Computational Analysis of Tandem Micro-Vortex Generators for Supersonic Boundary Layer Flow Control

Abstract

Micro-vortex generators (MVGs) are widely utilized as passive devices to control flow separation in supersonic boundary layers by generating ring-like vortices that mitigate shock-induced effects. This study employs large eddy simulation (LES) to investigate the flow structures in a supersonic boundary layer (Mach 2.5, Re = 5760) controlled by two MVGs installed in tandem, with spacings varying from 11.75 h to 18.75 h (h = MVG height), alongside a single-MVG reference case. A fifth-order WENO scheme and third-order TVD Runge–Kutta method were used to solve the unfiltered Navier–Stokes equations, with the Liutex method applied to visualize vortex structures. Results reveal that tandem MVGs produce complex vortex interactions, with spanwise and streamwise vortices merging extensively, leading to a significant reduction in vortex intensity due to mutual cancellation. A momentum deficit forms behind the second MVG, weakening that from the first, while the boundary layer energy thickness doubles compared to the single-MVG case, indicating increased energy loss. Streamwise vorticity distributions and instantaneous streamlines highlight intensified interactions with closer spacings, yet this complexity diminishes overall flow control effectiveness. Contrary to expectations, the tandem configuration does not enhance boundary layer control but instead weakens it, as evidenced by reduced vortex strength and amplified energy dissipation. These findings underscore a critical trade-off in tandem MVG deployment, suggesting that while vortex interactions enrich flow complexity, they may compromise the intended control benefits in supersonic flows, with implications for optimizing MVG arrangements in practical applications.

1. Introduction

Shock-wave boundary-layer interactions (SWBLI) pose a formidable barrier to optimizing the performance of aircraft and propulsion systems at supersonic speeds, yielding a cascade of undesirable consequences. These encompass degraded flow quality arising from extensive flow separation, which precipitates unstable and distorted flow patterns, elevated aerodynamic drag, engine unstart, pronounced total pressure losses, and intensified wall heating. These effects not only undermine operational efficiency but also present enduring design challenges that continue to engage the aerospace research community. To address these persistent issues, micro-vortex generators (MVGs) have emerged as a favored approach for controlling flow separation. These small, passive devices, typically occupying 10% to 80% of the boundary layer thickness, are significantly more compact than conventional vortex generators. Despite their modest stature, MVGs effectively alleviate flow separation with a negligible drag penalty, offering practical benefits such as ease of installation, lower drag, non-intrusiveness, and remarkable durability owing to their miniature size and passive operation.

The inception of conventional vortex generators dates to the 1980s, when they were introduced as high-lift devices to enhance aircraft flap performance. In subsonic regimes, their ability to improve aerodynamic efficiency has been well-established. More recently, their application has transitioned to supersonic flows, targeting the intricate challenges of SWBLI, particularly flow separation. Contemporary research has significantly advanced the understanding and application of MVGs in high-speed contexts. A 2022 study investigated the combined use of MVGs with secondary recirculation jets to control SWBLI at Mach 2.9, employing Reynolds-averaged Navier–Stokes (RANS) simulations to achieve up to an 88.96% reduction in separation bubble volume [1]. Similarly, a 2023 investigation into noise reduction in supersonic nozzles demonstrated that dual MVG arrays could reshape shock-cell structures, suggesting potential for broader flow interaction benefits [2]. A 2021 study on supersonic jet noise suppression further revealed MVG sensitivity to placement, weakening downstream shock cells and enhancing flow control efficacy [3]. Additional recent efforts include a 2022 analysis of streamwise MVG positioning in supersonic flow, showing its influence on separation control at Mach 2.5 [4], and a 2020 examination of tandem MVG effectiveness in high-speed flows, providing preliminary insights into multi-device interactions [5]. These studies build on earlier reviews, such as Lu et al.’s 2012 synthesis of MVG effects in high-speed flows [6] and Zhang et al.’s 2015 exploration of vortex trajectories with a dissymmetric micro-ramp at Mach 2.5 [7].

