# The Length and Time Scales of Transitional SBLIs

Nikhil Mahalingesh, Sébastien Piponniau, Pierre Dupont

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Nikhil Mahalingesh, Sébastien Piponniau, Pierre Dupont. The Length and Time Scales of Transitional SBLIs. Pawel Flaszynski; Holger Babinsky; Piotr Doerffer. Towards Effective Flow Control and Mitigation of Shock Effects in Aeronautical Applications, 201, Springer Nature Switzerland, pp.105-125, 2025, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 978-3-031-86604-3-10.1007/978-3-031-86605-0_6. hal-05236159

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# The Length and Time Scales of Transitional SBLIs

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Nikhil Mahalingesh<sup>D</sup>, Sébastien Piponniau<sup>D</sup>, and Pierre Dupont<sup>D</sup>

**Abstract** Shock-wave boundary layer interaction is a commonly found flow phenomenon in transonic and supersonic aerodynamics. However, interactions involving laminar boundary layers have received relatively less interest, due to the complexity of transition. As a result, experiments were performed to study laminar boundary layers and their interaction with shock-waves. The Mach number was 1.65 and the unit Reynolds number was varied between 5.6 and 11 million m⁻¹. Pitot probes and hot-wire anemometry were employed for flow measurements. Experiments of the transition process of a natural laminar boundary layer captured the modal growth mechanisms of the primary instability, and a new time scale was found in the latter stages of the transition process. A new multi-sensor hot-wire probe was developed to study this new time scale, which revealed strange physical properties. Experiments of transitional SBLIs were performed on a 6° and a 10° compression ramp. A new non-dimensional parameter was developed for scaling the strength of the imposed shock, that was able to reconcile the large scatter in a diverse collection of length scales of transitional interactions. Measurements of the boundary layer transitional mechanisms over the interaction showed an accelerated growth over the separated shear layer, but surprisingly the growth of sub-harmonic instabilities was bypassed at reattachment. Finally, low-frequency unsteadiness at separation was found at Strouhal number of 0.05, similar to other studies on transitional interactions. A possible link between the presence of non-linearities over the separated shear layer and the low-frequency unsteadiness was found.

**Keywords** Transonic flow · Shock wave—boundary layer interaction · Boundary layer transition · Low-frequency unsteadiness · Measurements

<footnote>N. Mahalingesh · S. Piponniau (✉) · P. Dupont
CNRS, IUSTI, Aix Marseille University, Marseille, France
e-mail: sebastien.piponniau@univ-amu.fr</footnote>

<footnote>N. Mahalingesh
e-mail: nikhilmahalingesh@gmail.com</footnote>

<footnote>P. Dupont
e-mail: pierre.dupont@univ-amu.fr</footnote>
# 1 Introduction

Shock wave Boundary Layer Interaction (SBLI) is a classical flow phenomenon of high speed aerodynamics, which are encountered in various practical situations such as engine inlets, transonic wings, rocket nozzles, compressors, etc. SBLIs generally occur when a boundary layer confronts an adverse pressure gradient, which can result in a number of unfavourable effects including separation of the boundary layer, increased viscous dissipation, intense thermal loading, structural fatigue, and also unsteadiness of the associated shock-waves. These issues can lead to dangerous loss of control or even catastrophic failure of aerospace components, thus restricting the design and envelope of air-borne vehicles. Consequently, SBLIs have been the subject of research since the 1940s.

Experiments and numerical simulations have been performed for various geometrical configurations (such as oblique shock impingement, compression ramps, aerofoils, compressor and/or turbine blades, etc.) and for a range of flow parameters, including various Mach numbers, Reynolds numbers, and flow deflections. These studies have made great strides in the understanding of the length and time scales of SBLIs. However, limitations from an experimental and computational point of view, had restricted much of the focus on the interaction between shock waves and turbulent boundary layers. And hence, attention to the interaction between shock waves and laminar boundary layers had been relatively lower. Recent advancements in experimental techniques and computational resources, together with emerging focus on sustainability and increased efficiency, have renewed interest on laminar interactions.

The European TFAST (Transition Location Effect on Shock Wave Boundary Layer Interaction) project was one such initiative, with the objective to understand the interaction between shock waves and laminar boundary layers in more detail [38]. As part of this project, experiments and numerical simulations were performed at various academic and industrial institutions across Europe for a period of five years between 2012 and 2017. Although significant progress was made as a result of this project, several open questions were left unanswered, including the complex effects of boundary layer transitional mechanisms on the spatial and temporal scales of such interactions [11].

