4.1. The development of the turbulent front and perturbations

The fundamental properties of the shear-less turbulent front are examined through the lateral profiles of velocity statistics, with results presented for $y \geqslant 0$ due to statistical symmetry at $y = 0$ . Figure 4( $a$ ) showcases the lateral profiles of turbulent kinetic energy $k_T = \langle {u'}^2 + {v'}^2 + {w'}^2 \rangle / 2$ in R2200. In the central region, $k_T$ exhibits a uniform distribution and decays over time. For $y/L_0 \gt 1$ , $k_T$ increases with time owing to the spatial development of the turbulent front, which reduces the uniform $k_T$ region.

Figure 4( $b$ ) illustrates the skewness and flatness of the lateral velocity component, defined as $S_v = \langle {v'}^3 \rangle / v_{rms}^3$ and $F_v = \langle {v'}^4 \rangle / v_{rms}^4$ , respectively. For reference, the skewness and flatness values of a Gaussian probability distribution are $ S_v = 0$ and $ F_v = 3$ , respectively. In the central region, the values are approximately $ S_v \approx 0$ and $ F_v \approx 3$ . When two turbulent flows with different scales interact, such as in a shear-less turbulent mixing layer formed between different grid turbulences generated by two grids, the intermittency of large-scale motions can lead to increases in $S_v$ and $F_v$ (Veeravalli & Warhaft Reference Veeravalli and Warhaft1989; Kang & Meneveau Reference Kang and Meneveau2008; Matsushima et al. Reference Matsushima, Nagata and Watanabe2021; Nakamura et al. Reference Nakamura, Matsushima, Zheng, Nagata and Watanabe2022). These studies have shown that large-scale motions from a high turbulent kinetic energy region intermittently penetrate into a low turbulent kinetic energy region. The current results with one side remaining irrotational also highlight increases in $S_v$ and $F_v$ with peaks occurring in the range $1.3 \lt y/L_0 \lt 1.4$ . The development of the flow causes an outward shift in the locations of these peaks. The present DNS, initiated by inserting HIT into quiescent fluid, effectively reproduces the large-scale intermittency characteristic of this flow. Here, $ S_v \gt 0$ is associated with large-scale turbulent motions penetrating into the non-turbulent region, where velocity fluctuations are primarily governed by irrotational motion induced by turbulence (da Silva & Pereira Reference da Silva and Pereira2008). Even in the non-turbulent region at $ y/L_0 \approx 2$ , $ S_v$ maintains a small but non-zero value, indicating the influence of intermittent turbulent motion that causes $ S_v \gt 0$ .

Figure 5.

Temporal variations of turbulent kinetic energy in the perturbation region, $ k_T(\tau )$ , normalised by its initial value, $ k_T(\tau =0)$ . Results from the R350 series with external perturbations are compared for different wavelengths $ \lambda _f/\eta _P$ .

The perturbations introduced at a given time instance subsequently decay over time. Figure 5 illustrates the decay of perturbations by plotting the turbulent kinetic energy of external perturbations as a function of time. The R350 series is compared for different wavelength values. Here, the turbulent kinetic energy $ k_T$ is evaluated by calculating the spatial average of $ (u^2 + v^2 + w^2)/2$ over $ |y/L_0| \gt 2.0$ , which is always outside the turbulent front. Small-scale perturbations tend to decay rapidly with time. For $ \lambda _f/\eta _P = 8$ –16, $ k_T$ reaches a minimum value and then begins to increase, driven by velocity fluctuations from large-scale motions induced by the turbulent front. Thus, the direct effects of small-scale perturbations are significant only at early times. As the wavelength increases, the decay of perturbation energy slows, allowing the external perturbations to continuously influence flow evolution.

4.2. External perturbation effects on the outer boundary of scalar mixing layer

The effects of external perturbations on the outer boundary of the scalar mixing layer, delineated by the scalar isosurface, are examined herein. Previous DNS studies indicate that small-scale shear layers predominantly manifest within the TNTI layer, forming at the outer boundary of the turbulent front (Hayashi et al. Reference Hayashi, Watanabe and Nagata2021b ). The external perturbations, particularly those with the unstable wavelength of small-scale shear instability, are anticipated to significantly influence the evolution of shear layers near this boundary. Figure 6 visualises the evolution of shear layers near the boundary, viewed from the direction of $y \gt 0$ . The triple decomposition of the velocity gradient tensor is employed to quantify the intensities of shear and rigid-body rotation, denoted by $I_S$ and $I_R$ respectively (Hayashi et al. Reference Hayashi, Watanabe and Nagata2021a ). The shear layers and vortex tubes are identified using $I_S$ and $I_R$ , respectively. While the perturbations contribute to vorticity outside the scalar mixing layer, they are too weak to be recognised as shear layers or vortex tubes. When no perturbations are added in R350, as shown in figure 6( $a$ ), the shear-layer labelled as “S” evolves over time without producing any vortex tubes. In contrast, the introduction of a perturbation with the unstable wavelength in R350L30e leads to the formation of a vortex tube, labelled as “V” in figure 6( $b$ ). This promotion of vortex formation is absent in R350L210e, which uses a larger perturbation wavelength, as visualised in figure 6( $c$ ). Similarly, no vortex formation is promoted in R350L8e (not shown here), which employs a much smaller wavelength than the unstable one. External perturbations introduced outside the turbulent front prove effective in exciting small-scale shear instability near the boundary only when the wavelength approximates 30 times the Kolmogorov scale, which aligns with the unstable mode of the shear layers. This finding is consistent with numerical experiments involving internal perturbations (Watanabe & Nagata Reference Watanabe and Nagata2023; Watanabe Reference Watanabe2024 Reference Watanabe a ), where artificial velocity perturbations of approximately 30 times the Kolmogorov scale, when superimposed inside turbulence, cause many small-scale shear layers to become unstable and produce vortex tubes.

Figure 6.

Temporal evolutions of small-scale shear layers and vortex tubes at different time instances ( $\tau /\tau _{\eta P}=1.3$ , 3.7, 5.5 and 7.3) progressing from left to right in each panel: ( $a$ ) R350, ( $b$ ) R350L30e and ( $c$ ) R350L210e. The shear layers (grey) and vortex tubes (orange) are visualised respectively by the isosurfaces of shear intensity $I_S=1.5\langle I_S\rangle$ and rigid-body-rotation intensity $I_R=3\langle I_R\rangle$ , with averages used as reference values taken at the flow centre. The visualised domain is a cube of size $L_0^3$ , with the walls coloured by $I_S$ .

Figure 7.

( $a$ ) Growth rate $\dot {V}_S$ of the scalar mixing layer contributed from the low-scalar side in the series of R350. ( $b$ ) Increase ratio in $\dot {V}_S$ due to external perturbations, denoted as $\Delta \dot {V}_S$ .

