated by the pressure–strain term $\hat {\varPhi }_{12}$ and production term $\hat {P}_{12}$ with peaks occurring at nearly the same time with the production $\hat {P}$. The viscous dissipation $\hat {\epsilon }_{12}$ is negligibly small and increases slightly as the convective Mach number increases.

Figure 18. Time evolution of $\hat {P}_{12}/\Delta U^3$, $\hat {\epsilon }_{12}/\Delta U^3$, $\hat {\varPhi }_{12}/\Delta U^3$ at (a) $M_c=0.2$, (b) $M_c=0.8$ and (c) $M_c=1.8$. The legend is the same for all plots. The dashed vertical lines mark the self-similar time periods.

4.4. Turbulence structure and length scale

As already mentioned, the contribution of dilatational component to overall Reynolds stress is quite limited. We therefore believe that the transportation of TKE is dominated by the solenoidal velocity field which is characterised by small-scale vortical structures and energetic large-scale velocity structures. According to Kolmogorov's turbulence theory (Kolmogorov Reference Kolmogorov1941), the characteristic length scale of the smallest eddies is given by the Kolmogorov length scale $\eta$ where most of the kinetic energy is dissipated by viscosity. Figure 19(a) shows the temporal evolution of the normalised centreline Kolmogorov length scale $\eta _{y0}/\Delta x$ for cases with convective Mach numbers $M_c = 0.2$, 0.8 and 1.8. We observe that $\eta _{y0}/\Delta x$ at $y=0$ reaches a minimum value during the transition region and then gradually increases as the development of turbulent flow. As the convective Mach number increases, the centreline Kolmogorov length scale $\eta _{y0}/\Delta x$ at $M_c = 1.8$ becomes twice as large as that at $M_c = 0.2$. From figure 19(b), it can be seen that the $\eta /\Delta x$ extracted in the self-similar region is approximately constant inside the turbulent mixing region, which is consistent with the previous results of compressible turbulent mixing layer by Vaghefi & Madnia (Reference Vaghefi and Madnia2015) and incompressible planar jet by Watanabe et al. (Reference Watanabe, Zhang and Nagata2018). Outside the mixing layer, $\eta /\Delta x$ decreases as the convective Mach number increases, due to the fluctuations induced by the shocklets here.

Figure 19. (a) Time evolution of the centreline Kolmogorov length scale $\eta _{y0}/\Delta x$ and (b) mean profile of $\eta /\Delta x$ at $M_c=0.2, 0.8$ and $1.8$. The dashed vertical lines mark the self-similar time periods in (a).

In figure 20, we compare the visualisations of small-scale vortical structures in transition region at $M_c = 0.2$ and $1.8$, where the structures are displayed by the iso-surfaces of the second invariant of the instantaneous velocity gradient tensor $\boldsymbol{\mathsf{Q}}$. At lower convective Mach number $M_c = 0.2$, the vortices are highly three-dimensional, with their diameter of the same order of the Kolmogorov scale. There are plentiful shapes of small-scale vortices, including hairpin vortices, rib vortices and complex helical vortices, giving rise to much random orientation of vortical structures. In contrast, the flow field is dominated by more regular and smooth streamwise elongated vortices at $M_c = 1.8$ and $\tau =250$. This phenomenon is consistent with previous studies of small-scale vortical structures in compressible mixing layer, where it is generally believed that the vortical structures are more aligned in the streamwise direction at higher $M_c$, particularly in the transition region (Balaras et al. Reference Balaras, Piomelli and Wallace2001; Fathali et al. Reference Fathali, Meyers, Rubio, Smirnov and Baelmans2008; Arun et al. Reference Arun, Sameen, Srinivasan and Girimaji2019; Li et al. Reference Li, Peyvan, Ghiasi, Komperda and Mashayek2021).

Figure 20. Small-scale vortical structures are visualised by the iso-surfaces of the second invariant $Q$ of the velocity gradient tensor and coloured based on local values of $u^{\prime \prime }$: (a) $Q=20$, $\tau =125$ and $M_c=0.2$; (b) $Q=2$, $\tau =250$ and $M_c=1.8$.

