4.1. Turbulent kinetic energy

Figures 3(a)–3(c) present the contributions of the low-speed LSSs to the turbulent kinetic energy $\bar {K}_{i,{n}}/(\rho _\infty \Delta U^2)$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The components of the turbulent kinetic energy are normalized by $\rho _\infty \Delta U^2$. Previous studies have demonstrated the symmetry of $\bar {K}_{i}$ about the centre plane of the mixing layers (Rogers & Moser Reference Rogers and Moser1994; Pantano & Sarkar Reference Pantano and Sarkar2002; Wang et al. Reference Wang, Wang and Chen2022). However, figures 3($a$)–3($c$) show distinct upward shifts in the peak positions of $\bar {K}_{i,{n}}/(\rho _\infty \Delta U^2)$, suggesting that the contributions of the low-speed LSSs are dominant in the upper half of the mixing layer. This observation is qualitatively similar to the results of Lee et al. (Reference Lee, Ahn and Sung2015) in wall-bounded turbulence, but the distribution in the vertical direction in wall turbulence is more complex. Lee et al. (Reference Lee, Ahn and Sung2015) observed that low-speed LSSs contribute more to turbulent kinetic energy in the outer region, while high-speed LSSs contribute more in the near-wall region. In mixing layers, the low-speed LSSs contribute more to turbulent kinetic energy in the upper half-region, while high-speed LSSs contribute more in the lower half-region.

Figure 3.

(a–c) Contributions of the low-speed LSSs to turbulent kinetic energy $\bar {K}_{i,{n}}/(\rho _\infty \Delta U^2)$ and (d–f) conditionally averaged turbulent kinetic energy $\hat {K}_{i,{n}}/(\rho _\infty \Delta U^2)$ at $M_c=0.2$, $0.8$ and $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

Figures 3($d$)–3( $f$) show the conditionally averaged turbulent kinetic energy conditioned on the low-speed LSSs $\hat {K}_{i,{n}}/(\rho _\infty \Delta U^2)$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. Compared with the profiles of $\bar {K}_{i,{n}}/(\rho _\infty \Delta U^2)$, the profiles of $\hat {K}_{i,{n}}/(\rho _\infty \Delta U^2)$ are similar in the central region of the mixing layer, but are significantly amplified at the edges, especially for the streamwise and vertical components of turbulent kinetic energy. At $M_c = 0.2$, $\hat {K}_{2,{n}}/(\rho _\infty \Delta U^2)$ exhibits two plateaus, located at $-0.8 < y/\delta _\omega < -0.6$ and $0.6 < y/\delta _\omega < 0.8$, respectively. Surprisingly, when $y/\delta _\omega >0.7$, the vertical turbulent kinetic energy exceeds the streamwise turbulent kinetic energy. As the convective Mach number increases from $M_c= 0.2$ to $1.8$, the magnitudes of $\hat {K}_{i,{n}}/(\rho _\infty \Delta U^2)$ become smaller in both the central region and edges of the mixing layer, demonstrating the significant suppression of compressibility on the turbulent kinetic energy. Interestingly, their peak positions remain insensitive to the convective Mach number.

From the perspective of turbulent structures, the turbulent kinetic energy at the central region of the mixing layer primarily originates from LSSs, while near the edges of the mixing layer, it mainly comes from small-scale structures. The amplification of turbulent kinetic energy near the edges of the mixing layer is consistent with the amplitude modulation effect exerted by the LSSs onto the near-edge small-scale structures. This effect has been extensively studied in wall turbulence (Marusic et al. Reference Marusic, Mathis and Hutchins2010; Mathis et al. Reference Mathis, Hutchins and Marusic2011a; Agostini & Leschziner Reference Agostini and Leschziner2014) and has also been confirmed in free shear layers (Bandyopadhyay & Hussain Reference Bandyopadhyay and Hussain1984; Fiscaletti et al. Reference Fiscaletti, Ganapathisubramani and Elsinga2015; Wang et al. Reference Wang, Wang and Chen2022, Reference Wang, Guo, Wang and Chen2024), including turbulent wakes, mixing layers and jets. Talluru et al. (Reference Talluru, Baidya, Hutchins and Marusic2014) reported that in wall turbulence, all three components of velocity are modulated in a very similar manner and the amplitude modulation is relatively uniform across all three velocity components. However, according to figures 3($d$)–3($f$), the amplitude modulation in the mixing layers is not uniform across all three velocity components. The vertical velocity exhibits stronger amplitude modulation compared with the streamwise and spanwise components, with the spanwise velocity showing the least significant amplitude modulation. Meanwhile, we infer that the degree of amplitude modulation decreases with increasing convective Mach number in compressible mixing layers.

4.2. Production

Figure 4($a$) shows the contribution of low-speed LSSs to the production of turbulent kinetic energy $\bar {P}_{n}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The budget terms are normalized by the average dissipation at the centreline $\bar {\epsilon }_{c}$. The production contribution of low-speed LSSs $\bar {P}_{n}/\bar {\epsilon }_{c}$ is unimodal with the peak located in the upper half of the mixing layer ($y>0$). Before reaching the edge of the mixing layer, when $y/\delta _\omega >0.7$ or $y/\delta _\omega <-0.6$, $\bar {P}_{n}/\bar {\epsilon }_{c}$ is nearly zero. This observation highlights that the dominant contribution of the low-speed LSSs to the production comes from the upper half of the mixing layer. Here, $\bar {P}_{n}/\bar {\epsilon }_{c}$ for different Mach numbers do not completely overlap, displaying a separation of approximately 10 % to 20 %. For $y/\delta _\omega >-0.25$, the production contribution of low-speed LSSs is highest at $M_c=0.2$, moderate at $M_c=1.8$ and lowest at $M_c=0.8$, indicating a non-monotonic variation with $M_c$. Conversely, for $y/\delta _\omega <-0.25$, $\bar {P}_{n}/\bar {\epsilon }_{c}$ increases with increasing $M_c$.