Figure 7( $a$ ) displays the temporal variation of the surface-integrated propagation velocity of the scalar isosurface, $ \dot {V}_S = [ v_P ]_S$ , which represents the growth rate of the scalar mixing layer contributed from the low-scalar-value sides. Here, the integral is taken for the scalar isosurfaces on both flow sides. The effects of perturbations are compared across the R350 series. The growth rate is significantly enhanced by perturbations with the unstable wavelength of shear layers in R350L30e compared with the baseline case R350. A subtle increase is observed for R350L210e with a larger wavelength, while the growth rate remains largely unaffected in the case of a smaller wavelength. All cases adopt the same perturbation amplitude, illustrating that perturbations efficiently increase the growth rate when the wavelength matches the unstable mode of small-scale shear layers. Figure 7( $b$ ) shows the relative change in $ \dot {V}_S$ , denoted as $ \Delta \dot {V}_S$ . Throughout the paper, $ \Delta f$ is defined as $ (f_1 - f_2) / f_2$ , where $ f_1$ and $ f_2$ are statistical quantities in simulations with and without perturbations, respectively. The perturbation effect becomes increasingly pronounced in $\dot {V}_S$ with time, with increments due to perturbations exceeding 10 % from the baseline cases in R350L30e and R2200L30e. The increases in $ \dot {V}_S$ in R350L30e and R2200L30e are almost identical despite differing Reynolds numbers. These two cases with the unstable wavelength of shear layers adopt the same normalised perturbation amplitude $ u_f/u_{\eta P} = 1.4$ , and the perturbation in R2200L30e with smaller $u_{\eta P}$ has less kinetic energy than that in R350L30e. Therefore, even a weaker perturbation can lead to the same enhancement effect on the growth rate of the scalar mixing layer at a higher Reynolds number. The peak in $ \Delta \dot {V}_S$ is attained at $ \tau /\tau _{\eta P} \approx 11$ in both R350L30e and R2200L30e, suggesting that the time scale for enhancing the scalar mixing layer growth is related to the Kolmogorov time scale. This temporal evolution aligns with the visualisation of the unstabilised shear layer in figure 6( $b$ ), where vortex formation due to the perturbation occurs over a span of about $ 10\tau _{\eta P}$ . These scalings based on $ u_{\eta P}$ and $ \tau _{\eta P}$ relate to the scaling of small-scale shear layers, whose characteristic velocity and length scales are $ u_{\eta }$ and $ \eta$ , respectively (Watanabe et al. Reference Watanabe, Tanaka and Nagata2020; Fiscaletti et al. Reference Fiscaletti, Buxton and Attili2021).

Figure 8.

( $a$ ) Relative change in the scalar isosurface area $A$ due to external perturbations, denoted as $\Delta A$ . ( $b$ ) Production and destruction terms of the normalised surface area $A'$ and their balance $\dot {A}'$ , described by (3.4), in R350 and R350L30e.

The growth rate $\dot {V}_S$ is expressed as $\dot {V}_S = A V_P$ , where $A$ is the isosurface area and $V_P$ is the mean propagation velocity. Figure 8( $a$ ) illustrates the relative change in $A$ due to perturbations, denoted as $\Delta A$ . The temporal variation of $\Delta A$ closely mirrors that of $\Delta \dot {V}_S$ , with the surface area also increasing by more than 10 %, comparable to the peak in $\Delta \dot {V}_S$ , when the perturbations feature the unstable wavelength of small-scale shear layers. Thus, the increase in $\dot {V}_S$ is predominantly driven by the perturbation effect on $A$ , although $V_P$ also experiences a slight increase due to the perturbation (the increase is less than 1.5 % and is not shown in the figure). Consequently, the rapid growth of the scalar mixing layer in cases R350L30e and R2200L30e is attributed to the enlarged area of the outer boundary of the scalar mixing layer, facilitating efficient scalar transfer from the inside to the outside, thereby expanding the scalar mixing layer.

The increases in the growth rate, $\Delta \dot {V}_S$ , and the surface area, $ \Delta A$ , are similar for R350L30e and R2200L30e, both of which use the same normalised perturbation parameters, $ \lambda _f/\eta _P = 30$ and $ u_f/u_{\eta P} = 1.4$ . Watanabe & Nagata (Reference Watanabe and Nagata2023) investigated the effects of perturbations on decaying isotropic turbulence using the same approach as the present study. Perturbations with $ \lambda _f/\eta _P = 30$ resulted in an increase in the number of vortices generated by the instability of small-scale shear layers. The increase ratio remained consistent across different Reynolds number cases when $ u_f/u_{\eta P}$ was held constant. This behaviour aligns with the comparable increase in $\dot {V}_S$ and $ A$ observed for R350L30e and R2200L30e.

The increases in the growth rate, $\Delta \dot {V}_S$ , and the surface area, $ \Delta A$ , are similar for R350L30e and R2200L30e, both of which use the same normalised perturbation parameters, $ \lambda _f/\eta _P = 30$ and $ u_f/u_{\eta P} = 1.4$ . Watanabe & Nagata (Reference Watanabe and Nagata2023) investigated the effects of perturbations on decaying isotropic turbulence using the same approach as the present study. Perturbations with $ \lambda _f/\eta _P = 30$ resulted in an increase in the number of vortices generated by the instability of small-scale shear layers. The increase ratio remained consistent across different Reynolds number cases when $ u_f/u_{\eta P}$ was held constant. This behaviour aligns with the comparable increase in $\dot {V}_S$ and $ A$ observed for R350L30e and R2200L30e.

A similar influence of the promoted instability was reported for turbulent entrainment, whose rate increases when the small-scale shear layers become unstable by internal perturbations (Watanabe Reference Watanabe2024a ). The entrainment rate is defined as the surface integral of the propagation velocity of the enstrophy isosurface (Holzner & Luthi Reference Holzner and Lüthi2011). Enhanced entrainment is attributed to an enlarged isosurface area and an increased mean propagation velocity of the enstrophy isosurface, both contributing comparably to the increase in the entrainment rate. In contrast, for the growth rate of the scalar mixing layer, the increase in surface area predominates. Therefore, the enhancements in the entrainment rate and the growth rate of the scalar mixing layer are driven by different mechanisms. The small-scale shear instability promoted by internal perturbations amplifies enstrophy production due to vortex stretching (Watanabe & Nagata Reference Watanabe and Nagata2023), which potentially enhances enstrophy diffusion from turbulent to non-turbulent regions, thereby increasing the mean propagation velocity of the enstrophy isosurface.

The mechanism by which the perturbation increases the surface area can be examined with (3.4). Figure 8( $b$ ) presents a comparison between R350 and R350L30e in the temporal variations of the time derivative of the normalised surface area $\dot {A}' = dA'/dt$ and the two terms $P_A$ and $D_A$ in (3.4). The results for R350 indicate that $P_A \gt 0$ and $D_A \lt 0$ contribute to the production and destruction of surface area, respectively. The roles of $P_A$ and $D_A$ align with those observed for the scalar isosurface in a turbulent mixing layer (Blakeley et al. Reference Blakeley, Olson and Riley2023). Even without perturbations, $P_A$ exceeds $D_A$ , resulting in an increase in surface area over time. The introduction of perturbations that lead to small-scale shear instability affects both terms significantly. Initially, $P_A$ increases compared with the baseline case, followed by a rise in the magnitude of $D_A$ . The faster increase in $P_A$ leads to an increase in the surface area with large $\dot {A}'$ in R350L30e. It is suggested that the enhanced shear instability excites small-scale turbulent motions (Watanabe & Nagata Reference Watanabe and Nagata2023), which in turn amplify the velocity gradients contributing to $P_A$ . Consequently, the increase in the surface area is explained by the surface deformation at small scales. This deformation results in a more complex geometry of the surface, affecting the destruction term $D_A$ , defined with the divergence of the surface-normal vector $\partial n_i/ \partial x_i$ , at a later time.