The effects of the small-scale vortical structures on the anisotropy of viscous dissipation are further investigated by scrutinising flow variables in the $y$–$z$ planes, as shown in figure 21. In these $y$–$z$ slices at $x=0.5L_x$, the black solid lines represent the iso-lines of viscous dissipation, whereas the contours show the velocity fluctuation. For case M02, the distribution of streamwise viscous dissipation ${\epsilon _1}$ is chaotic and fragmental, as shown in figure 21(a). Here ${\epsilon _2}$ and ${\epsilon _3}$ are not shown because they resemble ${\epsilon _1}$. For case M18, the mixing region is packed with mushroom-like structures of positive and negative streamwise velocity fluctuations, with strong streamwise viscous dissipation regions wrapping around them. Comparing figures 21(b) and 21(c), the lift-up of low-speed fluid and the drop-down of high-speed fluid are clearly presented. The prominent mushroom-like structures are created by the interaction of a pair of streamwise counter-rotating vortices, as presented in a enlarged view in figure 21(e), with the vectors showing the velocity fluctuations in the vertical $v^{\prime \prime }$ and spanwise $w^{\prime \prime }$ directions. This is similar to several previous studies on flat-plate boundary layers (Strand & Goldstein Reference Strand and Goldstein2011; Buffat et al. Reference Buffat, Penven, Cadiou and Montagnier2014; Zhao & Sandberg Reference Zhao and Sandberg2020). From figures 21(c) and 21(d), we can see that the iso-lines of ${\epsilon _2=0.005}$ and ${\epsilon _3=0.005}$ are more sparse, with smaller length scales in the $y$–$z$ plane, as compared with the iso-lines of ${\epsilon _1=0.01}$, revealing that the viscous dissipation of ${K}_2$ and ${K}_3$ is much weaker than that of ${K}_1$ for $M_c=1.8$ in the transition region, which confirms the results in figure 16.

Figure 21. Contours of instantaneous velocity fluctuations $u_i^{\prime \prime }$ along with stronger events of viscous dissipation indicated by black solid lines: (a) ${\epsilon _1}/\Delta U^3=0.03$, $-0.3< u^{\prime \prime }<0.3$ at $M_c=0.2$ and $\tau =125$; (b) ${\epsilon _1}/\Delta U^3=0.01$, $-0.3< u^{\prime \prime }<0.3$, (c) ${\epsilon _2}/\Delta U^3=0.005$, $-0.2< v^{\prime \prime }<0.2$, and (d) ${\epsilon _3}/\Delta U^3=0.005$, $-0.2< w^{\prime \prime }<0.2$ at $M_c=1.8$ and $\tau =250$. (e) Enlarged view of the region in red-lined rectangular box in plot (b), showing the velocity vectors of $v$ and $w$.

We are now going to focus on the large-scale flow structures of the mixing shear layer. Figures 22 and 23 show the instantaneous fields of the streamwise velocity $u^{\prime \prime }$ and density fluctuations $\rho ^{\prime }$ in the horizontal plane at the centre of the shear layer ($y = 0$) at different time instants for cases M02 and M18, respectively. For the nearly incompressible case with $M_c=0.2$, the density fields exhibit a typical pattern of spanwise Kelvin–Helmholtz rollers that merge and eventually fill up the whole computational domain, which is similar to the situation reported in incompressible (Rogers & Moser Reference Rogers and Moser1994; Balaras et al. Reference Balaras, Piomelli and Wallace2001) and weakly compressible (Mungal Reference Mungal1995; Pirozzoli et al. Reference Pirozzoli, Bernardini, Marié and Grasso2015) mixing shear layers. The streamwise elongated large-scale structures of $u^{\prime \prime }$ with an imposed low-frequency spanwise meander can also be observed in the transition region at $\tau =375$. At late time $\tau =750$, we can see that the high-speed regions tend to propagate abreast along the spanwise direction. For the strongly compressible case with $M_c=1.8$, no clear hints of spanwise rollers can be observed both in the fields of the streamwise velocity $u^{\prime \prime }$ and density fluctuations $\rho ^{\prime }$. Although large-scale low- and high-speed regions form at an early time of $\tau =125$, they are highly elongated in the streamwise direction and more slender in the spanwise direction.

Figure 22. Instantaneous fields of (a),(c),(e) $u^{\prime \prime }$ and (b),(d),(f) $\rho^{\prime}$ in the central $x$–$z$ plane for case M02: (a),(b) $\tau =100$; (c),(d) $\tau =375$; and (e),(f) $\tau =750$.

Figure 23. Instantaneous fields of (a),(c),(e) $u^{\prime \prime }$ and (b),(d),(f) $\rho ^{\prime }$ in the central $x$–$z$ plane for case M18: (a),(b) $\tau =125$; (c),(d) $\tau =500$; and (e),(f) $\tau =2000$.

