2.1 Facility

All experiments were carried out in the HEG (High Enthalpy shock tunnel Göttingen) of the German Aerospace Center (DLR). HEG is a large-scale reflected-shock wind tunnel, making use of free-piston compression to generate the driver conditions necessary for simulating high-speed flows. HEG is capable of reproducing a wide range of flow conditions, but is limited in terms of test duration to at most a few milliseconds. Further information regarding the operating principle of and conditions achievable in HEG is provided, for example, in Hannemann (Reference Hannemann2003).

Table 1.

Typical facility reservoir (subscript 0) and computed free stream (subscript $\infty$ ) properties from the test conditions employed in the experiments of the present study.

The present investigation included both relatively low-enthalpy conditions ( $h_{0}=3.1{-}3.3~\text{MJ}~\text{kg}^{-1}$ ) and a single high-enthalpy condition ( $h_{0}=11.9~\text{MJ}~\text{kg}^{-1}$ ). At low enthalpy, various unit Reynolds numbers in the range from 2.6 to $6.5\times 10^{6}~\text{m}^{-1}$ were achieved by adjusting the reservoir pressure. These conditions are intended to simulate Mach-8 flight at different altitudes. Representative reservoir and free stream properties at all conditions discussed in this work are provided in table 1. Note that the naming convention adopted in the present article is based on reservoir pressure; to prevent confusion with previously published work, the internal HEG numbering convention for each condition is also provided in table 1. The reservoir pressures were measured directly, while the enthalpy was calculated from incident shock-speed measurements. Free stream conditions are derived from simulations of the nozzle flow using the DLR TAU code (Gerhold et al. Reference Gerhold, Friedrich, Evans and Galle1997); these have been compared with extensive calibration-rake measurements to ensure accuracy. The single-shot uncertainty at low enthalpy is estimated as 5 % in $p_{0}$ and 4 % in $h_{0}$ , with run-to-run repeatability of the same order (Laurence et al. Reference Laurence, Karl, Martinez Schramm and Hannemann2013). The corresponding uncertainties in free stream properties are estimated as 5 % in $p_{\infty }$ and ${\it\rho}_{\infty }$ , 3 % in $T_{\infty }$ , 2 % in $u_{\infty }$ , and 0.3 % in $M_{\infty }$ (Laurence et al. Reference Laurence, Karl, Martinez Schramm and Hannemann2013). At high enthalpy, the uncertainty in free stream conditions is somewhat higher but also more difficult to quantify due to difficulties in accurately modelling the thermochemical processes taking place during the nozzle expansion.

Typical reservoir pressure traces for the conditions employed are presented in figure 1. For the low-enthalpy conditions (A–C), the traces show that, after a start-up period lasting 2– $3~\text{ms}$ , quasi-steady conditions are attained and persist for approximately $3~\text{ms}$ ; the variation in pressure during this time is of the order of 3.5 % (standard deviation about the mean). The test time is terminated by the arrival of expansion waves from the diaphragm burst, resulting in a steadily decreasing reservoir pressure thereafter. For condition D (high enthalpy), the test time is limited to approximately $1~\text{ms}$ .

Figure 1.

Typical reservoir pressure traces for the test conditions detailed in table 1: (a) low enthalpy (conditions A–C); (b) high enthalpy (condition D). The approximate test times are indicated by the vertical lines.

2.2 Model and instrumentation

The model for this study was a slender 7°-half-angle cone, mounted at zero incidence ( $\pm 0.01^{\circ }$ ). The nose section was interchangeable: with a sharp nose, the cone length was $1100~\text{mm}$ . Although a blunted nose of $2.5~\text{mm}$ radius was used in all experiments described here, for clarity we will refer to the distance along the cone surface, $s$ , measured from the extrapolated sharp nose. As shown in figure 2, the lower part of the cone was fitted with a surface insert constructed of a carbon-fibre-reinforced carbon ceramic material. This material has a natural open porosity (approximately 15 %) with a pore size on the scale of ${\sim}30~{\rm\mu}\text{m}$ , and was expected to lead to delayed transition in comparison to the smooth surface (Fedorov et al. Reference Fedorov, Malmuth, Rasheed and Hornung2001; Rasheed et al. Reference Rasheed, Hornung, Fedorov and Malmuth2002; Fedorov et al. Reference Fedorov, Shiplyuk, Maslov, Burov and Malmuth2003, Reference Fedorov, Kozlov, Shiplyuk, Maslov and Malmuth2006). Relevant results obtained with surface instrumentation as well as characterization of the ultrasonically absorptive properties of the ceramic material are described in earlier publications (Wagner et al. Reference Wagner, Hannemann and Kuhn2013a ,Reference Wagner, Hannemann, Wartemann and Giese b ; Wagner, Hannemann & Kuhn Reference Wagner, Hannemann and Kuhn2014).

