Reattachment vortices in hypersonic compression ramp flow: an input-output analysis
We employ global input-output analysis to quantify amplification of exogenous disturbances in compressible boundary layer flows. Using the spatial structure of the dominant response to time-periodic inputs, we explain the origin of steady reattachment and Görtler-like vortices in laminar hypersonic flow over a compression ramp. Our analysis of the resulting shock/boundary layer interaction reveals that the vortices arise from selective amplification of upstream perturbations. These vortices cause heat streaks with a specific spanwise wavelength near flow reattachment and they can trigger transition to turbulence. The streak wavelength predicted by our analysis compares favorably with observations from two different hypersonic compression ramp experiments.
Compression corners are commonly encountered in intakes, control surfaces, and junctions. High speed flow on a compression corner is a canonical case of shock/boundary layer interaction (SBLI) (Simeonides & Haase, 1995) involving flow separation and reattachment with a shock system. Even though the compression ramp geometry is homogeneous in the spanwise direction, experiments (Roghelia et al., 2017) and numerical simulations (Navarro-Martinez & Tutty, 2005) show that the flow over it exhibits three-dimensionality in the form of streamwise streaks near reattachment. The streaks are associated with persistent large local peaks of heat transfer; they can destabilize the boundary layer and cause transition (Simeonides & Haase, 1995; Roghelia et al., 2017).
Recently, Roghelia et al. (2017) and Chuvakhov et al. (2017) investigated hypersonic compression ramp flows using temperature sensitive paint (TSP) and infrared (IR) imaging. These techniques were employed to study the formation of streamwise vortices and reattachment heat flux patterns. The observed structures are often attributed to Görtler-like vortices (Inger, 1977; Simeonides & Haase, 1995; Chuvakhov et al., 2017). These vortices develop from upstream perturbations (Hall, 1983) which are amplified by centrifugal forces that arise from concave streamline curvature near reattachment (Dwivedi et al., 2017). However, in most compression ramp studies, the comparison with theory of Görtler instability on curved walls is qualitative and does not account for the dynamics in the separation bubble. Zhuang et al. (2017) used nano-tracer planar laser scattering to visualize a Mach turbulent boundary layer turning on a compression ramp. They found that Görtler-like vortices not only appear after reattachment but also in the separation bubble. It is thus important to understand the effect of the recirculation bubble dynamics on the formation and amplification of streamwise vortices.
To include the effect of the separated flow, Sidharth et al. (2018) carried out a global stability analysis and discovered a 3D global instability in the separation bubble, which results in temperature streaks post-reattachment. The spanwise wavelength of the global instability scales with the recirculation length (Sidharth et al., 2017). This is in contrast to the spanwise wavelength observed for Görtler-like vortices (Chuvakhov et al., 2017; Roghelia et al., 2017; Navarro-Martinez & Tutty, 2005), which scale with the reattachment boundary layer thickness, indicating that the global instability is not responsible for their formation. To characterize the role of external perturbations in the formation of these vortices, we consider compression ramp flows that do not exhibit 3D global instability. External perturbations are amplified as they pass through the flow field and we utilize global input-output (I/O) analysis to quantify this amplification.
The I/O analysis evaluates the response (outputs) of a dynamical system to external perturbation sources (inputs). For time-periodic inputs, the transfer function maps the input forcing to output responses; see Figure 1 for an illustration. For small perturbations, the transfer function can be obtained by linearizing the compressible Navier-Stokes (NS) equations around a laminar base flow. The I/O approach has previously been employed to quantify amplification and study bypass transition mechanisms in incompressible channel flows (Jovanović, 2004; Jovanović & Bamieh, 2005), boundary layer flows (Brandt et al., 2011; Fosas de Pando & Schmid, 2017; Cook et al., 2018; Ran et al., 2018), and jets (Jeun et al., 2016). In this paper, the input-output pair representing the dominant response of the linearized dynamics is computed to quantify the amplification of upstream perturbations in shock/boundary layer interaction.
Our presentation is organized as follows. In § 2, we present the linearized model and provide a brief summary of the I/O formulation. In § 3, we compute the amplification in attached supersonic flat-plate boundary layers and verify our method against state-of-the-art approaches. In § 4, we evaluate the dominant input-output 3D perturbation fields around a 2D laminar hypersonic base flow on a compression ramp to explain the origin of streamwise vortices near reattachment. We conclude our presentation in § 5.
2 Formulation and background
The compressible NS equations for perfect gas in conservative form are given by
where is the flux vector and is the vector of conserved variables representing mass, momentum, and total energy per unit volume of the gas (Candler et al., 2015). We decompose the state vector into a steady base component and a perturbation component , . The evolution of small perturbations is then governed by the linearized flow equations,
where represents the compressible NS operator resulting from linearization of (1) around the base flow . A second order central finite volume discretization (Sidharth et al., 2018) is used to obtain the finite dimensional approximation of Eq. (2),
which describes the dynamics of spatially discretized perturbation vector .
