2.1. The SML experimental platform

Experiments were conducted on the SML experimental platform (figure 1) in the Hypersonic Aero-Optical Effects Laboratory (HAOEL) at the National University of Defense Technology. This platform is an air-breathing, direct-connect wind tunnel with test section size 250 mm $\times$ 50 mm $\times$ 100 mm ( $x$ , $y$ , $z$ ) and atmospheric air as the working fluid. It consists of the contraction section, stabilising section, nozzle section, test section, diffuser section and vacuum tank. To facilitate adjustment of the static pressure ratio between the incoming flows in the high- and low-speed regions, a total pressure adjustor and a honeycomb are installed on the low-speed side of the stabilising section. The nozzles were designed using the B-spline curve method, with Mach numbers $M_1 = 2.60$ and $M_2 = 3.60$ , respectively. The operating parameters adopted in the experiments are listed in table 1, where the velocity ratio is $r=U_1/U_2\approx 0.89$ , the density ratio is $s=\rho _1/\rho _2\approx 0.66$ , and the convective Mach number is $M_{ {C}}\approx 0.17$ , indicating that the SML is weakly compressible.

Table 1.

Operating parameters of the platform.

Table 1. Long description

A table comparing operating parameters of a platform for two different flow speeds. The table has two rows and eleven columns. The columns are labeled as M, p s (kPa), p 0 (kPa), T s (K), T 0 (K), a (m s superscript negative 1), U (m s superscript negative 1), rho (kg m superscript negative 3), and Re (m superscript negative 1). Row 1: Flow 1 (low speed) with values 2.60, 1.15, 23.02, 128.83, 303.15, 227.57, 591.69, 3.12 times 10 superscript negative 2, 2.11 times 10 superscript 6. Row 2: Flow 2 (high speed) with values 3.60, 1.15, 101.33, 84.35, 303.15, 184.15, 662.94, 4.76 times 10 superscript negative 2, 5.62 times 10 superscript 6.

Figure 1.

Schematic diagram of the SML experimental platform in the HAOEL.

2.2. The MVG array

The flow control device adopted in this study is the MVG, as shown in figure 2. This device is designed based on the classic Anderson MVG (Anderson et al. Reference Anderson, Tinapple and Surber2006). As an effective flow control technique, Anderson’s configuration has been widely applied to the flow control of shock wave–boundary layer interaction (Giepman et al. Reference Giepman, Schrijer and van Oudheusden2014; Schreyer et al. Reference Schreyer, Sahoo, Williams and Smits2021), forward step (Zhang et al. Reference Zhang, Li, Zuo, Qin, Cheng, Feng and Bao2016), compression corner (Chen et al. Reference Chen, Kong, Li and Yang2019) and mixing enhancement (Li et al. Reference Li, Li, Yang, Zhu, Chen and Li2019). Compared with the Anderson configuration, the MVGs used in this study are partially truncated at the trailing edge, as in the configuration adopted by Verma & Chidambaranathan (Reference Verma and Chidambaranathan2015), and employ an array design. This design is derived from the engineering requirements of the HAOEL for thermal protection of optical dome windows and suppression of aero-optical effects. In recent years, experimental studies by the HAOEL (Ding et al. Reference Ding, Yi, Zhao, Xiong and Ouyang2019a ; Xia et al. Reference Xia, Ding and Yi2023; Sun et al. Reference Sun, Ding, Liu and Yi2024) have confirmed that the MVGs adopted in this study mitigate the aero-optical effects of the optical dome, and improve the cooling efficiency of the SCF. Ding et al. (Reference Ding, Yi, Xu and Zhao2021) pointed out that the breakdown of large-scale K–H vortices, or more precisely, the reduction in the SML’s growth rate, may be the reason for the above capabilities of MVGs. However, the intrinsic mechanism by which MVGs suppress the SML’s growth rate remains unclear, which is the main motivation of this study. To this end, four geometrically scaled MVGs with different trailing-edge heights ( $h = 0.5, 0.8, 1.0, 1.2$  mm) were designed to conduct flow visualisation experiments. The boundary layer thickness at the end of the splitter is $\delta = 1.25$  mm (see § 2.4), which means that the MVGs are submerged under the boundary layer. For the convenience of description, the SMLs controlled by these four MVGs will be named $\rm {CSML}_{05}$ , $\rm {CSML}_{08}$ , $\rm {CSML}_{10}$ and $\rm {CSML}_{12}$ , respectively.

Figure 2.

(a) Photograph of the $h = 1$  mm MVG. (b) Mounting position of the MVG. (c) Geometry of the MVG.

2.3. The NPLS visualisation

The NPLS technique is a non-intrusive, high-resolution flow field visualisation method that relies on Rayleigh scattering signals from nano-tracers (Zhao et al. Reference Zhao, Yi, Tian and Cheng2009; Yi et al. Reference Yi, Zhao, Tian, He and Chen2013; He et al. Reference He, Yi and Lu2017) and has been widely applied to the measurement of supersonic and hypersonic flow fields (Yi et al. Reference Yi, He, Zhao, Tian and Cheng2009, Reference Yi, Zhu, Zhi and Yu2015; Chen et al. Reference Chen, Yi, He, Tian and Zhu2012; Zhu et al. Reference Zhu, Yi, Chen and Tian2012; Zhuang et al. Reference Zhuang, Tan, Huang, Liu and Zhang2018; Gang et al. Reference Gang, Yi and Niu2021; Xia et al. Reference Xia, Ding, Sun, Yi, Liu, Huo and Luo2024; Xu et al. Reference Xu, Ye, Yi, Xue and Chen2024). The composition of the NPLS system is illustrated in figure 3. In the present study, $\rm {TiO}_2$ with nominal diameter 18 nm and bulk density 300 $\rm {kg\,m^{-3}}$ was selected as the tracer. According to Zhao et al. (Reference Zhao, Yi, Tian and Cheng2009), the relaxation time of the $\rm {TiO}_2$ tracer is 66.3 ns. An Imperx B2020 charge-coupled device (CCD) camera was employed, featuring resolution 2048 $\times$ 2048 pixels, a 12-bit dynamic range, and sampling frequency 100 Hz. The flow field was illuminated using a Beamtech Vlite-380 dual-cavity Nd: YAG laser at 532 nm, with pulse duration 6 ns and pulse energy 350 mJ. Because the laser pulse duration is much shorter than the camera exposure time, the effective exposure time is essentially the laser pulse duration 6 ns. During this effective exposure time, the mainstream flow travels approximately 4 ${\unicode[Arial]{x03BC}}$ m downstream, which is much smaller than the spatial resolution of the NPLS system. Therefore, the resulting imaging blurring is negligible (Zhuang et al. Reference Zhuang, Tan, Huang, Liu and Zhang2018).

Figure 3.

Schematic diagram of the composition of the NPLS system.

Figure 4.

Instantaneous NPLS greyscale images of the SML, resolution 108.3 ${\unicode[Arial]{x03BC}}$ m pixel⁻¹, $h = 1$  mm: (a) FSML, (b) $\rm {CSML}_{05}$ tip, (c) $\rm {CSML}_{05}$ notch, (d) $\rm {CSML}_{08}$ tip, (e) $\rm {CSML}_{08}$ notch, (f) $\rm {CSML}_{10}$ tip, (g) $\rm {CSML}_{10}$ notch, (h) $\rm {CSML}_{12}$ tip, (i) $\rm {CSML}_{12}$ notch.

2.4. Flow visualisation results

Figure 4 presents the instantaneous NPLS images (resolution 108.3 ${\unicode[Arial]{x03BC}}$ m pixel⁻¹) of the SML, with the $x$ -axes and $y$ -axes non-dimensionalised by the trailing-edge height of the MVGs, $h = 1$  mm. The darker regions in the images represent the low-speed flow ( $M_1 = 2.60$ ), and the brighter regions represent the high-speed flow ( $M_2 = 3.60$ ). This contrast arises because only the high-speed flow was seeded. In the initial region of the SML, the mixing interface is smooth, proving that the boundary layers on the splitter are laminar. The thickness of this boundary layer is estimated to be half of the mixing interface’s thickness, i.e. approximately 1.25 mm. Figure 4(a) shows the spatial development of the FSML. After the high-speed and low-speed flows separate from the trailing edge of the splitter, a laminar region approximately 30 mm in length emerges. Subsequently, under the strong shear stress induced by the velocity gradient between the high-speed and low-speed flows, several anticlockwise-rolling, upstream-inclined, and continuously growing K–H vortices emerge in the FSML. As the FSML evolves downstream, the K–H vortices become unstable and break down, eventually transitioning to fully developed turbulence.