Figure 4.

($a$) Contributions of the low-speed LSSs to production $\bar {P}_{n}/\bar {\epsilon }_{c}$ and ($b$) conditionally averaged production $\hat {P}_{n}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$.

Figure 4($b$) shows the conditionally averaged production term conditioned on the low-speed LSSs $\hat {P}_{n}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The shapes of $\bar {P}_{n}/\bar {\epsilon }_{c}$ and $\hat {P}_{n}/\bar {\epsilon }_{c}$ are similar to each other. Unlike $\hat {K}_{i,{n}}/(\rho _\infty \Delta U^2)$, $\hat {P}_{n}/\bar {\epsilon }_{c}$ is not amplified at the edges of the mixing layer, indicating that the small-scale structures hardly generate turbulent kinetic energy. Overall, the production term is highest at $M_c = 0.2$, moderate at $M_c = 1.8$ and lowest at $M_c = 0.8$, indicating a non-monotonic relationship with $M_c$, except for $0.3 < y/\delta _\omega <0.6$. When normalized by $\rho \infty \Delta U^3/\delta _\omega$, the production of the TKE $\bar {P}_{n}$ clearly shows a monotonic decrease with increasing $M_c$, consistent with previous studies (Pantano & Sarkar Reference Pantano and Sarkar2002; Vaghefi Reference Vaghefi2014), although this result is not presented in this paper.

Figures 5($a$) and 5($b$) show two-dimensional conditionally averaged velocity and production based on the condition of $u^{\prime \prime }<-0.15\Delta U$ at convective Mach numbers $M_c= 0.2$ and $1.8$. The velocity is normalized by $\Delta U$. The conditionally averaged streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) demonstrates elongated forward-leaning LSSs, extending a distance of $2\delta _\omega$ in the streamwise direction. These structures exhibit a strong negative correlation with the conditionally averaged vertical velocity $\breve {v}_{n}/\Delta U$ (dashed lines), indicating that low-speed LSSs are predominantly moving upwards. The spatial distribution of the production is inherently similar to the shape of the LSSs, confirming that LSSs contribute favourably to the turbulence production. The production mainly occurs at the centre of the LSSs, and decreases as the convective Mach number increases from $M_c= 0.2$ to $1.8$.

Figure 5.

Two-dimensional conditionally averaged production $\breve {P}_{n}/\bar {\epsilon }_{c}$ (contours), streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) and vertical velocity $\breve {v}_{n}/\Delta U$ (dashed lines), and based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at ($a$) $M_c=0.2$, ($b$) $M_c=1.8$. Conditionally averaged negative component of production $\breve {P}^-_{n}/\bar {\epsilon }_{c}$ at ($c$) $M_c=0.2$, ($d$) $M_c=1.8$.

We decompose the kinetic energy production ${P}$ into a positive component ${P}^+$ and a negative component ${P}^-$: ${P}={P}^+ + {P}^-$. Here, ${P}^+ = 0.5(P+|P|)$ and ${P}^- = 0.5(P-|P|)$. The positive production is contributed by negative Reynolds shear stress corresponding to Q2 and Q4 events (or ejection and sweep events) in quadrant analysis, while the negative production is contributed by positive Reynolds shear stress corresponding to Q1 and Q3 events (or positive quadrant events) (Pope Reference Pope2000; Wallace Reference Wallace2016). On average, the low-speed LSSs move upwards, and the high-speed LSSs move downwards; namely, they contribute most of the positive kinetic energy production. In figures 5($c$) and 5($d$), we present the conditionally averaged negative component of production $\breve {P}^-_{n}/\bar {\epsilon }_{c}$, which is significantly smaller than the total production $\breve {P}_{n}/\bar {\epsilon }_{c}$. At $M_c=0.2$, negative production events are mainly concentrated within the upper-right region of the low-speed LSSs. Conversely, at $M_c=1.8$, these negative production events primarily emerge within the lower-left region of the low-speed LSSs.

4.3. Dissipation

Figure 6($a$) shows the contribution of low-speed LSSs to the dissipation of turbulent kinetic energy $\bar {\epsilon }_{n}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The dissipation contribution of low-speed LSS $\bar {\epsilon }_{n}/\bar {\epsilon }_{c}$ is unimodal peaking around $y/\delta _\omega \approx 0.1$. This observation indicates that the dissipation induced by low-speed LSSs tends towards the upper portion of the mixing layer. Unlike the contribution of low-speed LSSs to the production term, $\bar {\epsilon }_{n}/\bar {\epsilon }_{c}$ remains non-zero near the edge of the mixing layer, extending until $y/\delta _\omega >0.8$ or $y/\delta _\omega <-0.8$. In the profile of conditionally averaged dissipation $\hat {\epsilon }_{n}/\bar {\epsilon }_{c}$, the dissipation near the edge of the mixing layer is significantly amplified, as shown in figure 6($b$). It is evident that in regions $-1.0 < y/\delta _\omega < -0.5$ and $0.5 < y/\delta _\omega < 1.0$, there are noticeable bumps in $\hat {\epsilon }_{n}/\bar {\epsilon }_{c}$, with the bump being more prominent in the upper region. This observation confirms that the amplification of turbulent kinetic energy near the edges of the mixing layer is due to small-scale structures, as shown in figures 3($d$)–3($f$). As the convective Mach number increases, these bumps of $\hat {\epsilon }_{n}/\bar {\epsilon }_{c}$ rapidly decrease but still persist.

Figure 6.

