Transient growth analysis of oblique shock wave/boundary-layer interactions at Mach 5.92

TL;DR

This study investigates how transient growth mechanisms in high-speed boundary layers interacting with oblique shock waves can lead to flow perturbation amplification, highlighting non-modal effects relevant for hypersonic flow transition.

Contribution

It introduces a power-iteration approach to quantify transient growth in compressible boundary layers with shock interactions, revealing the significant impact of shock waves on flow perturbation amplification.

Findings

Oblique waves dominate transient response without shock.

Oblique shock waves increase transient growth significantly.

Base flow deceleration near reattachment amplifies streaks.

Abstract

We study physical mechanisms that trigger transient growth in a high-speed spatially-developing laminar boundary layer that interacts with an oblique shock wave. We utilize an approach based on power-iteration, with the global forward and adjoint linearized equations, to quantify the transient growth in compressible boundary layers with flow separation. For a Mach 5.92 boundary layer with no oblique shock wave, we show that the dominant transient response consists of oblique waves, which arise from the inviscid Orr mechanism, the lift-up effect, and the first-mode instability. We also demonstrate that the presence of the oblique shock wave significantly increases transient growth over short time intervals through a mechanism that is not related to a slowly growing global instability. The resulting response takes the form of spanwise periodic streamwise elongated streaks and our analysis of the linearized inviscid transport equations shows that base flow deceleration near the reattachment location contributes to their amplification. The large transient growth of streamwise streaks demonstrates the importance of non-modal effects in the amplification of flow perturbations and identifies a route for the emergence of similar spatial structures in transitional hypersonic flows with shock wave/boundary-layers interaction.

Figures

Equations

$$\begin{array}[]{rcl}\dfrac{\partial{p}}{\partial{t}}\;+\;\bm{u\cdot\nabla}p\;+\;\rho{a^{2}}\bm{\nabla\cdot{u}}&=&\dfrac{1}{Re}\left(\dfrac{1}{M_{\infty}^{2}Pr}\,\bm{\nabla\cdot}(\mu\bm{\nabla}T)\;+\;(\gamma-1)\phi\right),\\[7.11317pt] \dfrac{\partial\bm{u}}{\partial{t}}\;+\;\dfrac{1}{\rho}\,\bm{\nabla}p+\bm{u\cdot\nabla{u}}&=&\dfrac{1}{Re}\,\dfrac{1}{\rho}\,\bm{\nabla\cdot\uptau},\\[7.11317pt] \dfrac{\partial{s}}{\partial{t}}\;+\;\bm{u\cdot\nabla}s&=&\dfrac{1}{Re}\dfrac{1}{\rho{T}}\left(\dfrac{1}{(\gamma-1)M_{\infty}^{2}Pr}\,\bm{\nabla\cdot}(\mu\bm{\nabla}T)\;+\;\phi\right).\end{array}$$

$$\text{uptau}=\mu \left(\nabla \mathbf{u}+(\nabla \mathbf{u}{)}^{\mathrm{T}}-\frac{2}{3}(\nabla \cdot \mathbf{u})\mathbf{I}\right).$$

$$\mu (T)=T^{3/2}\frac{1+T_s/T_{\infty }}{T+T_s/T_{\infty }}.$$

$$\frac{\partial \mathbf{q}}{\partial t}=A\mathbf{q}\Rightarrow \mathbf{q}(t)=e^{At}\mathbf{q}(0),$$

$$E=\frac{\bar{\rho }{u}_i^{\prime }{u}_i^{\prime \ast }}{2}+\frac{M_{\infty }^2|p^{\prime }{|}^2}{2}+\frac{(\gamma -1)M_{\infty }^2|s^{\prime }{|}^2}{2},$$

$$G(t)=\underset{\mathbf{q}(0)\ne 0}{\max }\frac{||\mathbf{q}(t)|{|}_E^2}{||\mathbf{q}(0)|{|}_E^2},$$

$$\sigma (v_g)=\underset{t\to \infty }{\lim }\frac{\partial }{\partial t}\left(t^{1/2}A(x_0+v_{g}t,t)\right)\approx \frac{\ln (A(x_0+v_g{t}_2,t_2)/A(x_0+v_g{t}_1,t_1)}{t_2-t_1}+\frac{\ln (t_2/t_1)}{2(t_2-t_1)},$$

$$\begin{array}[]{rcl}\omega_{i}&=&\sigma(v_{g})\;-\;v_{g}\,\dfrac{\sigma(v_{g}+\delta{x}/t_{2})\,-\,\sigma(v_{g})}{\delta{x}/t_{2}},\\[9.95863pt] \alpha(v_{g})&=&\dfrac{\phi(x_{o}+v_{g}t_{2}+\delta{x},t_{2})\,-\,\phi(x_{o}+v_{g}t_{2},t_{2})}{\delta{x}},\end{array}$$

$$\frac{\partial E}{\partial t}=\mathcal{P}+\mathcal{S}+\mathcal{V}-\mathcal{T}.$$

$$\frac{D{u}^{\prime }}{Dt}\approx -\frac{1}{\bar{\rho }}\frac{\partial \bar{U}}{\partial x}(\rho u{)}^{\prime }-\frac{1}{\bar{\rho }}\frac{\partial \bar{U}}{\partial y}(\rho v{)}^{\prime },$$

$$\frac{1}{2}{\text{bigintsss}}_0^{y_0,z_0}\frac{D|u^{\prime }{|}^2}{Dt}\mathrm{d}y\mathrm{d}z\approx -{\text{bigintsss}}_0^{y_0,z_0}\frac{1}{\bar{\rho }}\frac{\partial \bar{U}}{\partial x}{u}^{\prime \ast }(\rho u{)}^{\prime }\mathrm{d}y\mathrm{d}z-{\text{bigintsss}}_0^{y_0,z_0}\frac{1}{\bar{\rho }}\frac{\partial \bar{U}}{\partial y}{u}^{\prime \ast }(\rho v{)}^{\prime }\mathrm{d}y\mathrm{d}z,$$