Figure 9.

( $a$ ) Plots of the number of boxes, $N_\Delta$ , required to cover the scalar isosurface at $\tau /\tau _{\eta P}=13$ for R350, R350L30e, R2200 and R2200L30e, with box size $\Delta$ . ( $b$ ) Temporal variations of the fractal dimension $D_f$ , evaluated by fitting the power law $N_\Delta \sim \Delta ^{-D_f}$ .

The scalar isosurface in turbulence exhibits fractal properties, crucial for understanding the scaling of its surface area. This study evaluates the fractal dimension $D_f$ of the scalar isosurface using a three-dimensional box-counting method, which counts the number of cubic boxes, $N_\Delta$ , required to cover the surface as a function of the box size $\Delta$ . The calculation of $N_\Delta$ is performed separately for the scalar isosurfaces on the $y\gt 0$ and $y\lt 0$ sides. The fractal dimension is determined as the power-law exponent in the plot of $N_\Delta (\Delta )$ . Figure 9( $a$ ) compares $N_\Delta (\Delta )$ between the baseline cases and the perturbed cases with the unstable wavelength of small-scale shear layers. The lines represent the power law $N_\Delta \sim \Delta ^{-D_f}$ , evaluated using a least squares method. For both Reynolds numbers, deviations in the perturbed cases from the baselines become more prominent as the scale decreases. Specifically, $N_\Delta$ increases at small scales when the perturbations are introduced, with an increase of approximately 10 % for $\Delta /\eta \lt 15$ from the baseline. These results indicate that the promoted small-scale shear instability leads to a more complex geometry of the scalar isosurface at smaller scales. This complexity is consistent with the enhanced effects of surface deformation by velocity gradients observed in figure 8( $b$ ).

The fractal dimension $ D_f$ is determined by fitting a power-law function to the plot of $ N_\Delta (\Delta )$ . The fractal dimension can slightly vary depending on the fitting range. However, the analysis applies the fitting to all data points shown in figure 9( $a$ ). This approach is chosen to focus primarily on examining the perturbation effects on $ D_f$ rather than pinpointing the precise values of $ D_f$ . Figure 9( $b$ ) illustrates the temporal variations of $ D_f$ . In the absence of perturbations, $ D_f \approx 2.06$ and 2.12 are observed for R350 and R2200, respectively. The introduction of perturbations with the unstable wavelength of small-scale shear layers in R350L30e and R2200L30e results in an increase in $ D_f$ . Such increases are not observed for perturbations with other wavelengths in R350L8e and R350L210e. At $ \tau /\tau _{\eta P} = 15$ , $ D_f$ reaches approximately 2.11 in R350L30e and 2.16 in R2200L30e, which represent an increase of approximately 2 % over the baseline cases. While this relative increase in $ D_f$ may not seem as substantial as those observed for other quantities, such as $ \Delta \dot {V}_S$ , it is critically important for the surface area $ A$ . This is because $ D_f$ appears as an exponent in the scaling of $ A$ , expressed as $ A \sim \mathcal {L}^{D_f-2}$ , where $ \mathcal {L}$ represents the ratio between the length scale of large-scale motions and the Kolmogorov scale (Meneveau & Sreenivasan Reference Meneveau and Sreenivasan1990). Therefore, even a slight increase in $ D_f$ can have a significant impact on the surface area, enhancing the scalar transfer and mixing near the boundary.

The fractal dimension $ D_f$ is determined by fitting a power-law function to the plot of $ N_\Delta (\Delta )$ . The fractal dimension can slightly vary depending on the fitting range. However, the analysis applies the fitting to all data points shown in figure 9( $a$ ). This approach is chosen to focus primarily on examining the perturbation effects on $ D_f$ rather than pinpointing the precise values of $ D_f$ . Figure 9( $b$ ) illustrates the temporal variations of $ D_f$ . In the absence of perturbations, $ D_f \approx 2.06$ and 2.12 are observed for R350 and R2200, respectively. The introduction of perturbations with the unstable wavelength of small-scale shear layers in R350L30e and R2200L30e results in an increase in $ D_f$ . Such increases are not observed for perturbations with other wavelengths in R350L8e and R350L210e. At $ \tau /\tau _{\eta P} = 15$ , $ D_f$ reaches approximately 2.11 in R350L30e and 2.16 in R2200L30e, which represent an increase of approximately 2 % over the baseline cases. While this relative increase in $ D_f$ may not seem as substantial as those observed for other quantities, such as $ \Delta \dot {V}_S$ , it is critically important for the surface area $ A$ . This is because $ D_f$ appears as an exponent in the scaling of $ A$ , expressed as $ A \sim \mathcal {L}^{D_f-2}$ , where $ \mathcal {L}$ represents the ratio between the length scale of large-scale motions and the Kolmogorov scale (Meneveau & Sreenivasan Reference Meneveau and Sreenivasan1990). Therefore, even a slight increase in $ D_f$ can have a significant impact on the surface area, enhancing the scalar transfer and mixing near the boundary.

The scalar isosurfaces in the baseline cases essentially delineate the TNTI layer, a subject of considerable interest in prior studies. It has been established that the enstrophy isosurface near the outer boundary of the TNTI layer closely aligns with the scalar isosurface at $Sc=1$ , in terms of both geometry and location, particularly when the scalar threshold is appropriately selected (Gampert et al. Reference Gampert, Boschung, Hennig, Gauding and Peters2014; Watanabe et al. Reference Watanabe, Zhang and Nagata2018). Therefore, the values of $D_f$ observed for R350 and R2200 should be comparable to those reported for the TNTI layer. The $ D_f$ value associated with Kolmogorov scaling, 2.33, has been observed in fully developed intermittent turbulent flows (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016, Reference Mistry, Dawson and Kerstein2018; Zhuang et al. Reference Zhuang, Tan, Huang, Liu and Zhang2018; Breda & Buxton Reference Breda and Buxton2019; Abreu et al. Reference Abreu, Pinho and da Silva2022; Xu & Long Reference Xu, Long and Wang2023). Other studies have reported smaller $ D_f$ values, around 2.1–2.2 (Zhang et al. Reference Zhang, Watanabe and Nagata2023; Er et al. Reference Er, Laval and Vassilicos2023), which have also been observed in the developing regions of jets (Breda & Buxton Reference Breda and Buxton2019; Xu & Long Reference Xu, Long and Wang2023). The present DNS results yield $ D_f = 2.05{-}2.12$ , which is smaller than 2.33. These $ D_f$ values are discussed in relation to the variations noted in previous studies. As shown in figure 9( $b$ ), $ D_f$ remains nearly constant over time when no perturbations are introduced, indicating that the transient behaviour associated with turbulence development, observed in the near field of jets, is irrelevant in this context. The TNTI layer comprises two sublayers (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Taveira & da Silva Reference Taveira and da Silva2014; van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014). The outer sublayer, known as the viscous superlayer, is characterised by a vorticity dynamics dominated by viscous effects. The inner sublayer, referred to as the turbulent sublayer, is where inviscid vortex stretching occurs actively. The location of the enstrophy isosurface within the layer varies depending on the threshold selected. Consequently, the fractal dimension $D_f$ of the enstrophy isosurface also increases as the threshold becomes larger. In a turbulent jet, for example, $D_f$ is reported as small as 2.1 when a low enstrophy threshold is used such that the surface marks the irrotational boundary between the TNTI layer and the irrotational fluid (Er et al. Reference Er, Laval and Vassilicos2023). This value is consistent with the present values of $D_f$ . Furthermore, the present study notes that $D_f$ increases with the Reynolds number, a trend similarly reported for the enstrophy isosurface within the TNTI layer of turbulent boundary layers (Zhang et al. Reference Zhang, Watanabe and Nagata2023). In addition, the scalar isosurface in isotropic turbulence often shows a decreasing trend in fractal dimension as the isoscalar value deviates from the mean scalar. When the isoscalar value significantly differs from the mean scalar value, the fractal dimension is approximately 2 (Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020). In the current study, the scalar isosurface chosen to represent the outer boundary of the scalar mixing layer also substantially deviates from the mean scalar value within the layer. The present values of $D_f$ close to 2 align with these observations of large scalar-fluctuation values in isotropic turbulence.