Foundational research has provided a robust platform for these advancements. Anderson et al. delivered standardized analyses of MVG performance [8], while Babinsky et al. conducted meticulous experiments across various MVG designs to evaluate their control effects [9]. Experimental investigations have further enriched this domain: Sun’s particle image velocimetry (PIV) study elucidated the three-dimensional instantaneous flow behind a micro-ramp in a supersonic boundary layer, capturing intricate wake dynamics [10]. Wang’s empirical work, employing nano-tracer planar laser scattering (NPLS) and PIV in a low-noise supersonic wind tunnel, furnished detailed flow field data [11]. Computational efforts have paralleled these experiments, utilizing diverse methodologies to deepen insights and inform design. High-order simulations by Rizzetta and Visbal addressed compression corner flows via implicit large-eddy simulation (LES) [12], while Kaenal et al. applied LES to ramp flows using an approximate deconvolution model [13]. Li and Liu identified large-scale vortex rings behind MVGs through implicit LES with a fifth-order WENO scheme [14], and Xue et al. contrasted the simpler wake structures of micro-vanes with the complex, ring-like vortex trains of micro-ramps [15]. Wang et al.’s LES at Mach 2.7 explored wake organization downstream of ramp-type MVGs, highlighting their fluid redistribution role [16]. Recent studies, such as Wu et al.’s 2022 investigation of MVG streamwise location effects on SWBLI [4] and Nilavarasan et al.’s 2022 work on ramped vanes for flow separation control at Mach 2.0 [17], further underscore the evolving utility of MVGs. Wang investigated the control effect of a vane-shaped micro vortex generator (VG) on the inception and development of tip vortex cavitation [18]. In our previous study [19,20,21], MVG-controlled supersonic boundary layers under different Mach numbers (1.5–4.5) are simulated. Detailed vortex structures were captured and investigated. Our LES results confirmed that SWBLI with ring-like vortices generated by the MVG influence the ramp shock wave intensely. A V-shaped separation zone was discovered on the wall boundary. It was also confirmed that the interaction between ring-like vortices and ramp shock wave is the mechanism for the formation of this V-shaped separation zone and the flow separation reduction at the ramp corner. The vortex structure, especially the ring-like vortices generated by MVG, is the key to the supersonic boundary layer control.

Despite this extensive body of work, a critical limitation persists: most studies have focused on the flow field of a single MVG, or a single row of MVGs arranged laterally. In practical engineering scenarios, multi-row MVG configurations are commonplace to address larger-scale flow control demands, yet the dynamics of tandem arrangements—where one MVG follows another in the streamwise direction—remain underexplored. Recent literature offers limited insights into such setups. Lu et al.’s 2010 experimental study of near-wake interactions in an MVG array measured flow between two MVGs but did not address tandem configurations explicitly [22]. A 2017 study of supersonic flow over a backward-facing step with three MVGs provided valuable data, yet their arrangement was not specified as tandem [23]. Broader investigations, such as Sajeev et al.’s 2020 analysis of tandem MVGs in high-speed flows [5], suggest interaction effects but lack the specificity of sequential placement. Similarly, Giepman et al.’s 2016 study on wake properties of micro-ramps at varying Mach and Reynolds numbers [24] and Sun et al.’s 2012 PIV analysis of three-dimensional flow past a micro-ramp [25] focus on single devices or arrays, not tandem setups. This dearth of research on tandem MVG configurations highlights a significant gap. Understanding how sequential MVGs interact, influence flow structures, and mitigate SWBLI is essential for optimizing their design and deployment in real-world applications.

This study confronts this gap by conducting a comprehensive numerical simulation of supersonic boundary layer flow controlled by two MVGs installed in tandem. By examining the flow structures and interactions induced by this configuration, we aim to clarify the mechanisms through which tandem MVGs modify the boundary layer, reduce separation, and alter shock-wave dynamics. This investigation extends the proven efficacy of single MVGs to explore their synergistic effects in a tandem arrangement, offering novel insights into their potential for enhanced flow control. Such knowledge is pivotal for advancing MVG array designs, enabling more effective strategies to counteract SWBLI’s adverse effects in high-speed aerospace systems.