To quantify the features of streamwise elongated large-scale structures in the mixing shear layer, the two-point correlation $R_{uu}$ (defined in (2.19)) at three different time instants in the self-similar region is shown in figure 24 at $M_c=0.2$ and 1.8. Ten contour levels ranging from 0.1 to 1 with increments of 0.1 are shown and each direction is normalised by the local vorticity thickness $\delta _\omega$. The undulation of the lower $R_{uu}$ level is caused by the decreasing sample size away from the origin. We find that the three sets of contours (at three different time instants) in the self-similar region show very good collapse when scaled by vorticity thickness $\delta _\omega$ for both $M_c=0.2$ and $1.8$, which confirms that the self-similar state of the mixing layer is dominated by large-scale structures (Vreman et al. Reference Vreman, Sandham and Luo1996; Pantano & Sarkar Reference Pantano and Sarkar2002). In figures 24(a) and 24(b), the correlation decays to a value of 0.1 at a distance larger than $\delta _\omega$, which indicates that there is significant streamwise elongated spatial coherence. The fusiform-like correlation contours are inclined with respect to the local streamwise direction with an inclination angle decreasing from $18^\circ$ at $M_c=0.2$ to $12^\circ$ at $M_c=1.8$, which is similar to the experimental results reported by Messersmith & Dutton (Reference Messersmith and Dutton1996) and Kim, Elliott & Dutton (Reference Kim, Elliott and Dutton2020). Figures 24(c),(d) and 24(e),(f) show $R_{uu}$ in the $x$–$z$ and $y$–$z$ planes, respectively. We find that the correlation contours are compact in spanwise direction and moderately elongated in the vertical direction, indicating a relatively small spanwise length scale.

Figure 24. Two-point correlations of the streamwise velocity fluctuation $u^{\prime \prime }$ in the (a),(b) $x$–$y$, (c),(d) $x$–$z$ and (e),(f) $y$–$z$ planes at three different time instants: (a),(c),(e) $M_c=0.2$ and (b),(d),(f) $M_c=1.8$. The contour levels range from 0.1 to 1 with increments of 0.1.

In general, the streamwise and spanwise length scales of large-scale structures are extracted from the two-point correlation functions of the streamwise velocity fluctuations with a nominal threshold. A higher threshold leads to smaller and more stable coherence length (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2014). Here, it is found that the threshold strongly influences the streamwise correlation length, so it is difficult to determine the compressibility effect on the length scale. For example, when we chose a larger value of the threshold, $R_{uu} \ge 0.3$, the streamwise length scale was found to be larger at $M_c=0.2$, in contrast to the situation of the threshold value 0.1. The compressibility effect on the length scale of large-scale structures is further investigated by inspecting the integral length scales, as shown in figure 25. We can see that the normalised integral length scales $l_x/\delta _\omega$ and $l_x/\delta _\omega$ rapidly decrease in the transition region from a very large initial value, then followed by a nearly constant length scale until the end of the simulation, confirming again the self-similarity of large-scale structures. This observation indicates that the current computational domain is large enough to solve the large-scale turbulence. Otherwise, the growth of large-scale structures would be restricted resulting in the decrease of integral length scale (Vreman et al. Reference Vreman, Sandham and Luo1996; Vadrot et al. Reference Vadrot, Giauque and Corre2021). In the self-similar region, the streamwise length scale $l_x/\delta _\omega$ increases significantly with the convective Mach number, indicating that the large-scale structures of $R_{uu}$ became increasingly elongated in the streamwise direction. As the convective Mach number increases, the spanwise integral length scale is almost unchanged with $l_z/\delta _\omega \approx 0.3$, whereas a clear decrease of the spanwise size of large scale structures can be found in figure 24. We also find that the spanwise integral length scale agrees very well with the previous study of a compressible spatially developing mixing layer (Pirozzoli et al. Reference Pirozzoli, Bernardini, Marié and Grasso2015).

Figure 25. Evolutions of the normalised integral length scales in the streamwise direction $l_x/\delta _\omega$ and in the spanwise direction $l_z/\delta _\omega$ for (a) $M_c=0.2$ and (b) $M_c=1.8$. The dashed vertical lines mark the self-similar time periods.

