Boundary layer separation and friction drag form key delimiting phenomena that subvert the aerial platforms from achieving greater efficiency and accessing wider operation envelope. Both these phenomena are significantly aggravated in supersonic platforms due to the interactions between shock waves with the boundary layer that develops over the vehicle surface and within the engines. The present work demonstrates a new paradigm that leverages the native or programmable material properties of the aerostructures to engender simultaneous reduction in the separation scales and mitigation of skin friction drag. As a first step toward realizing this paradigm, the present work demonstrates, for the first time, the simultaneous skin friction drag mitigation in a Mach 2.5 boundary layer and control of shock induced boundary layer separation, both using viscoelastic implants placed under the flow. It is experimentally demonstrated that the appropriately chosen viscoelastic materials can simultaneously reduce the skin friction coefficient at the measurement location by 11% and mitigate the size of a large-scale separated flow by up to 28%. The reported performance matches the current generation flow effectors in both separation scale and skin friction mitigation. The present study opens a new application space for soft/programmable materials in high speed aerial vehicles.

A viscous boundary layer that develops over moving platforms plays a prominent role in determining the aerodynamic drag and performance metrics of the vehicles.1 The boundary layer is often turbulent in high speed vehicles that results in elevated drag, mostly in the form of skin friction. Furthermore, the obstructions presented to the flow along the vehicle surface and the pressure gradients presented to the boundary layer cause them to separate from the parent surface that further amplifies the drag in the form of pressure drag.1,2 While the flow separation is an inhibitory phenomenon at all speeds, the presence of the accompanying separation shock at high speeds causes debilitating large amplitude unsteady loads that necessitated heavier structures. Decades long investigations made to illuminate the governing physics of turbulent boundary layers and boundary layer separation are documented in multiple review articles3–7 and textbooks.2 

Two approaches have been successfully adopted to mitigate turbulent skin friction drag: (1) altering near-wall streak formation dynamics and (2) repositioning turbulent streaks away from the wall. A variety of flow devices, both active and passive control types, have been developed to leverage the above-mentioned approaches to mitigate skin friction drag.8,9 Some of the most common devices in high speed flows that use passive control, such as using riblets,10,11 surface texturing, etc.; similarly, mass injection and boundary layer suction are a couple of examples of active control strategies, whereas the passive approaches have a relatively narrow flow conditions where they are effective, the active methods demand additional implementation infrastructure in return for a larger effectiveness envelope.

Mitigating shock induced separation has also been successfully demonstrated using passive devices, such as vortex generators,12–18 cavities,19 vanes, etc., as well as active methods like plasma discharges,20–22 steady/pulsed jet injection,23–28 shock control bump,29–31 and surface morphing,32–34 to name a few.25,35 The operating principle of many of these devices is to energize the boundary layer that makes them more resistant to separation; however, this invariably results in greater skin friction drag. Other devices, such as the shock control bumps and morphing, generate a pre-compression shock system that weakens the compression delivered by the main shock; these devices, however, did not make a reported impact on the overall skin friction drag. Thus, at present, no device has been shown to simultaneously reduce the skin friction and mitigate the separation scale in high speed flows. The present study fills this void by employing viscoelastic material implants that behave as passive flow effectors to mitigate both shock-induced flow separation and boundary layer skin friction.

There has been a decadal work on the use of compliant materials in drag reduction in liquid flows36 and the recent works of Zhang et al.37 illuminated how the turbulent events can interact with the compliant material dynamics. The interest in the application of viscoelastic materials for high-speed aerodynamic control stems from Pham et al.,38 who demonstrated the potential of the separation scale mitigation in supersonic flows with the introduction of a viscoelastic material (polyurethane rubber with a shore hardness of 60A) implanted beneath the flow. The rubber implant was demonstrated to mitigate the mean separation scale by an average of 25% over separation sizes between two and four boundary layer thicknesses. A further 30% reduction in the unsteady pressure loading due to shock oscillations was also reported in that study, which mitigates aerostructure fatigue. While the results are encouraging, the separation sizes studied in Pham et al. were rather modest compared to many of the operating points of a vehicle's flight envelope; this raises the question of scalability of this approach with large separation size. Furthermore, Pham et al. did not address if there is additional skin friction drag penalty with using the viscoelastic implants as observed in other traditional separation control actuators. The present work addresses both these open questions by investigating the shock induced separation as well as the turbulent boundary layer development over viscoelastic materials at practically relevant scales.

