Direct numerical simulation of supersonic boundary layer transition induced by gap-type roughness

Abstract

The transition of the supersonic boundary layer induced by roughness is a highly intricate process. Gaining a profound understanding of the transition phenomena and mechanisms is crucial for accurate prediction and control. In this study, to delve into the flow mechanisms of a transition in a supersonic boundary layer induced by the medium gap-type roughness, direct numerical simulation is employed to capture and analyze the transition process. Research indicates that as the flow over the flat plate passes the gap, the spanwise convergence effect leads to the formation of both upper and lower counter-rotating vortex pairs. As the flow progresses, these counter-rotating vortex pairs in the central region exhibit attenuation, with streamwise vortices developing on both sides. At a certain downstream distance, the boundary layer becomes unstable, triggering the formation of streamwise vortex legs. These streamwise vortex legs undergo further evolution, transforming into hairpin vortices and leg-buffer vortices. The formation of the central low-speed zone downstream of the roughness element is mainly attributed to the lift-up effect of the low-speed flow propelled by the central counter-rotating vortex pairs. The low-speed streaks on both sides are primarily influenced by the streamwise vortices. Through a meticulous analysis of the turbulent kinetic energy distribution and its generation mechanisms during the transition phase, this study infers that the primary sources of turbulent kinetic energy are the hairpin vortices, leg-buffer vortices, and their consequent secondary vortices. Combined with modal analysis, the study further elucidates the generation and breakdown of hairpin and leg-buffer vortices.

1 Introduction

The investigation of the transition from a laminar to a turbulent boundary layer is one of the most intricate and enduring challenges in classical physics [1]. For instance, in the inlet flow of air-breathing supersonic vehicles (like the X-43A [2]), transition can be prematurely induced by attaching roughness elements to the front of the inlet. Such an approach is proved to be effective in mitigating, or even suppressing, local flow separation caused by shockwave/boundary layer interactions, thereby significantly enhancing start performance of the inlets. However, the intricate nature of high-speed flow transitions poses challenges in the design and implementation of these roughness elements [3]. Hence, it is critical to understand the mechanisms of supersonic boundary layer transition induced by roughness. This understanding is not only of academic significance but also vital for the precise prediction and effective control of transitional flows in engineering applications.

Currently, research on roughness-induced supersonic boundary layer transition has amassed abundant results, offering divergent explanations for the transition mechanisms instigated by different roughness types. The comprehensive reviews by Reda [4] and Yang et al. [5] have extensively summarized the research on high-speed flow transitions induced by surface roughness. These reviews provide detailed analyses of experimental data across various transitional flows. Klebanoff and Tidstrom [6] suggested, based on their study of transition induced by two-dimensional roughness, that the roughness does not directly introduce disturbances into the boundary layer. Instead, their primary function appears to be the amplification or destabilization of pre-existing disturbances. In contrast, the impact of three-dimensional roughness on the boundary layer transition significantly diverges from that of its two-dimensional counterpart. Cossu and Brandt [7] have demonstrated that small three-dimensional roughness elements, rather than inducing flow separation, give rise to streamwise velocity streaks. Intriguingly, these streaks can exert a stabilizing influence on the Tollmien-Schlichting waves, thereby delaying the natural transition to turbulence. Lu et al. [8, 9] and Fransson et al. [10, 11] further substantiated the transition delay effects of small roughness and conducted a parametric study on the shielding band associated with this delay phenomenon. However, when the size of the roughness exceeds a certain threshold, the velocity streaks induced by three-dimensional roughness may become unstable. These streaks are prone to nonlinear interactions [12] or may experience secondary instability [13], thereby significantly promoting the laminar-to-turbulent transition.

Lu et al. [14] and Landahl [15] provided explanations on the formation of high-speed and low-speed streaks induced by three-dimensional roughness. As the flow interacts with the roughness, counter-rotating vortex pairs are generated, transferring high-speed flow from the outer to the near-wall boundary layer and vice versa for low-speed flow. This interaction leads to the formation of high- and low-speed streaks. Studies by De Tullio [16], Choudhari [17], and Loiseau et al. [18] highlight the crucial role of low-speed streaks in promoting the growth of unstable modes downstream of the roughness. Experimental investigations by Asai et al. [19] into the instability characteristics of these streaks revealed that sinuous mode development associates with the instability in spanwise high-shear layers, whereas the varicose mode stems from the Kelvin-Helmholtz instability [20] in strong shear layers along the wall-normal direction.

Under the same flow conditions, distributed roughness elements can cause an earlier transition in comparison to isolated roughness [4]. Corke et al. [21] found that, relative to the smooth flat plate flow, low-inertia flows within the gaps of distributed roughness are more susceptible to disturbances from the free stream. Muppidi and Mahesh [22] found that the obstructive effect of distributed roughness leads to a deceleration of the near-wall flow, thereby forming a strong shear layer. Meanwhile, the raised sections of the roughness generate an upward pulsating motion, resulting in the formation of the streamwise vortices. The continuous generation of streamwise vortex structures, interacting with the shear layer, triggers the transition of the boundary layer. The research by Lefieux et al. [23] further substantiate this viewpoint.