The current work is an extension of those experiments, being performed as part of the subsequent European project, called TEAMAero (Towards Effective Flow Control and Mitigation of Shock Effects In Aeronautical Applications). During the previous TFAST project, the interaction between an externally generated oblique shock wave and a laminar boundary layer was studied experimentally in the current experimental facilities of the IUSTI laboratory. To complement these experiments, the focus was shifted towards the interaction between a laminar boundary layer and a compression ramp in the current work. The main objective was to further develop the general understanding of transitional SBLIs, building on the findings of the TFAST project.
This chapter is intended to be a brief summary of the work carried out as part of the Ph.D. program at the IUSTI laboratory of Aix-Marseille University. Hence, the reader is referred to the Ph.D. thesis manuscript for a more detailed description of the literature review, experimental methodology, verification, validation, and detailed analyses of the results [37].

# 2 Experimental Methodology

## 2.1 Experimental Setup

The experiments were performed at the IUSTI laboratory of Aix-Marseille University and CNRS, France. The supersonic wind tunnel was a closed-loop system which could be continuously operated for several hours without significant drift in free-stream properties (an increase of 1K/hr and a maximum change in total pressure of ±0.5 mbar). Experiments were performed in the S8 test section, where a symmetric converging-diverging nozzle accelerated the flow to a Mach number of 1.65, corresponding to a free-stream velocity ($u_∞$) of 464 m/s (Fig. 1).

The total temperature was maintained at ambient conditions (approximately 295 K, depending on weather conditions), while the total pressure ($p_t$) of the free-stream could be varied. The current experiments were performed for total pressures of 0.4, 0.6 and 0.8 atm (Table 1).

Downstream of the nozzle, the test section was 105 mm in height and 170 mm in span-wise width. The geometric models of the flat plate and the two compression ramps were similar in construction with a sharp leading edge, total length of 280 mm

<figure><img src="image_3.png" /><figcaption>**Fig. 1** S8 Nozzle and test section</figcaption></figure>

<table><caption>**Table 1** Operating conditions of the experiments</caption><thead><tr><th>p<sub>t</sub> [atm]</th><th>M</th><th>Re<sub>u</sub> (× 10<sup>6</sup>) [m<sup>-1</sup>]</th><th>Re<sub>c</sub> (× 10<sup>6</sup>)</th><th>σ<sub>ρu</sub> [%]</th><th>σ<sub>u</sub> [%]</th><th>σ<sub>p</sub> [%]</th></tr></thead><tbody><tr><td>0.4</td><td>1.64</td><td>5.61</td><td>0.65</td><td>0.07</td><td>0.04</td><td>0.16</td></tr><tr><td>0.6</td><td>1.64</td><td>8.37</td><td>0.96</td><td>0.06</td><td>0.03</td><td>0.13</td></tr><tr><td>0.8</td><td>1.65</td><td>11.01</td><td>1.27</td><td>0.05</td><td>0.03</td><td>0.11</td></tr></tbody></table>
and spanning the entire width of the test section. The location of the corner of the ramp ($x_c$) was 115 mm from the leading edge for both ramps. Reynolds number based on the location of the corner is shown in Table 1 for different total pressures of the free-stream. The two models were placed at a height of 25 mm from the floor, using supports near the span-wise edges of the wind tunnel. The floor of the test section was modified to have an additional depth of 10 mm to alleviate chocking of the secondary flow underneath the models.

Pictures and schematics of the models are shown in Figs. 2, 3 and 4. It is to be noted here that the adiabatic wall temperature for these models nearly reached ambient conditions (total temperature of the freestream), considering a recovery factor: $r \approx Pr^{1/2} \approx 0.84$ for a fully laminar boundary layer and $r \approx Pr^{1/3} \approx 0.89$ for a fully turbulent boundary layer [3], where Pr is the Prandtl number.

The mean flow in the test section was found to be quite uniform, with static pressure varying within 1% over the entire test section, and the Mach number evolution showed a similar behaviour, with a trivial deceleration from 1.64 to 1.63 over the test section. Hence, it was concluded that the mean flow was uniform within reasonable limits of pressure and velocity.

The free-stream noise at the exit of the nozzle was measured using the classical single sensor hot-wire anemometer. Table 1 shows the measured turbulence intensities

<figure><img src="image_4.png" /><figcaption>**Fig. 3** 6° compression ramp geometrical configuration</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4** 10° compression ramp geometrical configuration</figcaption></figure>

in terms of non-dimensional fluctuations (root mean square) of mass-flux, veloc-
ity and pressure. The free-stream noise was found to be hypo-turbulent in nature,
with low turbulence intensities for all operating conditions. Thus, ensuring that the
laminar boundary layer would not undergo bypass transition [5, 12].