Figure 2.

Sketch of the slender cone model with porous insert (not to scale).

The cone was equipped with various surface sensors on both the smooth and porous sides: thermocouples to provide approximate transition locations, Kulite pressure transducers for mean measurements and fast-response PCB pressure transducers for measuring instabilities. A line of thermocouples lay along each of the two rays corresponding to the plane-of-visualization of the schlieren set-up described in the following subsection (i.e. the uppermost and lowermost rays). The fast-response pressure transducers were offset in the circumferential direction relative to this plane.

In figure 3 we show the heat-flux profiles derived from thermocouple data for the three low-enthalpy test conditions on both the smooth and porous sides. Transition is indicated by the sharp rise in heat flux in all cases; as expected, the transition location moves forward with increasing unit Reynolds number. The transition-delaying effect of the porous surface is evident from the shifting downstream of the heat-flux rise relative to the smooth surface for each condition, as reported previously in Wagner et al. (Reference Wagner, Hannemann and Kuhn2013a ,Reference Wagner, Hannemann, Wartemann and Giese b ). For the high-enthalpy condition (D), the thermocouple measurements indicated that the boundary layer was laminar over the entire cone length on both the smooth and porous sides. For the purposes of calculating the wall temperature ratio, the small surface temperature rise during the short test time can be neglected and the wall can be assumed to remain at room temperature.

Figure 3.

Typical heat-flux profiles on the (a) smooth and (b) porous sides of the transition cone at the various low-enthalpy test conditions: $\triangle$ , condition A; $\circ$ , condition B; ▫, condition C.

2.3 Schlieren visualization technique

A conventional (non-focusing) Z-fold schlieren arrangement was employed for all visualizations described herein. The light source was a Cavilux Smart pulsed-diode laser, which provided pulses of 20–40 ns duration at 690 nm. Two 1.5-m focal length spherical mirrors collimated the light beam to pass through the test section and then refocused it on the opposite side. Images were recorded using either a Phantom v641 or v1210 high-speed CMOS camera, the latter for condition-D experiments at which higher second-mode frequencies were expected. A horizontal knife-edge cutoff was used in all cases, meaning density gradients approximately normal to the cone surface were visualized. A bandpass filter was placed in front of the camera to minimize the influence of test gas luminosity, which was expected to become significant especially at the high-enthalpy condition.

Wavepacket propagation speeds were determined using the image correlation methodology described in Laurence et al. (Reference Laurence, Wagner, Hannemann, Wartemann, Lüdeke, Tanno and Itoh2012, Reference Laurence, Wagner and Hannemann2014). In order to overcome the aliasing that arises in the correlation curves if the frame rate is lower than the dominant instability frequency, we employed the pulse-bursting technique described in the later of these two works. The essence of the technique, as illustrated in figure 4, is that the camera exposure time is set to the maximum value for the given frame rate and pairs of light source pulses are used, the first pulse catching the end of one gate period and the second the beginning of the next. In this way, the maximum disturbance frequency for which the propagation speed can be resolved unambiguously (i.e. without aliasing) increases from the camera frame rate to $1/{\rm\Delta}t$ , where ${\rm\Delta}t$ is the pulse separation. If two pulse pairs with different separation are used, this maximum frequency can be increased even further because the aliasing peaks will occur at different frequencies (see Laurence et al. Reference Laurence, Wagner and Hannemann2014 for further details). In the present campaign, such four-pulse bursts were used for all experiments. The pulse pattern, frame rate and resolution depended on the experimental condition; details are provided in table 2. Typically ${\sim}200$ images were gathered during the test time in a given experiment.