The experiments were performed in NCSU's supersonic wind tunnel facility at Mach 2.5. The wind tunnel is a blowdown-type facility with 8 s of run time that provided freestream pressure with less than 2% drift. The test section measures 150 × 150 mm2 in cross section and 650 mm in length. More details about the tunnel operation can be found in earlier works on this facility.39–42 The test article [schematic in Fig. 1(a)] was made of a steel plate [355.6 mm (l) × 101.6 mm (w) × 10.5 mm (t)] with a sharp leading-edge and two side fences (25 mm tall) that extended along the length of the plate. These side fences helped generate large separation sizes and also engendered a three-dimensional separated flow with a representative topology as found in supersonic engine inlets and other internal flows. The test article was placed in the freestream at nominally zero pitch and yaw angles for all the experiments. This allowed a fresh boundary layer to develop along the plate and side fences. The boundary layer naturally transitioned to turbulence within a few mm from the plate leading edge. A compression ramp was placed 300 mm from the plate leading edge to generate flow separation. The ramp leading edge was taken at x = 0 and positive along the freestream direction. Two ramp angles (20° and 24°) were implemented to produce separation sizes of 8 and 12 boundary layer thicknesses (δ), respectively. The ramp was removed when the boundary layer skin friction investigations were made. Multiple campaigns of mean pressure imaging over the surface without adding a compression ramp evidenced that the boundary layer develops under nominally zero pressure gradient. Integral boundary layer properties measured at x = −50 mm are summarized in Table I.

FIG. 1.

(a) Schematic of the experimental setup and (b) mean surface undulation map [w(x, z)] of the viscoelastic implant obtained using a 3D profilometer.

FIG. 1.

(a) Schematic of the experimental setup and (b) mean surface undulation map [w(x, z)] of the viscoelastic implant obtained using a 3D profilometer.

Close modal
TABLE I.

Boundary layer integral properties over various viscoelastic materials.

Config.δ (mm)δ* (mm)θ (mm)H0Cf × 10−3ysonic (mm)ps/p FIT
Rigid 4.4 0.5 0.39 1.29 3.05 0.06 1.66 
60 A 4.4 0.59 0.45 1.32 3.0 (−1.6%) 0.077 1.66 
20 A 5.0 0.79 0.59 1.33 2.85 (−6.7%) 0.11 1.65 
00–10 5.7 0.9 0.7 1.34 2.7 (−11.5%) 0.126 1.63 
Config.δ (mm)δ* (mm)θ (mm)H0Cf × 10−3ysonic (mm)ps/p FIT
Rigid 4.4 0.5 0.39 1.29 3.05 0.06 1.66 
60 A 4.4 0.59 0.45 1.32 3.0 (−1.6%) 0.077 1.66 
20 A 5.0 0.79 0.59 1.33 2.85 (−6.7%) 0.11 1.65 
00–10 5.7 0.9 0.7 1.34 2.7 (−11.5%) 0.126 1.63 

A 4 mm (t) recess [250 mm (l) × 57 mm (w)] was made on the plate to allow viscoelastic materials to be implanted into the plate; the depth of the recess was sufficient to realize the bulk material properties of the inserted materials. Polyurethane rubbers with different hardnesses: 00 Shore 10, A Shore 20, and A Shore 60 (Young's modulus: 0.3, 0.8, and 2.2 MPa, respectively, see Table II) formed the viscoelastic material implants. The rubbers used were two-part liquid pour rubbers that naturally adhered to the steel during the curing process. After the rubber was poured into the recess, the model was placed in a vacuum oven to go through a degassing process before curing. The mean undulations of the surface [w(x, z)] after curing were mapped using a 3D profilometer (Filmetrics Inc., model: Profilm3D). The undulation map over 1.5  × 1.5 mm2 area shown in Fig. 1(b) reveals a smooth finish and the spatial r.m.s. undulation collecting w(x, z) across the spatial domain was 0.49 μm, which is well below the roughness limit of the boundary layer.

TABLE II.

Young’s modulus (Y) of the different rubber materials used in the present study.