The flow deflection (φ) of the two compression ramps were chosen such that direct comparisons could be drawn with the oblique shock reflection experiments from the previous European project TFAST [11].

**2.2 Experimental Techniques**

A classical Pitot probe was used to make measurements of the mean flow field. The tip of the probe was 0.3 mm in height with a opening of 0.15 mm in height, which measured the mean stagnation pressure of the flow. Flow properties such as Mach number, pressure, temperature, velocity and density were determined using standard equations.

To address temporal properties of the flow, hot-wire anemometry was used. The wire was made of platinum and tungsten, with a diameter of 2.5 µm and length of 0.5 mm. The Streamline amplifier from Dantec Dynamics was used in constant temperature mode and the bridge was operated in a symmetrical configuration. An overheat ratio of 0.8 was used and the anemometer was sensitive to voltage fluctuations up to 150 kHz. The acquisition frequency was 400 kHz and the signal was acquired for 10 s. The National Instruments NI6133 analog-to-digital signal acquisition system was used to save the data with 14 bits of resolution, with a low pass filter of 300 kHz before acquisition.

Figure 5 shows the two types of hot-wire probes that were used for this study. The standard straight probe, which was suitable for measurements made in the external flow (Fig. 5a). In the potential region, the probe received fluctuations radiated by the boundary layer [8–10] and the general noise of the wind tunnel [7] (Fig. 6).
<figure><img src="image_6.png" /><figcaption>**Fig. 5** Types of hot-wire probes</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 6** Illustration of the hot-wire measurements (blue circles) made inside the boundary layer</figcaption></figure>

Additionally, a boundary layer hot-wire probe was also used (Fig. 5b). Boundary layer probes have historically been used for boundary layer research in both low-speed [1, 2] and high-speed [4, 6] experimental facilities, without significant effect on the flow field.

The non-dimensional fluctuations of hot-wire voltage was directly related to the non-dimensional fluctuations of mass-flux through the King's law coefficient [13]. Hence, the data processing and analyses were directly performed on the measured voltage of the hot-wire. The measured time signals were de-trended, and the mean was subtracted before being transformed into Fourier space, and Power Spectral Density (PSD) was estimated using Welch's method. The time signals were broken into segments of 2¹⁶ points, with an overlap of 50%. The Hamming window was applied to the overlapping segments and the periodogram was calculated by computing the discrete Fourier transform. The individual periodograms were averaged to reduce the variance of the measurements, and estimates of the power spectral density were obtained. Fully converged spectral statistics were obtained using a total of 2²²
<figure><img src="image_8.png" /><figcaption>**Fig. 7** Trishula probe</figcaption></figure>

sampled points. The PSD was normalized by the square of the mean voltage at each location.

A new probe was developed using three hot-wire sensors (named as Trishula). The three sensors were placed on the vertices of an isosceles triangle on the stream-wise span-wise plane (Fig. 7a). The spacing between the sensors were chosen be as close as possible without mutual interference from each other (Fig. 7b). This probe was developed to resolve characteristics of 3D plane waves in the flow such as spatial orientation with respect to the stream-wise direction, span-wise wavelengths, phase, and group velocities. The reader is referred to the Ph.D. thesis manuscript for more details about the design, construction and working principle of this new probe [37].

# 3 Results: Boundary Layer Transition at Zero Pressure Gradient

## 3.1 Evolution of Time Scales Along the Boundary Layer

Time signals of the laminar boundary layer were measured for different total pressures and at various stream-wise locations on the flat plate using the single-sensor boundary layer probe hot-wire probe. Figure 8 shows the estimates of the PSD represented in pre-multiplied and normalized format to highlight the dominant frequencies at each stream-wise location. Frequency is shown on the vertical axis, represented in non-dimensional angular frequency ($\omega^* = 2\pi f/(Re_u u_\infty)$). Reynolds number is shown on the horizontal axis, based on the local Blasius reference length scale ($R = \sqrt{Re_x}$).

Inside the boundary layer, a coherent time scale was observed in the High Frequency (HF) range, over the first part of the flat plate. These time scales were not observed outside the boundary layer (some spots of HF scales corresponded to the general background noise of the wind tunnel). Hence, these HF scales did not radiate any fluctuations outside the boundary layer.
<figure><img src="image_9.png" /><figcaption>**Fig. 8** Space-time evolution of PSD represented in terms of $\omega^*$</figcaption></figure>

Figure 8 also compares the contours of PSD from the current hot-wire measurements with the neutral map of the most unstable frequencies found by [14].