($a$) Contributions of the low-speed LSSs to dissipation $\bar {\epsilon }_{n}/\bar {\epsilon }_{c}$ and ($b$) conditionally averaged dissipation $\hat {\epsilon }_{n}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$. ($c$) $\bar {\epsilon }_{n}$ is normalized by $\rho _\infty \Delta U^3/\delta _\omega$ with the error bar representing discrepancy among the maximum and minimum dissipation components.

When normalized by $\rho _\infty \Delta U^3/\delta _\omega$, the dissipation decreases rapidly with increasing convective Mach number, as shown in figure 6($c$). This trend is in good agreement with previous results reported by Vaghefi (Reference Vaghefi2014). However, Pantano & Sarkar (Reference Pantano and Sarkar2002) reported that the dissipation changes relatively little with varying convective Mach numbers. The error bar in figure 6($a$) represents the discrepancy between the streamwise and vertical dissipation components, with each representing the maximum and minimum dissipation component, respectively. It is clear to see that the dissipation becomes more anisotropic in component as the convective Mach number increases from $M_c= 0.2$ to $1.8$, indicating the anisotropic suppression effect of compressibility on the mixing process.

Figure 7 shows two-dimensional conditionally averaged dissipation $\breve {\epsilon }_{i,{n}}/\bar {\epsilon }_{c}$ and streamwise velocity $\breve {u}_{n}/\Delta U$ based on the condition of $u^{\prime \prime }<-0.15\Delta U$ at convective Mach numbers $M_c= 0.2$ and $1.8$. The dissipation concentrates in the upper regions of the low-speed LSSs due to the modulation effect exerted by the LSSs on the small-scale structures, wherein the small-scale vortical structures are activated at the top of low-speed LSSs. This modulation effect was initially identified and extensively studied in wall turbulence (Marusic et al. Reference Marusic, Mathis and Hutchins2010; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Chan & Chin Reference Chan and Chin2022) and was observed in compressible mixing layers (Wang et al. Reference Wang, Wang and Chen2022). At $M_c=0.2$, the dissipation is concentrated in the upper part of the LSSs, exhibiting downstream skewness along the flow, with similar distributions and strengths across its components. At $M_c=1.8$, dissipation within the LSSs exhibits noticeable component anisotropy. The dissipation in the flow direction predominates significantly over the other two components, while the spanwise component remains intermediate, and the vertical component is the smallest.

Figure 7.

Two-dimensional conditionally averaged components of dissipation $\breve {\epsilon }_{i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at ($a$–$c$) $M_c=0.2$ and ($d$–$f$) $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

4.4. Conversion between kinetic and internal energies

Figure 8($a$) shows the contribution of low-speed LSSs to the pressure-dilatation term $\bar {\varPhi }_{n}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The pressure-dilatation term represents the energy exchange between turbulent kinetic energy and internal energy, and is significantly smaller than the dissipation term. At convective Mach number $M_c= 0.2$, the pressure-dilatation term $\bar {\varPhi }_{n}/\bar {\epsilon }_{c}$ is negative almost everywhere with its minimum at $y/\delta _\omega = -0.2$, indicating that the mean pressure-dilatation converts turbulent kinetic energy into internal energy within the LSSs. As the convective Mach number increases from $M_c = 0.2$ to $1.8$, the magnitude of the minimum value of $\bar {\varPhi }_{n}/\bar {\epsilon }_{c}$ increases, and the location of this minimum shifts slightly downwards. At higher convective Mach numbers $M_c = 0.8$ and $1.8$, $\bar {\varPhi }_{n}/\bar {\epsilon }_{c}$ becomes positive for $y/\delta _\omega > 0.2$, peaking at $y/\delta _\omega = 0.5$. This observation implies that pressure-dilatation converts internal energy into the turbulent kinetic energy at the upper region of low-speed LSSs where the small-scale structures are concentrated. The contribution of low-speed LSSs to mean pressure-dilatation $\bar {\varPhi }_{n}/\bar {\epsilon }_{c}$ increases with enhanced compressibility, but it still remains almost two orders of magnitude smaller than the production and dissipation terms, and has usually been neglected in previous literature (Pantano & Sarkar Reference Pantano and Sarkar2002; Arun et al. Reference Arun, Sameen, Srinivasan and Girimaji2019).

Figure 8.

($a$) Contributions of the low-speed LSSs to pressure-dilatation $\bar {\varPhi }_{n}/\bar {\epsilon }_{c}$ and ($b$) conditionally averaged pressure-dilatation $\hat {\varPhi }_{n}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$.

However, the local pressure-dilatation can reach the same order of magnitude as the dissipation term near the edges of the mixing layer, as shown in figure 8($b$), where the conditionally averaged pressure-dilatation term conditioned on the low-speed LSSs $\hat {\varPhi }_{n}/\bar {\epsilon }_{c}$ obtains a minimum of $-0.15$ at $y/\delta _\omega =0.9$ at the highest convective Mach number $M_c = 1.8$, even exceeding the magnitude of dissipation at the corresponding locations. It is clearly observed that at $M_c = 1.8$, the pressure-dilatation fluctuates significantly near the upper edge of the mixing layer. Additionally, beyond the upper edge of the mixing layer, at $y/\delta _\omega > 1.0$, there is still noticeable negative pressure-dilatation, whereas near the lower boundary of the mixing layer, the magnitude of pressure-dilatation is much smaller. This is likely related to the formation of shock waves. Rossmann, Mungal & Hanson (Reference Rossmann, Mungal and Hanson2002) and Zhou, He & Shen (Reference Zhou, He and Shen2012) discovered that the shock formation mechanism closely resembles the scenario of high-speed flow around a bluff body. In other words, the upward-moving low-speed LSSs act like bulges of turbulent fluid protruding into the high-speed free stream, leading to the formation of shock waves propagating outwards from the surfaces of these LSSs. The formation of shock waves results in a dramatic conversion of turbulent kinetic energy into internal energy.