Introducing various types of roughness in the front of the model results in diverse effects on the locations and characteristics of supersonic boundary layer transition [24]. Researchers used roughness to induce forced transitions in supersonic inlets and achieved good results [25]. Moreover, due to reasons such as design, deformation, and surface ablation, various types of roughness structures inevitably exist on the surface of the structure [26], resulting in different effects on the stability of boundary layer flow. Existing research has predominantly focused on the transition induced by isolated roughness, with limited investigation into the transition mechanisms induced by distributed roughness. To investigate the flow characteristics at the troughs and gaps of roughness and the impact of the gaps on transition properties in flat plate flows, we have devised a new form of distributed roughness. This innovative roughness is termed as gap-type roughness, characterized by a gap structure in the middle of the three-dimensional rough strip. Currently, numerical simulations and experimental studies on roughness with this configuration are not found in the literature, thus rendering this research of high academic value.

The primary aim of this study is to delve into the flow mechanisms of gap-type roughness induced transition in supersonic boundary layer. We intend to establish a comprehensive understanding of this complex phenomenon, serving as a vital reference for future experimental and numerical studies. Additionally, the findings are expected to provide a theoretical basis for accurate transition prediction and shaping design in engineering applications involving such roughness structures. To achieve these objectives, this study employs a direct numerical simulation to conduct a mechanistic study on the supersonic boundary layer transition induced by the gap-type roughness. It comprehensively analyzes the characteristics of various complex vortex structures within the transition, elucidates their origins, and underscores their significant roles in the transition process. The study summarizes the mode characteristics of the transitional flow and employs modal decomposition techniques for an in-depth analysis of the modal development features during the transition process.

2 Computation set-up

2.1 DNS computational domain set-up

The computational conditions employed in this study are summarized in Table 1, where ${\delta }_{in}$ represents the laminar boundary layer thickness at the inlet, and ${Re}_{\theta }$ denotes the momentum thickness Reynolds number of the boundary layer profile at $x=7.0$ in. The same free-stream conditions have also been utilized in the direct numerical simulations by Pirozzoli et al. [27] and Li et al. [28].

Table 1 The computational free-stream flow parameters

Figure 1 illustrates the detailed configuration of the computational domain. The computational domain size is $L_x \times L_y \times L_z=5.0 \text { in} \times 0.24 \text { in} \times 0.175 \text { in}$. To reduce computational grid requirements and mitigate the influence of the starting shockwave on the boundary layer, the computational inlet is chosen at $x_0=4\text{ in}$ (with the origin at the leading edge of the flat plate), and the two-dimensional laminar solution obtained by the compressible Blasius equations is utilized as the inlet profile. A pressure outlet condition is employed for the outlet condition. The wall is subjected to a non-slip isothermal condition, characterized by the boundary layer equation ${\partial }_p/{\partial }_n = 0$ (where n is the wall-normal direction). For the upper boundary of the computational domain, a far-field boundary condition based on the Riemann invariants is implemented, while periodic conditions are applied to the lateral boundaries.

Fig. 1

Computational domain configuration

Figure 2 illustrates the shape and parameters of the roughness element. Usually, based on the roughness height, roughness elements can be classified into three categories: 1) small roughness ($h/\delta<<1$); 2) medium roughness ($h/\delta \sim 1/2$); 3) large roughness ($h/\delta \sim 1$) [29]. In this paper, we focus on the effects of the medium-height gap-type roughness on the stability and transition of supersonic laminar boundary layers. The roughness has a three-dimensional spanwise strip shape, with a smooth-edged gap located in the middle of the strip. The flow behind the roughness becomes spontaneously unsteady, and the turbulence is obtained without external forcings. When the flow passes to the roughness, it can pass through the gap in the middle while encountering resistance on both sides, which is in contrast to the flow around isolated roughness simulated in the authors’ previous studies [8, 9]. The gap-type roughness is of the following form [29]:

$$\begin{aligned} y(x)= & {} \frac{1}{2} h\left[ \tanh \sigma _{x} \frac{x-x_{R}-0.5 R_{x}}{\Delta _{x}}-\tanh \sigma _{x} \frac{x-x_{R}+0.5 R_{x}}{\Delta _{x}}\right] ,\nonumber \\ y_{3 d}(x, z)= & {} \frac{1}{2} y(x)\left[ \tanh \sigma _{z} \frac{z-z_{R}-0.5 R_{z}}{\Delta _{z}}-\tanh \sigma _{z} \frac{z-z_{R}+0.5 R_{z}}{\Delta _{z}}\right] . \end{aligned}$$

(1)

Where $y_{3d}$ is the wall-normal coordinate of the roughness surface, ($x_R$, $z_R$) are the streamwise and spanwise coordinates of the gap center, and h is defined as the height of the roughness. $G_x$ is the streamwise width of the roughness, and $G_z$ is the spanwise width of the gap. $\sigma _{x}, \sigma _{z}$ and $\Delta _{x}, \Delta _{z}$ are parameters controlling the smooth transition at the edges of the roughness and the gap. The location of the roughness is the same as the disturbance location in the experiments [27]. The specific values for each parameter are listed in Table 2, with all length units given in inches.