Frequencies inside the red curve correspond to the most amplified modes according to stability theory and frequencies outside the red curve were expected to be damped. Accordingly, the non-dimensional angular frequency of the most unstable mode decreases with increasing Reynolds number. This trend was well captured by the hot-wire measurements inside the boundary layer, and there was reasonable agreement between theory and experiments to suggest that these HF are consistent with the compressible TS waves. At low supersonic speeds, these self-excited modes of the laminar boundary layer are three-dimensional in nature i.e. they are oriented at an angle with respect to the free-stream flow direction and are referred to as Oblique
Modes (OM) [15]. The laminar boundary layer is receptive to small disturbances in the free-stream, which trigger these naturally unstable eigen modes. OMs have been studied extensively by experimental [4, 16, 17] and numerical [14, 18, 19] studies for a range of Mach and Reynolds numbers.

Apart from the HF scales, a second time scale was observed in the range of Medium Frequencies (MF) near the end of the flat plate at *R* ≈ 1300, from both inside and outside the boundary layer. In fact, Fig. 8 highlighted that these MF scales were approximately equal to half of the HF scales. This suggested that these MF time scales were possibly the sub-harmonics of the HF OMs from the first part of the laminar boundary layer.

## 3.2 Evolution of Amplitudes Along the Boundary Layer

Figure 9 shows the evolution of the band-passed non-dimensional amplitude of hot-wire fluctuations for different Reynolds numbers, where $A^* = 1$ corresponded to the background noise of the wind tunnel. Inside the boundary layer, both time scales exhibited exponential growth over the entire flat plate. It was observed that the HF scales had a higher amplitude than the MF for the first part of the measurements. Nevertheless, both scales showed the same exponential growth rate up to $R \approx 1200$. This growth rate in the first part of the flat plate was compared with predictions from stability theory, and a reasonable agreement was found [37].

In the second part of the flat plate, it seems that the growth rate of the MF was higher than the HF and subsequently, the two curves meet at $R \approx 1300$. From this point on, both time scales exhibited a much higher growth rate than the first part of the flat plate, reaching amplitudes of nearly 200 times the free-stream noise of the

<figure><img src="image_10.png" /><figcaption>**Fig. 9** Stream-wise evolution of the non-dimensional amplitude of the hot-wire measurements</figcaption></figure>
wind tunnel. This high growth rate in the second part of the flat plate, dominated by
MFs, suggested that non-linear interactions were taking place inside the boundary
layer.

**3.3 Dispersion Relationship of MF**

In order to uncover the nature of these mysterious MF time scales, measurements
were made using the three sensor hot-wire probe over the flat plate in the external
flow. Using the phase difference between the three sensors, the wavenumber (k) and
angle of orientation (ψ) were obtained for the MFs. Given the large spacing (with
respect to the boundary layer thickness) between the sensors of this probe, such
measurements could be susceptible to spatial aliasing (Fig. 10).

An anti-aliasing analysis was performed using synthetic signals with different combinations of wavenumbers and angular orientation. The results of this anti-aliasing analysis is shown in Fig. 10, which highlighted that the MFs could have multiple possible input combinations of (k, ψ) (shown in grey contour lines). A summary of the different possibilities is shown in Table 2. Family I corresponds to actual (or real) measurements by the Trishula probe, whereas families II, III and IV correspond to results from the (synthetic) anti-aliasing analysis.

It was observed that the families III and IV obtained from the anti-aliasing analysis
of the current experiments were very similar to previous experiments from literature
[20], where it was shown that a phase synchronization between the HF OMs and
its sub-harmonics resulted in a resonance phenomenon. The synchronization of the
wavenumbers in the current experiments (from families III and IV) and in the exper-
iments of [20] are demonstrated in Table 3. Further, the wavenumber and angular
orientation of OM (for equivalent Mach and Reynolds numbers) from LST analy-

<figure><img src="image_11.png" /><figcaption>**Fig. 10** Synthetic data analysis of the multi-sensor probe. (legend: ×: Measured Point, solid black line: Aliasing limit, grey contour lines: low residuals)</figcaption></figure>
<table><caption>**Table 2** Possible properties of MF obtained from synthetic analysis</caption><thead><tr><th></th><th>k<sub>x</sub><sup>*</sup></th><th>k<sub>z</sub><sup>*</sup></th><th>ψ</th><th>u<sub>p</sub>/u<sub>∞</sub></th></tr></thead><tbody><tr><td>I</td><td>0.0273</td><td>0.0067</td><td>14°</td><td>0.25</td></tr><tr><td>II</td><td>0.1553</td><td>0.0565</td><td>20°</td><td>0.04</td></tr><tr><td>III</td><td>0.0283</td><td>±0.0776</td><td>±70°</td><td>0.09</td></tr><tr><td>IV</td><td>0.0287</td><td>±0.1628</td><td>±80°</td><td>0.04</td></tr></tbody></table>