The increase in surface area and the associated complex geometry resulting from small-scale shear instability align well with experimental findings concerning the scalar interface of jets and wakes in background turbulence (Kohan & Gaskin Reference Kohan and Gaskin2022; Chen & Buxton Reference Chen and Buxton2023). The background turbulence in these studies is fully developed and is characterised by a wide range of motion scales, which is different from the current single-scale perturbations. The effects of this background turbulence on the small-scale geometry could potentially be related to the shear instability observed in this study. However, drawing a definitive conclusion at this stage is challenging due to the multi-scale nature of the background turbulence. As discussed in § 2.3, background turbulence in most experiments typically has a characteristic length scale comparable to the integral scale of primary turbulence. Further experimental studies employing small-scale background turbulence are required to observe the effects of promoted small-scale shear instability on the scalar interface. The scalar interface of the boundary layer subjected to free-stream turbulence has also been investigated using DNS (Wu et al. Reference Wu, Wallace and Hickey2019). The geometry of the interface differs between transitional and fully developed boundary layers, which exhibit distinct vortical structures near the edge of the boundary layer. The influence of free-stream turbulence on the interface is expected to depend on the state of the primary turbulence.

Figure 10.

Lateral profiles of scalar statistics in R350, R350L8e, R350L30e and R350L210e: ( $a$ ) mean scalar $\langle \phi \rangle$ and ( $b$ ) mean scalar dissipation rate $\langle \varepsilon _\phi \rangle$ .

Figure 11.

Lateral profiles of scalar statistics in R2200 and R2200L30e: ( $a$ ) mean scalar $\langle \phi \rangle$ ; ( $b$ ) mean scalar dissipation rate $\langle \varepsilon _\phi \rangle$ ; ( $c$ ) r.m.s. scalar fluctuations $\phi _{rms}$ ; ( $d$ ) lateral turbulent scalar flux $\langle v'\phi ' \rangle$ .

4.3. External perturbation effects on passive-scalar statistics

Figure 10( $a$ ) compares the mean scalar distribution between the perturbed and baseline cases. Over time, the scalar mixing layers on both sides develop and eventually reach the flow centre, as confirmed by $\langle \phi \rangle \lt 1$ at $y=0$ . The scalar mixing layer expands in the $y$ direction, resulting in an increase in $\langle \phi \rangle$ over time at larger $y$ values. Notably, the mean scalar distribution remains largely unaffected by perturbations. Figure 10( $b$ ) provides a comparison for the mean scalar dissipation rate, $\langle \varepsilon _\phi \rangle = \langle D \nabla \phi \cdot \nabla \phi \rangle$ . In the case of R350L30e, where small-scale shear instability is excited, a slight increase in the dissipation peak is observed at $\tau /\tau _{\eta P} = 6$ and 15. The development of the mean scalar is dominated by turbulent scalar flux, largely contributed by large-scale turbulent motions. The negligible effects of perturbations on $\langle \phi \rangle$ suggest that the promoted shear instability does not significantly alter large-scale motions. However, the slight increase in $\langle \varepsilon _\phi \rangle$ suggests that the scalar distribution at small scales is affected by the instability.

Figure 11 presents the results for higher $Re_{L0}$ cases, R2200 and R2200L30e, including the r.m.s. scalar fluctuations, $\phi _{rms} = \sqrt {\langle {\phi '}^2 \rangle }$ , and the lateral turbulent scalar flux, $\langle v'\phi ' \rangle$ . The perturbation effects are consistent across both $Re_{L0}$ cases. The mean scalar profile remains largely unaffected by the perturbations, while a slight increase in the mean scalar dissipation rate is observed in R2200L30e compared with the baseline case of R2200. Additionally, the r.m.s. scalar fluctuations and turbulent scalar flux exhibit negligible differences between the perturbed and unperturbed cases, as shown in figures 11( $c$ ) and ( $d$ ). Figure 11( $a$ ) includes an enlarged plot of the mean scalar, $\langle \phi \rangle$ , near the location where $\langle \phi \rangle \approx 0.02$ . While the scalar isosurface propagation of $\phi = 0.02$ is enhanced by external perturbations in R350L30e, the mean scalar profile around $\langle \phi \rangle \approx 0.02$ remains largely unaffected by the perturbations.

External perturbations are expected to have more pronounced influences near the boundary than in the core region of the scalar mixing layer. This behaviour near the boundary cannot be captured by the conventional statistics defined by spatial averages, which tend to focus more on the bulk properties of the flow. This issue has been discussed in previous studies of the TNTI layer. Conditional averages of various flow variables, such as velocity and enstrophy, taken relative to the interface location, exhibit a pronounced gradient near the interface (Bisset et al. Reference Bisset, Hunt and Rogers2002). However, this gradient is not visible in conventional statistics, defined using time averages in statistically steady flows or spatial averages in homogeneous directions. The TNTI layer is thin, with a typical thickness of approximately 10–20 times the Kolmogorov scale, and occupies only a small fraction of the flow region (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014). Consequently, the TNTI layer contributes minimally to conventional statistics, which are predominantly influenced by flow regions far away from the TNTI layer. Therefore, the effects of external perturbations on the boundary region of the scalar mixing layer are expected to be more evident in interface-based statistics than in the conventional statistics examined in figures 10 and 11. For this reason, the present exploration is further extended by employing conditional statistics near the outer boundary of the scalar mixing layer. The analysis of perturbation effects using conditional statistics near the outer boundary focuses primarily on cases R2200 and R2200L30e, with the latter featuring perturbations with the unstable wavelength of small-scale shear layers. The dependence on perturbation wavelength and Reynolds number is later discussed with all cases of R350 and R2200.

Figure 12.

Conditional profiles of mean scalar $\langle \phi \rangle _I$ and mean scalar dissipation rate $\langle \varepsilon _\phi \rangle _I$ at $\tau =0$ in R2200.