This paper is organized as follows: Section 2 delineates the case setup and numerical methods employed; Section 3 presents the numerical results with in-depth discussions; and Section 4 provides concluding remarks, outlining implications and future research directions.

2. Case Setup and Numerical Methods

2.1. Case Setup

The simulation domain and MVG configuration are illustrated in Figure 1. In this study, the trailing edge of the MVG is inclined at a 70° angle to simplify grid generation [19], while other dimensions align with those reported in the experiments by Babinsky et al. [9]. A high-quality, orthogonal, wall-refined mesh was constructed, as depicted in Figure 2. More details of the mesh can be found in our previous work [19]. Consistent with the experimental findings of Babinsky et al. [9] and our prior simulations, the inlet boundary layer thickness, ${\delta }_0$, is set to $2 \mathrm{h}$, where h denotes the MVG height. LES were conducted for three cases (Case 1 to Case 3) featuring tandem MVGs in a supersonic boundary layer, with varying separations between the devices. The spacing, $∆Z$, ranges from $11.75 \mathrm{h}$ to $18.75 \mathrm{h}$, selected to promote significant interaction between the vortex structures generated by the two MVGs. For reference, an additional case (Case 0) with a single MVG was simulated. Details of the spacing for each case are provided in

Table 1. Configuration of All Cases.

	Case 0	Case 1	Case 2	Case 3
$∆Z$	-	$11.75 \mathrm{h}$	$15.25 \mathrm{h}$	$18.75 \mathrm{h}$
$n_{streamwise}$	860	860	930	1000

. The mesh dimensions in the spanwise (X) and wall-normal (Y) directions are fixed at $n_{spanwise}\times n_{normal}=149\times 342$, while streamwise grid sizes ($n_{streamwise}$) vary by case, as specified in Table 1.

2.2. Numerical Methods

To clarify the underlying mechanisms and achieve a thorough understanding of MVG dynamics, the use of high-order direct numerical simulation (DNS) and LES is essential [12]. The governing equations are expressed in non-dimensional form using the conservative formulation of the Navier–Stokes equations, as detailed below:

$${\displaystyle \frac{\partial \mathit{Q}}{\partial t}}+{\displaystyle \frac{\partial \mathit{F}}{\partial x}}+{\displaystyle \frac{\partial \mathit{G}}{\partial y}}+{\displaystyle \frac{\partial \mathit{H}}{\partial z}}={\displaystyle \frac{1}{Re}}\left({\displaystyle \frac{\partial {\mathit{F}}_{\mathit{v}}}{\partial x}}+{\displaystyle \frac{\partial {\mathit{G}}_v}{\partial y}}+{\displaystyle \frac{\partial {\mathit{H}}_v}{\partial z}}\right),$$

(1)

where the vector of conserved quantities $\mathit{Q}$, inviscid flux vector $\mathit{E},\mathit{F}$, and $\mathit{G}$, and viscous flux vector ${\mathit{E}}_v, {\mathit{F}}_v$ and ${\mathit{G}}_v$ are

$$\mathit{Q}=\left(\begin{array}{c}\rho \\ \rho u\\ \rho v\\ \rho w\\ e\end{array}\right),\mathit{F}=\left(\begin{array}{c}\rho u\\ \rho u^2+p\\ \rho uv\\ \rho uw\\ \left(e+p\right)u\end{array}\right), \mathit{G}=\left(\begin{array}{c}\rho v\\ \rho uv\\ \rho v^2+p\\ \rho vw\\ \left(e+p\right)v\end{array}\right), \mathit{H}=\left(\begin{array}{c}\rho w\\ \rho uw\\ \rho vw\\ \rho w^2+p\\ \left(e+p\right)w\end{array}\right)$$