As shown in figures 4(b) and 4(c), compared with the FSML, the K–H vortices in $\rm {CSML}_{05}$ begin to break down, but the flow field does not change significantly. As illustrated in figures 4(d) and 4(e), the breakdown of the K–H vortices in $\rm {CSML}_{08}$ becomes more significant, the position where the SML transitions to turbulence advances, and the size of the vortical structures decreases in the turbulence section. As presented in figures 4(f) and 4(i), the K–H vortices are completely broken down, and the large-scale vortical structures disappear in $\rm {CSML}_{10}$ and $\rm {CSML}_{12}$ . It can also be found from figure 4 that the flow patterns after the tips and notches have certain differences. Studies have shown that MVGs can induce counter-rotating streamwise vortex pairs (Verma & Chidambaranathan Reference Verma and Chidambaranathan2015; Ye et al. Reference Ye, Schrijer and Scarano2016; Tambe et al. Reference Tambe, Schrijer and van Oudheusden2021), resulting in downwash at the notches and upwash at the tips, thereby inducing different flow structures at the tips and notches. In combination with the numerical simulation results presented in the subsequent sections, we will further elaborate on the differences downstream of the tips and notches.

Figure 5 shows the time-averaged NPLS images of the SML, from which we can clearly see the spatial distribution of greyscale fluctuations, which, to some extent, reflect the density fluctuations of the SML. The density fluctuations in the mainstream on both sides are weak, resulting in relatively uniform greyscale; the density fluctuations in the centre of the SML are strong, producing chaotic greyscale. In order to quantitatively extract the thickness of the SML from figure 5, the greyscale fluctuation in the images is defined as

(2.1) \begin{equation} I_{\mathit{rms}}^{x,y} = \sqrt {\frac {1}{N}\sum _{t=1}^{N} \left ( I^{t,x,y} - \overline {I^{x,y}} \right )^2},\end{equation}

where $x$ and $y$ represent Cartesian coordinates, $t$ is time, $I$ is greyscale, $N$ is the number of pictures collected in an experiment, and the overbar represents the time-averaged value.

Figure 5.

Time-averaged NPLS greyscale images of the SML, resolution 108.3 ${\unicode[Arial]{x03BC}}$ m pixel⁻¹, $h = 1$  mm, 300 samples used for each case: (a) FSML, (b) $\rm {CSML}_{05}$ tip, (c) $\rm {CSML}_{05}$ notch, (d) $\rm {CSML}_{08}$ tip, (e) $\rm {CSML}_{08}$ notch, (f) $\rm {CSML}_{10}$ tip, (g) $\rm {CSML}_{10}$ notch, (h) $\rm {CSML}_{12}$ tip, (i) $\rm {CSML}_{12}$ notch. The upper and lower boundaries are depicted by yellow dashed lines.

Figure 5. Long description

Panel A: The image shows a greyscale visualization of the supersonic mixing layer (SML) with upper and lower boundaries marked by yellow dashed lines. The x-axis is labeled ‘x/h’ and ranges from 20 to 220, while the y-axis is labeled ‘y/h’ and ranges from -15 to 15. The image includes a schematic of the tip configuration. Panel B: The image shows a greyscale visualization of the SML with the tip configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The root mean square (rms) value of the boundary layer thickness is 0.107. Panel C: The image shows a greyscale visualization of the SML with the notch configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.109. Panel D: The image shows a greyscale visualization of the SML with the tip configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.088. Panel E: The image shows a greyscale visualization of the SML with the notch configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.087. Panel F: The image shows a greyscale visualization of the SML with the tip configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.079. Panel G: The image shows a greyscale visualization of the SML with the notch configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.082. Panel H: The image shows a greyscale visualization of the SML with the tip configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.067. Panel I: The image shows a greyscale visualization of the SML with the notch configuration, with the x-axis labeled ‘x/h’ ranging from 20 to 220 and the y-axis labeled ‘y/h’ ranging from -15 to 15. The rms value of the boundary layer thickness is 0.067.

Figure 6.

(a) The greyscale fluctuation curve of cross-section $\mathrm{AA}^{\prime }$ in figure 5 and the definition of thickness, $\delta _{I,\mathit{rms}}$ , of the SML. (b) Plot of $\delta _{I,\mathit{rms}}^{\prime }$ versus MVG height.

We define the thickness of the SML, $\delta _{I,\mathit{rms}}$ , based on the greyscale fluctuations in the NPLS images. This thickness is defined to eliminate the ambiguity associated with ‘visual’ thickness (Brown & Roshko Reference Brown and Roshko1974; Pantano & Sarkar Reference Pantano and Sarkar2002; Zubair & Catrakis Reference Zubair and Catrakis2007), which is often approximated as twice the vorticity thickness under incompressible conditions (Papamoschou & Roshko Reference Papamoschou and Roshko1988). Figure 6(a) demonstrates the greyscale fluctuation curve of the SML at the $\mathrm{AA}^{\prime }$ cross-section in figure 5(a). On the high-speed and low-speed regions of the curve, the greyscale fluctuations are relatively stable and can be defined as $I_{ {rms}}^1$ and $I_{ {rms}}^2$ , respectively. The middle of the curve exhibits the greatest greyscale fluctuations, and thus represents the mixing zone. The upper and lower boundaries ( $y_1$ and $y_2$ ) of the SML are defined as positions where $ (I_{\mathit{rms}}-I_{\mathit{rms}}^1 )/I_{\mathit{rms}}^1 = 0.05$ and $ (I_{\mathit{rms}}-I_{\mathit{rms}}^2 )/I_{\mathit{rms}}^2 = 0.05$ , respectively. Then the local thickness of the SML is $\delta _{I,\mathit{rms}}=\|y_1-y_2\|$ . In each time-averaged NPLS image in figure 5, the upper and lower boundaries are marked with yellow dashed lines, and the average growth rate of the SML, $\delta _{I,\mathit{rms}}^{\prime }=\mathrm{d}\delta _{I,\textit{rms}}/\mathrm{d}x$ , is given. These results are drawn in figure 6(b). The results show that all MVGs inhibited the growth rate of the SML, and $\delta _{I,\mathit{rms}}^{\prime }$ became smaller with increasing trailing-edge height of MVGs. Moreover, there is little difference in growth rates between tips and notches, indicating that the control effect of MVGs on growth rate is uniform across the spanwise direction. In order to clarify the internal mechanism of MVGs inhibiting the growth rate of the SML, we select a typical case, namely MVGs with $h = 1$  mm, to carry out numerical simulation research.

3.1. Governing equations and numerical methods

The unsteady, three-dimensional, compressible Navier–Stokes equations in conservative form (Lu et al. Reference Lu, Yuan and Dawes2012) are solved for the spatially evolving mixing layer and written as follows:

(3.1) \begin{equation} \frac {{\partial S}}{{\partial t}} + \boldsymbol{\nabla }\boldsymbol{\cdot }\left ( {{{\boldsymbol{F}}_c} - {{\boldsymbol{F}}_d}} \right ) = 0, \end{equation}

where

(3.2) \begin{equation} S = \left [ {\begin{array}{*{20}{c}} \rho \\[3pt] {\rho {u_1}} \\[3pt] {\rho {u_2}} \\[3pt] {\rho {u_3}} \\[3pt] {\rho E} \end{array}} \right ]\!,\quad {\boldsymbol{F}_{c,i}} = \left [ {\begin{array}{*{20}{c}} {\rho {u_i}}\\[3pt] {\rho {u_i}{u_1} + p{\delta _{i1}}} \\[3pt] {\rho {u_i}{u_2} + p{\delta _{i2}}} \\[3pt] {\rho {u_i}{u_3} + p{\delta _{i3}}} \\[3pt] {\rho {u_i}H} \end{array}} \right ]\!,\quad {\boldsymbol{F}_{d,i}} = \left [ {\begin{array}{*{20}{c}} 0 \\[3pt] {{\sigma _{i1}}} \\[3pt] {{\sigma _{i2}}} \\[3pt] {{\sigma _{i3}}} \\[3pt] {{u_{\!j}}{\sigma _{ij}} + {q_i}} \end{array}} \right ]\!. \end{equation}

Here, $S$ is the conservative variable, ${\boldsymbol{F}}_c$ and ${\boldsymbol{F}}_d$ denote the convective and diffusive flux vectors, respectively, $u$ is velocity, $\rho$ is density, $p$ is pressure, $E$ is total energy, $H$ is total enthalpy, $\sigma _{ij}$ is viscous stress, $q_i$ is heat flux, $\delta _{ij}$ is the Kronecker symbol, and indices $i,j=1,2,3$ denote the three spatial directions.