Figure 12 displays conditional profiles of the mean scalar, $\langle \phi \rangle _I$ , and the mean scalar dissipation rate, $\langle \varepsilon _\phi \rangle _I$ , in R2200 without perturbations. The profiles exhibit typical behaviours observed in intermittent flows, similar to those studied for understanding mixing near the TNTI layer (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009; Gampert et al. Reference Gampert, Boschung, Hennig, Gauding and Peters2014; Watanabe et al. 2015b; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016; Zhang et al. Reference Zhang, Watanabe and Nagata2019; Kohan & Gaskin Reference Kohan and Gaskin2020). In these profiles, $\langle \phi \rangle _I$ sharply decreases to zero within a thin interfacial layer approximately $10\eta$ thick, while exibiting a slow increase in the core region of the scalar mixing layer, where $\zeta _I/\eta \lesssim -10$ . The scalar dissipation rate peaks within this interfacial layer due to the scalar gradient across the layer. To delve deeper into the temporal variations of these conditional statistics, spatial averages are also taken within defined subregions labelled as I1, I2 and C1–C3 in figure 12. The outer part of the interfacial layer, $-3.0 \leqslant \zeta _I / \eta \leqslant 0$ , is designated as I1, while the region containing the peak in the scalar dissipation rate, $-6.0 \leqslant \zeta _I / \eta \leqslant -3.5$ , is defined as I2. These ranges of $\zeta _I$ are chosen based on the location of the dissipation peak. The core regions of the scalar mixing layer are denoted by C1, C2 and C3, sequentially moving away from the interfacial layer from C1 to C3. The core region is statistically nearly homogeneous, as the shear-free turbulent front originates from HIT. This contrasts with I1 and I2, where the mean scalar rapidly decreases from the inside to the outside of the scalar mixing layer. Due to the small spatial variations of statistics with $\zeta _I$ in the core region, C1, C2 and C3 are separated by a constant spacing of $10\eta$ , ranging from $\zeta _I / \eta = -10$ to $-40$ . The region between I2 and C1 is excluded from this temporal analysis to ensure a clear separation between the two subregions. Averages within each subregion are denoted by $\overline {f}$ . The r.m.s. scalar fluctuation within each subregion is calculated as $\phi _{rms} = (\overline {\phi ^2} - \overline {\phi }^2)^{1/2}$ . The mean scalar dissipation rate is denoted by $\overline {\varepsilon _\phi }$ . Together, these two quantities define the mixing time scale $\tau _\phi = \phi _{rms}^2 / \overline {\varepsilon _\phi }$ , which provides insights into how quickly the scalar is mixed within these regions (Fox Reference Fox2003). These subregions are used exclusively to examine the temporal variations of perturbation effects, while the $\zeta _I$ dependence of statistics is also captured in the conditional averages presented as functions of $\zeta _I$ .

Figure 13.

External perturbation effects on conditional statistics at $\tau /\tau _{\eta P}=13$ in R2200 and R2200L30e: ( $a$ ) mean scalar and ( $b$ ) r.m.s. scalar fluctuation and mean scalar dissipation rate.

Figure 13( $a$ ) compares the conditional mean scalar profiles between R2200 and R2200L30e at $\tau /\tau _{\eta P}=13$ , an instance at which the development of the scalar mixing layer has already been influenced by promoted small-scale shear instability in R2200L30e. The perturbation in R2200L30e results in a smaller mean scalar compared with R2200. The faster growth of the scalar mixing layer in R2200L30e is linked to enhanced molecular diffusion, which is more effective in transferring scalar outward due to the increased surface area. Additionally, promoting small-scale instability also accelerates the growth of the turbulent front through entrainment (Watanabe Reference Watanabe2024 Reference Watanabe a ), which draws the outer fluid with smaller $\phi$ values into the turbulent front. These changes due to the perturbations explain the reduced conditional mean scalar observed in R2200L30e. This reduction in the conditional mean scalar is consistent with findings from studies of TTI layers in a turbulent wake developing in free-stream turbulence. The increasing effects of free-stream turbulence tend to decrease the conditional mean scalar near the interface (Kankanwadi & Buxton Reference Kankanwadi and Buxton2020).

Figure 13( $b$ ) examines the r.m.s. scalar fluctuations $\phi _{rms} = (\langle \phi ^2 \rangle _I - \langle \phi \rangle _I^2)^{1/2}$ and the mean scalar dissipation rate $\langle \varepsilon _\phi \rangle _I$ at the same time instance as figure 13( $a$ ). The perturbations lead to an increase in $\phi _{rms}$ , explained by a greater amount of entrained fluid with scalar values $\phi \approx 0$ that differ from those within the scalar mixing layer. Interestingly, the peak in the mean scalar dissipation rate decreases with the perturbations. The scalar gradient within the interfacial layer, dominated by the normal component at the scalar isosurface, is well captured by the mean scalar profile along $\zeta _I$ , taken in the surface-normal direction. Consequently, the lower mean scalar in the scalar mixing layer in the perturbed case correlates with a smaller dissipation peak within the interfacial layer. Conversely, the scalar dissipation in the core region, where $\zeta _I/\eta \lesssim -10$ , is increased by the perturbations, as expected from the increase in the r.m.s. scalar fluctuations. These conditional statistics reveal more significant perturbation effects on both the mean scalar and mean scalar dissipation rate than conventional statistics in figure 11. The external perturbations predominantly influence the scalar and flow fields near the boundary of the scalar mixing layer. Conventional statistics fail to capture these perturbation effects near the boundary, as they are obscured by contributions from flow regions irrelevant to the boundary region.

Figure 14.

( $a$ ) Conditional averages of kinetic energy $\langle k\rangle _I$ and its dissipation rate $\langle \varepsilon \rangle _I$ and ( $b$ ) Kolmogorov scale defined with the conditional statistics, $\eta _I$ , at $\tau /\tau _{\eta P}=13$ in R2200 and R2200L30e.

The impact of perturbations on the flow field is analysed through conditional averages of kinetic energy and its dissipation rate, defined as $\langle k \rangle _I = \langle (u^2 + v^2 + w^2)/2 \rangle _I$ and , respectively. Figure 14( $a$ ) presents a comparison of these quantities between R2200 and R2200L30e at $\tau /\tau _{\eta P}=13$ . In the core region of the scalar mixing layer ( $\zeta _I/\eta \lt -10$ ), both kinetic energy and its dissipation rate are reduced due to the perturbations. This reduction is consistent with the enhanced entrainment of outer fluids, which typically possess lower kinetic energy compared with the turbulent front (Taveira & da Silva Reference Taveira and da Silva2013). The perturbations also appear to have a minimal effect on $\langle k \rangle _I$ outside the scalar mixing layer, where large-scale motions induced by the turbulent front dominate (da Silva & Pereira Reference da Silva and Pereira2008). This observation aligns with the nature of the perturbations, which have a small amplitude and decay over time, limiting their ability to significantly alter the large-scale flow. Even without perturbations, the region outside the scalar mixing layer retains non-negligible kinetic energy and dissipation. In R2200, the boundary of the scalar mixing layer approximates the interface separating turbulent and non-turbulent regions (Gampert et al. Reference Gampert, Boschung, Hennig, Gauding and Peters2014; Watanabe et al. Reference Watanabe, Zhang and Nagata2018). The kinetic energy of this irrotational motion induced by turbulence decays with distance from the interface, as theoretically predicted (Phillips Reference Phillips1955; Teixeira & da Silva Reference Teixeira and da Silva2012; Zecchetto et al. Reference Zecchetto, Xavier, Teixeira and da Silva2024). This decay is reflected in the decreases of $\langle k \rangle _I$ and $\langle \varepsilon \rangle _I$ with increasing $\zeta _I$ , consistent with observations in other intermittent turbulent flows (Taveira et al. Reference Taveira and da Silva2013; Watanabe et al. 2016b; Buxton et al. Reference Buxton, Breda and Dhall2019).