(2)

$${\mathit{F}}_v=\left(\begin{array}{c}0\\ {\sigma }_{xx}\\ {\sigma }_{xy}\\ {\sigma }_{xz}\\ (u{\sigma }_{xx}+v{\sigma }_{xy}+w{\sigma }_{xz}+{\displaystyle \frac{1}{\left(\gamma -1\right)Pr{M}_{\infty }^2}}k\left(T\right){\displaystyle \frac{\partial T}{\partial x}}\end{array}\right),$$

(3)

$${\mathit{G}}_v=\left(\begin{array}{c}0\\ {\sigma }_{xy}\\ {\sigma }_{yy}\\ {\sigma }_{yz}\\ (u{\sigma }_{xy}+v{\sigma }_{yy}+w{\sigma }_{yz}+{\displaystyle \frac{1}{\left(\gamma -1\right)Pr{M}_{\infty }^2}}k\left(T\right){\displaystyle \frac{\partial T}{\partial y}}\end{array}\right),$$

(4)

$${\mathit{H}}_v=\left(\begin{array}{c}0\\ {\sigma }_{xz}\\ {\sigma }_{yz}\\ {\sigma }_{zz}\\ (u{\sigma }_{xz}+v{\sigma }_{yz}+w{\sigma }_{zz}+{\displaystyle \frac{1}{\left(\gamma -1\right)Pr{M}_{\infty }^2}}k\left(T\right){\displaystyle \frac{\partial T}{\partial z}}\end{array}\right).$$

(5)

The components of viscous stress are:

$$\begin{array}{c}{\sigma }_{xx}={\displaystyle \frac{2}{3}}\mu \left(T\right)\left(2{\displaystyle \frac{\partial u}{\partial x}}-{\displaystyle \frac{\partial v}{\partial y}}-{\displaystyle \frac{\partial w}{\partial z}}\right), {\sigma }_{yy}={\displaystyle \frac{2}{3}}\mu \left(T\right)\left(-{\displaystyle \frac{\partial u}{\partial x}}+2{\displaystyle \frac{\partial v}{\partial y}}-{\displaystyle \frac{\partial w}{\partial z}}\right),\\ {\sigma }_{zz}={\displaystyle \frac{2}{3}}\mu \left(T\right)\left(-{\displaystyle \frac{\partial u}{\partial x}}-{\displaystyle \frac{\partial v}{\partial y}}+2{\displaystyle \frac{\partial w}{\partial z}}\right), {\sigma }_{xy}=\mu \left(T\right)\left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right),\\ {\sigma }_{xz}=\mu \left(T\right)\left({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}\right), {\sigma }_{yz}=\mu \left(T\right)\left({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}}\right).\end{array}$$

(6)

We can write the governing equations in curvilinear coordinates as

$${\displaystyle \frac{\partial \widehat{\mathit{Q}}}{\partial t}}+{\displaystyle \frac{\partial \widehat{\mathit{F}}}{\partial \xi }}+{\displaystyle \frac{\partial \widehat{\mathit{G}}}{\partial \eta }}+{\displaystyle \frac{\partial \widehat{\mathit{H}}}{\partial \zeta }}={\displaystyle \frac{1}{Re}}\left({\displaystyle \frac{\partial {\widehat{\mathit{F}}}_v}{\partial \xi }}+{\displaystyle \frac{\partial {\widehat{\mathit{G}}}_v}{\partial \eta }}+{\displaystyle \frac{\partial {\widehat{\mathit{H}}}_v}{\partial \zeta }}\right)$$