For a Newtonian fluid satisfying Stokes’ hypothesis, the viscous stress is given by

(3.3) \begin{equation} \sigma _{ij} = \mu \left ( {\frac {{\partial {u_i}}}{{\partial {x_{\!j}}}} + \frac {{\partial {u_{\!j}}}}{{\partial {x_i}}} - \frac {2}{3}\frac {{\partial {u_k}}}{{\partial {x_k}}}{\delta _{ij}}} \right )\!. \end{equation}

The heat flux $q_i$ is computed via Fourier’s law of heat conduction:

(3.4) \begin{equation} {q_i} = - \kappa \frac {{\partial T}}{{\partial {x_i}}}. \end{equation}

The air in high- and low-speed regions is considered as an ideal gas. The ideal gas equation of state is given by

(3.5) \begin{equation} P = \rho {R_{ {g}}}T,\quad e = {C_v}T,\quad {C_v} = \frac {{{R_g}}}{{\gamma - 1}},\quad \kappa = \frac {\mu C_p}{Pr}, \end{equation}

where viscosity coefficient $\mu$ and thermal conductivity coefficient $\kappa$ are determined by Sutherland’s law (Sutherland Reference Sutherland1893), $R_g = 287.05$ J kg⁻¹ K⁻¹ is the gas constant of air, $\textit {Pr} = 0.72$ is the Prandtl number, $\gamma = 1.4$ is the specific heat ratio, $C_p = 1005$ J kg⁻¹ K⁻¹ is the specific heat capacity at constant pressure, and $C_v = 717.875$ J kg⁻¹ K⁻¹ is the specific heat capacity at constant volume.

The space–time expansion of the kinetic energy preserving flux reconstruction (STE-KEP-FR) method is employed to solve the governing equation. This method was based on a high-order flux reconstruction scheme (Huynh Reference Huynh2007), which has features of high cohesion and low coupling. The STE-KEP-FR method demonstrates strong suitability for GPU computation and high-order accuracy, enabling high-resolution LES/DNS. A space–time expansion with a local time-stepping framework was applied, which results in no explicit/implicit iteration, completely matrix-free computation, and avoidance of convergence problems in complicated simulations. To enhance nonlinear stability, a split-form kinetic preserving method was applied to deal with the convective term. Lax–Friedrichs flux splitting is used for the convective flux. The viscous flux is directly taken as the average of the fluxes on both sides along the outer normal. The time advance adopts the framework of prediction and correction, where the prediction step uses the continuous extension Runge–Kutta method, and the correction step uses high-precision time integration. The STE-KEP-FR method maintains the high-order accuracy of local reconstruction, and requires no explicit integration, achieving pure high-order accuracy in the difference form (not a difference scheme), and relying on a simple linear combination, thereby greatly reducing computational cost.

The STE-KEP-FR method has been successfully applied to aerofoils (Lu Reference Lu2012; Lu et al. Reference Lu, Yuan and Dawes2012), flow around cylinders (Liu et al. Reference Liu, Lu and You2013), and slat tonal noise analysis (Zhang et al. Reference Zhang, Tan and Yao2026). However, the application of the STE-KEP-FR method in the SML remains scarce. To the best of the authors’ knowledge, the present work is the first to employ the STE-KEP-FR method in the SML simulation. This method applies to the SML because the latter is a free shear layer, akin to the wakes behind aerofoils and cylinders. To further confirm the validity of this method, we perform a grid independence study in § 3.3, and validate the velocity, density and Reynolds stress profiles against existing experimental and numerical data.

Figure 7.

Schematic diagram of computational domains and boundary conditions.

3.2. Case set-up

The parameters of the numerical simulation are consistent with the experiments. Figure 7(a) defines the computational domain of the FSML, with dimensions 1060 mm $\times$ 150 mm $\times$ 100 mm in the $x$ , $y$ and $z$ directions, and a total solution points $1.38\times 10^8$ . The upper and lower regions correspond to low-speed and high-speed flows, respectively, separated by a splitter. The centre of the splitter’s trailing edge is defined as the origin. The inlets are set as total pressure inlets, with no additional disturbances applied. Thus a laminar boundary layer develops on the splitter. The splitter length is set to 60 mm such that the boundary layer thickness at the trailing edge of splitter matches the experimental value. The upper and lower boundaries of the computational domain are set as pressure far fields, with the flow parameters matching those at the inlet. The boundaries in the spanwise ( $z$ ) direction adopt periodic conditions. The outlet is set as a pressure outlet ( $p_{{s,\textit{outlet}}} = 10$  Pa).

Figure 7(b) defines the computational domain of the CSML, with dimensions 1060 mm $\times$ 150 mm $\times$ 14.4 mm in the $x$ , $y$ and $z$ directions. Given data reliability and computational costs, the spanwise width of the computational domain for CSML is set to 14.4 mm, rather than 100 mm for FSML. The solution points of CSML are $2.57\times 10^8$ , approximately 1.86 times those of FSML, approaching the workstation’s maximum capacity. In addition, since the left- and right-hand boundaries are set as periodic conditions, the influence of the spanwise width on the computational results can be neglected. The MVGs are installed on the low-speed side of the splitter, and their trailing edges are tangent to each other. The trailing-edge height, trailing-edge width and leading-edge width are 1.0, 0.8 and 2.3 mm, respectively. The nominal boundary layer thickness at the location of the MVGs is $\delta _{99} = 1.25$  mm, and the MVGs are submerged under the nominal boundary layer thickness.

3.3. Case validation

The objective of this subsection is to validate the accuracy of our simulation by comparing the current results against existing experimental and numerical studies. It is generally accepted that after an initial transition period, the SML enters the self-similar region characterised by linear thickness growth. Analysis of the spatial evolution of the average velocity profile indicates that the momentum deficit region disappears beyond $x = 600$  mm (see § 4.4.1), which is taken as the start of the self-similar region. To ensure the self-similarity of the SML, numerical results are validated in regions with $x \gt 600$  mm.

3.3.1. Grid independence inspection

Four distinct grid sets were used to simulate the FSML for grid independence inspection. The grids are evenly spaced along the streamwise ( $x$ ) and spanwise ( $z$ ) directions, while clustering is applied in the transverse ( $y$ ) direction around the core region of the SML. Due to the use of the STE-KEP-FR method, each grid element contains multiple solution points, which are implemented as

(3.6) \begin{equation} N_{ {p}} = \left ( {{O_{ {p}}} + 1} \right )^3 \!, \end{equation}

where $N_{\mathit{p}}$ denotes the number of solution points per grid element, and $O_{\mathit{p}}$ represents the solver’s order.

A third-order precision solver, corresponding to 64 solution points per grid element, was used here. Accordingly, the solution points of four grid sets were $N_{node}=3.22\times 10^7$ , $6.46\times 10^7$ , $1.38\times 10^8$ and $1.54\times 10^8$ . Figure 8 presents the normalised mean velocity profiles and power spectral density (PSD) curves of the FSML’s density at different solution points. As shown, the normalised mean velocity profiles exhibit substantial overlap across the four cases, indicating that the present algorithm achieves satisfactory computational accuracy for mean quantities in the FSML. However, significant discrepancies are observed in the PSD curves, except for those of $N_{node}=1.38\times 10^8$ and $1.54\times 10^8$ . In addition, under all cases, the $k^{-5/3}$ law in the PSD curve is consistent with the energy spectrum of Kolmogorov’s inertial subrange. The $k^{-12.7}$ law indicates that the calculated minimum vortex scale lies in the dissipation range. Considering computational accuracy and cost, the $N_{node}=1.38\times 10^8$ grid set is adopted for the following analysis.

Figure 8.

Grid independence inspection results at $x=650$ mm: (a) normalised mean velocity profile; (b) normalised PSD of $\rho$ .

Figure 8. Long description

Panel A: A line graph shows the normalized mean velocity profile. The horizontal axis is labeled y/δω and ranges from -1.5 to 1.5. The vertical axis is labeled (U - U1)/ΔU and ranges from 0 to 1.0. Multiple lines are plotted, representing different node counts. The lines overlap significantly, indicating similar trends across different node counts. Panel B: A line graph displays the normalized power spectral density (PSD) of ρ. The horizontal axis is labeled f (Hz) and ranges from 10^3 to 10^6. The vertical axis is labeled PSD of ρ and ranges from 10^-6 to 10^0. Multiple lines represent different node counts, with a legend indicating specific values. The lines show a general trend of decreasing PSD with increasing frequency, with notable slopes of -5/3 and -12.7.