Figure 14( $b$ ) presents the Kolmogorov scale $\eta _I=(\nu ^3/\langle \varepsilon \rangle _I)^{1/4}$ calculated as a function of $\zeta _I$ . In the core region, where $\zeta _I/\eta \lt -10$ , $\eta _I$ is hardly influenced by the perturbations. The changes in $\langle \varepsilon \rangle _I$ are not significant enough to substantially affect the local Kolmogorov scale within the scalar mixing layer. The Kolmogorov scale increases for $\zeta _I \gt 0$ , corresponding to the region outside the scalar mixing layer. This increase is directly associated with the decay in the kinetic energy dissipation rate discussed above. In studies of the TNTI layer in turbulent boundary layers, it has been shown that the area of the surface delineating the outer boundary of the TNTI layer scales as $\delta _{BL}^2(\delta _{BL}/\eta _I)^{D_f-2}$ , where $\delta _{BL}$ is the boundary-layer thickness and $\eta _I$ is taken from the turbulent core region near the TNTI layer (Zhang et al. Reference Zhang, Watanabe and Nagata2023). Perturbations have minimal impact on $\eta _I$ in the core region of the scalar mixing layer. Additionally, the lateral profile of mean scalar remains largely unaffected by the perturbations, indicating that the length scale of large-scale motions is also not influenced by perturbations. Thus, the changes in characteristic length scales due to perturbations contribute minimally to the observed increase in the scalar isosurface area. This increase is instead dominated by the rise in the fractal dimension, highlighting how geometric complexity, rather than changes in turbulent scales, primarily drives the expansion of the surface area in this case.

Figure 15.

Temporal variations of relative changes at different positions along the $\zeta _I$ coordinate in R2200L30e: ( $a$ ) r.m.s. scalar fluctuation, denoted as $\Delta \phi _{rms}$ , and ( $b$ ) mean scalar dissipation rate, denoted as $\Delta \overline {\varepsilon _\phi }$ .

Figure 15( $a$ ) illustrates the relative changes in r.m.s. scalar fluctuation, denoted as $\Delta \phi _{rms}$ , caused by perturbations in R2200L30e. The results, evaluated across five subregions depicted in figure 12, are plotted over time. Notably, the perturbations result in an increase in $\phi _{rms}$ across all positions near the outer boundary. The increase in $\Delta \phi _{rms}$ initially begins within the interfacial layer, specifically in I1, which represents the outer part of this layer. It is important to note that I1 has very small $\phi _{rms}$ , and therefore its increase, indicated by a large $\Delta \phi _{rms}$ , is not distinctly visible in figure 13( $b$ ). Within the core region of the scalar mixing layer, the increase in $\phi _{rms}$ is most pronounced in C1 and diminishes towards C3.

Figure 15( $b$ ) shows the relative changes in the mean scalar dissipation rate, $\Delta \overline {\varepsilon _\phi }$ , for each subregion. The temporal variations of $\Delta \overline {\varepsilon _\phi }$ align with those of $\Delta \phi _{rms}$ . The increase in $\overline {\varepsilon _\phi }$ also starts from I1, near the boundary of the scalar mixing layer. The mean scalar dissipation rate does not increase within I2, which is the peak location of the conditional mean scalar dissipation rate. Here, $\overline {\varepsilon _\phi }$ slightly decreases over time, as indicated by a negative $\Delta \overline {\varepsilon _\phi }$ . However, near the interfacial layer in C1, there is a significant increase in $\overline {\varepsilon _\phi }$ , with a relative increase of approximately 50 %. This effect progressively diminishes moving away from the boundary, from C1 to C3. These observations from figure 15 underscore that while perturbations enhance scalar fluctuations and dissipation rates near the boundary, their influence is not uniform across the scalar mixing layer. The scalar mixing processes near the outer boundary are particularly sensitive to these external perturbations.

Figure 16.

Temporal variations of relative changes in the mixing time scale at different positions along the $\zeta _I$ coordinate in R2200L30e. The symbols are the same as in figure 15.

Figure 16 presents the relative changes in the mixing time scale, denoted as $\Delta \tau _\phi$ . In subregions I1 and I2, which are within the interfacial layer, $\tau _\phi$ increases under the influence of the perturbations, indicating a slower mixing process due to the diffusive effects. Here, $\tau _\phi$ defines a local time scale, which does not take into account the spatial distribution of the scalar. Despite the locally slower mixing, the overall efficacy of outward scalar diffusion in the interfacial layer is enhanced due to the enlarged surface area, as evidenced by the increased growth rate of the scalar mixing layer. Conversely, in the core region of the scalar mixing layer, a locally faster mixing is observed, characterised by smaller $\tau _\phi$ or negative $\Delta \tau _\phi$ .

Figure 17.

Temporal variations of relative changes in the mean scalar dissipation rate $\Delta \overline {\varepsilon _\phi }$ in C1, across different perturbation wavelengths and Reynolds numbers.

Figure 17 compares the relative changes in the mean scalar dissipation rate within subregion C1 among 4 cases. Regardless of the Reynolds numbers, perturbations with the unstable wavelength of small-scale shear layers result in a significant increase in the scalar dissipation rate. In both R350L30e and R2200L30e, $\Delta \overline {\varepsilon _\phi }$ reaches 50 %, showing substantial enhancement even with a smaller perturbation energy in R2200L30e. The rate of increase in $\Delta \overline {\varepsilon _\phi }$ is similar for both cases when time is normalised by the Kolmogorov time scale, suggesting that a similar enhancement of the scalar dissipation rate is achievable at different Reynolds numbers. In contrast, perturbations at other wavelengths are less effective, as shown by the small variations in $\Delta \overline {\varepsilon _\phi }$ .

Figure 18.

The wavelength dependence of perturbation effects on ( $a$ ) the growth rate of the scalar mixing layer, $\Delta \dot {V}_S$ , and ( $b$ ) the mean scalar dissipation rate near the interface in C1, $\Delta \overline {\varepsilon _\phi }$ . Results are compared across the series of R350 with external perturbations for all tested wavelength values.