In the stratified shear layer, the turbulent structures and flow dynamics have been found to resemble those in wall turbulence (Watanabe et al. Reference Watanabe, Riley, Nagata, Matsuda and Onishi2019; Watanabe & Nagata Reference Watanabe and Nagata2021). The universality of the elongated large-scale structures can be examined in compressible temporally evolving mixing layers and turbulent boundary layers. The spanwise integral length scale is comparable to that observed in the logarithmic region of turbulent boundary layers (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Pirozzoli Reference Pirozzoli2012), where $l_z/\delta \approx 0.2\unicode{x2013}0.3$ ($\delta$ is the thickness of the wall boundary layer). Figure 26 compares the two-point correlation of $u^{\prime \prime }$ in the present mixing shear layer with results reported in previous studies of the flat-plate boundary layers. We find that the two-point correlation at $M_c=0.2$ is in excellent agreement with the outer part of subsonic flat-plate boundary layers experimentally studied by Bross et al. (Reference Bross, Scharnowski and Kähler2021) with the centre of the contours located at $y/\delta = 0.2$. Near the wall (indicated by a red solid line), the contour line is stretched forming an extended tail on the left side and compacted toward the centre of the structures. For the case with $M_c=1.8$, the two-point correlation at lower level decays faster than that in supersonic flat-plate boundary layers measured by Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) with the centre of the contours located at $y/\delta = 0.3$, and the contour lines collapse well at high level of $R_{uu} \ge 0.6$. We infer that the disparity of the two-point correlation at $M_c=1.8$ is caused by the arbitrary selection of the wall-normal location at which the correlation is computed in the boundary layer, due to the three-dimensional geometric properties of the large scale structures (Sillero et al. Reference Sillero, Jiménez and Moser2014; Kevin, Monty & Hutchins Reference Kevin, Monty and Hutchins2019b). As suggested by Kevin, Monty & Hutchins (Reference Kevin, Monty and Hutchins2019a), the centre height $y/\delta = 0.2$ corresponds to the centre of the mean large-scale structures in the boundary layer, so we can expect a better comparison than that in figure 26(b).

Figure 26. Comparisons of the two-point correlation of $u^{\prime \prime }$ in the mixing shear layer with that in the boundary layers (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Bross et al. Reference Bross, Scharnowski and Kähler2021). The red solid lines indicate the wall of the boundary layer. The red + symbols show the centre points located with a distance of (a) $0.2\delta$ and (b) $0.3\delta$ away from the wall. The levels of contour lines are 0.2, 0.4, 0.6 and 0.8.

In wall-bounded turbulence, the large-scale structures interact with the small-scale structures in two ways, namely the superposition and amplitude-modulation effects (Marusic et al. Reference Marusic, Mathis and Hutchins2010; Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2011; Agostini & Leschziner Reference Agostini and Leschziner2014). A top view of high- and low-speed large-scale structures and small-scale vortical structures in the upper half-domain is given in figure 27 for $M_c=1.8$. It is easy to identify two highly elongated low-speed (blue) regions flanked on either side in the spanwise direction by high-speed (red) regions, as shown in figure 27(b). Comparing figures 20(b) and 27(a), it is clear that the tendency of the vortical structures to align in the streamwise direction at high convective Mach number are much weaker at later times, which is consistent with the previous numerical analysis by Arun et al. (Reference Arun, Sameen, Srinivasan and Girimaji2019). We can see that the small-scale vortical structures have an apparent preference for clustering into the low-speed regions, and are more sparse in the high-speed regions. The iso-surface of low-speed regions acts as an envelope of the clustering small-scale structures. The observations are consistent with previous studies on wall turbulence (Marusic et al. Reference Marusic, Mathis and Hutchins2010; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Chan & Chin Reference Chan and Chin2022).

Figure 27. Top view of coherent structures in the upper half-domain $y>0$ for $M_c=1.8$ at $\tau =1250$. (a) Small-scale vortical structures visualised by the iso-surfaces of the second invariant of the velocity gradient tensor $Q =2$ and coloured based on local values of $u^{\prime \prime }$. (b) The high- and low-speed large-scale structures presented by the iso-surfaces of $u^{\prime \prime } = 0.3$ (blue) and $-$0.3 (red), respectively.