Figure 4.

The burst method used for schlieren visualization in the present experiments, whereby a four-pulse pattern is programmed into the laser light source and used in conjunction with an extended camera exposure period; $f_{camera}$ is the camera frame rate.

Table 2.

Camera parameters used for the different experimental conditions. The frame rate column is the base camera frequency upon which the pulse pattern was imposed. In the pulse spacing column, the first three numbers are ${\rm\Delta}t_{12}$ , ${\rm\Delta}t_{23}$ and ${\rm\Delta}t_{34}$ (see figure 4), respectively; the number in parentheses is the delay between pulse 4 in one pattern and the pulse 1 in the next.

The maximum disturbance frequency that can be measured with the present technique is constrained by two factors. First, the standard Nyquist criterion applies here in terms of wavenumber, i.e. since each pixel acts as a measurement point, the maximum wavenumber (in $\text{mm}^{-1}$ ) that can be resolved is $sf/2$ , where $sf$ is the image scale factor in pixels per mm from table 2. In all cases this was much larger than the wavenumbers of the relevant boundary-layer structures. The second frequency constraint is associated with the correlation-based determination of the disturbance propagation speed (as outlined above), which is used to convert wavenumber to frequency. This does not impose a strict upper frequency limit since, even if aliasing peaks are present in the correlation curves, a priori knowledge of the propagation speed (i.e. that it is approximately 80–90 % of the edge velocity) can be used to identify the correct peak. Nevertheless, the structural evolution of wavepackets may introduce errors in the speed determination if the propagation distance between images is larger than a few wavelengths; as such, an approximate upper limit of $n/{\rm\Delta}t_{min}$ , where $n$ is 3–4 and ${\rm\Delta}t_{min}$ is the minimum time between any two pulses in the pattern, can be assumed. For all conditions in table 2, we see that this is above 1 MHz.

Throughout the following sections, we present frequency spectra of boundary-layer disturbances; these were derived as follows. First, the image of interest $I_{i}$ was normalized by a reference image consisting of the mean of a 41-image sequence centred about $I_{i}$ , thus removing the effect of non-uniformities in the light source and mean boundary-layer intensity profile. Wavenumber spectra were then derived by computing Fourier transforms of rows of pixels at a constant height above the cone surface, with windowing where appropriate. Finally, these wavenumber spectra were converted into frequency spectra using the mean phase velocity calculated previously from image correlations.

Two points regarding the schlieren set-up merit a further brief discussion here. First, as the arrangement was non-focusing, the measurements are necessarily line of sight. This will not cause any ambiguity in terms of phase, since second-mode disturbances are primarily two-dimensional (Demetriades Reference Demetriades1974). It will, however, potentially lead to ambiguities in the wall-normal distributions, since the visualized disturbance may be circumferentially displaced from the vertical plane lying through the cone axis. In Laurence et al. (Reference Laurence, Wagner and Hannemann2014), we noted that the presence of a near-wall peak in the wall-normal power distribution could be used to distinguish if an individual disturbance lay close to this plane; however, ensemble-averaged wall-normal distributions cannot be reliably obtained because of this limitation. Furthermore, disturbances other than the second mode are in general three-dimensional and will not be properly resolved by the line-of-sight technique. Second, as the schlieren was uncalibrated and nonlinear (the light source was circular), the calculation of amplification rates is not possible; moreover, this lack of calibration introduces an additional, more subtle problem. Although the flow-off intensity distribution was close to uniform for most sequences, for flow-on images, the unsteady schlieren signal caused by the passage of second-mode disturbances is superimposed on the spatially non-uniform (especially in the wall-normal direction) intensity profile introduced by the mean boundary layer. In the analysis that follows, we compensated for this to first order by normalizing the unsteady intensity signal from each pixel by the time-averaged intensity; nevertheless, because of the lack of calibration of the system, the effective schlieren sensitivity will vary according to the height above the cone surface (especially in the near-wall region, where the mean density gradients are relatively large compared to the bulk of the boundary layer). Although we expect this effect to be generally modest, caution should be exercised particularly in making quantitative comparisons between the disturbance strength in the near-wall and outer regions.