3.3.2. Verification of numerical results

To verify the reliability of the simulated SML in this study, this subsubsection examines the statistical characteristics from three aspects: velocity profiles, density profiles and Reynolds stress distributions. The mean velocity profiles shown in figure 9(a) agree well with previous studies (Spencer & Jones Reference Spencer and Jones1971; Bell & Mehta Reference Bell and Mehta1990; Goebel et al. Reference Goebel, Dutton, Krier and Renie1990; Olsen & Dutton Reference Olsen and Dutton2002; Barre & Bonnet Reference Barre and Bonnet2015; Mathews et al. Reference Mathews, Wang, Wang and Jumper2019). The normalised velocity profiles at different streamwise positions collapse well in the self-similar region. Figure 9(b) shows the normalised average density profile, which is consistent with the results of Pantano & Sarkar (Reference Pantano and Sarkar2002). The slight difference observed in figure 9 may be due to the difference in density ratios and compressibility of different studies.

Figure 9.

Statistical characteristics of the SML: (a) normalised velocity profiles; (b) normalised density profiles.

Figure 10 demonstrates the normalised Reynolds stress. It is revealed that both the profile shapes and peak values of the normal and shear Reynolds stresses align well with published data (Spencer & Jones Reference Spencer and Jones1971; Bell & Mehta Reference Bell and Mehta1990; Rogers & Moser Reference Rogers and Moser1994; Pantano & Sarkar Reference Pantano and Sarkar2002; Mathews et al. Reference Mathews, Wang, Wang and Jumper2019). The notable peak in $\langle w^{\prime }w^{\prime }\rangle / (\Delta U )^2$ deviates from prior studies, which may be attributed to stronger spanwise velocity fluctuations under compressible conditions. These verifications confirm that the STE-KEP-FR method provides a reliable representation of the SML during the self-similar stage.

Figure 10.

Normalised Reynolds stress profiles of the SML.

Figure 10. Long description

Panel A: A line graph shows the normalized Reynolds stress profile for the streamwise component. The x-axis is labeled 'y/δω' and the y-axis is labeled '<u'u'>(ΔU)^2'. Multiple datasets are plotted with different symbols and colors. Panel B: A line graph displays the normalized Reynolds stress profile for the wall-normal component. The x-axis is labeled 'y/δω' and the y-axis is labeled '<v'v'>(ΔU)^2'. Multiple datasets are plotted with different symbols and colors. Panel C: A line graph presents the normalized Reynolds stress profile for the spanwise component. The x-axis is labeled 'y/δω' and the y-axis is labeled '<w'w'>(ΔU)^2'. Multiple datasets are plotted with different symbols and colors. Panel D: A line graph shows the normalized Reynolds stress profile for the shear component. The x-axis is labeled 'y/δω' and the y-axis is labeled '<u'v'>(ΔU)^2'. Multiple datasets are plotted with different symbols and colors. Each panel includes a legend identifying the sources of the data.

4.1. Characteristics of instantaneous and time-averaged flow fields

Figure 11 presents streamwise slices of the instantaneous density fields of FSML and CSML. The slice height is $H = 50$  mm. The regions $x/H\in [{0,1.5} ]$ marked with the black dashed boxes have been enlarged for clarity. The streamwise slices capture the spatial evolution of the SML from laminar to turbulent flow, along with the details of entrainment and development of K–H vortices. Owing to unavoidable initial perturbations in the experiments, the SML measured using the NPLS system develops more rapidly than the numerical simulation results. The numerical simulations provide a clearer view of the expansion wave system at the trailing edge of the splitter. Furthermore, because the upper and lower boundaries of the computational domain are set as the pressure far field, reflected shock waves induced by the tunnel wall do not appear.

Figure 11.

Streamwise slices of instantaneous LES density fields, where the $x$ -axes and $y$ -axes are normalised by $H = 50$  mm: (a) FSML, (b) CSML tip, (c) CSML notch.

As shown in figure 11(a), following separation from the trailing edge of the splitter, the boundary layers on the high- and low-speed regions initially undergo a laminar mixing procedure over a short distance. Subsequently, under sustained velocity shear, the K–H instability progressively amplifies, destabilising the SML, and generating K–H vortices. These vortices rotate and merge continuously to increase in size. At the downstream region, the K–H vortices destabilise and break up into small-scale turbulence. Figures 11(b) and 11(c) show that under MVG control, the ‘visual’ thickness of the SML decreases, and K–H vortices are replaced by small-scale turbulence. From the enlarged view, the SML can be divided into high-speed, low-speed and middle layers in the near-field region downstream of the MVGs. The middle layer forms through the convergence of laminar boundary layers on the high-speed and low-speed sides of the splitter. In the near-field region, the flow field structures differ downstream of the tips and notches. The middle layer behind the tips is thicker. This is because the counter-rotating vortical pairs induced by MVGs (Lin Reference Lin1999; Ye et al. Reference Ye, Schrijer and Scarano2018; Tambe et al. Reference Tambe, Schrijer and van Oudheusden2021; Wu et al. Reference Wu, He, Yu and Liu2022; Cheawchan et al. Reference Cheawchan, Mohamed, Ng and New2023) experience upwash and downwash at the tips and notches, respectively. The vortex structures behind the tips and notches are generally consistent when $x/H \gt 1.5$ , indicating that the adjacent counter-rotating vortex pairs have merged and developed uniformly.

Figure 12 shows the spanwise slices of the instantaneous and time-averaged density fields of FSML and CSML. The $x$ -axes are normalised by $H= 50$  mm, and the $y$ -axes are normalised by $W_1 = 100$  mm and $W_2 = 14.4$  mm for FSML and CSML, respectively. From figure 12(a), two-dimensional ‘rollers’ (spanwise vortices) spanning the entire spanwise region are observed, which are typical structures in the weakly compressible SML (Tan et al. Reference Tan, Zhang and Lv2018; Zhang et al. Reference Zhang, Zhu, Yi and Wu2019; Wang et al. Reference Wang, Wang, Lei and Feng2022), and reflect the rolling and development of K–H vortices. As the FSML progresses downstream, the ‘rollers’ gradually break up, with the fragmentation of K–H vortices and the strengthening of the flow’s three-dimensional characteristics. From figure 12(b), regular streamwise density stripes appear in region I ( $x/H\in [{0,1} ]$ ) of the CSML. The fact that each MVG induces two density stripes indicates the presence of streamwise vortices. As seen in figures 12(c) and 12(d), the time-averaged density field of the FSML is uniform in the spanwise direction, while that of the CSML is characterised by streamwise density stripes. As shown in figure 12(d), adjacent density stripes remain independent in region I, and intertwine and merge in the spanwise direction in region II ( $x/H\in [{1,3.5} ]$ ). When $x/H \gt 3.5$ , the intertwining and merging of density stripes finish, and the density distribution becomes uniform in the CSML.

Figure 12.

Spanwise slices (at $y=0$ ) of (a,b) instantaneous and (c,d) time-averaged LES density fields. The $x$ -axes are normalised by $H=50$  mm, and the $z$ -axes are normalised by $W_1 = 100$  mm and $W_2 = 14.4$  mm. The plots are for (a,c) FSML, (b,d) CSML.

Figure 12. Long description

Panel A: A heat map shows instantaneous density fields for FSML. The x-axis is normalized by 1.0 millimeters, and the z-axis is normalized by 1.0 millimeters. The density values range from 0.02 to 0.04 kilograms per cubic meter, with a color gradient from blue to red. Panel B: A heat map shows instantaneous density fields for CSML. The x-axis is normalized by 2.0 millimeters, and the z-axis is normalized by 2.0 millimeters. The density values range from 0.02 to 0.04 kilograms per cubic meter, with a color gradient from blue to red. Panel C: A heat map shows time-averaged density fields for FSML. The x-axis is normalized by 1.0 millimeters, and the z-axis is normalized by 1.0 millimeters. The density values range from 0.02 to 0.04 kilograms per cubic meter, with a color gradient from blue to red. Panel D: A heat map shows time-averaged density fields for CSML. The x-axis is normalized by 2.0 millimeters, and the z-axis is normalized by 2.0 millimeters. The density values range from 0.02 to 0.04 kilograms per cubic meter, with a color gradient from blue to red.