Full text

I INTRODUCTION

Shock wave/boundary-layer interactions (SWBLI) are commonly encountered in high-speed flows over complex geometries that involve intakes, control surfaces, and junctions. An oblique shock wave impinging on the flat-plate boundary layer is a canonical setup used to investigate the resulting flow separation and reattachment in SWBLI Currao et al. (2020); Sandham et al. (2014). Despite the spanwise homogeneity of the flat-plate geometry, experiments Currao et al. (2020) and numerical simulations Sandham et al. (2014); Pagella et al. (2002) demonstrate the emergence of three-dimensional streamwise streaky flow structures near the reattachment location before transition to turbulence.

Recent studies of two-dimensional laminar SWBLI Chuvakhov and Radchenko (2020); Cao et al. (2019) show that such configurations can significantly amplify external three-dimensional disturbances. Previous analyses focused on amplification of flow perturbations that arise from streamwise curvature of streamlines in the region where the separated flow reattaches to the wall Navarro-Martinez and Tutty (2005); Priebe et al. (2016). Recent experimental observations also highlight the presence of three-dimensional structures inside the recirculation bubble Zhuang et al. (2017). In this paper, we consider the role of the entire recirculation bubble in the amplification of flow perturbations.

Several studies investigated the effect of the separated flow by carrying out global stability analysis Theofilis et al. (2000); Theofilis (2011). These references show that 2D SWBLI that arise from impinging oblique shock waves Hildebrand et al. (2018a); Robinet (2007) and compression corners Sidharth et al. (2018) exhibit an intrinsic 3D global instability inside the separation bubble as the strength of interactions increase. However, in contrast to recent experimental observations Chuvakhov et al. (2017), the wavelength predicted by global stability analysis scales with the length of the recirculation zone Sidharth et al. (2017), thereby suggesting that other amplification mechanisms may play a more prominent role.

In the absence of global instabilities, previous studies Butler and Farrell (1992); Andersson et al. (1999); Jovanović and Bamieh (2005) showed that non-modal amplification can lead to subcritical transition. Despite the asymptotic decay of the flow perturbations, non-orthogonality of the associated eigenmodes can induce significant transient amplification and trigger non-linear interactions that ultimately break down into turbulence Schmid and Henningson (2001). This amplification is closely related to high sensitivity of laminar flow to external disturbances Jovanović (2004); Jovanović and Bamieh (2005); Schmid (2007); Jovanović (2020) and it can be quantified using frequency domain input-output analysis. Recent work on SWBLI over compression ramps Dwivedi et al. (2019) and double wedges Dwivedi et al. (2018) demonstrated significant amplification of steady upstream vortical disturbances around globally stable laminar flow.

Another common approach to quantifying non-modal amplification is through computation of maximum growth of perturbations over a given interval in time or space Schmid and Henningson (2001). Most previous studies of compressible flows evaluate the initial conditions, which result in optimal temporal growth at a given spatial location Hanifi et al. (1996); Bitter and Shepherd (2015); Bitter (2015). For perturbations of a given frequency, a similar approach based on the linearized boundary layer equations provides a method for quantifying the optimal spatial amplification in spatially evolving flows Luchini (2000); Tumin and Reshotko (2003); Zuccher et al. (2006); Tempelmann et al. (2012); Paredes et al. (2016). For SWBLI, this methodology was recently utilized to evaluate the spatial growth of streamwise streaks in the reattaching boundary layers in the absence of flow recirculation Dwivedi et al. (2017) .

In the present work, we utilize the global linear system to evaluate the optimal spatio-temporal growth of canonical SWBLI configuration in which an oblique shock wave impinges on a flat plate boundary layer at Mach $5.92$. We first demonstrate the validity of our approach by evaluating amplification mechanisms in a spatially-evolving hypersonic boundary layer. In spite of the presence of a slowly growing global instability in the oblique SWBLI, we show that significant transient growth arises over short time intervals through a mechanism that is not related to global instability. The dominant response takes the form of streamwise streaks, which are ubiquitous in experiments and numerical simulations. To uncover physical mechanisms responsible for this significant increase in transient growth, we analyze the inviscid transport of the optimal wavepacket in the flow. Similar to SWBLI setups without a global instability Dwivedi et al. (2019), our analysis demonstrates that the base flow deceleration plays a dominant role in amplification of streamwise-streaky flow structures.

Our presentation is organized as follows. In Section II, we formulate the problem, describe the governing equations and the numerical method, and summarize the approach that we use to compute the spatial structure of the optimal initial conditions and the resulting temporal growth envelopes. In Section III.1, we investigate the importance of convective instabilities in a Mach $5.92$ spatially-developing boundary layer and, in Section III.2, we examine how the formation of a recirculation bubble in SWBLI changes the spatio-temporal response of the linearized flow equations. In Section IV, we conduct a wavepacket analysis to show the emergence of oblique waves in the absence of SWBLI and streamwise streaks in the presence of the shock. We also uncover physical mechanisms responsible for transient growth by examining the inviscid transport equation. We conclude our presentation in Section V.