In the present work, we are interested in quantifying the amplification of exogenous disturbances in boundary layer flows (Schmid, 2007). To accomplish this objective, we augment the evolution model (3) with external excitation sources
where is a spatially distributed and temporally varying disturbance source (input) and is the quantity of interest (output). In Eq. (4) the matrix specifies how the input enters into the state equation, while the matrix extracts the output from the state . An I/O relation is obtained by applying the Laplace transform to (4),
where denotes the initial condition and is the complex number. Equation (5) can be used to characterize both the unforced (to initial condition) and forced (to external disturbances) responses of the flow perturbations.
In boundary layer flows, the linearized flow equations are typically stable. Thus, for a time-periodic input with frequency , , the steady-state output of a stable system (4) is given by , where and is the frequency response
Here, denotes the complex-conjugate transpose, and are unitary matrices, and is the rectangular diagonal matrix of the singular values . The columns of the matrix represent the input forcing directions that are mapped through the frequency response to the corresponding columns of the matrix ; for , the output is in the direction and the amplification is determined by the corresponding singular value . For a given temporal frequency , we use a matrix-free approach (Dwivedi et al., 2018) to compute the largest singular value of . Note that denotes the largest induced gain with respect to a compressible energy norm (Hanifi et al., 1996), where () identify the spatial structure of the dominant I/O pair.
3 Validation: supersonic flat plate boundary layer
Before analyzing the amplification of disturbances in a hypersonic flow involving shock/boundary layer interaction, we apply I/O analysis to compute amplification in a supersonic flow over a flat plate. Our computations are verified against conventional approaches to demonstrate the agreement for canonical problems. Two amplification mechanisms are considered: two-dimensional unsteady acoustic amplification (Ma & Zhong, 2003) and three-dimensional steady lift-up amplification (Zuccher et al., 2005).
3.1 Two-dimensional unsteady perturbations: acoustic amplification
Local spatial instabilities corresponding to acoustic perturbations dominate the transition in high speed flat plate boundary layers (Fedorov, 2011). Using local spatial linear stability theory (LST) and direct numerical simulations (DNS), Ma & Zhong (2003) showed that perturbation frequencies result in spatial growth due to the local instability over a part of the domain. We consider I/O analysis at and compare the region of growth with that predicted from LST (Ma & Zhong, 2003); see Figure 2(a) for geometry. The base flow is computed using the finite volume compressible flow solver US3D (Candler et al., 2015) with cells in the wall-normal and cells in the streamwise direction. This resolution yields grid-insensitive I/O results.
As shown in Figure 2(a), we use the matrix in Eq. (4) to localize the disturbance input at downstream of the leading edge. This choice allows us to avoid large streamwise gradients in the base flow in the vicinity of the leading edge. The slow streamwise variation of the base flow implies that LST is approximately valid downstream of this location. Furthermore, this location is sufficiently upstream of the neutral point of the acoustic instability (). This ensures that any non-modal growth arising from the Orr-mechanism (Dwivedi et al., 2018) decays before the spatial growth rate of the local acoustic instability becomes positive. Sponge regions are used at the top and right boundaries to model non-reflecting radiation boundary conditions. We have verified independence of our results on the strength and the location of the sponge zones.
The output of interest is chosen to be the perturbation field in the entire domain, i.e., . Figure 2(b) shows the spatial structure of pressure perturbation in the principal output mode . We compute the local spatial growth rate from pressure at the wall , . Figure 2(c) shows that our I/O analysis correctly identifies the region of spatial instability and predicts growth rates that are close to those resulting from LST (Ma & Zhong, 2003). The difference can be attributed to the fact that LST does not account for the spatially-growing nature of the base flow.
3.2 Three-dimensional steady perturbations: lift-up mechanism
The spatially-developing boundary layer also supports significant growth of perturbations that are not related to a dominant eigenmode of the linearized dynamical generator. For example, the steady 3D streak-like perturbations, that result from the lift-up mechanism (Ellingsen & Palm, 1975), play an important role in transition induced by distributed surface roughness (Reshotko, 2001). Zuccher et al. (2005) used the linearized boundary layer (BL) equations to compute spatial transient growth and analyze this mechanism. For the verification purposes, we compare the spanwise wavelength of the maximally amplified streaks resulting from the linearized BL equations and the I/O analysis. We specifically consider the conditions in Zuccher et al. (2005) corresponding to a boundary layer on a 2D flat plate in a supersonic free-stream.