(7)

where

$$\begin{array}{c}\widehat{\mathit{Q}}={\displaystyle \frac{\mathit{Q}}{J}}, \widehat{\mathit{F}}={\displaystyle \frac{{\xi }_x\mathit{F}+{\xi }_y\mathit{G}+{\xi }_z\mathit{H}}{J}}, \widehat{\mathit{G}}={\displaystyle \frac{{\eta }_x\mathit{F}+{\eta }_y\mathit{G}+{\eta }_z\mathit{H}}{J}}, \widehat{\mathit{H}}={\displaystyle \frac{{\zeta }_x\mathit{F}+{\zeta }_y\mathit{G}+{\zeta }_z\mathit{H}}{J}},\\ {\widehat{\mathit{F}}}_v={\displaystyle \frac{{\xi }_x{\mathit{F}}_v+{\xi }_y{\mathit{G}}_v+{\xi }_z{\mathit{H}}_v}{J}}, {\widehat{\mathit{G}}}_v={\displaystyle \frac{{\eta }_x{\mathit{F}}_v+{\eta }_y{\mathit{G}}_v+{\eta }_z{\mathit{H}}_v}{J}}, {\widehat{\mathit{H}}}_v={\displaystyle \frac{{\zeta }_x{\mathit{F}}_v+{\zeta }_y{\mathit{G}}_v+{\zeta }_z{\mathit{H}}_v}{J}}.\end{array}$$

(8)

The Jacobian $J$ of the coordinate transformation between the curvilinear $\left(\xi ,\eta ,\zeta \right)$ and Cartesian $\left(x,y,z\right)$ frames is:

$$J={\displaystyle \frac{1}{\left|\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& x_{\xi }\end{array}& \begin{array}{cc}0& 0\\ x_{\eta }& x_{\zeta }\end{array}\\ \begin{array}{cc}0& y_{\xi }\\ 0& z_{\xi }\end{array}& \begin{array}{cc}{y}_{\eta }& y_{\zeta }\\ y_{\eta }& z_{\zeta }\end{array}\end{array}\right|}},$$

(9)

and

$$\left(\begin{array}{ccc}{\xi }_x& {\xi }_y& {\xi }_z\\ {\eta }_x& {\eta }_y& {\eta }_z\\ {\zeta }_x& {\zeta }_y& {\zeta }_z\end{array}\right)=J\left(\begin{array}{ccc}{y}_{\eta }{z}_{\zeta }-y_{\zeta }{z}_{\eta }& z_{\eta }{x}_{\zeta }-z_{\zeta }{x}_{\eta }& x_{\eta }{y}_{\zeta }-x_{\zeta }{y}_{\eta }\\ y_{\zeta }{z}_{\xi }-y_{\xi }{z}_{\zeta }& z_{\zeta }{x}_{\xi }-z_{\xi }{x}_{\zeta }& x_{\zeta }{y}_{\xi }-x_{\xi }{y}_{\zeta }\\ y_{\xi }{z}_{\eta }-y_{\eta }{z}_{\zeta }& z_{\xi }{x}_{\eta }-z_{\eta }{x}_{\zeta }& x_{\xi }{y}_{\eta }-x_{\eta }{y}_{\zeta }\end{array}\right).$$

(10)

The reference values for length, density, velocity, temperature, and pressure are ${\delta }_{in}, {\rho }_{\infty }, U_{\infty }, T_{\infty }$ and ${\rho }_{\infty }{U}_{\infty }^2$ respectively, where ${\delta }_{in}$ is the inflow displacement thickness. The Mach number $M_{\infty }$ and Reynolds number $Re$ are expressed as:

$$M_{\infty }={\displaystyle \frac{U_{\infty }}{\sqrt{\gamma R{T}_{\infty }}}}, Re={\displaystyle \frac{{\rho }_{\infty }{U}_{\infty }{\delta }_{in}}{{\mu }_{\infty }}},$$

(11)

where $R$ is the ideal gas constant, $\gamma$ the ratio of specific heats, and ${\mu }_{\infty }$ the viscosity. To complete the system, the ideal gas law will be applied as follows,

$$p=\rho RT.$$

(12)