4.2. Dominant vortical structures and their DMD modes

From the perspectives of instantaneous and time-averaged flow field structures, the FSML and CSML are dominated by spanwise ‘rollers’ and streamwise density stripes, respectively. To elucidate the dominant vortical structures of the FSML and CSML, the $Q$ -criterion was employed to extract the instantaneous vortex structures within the flow field, as illustrated in figure 13. Under the influence of ${\partial u}/{\partial y}$ , the FSML is characterised by spanwise vortices, which originate from the rotation and pairing of $\Lambda$ , hairpin and K–H vortices (Winant & Browand Reference Winant and Browand1974; Zhang et al. Reference Zhang, Zhu, Yi and Wu2019). The spatial growth of the FSML is primarily realised through K–H instability. In the CSML, the streamwise vortices become the dominant structures. The spatial growth of the CSML is achieved through the pairing and merging of adjacent streamwise vortices alongside K–H instability.

Figure 13.

Instantaneous vortex structures of (a) FSML and (b) CSML using $Q$ -criteria. Here, $Q=5\times 10^6$ , coloured by velocity magnitude, the streamwise lengths of flow fields are $3H = 150$  mm, and the spanwise widths of flow fields are $W_1/2 = 50$  mm and $W_2 = 14.4$  mm.

To quantitatively assess the intensity of vortices in the FSML and CSML, we introduce the circulation $\varGamma$ (Lin Reference Lin1999; Tambe et al. Reference Tambe, Schrijer and van Oudheusden2021; Wu et al. Reference Wu, He, Yu and Liu2022), defined as the surface integral of vorticity. The dimensionless circulation is calculated as

(4.1) \begin{equation} {\varGamma _x} = \frac {1}{S}\int _S {\left | {{\omega _x}} \right |}\, {\text{d}}S,\quad{\varGamma _y} = \frac {1}{S}\int _S {\left | {{\omega _y}} \right |}\, {\text{d}}S,\quad{\varGamma _z} = \frac {1}{S}\int _S {\left | {{\omega _z}} \right |}\, {\text{d}}S, \end{equation}

where $\omega$ is the vorticity, and $S$ is the surface area of transverse slices at a given $y$ – $z$ plane.

Figure 14 depicts the variation of circulation along the streamwise direction. For the FSML, $\varGamma _z$ gradually diminishes during the early development stage ( $x=0{-}200h$ ), while $\varGamma _x$ and $\varGamma _y$ initially remain near zero, then increase slowly, which confirms that the FSML in region $x=0{-}200h$ is dominated by spanwise vortices. When $x\gt200h$ , $\varGamma _x$ , $\varGamma _y$ and $\varGamma _z$ tend to be consistent, indicating the development of three-dimensional turbulence. Compared with the FSML, $\varGamma _{x}$ and $\varGamma _{y}$ surge from near zero to 0.01 s $^{-1}$ and 0.02 s $^{-1}$ , respectively, during the early development stage of the CSML. In contrast, $\varGamma _{z}$ shows little variation in amplitude or spatial distribution, declines gradually, and remains as the dominant vorticity component. This indicates that the streamwise vortices induced by MVGs weaken energy concentration in the spanwise direction, and promote energy redistribution across the $x$ , $y$ and $z$ directions, which alters the energy transport path of the FSML. When $x\gt200h$ , the SML enters a relatively developed stage, where $\varGamma _x$ , $\varGamma _y$ and $\varGamma _z$ tend to be consistent again.

Figure 14.

Circulation change of (a) the FSML and (b) the CSML along the streamwise direction, where the $x$ -axes are normalised by the height of the MVGs, $h = 1$  mm.

The DMD method decomposes complex flows into representations based on simple spatio-temporal coherent structures, characterised by its equation-free and data-driven nature (Schmid Reference Schmid2010; Kutz et al. Reference Kutz, Brunton, Brunton and Proctor2016). It can effectively extract the principal modes of complex unsteady flows, yielding the corresponding frequencies and growth rates. It treats flow evolution as a linear dynamical process, and analyses snapshots of the flow field to derive low-order modes $\phi _i$ and their corresponding eigenvalues $\lambda _i$ and amplitudes $b_i$ . If the sampling frequency of snapshots is $f_{\mathit{snapshot}}$ , then the real and imaginary parts of $\mathrm{log} (\lambda _i f_{\mathit{snapshot}} )$ correspond to the growth rate and frequency of modes.

To elucidate the mechanism governing the transition from spanwise vortices to streamwise vortices in the CSML, the normal velocity $V$ was selected to conduct DMD analyses in regions I and II depicted in figure 12. For both the FSML and CSML, DMD analyses were performed on 200 flow fields acquired at 1 MHz. Following Schmid (Reference Schmid2011), the DMD modes are ranked in descending order of the absolute value of amplitudes, $|b_i|$ . The data reconstruction loss rate is defined as

(4.2) \begin{equation} \varepsilon = \frac {|V_{\mathit{DMD}} - V_{\mathit{original}}|}{V_{\mathit{original}}} \times 100\,\%, \end{equation}

where $V_{\mathit{DMD}}$ and $V_{\mathit{original}}$ are reconstructed and original velocity fields, respectively.

Figure 15 displays the normal velocity distribution of the SML at a given time, along with the DMD reconstruction results. To determine the number of modes $q_{\mathit{re}}$ used for reconstruction, $\varepsilon$ is plotted as a function of $q$ in figures 15(ai–aiv). The horizontal dashed lines mark the threshold of $\varepsilon = \varepsilon _{\textit{min}} + 0.05\,\Delta \varepsilon$ , where $\Delta \varepsilon = \varepsilon _{\textit{max}} - \varepsilon _{\textit{min}}$ . The arrows indicate the position of $q_{\mathit{re}}$ , which is decided by $ [ \varepsilon - ( \varepsilon _{\textit{min}} + 0.05\, \Delta \varepsilon ) ]_{\textit{min}}$ . For $q\lt q_{\mathit{re}}$ , $\varepsilon$ decreases rapidly with $q$ . Further increasing $q$ beyond $q_{\mathit{re}}$ yields diminishing returns for $\varepsilon$ reduction. It should be noted that the magnitude of $\varepsilon$ in region I of the CSML is substantially smaller than that in the other three regions in figure 15. The choice of $q_{\mathit{re}}=33\ (\varepsilon \approx 0.97\,\%)$ already yields a reconstruction comparable to that of $q_{\mathit{re}}=116$ . Therefore, $q_{\mathit{re}}=33$ is adopted for region I of the CSML. Compared to the FSML, the number of DMD modes for reconstruction in region I of the CSML is reduced by more than $50\,\%$ . This is because the streamwise vortices generated by MVGs exhibit good spatial stability in region I. Meanwhile, the number of DMD modes required for reconstruction in region II of the CSML is slightly higher than in the FSML, because the flow field in region II of the CSML is more complex.

Figure 15.

(a) Data reconstruction loss rate as a function of number of modes. (b) Original normal velocity fields. (c) The DMD-reconstructed normal velocity fields. In (b) and (c), the $x$ -axes are normalised by $H = 50$  mm, and $z$ -axes are normalised by $W_1 = 100$  mm and $W_2 = 14.4$  mm. The numbers i, ii, iii and iv in the labels indicate FSML region I, FSML region II, CSML region I and CSML region II, respectively.

Figures 16(ai–aiv) present the eigenvalue distributions of all modes of the normal velocity fluctuation in the SML. The horizontal and vertical axes denote the real and imaginary parts, respectively. Eigenvalues inside the unit circle have negative growth rates and correspond to stable modes, while those outside the unit circle have positive growth rates and correspond to unstable modes. In both the FSML and CSML, the modes are stable. To identify the dominant modes, the modes are ranked in descending order according to their modal energy

(4.3) \begin{equation} E_i = \sum _{j=1}^{N} \left | b_i \lambda _i^{j-1} \right |^2 \!, \end{equation}

Figure 16.

(a) Eigenvalue distributions of normal velocity fluctuations, where the dashed line represents the unit circle; the blue, yellow and grey circles denote the first dominant modes, the second dominant modes, and the remaining stable modes, respectively. (b) First dominant DMD modes. (c) Second dominant DMD modes. For the dominant modes shown in (b) and (c), the $x$ -axes are normalised by $H=50$  mm, and $z$ -axes are normalised by $W_1 = 100$  mm and $W_2 = 14.4$  mm. The numbers i, ii, iii and iv in the labels indicate FSML region I, FSML region II, CSML region I and CSML region II, respectively.