The above results confirm that the growth rate of the scalar mixing layer and small-scale scalar mixing near the boundary are enhanced by perturbations with a wavelength of approximately $30\eta _P$ . The wavelength dependence of these effects is thoroughly investigated by examining the series of R350 with external perturbations with varying wavelength parameters. Figure 18( $a$ ) plots the relative change in the growth rate, $\Delta \dot {V}_S$ , as a function of wavelength $\lambda _f$ , at three time instances, while figure 18( $b$ ) presents the corresponding results for the mean scalar dissipation rate near the boundary, $\Delta \overline {\varepsilon _\phi }$ , calculated in C1. Temporal variations for selected $\lambda _f$ cases have been shown in figures 7( $b$ ) and 17. The results indicate peaks in $\Delta \dot {V}_S$ and $\Delta \overline {\varepsilon _\phi }$ for $28 \lesssim \lambda _f/\eta _P \lesssim 35$ . This range aligns with the most unstable wavelength predicted by the shear-layer model and the statistics of shear layers in isotropic turbulence (Watanabe & Nagata Reference Watanabe and Nagata2023). The peak wavelength slightly increases over time, likely due to the increase in the Kolmogorov scale, which rises by 30 % from the perturbation introduction to the end of the simulation. The effects of perturbations diminish as $\lambda _f$ deviates from the peak wavelength, with smaller $\lambda _f$ resulting in a more rapid decline compared with larger $\lambda _f$ . This behaviour is consistent with the faster decay of small-scale perturbations in figure 5. Perturbations with wavelengths slightly different from the peak wavelength still enhance the growth rate and scalar dissipation rate. For individual shear layers, the most unstable wavelength depends on the thickness. While the probability density function of shear-layer thickness peaks at $4\eta$ in isotropic turbulence, the full width at half maximum is also approximately $4\eta$ (Watanabe & Nagata Reference Watanabe and Nagata2023), leading to variations in the most unstable wavelength. Consequently, perturbations remain effective in influencing the shear layers even when the wavelength slightly deviates from $\lambda _f=30\eta _P$ .

Figure 19.

Relative changes due to perturbations introduced in different flow regions in R350L30e, R350L30i and R350L30ei: ( $a$ ) the growth rate of the scalar mixing layer, $\dot {V}_S$ and ( $b$ ) the area $ A$ of the scalar isosurface.

4.4. Dependence on perturbation locations

The effects of perturbations on passive-scalar statistics are compared across different perturbation locations in the R350L30e, R350L30i and R350L30ei cases, where perturbations with the unstable wavelength of small-scale shear layers are applied. For internal perturbations, changes in the flow field due to perturbation-induced shear instability, such as the number of vortices and other turbulent statistics, have been detailed for decaying HIT in our previous study (Watanabe & Nagata Reference Watanabe and Nagata2023).

Figure 19( $a$ ) compares the relative changes in the growth rate of the scalar mixing layer, $\Delta \dot {V}_S$ , among R350L30e, R350L30i and R350L30ei. All cases exhibit an enhanced growth rate, with the perturbation effects appearing more rapidly when introduced inside the turbulent region, as seen in R350L30i and R350L30ei. The promoted shear instability, observed in figure 6, occurs within the turbulent front. In R350L30i and R350L30ei, the perturbations are introduced within the turbulent region, where the shear layers are present, and are expected to immediately influence the evolution of the shear layers. External perturbations do not alter the shear layers early on because they are introduced outside where no shear layers exist. Perturbations in R350L30e and R350L30i are bounded by the scalar isosurface used to evaluate $\dot {V}_S$ . The differing timescales in the increase in $\dot {V}_S$ between these cases confirm that the perturbations themselves do not directly cause the increase in growth rate. Instead, it occurs through the excitement of small-scale shear instability, as visualised in figure 6( $b$ ). All cases also show a decrease in $\Delta \dot {V}_S$ after the peak, but the initial increase lasts longer for external perturbations in R350L30e. Internal perturbations evolve within the turbulent flow. Although internal perturbations introduce a peak in the kinetic energy spectrum at the perturbation wavelength, this peak decays rapidly in turbulence due to energy transfer toward smaller scales and energy dissipation (Watanabe & Nagata Reference Watanabe and Nagata2023). Conversely, external perturbations introduced outside the turbulence can maintain their length scale, leading to more pronounced effects even later. This difference may lead to a larger peak value of $\Delta \dot {V}_S$ for external perturbations in R350L30e compared with internal perturbations in R350L30i. Furthermore, the most significant increase in the growth rate is observed in R350L30ei, where perturbations are introduced across the entire flow region. The timing of the peak in $ \Delta \dot {V}_S$ lies between those of R350L30i and R350L30e, as both internal and external perturbations influence the growth rate in R350L30ei.

Figure 19( $b$ ) shows the relative change in the area, $\Delta A$ , of the scalar isosurface. Unlike the external perturbations used in R350L30e, the peak values of $\Delta A$ and $\Delta \dot {V}_S$ differ when perturbations are introduced inside the turbulent region (R350L30i and R350L30ei). In these cases, the peaks of $\Delta \dot {V}_S$ are larger than those of $\Delta A$ , indicating that internal perturbations also enhance the local propagation velocity of the outer boundary of the scalar mixing layer.

Figure 20.

Temporal variations in the fractal dimension $D_f$ of the scalar isosurface defining the outer boundary of the scalar mixing layer in R350, R350L30e, R350L30i and R350L30ei.

Figure 20 presents the fractal dimension $D_f$ for R350, R350L30e, R350L30i and R350L30ei. Regardless of the perturbation location, the perturbations result in an increase in $D_f$ , indicating a more complex surface geometry. Similar to the observations for $\Delta \dot {V}_S$ and $\Delta A$ , the effects of internal perturbations in R350L30i diminish more rapidly compared with external ones. Initially, internal perturbations cause a greater increase in $D_f$ , which correlates with a larger initial increase in the growth rate of the scalar mixing layer. Perturbations introduced across the entire flow region have the most significant impact on the fractal dimension of the scalar isosurface. The correlation between the increases in $D_f$ and $\dot {V}_S$ further underscores that the enhanced surface complexity, driven by shear instability, plays a significant role in the increased growth rate of the scalar mixing layer.

Figure 21.

Production and destruction terms of the normalised surface area $A'$ , as described by (3.4), in R350, R350L30e, R350L30i and R350L30ei: ( $a$ ) production ( $P_A$ ) and destruction ( $D_A$ ) terms of the isosurface area; and ( $b$ ) their balance ( $\dot {A}'$ ).

Figure 21( $a$ ) illustrates the temporal variations of the production ( $P_A$ ) and destruction ( $D_A$ ) terms in the isosurface-area equation (3.4). When perturbations are introduced, $P_A$ increases rapidly. Then, the increase in $D_A$ follows with a delay relative to the increase in $P_A$ . Consequently, the isosurface area increases when perturbations are introduced. This is further confirmed in figure 21( $b$ ), which compares $\dot {A}'=P_A+D_A$ across the four simulation cases. In R350L30i and R350L30ei, where perturbations are introduced within the turbulent region, $\dot {A}'$ shows a rapid increase compared with R350L30e with external perturbations, resulting in a faster initial growth rate of the scalar mixing layer.

Figure 22.

Conditional profiles of ( $a$ ) mean scalar $\langle \phi \rangle _I$ and ( $b$ ) r.m.s. scalar fluctuation $\phi _{rms}$ and mean scalar dissipation rate $\langle \varepsilon _\phi \rangle _I$ at $\tau /\tau _{\eta P}=7.5$ in R350, R350L30e, R350L30i and R350L30ei.