4.5. Inter-component energy transfer

The pressure-strain components are responsible for redistributing turbulent kinetic energy from the streamwise direction to the vertical and spanwise directions on average in turbulent mixing layers (Pantano & Sarkar Reference Pantano and Sarkar2002; Arun et al. Reference Arun, Sameen, Srinivasan and Girimaji2019). Figures 9($a$), 9($c$) and 9($e$) show the contribution of low-speed LSSs to the pressure-strain components $\bar {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The three pressure-strain components $\bar {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ are of the same order of magnitude as the dominant budget terms including production and dissipation. The profiles of all pressure-strain components $\bar {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ exhibit a unimodal shape with their peaks located in the upper half of the mixing layer, demonstrating that the contribution of the low-speed LSSs to the component redistribution of turbulent kinetic energy concentrates in the upper half of the mixing layer. The redistribution predominantly occurs within the region $-0.5< y/\delta _\omega <0.7$, where the streamwise component $\bar {\varPhi }_{1,{n}}/\bar {\epsilon }_{c}$ is negative, while both $\bar {\varPhi }_{2,{n}}/\bar {\epsilon }_{c}$ and $\bar {\varPhi }_{3,{n}}/\bar {\epsilon }_{c}$ components are positive, with their magnitudes being nearly identical to each other. This result indicates that the energy loss from the streamwise component is equally redistributed to the vertical and spanwise components.

Figure 9.

($a$–$c$) Contributions of the low-speed LSSs to pressure-strain $\bar {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ and ($d$–$f$) conditionally averaged pressure-strain $\hat {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

Near the edge of the mixing layer, $\bar {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ is obviously non-zero for $y/\delta _\omega < -0.5$ and $y/\delta _\omega > 0.7$. Similar to the dissipation, the volume fraction corresponding to the pressure redistribution process near the mixing layer edge is small, but the magnitude of the pressure-strain components is not negligible. This is evident in the conditionally averaged pressure-strain components $\hat {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ shown in figures 9($d$)–9($f$). Within region $-1.0 < y/\delta _\omega < -0.5$, both $\hat {\varPhi }_{1,{n}}/\bar {\epsilon }_{c}$ and $\hat {\varPhi }_{3,{n}}/\bar {\epsilon }_{c}$ exhibit a negative peak, while $\hat {\varPhi }_{2,{n}}/\bar {\epsilon }_{c}$ shows a positive peak, even reaching twice the magnitude of its peak observed in the central region of the mixing layer at low convective Mach number $M_c= 0.2$. These peaks are all located around $y/\delta _\omega \approx -0.85$. The observation indicates that both streamwise and spanwise turbulent kinetic energy are simultaneously converted to the vertical direction near the edge of the mixing layer. This can be attributed to the dynamics of the TNTI. A key characteristic of the TNTI is the well-known vorticity jump (or strong velocity gradient) across the interface, which resembles a shear layer that can become unstable under certain conditions. This unstable shear layer can cause the formation of secondary Kelvin–Helmholtz instabilities, which are linked to the nibbling process at the TNTI (Philip et al. Reference Philip, Meneveau, de Silva and Marusic2014; Reuther & Kähler Reference Reuther and Kähler2020). The secondary Kelvin–Helmholtz instabilities redistribute turbulent kinetic energy by converting the horizontal components into the vertical component. Similarly, at the upper edge of the mixing layer, within the region $0.5 < y/\delta _\omega < 1.0$, the exchange of streamwise and spanwise turbulent kinetic energy to the normal direction also occurs, albeit with relatively weaker intensity. It is interesting to note that the peaks of the conditionally averaged pressure-strain $\hat {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ near the edges of the mixing layer rapidly decrease with the increase in convective Mach number, whereas the peaks in the central region of the mixing layer remain insensitive to the convective Mach number. This observation may be due to the fact that the pressure redistribution process occurs at both large and small scales. The large scales contribute more significantly to the overall redistribution, while the small scales exhibit more intense activity (Lee & Moser Reference Lee and Moser2019; Wang et al. Reference Wang, Wang, Li and Chen2021). The increase in compressibility noticeably suppresses the pressure redistribution process associated with the small-scale structures. When normalized by $\rho _\infty \Delta U^3/\delta _\omega$, the pressure-strain terms exhibit a significant decrease with increasing Mach number, and are omitted here for the sake of brevity. In agreement with previous investigations, the decreasing of the pressure-strain term with increasing $M_c$ leads to inhibited energy redistributing from the streamwise to cross-stream fluctuations, to the reduced turbulence production and, finally, to reduced turbulence levels as well as reduced growth rate of the shear layer (Vreman et al. Reference Vreman, Sandham and Luo1996; Pantano & Sarkar Reference Pantano and Sarkar2002; Mahle Reference Mahle2007).