Fig. 2

The gap-type roughness element

Table 2 Gap-type roughness geometry parameters

Table 3 is the calculational grid information for the gap-type roughness induced transition. The grid points within the computational domain are $N_x \times N_y \times N_z= 2240 \times 96 \times 256$. The grid resolution is $\Delta x_{min}^{+} \times \Delta y_{min}^{+} \times \Delta z_{min}^{+}=13.2 \times 0.82 \times 5.58$ (the dimensionless grid size is defined as $\Delta x_{min}^{+}=\sqrt{\tau _{w} \rho _{w}} \Delta x_{min} / \mu _{w}$, where ${\tau }_w$ is the wall shear stress and ${\Delta }x_{min}$ represents the grid size). The streamwise grid is uniformly distributed from upstream of the roughness ($x=4.4\text{ in}$) to the fully developed turbulent region ($x=7.5\text{ in}$), the spanwise grid is also uniformly distributed, while the wall-normal grid is refined close to the wall to meet the requirements for accurately simulating the boundary layer flow. The specific distribution of the wall-normal grid coordinates is obtained using the hyperbolic sine function [30] to ensure sufficient continuity and the required number of boundary layer grid points. At the end of the domain, there is a sponge region [31] where the grid is gradually coarsened in the streamwise direction in order to dissipate spurious wave reflections generated at the outlet boundary. Figure 3 illustrates a schematic of the local grid around the wall boundary and symmetry plane of the gap-type roughness in three dimensions.

Table 3 Calculational grid dimensions for the transition

Fig. 3

Grid distributions over the gap-type roughness

2.2 Validations on the code and results

For this study, the unsteady three-dimensional Navier-Stokes equations for compressible flow are utilized as the governing equations. The OpenCFD finite-difference code, developed by Li et al. [32], is employed for solving these equations. Its reliability has been affirmed in several studies [8, 9, 28]. Spatial discretization of convective terms is achieved using the WENO-SYMBO method [33] coupled with the Steger-Warming splitting scheme [34], while viscous terms are computed through an eighth-order central-differencing approach. Time advancement is handled by a third-order total variation diminishing Runge-Kutta method [35], with a physical time step of $\Delta t = 1.30 \times 10^{-7} \text{ s}$. For statistical requirements, flow field samples spanning about two non-dimensional time units (defined as $L_x / U_{\text {ref}}$) are collected, with a sampling interval of $0.001 L_x / U_{\text {ref}}$. All physical quantities in this study are presented in a dimensionless format, normalized by free-stream values.

To validate the inlet boundary profile, Fig. 4 presents the DNS velocity profiles at different streamwise locations on the symmetry plane upstream of the roughness ($4.0\text{ in}<x<4.55\text{ in}$), compared with the similarity solutions for compressible laminar boundary layers. As is apparent, the computational results closely match the laminar similarity solution upstream of $x=4.3\text{ in}$. However, as the flow approaches the roughness, the flow becomes impeded, and the computational results gradually deviate from the laminar similarity solution.

Fig. 4

Velocity profiles on the symmetry plane of laminar section at different streamwise locations

In order to verify whether the chosen spanwise width of the computational domain in this study is sufficient, the spatial correlation function is employed to quantify the correlation of turbulent fluctuation quantities at different spanwise distances. The correlation function is defined as follows [27]:

$$\begin{aligned} R_{\alpha \alpha }\left( r_{z}\right) =\sum \limits _{k=1}^{(N_z-1)} \overline{\alpha _{k} \alpha _{k+k_{r}}}, \quad k_{r}=0,1,2, \ldots , k-1. \end{aligned}$$

(2)

Where $\alpha$ represents the fluctuating quantity, chosen here as the streamwise, wall-normal, and spanwise components of the fluctuating velocity. $N_z$ denotes the number of grid points in the spanwise direction, and $r_z=k_r{\Delta }_z$. Figure 5 presents the distribution of spanwise correlation coefficients for the fluctuating velocity components at wall-normal locations of $y^{+}=20 \text{ and } y^{+}=100$ under $x=6.0\text{ in}$. Overall, spanwise correlation coefficients reduce rapidly within a spanwise distance of $0.1L_z$, and get close to zero after going through half of the domain. This indicates that the chosen spanwise width in this study adequately satisfies the requirements for spanwise flow correlations, validating the use of periodic boundary conditions.