Figure 16. Long description

Panel A: Four scatter plots show eigenvalue distributions of normal velocity fluctuations. The dashed line represents the unit circle. Blue, yellow, and grey circles denote the first dominant modes, the second dominant modes, and the remaining stable modes, respectively. Panel B: Two heatmaps display the first dominant DMD modes. Panel C: Two heatmaps show the second dominant DMD modes. The x-axes are normalized by millimeters, and the z-axes are normalized by millimeters. The numbers i, ii, iii, and iv in the labels indicate different regions of the supersonic mixing layer.

where $i=1{-}q_{\mathit{re}}$ , and $N=200$ is the number of snapshots used for the DMD decomposition. The top two modes with the largest $E_i$ are identified as the dominant modes. The first dominant modes are indicated by blue circles, the second by yellow circles, with the corresponding frequencies annotated. The characteristic frequencies of the dominant modes in the CSML are notably higher than those in the FSML, indicating an enhanced dominance of small-scale flow structures under MVG control.

Figures 16(b) and 16(c) depict the first and second dominant DMD modes of normal velocity fluctuations, respectively. As shown in figures 16(bi,bii,ci,cii), both the first and second dominant modes of the FSML exhibit spanwise periodic structures. The formation of these large-scale spanwise vortices is attributed to the K–H instability. As depicted in figures 16(biii) and 16(ciii), the dominant modes of CSML in region I exhibit streamwise-aligned periodic structures, verifying that the flow field behind the MVGs (the centrelines of the MVG tips are annotated with dashed lines) is dominated by streamwise vortices. The dominant modes in region I exhibit ‘comb-like’ patterns, reflecting strong spatial similarity across all MVG tips. The term ‘comb-like’ derives from the fact that the positive and negative modes are consistent in shape and size, and regularly aligned on opposite sides of the tips’ centrelines. Furthermore, the positive and negative modes behind adjacent tips show spatial phase difference and alternate along the streamwise direction, which is the feature of travelling waves (Wang et al. Reference Wang, Qu, Wang, Sun and Bai2023). As depicted in figures 16(biv) and 16(civ), the second dominant modes behind the tips begin to spread laterally at $x/H = 1$ , and progressively align and coalesce, which causes the phase difference gradually diminish. When $x/H \gt2$ , large-scale spanwise vortices similar to those in the FSML gradually form, thereby restoring K–H instability dominance. The blurred white patterns observed in figures 16(b) and 16(c) may result from the superposition of two or more modes with similar frequencies. This interference causes local enhancement of velocity fluctuations in certain regions and offset in others, ultimately manifesting as blurred stripes.

In summary, the introduction of MVGs shifts the dominant vortical structures in the initial stage of the SML from spanwise to streamwise vortices, although spanwise vortices regain dominance further downstream. The interaction between streamwise and spanwise vortices dominates the spatial growth of the CSML, which is in contrast to the sole dominance of K–H vortices in the FSML.

4.3. Spatial growth of the SML dominated by spanwise and streamwise vortices

To reveal the differences in spatial growth rates between the SML dominated by spanwise vortices and streamwise vortices, the growth of vorticity thickness along the streamwise direction was computed. Vorticity thickness (Papamoschou & Roshko Reference Papamoschou and Roshko1988; Dimotakis Reference Dimotakis1991b ; Ragab & Sheen Reference Ragab and Sheen1992; Barre & Bonnet Reference Barre and Bonnet2015) is defined as

(4.4) \begin{equation} \delta _\omega = \frac {{\Delta U}}{{{{\left ( {{{{\text{d}}\bar U} \mathord {\left /{\vphantom {{{\text{d}}\bar U} {{\text{d}}y}}}\right .} {{\text{d}}y}}} \right )}_{\textit{max}}}}}, \end{equation}

where $\Delta U = {U_2} - {U_1}$ is the velocity difference between the high-speed and low-speed flows.

Figure 17 illustrates the evolution of vorticity thickness. The dash-dotted and dashed lines represent the fitted curves for vorticity thickness and the corresponding growth rates in regions I and II, respectively. The vorticity thickness grows linearly in both regions. The growth rates of the SML in region I are relatively low because the high-speed and low-speed layers are separated by the middle layer and have not yet fully interacted (figure 11), maintaining the SML in a laminar growth stage. The growth rates of the FSML and CSML in region I are $\mathrm{d}\delta _\omega /\mathrm{d}x|_{\mathit{FSML}}^{\mathit{I}} = 0.0056$ and $\mathrm{d}\delta _\omega /\mathrm{d}x|_{\mathit{FSML}}^{\mathit{II}} = 0.0037$ , respectively. The growth rate decreases 34 % under MVG control.

Figure 17.

The evolution of vorticity thickness versus streamwise positions, where the $x$ -axes and $y$ -axes are normalised by $H = 50$  mm: (a) FSML, (b) CSML.

After the SML develops into region II, the growth rate of the vorticity thickness increases significantly. For the FSML, $\mathrm{d}\delta _\omega /\mathrm{d}x|_{\mathit{FSML}}^{\mathit{II}} = 0.0178$ , which is 3.18 times $\mathrm{d}\delta _\omega /\mathrm{d}x|_{\mathit{FSML}}^{\mathit{I}}$ ; for the CSML, $\mathrm{d}\delta _\omega /\mathrm{d}x|_{\mathit{CSML}}^{\mathit{II}} = 0.0153$ , which is 4.14 times $\mathrm{d}\delta _\omega /\mathrm{d}x|_{\mathit{CSML}}^{\mathit{I}}$ . The pronounced increase in the growth rate of the SML in region II can be attributed to the interaction between the high-speed and low-speed layers (figure 11). The SML grows rapidly under the influence of the K–H instability. Meanwhile, the growth rate of the CSML is 14 % lower than that of the FSML. Overall, the growth rate of the CSML is lower than that of the FSML, indicating that the spatial growth rate of the CSML (dominated by streamwise vortices) is lower than that of the FSML (dominated by spanwise vortices).

Using (1.2), along with the velocity ratio and density ratio of the FSML, the growth rate of the incompressible mixing layer was calculated as $\delta '^{\mathit{in}}= 0.0186$ . This value was used to normalise the numerical results of the growth rates, which are presented in figure 18. To validate the numerical results, the growth rate data in figure 5 are also marked, where $\delta _{\omega }\approx 0.17\delta _{I,\mathit{rms}}$ under current weakly compressible conditions. Although the growth rates obtained from the experiment and simulation are not fully consistent, the trend of MVG-induced growth rate suppression is the same. For comparison, a series of previous experimental and simulation results (Papamoschou & Roshko Reference Papamoschou and Roshko1988; Elliott & Samimy Reference Elliott and Samimy1990; Goebel & Dutton Reference Goebel and Dutton1991; Clemens & Mungal Reference Clemens and Mungal1992; Slessor et al. Reference Slessor, Bond and Dimotakis1998; Rossmann et al. Reference Rossmann, Mungal and Hanson2002; Fu & Li Reference Fu and Li2006; Barre & Bonnet Reference Barre and Bonnet2015; Kim et al. Reference Kim, Elliott and Dutton2020), as well as the growth rate prediction curve proposed by Dimotakis (Reference Dimotakis1991b ), are also shown in figure 18. The growth rate of the FSML is in good agreement with the predictions of Dimotakis (Reference Dimotakis1991b ). The growth rate of the CSML is significantly lower than that of the FSML.

Figure 18.

Growth rates of the SML in current and previous studies.

Figure 18. Long description

A scatter plot representing the relationship between the normalized vorticity thickness and the convective Mach number. The horizontal axis represents the convective Mach number, while the vertical axis represents the normalized vorticity thickness. The plot includes several data points from different studies, indicated by various symbols and colors. A dashed trend line is present, showing a general decrease in normalized vorticity thickness with increasing convective Mach number. The data points are clustered around the trend line, with some scatter, indicating the variability in the data.

Numerous studies have confirmed that the growth rate of the SML decreases with increasing $M_{ {C}}$ , primarily due to the reduction of the TKE production term (Sarkar Reference Sarkar1995; Vreman et al. Reference Vreman, Sandham and Luo1996; Pantano & Sarkar Reference Pantano and Sarkar2002; Kim et al. Reference Kim, Elliott and Dutton2020; Wang et al. Reference Wang, Wang, Lei and Feng2022). Specifically, Vreman et al. (Reference Vreman, Sandham and Luo1996) established a direct relationship between the growth rate of momentum thickness and the integral of the TKE production term, indicating that their reduction is equivalent. In general, a high growth rate is accompanied by strong TKE, which increases scalar transport and mixing efficiency, thereby enhancing the SML’s growth rate. However, it remains unclear whether the reduction in the TKE production term is responsible for inhibiting the growth rate of the SML dominated by streamwise vortices. Next, we will further investigate the reasons for this inhibition from the perspectives of mean flow characteristics and the TKE transport equation.