4.1 Wavepacket appearance

In figure 11 we show two image pairs from the condition-B porous experiment in each of which pair the propagation of a wavepacket is visible; the two images in each case are separated by $31.7~{\rm\mu}\text{s}$ . The images of the lower pair are zoomed-in versions of those at $t_{i}+155.8$ and $187.4~{\rm\mu}\text{ s}$ in figure 7. For brevity in the following discussion, we refer to the two images in the upper pair (pair 1) as 1 : 1 and 1 : 2, and those in the lower pair (pair 2) as 2 : 1 and 2 : 2. The wavepacket in pair 1 is at an earlier stage of development than that in pair 2: the structures in image 1 : 2 are similar to those of 2 : 1, and in 2 : 2 the structures are becoming chaotic and indistinct, i.e. the wavepacket is close to breaking down. In image 1 : 1, the disturbance exhibits the characteristic oblique ‘rope-like’ appearance noted earlier (and observed by other researchers) with the visible features lying mainly within the upper part of the boundary layer. Later, in image 1 : 2 (and also in 2 : 1), the peaks of the rope-like structures appear to fold over, creating a more interwoven appearance reminiscent of plaits or braids. In the centre of the wavepackets, these peaks extend above the visual edge of the boundary layer. For convenience, we refer hereinafter to wavepackets in this later stage of development as ‘plaited’ and those with oblique structures as in image 1 : 1 as ‘rope-like’, though this distinction is necessarily subjective to some extent. In image 2 : 2, even as the bulk of the wavepacket takes on a chaotic, turbulent appearance, some of the stronger regular structures are seen to persist in the centre of the disturbance. A similar persistence of the second-mode structures was observed in the pressure measurements of Casper et al. (Reference Casper, Beresh and Schneider2014), but in this earlier work the rope-like structures were observed at the extremes of the breaking down wavepacket, with the centre developing into turbulence. The rope-like and plaited stages were regularly observed in the evolution of wavepackets in both this and other sequences: in figure 5, for example, the disturbance at $t_{i}+28.1~{\rm\mu}\text{s}$ is rope-like, whereas that at $t_{i}+155.0~{\rm\mu}\text{s}$ has a more plaited appearance. In some cases no distinct plaited state could be discerned in the development, the wavepacket being rope-like in one image pair and already breaking down in the next. This may indicate that the plaited state is sometimes bypassed, or simply that it is short lived.

We also observed that the visualized disturbance strength just prior to breakdown could vary significantly, though this could not be quantified in a meaningful way because of the lack of calibration of the schlieren set-up. While this might indicate that the disturbance amplitude at saturation was varying, an alternative (and we believe, more likely) explanation is that the spanwise extent, and thus the effective integrated path length, of the various wavepackets differed.

Figure 12.

(a,b) Schlieren visualizations from a smooth-wall condition-A experiment showing two distinct wavepackets, the first (a) in a ‘rope-like’ state, the second (b) in a ‘plaited’ state; (c,d) the power spectral distributions versus wall-normal distance of these two wavepackets (a  $\rightarrow$  c; b $\rightarrow$ d).

4.2 Evolution of disturbance wall-normal distribution

In our earlier work (Laurence et al. Reference Laurence, Wagner and Hannemann2014), when the power spectrum of a second-mode disturbance was plotted at different heights above the cone surface, a distinctive double peak at the location of the fundamental frequency was noted, with local maxima both at the wall and inside the boundary layer. Such an example from a condition-A smooth-wall sequence is shown in figure 12 (upper visualization and left panel). Wavepackets showing such wall-normal distributions were seen regularly throughout this sequence and were primarily in a rope-like state. For disturbances that had evolved into a plaited state, however, the wall-normal distribution generally exhibited a subtle difference, as exemplified in the right panel of figure 12 (corresponding to the lower schlieren visualization). Here we see the development of a third peak close to the boundary-layer edge, with the remainder of the distribution remaining similar to that of the rope-like disturbance.