Fig. 5

Spanwise correlation coefficients at x = 6.0 in and two different wall-normal positions

Furthermore, Fig. 6a provides the mean velocity profile by van Driest transformation ($U^+_{vd}$) [36] at $x=7.0\text{ in}$, compared with theoretical formulations, DNS results from Pirozzoli [27] and Li [28]. Figure 6b displays time- and spanwise-averaged skin friction coefficients ($C_f$) along streamwise, offering DNS results and theoretical results for reference. The theoretical results are obtained from the formula of White et al. Considering differences of the transition on-set location, the reference curve abscissa is adjusted accordingly (shifted by about 0.3 in). Figure 6c illustrates the distributions of time-averaged physical quantities ($\rho u$, u, T) along the wall-normal direction at $x=7.0\text{ in}$. Figure 6d presents the Reynolds normal stress curve at $x=7.0\text { in}$. It can be observed that all curves align well with the reference results, which validates the accuracy and reliability of the direct numerical simulation of transonic flow over the flat plate.

In order to validate the grid convergence of the numerical simulation, a further refined grid was employed and the results obtained from the refined grid are depicted in Fig. 6. Minor differences between the results from the two grids indicate that the present grid satisfies grid convergence effectively.

Fig. 6

Grid convergence study and code validation

3 Results

3.1 Instantaneous flow characteristics

Figure 7 presents instantaneous streamwise velocity (u) contours captured at three different slices in the wall-normal direction ($y^{+}=5, y^{+}=10 \text{ and } y^{+}=100$). Analysis of these figures reveals that at $y^{+}=5 \text{ and } y^{+}=10$, two symmetric high-speed streaks form after the flow passes over the roughness. The two streaks expand outward on both sides with a low-speed flow region in between. At $y^{+}=100$, there is a noticeable intensification of both the high-speed streaks and the low-speed region along the downstream, reflecting the strong impact of streamwise vortices. Moreover, new low-speed streaks generate on the outer edges of the high-speed streaks, with less intensity compared to the central low-speed region but extending longitudinally over a greater distance. As the flow develops to $x = 5.3\text{ in}$, the central low-speed region and the adjoining high-speed streaks begin to become unstable, forming a “$\Lambda$” shaped structure that quickly breaks down. The flow initiates the transition to turbulence. At $x = 5.6\text{ in}$, clear low-speed streaks appear on both sides of the central region. Beyond $x = 5.8\text{ in}$, the flat plate flow rapidly undergoes extensive transition, particularly in the near-wall region, leading to the emergence of numerous small-scale turbulent structures. These significantly intensified small-scale structures, characterized by alternating high- and low-speed regions, signify that a lot of external high-speed flows enter into the boundary layer. This observation demonstrates the strong momentum exchange capabilities inherent in the turbulent flow.

Fig. 7

Slices of $y^{+}=5, y^{+}=10\ \text {and}\ y^{+}=100$, colored with instantaneous streamwise velocity u

In order to further figure out the evolution of streak structures during this transition process, Fig. 8 illustrates the distribution of streak intensity at two different time instants, accompanied by the streamwise skin friction coefficient ($C_f$) curve for reference. The intensity of the streak structures is quantified utilizing the streak strength function [13], with the specific formula delineated in Eq. (3). It can be observed that upstream of the roughness, the streak intensity can be negligible. However, after passing over the roughness, the streak intensity significantly increases and then goes through a plateau period, which should be due to the influence of streamwise vortices. To $x = 5.2 \text{ in}$, the streak intensity undergoes pronounced oscillations, and the on-set of streak oscillations corresponds precisely to the rapid-increased starting position of skin friction coefficients. Moreover, the streak amplitude exceeds 40% of the free-stream flow velocity, meeting the conditions for the instability development of streak intensity as outlined by Andersson et al. [13]. The streak intensity peaks around $x = 5.7\text{ in}$ (approaching 60% of the freestream velocity), slightly earlier than the peak in the friction coefficient. After entering the turbulent region, the streak intensity gradually decreases, but the streak amplitude generally maintains around 40% of the free-stream velocity.

$$\begin{aligned} A_{s t}(x)=\frac{1}{2}\left[ \max _{y, z}\left( u\left(x, y,\vphantom{z_{0}} z\right)-u\left( x, y, z_{0}\right) \right) -\min _{y, z}\left( u\left(x, y,\vphantom{z_{0}} z\right)-u\left( x, y, z_{0}\right) \right) \right] . \end{aligned}$$

(3)

Fig. 8

Instantaneous streak strength and streamwise skin friction coefficient distributions