Figure 10 shows two-dimensional conditionally averaged pressure-strain $\breve {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ and velocity $\breve {u}_{n}/\Delta U$ based on the condition of $u^{\prime \prime }<-0.15\Delta U$ at convective Mach numbers $M_c= 0.2$ and $1.8$. Although, on average, energy is evenly redistributed from streamwise to vertical and spanwise directions, the shapes of $\breve {\varPhi }_{2,{n}}/\bar {\epsilon }_{c}$ and $\breve {\varPhi }_{3,{n}}/\bar {\epsilon }_{c}$ in low-speed LSSs are completely different. This observation indicates a highly non-uniform redistribution of turbulent kinetic energy from the streamwise to the vertical and spanwise directions at the same location within LSSs. Far away from the centre of low-speed LSSs and biased towards the upstream direction, $\breve {\varPhi }_{1,{n}}/\bar {\epsilon }_{c}$ and $\breve {\varPhi }_{3,{n}}/\bar {\epsilon }_{c}$ display a large-scale region with strikingly similar shapes and large magnitude. In this region, $\breve {\varPhi }_{1,{n}}/\bar {\epsilon }_{c}$ takes on a negative value, while $\breve {\varPhi }_{3,{n}}/\bar {\epsilon }_{c}$ is positive. Thus, the streamwise turbulent kinetic energy $K_1$ is predominantly redistributed to the spanwise direction $K_3$. Near the centre of the LSSs and in a smaller region located above, there is a significant transfer of energy from $K_1$ to $K_2$, as shown in figures 10($b$) and 10($e$). Below the centre of the LSSs, at $(r_x/\delta _\omega, r_y/\delta _\omega )\approx (0.0,-0.2)$, $\breve {\varPhi }_{3,{n}}/\bar {\epsilon }_{c}$ is negative, while $\breve {\varPhi }_{1,{n}}/\bar {\epsilon }_{c}$ and $\breve {\varPhi }_{2,{n}}/\bar {\epsilon }_{c}$ are positive, with relatively small magnitudes. Therefore, a small portion of turbulent kinetic energy is redistributed from $K_3$ to $K_1$ and $K_2$, primarily to $K_2$. The non-uniform redistribution shown in figures 9 and 10 is related to the pressure response caused by different scales. In the central region of the mixing layer, energy redistribution primarily occurs at larger scales, where the streamwise turbulent kinetic energy is transferred into the spanwise and vertical directions. Near the edges of the mixing layer, energy redistribution takes place at smaller scales, where turbulent kinetic energy shifts from the streamwise and spanwise components to the vertical component.

Figure 10.

Two-dimensional conditionally averaged components of pressure-strain $\breve {\varPhi }_{i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at ($a$–$c$) $M_c=0.2$ and ($d$–$f$) $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

To further investigate the turbulent kinetic energy exchange among different components, we decompose the pressure-strain terms into positive and negative components: $\varPhi _i=\varPhi ^+_i + \varPhi ^-_i$, where the positive component $\varPhi _i^+ = 0.5(\varPhi _i+|\varPhi _i|)$ and the negative component $\varPhi ^-_i = 0.5(\varPhi _i-|\varPhi _i|)$. The two-dimensional conditionally average of the positive and negative pressure-strain components are shown in figures 11 and 12 for $M_c=0.2$ and $1.8$, respectively. At low Mach number $M_c=0.2$, the contours of $\breve {\varPhi }^+_{i,{n}}/\bar {\epsilon }_{c}$ and $\breve {\varPhi }^-_{i,{n}}/\bar {\epsilon }_{c}$ exhibit strikingly similar shapes, featuring a maximum value skewed towards the upper right side of the LSSs, with $\breve {\varPhi }^+_{1,{n}}/\bar {\epsilon }_{c}$ having smaller magnitude. This observation indicates that each turbulent kinetic energy component not only receives energy from the other two directions, but also contributes energy to them simultaneously. At high Mach number $M_c=1.8$, the magnitudes of $\breve {\varPhi }^+_{i,{n}}/\bar {\epsilon }_{c}$ and $\breve {\varPhi }^-_{i,{n}}/\bar {\epsilon }_{c}$ decrease and become more anisotropic compared with that at $M_c=0.2$. Here, $\breve {\varPhi }^-_{1,{n}}/\bar {\epsilon }_{c}$, $\breve {\varPhi }^+_{2,{n}}/\bar {\epsilon }_{c}$ and $\breve {\varPhi }^+_{3,{n}}/\bar {\epsilon }_{c}$ display two distinct peak regions in their contours, and are significantly larger than their respective counterparts. The transfer of energy from $K_1$ to $K_2$ predominantly occurs in a smaller-scale region at the upper right of the large-scale centre (at $r_x=0, r_y=0$), while the transfer from $K_1$ to $K_3$ takes place in a larger-scale region to the upper left of the large-scale centre. The exchange of turbulent kinetic energy between other components is notably reduced at $M_c=1.8$.

Figure 11.

Two-dimensional conditionally averaged ($a$–$c$) positive part of pressure-strain $\breve {\varPhi }^+_{i,{n}}/\bar {\epsilon }_{c}$, ($d$–$f$) negative part of pressure-strain $\breve {\varPhi }^-_{i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at $M_c=0.2$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

Figure 12.

Two-dimensional conditionally averaged ($a$–$c$) positive part of pressure-strain $\breve {\varPhi }^+_{i,{n}}/\bar {\epsilon }_{c}$, ($d$–$f$) negative part of pressure-strain $\breve {\varPhi }^-_{i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at $M_c=1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

4.6. Spatial diffusion

Figures 13($a$)–13($c$) show the contribution of low-speed LSSs to the turbulent diffusion of turbulent kinetic energy $\bar {T}_{t,i,{n}}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. Turbulent diffusion exhibits a clear anisotropy. The streamwise component of turbulent diffusion is the largest, followed by the vertical component and the spanwise component is the smallest. The contribution of low-speed LSSs to the streamwise turbulent diffusion term, $\bar {T}_{t,1,{n}}/\bar {\epsilon }_{c}$, resembles a wave shape, with a phase shift upwards by approximately $0.15/\delta _\omega$. The peak of this wave is located at $y/\delta _\omega = 0.4$, while the trough is positioned around $y/\delta _\omega = -0.15$. This observation indicates that within low-speed LSSs, $\bar {T}_{t,1,{n}}/\bar {\epsilon }_{c}$ transports streamwise turbulent kinetic energy $K_1$ upwards, whereas within high-speed LSSs, it transports $K_1$ downwards (not shown). It is interesting to note that the growth of $\bar {T}_{t,1,{n}}/\bar {\epsilon }_{c}$ from its trough to peak is nearly linear. Unlike the streamwise component, the vertical component of turbulent diffusion $\bar {T}_{t,2,{n}}/\bar {\epsilon }_{c}$ transports vertical turbulent kinetic energy $K_2$ from the centre of the mixing layer to both the upper and lower edges of the mixing layer simultaneously, although the downward transport is relatively minor. The spanwise component of turbulent diffusion $\bar {T}_{t,3,{n}}/\bar {\epsilon }_{c}$, however, behaves similarly to the vertical component, as it transports spanwise turbulent kinetic energy $K_3$ from the centre of the mixing layer to both the upper and lower edges of the mixing layer.