II PROBLEM FORMULATION

II.1 Flow configuration

A spatially-developing boundary layer and an oblique shock-wave/boundary-layer interaction are considered in this paper. For both of these flow configurations, the free-stream conditions match experiments performed in the ACE Hypersonic Wind Tunnel at Texas A&M University Semper et al. (2012). As shown in Figure 1, a laminar boundary layer enters at the left boundary and travels over a flat plate situated along the bottom boundary. We examine a unit Reynolds number $4.6\times 10^{6}$ m*-1* or $Re=\rho_{\infty}u_{\infty}\delta^{*}_{\text{in}}/\mu_{\infty}=9660$ based on the undisturbed boundary-layer displacement thickness $\delta^{*}_{\text{in}}=2.1$ mm at the left boundary. Here, $\rho_{\infty}$, $u_{\infty}$, and $\mu_{\infty}$ denote the free-stream density, velocity, and dynamic viscosity, respectively. A bow shock that persists throughout the domain is produced by the leading edge of the flat plate (which is upstream of the left boundary). For the SWBLI, which is depicted in Figure 1, an incident oblique shock wave enters the computational domain through the left boundary well above the bow shock and the boundary layer. This incident shock propagates at an angle $\theta_{1}$ and its interaction with the bow shock changes the angle to $\theta_{2}$. Upstream of the bow and incident shocks, the free-stream Mach number, temperature, and pressure are $M_{\infty}=5.92$, $T_{\infty}=53.06$ K, and $p_{\infty}=308.2$ Pa. After the shock/shock interaction, the incident oblique shock wave strikes the boundary layer, thereby causing separation from the wall and formation of a recirculation bubble.

II.2 Governing equations

We use the compressible Navier-Stokes equations to model the dynamics of a spatially-developing boundary layer and an oblique SWBLI. These equations govern the spatio-temporal evolution of the state $[\,p~{}\,\bm{u}^{\mathrm{T}}~{}\,s\,]^{\mathrm{T}}$, where $p$, $\bm{u}$, and $s$ are the nondimensional pressure, velocity, and entropy, respectively Sesterhenn (2000). After nondimensionalization with respect to the displacement thickness $\delta^{*}_{\text{in}}$, free-stream velocity $u_{\infty}$, density $\rho_{\infty}$, and temperature $T_{\infty}$, the equations take the form

[TABLE]

We define the free-stream Mach number as $M_{\infty}=U_{\infty}/a_{\infty}$, where $a_{\infty}=\sqrt{\gamma{p_{\infty}/}\rho_{\infty}}$ denotes the free-stream speed of sound. The equation of state for an ideal fluid is $\gamma{M_{\infty}^{2}}p=\rho{T}$. Furthermore, $\gamma=1.4$ is the assumed constant ratio of specific heats. We define entropy as $s=\ln(T)/[(\gamma-1)M_{\infty}^{2}]-\ln(p)/(\gamma{M_{\infty}^{2}})$ so that $s=0$ when $p=1$ and $T=1$ Nichols and Lele (2011a).

The viscous stress tensor $\bm{\uptau}$ is written in terms of the identity matrix $\bm{I}$, velocity vector $\bm{u}$, and dynamic viscosity $\mu$ to yield

[TABLE]

We define the viscous dissipation as $\phi=(1/\mu)\,\bm{\uptau\!:\!\nabla{u}}$, where the operator $\bm{:}$ represents a scalar product of two tensors, $\bm{A:B}=\mathrm{trace}\,(\bm{A}^{\mathrm{T}}\bm{B})$. Further, the second viscosity coefficient is set to $\lambda=-2\mu/3$. In order to compute the dynamic viscosity $\mu$, Sutherland’s law is used with $T_{s}=110.3~{}\mathrm{K}$,

[TABLE]

The Prandtl number is set to a constant value $Pr=c_{p}\mu/\kappa=0.72$, where $\kappa$ and $c_{p}$ are the coefficients of heat conductivity and specific heat at constant pressure, respectively.

To study the behavior of small fluctuations around various base flows, System (1) is linearized by decomposing the state into a sum of the steady $\bar{\bm{q}}$ and fluctuating $\bm{q}$ parts. The linearized Navier-Stokes (LNS) equations take the following form Hildebrand et al. (2018b),

[TABLE]

where $A$ is the Jacobian operator resulting from the linearization of System (1) around the base state $\bar{\bm{q}}$. As discussed below, the transient growth analysis also requires the adjoint of the Jacobian operator, $A^{\text{H}}$, where we compute $A^{\text{H}}$ using a discrete approach. For a detailed discussion of adjoint operators see Luchini and Bottaro (2013).