Instantaneous streamwise velocity contours at different slices along the streamwise direction are illustrated in Fig. 9, encompassing eight intermediate slices from the roughness location ($x = 4.55 \text{ in}$) to the point of complete transition ($x = 5.95\text{ in}$). These figures intuitively depict the entire process from the stable laminar state to the onset of instability and ultimately complete transition. At $x = 4.75\text{ in}$, the boundary layer exhibits a “W”-shaped distribution in the spanwise direction due to the influence of the gap. This distribution is characterized by a protruding low-speed flow in the central region, flanked by sinking high-speed flows on either side. As the flow further develops downstream, due to the formation and evolution of the counter-rotating vortex pairs, the central low-speed region takes on a mushroom-like structure. The top of the mushroom-like structure breaks at $x = 5.35 \text{ in}$, and the flow then transitions to a turbulent state. Analyzing the regions on both sides, the boundary layer exhibits significant curvature in the spanwise direction, attributable to the lift-up effect caused by streamwise vortices. By $x = 5.75 \text{ in}$, the flat plate flow has essentially broken into smaller-scale turbulent structures. Notably, the transition process initially progresses through an extended slow-development phase. But once the laminar flow breaks down, the transition enters a rapid-development stage (from $x = 5.35 \text{ in}$ to $x = 5.75\text{ in}$) and rapidly completes. This observation aligns with the conclusion drawn from the friction coefficient curve.

Fig. 9

Slices at different streamwise locations, colored with instantaneous streamwise velocity u

For an in-depth exploration of the formation of the central counter-rotating vortex pairs and the development of high- and low-speed streaks downstream of the roughness, Fig. 10 plots two different streamwise slices near the roughness, colored with streamwise velocity (u) and spanwise velocity (w). At the gap location ($x = 4.55 \text{ in}$), the flow converges from both sides toward the center. This convergence results in the formation of two counter-rotating vortex pairs, $V_1$ and $V_2$, situated above and below the lower saddle point, respectively. The origination of these vortex pairs is attributed to the separation of the spanwise converging flow. The saddle and separation points are marked by red dots in Fig. 10b. Combining with streamwise velocity isolines, we can see that the upper counter-rotating vortex pair $V_1$ elevates the low-speed flow upward, while the lower vortex pair $V_2$ transports the relatively high-speed flow from the upper to the lower layers. Moving downstream to $x = 4.57 \text{ in}$, as seen in Fig. 10d, both counter-rotating vortex pairs ($V_1$ and $V_2$) significantly weaken, with $V_2$ nearing complete dissipation. However, in the regions flanking the central area, a new streamwise vortex pair, $V_3$, emerges, exhibiting a symmetrical distribution. The formation of the new streamwise vortex pair is mainly due to the opposite-directional spanwise flow at the bottom and middle of the boundary layer, which further leads to shear interactions. Under the influence of the streamwise velocity, these vortices exhibit a helical converging streamline pattern.

Fig. 10

Slices of $x = 4.55 \text{ in}$ and $x = 4.57 \text { in}$, colored with streamwise velocity (u) and spanwise velocity (w). Lines indicate streamwise velocity isolines in (a), (c) and indicate 2D streamlines in (b), (d)

Figure 11 illustrates the distribution of the vorticity components at three different streamwise slices, accompanied by two-dimensional streamlines. $w_x$ is the streamwise vorticity component and $w_y$ is the vorticity component in the normal direction. Observations reveal that as the flow progresses downstream of the roughness, the spanwise converging effect in the central region begins to diminish, and the counter-rotating vortex pairs $V_1$ and $V_2$ gradually weaken. Meanwhile, the flow in the bottom layer diverges towards both sides, and the streamwise vortex pair $V_3$ continues to develop and strengthen. At $x = 4.6 \text{ in}$, the vortex pair $V_1$ shows a noticeable reduction in both strength and size, while the vortex pair $V_2$ has entirely disappeared. As the flow advances further downstream, the streamwise vortex pair $V_3$ significantly strengthens and extends outward. It is noteworthy that the central vortex pair $V_1$ begins to regain strength and size. Analyzing the trends in vortex components $w_x$ and $w_y$, we can know that at the gap, the vortex pair $V_1$ is essentially a streamwise vortex pair, with the component $w_x$ being overwhelmingly dominant. Progressing downstream, $w_x$ gradually weakens, while $w_y$ rapidly increases. By $x = 4.7 \text{ in}$, the vortex pair $V_1$ is dominated by the $w_y$ component. A similar trend also occurs in the streamwise vortex pair $V_3$. The decrease in $w_x$ is primarily attributable to the diminishing spanwise converging effect, while the increase in $w_y$ results from the lift-up effect exerted by vortex pairs. The lower-speed flow is pushed upward; meanwhile, more high-speed flows from both sides enter to the boundary layer, leading to an increase in the spanwise velocity difference. As a consequence, the vortex pair gradually strengthens. In summary, the low-speed streaks in the central region downstream of the roughness are primarily caused by the lift-up effect of the counter-rotating vortex pair $V_1$, lifting the low-speed flow upward, while the low-speed streaks on both sides are induced by the streamwise vortex $V_3$.