<table><caption>**Table 3** Comparison of phase synchronization between sub-harmonics</caption><thead><tr><th></th><th>k<sub>x</sub><sup>1</sup> + k<sub>x</sub><sup>2</sup> = k<sub>x</sub><sup>3</sup></th><th>k<sub>z</sub><sup>1</sup> + k<sub>z</sub><sup>2</sup> = k<sub>z</sub><sup>3</sup></th><th>ψ<sup>3</sup></th></tr></thead><tbody><tr><td>Current experiments</td><td>0.0283 + 0.0287 = 0.0570</td><td>-0.0776 + 0.1628 = 0.0852</td><td>56°</td></tr><tr><td>Kosinov et al. [20]</td><td>0.0368 + 0.0308 = 0.0676</td><td>-0.0849 + 0.1980 = 0.1131</td><td>59°</td></tr><tr><td>Mack [15]</td><td>0.05</td><td>0.0714</td><td>55°</td></tr></tbody></table>

sis of [15] are also shown, and very good agreement was found between the three results. These measurements confirmed that the MF observed here corresponded to sub-harmonic modes of the OM, and phase synchronization was possible between the sub-harmonics and the OM, leading to non-linear interactions.

DNS studies have shown that this sub-harmonic resonance phenomenon can accelerate or “enrich” the oblique breakdown transition scenario [21, 22]. The non-linear regime of the transition process had received relatively less attention from an experimental point of view. And, the current experiments used novel measurement techniques which provided a detailed description of this phenomenon involving the sub-harmonic modes of the boundary layer, including their frequency, amplitude, wave-numbers, angular orientation and phase velocity.

# 4 Results: Length Scales of Transitional SBLIs

The interaction between a laminar boundary layer and a compression ramp has been well documented for a number of flow deflections, Mach, and Reynolds numbers [23]. Pitot measurements were used to verify and validate the two-step pressure rise over the separated SBLI. Comparisons were made with the inviscid pressure rise due to the compression ramp, and also the pressure rise at separation was verified with free-interaction theory [24]. These verifications and validations are described in more detail in the Ph.D. thesis manuscript [37].

Additionally, Pitot probe measurements provided quantitative insight into the upstream influence of the SBLI through the length of interaction, which was defined as the stream-wise distance between the corner of the ramp and the mean location of boundary layer separation (measured at the wall). It was observed that the length of interaction decreased when Reynolds number was increased, for both the ramps. And
comparing between the two ramps, the length of interaction was larger for higher ramp angles.

An attempt was made to compile a number of experimental measurements of the lengths of interaction for transitional SBLIs. Particularly, experiments of oblique shock reflections and compression ramps were collected. This compilation had a collection of independent experiments performed over the past 70 years, in different wind tunnel facilities, and with a wide range of operating conditions. It is important to note that such a compilation was made for the first time for laminar and transitional SBLIs. The reader is referred to the Ph.D. thesis manuscript regarding the details of this compilation [37].

For comparisons to be made between different geometries (e.g. oblique shock reflections and compression ramps), there was a need to utilise a common length scaling for both geometries. A common length scaling was developed for turbulent SBLIs, based on the mass flow deficit between the outgoing and incoming boundary layer [25]. Although this scaling was developed for turbulent SBLIs, it should also be applicable to transitional SBLIs, given that this formulation was developed on a mass conservation basis. Additionally, a non-dimensional separation criterion was also proposed [25], comparing the overall pressure rise across the interaction with the dynamic pressure of the freestream.

Figure 11 shows the compilation of the transitional SBLI experiments represented using the scaling proposed above [25]. The reader is referred to the Ph.D. thesis manuscript for the legend of this compilation [37]. It was observed that this particular set of scaling parameters resulted in different trends in the data, and this variety in the trends was greater than the uncertainty associated with the measurement techniques.

The reason why such a scaling worked for the compilation of turbulent SBLIs [25], was because Reynolds number had a weak effect on the separation criterion of turbulent SBLIs. However, it was clear that from the current experiments that Reynolds

<figure><img src="image_12.png" /><figcaption>**Fig. 11** Length of interaction based on the mass-balance scaling [25] with $k = 9$ (legend: [37])</figcaption></figure>
number had a significant effect on the pressure required to separate a laminar boundary layer. And the inability of the shock strength scaling to take into account this Reynolds number effect, possibly led to this scatter observed in transitional SBLIs.