Shore hardnessY (MPa)
60 A 2.2 
20 A 0.8 
00–10 0.3 
Shore hardnessY (MPa)
60 A 2.2 
20 A 0.8 
00–10 0.3 

Surface streakline visualization provides a qualitative picture of wall shear and serves to delineate the major surface topological features of shock-induced separation unit. This visualization method involves applying the area of interest with a viscous oil dissolved with a fluorescent dye. The differential wall shear at various regions of the flow helps visualize the separation and reattachment loci and different vortices present in the flow unit. In this work, a 400 nm light illuminated the pigment laden oil and the red fluorescence emission from the pigment was recorded as a movie sequence using a DSLR camera (Nikon Inc., Model: D80) fitted with a F/1.4 85 mm lens and a 600 nm lowpass filter. One hundred frames that covered a duration of 3.33 s of run time was averaged to present the mean streakline patterns.

Pressure sensitive paint (PSP) was used to quantify the mean surface pressure fields and add a redundant quantification of the shock-induced separation locus. A commercial paint mixture (ISSI Inc., Model: uniFIB) was spray painted on the model surface to obtain a very thin coat (average thickness measured to be 12.5 μm) and the surface finish over a similar paint was measured to have an r.m.s. roughness of 1.5 μm. The painted model surface was illuminated by two ultraviolet lamps (400 nm wavelength). A sCMOS camera (Photron Inc., NOVA-S16) fitted with a F/1.4 85 mm lens and a 600 nm bandpass filter collected the fluorescence of the paint. The field of view was centered around the mid-span of the compression ramp (z = 0) and covered 160 mm (x) × 120 mm (z), with a digital resolution of 0.25 mm/pixel. Mean pressure fields were calculated from the PSP images using intensity normalization method described in Refs. 43 and 44; the uncertainty in mean pressure was 1%.

Particle image velocimetry (PIV) was employed to obtain the off-surface flow velocity from which the skin friction coefficients were calculated. The method involved illuminating the externally injected neutrally buoyant tracer particles (silicon oil droplets of 0.2 − 0.3 μm agglomerated size) by two time separated laser sheets (λ = 532 nm) and imaging the resulting laser scattering by a frame straddling interline CCD camera (PCO Inc., model: PCO2000). The laser and camera timings were controlled and synchronized to within 1 ps using a low jitter delay generator (Stanford Research Systems, Inc., model: DG645). The field of view measured was 73.5  × 73.5 mm2 centered around x = −25 mm with a digital resolution of 0.072 mm/pixel.

The processing of the particle scattering images to render the velocity fields was performed using DaVis 8.4 software. Multi-pass processing was performed using a 50% overlap followed by outlier detection resulted in nominal vector dropouts of <10% for all datasets. These removed vectors were replaced through interpolation and a smoothing filter. The spacing between successive velocity vectors was 0.56 mm.

Surface streakline visualization of the separated flow is first compared between rigid surface and rubber implant to obtain a global picture of the individual flow units. Figure 2(a) presents a side by side comparison of the mean surface streakline field of the separated flow over the rigid and rubber implant (20A shore hardness) generated by the 24° compression ramp. The separation locus is identified as the band of pigment accumulation caused by the absence of wall shear stress at separation. The separation locus is noticeably curved in both rigid and rubber surfaces due to the influence of corner vortices (“CV”), which entrain the fluid from the separation bubble. A strong reversed flow composed of two counter-rotating vortices that are symmetrically placed about the mid span dominated the separation flowfield. Unfortunately, the pigment could not adhere with enough density to create high resolution patterns in the reattachment region. Nonetheless, observing the patterns closely, it was observed that the region of the reattachment was quite three dimensional and was indeed dominated by the corner flows. The resulting reattachment locus (“R”) is also curved in a manner that can be described as a mirror image of the separation locus. Thus, the overall flow topology is identical between the rigid plate and rubber implant other than a slightly less curved separation contour with the rubber insert compared to the rigid plate, which is more evident in the pressure contours of Fig. 2(b).

The mid-span location of the separation and reattachment (xS and xR) are used for making comparisons between the rigid and viscoelastic implants. The separation size, Lsep, defined as distance between mid-span xS and xR was 51 mm (≈ −12δ) for the rigid plate, which is over three times larger than the typical separation size that have been employed in the past to demonstrate separation control. The mid-span xS with rigid plate and 20A shore hardness rubber implant are x = −38 and x = −31 mm, respectively; this demonstrates that the separation onset occurs 16% downstream. It is also observed that the xR of the rigid and rubber implant configurations are located at xR = +15 and xR = +13 mm, respectively. Thus, an overall reduction in the separation size of 17% was observed due to introduction of the rubber implant.