Fig. 11

Different streamwise slices, colored with vortex components $w_x$ and $w_y$. Lines indicate 2D streamlines

To further clarify the transition mechanism in the central region, Fig. 12 illustrates a detailed visualization of the instantaneous vortex structures of the flat plate transition flow, depicted using the Q criterion and colored with the normal velocity fluctuation ($v'$). The Q criterion is the second invariant of $\nabla u$ calculated by Eq. (4):

$$\begin{aligned} Q=-\frac{1}{2}\left[ \left( \frac{\partial u}{\partial x}\right) ^{2}+\left( \frac{\partial v}{\partial y}\right) ^{2}+\left( \frac{\partial w}{\partial z}\right) ^{2}\right] -\frac{\partial u}{\partial y} \frac{\partial v}{\partial x}-\frac{\partial u}{\partial z} \frac{\partial w}{\partial x}-\frac{\partial v}{\partial z} \frac{\partial w}{\partial y}. \end{aligned}$$

(4)

Fig. 12

Instantaneous vortex structures ($Q^*=1000$, colored with $v'$)

The transition within the central region is observed to commence at $x = 5.3\text{ in}$, characterized by the formation of large-scale hairpin vortices that quickly break down into a series of smaller-scale vortex structures. From the side view and top view, it is noticeable that in the initial stage of transition, there formed a contiguous series of hairpin vortices and leg-buffer vortices [37]. Notably, the leg-buffer vortices, positioned between adjacent hairpin vortices, exhibit rotational directions and developmental trends that are diametrically opposite to those of the hairpin vortices. With the flow developing downstream, two streamwise vortex legs converge towards the central plane, overlap at the head and form fully developed hairpin vortices. These hairpin vortices gradually elevate, creating different tilt angles $\alpha$ with the wall, as shown in Fig. 12b. Within these hairpin vortices, the low-momentum flow (red region) is lifted upwards, while the high-speed flow from the outer side region (blue region) enters the near-wall region, creating a down-sweep effect. Conversely, the development of leg-buffer vortices is oriented towards the sidewalls. They gradually incline towards the wall (tilt angles $\beta$ as shown in Fig. 12b), exhibiting a “$\Lambda$” shape. The inner side of leg-buffer vortices brings high-speed flows into the near-wall region, while their outer side delivers low-speed flows into the upper layer. As the flow develops downstream, under the nonlinear disturbance of multiple vortices, these two large-scale vortex structures quickly break down, forming a multitude of smaller-scale flow structures.

Compared to the transition induced by the isolated roughness [29, 30], there are notable distinctions in the vortex structures in the two types of transition. When the flow passes over the isolated roughness, it becomes obstructed and flows to both sides, forming horseshoe vortices and edge vortices under strong shear. The edge vortices develop downstream to the counter-rotating vortex pairs. In contrast, the counter-rotating vortex pairs induced by gap-type roughness involve two reasons: the spanwise convergence effect of the flow upstream and at the gap, and the enhancement of normal vorticity brought by the downstream high-speed flow entering the boundary layer. Since the generation of high-low-speed streaks is related to the up-lift and down-sweep motions of counter-rotating vortex pairs, the distribution of high-low-speed streaks also differs between the two types of transition. Additionally, there are significant differences in the downstream hairpin vortices. The parallel streamwise vortex legs induced by the isolated roughness play a central role in the formation of transition, and therefore, the hairpin vortices exhibit a conventional parallel lift pattern. In contrast, the leg-buffer vortices induced by the gap-type roughness display a distinct “$\Lambda$” shape, followed by a positive-negative figure-eight crossing pattern. Finally, compared to the turbulent wedge diffusion induced by the isolated roughness, the generation of small-scale individual streamwise vortices on both sides leads to the stronger spanwise spreading of the turbulent region in the gap-type roughness transition, resulting in a shorter overall transition process.

In summary, based on the comprehensive analysis, the transition process can be divided into four stages. Stage one: At the gap, upper and lower counter-rotating vortex pairs are formed due to the converging effect of the spanwise flow. Stage two: After passing through the roughness, the central counter-rotating vortex pairs gradually weaken, and the streamwise vortex pair of both sides emerges and develops. Stage three: At a certain distance downstream, the boundary layer becomes unstable, triggering the formation of streamwise vortex legs which subsequently evolve into hairpin vortices and leg-buffer vortices. The flow in the central region enters the transition phase. Stage four: The combined effects of the central region flow and the low-speed streaks on both sides lead to the transition spreading across the spanwise direction. Ultimately, the flow transforms into a fully developed turbulent flow.

3.2 Average flow characteristics and mode analysis

The generation process of turbulent kinetic energy can characterize the transition of flow from laminar to fully turbulent. Therefore, in this section, we focus on the generation of turbulent kinetic energy to analyze the relationships between various vortex structures and the transition process. The turbulent kinetic energy generation term formula is given by Eq. (5).

$$\begin{aligned} P=-\overline{\rho u_{i}^{\prime \prime } u_{j}^{\prime \prime }} \frac{\partial \tilde{u}_{i}}{\partial x_{j}}. \end{aligned}$$

(5)