4.4. Mean flow characteristics

4.4.1. Velocity profile

Figure 19 depicts the evolution of the average velocity profile of the SML along the streamwise direction. The solid line represents the FSML, the dash-dotted line indicates the tip of the CSML, and the dashed line denotes the notch of the CSML. Due to the presence of the splitter, a momentum deficit region ( $\bar U - U_1\lt 0$ ) inevitably appears in the SML. As expected, this momentum deficit region, which stems from the boundary layer stripped from the splitter, persists on the low-speed side of the SML (Bell & Mehta Reference Bell and Mehta1990; Goebel et al. Reference Goebel, Dutton, Krier and Renie1990; Mehta Reference Mehta1991). Goebel et al. (Reference Goebel, Dutton, Krier and Renie1990) attributed this to the smaller velocity difference between the deficit region and the low-speed stream, causing the deficit on the low-speed side of the SML to decay more slowly. This region gradually diminishes as $x$ increases, and completely disappears when $x\geqslant 12H$ , indicating that the SML entered the self-similar stage. It is evident that the reduction in momentum deficit is faster in the FSML.

Figure 19.

The evolution of the average velocity profile of the SML along the streamwise direction, where the $y$ -axes are normalised by the initial displacement thickness $\delta _0^*$ : (a) $x = H$ , (b) $x = 2H$ , (c) $x = 3H$ , (d) $x = 4H$ , (e) $x = 6H$ , (f) $x = 12H$ .

The average velocity profile exhibits significant changes under MVG control. At $x = H$ , the momentum deficit behind the MVGs, especially the notch, decreases markedly. This is because the counter-rotating streamwise vortices induced by MVGs (Ye et al. Reference Ye, Schrijer and Scarano2016) enhance momentum and mass exchange between the low-speed and middle layers (figure 11), thereby increasing the velocity in the middle layer. At $x = 2H$ , the momentum deficit at the tip and notch becomes more consistent and is still weaker than that of the FSML. The velocity profiles at the tip and notch are nearly identical at $x \in [ {3H,12H} ]$ , indicating the coalescence of streamwise vortex pairs and a reduced spanwise variation in the velocity distribution. In addition, the momentum deficit of the CSML exceeds that of the FSML. In brief, the momentum deficit decays faster in the FSML. As $x$ increases to $12H$ , the flow field enters the self-similar region, where the velocity profile of FSML develops faster – a finding consistent with the growth rate results presented in § 4.3.

It can be inferred that the spatial development of the CSML lags behind that of the FSML, because the velocity profiles of the CSML increasingly approach those of the FSML with increasing $x$ , and the spanwise vortices regain dominance in the downstream region of the CSML (see § 4.2). Building on the analysis in § 2.4, it can be inferred that the height and mounting position of MVGs can significantly affect this process.

4.4.2. Reynolds stress profile

Figure 20 depicts the development of every component of the Reynolds stress along the streamwise direction. The solid line represents the FSML, the dash-dotted line indicates the tip of the CSML, and the dashed line denotes the notch of the CSML. Before the FSML enters the self-similar stage, $\langle {u^{\prime }u^{\prime }} \rangle$ and $\langle {v^{\prime }v^{\prime }} \rangle$ play a dominant role, which facilitates the increase of the centreline velocity and the growth of the thickness in the normal direction.

Figure 20.

The evolution of the Reynolds stress profile of the SML along the streamwise direction, where the $x$ -axes are normalised by the initial displacement thickness $\delta _0^*$ : (a) $x = H$ , (b) $x = 2H$ , (c) $x = 3H$ , (d) $x = 4H$ , (e) $x = 6H$ , (f) $x = 12H$ . The numbers i, ii, iii and iv in the labels indicate $\langle u^{\prime }u^{\prime }\rangle /(\Delta U)^2$ , $\langle v^{\prime }v^{\prime }\rangle /(\Delta U)^2$ , $\langle w^{\prime }w^{\prime }\rangle /(\Delta U)^2$ and $\langle u^{\prime }v^{\prime }\rangle /(\Delta U)^2$ , respectively.

Figure 20. Long description

The image contains six rows and four columns of line graphs, each labeled with a combination of letters and numbers. Panel (ai) to (aiv) in the first row show the Reynolds stress profile for one condition, while subsequent rows (bi) to (fiv) represent different conditions. Each graph has the y-axis labeled as y/δ0* and the x-axis labeled with different Reynolds stress components normalized by the initial displacement thickness. The graphs compare three different methods: FSML (red solid line), CSML tip (yellow dashed line), and CSML notch (blue dashed line). The trends in the graphs show how the Reynolds stress profiles evolve along the streamwise direction under different conditions and methods.

As for the CSML, Reynolds stress has undergone remarkable changes. At $x=H$ , the magnitude of $\langle v^{\prime }v^{\prime } \rangle$ decreases, while the magnitude of $\langle {w^{\prime }w^{\prime }}\rangle$ at the notch increases to 4.4 times that of the FSML. As shown in figure 15(biii), these trends result from the destabilisation and breakdown of streamwise vortices at $x=H$ , which enhances spanwise velocity fluctuations. Additionally, under MVG control, the Reynolds shear stress $\langle {u^{\prime }v^{\prime }}\rangle$ differs between the tip (mainly negative) and the notch (positive). This is because the counter-rotating vortex pairs induced by the MVGs upwash at the tip, and downwash at the notch. At $x=2H$ , following the breakdown of streamwise vortices, the Reynolds stresses in the CSML are significantly reduced, and the differences between the tip and notch diminish. Near $x=3H$ , the peak value of $\langle {u^{\prime }u^{\prime }}\rangle$ abruptly increases to approximately twice that of the FSML, which is attributed to the fact that the streamwise vortex bands behind MVGs coalesce into large-scale spanwise vortices, as shown in figure 15(biv). When $x\gt 4H$ , the Reynolds stress distributions at the tip and notch converge. Notably, $\langle {v^{\prime }v^{\prime }}\rangle$ is reduced at all streamwise locations, thereby suppressing the growth of the SML’s thickness along the $y$ -axis. Meanwhile, $\langle {w^{\prime }w^{\prime }}\rangle$ and $\langle {u^{\prime }v^{\prime }}\rangle$ are also inhibited to varying degrees by the MVGs.

4.5. The TKE transport

Returning to the question raised at the end of § 4.3, to clarify whether the reduction in the TKE production term is responsible for the inhibited growth rate of the CSML, we examine the TKE transport equation (Pantano & Sarkar Reference Pantano and Sarkar2002; Wang et al. 2022; Wang et al. 2025) as given by

(4.5) \begin{equation} \frac {\partial \overline {K}}{\partial t}+\frac {\partial (\overline {K}\tilde {u}_k )}{\partial x_k}=\overline {T}+\overline {P}+\overline {\epsilon }+\overline {\varPhi }+\overline {\varSigma }, \end{equation}

where $K= (\rho u_i^{\prime \prime }u_i^{\prime \prime } )/2$ is the TKE. The two terms on the left-hand side of (4.5) represent the temporal variation term and the convective term, respectively. The terms on the right-hand side are the transport terms (TT), including the TKE production term ( $P$ ), viscous dissipation term ( $\epsilon$ ), spatial diffusion term ( $T$ ), pressure-dilatation term ( $\varPhi$ ), and mass flux coupling term ( $\varSigma$ ). The TT have clear physical meanings, and serve as the primary mechanisms for energy exchange among mean kinetic energy, TKE and internal energy (Li et al. Reference Li, Huang, Yan, Li and Liao2021), which helps to reveal the differences in TKE exchange patterns within the FSML and CSML.

The TKE production term originates from the interaction between the time-averaged velocity gradient and the Reynolds stress, which characterises the rate of energy transferred from the mean flow to turbulent fluctuations via shear stress, and serves as the primary driving force sustaining turbulence. The viscous dissipation term represents the rate at which TKE fluctuations are converted into thermal energy via molecular viscosity, leading to irreversible energy loss in turbulence. The spatial diffusion term transports TKE from regions with high turbulence intensity to low turbulence intensity, thereby homogenising the spatial distribution of TKE. The pressure-dilatation term denotes the spatial transport of TKE induced by the coupling between turbulent pressure fluctuations and the gradient of velocity fluctuations. Pantano & Sarkar (Reference Pantano and Sarkar2002) demonstrated that the magnitude of $\varSigma$ is negligible compared with the other TT, even at large $M_{ {C}}$ . Therefore, the mass flux coupling term is not considered in the present study.