Figure 13.

($a$–$c$) Contributions of the low-speed LSSs to turbulent diffusion $\bar {T}_{t,i,{n}}/\bar {\epsilon }_{c}$ and ($d$–$f$) conditionally averaged turbulent diffusion $\hat {T}_{t,i,{n}}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

The conditionally averaged turbulent diffusion term $\hat {T}_{t,i,{n}}/\bar {\epsilon }_{c}$ is presented in figures 13($d$)–13($f$), where the local turbulent diffusion process is highlighted at the edges of the mixing layer. Here, $\hat {T}_{t,1,{n}}/\bar {\epsilon }_{c}$ peaks at $y/\delta _\omega = 0.7$, with an additional small trough appearing at $y/\delta _\omega = -0.7$. Additionally, $\hat {T}_{t,1,{n}}/\bar {\epsilon }_{c}$ exhibits linear growth from its minimum to maximum values. At $M_c = 0.2$ and $0.8$, the range of this linear growth region expands to almost $y/\delta _\omega = 0.7$. Furthermore, the range of linear growth widens with decreasing convective Mach number, in contrast to the behaviour of the linear stage of volume fraction, as shown in figure 2($a$). The two peaks of $\hat {T}_{t,2,{n}}/\bar {\epsilon }_{c}$ and $\hat {T}_{t,3,{n}}/\bar {\epsilon }_{c}$ have shifted to $y/\delta _\omega = \pm 0.8$. The vertical variation of $\hat {T}_{t,2,{n}}/\bar {\epsilon }_{c}$ is very large, especially at the edges of the mixing layer, where its peak near the upper edge even exceeds the peak of $\hat {T}_{t,1,{n}}/\bar {\epsilon }_{c}$. The magnitude of $\hat {T}_{t,3,{n}}/\bar {\epsilon }_{c}$ is noticeably smaller than the other two components. As the convective Mach number increases, the conditionally averaged turbulent diffusion term $\hat {T}_{t,i,{n}}/\bar {\epsilon }_{c}$ decreases rapidly near the edges of the mixing layer, while it remains relatively unchanged at the centre of the mixing layer.

More insight into the spatial distribution of the turbulent diffusion can be gained from its two-dimensional conditional average shown in figure 14. The streamwise turbulent diffusion features two large-scale regions of similar shapes, with positive values in the upper left and negative values in the lower right. This suggests that the streamwise turbulent diffusion $\breve {T}_{t,1,{n}}/\bar {\epsilon }_{c}$ transports streamwise turbulent kinetic energy from the lower right region of the low-speed LSSs to its upper left region. In other words, the direction of turbulent diffusion of $K_1$ is not vertical but inclined, with a skew angle of approximately $18^\circ$ at $M_c= 0.2$ and $15^\circ$ at $1.8$. This angle is measured between the line (black dashed line figure 14a,d) connecting the extrema (maximum and minimum points) of the two large-scale regions and the horizontal direction. The direction of turbulent diffusion $K_2$ also tends to be from the lower right region of the LSSs towards its upper left region, but with a weaker intensity. However, the direction of turbulent diffusion $K_3$ tends to be from the upper left region of the LSSs towards its lower right region, opposite to the direction of $K_1$ and $K_2$. The inclined distribution of the turbulent diffusion components in the $x\unicode{x2013}y$ plane causes positive and negative values to cancel each other out when averaged in the horizontal plane. As a result, the maximum positive and negative values of the turbulent diffusion components in the two-dimensional average shown in figure 14 are significantly larger than their counterparts in figures 13($d$)–13($f$), especially for the streamwise component of turbulent diffusion.

Figure 14.

Two-dimensional conditionally averaged components of turbulent diffusion $\breve {T}_{t,i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at ($a$–$c$) $M_c=0.2$ and ($d$–$f$) $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

Figures 15($a$)–15($c$) show the contribution of low-speed LSSs to the pressure diffusion of turbulent kinetic energy $\bar {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. Compared with the turbulent diffusion, the value of pressure diffusion is much smaller. The magnitudes of the three components of pressure diffusion $\bar {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ are comparable. The contribution of low-speed LSSs to the streamwise component of pressure diffusion $\bar {T}_{p,1,{n}}/\bar {\epsilon }_{c}$ also exhibits a wave shape, with its peak at $y/\delta _\omega = - 0.5$ and its trough at $y/\delta _\omega = 0.2$. Thus, it transports streamwise turbulent kinetic energy $K_1$ from the upper to the lower part of the mixing layer, in contrast to the role of $\bar {T}_{t,1,{n}}/\bar {\epsilon }_{c}$. The vertical pressure diffusion term $\bar {T}_{p,2,{n}}/\bar {\epsilon }_{c}$ exhibits a positive peak located at $y/\delta _\omega \approx 0$, flanked by negative peaks on either side, with the upper negative peak stronger than the lower one, indicating that $\bar {T}_{p,2,{n}}/\bar {\epsilon }_{c}$ transports vertical turbulent kinetic energy $K_2$ from the edges of the mixing layer towards its centre, in contrast to the role of $\bar {T}_{t,2,{n}}/\bar {\epsilon }_{c}$. The distribution of $\bar {T}_{p,3,{n}}/\bar {\epsilon }_{c}$ along the $y$-direction is relatively complex, with its sign changing multiple times. In the range of $-0.7< y/\delta _\omega < 0.5$, $\bar {T}_{p,3,{n}}/\bar {\epsilon }_{c}$ primarily transports $K_3$ from the upper to lower part of the mixing layer.