A grid with cells in the wall-normal and cells in the streamwise direction is used to compute the base flow and conduct I/O analysis. The input is localized to a plane at streamwise location , where denotes the plate length. As in the previous subsection, the output is the perturbation field in the entire domain. Due to homogeneity in the spanwise direction, 3D perturbations are of the form , where is the spanwise wavenumber. To capture the steady lift-up mechanism, we conduct the I/O analysis for . In Figure 3(a), we illustrate the -dependence of the gain resulting from the I/O analysis and the value of at which the largest spatial transient growth takes place (Zuccher et al., 2005). We attribute a slight difference in the location of maxima to different base flow profiles; while we use a numerically computed 2D base flow, Zuccher et al. (2005) used an analytical self-similar base flow profile. The input (shown in Figure 3(c)) consists of streamwise vortical perturbations and the output consists of a rapid development of streamwise velocity streaks. The algebraic nature of the growth (as opposed to exponential) is illustrated in Figure 3(b) for different . As expected, a large initial transient growth is followed by eventual downstream decay.
The above results show that I/O analysis correctly captures the physical mechanisms responsible for amplification in canonical supersonic flows. As we demonstrate in the next section, this analysis can also be readily applied to study the early stages of transition in complex hypersonic compression ramp flow with shock/boundary layer interaction.
4 Input-output analysis of hypersonic compression ramp flow
Streamwise streaks in wall temperature are often observed in compression ramp experiments. Although they are typically attributed to the centrifugal amplification that arises from concave flow curvature near reattachment (Navarro-Martinez & Tutty, 2005; Chuvakhov et al., 2017), quantifying amplification in the presence of a recirculation bubble is an open challenge. Herein, we employ the I/O framework to study the amplification of spanwise periodic upstream disturbances in the compression ramp flow and explain the origin of heat streaks at reattachment.
Recently, Roghelia et al. (2017) and Chuvakhov et al. (2017) reported multiple hypersonic compression ramp experiments in two different facilities with matched freestream Mach and Reynolds numbers. Temperature sensitive paint (TSP) and infrared thermography measurements of reattachment heat-flux wall patterns revealed quantitatively similar streaks. The effects of freestream Reynolds number and leading edge radius on the spanwise wavelength of the streaks were also reported. In Inger (1977); Navarro-Martinez & Tutty (2005), scaling of with the incoming boundary layer was analyzed. Our objective is to identify the streak wavelength that is selected by the shock/boundary layer interaction.
We consider the experiments performed in the UT-1M Ludwig tube (Chuvakhov et al., 2017) at Mach 8. The geometry consists of an isothermal flat plate with a sharp leading edge and wall temperature , followed by an inclined ramp with ; see Figure 4(a). Table 1 summarizes the two free-stream conditions that we considered in our study.
Figure 4(b) provides comparison of the experimental schlieren image with the 2D base-flow density gradient magnitude field that is computed using US3D (Candler et al., 2015). Apart from the fact that the experimental schlieren contains the side-wall shock, the overall agreement is excellent. Figure 4(c) uses variation of the Stanton number St along the surface (Schlichting & Gersten, 2016) to compare the heat fluxes resulting from experiment and simulations with different grid resolutions. Except near separation and post-reattachment regions, the 2D flow captures the heat flux trends correctly. The observed discrepancy can be attributed to the presence of 3D flow structures in the experiments. Since the flow is globally stable with respect to 3D perturbations (Sidharth et al., 2018), we conjecture that these flow structures arise from non-modal amplification of 3D perturbations around the 2D base flow. To verify our hypothesis, we employ global I/O analysis to quantify the amplification of exogenous disturbances and understand mechanisms that trigger transition in a hypersonic compression ramp flow.
Because of the persistence of heat streaks in the experimental TSP images, we confine our analysis to amplification of inputs with zero temporal frequency (i.e., we set ). Our I/O analysis of steady disturbances reveals the importance of incoming streamwise vortical perturbations as inputs which lead to reattachment streaks as the response. This observation holds for all values of that we consider in our study. We also note that responses to unsteady low-frequency inputs (with ) are nearly identical to the steady case (Dwivedi et al., 2018) and are omited for brevity. The I/O analysis is conducted on a grid with cells in the streamwise and cells in the wall-normal direction (labeled as G3 in Figure 4(c)). Via the proper selection of the matrix in Eq. (4), we restrict the inputs to the domain prior to separation (i.e., ) and choose the perturbation field in the entire domain as the output, . Numerical sponge boundary conditions are applied near the leading edge () and the outflow ().
Figure 5(a) shows the dependence of the gain G on . For both Reynolds numbers, the amplification curve achieves its maximum for a particular value of . This indicates that SBLI selectively amplifies upstream perturbations with a specific spanwise wavelength. For both free-stream conditions, the experimental estimates of resulting from the TSP images in Figure 5(b) agree well with the predictions of I/O analysis. Both approaches identify for maximally amplified streaks, where is the reattachment boundary layer thickness. This value is also consistent with previous studies (Inger, 1977; Navarro-Martinez & Tutty, 2005). We finally note that, in the presence of random upstream perturbations, corresponding to the peak in the amplification curve is predicted to be the dominant wavelength of the reattachment streaks (Nichols, 2018).