II.3 Power iteration

We employ an iterative approach, based on power iteration Schmid (2007), to conduct the transient growth analysis. For compressible flows, the fluctuations’ energy is defined as Chu (1965); Hanifi et al. (1996),

[TABLE]

where ∗ denotes the complex conjugate. This expression is obtained by eliminating conservative compression work transfer terms and the energy is induced by the inner product $(\bm{q}_{1},\bm{q}_{2})_{E}=\int\bm{q}_{2}^{\mathrm{H}}W\bm{q}_{1}\,\mathrm{d}x\,\mathrm{d}y$, where the weighting matrix is given by $W=(1/2)\,\mathrm{diag}[M_{\infty}^{2},\bar{\rho},\bar{\rho},\bar{\rho},(\gamma-1)M_{\infty}^{2}]$. The energy growth at time $t$ is determined by,

[TABLE]

where $\bm{q}(0)$ and $\bm{q}(t)$ represent the initial and final states. Since $\bm{q}(0)$ depends on the time interval on which optimization is performed, $G(t)$ determines an envelope of all possible optimal responses. For a given interval $(0,t]$, we compute the maximum transient growth by alternating between integration of the governing equations forward in time and integration of the adjoint equations backward in time [Schmid, 2007,Guégan et al., 2006]. Starting from a random initial condition, this approach converges to the principal eigenfunction of the direct-adjoint system, which is equivalent to the principal singular function of the direct system in Eq. 4.

II.4 Numerical method

The base flow computations for the spatially developing boundary layer and the oblique SWBLI are carried out using US3D Candler et al. (2015a), a compressible finite volume flow solver. We apply symmetry boundary conditions along the side of the domain, thereby ensuring that the base flow remains two dimensional. For the transient growth calculations, we use the formulation by Sesterhenn Sesterhenn (2000) but our numerical method differs from Sesterhenn (2000). We use the fourth-order centered finite differences applied on a stretched mesh to discretize the LNS equations. This yields a large sparse matrix Nichols and Lele (2011a) and time integration is performed using an implicit first-order Euler method.We verify the resulting temporal evolution using an explicit fourth-order Runge-Kutta method and find that there is almost no difference. The inversion step is computed via the lower-upper (LU) decomposition of the shifted sparse matrix using the parallel SuperLU package Li and Demmel (2003). A numerical filter is used to add minor amounts of scale-selective artificial dissipation to damp spurious modes associated with the smallest wavelengths allowed by the mesh Durran (1999). We introduce the numerical filter by adding terms of the form $\epsilon(\partial^{4}\bm{q}/\partial{x^{4}})$ and $\epsilon(\partial^{4}\bm{q}/\partial{y^{4}})$. The value of $\epsilon$ is selected to be as small as possible (e.g., $\epsilon=0.0125$) in order to minimize the impact of this artificial dissipation while maintaining numerical stability. Furthermore, we check that our results are not sensitive to this choice by repeating our calculations with different values of $\epsilon$. Sponge layers are employed at the top, left, and right boundaries of the nonparallel boundary layer and oblique SWBLI to absorb outgoing information with minimal reflection Mani (2010).

III TRANSIENT GROWTH ANALYSIS

III.1 Spatially-developing boundary layer

In our previous study Hildebrand et al. (2018b), we verified our numerical method for computing the optimal transient growth of parallel high-speed compressible boundary layers Bitter and Shepherd (2015); Hanifi et al. (1996). The largest transient response in such flows is triggered by the lift-up mechanism Landhal (1977, 1980) similar to that observed in the low-speed incompressible wall-bounded shear flows Schmid and Henningson (2001); Jovanović (2020). Acoustic disturbances are also significantly amplified in spatially developing boundary layers Bitter (2015) and we utilize global transient growth analysis to examine the importance of different amplification mechanisms.

We use US3D to compute the steady two-dimensional flow over an adiabatic wall. The free-stream conditions presented in the previous section correspond to that of the SWBLI without the shock. The numerical domain extends 235$\delta^{*}_{\text{in}}$ in the streamwise direction and 36$\delta^{*}_{\text{in}}$ in the wall-normal direction. We utilize a Cartesian mesh to discretize this domain. The mesh is stretched in the wall-normal direction with $y^{+}=0.6$, and it is uniformly spaced in the streamwise direction. A total of $n_{x}=500$ and $n_{y}=420$ grid points are used to resolve this domain in the streamwise and wall-normal directions, respectively. At the inflow, we impose a boundary-layer profile computed in an earlier study Shrestha et al. (2016). Furthermore, we impose a characteristic-based supersonic outlet boundary condition along the top and right edges of the domain. This boundary condition allows the shock waves to exit the domain without reflection. The base flow simulations are carried until the numerical residual is of the order of machine precision. Figure 2 shows base flow contours of the spatially-developing boundary layer in the absence of the incident oblique shock.

The optimal transient growth of the spatially-developing boundary layer is computed by setting all perturbations at the top, left, and right boundaries to zero with the help of three sponge layers Mani (2010). The iterative process is initialized using a random flow field with unit energy. Figure 3 shows the transient growth as a function of $t$, nondimensional spanwise wavenumber $\beta$, and length $L_{x}$ of the streamwise domain. The transient growth becomes larger as the streamwise extent of the domain increases, thereby indicating that the spatially-developing boundary layer is convectively unstable. In other words, even though the linearized system is globally stable, localized upstream perturbations grow substantially as they convect downstream with the flow Cossu and Chomaz (1997); Nichols and Lele (2011b). Figure 4 displays the dependence on the spanwise wavenumber of the largest transient energy growth, $G_{\max}=\max_{t}G(t)$, for five different streamwise domain lengths. The largest transient growth occurs for the longest domain and is equal to $5.63\times 10^{3}$ for $\beta=0.38$, $t=350$, and $L_{x}=470$.