Figure 15.

($a$–$c$) Contributions of the low-speed LSSs to pressure diffusion $\bar {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ and ($d$–$f$) conditionally averaged pressure diffusion $\hat {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

The conditionally averaged pressure diffusion term $\hat {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ is presented in figures 15($d$)–15($f$), where the local pressure diffusion process is highlighted at the edges of the mixing layer. Here, $\hat {T}_{p,1,{n}}/\bar {\epsilon }_{c}$ peaks at $y/\delta _\omega = -0.8$. Additionally, $\hat {T}_{p,2,{n}}/\bar {\epsilon }_{c}$ is nearly symmetric about $y/\delta _\omega = 0$, exhibiting two remarkable negative peaks at $y/\delta _\omega = \pm 0.8$, with their magnitudes nearly equivalent to the maximum peak of $\hat {T}_{t,2,{n}}/\bar {\epsilon }_{c}$ (shown in figure 13$e$). At the edges of the mixing layer, $\hat {T}_{p,3,{n}}/\bar {\epsilon }_{c}$ exhibits two positive peaks, which are slightly larger than the two peaks of $\hat {T}_{t,3,{n}}/\bar {\epsilon }_{c}$ (shown in figure 13$f$) at the edges. As the convective Mach number increases, the conditionally averaged pressure diffusion term $\hat {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ decreases rapidly near the edges of the mixing layer.

Figure 16 presents the two-dimensional conditional average of the pressure diffusion $\breve {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ at $M_c=0.2$ and $1.8$. On the left side of the centre of low-speed LSSs, there exists a large-scale region where the streamwise component of pressure diffusion is negative, while on the right side, there is a large-scale region with positive streamwise pressure diffusion. The black dashed line connecting the extrema (maximum and minimum points) of the two large-scale regions is almost horizontal, as shown in figures 16($a$) and 16($d$). This suggests that within low-speed LSSs, the streamwise pressure diffusion component $\breve {T}_{p,1,{n}}/\bar {\epsilon }_{c}$ primarily transports $K_1$ from left to right, and its non-uniformity nearly leads to the transport of $K_1$ from top to bottom, as shown in figure 15($a$). The distribution of streamwise pressure diffusion components in the $x\unicode{x2013}y$ plane leads to a more pronounced effect of positive and negative values cancelling each other out when averaged in the horizontal plane. The two-dimensional distributions of the vertical and spanwise components of pressure diffusion are relatively complex, but the overall trend is to transport the vertical and spanwise turbulent kinetic energy downwards. At higher convective Mach number $M_c=1.8$, the pressure diffusion occurs within a smaller region.

Figure 16.

Two-dimensional conditionally averaged components of pressure diffusion $\breve {T}_{p,i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at ($a$–$c$) $M_c=0.2$ and ($d$–$f$) $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

The three components of viscous diffusion $\bar {T}_{v,i,{n}}$ are each an order of magnitude smaller than turbulent diffusion and pressure diffusion across all three convective Mach numbers; therefore, they are not further analysed here for brevity.

4.7. Mass flux coupling

According to the definition of the mass flux coupling term (3.3f), its ensemble average is very small because the Favre average of $u^{\prime \prime }$ is zero (Pantano & Sarkar Reference Pantano and Sarkar2002). The conditional averages of $u^{\prime \prime }$ and $v^{\prime \prime }$ are of the order of $\Delta U$, as shown in figures 5($a$) and 5($b$), while the conditional average of $w^{\prime \prime }$ remains negligible. Consequently, the conditional average of the mass flux coupling term within LSSs is not zero. Figures 17(a) and (17b) show the contribution of low-speed LSSs to the mass flux coupling term $\bar {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The contribution of low-speed LSSs to the streamwise mass flux coupling term $\bar {\varSigma }_{1,{n}}/\bar {\epsilon }_{c}$ exhibits a positive peak at $y/\delta _\omega =-0.25$ and it displays a negative trough at $y/\delta _\omega =0.25$, with the magnitudes of its peak and trough being almost equal. Considering the symmetry of high- and low-speed LSSs, $\bar {\varSigma }_{1,{p}}/\bar {\epsilon }_{c}$ and $\bar {\varSigma }_{1,{n}}/\bar {\epsilon }_{c}$ are symmetrical about the origin point, which means that they nearly cancel each other out, resulting in $\bar {\varSigma }_{1}$ being nearly zero. The characteristics of $\bar {\varSigma }_{2,{n}}/\bar {\epsilon }_{c}$ are very similar to those of $\bar {\varSigma }_{1,{n}}/\bar {\epsilon }_{c}$, except the magnitudes of the peaks and troughs are somewhat smaller. These results indicate that $\bar {\varSigma }_{1,{n}}/\bar {\epsilon }_{c}$ and $\bar {\varSigma }_{2,{n}}/\bar {\epsilon }_{c}$ transport streamwise and vertical turbulent kinetic energy downwards, respectively.

Figure 17.

(a,b) Contributions of the low-speed LSSs to mass flux coupling $\bar {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ and (c,d) conditionally averaged mass flux coupling $\hat {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$. (a,c) Streamwise and (b,d) vertical components.