Moreover, we examined the conditionally averaged flow fields, $\langle u_i^{\prime \prime }\mid u^{\prime \prime }<0 \rangle$ and $\langle \omega \mid u^{\prime \prime }<0 \rangle$, obtained by ensemble averaging $u_i^{\prime \prime }$ and $\omega$ around any point in the low-speed regions and at the centreplane $y=0$, where $\omega$ is the vorticity magnitude. Figure 28 displays the conditionally averaged flow field in the $x$–$y$ and $y$–$z$ planes for $M_c=0.2$ and $1.8$. For both low and high convective Mach numbers, the large-scale low-speed structures are lifted upwards by a pair of counter-rotating quasi-streamwise vortices flanking the large-scale structures, as shown in figures 28(b) and 28(d), resembling the typical counter-rotating roll modes found in wall turbulence (Marusic et al. Reference Marusic, Mathis and Hutchins2010; Kevin et al. Reference Kevin, Monty and Hutchins2019a). These quasi-streamwise roll modes in the mixing shear layer are inclined slightly to the horizontal, which can be inferred from the orientation of velocity vectors in large-scale low-speed structures, as depicted in figures 28(a) and 28(c). The spanwise distance between two centres of the counter-rotating quasi-streamwise vortices clearly decreases as the convective Mach number increases, which is consistent with previous study of a compressible spatially developing mixing layer (Pirozzoli et al. Reference Pirozzoli, Bernardini, Marié and Grasso2015). At $M_c=0.2$, we can see a clear sign of flow organisation into spanwise rollers located at both ends of conditionally averaged low-speed structures, which almost disappear at the highest convective Mach number $M_c=1.8$, indicating that the effect of compressibility can suppress the Kelvin–Helmholtz instability.

Figure 28. Conditional average of flow field conditioned to the low-speed structures, $u^{\prime \prime } < 0$, in the (a),(c) $x$–$y$ and (b),(d) $y$–$z$ planes: (a),(b) $M_c=0.2$ and (c),(d) $M_c=1.8$. The colour contours display $\langle \omega \mid u^{\prime \prime }<0 \rangle$ and $\langle u^{\prime \prime }\mid u^{\prime \prime } < 0 \rangle$ is plotted with solid black contour lines representing levels range from $-$0.2 to $-$0.05 (from inside to outside) with increments of 0.05. Vectors represent conditional-averaged velocity components with the longest arrow measuring 0.2. The $+$ (red) symbols show the centre location of roll modes.

Figure 28 also shows the distribution of the conditionally averaged vorticity intensity, $\langle \omega \mid u^{\prime \prime }<0 \rangle$. We find that, in the large-scale low-speed structures, the local maximum of $\langle \omega \mid u^{\prime \prime }<0 \rangle$ occurs at a higher distance from the centreline, confirming the clustering of small-scale vortical structures within the upper part of large-scale low-speed structures, as shown in figure 27. From figures 28(a) and 28(c), we can see that the clustering of small-scale vortical structures is directly associated with high-shearing motions presented by the longer arrow there. Taking into account the antisymmetry of the base flow $\tilde {u}_i(y) = -\tilde {u}_i(-y)$, the large-scale high- and low-speed structures statistically exhibit central symmetry patterns, therefore, the results on high-speed structures are not shown for brevity.

According to the present results, we propose a conceptual model for higher convective Mach number $M_c \ge 0.8$ to qualitatively depict the dynamics of the streamwise elongated large-scale structures and their interaction with small-scale vortical structures in the self-similar region, as illustrated in figure 29 viewed on a $y$–$z$ plane. This conceptual model resembles the outer part of the well-known model of wall-bounded turbulent flow proposed by Marusic et al. (Reference Marusic, Mathis and Hutchins2010). In this conceptual model, the high- (red) and the low-speed (blue) large-scale structures alternatively arrange in the spanwise direction with the associated counter-rotating roll modes (black circles) inducing the rising of the low-speed structures and the sinking of the high-speed structures. The embedding of the low-speed structures into the high-speed free stream results in a high-shear region promoting the small-scale vortical structures on top of the low-speed structures. Similarly, the small-scale vortical structures are activated at the bottom of high-speed large-scale structures. In the mixing layer, the large-scale high- and low-speed structures exhibit central symmetry about their symmetric centre on centreplane, whereas in the wall-bounded turbulence the near-wall part of the large-scale structures interact with the near-wall turbulence, through superposition and modulation effects (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009; Marusic et al. Reference Marusic, Mathis and Hutchins2010; Mathis et al. Reference Mathis, Hutchins and Marusic2011). It should be noted that, at low convective Mach number, the large-scale high- and low-speed structures and the spanwise rollers coexist, which deserves further study.

Figure 29. Schematic of large-scale coherent structures and its interaction across the mixing shear layer for higher convective Mach number $M_c \ge 0.8$. The red and blue regions indicate the high- and low-speed large-scale structures, respectively. The black circles is the counter-rotating roll modes. The gray contours show the clustering of small-scale vortical structures. The red and black dashed lines represent centreplane and boundaries of mixing layer, respectively.