The TT are defined by the following equations (Wang et al. Reference Wang, Wang, Lei and Feng2022):

(4.6) \begin{align} \overline {T}=\overline {T}_t+\overline {T}_p+\overline {T}_v, \\[-28pt] \nonumber \end{align}

(4.7) \begin{align} \overline {T}_t=-\frac {1}{2}\frac {\partial \overline {\rho u_i^{\prime \prime }u_i^{\prime \prime }u_k^{\prime \prime }}}{\partial x_k},\quad\overline {T}_p=-\frac {\partial \overline {p^{\prime }u_i^{\prime \prime }}}{\partial x_k}\delta _{ik},\quad\overline {T}_v=\frac {1}{\textit {Re}}\frac {\partial \overline {\sigma _{ik}^{\prime }u_i^{\prime \prime }}}{\partial x_k}, \\[-28pt] \nonumber \end{align}

(4.8) \begin{align} \overline {P}=-R_{ik}\frac {\partial \tilde {u}_i}{\partial x_k}, \\[-28pt] \nonumber \end{align}

(4.9) \begin{align} \overline {\epsilon }=-\frac {1}{\textit {Re}}\overline {\sigma _{ik}^{\prime }\frac {\partial u_i^{\prime \prime }}{\partial x_k}}, \\[-28pt] \nonumber \end{align}

(4.10) \begin{align} \overline {\varPhi }=\overline {p^{\prime }\frac {\partial u_i^{\prime \prime }}{\partial x_i}}, \\[-28pt] \nonumber \end{align}

(4.11) \begin{align} \overline {\varSigma }=\overline {u_i^{\prime \prime }}\left (\frac {1}{\textit {Re}}\frac {\partial \overline {\sigma }_{ik}}{\partial x_k}-\frac {\partial \overline {p}}{\partial x_i}\right )\!, \\[0pt] \nonumber \end{align}

where $\bar {\phi }$ is the Reynolds average over 3000 snapshots sampled at 1 ${\unicode[Arial]{x03BC}}$ s intervals, and $\tilde {\phi }=\overline {\rho \phi }/\overline {\rho }$ is the Farve average of $\phi$ . The Reynolds fluctuations and Favre fluctuations are $\phi ^{\prime }=\phi -\overline {\phi }$ and $\phi ^{\prime \prime }=\phi -\tilde {\phi }$ , respectively.

The turbulent stress tensoris defined by

(4.12) \begin{equation} R_{ij}=\overline {\rho }\,\widetilde {u_i^{\prime \prime }u_{\!j}^{\prime \prime }}=\overline {\rho u_i^{\prime \prime }u_{\!j}^{\prime \prime }}. \end{equation}

Figure 21 shows the TT at different streamwise locations of the SML, with the dominant distribution region of the TT highlighted by grey shading. Consistent with previous studies, $P$ , $\epsilon$ , $T$ and $\varPhi$ reach their maxima near the centre of the SML and decrease towards the high-speed and low-speed regions. This is because turbulent diffusion transfers TKE from the centre to the edges of the SML (Rogers & Moser Reference Rogers and Moser1994; Pantano & Sarkar Reference Pantano and Sarkar2002; Wang et al. Reference Wang, Xia, Zhao, Wang and Liu2025). Furthermore, the magnitudes of the TT decrease with increasing streamwise distance. For example, the peak value of $P$ at $x = 8H$ is reduced by 96–98 % relative to that at $x = H$ . This occurs for two reasons: first, large-scale vortices gradually break down into small-scale vortices due to continuous viscous dissipation; second, the increase in SML velocity weakens the shear stress between the SML and the free streams, ultimately leading to a reduction in the TT.

Figure 21.

The evolution of TKE terms of the SML along the streamwise direction, where the $y$ -axes are normalised by the initial displacement thickness $\delta _\omega ^0$ , and the $x$ -axes are normalised by $(\Delta U)^3/\delta _\omega ^0$ : (a) $x = H$ , (b) $x = 2H$ , (c) $x = 4H$ , (d) $x = 8H$ . The numbers i, ii, iii in the labels indicate FSML, CSML tip and CSML notch, respectively.

During the initial development stage of the FSML ( $x=H$ ), as shown in figure 21(a), the profiles of $P$ , $\epsilon$ , $\varPhi$ and $T$ exhibit double peaks. This phenomenon arises because the initial development of the FSML is dominated by spanwise vortices, and both the high- and low-speed layers undergo strong shear interactions with the middle layer (see the enlarged view of figure 11 a), thereby producing two peaks in the TT, as indicated by the two dash-dotted lines $y/\delta _\omega ^0\approx -0.17$ and $y/\delta _\omega ^0\approx 0.10$ in figure 21(a). After applying MVG control, the profiles of $P$ , $\epsilon$ , $\varPhi$ and $T$ change from double peaks to a single peak. At the tip (figure 21 aii), the single peak appears at $y/\delta _\omega ^0\approx 0.36$ ; at the notch (figure 21 aiii), the peaks of $P$ and $T$ occur at $y/\delta _\omega ^0\approx 0.22$ , while the peaks of $\epsilon$ and $\varPhi$ appear at $y/\delta _\omega ^0\approx 0.15$ . As shown in figures 11(b) and 11(c), the lower peak locations of the TKE TT at the notch are generated from the lower position of the middle layer. These phenomena indicate that the streamwise vortices induced by the MVGs redistribute TKE within the SML. Since the MVGs mounted on the low-speed side generate regular streamwise vortex bands, the dominant vortical structures transition from spanwise to streamwise vortices. Consequently, shear process is concentrated around the streamwise vortex bands, whereas TKE transport in the high-speed region of the SML is suppressed. Furthermore, compared with the FSML, the peak value of $P$ increases by $51\,\%$ at the CSML’s tip, and decreases by $24\,\%$ at the CSML’s notch. This disparity reflects the spatially non-uniform TKE transport capacity of the streamwise vortex bands.

At $x=2H$ , after the initial development stage, the double peaks of $P$ , $\epsilon$ , $\varPhi$ and $T$ in the FSML collapse into a single peak at $y/\delta _\omega ^0\approx -0.20$ (figure 21 bi), which is attributed to the stronger shear stress at the high-speed side. Under MVG control, as shown in figures 21(bii) and 21(biii), the magnitudes of the TT in the CSML are significantly reduced, and the distribution width of the TT in the transverse direction is markedly narrowed. This trend echoes the changes in the Reynolds stress variations shown in figure 20(b). Collectively, these observations confirm that the TKE production, transport, dissipation and pressure-dilatation are strongly suppressed, which can be attributed to the breakdown of large-scale K–H vortices. Furthermore, the discrepancy in the profiles of TT between the tip and the notch diminishes. As shown in figure 12(d), this reduction is attributed to the merging of the streamwise vortex bands, which enhances the spanwise homogeneity of the flow field.

As the SML develops further downstream ( $x\geqslant 4H$ ), the flow field of CSML becomes uniform in the spanwise direction and the spanwise vortices take over as the dominant structures, as established in §§ 4.1 and 4.2. Unsurprisingly, the discrepancy in the profiles of the TT between the tip and the notch is largely eliminated, as illustrated in figures 21(c) and 21(d). At $x=4H$ , the peak values of TT in the CSML exceed those in the FSML. Specifically, the peak values of $P$ , $\epsilon$ , $\varPhi$ , and $T$ are $338\,\%$ , $313\,\%$ , $518\,\%$ , and $236\,\%$ higher, respectively, than those of the FSML. This indicates that the rates of TKE production, transport, dissipation, and pressure-dilatation are substantially elevated at the centre of the CSML. At $x=8H$ , compared with the FSML, the peak values of $P$ , $T$ and $\varPhi$ in the CSML increase by $46\,\%$ , $243\,\%$ and $14\,\%$ , respectively, whereas the peak of $\epsilon$ decreases by $16\,\%$ . This indicates that the difference in TKE transport capacity between the centres of the CSML and FSML has reduced. The peak values of TT in the CSML exceed those in the FSML because within $x\in [4H,8H]$ , the SML has not yet reached the self-similar region, and the momentum deficit in the CSML is more pronounced (figure 19). Nevertheless, the profiles of the TT are still narrowed in the spanwise direction of the CSML.

In summary, under the modulation of the MVG-induced streamwise vortex bands, the distributions of TKE production, transport, dissipation and pressure-dilatation are narrowed in the transverse direction. This suppresses the outward transport of TKE towards the high- and low-speed sides, ultimately reducing the growth rate of the SML.