In this study, the implicitly implemented LES is adopted for simulation, which used the intrinsic dissipation of the numerical method to dissipate the turbulent energy accumulated at the unresolved scales with high wave numbers [12,20]. Our LES approach employs the unfiltered Navier–Stokes equations, solved using a fifth-order bandwidth-optimized WENO scheme for spatial discretization and an explicit third-order TVD-type Runge-Kutta scheme for time marching. Simulations were performed at $M_{\infty }=2.5$ and $Re=5760$. At the wall boundary, adiabatic, zero-pressure-gradient, and no-slip conditions are imposed, while non-reflecting boundary conditions are applied at the upper boundary to prevent wave reflection. Periodic conditions are specified at the front and rear spanwise boundaries. To enhance the analysis of vortex interactions—particularly the ring-like vortices induced by the MVG—a laminar inlet profile is adopted. A non-reflecting boundary condition is also implemented at the outflow boundary to maintain solution stability. Our supersonic CFD solver was validated through a series of benchmark cases and the supersonic ramp flows. Our numerical results for the MVG-controlled boundary layer were also found to be consistent with the existing experimental and numerical results. The details of the numerical approaches and the validations/comparisons can be found in ref. [20].

3. Numerical Results

The vortex structures across all four cases are depicted in Figure 3. The length was scaled by the inlet boundary layer thickness, ${\delta }_0$. To effectively capture the complex vortex structures within MVG-controlled supersonic boundary layers, this study employs the newly developed Liutex vortex identification method [26], which can capture the vortex core precisely. An iso-surface of Liutex = 0.3 is utilized in Figure 3 to highlight larger-scale vortices while preserving their prominence. Shock waves ahead of the MVGs are visualized using iso-surfaces of the pressure gradient. The results reveal that both streamwise and spanwise vortices are generated simultaneously behind the two MVGs in tandem. The vortex structure produced by the upstream MVG remains largely unaffected by the compression shock wave originating from the downstream MVG’s leading edge yet exhibits significant interaction with the vortex structure from the second MVG upon reaching its upper region. In contrast, in the single-MVG case (Case 0), only small-scale vortex structures emanating from the lower boundary layer form beneath the larger vortices generated by the MVG, with no discernible interaction between them.

The interaction between vortex structures induces some distortion in the vortex system behind the second MVG, with smaller spacings amplifying this effect, though the fundamental vortex generation pattern and mechanism remain unchanged. Figure 4 offers a detailed view of the vortex structures behind the second MVG in Case 1, contoured by streamwise velocity, where distinct ring-like vortices are evident around the momentum deficit within the circled region. Figure 5 presents the spanwise vorticity (the spanwise component of $\nabla \mathit{V})$ distribution on the central YZ plane (X = 0) in Case 1, with continuous, regularly spaced red spots marking the tops of these ring-like vortices. The ring-like vortices from the second MVG form over a short distance before interacting with the lower portion of the vortex structure produced by the first MVG, leading to a larger-scale interaction downstream and, eventually, the full merging of the two systems. Compared to the region downstream of the first MVG, this interaction generates a greater variety of vortex scales in the lower boundary layer behind the second MVG, resulting in a notably intricate vortex structure system.

Figure 6 illustrates the vorticity distribution across four cross-sectional planes in Case 1, with similar patterns observed in Cases 2 and 3. These planes are positioned at distances of $2 \mathrm{h}$, $4 \mathrm{h}$, $6 \mathrm{h}$, and $8 \mathrm{h}$ downstream from the trailing edge of the second MVG, where $\mathrm{h}$ is the MVG height. The figure reveals the evolution of vortex structures generated by the two MVGs, transitioning from independent entities near the second MVG to fully intertwined systems farther downstream. Figure 7 presents enlarged three-dimensional vortex structures behind the second MVG for Case 1, 2, and 3. The results indicate that increasing the spacing between the MVGs delays the interaction of their vortex structures to a more downstream location. This delay likely stems from the extended development of the first MVG’s vortex structure, which experiences a slight upward shift within the boundary layer over a longer distance.