Figure 13 presents the distribution of the turbulent kinetic energy (K) and the turbulent kinetic energy production term (P) at two different wall-normal slices. As shown in Fig. 13a, at $y^+=10$, K is mainly concentrated within the inner of the fully turbulent zone. Ascending to $y^+=50$, K is primarily located in the high-speed streaks and the outer edge of the turbulent zone. This distribution precisely corresponds to the interaction areas of large-scale vortex structures, such as hairpin and streamwise vortices. It is worth noting that at $x=5.8 \text{ in}$, the serrated region on both sides aligns well with the transition section of the flat plate flow, confirming the analysis presented earlier. As depicted in Fig. 13b, P reaches its peak or high value at $y^+=10$. The turbulent kinetic energy is mainly generated in the turbulent development zone and the interior region, particularly in the areas where large-scale hairpin vortices and streamwise vortices are distributed. The emergence of two symmetrical peak streaks can be attributed to the lift-up/down-sweep effects exerted by the hairpin and leg-buffer vortices. In these regions, the nonlinear interactions of vortex structures are most intense and active, which cause the large-scale vortex structures to break down into smaller ones with a significant amount of turbulent fluctuations.

Fig. 13

Two different wall-normal slices, colored with turbulent kinetic energy (K) and its production term (P)

Figure 14 depicts the distribution of the turbulent kinetic energy production term (P) at different streamwise slices, with some portions showing two-dimensional streamlines. In Fig. 14, P is extremely weak at $x=5.2 \text{ in}$. This suggests that the central counter-rotating vortex pairs and the streamwise vortex pair on both sides contribute negligibly to P. This is primarily due to the diminished strength of the central counter-rotating vortex pairs, which fails to induce strong down-sweep events on the outer side. As the flow develops to $x=5.3\text{ in}$, there is a relatively noticeable P distributed around the central counter-rotating vortex pairs. Combining the analysis conducted earlier and the distribution of 2D streamlines, we can see that the strong turbulent kinetic energy generation region corresponds well to the region where hairpin vortex legs and leg-buffer vortices are forming. At $x=5.4\text{ in}$, the peak of P is located within the interior of the hairpin and leg-buffer vortex structures, where it undergoes vigorous lift-up and down-sweep motions. Meanwhile, substantial turbulent kinetic energy is also generated in the near-wall regions on both sides. This is associated with the influence of hairpin vortex legs and the secondary vortices they induce. The lift-up and down-sweep events during the secondary vortex formation can cause turbulent fluctuations, accompanied by intense shear motion, leading to the generation of K. With the flow developing downstream of $x=5.6\text{ in}$, K is primarily generated in the near-wall region, attributed to small-scale vortex structures. Additionally, significant P is observed on both sides, indicative of the emergence and gradual development of streamwise vortices. Upon completion of the transition, the peak of P is distributed extensively throughout the entire near-wall region, consistent with the characteristics of a fully developed turbulent flow.

Fig. 14

Different streamwise slices, colored with turbulent kinetic energy production term (P). Lines indicate 2D streamlines

In order to capture the low-dimensional dynamical features of the transition flow, the Proper Orthogonal Decomposition (POD) technique is employed for analysis. The study recorded 500 snapshots, with an sampling interval of $\Delta t= 2.60 \times 10^{-7}\text{ s}$.

Figure 15 presents the power spectral density (PSD) curves depicting pressure fluctuations at different streamwise locations along the centerline of the wall. In Fig. 15a, the y-axis is displayed in a logarithmic scale. 1500 points were sampled, and the sampling interval $\Delta t=1.30 \times 10^{-7} \text{ s}$, solved by the Burg method. From Fig. 15a, it can be observed that immediately downstream of the roughness, pressure fluctuations are weak and primarily characterized by high-frequency oscillations. This phenomenon can be attributed to numerical perturbations induced by the shockwave at the leading edge of the roughness. As the flow progresses downstream, the low-frequency component develops first, forming a dominant frequency feature around $x=5.05\text{ in}$. The dominant frequency gradually increases from approximately $f \approx 1.7 \times 10^5 \text{ Hz}$ to $f \approx 2.4 \times 10^5 \text{ Hz}$. Considering the down-sweep effect mentioned earlier downstream of the gap roughness, we note that after the elevation of the central counter-rotating vortex pair, the down-sweep effect weakens for the near-wall flow, and there is a near-wall flow acceleration in the central region, ultimately resulting in an increase in the peak frequencies at different positions of the PSD downstream. During this progression, the pressure fluctuations of the dominant frequency rapidly intensify as the flow develops downstream. Around $x=5.35\text{ in}$, multiple sub-frequencies appear, with some exhibiting harmonic characteristics indicative of a multiple relationship. Meanwhile, at $x=5.275\text{ in}$, the high-frequency component of pressure fluctuations gradually develops. This reflects the transition process in the flow, where high-frequency small-scale fluctuations start to develop. From Fig. 15b, it can be observed that as the flow further develops, the PSD of the dominant frequency reaches its peak at $x=5.5\text{ in}$. Thereafter, the PSD of the dominant frequency gradually diminishes, indicating a weakening of the corresponding flow structures. Additionally, at $x=5.65\text{ in}$ and $x=5.8\text{ in}$, multiple sub-frequency features emerge, and their intensities gradually decrease. These features may be related to the flow structures corresponding to the dominant frequency, but the specific mechanism awaits further analysis and confirmation.