To gain insight into the structure of the most amplified flow perturbations, we examine the spatial evolution of the streamwise velocity and helicity perturbations for three different values of . These respectively identify the flow structures with small, dominant, and large spanwise wavelengths. Figure 6(a) shows the streamwise velocity perturbation magnitude for different values of along the 2D steady flow streamline through the boundary layer edge. After initial amplification post-separation (S), the amplification curve exhibits two distinct profiles in the separated shear layer. For small wavelengths, , saturates in the separated shear layer, persists over the length of the bubble, and undergoes rapid amplification near the reattachment (R). In contrast, for , we observe a steady amplification in the bubble followed by a weak amplification near reattachment. For the wavelength that corresponds to the largest amplification, , the large trends are followed until reattachment and significant amplification is achieved in the separated flow. Near reattachment, starts to follow the profile corresponding to flow perturbations with small .
These amplification trend are further illustrated in Figure 6(b) by plotting the corresponding perturbation helicity, , for different values of . Here, is the base flow velocity vector, is the perturbation vorticity vector, and the perturbation helicity determines the projection of on the base flow streamline. We note that a contribution from the separation bubble increases with increase in . Thus, the selected wavelength represents incoming vortical perturbations that grow post-separation and feed on the centrifugal effects at reattachment. In what follows, we refer to these spatially amplifying perturbations as ‘reattachment vortices’.
We now demonstrate that this post-separation amplification in the reattachment vortices is associated with the growth of perturbations in the recirculation bubble. We compare and contrast this mechanism with Görtler vortices where the growth exclusively arises from centrifugal effects. To quantify the difference, we repeat the I/O analysis in the absence of perturbation development in the recirculation bubble for higher value of the Reynolds number. By introducing a numerical sponge zone within the bubble, we ensure that the perturbation growth is dominated by centrifugal effects. In this case, the I/O analysis reveals streamwise Görtler vortices as dominant outputs and the amplification map is shown in Figure 7(a). We see that eliminating the role of the bubble causes the spanwise wavelength that corresponds to the largest amplification to decrease from to . The difference between the Görtler and the reattachment vortices is illustrated by plotting the corresponding perturbation helicity. While the amplification in the separation bubble for the reattachment vortices is caused by streamwise vortical motions in the separation zone, Görtler vortices only consist of counter-rotating streamwise vortices that travel parallel to a surface (akin to an artificial wall with the curvature properties of the separation streamline).
Figure 7(b) shows the evolution with streamwise coordinate of and the Görtler number close to the separation streamline, where, and respectively denote the local viscous length scale and radius of curvature. The spatial growth of Görtler vortices is compared with that of reattachment vortices. Both experience strong amplification in regions of high but the difference is attributed to the amplification of reattachment vortices in the separated shear layer. We note that, for small values of , the I/O analysis of the shock/boundary layer interaction reveals Görtler vortices as the dominant response. Their negligible spatial growth in the separated shear layer was observed in Figure 6(b) in the presence of perturbation dynamics in the bubble. This is consistent with Figure 7(a), which shows almost identical gains for small .
In summary, we have quantified the amplification of 3D upstream perturbations in a 2D base flow with shock/boundary layer interaction. The I/O analysis demonstrates that vortical perturbations with a specific spanwise wavelength are most amplified. In addition to centrifugal effects near reattachment, perturbation evolution in the recirculation bubble also contributes to their amplification.
5 Concluding remarks
We have employed an input-output analysis to investigate amplification of disturbances in compressible boundary layer flows. Our approach utilizes global linearized dynamics to propagate flow perturbations and analyze the spatial structure of the dominant response. We have verified that I/O approach captures both acoustic and vortical spatial growth mechanisms on a supersonic flat plate boundary layer without any a priori knowledge of the perturbation form.
In an effort to explain steady heat streaks near reattachment, we have also examined the origin of Görtler-like vortices in shock/boundary layer interaction on Mach 8 flow over compression ramp. In spite of global stability, the I/O analysis predicts large amplification of incoming streamwise vortical perturbations with a specific spanwise length-scale. We have demonstrated that, in addition to centrifugal effects, perturbations in the separation bubble significantly contribute to amplification of streamwise reattachment vortices and that our predictions agree well with two recent experimental observations.
The I/O approach provides a useful framework to quantify the spatial evolution of external perturbations in shock/boundary layer interactions. Improved understanding of amplification mechanisms can provide important insight and pave the way for the development of predictive transition models in complex high speed flows.
We acknowledge support from the Air Force Office of Scientific Research (FA9550-12-1-0064 and FA9550-18-1-0422) and the Office of Naval Research (N00014-15-1-2522).
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