In order to introduce Reynolds number in the shock strength scaling, the concept of free-interaction theory [24] was revisited. The main idea behind this formulation was that the non-dimensional pressure rise at separation showed universal behaviour for all SBLIs (apart from the constant being different for turbulent and laminar SBLIs).

The original idea for the non-dimensional shock strength scaling [25], was to compare the overall increase in pressure across the interaction with the pressure rise across separation $(\Delta p/(\Delta p)_{sep})$. The current compilation showed that Reynolds number had a significant effect on the pressure rise at separation for laminar and transitional SBLIs, and this behaviour could be universally described by free-interaction theory.

Hence, a new scaling for the shock strength was proposed, based on the ratio of pressures, as opposed to comparing the pressure differences:

$$S_r^* = \frac{p_3/p_1}{(p/p_1)_{sep}} = \frac{p_3}{p_1} \left[ 1 + \frac{\gamma M^2}{2} F_p \sqrt{\frac{2 c_f}{(M^2 - 1)^{1/2}}} \right]^{-1} \quad (1)$$

Where $p_3/p_1$ is the overall pressure rise across the interaction, $(p/p_1)_{sep}$ is obtained from the free-interaction theory [24], $F_p = 1.5$ is the coefficient of free-interaction at plateau and $c_f$ is the skin-friction coefficient at the mean location of separation.

This new scaling seemed to collapse most of the data set, as shown in Fig. 12. The effect of Reynolds number appeared to be well captured by this new scaling as the data points did not fall on a vertical line as the previous scaling (Fig. 11) and instead of the different trends observed in the previous figure, the same linear relationship

<figure><img src="image_13.png" /><figcaption>Fig. 12 Compilation of length scales for different non-dimensional shock strengths, $S_r^*$</figcaption></figure>
(i.e. same slope) was obtained for most of the data points from the compilation in Fig. 12.

The non-dimensional nature of this scaling automatically adjusted the shock strength; for low shock strengths with no separation, the imposed adverse pressure ratio is lower than the pressure ratio required to separate the boundary layer and hence, $S_r^* < 1$. For incipient interactions involving intermittent separation, the imposed pressure ratio is close to the pressure ratio required to separate the boundary layer ($S_r^* \approx 1$). And finally $S_r^* > 1$ corresponded to a typical separated SBLI.

The reader is referred to the Ph.D. thesis manuscript for a more detailed description of this separation criterion, including the effect of Reynolds number and freestream noise of the wind tunnel.

# 5 Results: Boundary Layer Transition at Adverse Pressure Gradient

Figure 13 compares the hot-wire spectra measured over the separated region between the two ramps ($0 < X^* < 1$). The PSD at different stream-wise locations are shown in absolute pre-multiplied format, to highlight the difference in amplitude between different measurements.

The bottom horizontal axis shows the non-dimensional angular frequency ($\omega^*$) according to stability theory of laminar boundary layers, and the top horizontal axis shows the non-dimensional Strouhal number according to SBLI normalization (based on the length of interaction and the free-stream velocity). Each colour represents the spectra at a given stream-wise location, and hence, several spectra are shown corresponding to different locations over the bubble.

The spectra for the 6° ramp remained approximately the same over the interaction, with no growth of amplitudes for any frequency. However, on the 10° compression ramp, the growth of MF time scales was observed. These MF time scales were similar in frequency to the ones found in the non-linear regime of the flat plate measurements.

Based on these measurements, it was presumed that these MF time scales were the same type of structures found in the flat plate boundary layer, that arose due to non-linear interactions of the upstream OMs. The wavelength, convection velocities and direction of propagation of such time scales have been documented in Table 2. The reason as to why they are only observed on the 10° ramp and not on the 6° ramp might be related to the size of the separated region. As the length was longer for the 10° ramp, the height of the shear layer was also higher. The higher height might have moved the generalized inflection point higher, resulting in the destabilization of the KH instability [26]. It has been shown that boundary layer modes are convectively amplified through the KH mechanism [27, 28].

Figure 14 shows the non-dimensional amplitude of MF over the separated region for the both ramps, compared to the amplitude evolution of MF over the flat plate. The amplitudes of the 6° ramps did not show a large change over the separated region,
<figure><img src="image_14.png" /><figcaption>**Fig. 13** Evolution of hot-wire spectra over the separation bubble</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 14** Comparison of MF growth rate between the flat plate and the compression ramps</figcaption></figure>
while an exponential growth was observed for the 10° ramp. This suggested that the growth of these time scales was strongly dependent on the size of the separated region, possibly due to the higher location of the generalized inflection point.