Figure 8 depicts the prominent large-scale vortex structures in Case 1 obtained by increasing the Liutex value. The ring-like vortices generated by the second MVG vanish entirely due to their interaction with the upstream vortex system, and this intense interplay results in the formation of additional large vortex structures along the spanwise direction, causing rapid spanwise expansion of the overall vortex system. Figure 9 presents the instantaneous streamline distribution on two cross-sectional planes located 2 h and 4 h downstream from the trailing edge of the second MVG in Case 1. The flow field reveals the influence of two pairs of counter-rotating streamwise vortices within the vortex structure, with the central fluid being continuously drawn upward while fluid on either side is swept downward, amplifying the vortex interactions. Figure 10 illustrates the large-scale vortex structures across all four cases, visualized with a higher Liutex value and contoured by spanwise vorticity. In the absence of the second MVG’s influence (Case 0), the vortex structure from the first MVG develops with notable regularity. However, in the tandem configurations, decreasing the spacing between MVGs intensifies their mutual interaction, leading to a more pronounced cancellation of spanwise vorticity.

To investigate the influence of vortex structure interactions on boundary layer dynamics and flow control effectiveness, the time-averaged streamwise velocity distribution on the central plane for Case 1 is presented in Figure 11. A momentum deficit forms behind both MVGs; however, the deficit induced by the first MVG diminishes significantly upon reaching the second MVG’s position, with no notable enhancement in the combined deficit downstream. Figure 12 provides the energy thickness (a parameter that quantifies the distance from the surface where the boundary layer’s energy deficit is compensated by the main stream’s energy flux) distribution curves along the streamwise direction for all four cases, revealing that the addition of the second MVG doubles the overall energy loss compared to the single-MVG case. Nevertheless, variations in the spacing between the tandem MVGs exhibit only a minor impact on the energy thickness farther downstream.

To assess the vortex structure intensity following their interaction, the vorticity magnitude is integrated over cross-sections downstream of the second MVG. Figure 13 illustrates the dimensions and extent of these cross-sections within the flow field. To emphasize the large eddy structures produced by the MVGs, these cross-sections exclude the lower boundary layer and the spanwise margins of the flow domain. Figure 14 presents the streamwise distribution of integrated vorticity magnitude across all four cases. In the tandem MVG cases, the total vorticity within the cross-sections surges sharply after the second MVG’s vortex structure emerges (around $z=4 \mathrm{h}$), then declines rapidly following their interaction (around $z=5 \mathrm{h}$). Notably, farther downstream, the total vorticity falls below that of the single-MVG case, indicating a significant vortex cancellation process during the interaction of the vortex structures generated by the two MVGs.

As a discussion and summary of the numerical results, the tandem installation of two MVGs generates a highly complex vortex structure that governs the boundary layer arising from the interaction between the vortices produced by each MVG. This interaction leads to mutual cancellation, significantly reducing the overall vortex strength. Additionally, the energy thickness distribution indicates a doubling of boundary layer energy loss compared to a single MVG. Consequently, deploying two MVGs in tandem may not enhance control over the supersonic boundary layer but could instead diminish its effectiveness.

4. Conclusions

This study conducted a series of LES to examine supersonic boundary layer flow controlled by two MVGs installed in tandem. Across all cases, momentum deficits and ring-like vortices were consistently observed downstream of the second MVG. The spanwise and streamwise vortices generated by tandem MVGs underwent multiple merging events, resulting in a significant reduction in vortex intensity. This interaction, coupled with the momentum deficit behind the second MVG, doubled the kinetic energy loss within the boundary layer compared to a single-MVG configuration. Consequently, the tandem arrangement weakens the overall vortex structures, diminishing their effectiveness in controlling the supersonic boundary layer and suggesting a potential drawback to this configuration for flow control applications. In contrast, single MVG configurations, as discussed in the Introduction, typically exhibit stronger vortex structures, highlighting the unique challenges of tandem setups.