Figure 22( $a$ ) compares the conditional mean scalar at $\tau /\tau _{\eta P}=7.5$ . At this time, the growth rate of the scalar mixing layer has increased by approximately 7 % in R350L30e and R350L30i, and by approximately 25 % in R350L30ei, due to the perturbations. Perturbations lead to a decrease in the mean scalar, aligning with expectations from the enhanced growth rate of the scalar mixing layer. However, internal perturbations in R350L30i sustain a sharp mean scalar gradient within the interfacial layer. This region experiences molecular diffusion transferring scalar from inside to outside the scalar mixing layer, typically smearing the distinct scalar distribution jump. Previous studies have shown that small-scale turbulent scalar transport to the interfacial layer helps sustain this scalar jump (Watanabe et al. 2015a,Reference Watanabe, Mori, Ishizawa and Nagata b ). The mean scalar gradient similar to the baseline case could be maintained by small-scale turbulent motion in the turbulent core region, which is amplified by the shear instability induced by internal perturbations. This is confirmed below with a velocity and scalar cospectrum. In the R350L30ei case, where perturbations are introduced throughout the entire flow region, a significant increase in the growth rate of the scalar mixing layer is observed. This leads to a notable decrease in the mean scalar within the mixing layer, resulting in a less steep mean scalar gradient within the interfacial layer compared with R350L30i.

Figure 22( $b$ ) displays the r.m.s. scalar fluctuation ( $\phi _{rms}$ ) and mean scalar dissipation rate ( $\langle \varepsilon _\phi \rangle _I$ ) near the outer boundary. The effects of both internal and external perturbations manifest similarly in the core region of the scalar mixing layer, with increases in both $\phi _{rms}$ and $\langle \varepsilon _\phi \rangle _I$ . However, the reduction in the dissipation peak observed with external perturbations (R350L30e) does not occur with internal perturbations (R350L30i). This distinction is due to the internal perturbations maintaining a sharp scalar gradient within the interfacial layer in figure 22( $a$ ). Further detailed comparisons for these perturbations are provided below with the analysis within each subregion on $\zeta _I$ . Case R350L30ei with perturbations in the entire flow region exhibits the most significant increase in the r.m.s. scalar fluctuation ( $\phi _{rms}$ ) and mean scalar dissipation rate in the core region, corresponding to the most substantial rise in the growth rate of the scalar mixing layer.

Figure 23.

Cospectrum of lateral velocity and passive scalar, $C_{v\phi }$ , at $y/L_0=0.9$ in the scalar mixing layer at $\tau /\tau _{\eta P}=7.5$ for R350, R350L30i, R350L30e and R350L30ei. The cospectrum and wavenumber in the $z$ direction, $k_z$ , are normalised by the Kolmogorov length and velocity scale at the flow centre at the time of perturbation introduction.

Figure 23 examines the cospectrum of lateral velocity and passive scalar, $ C_{v\phi }$ , calculated with Fourier transform in the $ z$ direction. This cospectrum, when integrated over the wavenumber $ k_z$ , represents the turbulent scalar flux and is assessed for potential enhancements of the turbulent scalar transport due to internal perturbations. Here, $ C_{v\phi }$ is taken at $ y/L_0 = 0.9$ and $ \tau /\tau _{\eta P} = 7.5$ . The observed increase in $ C_{v\phi }$ at small scales in R350L30i and R350L30ei indicates enhanced scalar transport due to small-scale shear instabilities. However, the total turbulent flux, represented by $ \langle v'\phi ' \rangle$ , is primarily influenced by larger scales, where $ C_{v\phi }$ remains largely unaffected by perturbations. This finding implies that the observed changes in $ C_{v\phi }$ at smaller scales are not sufficient to alter the mean scalar profile across the scalar mixing layer significantly. However, the turbulent scalar transport near the outer boundary of the scalar mixing layer is predominantly driven by small-scale velocity fluctuations, which are represented as a velocity difference between two points with a small separation in the local scalar transport equation (Watanabe et al. 2014a, 2015b). Consequently, the enhanced $C_{v\phi }$ at small scales can still influence the mean scalar profile near the boundary where internal perturbations maintain a sharp mean scalar gradient in R350L30i. In contrast, external perturbations do not affect $C_{v\phi }$ because their influence is confined to the vicinity of the boundary. This aligns with the observation of a smaller mean scalar gradient near the boundary in R350L30e as shown in figure 22( $a$ ).

Figure 24.

Relative changes in the mean scalar dissipation rate, $\Delta \overline {\varepsilon _\phi }$ , across various subregions near the outer boundary of the scalar mixing layer in ( $a$ ) R350L30i and ( $b$ ) R350L30ei.

Figure 24( $a$ ) displays the relative changes in the mean scalar dissipation rate due to internal perturbations (R350L30i) in each subregion on the $\zeta _I$ coordinate. The results demonstrate an almost uniform increase in scalar dissipation rate of 20–30 % across C1–C3 compared with the baseline case (R350). This uniform effect contrasts with external perturbations, whose impact on the scalar dissipation rate drastically decreases from C1 to C3, as shown in figure 15( $b$ ). However, external perturbations typically lead to a more significant increase in the mean scalar dissipation rate, as also discussed above for the growth rate of the scalar mixing layer. Figure 24( $b$ ) shows the results for perturbations introduced throughout the entire flow region in R350L30ei. Inside the scalar mixing layer near the interface, the mean scalar dissipation rate increases significantly, with the increase ratio ranging from 50 % to 75 % across subregions C1 to C3. The perturbation effects diminish progressively from C1 to C3 with increasing distance from the boundary. This behaviour is consistent with the influence of external perturbations in R350L30e, shown in figure 15( $b$ ).

Figure 25.

Lateral profiles of mean scalar dissipation rate, $\langle \varepsilon _\phi \rangle$ , at $\tau /\tau _{\eta P}=7.5$ in R350, R350L30e, R350L30i and R350L30ei.

Figure 25 compares the lateral profiles of the mean scalar dissipation rate, $\langle \varepsilon_\phi \rangle$ , at $\tau /\tau _{\eta P} = 7.5$ . The profiles indicate that internal perturbations in R350L30i and R350L30ei influence small-scale mixing in the entire region, resulting in a significant increase in the mean scalar dissipation rate. In contrast, the effects of external perturbations in R350L30e are largely restricted to the vicinity of the outer boundary of the scalar mixing layer. As a result, the mean scalar dissipation rate, which is predominantly influenced by the internal region of the scalar mixing layer, is only minimally affected by external perturbations in figure 25. This distinction underscores how internal and external perturbations differentially influence turbulence. Exciting small-scale shear instability internally leads to widespread enhancements in small-scale mixing throughout the turbulent region, while external perturbations enhance mixing near the boundary. The enhanced scalar dissipation rate resulting from the perturbations has significant implications for the turbulent mixing mechanism. This type of instability can occur even without perturbations, while the introduction of perturbations accelerates the process, intensifying the effects. The observed perturbation effects imply the important role of small-scale shear instability in the rapid small-scale mixing facilitated by turbulent motions.