Additionally, the growth rate of the MF over the SBLI was nearly three times higher compared to their evolution on the flat plate. These results suggested that the separated region accelerated the growth of these MF scales.

Figure 15 compares the spectra of the hot-wire measurements made inside the boundary layer for the flat plate, 6° and 10° ramps. The last measurement point with the highest amplitude was chosen from the flat plate to be compared with the highest amplitude of the inside measurements on both ramps. This corresponded to the first point downstream of reattachment for both ramps. This comparison clearly highlighted the differences in amplitudes and frequencies of the boundary layer between the three geometries.

Surprisingly, in both ramp configurations, there were no evidences of MF inside the boundary layer, which exhibited strong growth over the separated region for the 10° ramp. Two dominant time scales were identified from the measurements made inside the boundary layer for both the ramps; HF, typically associated with OM of the laminar boundary layer were identified, in the same range of frequencies as observed from the flat plate measurements. Additionally, Very High Frequencies (VHF) were also identified, which were not found inside the natural boundary layer measurements on the flat plate. It was believed that the VHF were most likely the evidence of breakdown of coherent structures in the boundary layer, indicating that the boundary layer was undergoing transition to turbulence.

What was surprising to note was that the MF, which were found to be most dominant from the external measurements over the separated region (Figs. 13 and 14), was not the most dominant time scale inside the reattached boundary layer of the SBLI. It seemed that the reattached boundary layer was preferentially selective to

<figure><img src="image_16.png" /><figcaption>**Fig. 15** Comparison of the hot-wire spectra between the flat plate, 6° and 10° ramp</figcaption></figure>
higher frequency modes. Given that the growth of these MF was linked to a possible phase synchronization between the OM and the sub-harmonic MF (Table 3), perhaps the reattachment compression possibly ruined this synchronization and resonance condition could have been broken. Despite these large amplitudes inside the boundary layer, these hot-wire spectra were not typical of a boundary layer that was fully turbulent, where most of the energy would be associated with the turbulent boundary layer time scale (VHF). These results suggested that the boundary layer was not fully turbulent at reattachment.

# 6 Results: LF Unsteadiness of Compression ramp SBLIs

Figure 16 compares the absolute pre-multiplied PSD of the compression waves at separation for the 6° and 10° ramps. The spectra at separation in the current experiments were characterized by LF time scales ($0.04 \le St_L \le 0.08$), which were typically associated with the large-scale “breathing” motion of the recirculating region [29–32]. While the presence of LF was clearly identified for the 10° ramp, it was barely distinguishable for the 6° ramp. In fact, just looking at the spectra of the 6° ramp alone, without comparing with the spectra from the 10° ramp, it could be said that there was no discernable LF unsteadiness found in the 6° ramp. Hence, the imposed flow deflection had a significant influence on the relative amplitude of LF unsteadiness in the current experiments.

In addition to LF, a broad range of HF ($0.4 \le St_L \le 2$) was observed. These HF were in the same range of frequencies typically associated with OM from the laminar boundary layer. Although it was shown that these time scales do not radiate fluctuations outside the boundary layer, the compression waves at separation might

<figure><img src="image_17.png" /><figcaption>**Fig. 16** Comparison of PSD over the mean separation point for the two compression ramps</figcaption></figure>
be receptive to these time scales of the laminar boundary layer. Moreover, the back-
ground wind tunnel noise could also be locally amplified through the compression
waves, and the hot-wire could have picked up such fluctuations. Therefore, it was
difficult to determine the true nature of these frequencies without further experiments
and analyses.

Several hypotheses and models have been proposed for the origin of LF unsteadi-
ness. Several studies have reported the growth of convective instabilities (with char-
acteristic MF) over the shear layer arising due to non-linear interactions of the lam-
inar boundary layer modes [28, 30, 32, 33]. Additionally, a correlation was found
between the presence of non-linear interaction over the shear layer to the LF unsteadi-
ness of the transitional SBLI [34–36]. Hence, it was concluded that the growth of
new MF time scales over the separated shear layer of the 10° ramp (Fig. 13b), which
were possibly triggered by non-linear interactions of OM, might have played a role
in the LF unsteadiness observed at separation. The absence of non-linear interactions
on the 6° ramp did not result in the growth of MF time scales over the shear layer,
and coincidentally no discernible LF unsteadiness was observed for the 6° ramp
(Fig. 13a).

Unfortunately, a consolidated theory on the origin and mechanism of LF unsteadi-
ness associated with transitional SBLI could not be developed with the current mea-
surements. Nevertheless, the presence of new MF time scales and LF unsteadiness
pointed to a speculative link between these flow features. Further investigations were
necessary to understand this phenomenon in more detail.