FIG. 2.

Demonstration of the separation scale mitigation with the boundary layer developing over the viscoelastic surface: (a) side by side comparison of the mean surface streakline pattern surrounding the shock induced separation between the rigid and viscoelastic (20A shore hardness) surface and (b) corresponding mean surface pressure field.

FIG. 2.

Demonstration of the separation scale mitigation with the boundary layer developing over the viscoelastic surface: (a) side by side comparison of the mean surface streakline pattern surrounding the shock induced separation between the rigid and viscoelastic (20A shore hardness) surface and (b) corresponding mean surface pressure field.

Close modal

In addition to the surface streakline fields, the pressure fields beneath the separation until between the rigid and 20A are also compared to provide another surrogate (redundant) measurement of the separation locus onset. Per the free interaction theory, the ratio of pressure at separation to the freestream pressure, ps/p, is related to the incoming boundary layer skin friction coefficient Cf and the freestream Mach number M as

psp=1+F×γ×M2×Cf0.52×(M21)0.25.
(1)

In the above-mentioned equation the constants F = 4.2 and the specific heat ratio γ = 1.4 for air. A line contour corresponding to the theoretical pressure at the separation line ps/p based on the free interaction theory is presented. It is seen that the separation onset based on the wall pressure occurs at x = −36 and x = −31 mm in the mid-span location corresponding to the rigid plate and rubber implant configurations, respectively. This corresponds to a downstream shift in the separation onset of 14%, which is nearly identical to the value from streakline visualization. As such, no analytical formulation is available to identify the reattachment pressure, and hence the reattachment locus is not delineated using pressure fields.

Figure 3 presents the separation size reduction by the rubber implant (60A shore hardness) across different separation sizes. The datasets from the current study and Pham et al.38 are presented in Fig. 3. The separation locus was determined from the surface streakline patterns in both studies and consistent oil/dye mixture formulations were made in both the studies to make a reliable comparison of the separation scale mitigation. It can be observed that the effectiveness of rubber implant decreases with increasing separation size and the overall reduction in Lsep is between 25% and 14% across the separation sizes tested. A very similar performance variation was also observed using the 20 A shore hardness rubber implant.

FIG. 3.

Variation of the shock-induced separation scale percentage mitigation with increasing separation size.

FIG. 3.

Variation of the shock-induced separation scale percentage mitigation with increasing separation size.

Close modal

The literature studies on shock-induced separation mitigation are mostly restricted to separation sizes less than about 5δ. Within the available literature on traditional separation control actuators, the separation scale mitigation is between 10% and 20% for Lsep < 5δ. A notable exception to the separation scales tested is from Funderburk and Narayanaswamy,45 where the authors considered separation mitigation at larger scales. Funderburk and Narayanaswamy45 used a vortex generator array on a xS = −8δ (28 mm) separation and showed an 18% reduction in the separation size. These results show that the separation mitigation with the rubber implant is very comparable to the traditional devices. Furthermore, the viscoelastic implants are effective across a wide range of separation scales encountered during a flight envelope and make demonstrable reductions in separation size whose topology is very representative of practical settings.

Boundary layer velocity profiles over the rigid surface and various viscoelastic implant surfaces were measured using particle image velocimetry technique at x = −50 mm (averaged over −55 mm < x < −45 mm) with the compression ramp removed. Interesting trends can be observed in Fig. 4(a) for the boundary layer developing over the rubber implant. First, the overall boundary layer profiles for all the rubber implants adhere to the canonical equilibrium turbulent boundary layer profile; this was evidenced by our ability to fit standard compressible equilibrium boundary layer profiles to the experimental data without appreciable scatter [Sun and Childs46 fit shown in Fig. 4(a)]. This adherence evidence that having the boundary layer develop over the viscoelastic surface does not fundamentally alter the boundary layer's composite structure. Furthermore, the ability to fit standard profiles allows the determination of important integral properties of the boundary layer that is not otherwise possible because of the lack of experimental data in the near wall region 0 <y < 1.5 mm. Table I presents the different integral properties of interest to this study using the fit provided by Sun and Childs46 to complete the boundary layer profile in the near wall region.

FIG. 4.