In contrast to the impact of the streamwise domain length, the spanwise wavenumber corresponding to the maximum growth decreases. In fact, figure 5 shows that the spanwise wavelength (which is proportional to the inverse of the spanwise wavenumber) scales with the boundary-layer thickness at the right outlet of the domain. Thus, Figures 4 and 5 imply that a significant portion of the transient growth occurs near the streamwise end of the domain, and that this region selects the spanwise wavelength that supports the largest growth. This is a consequence of the square-root growth of the boundary layer in the presence of convective instabilities: the spanwise length scale of the local instability near the end of the domain dictates the global response because of the slow spatial development of the boundary layer Bitter (2015).

In Figure 6, we illustrate the real part of the normalized streamwise velocity component of the optimal initial and final states of the spatially-developing boundary layer with $L_{x}=235$ and $\beta=0.6$. The initial state is comprised of the streamwise elongated structures near the left inlet that are tilted against the mean shear of the boundary layer. We note that these streamwise structures have a shallow angle because of the high-speed nature of the flow. As time increases, these tilted streaks start to align with the mean shear thereby causing substantial growth via the inviscid Orr mechanism Schmid and Henningson (2001). In wall-bounded shear flows, these flow structures grow rapidly and robustly Butler and Farrell (1992). The final state in Figure 6(b) also consists of tilted streamwise perturbations along the mean shear that extend to the right outlet.

To examine whether the lift-up effect Landhal (1977) contributes to the transient response of the spatially-developing boundary layer, we extract wall-normal profiles of the entropy along with the streamwise, wall-normal, and spanwise velocity components of the optimal initial and final perturbations at $x=35$ and $x=185$, respectively. In Figure 7, the perturbations are normalized to have a unit maxima. We see that the wall-normal and spanwise velocities significantly contribute to the optimal initial condition, while the contributions of the streamwise velocity and the entropy are negligible. At the final time, however, the streamwise velocity is much larger than the other velocity components. This suggests that the lift-up mechanism indeed contributes to the optimal transient growth of the spatially-developing boundary layer. While initial streamwise vortices decay in time, streamwise elongated flow structures experience rapid and robust growth Bitter and Shepherd (2015). Along with the streamwise velocity perturbations, in Figure 7 we also notice that the entropy grows substantially with time. This is in concert with the observation that the lift-up effect has a strong impact on the optimal transient growth Tumin and Reshotko (2003). In summary, we conclude that it is most effective to excite the wall-normal and spanwise velocity components and that the most energy is carried by the streamwise velocity and the entropy perturbations.

Similar to the incompressible three-dimensional boundary layers Tempelmann et al. (2010), this section demonstrates the importance of both the inviscid Orr and the lift-up mechanisms in spatially-developing compressible boundary layers. However, as shown in Figures 3 and 4, the transient growth becomes larger as the streamwise extent of the domain increases. This observation suggests the presence of a convective instability. In Figure 8, we show the optimal three-dimensional final state of the Mach $5.92$ spatially-developing boundary layer in a longer domain with $\beta=0.38$ and $2L_{x}=470$. The final state consists of oblique streamwise structures that are reminiscent of first-mode instability. To corroborate the presence of the first-mode instability in the optimal transient response of the spatially-developing boundary layer, we closely examine a wavepacket response in Section IV (see Figure 15). Thus, in addition to the contributions arising from the inviscid Orr and lift-up effects the optimal transient response of the spatially-developing boundary layer also arises from the first-mode instability.

III.2 Shock-wave/boundary-layer interaction

As shown in the previous section, a flat-plate boundary layer at Mach 5.92 supports significant transient growth. In this section, we examine the impact of an oblique SWBLI, at the same conditions. We utilize the US3D compressible flow solver Candler et al. (2015b) to compute a steady-state two-dimensional base flow of an oblique shock-wave/laminar-boundary-layer interaction at Mach $5.92$. The incident oblique shock wave is introduced by modifying the inlet boundary-layer profile so that the Rankine-Hugoniot jump conditions are satisfied at the point where it enters the domain. We select this point so that the incident shock impinges upon the wall at a distance of $119\delta^{*}_{\text{in}}$ from the leading edge. In this study, we are only interested in an oblique shock angle of $\theta_{1}=13\degree$, which changes to an angle of $\theta_{2}=12.89\degree$ after relatively weak interaction with the upstream bow shock. All the other boundary conditions are identical to those described in Section III.1.

The domain extends $235\delta^{*}_{\text{in}}$ and $36\delta^{*}_{\text{in}}$ in the streamwise and wall-normal directions, respectively. This domain is discretized by a Cartesian mesh that is nonuniformly spaced in the wall-normal direction with $y^{+}=0.6$ and uniformly spaced in the streamwise direction. A total of $n_{x}=998$ and $n_{y}=450$ grid points are used to resolve this domain. Figure 9 illustrates color plots of nondimensional streamwise velocity and density perturbations around the SWBLI base flow. All length scales are non-dimensionalized with the boundary layer thickness at the inflow $\delta^{*}_{\text{in}}$. The incident oblique shock wave causes the boundary layer to separate from the wall at $x\approx 50$, leading to the formation of a separation shock and recirculation bubble. At the recirculation bubble apex, an expansion fan forms and extends up into the free-stream. Moreover, at $x\approx 155$, the flow reattaches to the wall and compression waves coalesce to form a second reflected shock wave.