The conditionally averaged mass flux coupling term $\hat {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ is presented in figures 17($c$) and 17($d$). The trough values of $\hat {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ are slightly larger than the peak values. At the edges of the mixing layer, the local values of the mass flux coupling term do not exhibit significant amplification, unlike the other budget terms. Here, $\hat {\varSigma }_{2,{n}}/\bar {\epsilon }_{c}$ shows a minor trough near the lower edge, indicating a small amount of vertical turbulent kinetic energy is transported upwards into the turbulent region. The spanwise mass flux coupling term is negligible and is omitted here for the sake of brevity. Figure 18 shows two-dimensional conditionally averaged mass flux coupling $\breve {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ based on the condition of $u^{\prime \prime }<-0.15\Delta U$ at convective Mach numbers $M_c= 0.2$ and $1.8$. Both the streamwise and vertical components of mass flux coupling exhibit a relatively strong negative region in the upper part of the LSSs and a weaker positive region in the lower part, qualitatively consistent with figures 17($c$) and 17($d$).

Figure 18.

Two-dimensional conditionally averaged mass flux coupling $\breve {\varSigma }_{i,{n}}/\bar {\epsilon }_{c}$ (contours) and streamwise velocity $\breve {u}_{n}/\Delta U$ (solid lines) based on low-speed events with $u^{\prime \prime }<-0.15\Delta U$ at (a,b) $M_c=0.2$ and (c,d) $1.8$. (a,c) Streamwise and (b,d) vertical components.

4.8. Total variation of turbulent kinetic energy in LSSs

Figures 19($a$)–19($c$) show the contribution of low-speed LSSs to the total variation of turbulent kinetic energy, $\bar {V}_{i,{n}}/\bar {\epsilon }_{c}$, at convective Mach numbers $M_c= 0.2$, $0.8$, $1.8$. The contribution of low-speed LSSs to the total variation of streamwise turbulent kinetic energy $\bar {V}_{1,{n}}/\bar {\epsilon }_{c}$ exhibits two distinct positive peaks: a larger one in the upper half and a smaller one in the lower half of the mixing layer. Additionally, $\bar {V}_{2,{n}}/\bar {\epsilon }_{c}$ exhibits a distinct double-peak distribution, while $\bar {V}_{3,{n}}/\bar {\epsilon }_{c}$ shows a positive peak only in the upper half of the mixing layer and is nearly zero in the lower half. Near the centre of the mixing layer, $\bar {V}_{2,{n}}/\bar {\epsilon }_{c}$ and $\bar {V}_{3,{n}}/\bar {\epsilon }_{c}$ are almost zero, whereas $\bar {V}_{1,{n}}/\bar {\epsilon }_{c}$ shows a small negative value, indicating a reduction in streamwise turbulent kinetic energy $K_1$ at this position. This suggests that even at the centre, the mixing layer has not achieved a completely self-similar state. As the convective Mach number increases, $\bar {V}_{i,{n}}/\bar {\epsilon }_{c}$ decreases and the peak of $\bar {V}_{2,{n}}/\bar {\epsilon }_{c}$ in the lower half almost disappears at $M_c= 1.8$.

Figure 19.

($a$–$c$) Contributions of the low-speed LSSs to the total variation of turbulent kinetic energy $\bar {V}_{i,{n}}/\bar {\epsilon }_{c}$ and ($d$–$f$) conditionally averaged total variation $\hat {V}_{i,{n}}/\bar {\epsilon }_{c}$ based on low-speed events with $u^{\prime \prime }<0$ at $M_c=0.2$, $0.8$ and $1.8$. (a,d) Streamwise, (b,e) vertical and (c,f) spanwise components.

The conditionally averaged total variation $\hat {V}_{i,{n}}/\bar {\epsilon }_{c}$ is presented in figures 15($d$)–15($f$). Near the upper edge of the mixing layer, the total variation of $K_1$ is very prominent, with a peak at $y/\delta _\omega \approx 0.7$. In contrast, near the lower edge, the total variation of $K_1$ is nearly zero and surprisingly negative at $M_c = 0.2$. The total variations of $K_2$ and $K_3$ both exhibit significant positive peaks near both edges of the mixing layer, with the peak at the upper edge being larger. The total variation of $K_2$ near the upper edge, at $y/\delta _\omega \approx 0.7$, is particularly notable, even exceeding the value of $K_1$. As the convective Mach number increases, the total variations of turbulent kinetic energy in all three directions decrease significantly at the edges. These results further confirm the amplitude modulation effect of LSSs on small-scale structures at the edges of the mixing layer, consistent with the findings shown in figures 3($d$)–3($f$). The level of amplitude modulation is strongest for the vertical velocity, followed by the streamwise velocity and weakest for the spanwise velocity. Additionally, the amplitude modulation decreases significantly with increasing Mach number. The entrainment process is usually associated with two mechanisms: nibbling, which is considered to be carried out by small-scale structures and plays the main role, and engulfment, which is mostly due to large-scale eddies (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2005; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Jahanbakhshi & Madnia Reference Jahanbakhshi and Madnia2016). We infer that with increasing convective Mach number, the level of amplitude modulation decreases significantly, resulting in weakened small-scale structures near the edge of the mixing layer. This leads to an evident increase in the normalized Kolmogorov length scale $\eta /\delta _\omega$, indicating a reduction in scale separation, as reported by Wang et al. (Reference Wang, Guo, Wang and Chen2024). Krug et al. (Reference Krug, Holzner, Lüthi, Wolf, Kinzelbach and Tsinober2015) proposed a relationship estimating the surface area $A$ of the TNTI, where $A \propto (\eta /\delta _\omega )^{-1/3}$. Based on this relationship, we can infer that the TNTI surface area decreases as the small-scale structures weaken. The reduction in TNTI surface area, in turn, decreases the mass flow rate associated with nibbling, as reported by Jahanbakhshi & Madnia (Reference Jahanbakhshi and Madnia2016).