(a) Collection of the measured boundary layer profiles over a rigid surface and different viscoelastic implant surfaces. The corresponding boundary layer profile fits from Sun and Childs44 are shown as black solid lines. (b) Variation of Cf with increasing material softness quantified by the Young's modulus. (c) Measured storage and loss modulus of the 20A rubber sample over frequency between 0 and 150 Hz.

FIG. 4.

(a) Collection of the measured boundary layer profiles over a rigid surface and different viscoelastic implant surfaces. The corresponding boundary layer profile fits from Sun and Childs44 are shown as black solid lines. (b) Variation of Cf with increasing material softness quantified by the Young's modulus. (c) Measured storage and loss modulus of the 20A rubber sample over frequency between 0 and 150 Hz.

Close modal

Among the different quantities of interest, the skin friction coefficient (Cf) is one of the most important since it directly relates to the skin friction drag and also dictates the pressure rise required to cause flow separation [see Eq. (1)]. Figure 4(b) and Table I show that Cf exhibits a consistent decrease with increasing softness of the rubber implant. For the softest rubber (lowest shore hardness), the Cf is lower than that of the rigid plate by 11.5%. To provide a comparison with more traditional approaches, optimized riblets have typically resulted in a Cf reduction of 7%47 and their effectiveness decreases with increasing velocity. Similarly, external blowing and other approaches to break up the outer scale structures have resulted between 10% and 15% reduction in Cf9 in low speed flows; a notable exception is Abbassi et al.48 where the authors have reported a 30% drag reduction. Other works that create body force using plasma actuators have produced up to 50% drag reduction at low velocities, and the performance significantly degrades above incompressible speeds. These comparisons evidence a similar performance with using soft surfaces as compared to traditional devices toward skin friction drag reduction at elevated speeds.

It should be remarked that the material property that was employed to scale the skin friction coefficient in Fig. 4(b) was the Young's modulus (storage modulus). However, this choice does not capture the complex physical interactions that arise because of the elastic and viscous properties of the viscoelastic materials. To address this issue, the storage and loss modulus of the 20A shore hardness rubber was measured using a dynamic mechanical analyzer. The measured loss modulus that captures the viscous (“liquid-like”) behavior in the rubber implants was less than 10% of the storage modulus [Fig. 4(c)], which shows that the interactions are dominated by the elastic (“solid-like”) behavior of the viscoelastic materials and the role of the viscous behavior of these implants is rather modest. In simple terms, the storage modulus determines the energy that can be stored in a material to be released at a later time as desire; one may envision an elastic panel that can store the mechanical energy by its deflection and release the energy once the deflecting force is removed. The loss modulus determines the amount of energy that is lost by the material. Once again taking the same analogy, one can envision deforming a viscous liquid by applying a mechanical force (energy); however, the liquid will never regain its original shape when the force is released. In other words, the mechanical energy is lost to the material. For the rubber inserts employed, the comparatively minor contributions from viscous interactions justify the use of the Young's modulus to scale the skin friction coefficient. Within the elastic behavior, there are several underpinning mechanisms of interactions between the viscoelastic surface and the boundary layer flow. Within the scope of the viscoelastic interactions, one can also entertain the possibility of the formation of transient bulging and depressions on the implant surface. The present work addressed this possibility by studying the laser scattering from the viscoelastic surfaces in the raw images used for PIV data processing. No obvious evidence of static or transient wall normal deformations were obtained in over 300 instantaneous scattering fields. Note, however, that the measurement resolution is rather coarse (100 μm) and there are possibilities that transient surface undulations of lower amplitudes are indeed present that were not measured. Furthermore, no noticeable residual undulations were visually observed at the end of each test run, which discounts macroscale static deformations to the material surface as a contributing factor. The viscoelastic surface/flowfield interaction mechanisms are as such poorly understood and are expected to lead to several non-dimensional parameters and functional relationships to obtain a full representation of the causative agents of skin friction coefficient mitigation. Detailed investigations of the turbulent processes over the viscoelastic surfaces are beyond the scope of this manuscript.

While the appreciable reduction of skin friction with increasing rubber softness by itself has application for drag mitigation over a broad scope of vehicle configurations, the fact that the viscoelastic implants of certain shore hardness cause a dual separation mitigation and skin friction drag reduction makes this application very unique from other traditional separation control devices that invariably cause additional drag by the control device. Thus, the physics-based optimal integration of soft material can lead to access to greater vehicle performance envelope and simultaneously improve their overall efficiency. In the following paragraphs, the governing physics that result in the separation scale mitigation is postulated.