We utilize the power iteration method with a random initial condition of unit energy to compute the optimal transient growth of the SWBLI in Figure 9. Time interval for computing the optimal growth is set to roughly $2$ flow-through times, i.e., $(0,450]$, where a flow-through time is defined as the time it takes a fluid particle that travels with the free-stream at Mach $5.92$ to traverse the entire streamwise length of the domain. Similar to Figure 4, we examine the dependence on the spanwise wavenumber $\beta$ of the largest transient growth in Figure 10. The peak transient growth of $1.36\times 10^{7}$ at $\beta=2.6$ is roughly four orders of magnitude larger than the transient growth seen in Figure 4 for a streamwise domain length of $L_{x}=235$. This agrees with previous studies that often find that the formation of a recirculation bubble in a boundary-layer flow substantially increases the transient growth Alizard et al. (2008); Cherubini et al. (2010). The spanwise wavenumber also increases from $\beta=0.6$ for a nonparallel boundary layer to $\beta=2.6$ for an SWBLI. As the spanwise wavenumber deviates from the preferential value of $\beta=2.6$, the transient growth decreases by several orders of magnitude.

The transient growth envelope of the SWBLI in Figure 9 with $\beta=2.6$ is shown in Figure 11. Along with the temporal envelope, we show growth curves that originate from three different initial conditions pertaining to the fixed time intervals with $t=150$, $t=214$, and $t=260$. We note that in Figure 11 each of the growth curves are tangential to the transient growth envelope at exactly one point and that the optimal transient growth of this SWBLI occurs at $t=214$.

Figure 12 shows the normalized spanwise velocity perturbation component of the 2D optimal initial and final states in a Mach $5.92$ SWBLI with $\theta_{1}=13\degree$. We also examine the spatial structure of spanwise velocity perturbations at intermediate times $t=150$ and $t=182$ during amplification. The evolution from the optimal initial condition corresponds to the transient growth curve with red dashes in Figure 11. By $t=150$, barely any transient growth is observed but, after $t=150$, there is a significant increase in the transient growth that persists until $t=214$. During the same time interval, in Figure 12 we observe a large spanwise velocity perturbation that arises, at least in part, from the shear layer on top of the recirculation bubble and convects downstream until it hits the right sponge layer. This observation is in agreement with previous studies which identified perturbations that arise from the shear layer and the decelerated zone in low-speed separated boundary-layer flows Alizard et al. (2008); Cherubini et al. (2010); $\mathring{\mathrm{A}}$kervik et al. (2007). When this large perturbation starts to decrease as it moves through the right sponge layer, the transient growth curve in Figure 11 starts to decrease.

To determine the relative contribution of different perturbation components to the overall energy growth, in Figure 13 we evaluate the average magnitude of the velocity components as well as entropy and pressure as the optimal response evolves in time. We compute this average by integrating and normalizing magnitudes of flow perturbations in the domain at $t=214$. The streamwise velocity perturbation has the most significant contribution to the optimal transient growth, which corresponds to the formation of streamwise elongated streaks downstream of the recirculation bubble.

Figure 14 shows the streamwise velocity component of the 3D optimal initial and final states of a Mach $5.92$ SWBLI. The initial state consists of streamwise elongated structures that start near the inflow and extend past the separation point at $x\approx 50$. As these structures evolve downstream, they result in the formation of the spanwise periodic streamwise streaks. To illustrate the relative location of these structures, we also plot the isosurface (in green) corresponding to the 2D recirculation zone. We note that the optimal transient response of this oblique SWBLI is very similar to the optimal frequency response observed in an SWBLI over a compression ramp that was identified via input-output analysis Dwivedi et al. (2018, 2019).

IV PHYSICAL MECHANISM OF THE OPTIMAL SWBLI RESPONSE

As demonstrated in Section III.1, the optimal spatio-temporal response in spatially-developing boundary layer consists of three-dimensional oblique flow structures. However, after we incorporate an incident oblique shock wave, streamwise streaks that experience significantly larger transient growth emerge. Herein, we investigate physical mechanisms responsible for large amplification by analyzing the spatio-temporal evolution of the optimal response and the dominant terms in the linearized inviscid transport equations.

IV.1 Wavepacket analysis

Figures 6, 12, and 14 show that the optimal spatio-temporal responses of both the spatially-developing boundary layer and the oblique SWBLI consist of wavepackets that amplify as they propagate downstream. Wavepackets occur in a variety of convectively-unstable flows including low-speed boundary layers $\mathring{\mathrm{A}}$kervik et al. (2008) and high-speed jets Nichols and Lele (2011b), and they have been linked to transient growth when viewed in the global framework Chomaz (2005). To quantify wavepackets in our calculations, we follow the procedure described by Gallaire & Chomaz Gallaire and Chomaz (2003). First, we define the disturbance amplitude $A(x,t)$ to be the square root of the Chu energy integrated in the wall-normal direction at each streamwise location $x$ and time $t$. At $t=0$, the amplitude $A(x,0)$ peaks at $x=x_{0}$, which we take as the initial position of the wavepacket. We then consider the spatio-temporal response of the linearized flow equations along rays of constant group velocity $v_{g}=(x-x_{0})/t$ resulting from this initial position. For the boundary layer with and without SWBLI, Figure 15 shows snapshots of the amplitude $A$ plotted against the group velocity $v_{g}$ at different instants of time throughout the optimal transient responses. For the boundary layer without SWBLI, Figure 15(a) reveals that amplification occurs for $v_{g}\in(0.7,0.9)$. This means that an observer moving in a frame of reference moving to the right with velocity $v_{g}=0.8$ will find that perturbations grow in time, thereby implying convective instability. For group velocities outside this range perturbations decay in time. The blue and red colors in this plot, respectively, indicate times before and after the wavepacket peak reaches the apex of the recirculation bubble. At early times, we observe amplification comparable to that in the entire boundary layer over the extent of the separating flow. Once the wavepacket reaches the bubble apex the amplification rate dramatically increases, suggesting the presence of a different physical mechanism for perturbation growth in the presence of SWBLI.