7 Conclusions

Experiments were performed in the supersonic wind tunnel of the IUSTI laboratory, to study the length and time scales of transitional SBLIs. Given that such SBLIs often accelerated the transitional mechanisms of laminar boundary layers, another important objective was to study this phenomenon in more detail. The Mach number of the experiments was fixed to 1.65 and measurements were performed at relatively “low” unit Reynolds numbers (5.61 ≤ *Re*<sub>u</sub> (× 10<sup>6</sup>) [m<sup>-1</sup>] ≤ 11.01). A variety of experimental techniques were employed including Schlieren, Shadowgraph, Pitot probes and hot-wire anemometry. A new multi-sensor hot-wire probe was developed over the course of this research, to reveal the spatio-temporal nature of laminar boundary layer instability waves.

Measurements were first made on a simple flat plate geometry, to understand the
transitional mechanisms of an undisturbed laminar boundary layer with zero pressure
gradient. The transition mechanism was found to be divided into two regimes. In
the first regime, modal mechanisms corresponding to the primary instability of the
laminar boundary layer was found. These linear time scales agreed very well with the
predictions of the most amplified frequencies (or OMs) from stability theory. In the
second regime, a new MF time scale was found to be the most dominant, with higher
growth rates compared to the previous regime. It was believed that this new time scale
arose due to non-linear interactions between the self-excited eigen modes of the first regime. Measurements of this time scale were made using the newly developed multi-sensor hot-wire probe, which revealed large stream-wise wavelengths and low phase velocities compared to OM. The span-wise nature of these modes was uncertain due to possible spatial aliasing of the wave. Nevertheless, evidence was found to indicate that this MF could be related to the sub-harmonic resonance scenario. However, it was found in literature that these sub-harmonics do not drive the boundary layer to transition, but rather accelerate the oblique breakdown transition process involving stationary stream-wise vortex modes.

Following the measurements on the flat plate, experiments of transitional SBLI were made on nominally 2D compression ramps. The canonical nature of the current transitional compression ramps were verified with comparisons with free-interaction theory. A compilation of lengths of interaction reported from various experiments in literature was made, and a large scatter was found in the relationship between the length of interaction and the imposed shock strength. A new non-dimensional scaling was developed for the shock strength that included the effects of Reynolds number through the free-interaction theory. In particular, it was found that the non-dimensional length of interaction scaled with the ratio of pressure as opposed to the difference in pressure across the interaction. In particular, the pressure ratio across the interaction normalized with the pressure ratio needed to separate the boundary layer was developed. Such a scaling dramatically reduced the scatter in the data set, and all the experiments exhibited nearly the same linear relationship between this new shock strength parameter and length of interaction.

The hot-wire measurements over the compression ramp SBLI shed light on the transitional mechanisms of the separated laminar boundary layer. MF, in a similar range as that on the flat plate, were found to be amplified over the separated shear layer of the 10° ramp. Such an amplification was not found over the 6° ramp. The growth rate of these MF over the 10° ramp was found to be nearly three times higher compared to their growth rate on the flat plate (i.e. laminar boundary layer without interaction). Such MF were also found by other studies on transitional SBLIs, which reported that these frequencies were characteristic of convectively amplified modes of the boundary layer. Surprisingly, downstream of the interaction, these MFs were not the most dominant, but most of the energy was associated with HF OMs of the laminar boundary layer. The strong compression at reattachment was found to be preferentially selective towards higher frequencies, which eventually drove the boundary layer to turbulence at some point downstream of reattachment. This also suggested that the boundary layer was not fully turbulent after reattachment, contrary to other studies on transitional SBLIs.

The LF dynamics of such transitional SBLIs was also studied using hot-wire anemometry. It was found that while the 10° ramp was clearly characterized by LFs ($St_L \approx 0.05$) at the compression waves at separation, such LF unsteadiness was not found on the 6° ramp. While a conclusive theory on the origin and mechanism of LF unsteadiness was not determined, a “coincidental” link between the amplification of MF time scales over the separated shear layer and the presence of LF unsteadiness was noted, indicating that non-linearities (the possible cause for the growth of MF
over the separated shear layer) possibly played a role in the origin and mechanism
of LF unsteadiness in such transitional SBLIs.

**Competing Interests** The authors have no conflicts of interest to declare that are relevant to the content of this chapter.

**Acknowledgements** The authors would like to acknowledge the technical support of Pierre Lan-
toine while performing the experiments in the wind tunnel.

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