The reducing trend in Cf and the boundary layer velocity profiles presented in Figs. 4(a) and 4(b) show that the boundary layer is less full (have lower overall momentum) with increasing rubber softness. As a result, the boundary layer is expected to separate more readily. The fact that a reduction in the separation scale is observed is rather counter intuitive to this general argument. In fact, the energization of the boundary layer has been the key strategy that many separation control devices that have been deployed. This suggests that a different mechanism is being accessed by the flowfield over the viscoelastic implants that results in the observed reduction in separation.

The underpinning mechanism is driven by the wall-normal location of the sonic point, i.e., the location where the local flow Mach number is unity. Table I shows that the sonic velocity location monotonically elevates with decreasing shore hardness and reaches a wall normal location that are 86% (20A rubber) and 110% (00–10 rubber) higher than that of the rigid plate. The higher elevation of the sonic point causes a greater dissipation of the separation shock to compression waves in the near wall region with decreasing shore hardness. The increase in shock dissipation, in turn, causes the wall pressure increase along the separation shock to spread over a greater streamwise distance and a corresponding decrease in the streamwise pressure gradient presented to the incoming boundary layer. The evidence of this mechanism can be seen in Fig. 5(a) that presents mid-span pressure evolution along the separation bubble. The pressure gradient in the vicinity of the separation shock foot where the onset of pressure increase occurs [marked in Fig. 5(a) and plotted in Fig. 5(b)] can be observed to monotonically decrease with decreasing shore hardness. Figure 5(b) also shows that the pressure gradient in the vicinity of the separation (averaged over 1.05 <pw/p< 1.30) over the rubber surface is lower than that of the rigid wall below a certain threshold shore hardness. For example, whereas the pressure gradient was lower than that of the rigid for the 20A and 00–10 rubber inserts, the gradient was greater than the rigid surface for a harder 60A rubber insert.

FIG. 5.

(a) Center span surface pressure variation with rigid and viscoelastic surfaces. Note that the minor variation in ps in the rubber materials was not included. (b) Evolution of the pressure gradient in the vicinity of the separation shock with the material Young's modulus. (c) Percentage change in the separation size with respect to the separation size over the rigid plate. The shock induced separation was generated using a using a 20° compression ramp in [(a) and (b)] and a 24° compression ramp in (c).

FIG. 5.

(a) Center span surface pressure variation with rigid and viscoelastic surfaces. Note that the minor variation in ps in the rubber materials was not included. (b) Evolution of the pressure gradient in the vicinity of the separation shock with the material Young's modulus. (c) Percentage change in the separation size with respect to the separation size over the rigid plate. The shock induced separation was generated using a using a 20° compression ramp in [(a) and (b)] and a 24° compression ramp in (c).

Close modal

Another consequence of the increased separation shock dissipation is that the onset of pressure increase occurs at a more upstream location. This monotonic trend can be observed in Fig. 5(a). Figure 5(a) also shows that the onset of the pressure rise occurs upstream of that of the rigid plate in 20A and 00–10 shore hardness rubber inserts. Interestingly, the pressure onset location for the 60A rubber insert was measurably downstream compared to the rigid plate, suggesting a threshold shore hardness to observe the upstream shift. The upstream shift of the pressure onset locations hastens the separation since, for a given pressure gradient near separation shock, the wall pressure reaches the separation pressure (pS) at a more upstream location when the pressure rise onset shifts upstream.

Finally, the reduction in Cf also slightly reduces the pressure at separation onset; Table I tabulates the ps/p for the different rubber implants used in this study based on the free interaction theory (FIT). The balance between these multiple effects ultimately determines the final outcome on the separation size, shown in Fig. 5(c) across different rubber inserts. For example, whereas the softest (00–10) rubber implant resulted in the lowest pressure gradient and the thickest incoming boundary layer; however, the earlier pressure rise onset caused an aggravation of the separation size [see Fig. 5(c)]. Therefore, the optimal choice for the rubber implant for separation mitigation is driven by the ability to predict both the pressure gradient near the vicinity of pressure rise, the boundary layer thickness, and the modified pressure onset location. This choice is expected to have a dependence on the inflow Mach number, separation strength, and the local wall shear in the rigid surface.

The authors have no conflicts to disclose.

James Walz: Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (lead); Software (equal); Validation (equal); Visualization (equal). Venkateswaran Narayanaswamy: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

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