As discussed in Gallaire and Chomaz (2003); Delbende et al. (1998), the amplification of wavepackets may be linked quantitatively to linear stability theory for locally parallel flows. In a frame of reference moving with group velocity $v_{g}$, the temporal growth rate is

[TABLE]

where the $t^{1/2}$-dependence accounts for the natural spread of the wavepacket. Here, $t_{1}$ and $t_{2}$ represent two time instants which are taken far enough apart to yield an accurate approximation Delbende et al. (1998). In the fixed frame, the growth rate and the streamwise wavenumber are determined by

[TABLE]

where $\phi$ denotes the phase angle of the resulting wavepacket, and $\delta x$ represents a small change in the streamwise position. The second term on the right-hand-side of the expression for $\omega_{i}$ in Eq. (8) accounts for spatial growth when converting between moving and fixed reference frames. At the wavepacket peak, this term is zero and the growth rate equals the maximum temporal growth rate $\omega_{i,\text{max}}$ which is obtained by solving the stability problem for the linearization around locally parallel flow over all real wavenumbers.

For the boundary layer without SWBLI, the wavepacket peak shown in Figure 15 gives $\omega_{i,\text{max}}=0.0076$ at the streamwise wavenumber $\alpha=0.11$ for the spanwise wavenumber $\beta=0.6$. Linear stability theory predicts $\omega_{i,\text{max}}=0.004$ and $\alpha=0.15$, for the Mack first-mode instability at the same spanwise wavenumber. Qualitative agreement between these results indicates that the Mack first-mode instability plays a key role in the optimal transient response consisting of oblique waves observed in the boundary layer without SWBLI. Similarly, for the wavepacket response in the presence of the oblique shock, we apply Eqs. (8) to the blue and the red curves in Figure 15(b). The blue curves correspond to the time instances before the wavepacket reaches the apex of the recirculation bubble. From these curves, we obtain a maximum temporal growth rate of $\omega_{i,\text{max}}=0.01$ with wavenumbers $\alpha=0.11$ and $\beta=2.6$. These initial perturbations are significantly less oblique than those obtained for the setup without the SWBLI. Linear theory predicts stability of the flow at this spanwise wavenumber and the observed growth rate $\omega_{i,\text{max}}=0.01$ is not related to the Mack-first mode. The peak associated with the red curves in Figure 15(b) for the time instants after the wavepacket crosses the recirculation bubble apex yields a growth rate of $\omega_{i,\text{max}}=0.0768$, which is almost an order of magnitude larger than the growth rate of the upstream response. These structures correspond to the long streamwise streaks that form downstream and the streamwise wavenumber $\alpha=0.029$ is much smaller than that observed upstream.

IV.2 Inviscid transport analysis

Our wavepacket analysis illustrates that the mechanism responsible for transient growth in SBLI is different from from the mechanism in a flat plate boundary layer. To investigate this further, we examine the equation that governs the evolution of Chu’s compressible energy $E$ Sidharth et al. (2018),

[TABLE]

Here, $\mathcal{P}$ represents the production of perturbations resulting from the base flow gradients, $\mathcal{S}$ denotes the source term that corresponds to the perturbation component of the inviscid material derivative, $\mathcal{V}$ accounts for the viscous dissipation, and $\mathcal{T}$ arises from advection of perturbations by the base-flow velocity. We evaluate the contribution of each term for the optimal response at $\beta=2.6$ as it amplifies in time (for $T\in[50,214]$). Figure 16 demonstrates that the production term $\mathcal{P}$ provides the dominant source of energy amplification. The inset in Figure 16 shows that, at the time when energy achieves its largest value ($t=190$), the production term $\mathcal{P}$ is active in the region near and after the reattachment location at $x=155$.

To investigate physical mechanisms which contribute to $\mathcal{P}$, we analyze the dominant terms in the corresponding linearized inviscid transport equations. As shown in Figure 13, the streamwise velocity component contributes the most to the kinetic energy. Motivated by this observation, we closely examine the spatio-temporal development of the streamwise momentum and identify amplification mechanisms that result from the interactions of flow perturbations with base flow gradients. After accounting for the dominant production terms, the transport of streamwise velocity is given by

[TABLE]

where $D/Dt={\partial}/{\partial t}\,+\,\bar{U}{\partial}/{\partial x}\,+\,\bar{V}{\partial}/{\partial y}$ quantifies the spatio-temporal evolution of $u^{\prime}$ as it is advected by the base flow $(\bar{U},\bar{V})$. The first term on the right-hand-side results in the growth of $u^{\prime}$ because of the streamwise deceleration of the base flow when ${\partial\bar{U}}/{\partial x}<0$ Sidharth et al. (2018), whereas the second term accounts for the growth that arises from the lift-up mechanism Landhal (1977, 1980). Multiplying Eq. (10) with $u^{\prime*}$ and integrating along the wall-normal and spanwise directions yields,

[TABLE]

where, $y_{0}=y_{\text{max}}$ is the wall-normal extent of the domain and $z_{0}=2\pi/\beta$ (here, $\beta=2.6$). We quantify the contribution of the streamwise velocity component to the energy of flow perturbations by examining the two terms on the right-hand-side of Eq. (11).