Fig. 15

Power spectral density curves of pressure fluctuations at different streamwise locations along the wall centerline ($y=0\text{ in}$)

In the streamwise direction, we selected instantaneous velocity fluctuation fields at three different streamwise locations $x=5.35\text{ in}$, 5.5 in, and 5.8 in, for the POD analysis, considering velocity fluctuation components in all three spatial directions. Figure 16 presents the POD modal energy distribution and the cumulative energy distribution at these three streamwise slices. It can be observed that the energy proportion carried by the first mode rapidly decreases as the flow develops downstream. At $x=5.35\text{ in}$, the first mode accounts for nearly 30% of the total energy, with cumulative energy of approximately the first 55 modes reaching 95%. However, by $x=5.8\text{ in}$, the energy proportion of the first mode drops to approximately 6%, and achieving a cumulative energy of 95% requires accumulating the first 250 modes. This indicates that there is a diminishing trend in the high-energy flow structures during the transition process. Based on the energy contribution, our analysis focuses on the first three POD modes. Figure 17 illustrates the frequency spectrum curves of the time coefficients for the first three POD modes. These curves reveal that, at $x=5.35\text{ in}$ and $x=5.5\text{ in}$, the first three modes predominantly exhibit a dominant frequency feature at approximately $f \approx 2.4 \times 10^5\text{ Hz}$. Mode 1 at $x=5.35\text{ in}$ and mode 2 at $x=5.5\text{ in}$ exhibit this feature most prominently, with some approximate harmonic features between modes, such as $f \approx 4.7 \times 10^5 \text{ Hz}$ and $f \approx 7.0 \times 10^5 \text{ Hz}$. However, at $x=5.8\text{ in}$, the dominant frequency modes mentioned earlier have substantially disappeared. Instead, the frequency characteristics of the first three modes exhibit closer proximity, showing a prominent low-frequency component and multiple characteristic frequencies.

Fig. 16

Distribution of POD modal energy and cumulative energy at three different streamwise slices

Fig. 17

Time coefficient frequency spectrum curves of the first three POD modes

Figure 18 presents the first three POD modes for the three different streamwise slices. Based on the analysis of the instantaneous flow field structure, it is evident that at $x=5.35\text{ in}$, the hairpin vortex legs and the leg-buffer vortices have formed. Therefore, mode 1 in Fig. 18a primarily reflects the velocity fluctuation in the shear regions above the inner side and below the outer side of these vortex structures. Mode 2 is concentrated in the near-wall layer beneath the secondary vortex and the head and mid-region of the hairpin vortex. Mode 3 more distinctly demonstrates the effects of additional secondary vortices. Comparatively, at $x=5.5\text{ in}$, the dominant mode 1 exhibits a significant spanwise distribution characteristic, indicating strong shear interactions among a series of secondary streamwise vortices. Mode 2 reflects the influence of large-scale streamwise vortex structures in the central region of the flow. As the flow develops to $x=5.8\text{ in}$, the strong pulsation regions for both mode 1 and mode 2 are mainly concentrated in the shear regions of the large-scale streamwise vortices on both sides. Meanwhile, mode 3 highlights the important role of spanwise distributed small-scale vortex structures of the turbulent zone, which is similar to mode 1 at $x=5.5\text{ in}$. This evolution in the dominant mode characteristics reflects the changes occurring within flow structures during the transition process.

Fig. 18

The first three POD modes at three different streamwise slices

4 Conclusions

In this study, we conducted direct numerical simulations to investigate the supersonic boundary layer transition induced by a gap-type roughness element. A comprehensive and in-depth analysis was carried out to explore the flow characteristics, formation mechanisms, and significant roles of the related flow structures during the transition process. Our investigation was conducted from three distinct perspectives: instantaneous flow characteristics, average flow characteristics, and modal analysis.

The results indicate that the transition process can be divided into four stages: the formation of counter-rotating vortex pairs at the gap, the formation and development of the streamwise vortex pair on both sides, the generation of hairpin vortices and leg-buffer vortices, and the spanwise expansion of the transition. The emergence of the central low-speed region downstream of the roughness is primarily due to the lift-up effect of the low-speed flow caused by the counter-rotating vortex pairs. On both sides, the low-speed streaks are predominantly influenced by the action of the streamwise vortex pair.

Moreover, we observe that the dynamic changes in vortex structures play a decisive role in the stability and transition characteristics of the supersonic boundary layer. Specifically, the hairpin vortices and the leg-buffer vortices, along with their induced secondary vortices, emerge as crucial factors in inducing boundary layer transitions and generating turbulent kinetic energy. The POD modal analysis within the transition region further confirms the pivotal role of these vortex structures during the transition process.