Figure 17(a) shows the streamwise variation of the positive contribution that arises from the production terms as the flow perturbation evolve with time. Initially, the perturbations consist of streamwise vortices in the separating shear layer and they amplify via the lift-up mechanism. As perturbations develop in time and interact with the recirculation bubble the amplification mechanism changes and the streamwise deceleration term becomes dominant. To compare the spatial amplification resulting from these mechanisms, we also compute the time-averaged production terms. Figure 17(b) illustrates that streamwise deceleration dominates the production of streamwise specific kinetic energy. This is in concert with a recent study of SWBLI over a compression ramp Dwivedi et al. (2019) which demonstrated that streamwise deceleration generates significant spatial amplification of streamwise streaks. Furthermore, inside the bubble, there is almost no contribution from the lift-up mechanism and its effects are only present near the separation point and significantly downstream of the reattachment point.

In addition to transient growth, our SWBLI configuration also supports a weak global instability that appears in the form of spanwise periodic streamwise streaks downstream of the recirculation bubble; also see Hildebrand et al. (2018a); Sidharth et al. (2018). While the streaky structure of the global instability bears some similarity to the transient growth investigated in this paper, the global instability occurs for significantly lower spanwise wavenumbers ($\beta=0.25$). Also, the weak growth rate of the global instability ($\omega_{i}=2.5\times 10^{-3}$) induces a significantly smaller amplification over the time intervals for which the largest transient growth occurs. This difference in the most-amplified wavelength between global instability and transient growth analysis is a consequence of significant non-modal effects in high-speed boundary layers Dwivedi et al. (2019). These effects make the SWBLI configuration extremely sensitive to external disturbances Schmid (2007); Jovanović (2020) and, in the presence of experimental imperfections, they can trigger non-linear interactions, destabilize the reattached boundary layer, and induce transition to turbulence.

In summary, the wavepacket analysis of Section IV.1 shows that while the optimal response in the absence of SWBLI consists of oblique waves, the streamwise elongated streaks emerge in the presence of the shock. Furthermore, our analysis of the inviscid transport equation in Section IV.2 identifies physical mechanisms for transient growth and provides valuable insights into differences in the presence and absence of the shock. Even though local dispersion relations in both flows are similar before the separation location, in the absence of SWBLI most growth appears at the end of the domain. This growth arises from Mack’s first mode instability and it is triggered by the oblique initial conditions that are tilted against the mean shear. In contrast, in the presence of the SBLI, amplification mostly arises from streamwise streaky perturbations as they convect to the reattachment location. In this case, the optimal initial condition takes the form of streamwise vortices. These are initially amplified by the lift-up mechanism and result in flow structures that can undergo considerable growth by streamwise compression.

V CONCLUDING REMARKS

We examined optimal transient growth in a high-speed compressible boundary layer interacting with an oblique shock wave. To capture the effect of flow separation and recirculation caused by SWBLI, we utilize global linearized flow equations to identify the spatial structure of optimal initial perturbations and study the spatio-temporal evolution of the resulting response. For a hypersonic boundary layer without SWBLI, we verified that our approach correctly captures the dominant amplification that arises from the inviscid Orr mechanism, the lift-up effect, and the first mode instability.

To investigate the potential of shock wave/boundary-layer interactions to initiate transition, we examine transient growth of a hypersonic flat plate boundary layer with an oblique shock impinging on it. While the flow supports weak global instability, our analysis predicts large transient growth over short time periods. We demonstrate that the optimal initial condition is given by spanwise periodic vortices which commonly appear in the presence of leading-edge imperfections and distributed roughness in experiments. These initial perturbations results in the formation of streamwise streaks which are ubiquitous in experimental studies that involve SWBLI.

We also uncover physical mechanisms responsible for transient growth by quantifying the temporal evolution of dominant flow structures in various regions of the flow field. This is done by evaluating the contribution of the base-flow gradients to the production of streamwise kinetic energy in the inviscid transport equations. Our analysis demonstrates that streamwise streaks that emerge through the lift-up mechanism are significantly amplified by streamwise deceleration in the separation bubble near reattachment. Similar observations in other high-speed flow configurations with SWBLI, including compression ramps and double wedges, suggest that streamwise deceleration provides a robust physical mechanism for amplification of flow perturbations that solely arise in compressible separated flows. Our ongoing effort focuses on examining the influence of the Mach number as well as the initial angle of the incident shock and the angle after the incident shock interacts with the bow shock on transition mechanisms and dominant flow structures.

Transient growth analysis provides a useful framework for quantifying non-modal amplification mechanisms in complex high-speed flows even in the presence of global instabilities. This approach can provide important physical insights about the transition to turbulence, especially in experimental facilities where test times may be limited and where it is difficult to precisely control possible sources of external excitation.

ACKNOWLEDGMENTS

Financial support from the Office of Naval Research (ONR) under Awards N00014-15-1-2522 and N00014-17-1-2496 and from the Air Force Office of Scientific Research (AFOSR) under Award FA9550-18-1-0422 is gratefully acknowledged. The views and conclusions contained herein are those of the authors and should not be interpreted as representing the official policies or endorsements, either expressed or implied, of the ONR, the AFOSR, or the US Government.