By examining the propagating wavepackets visualized in figure 11, we can gain further insight into the evolution of the wall-normal distribution. In figure 13 we show the distributions derived from the four images associated with the upper pair of figure 11 (i.e. the first and third distributions here correspond to images 1 : 1 and 2 : 1); in figure 14 the same is done with the four images associated with the lower pair of figure 11. The first distribution of figure 13(a) shows the double-peaked profile characteristic of the rope-like state, but already a slight relative growth in the region near the boundary-layer edge is visible. This continues in the subsequent image (b), with the strength of the third, near-edge peak now similar to that of the second peak at $y/{\it\delta}\approx 0.7$ . Growth in the near-wall part of the distribution is also apparent between the first two images, though this near-wall peak has almost vanished in the subsequent distribution (c). Redistribution of the disturbance energy towards the boundary-layer edge continues, but by the fourth image (d) the strength of the disturbance has waned and a spreading in frequency is also observed.

Figure 13.

Development of the wall-normal power spectral distribution of the wavepacket shown in the upper image pair of figure 11: (a) corresponds to the first image in the pair; the subsequent distributions are at time lags relative to the first of (b) 4.5, (c) 31.7 and (d) $34.5~{\rm\mu}\text{s}$ .

In figure 14, we see a similar evolution in the outer part of the boundary layer in the first two distributions (a,b), with the splitting of the disturbance energy between the two outer peaks even more pronounced. In this case the near-wall peak is still present, which may indicate that the wavepacket in the sequence of figure 13 was displaced slightly in the circumferential direction from the vertical imaging plane, obscuring the near-wall peak in the later images. Harmonics of the second mode near 750 kHz are clearly visible in these upper two distributions; more shall be said about this in the following subsection. In the lower two panels (c,d), we see that the strength of the fundamental has decreased markedly across the boundary layer as the wavepacket undergoes breakdown, though the triple-peaked distribution is still distinguishable. In the lower left panel (c), an additional strong component has appeared at a value of $y/{\it\delta}$ lying between the two outer peaks, but at a slightly lower frequency: 310 kHz versus 380 kHz. This is interesting in light of the observation of Stetson & Kimmel (Reference Stetson and Kimmel1993) that spectral dispersion can occur just prior to breakdown; that is, as the second mode attenuates, neighbouring frequencies can experience rapid growth. This new component has itself been substantially attenuated in the final panel (figure 14 d). Finally, we note that the strength of the first harmonic is little changed between figures 14(b) and 14(c) despite the decay of the fundamental peaks.

Figure 14.

Development of the wall-normal power spectral distribution of the wavepacket shown in the lower image pair of figure 11. (a) corresponds to the first image in the pair; the subsequent distributions are at time lags relative to the first of (b) 4.5, (c) 31.7, and (d) $34.5~{\rm\mu}\text{s}$ .

4.3 Harmonic development

In § 3, we saw that no harmonics were clearly discernible in the ensemble-averaged power spectra, whereas in the wall-normal distributions of figure 14, harmonic components are apparent at heights above the cone surface roughly coincident with the peaks in the fundamental. In examining the spectra of individual wave packets, harmonics were observed in some but not all cases, with plaited disturbances generally exhibiting stronger harmonics (as might be expected, given that they are at a later stage of development than rope-like waves). The development of nonlinear phenomena such as harmonics has been previously investigated using the bicoherence method (see, for example, Bountin, Shiplyuk & Maslov Reference Bountin, Shiplyuk and Maslov2008, Hofferth et al. Reference Hofferth, Humble, Floryan and Saric2013) - because of the lack of consistency in the appearance of harmonics in the present data, however, no such analysis was attempted here.

Figure 15.

(a,d) Schlieren images of a propagating wavepacket in a smooth-surface condition-A sequence. The temporal separation of the images is $28~{\rm\mu}\text{s}$ , which corresponds to a propagation distance of 59 mm. Below each schlieren visualization are bandpass-filtered versions, with the filter centred around the second-mode fundamental frequency (b,e) and first harmonic frequency (c,f). (g) Power spectra at the $y$ -location of maximum signal intensity for these two images: -▫- earlier; -▵- later. (h) Peak wall-normal distributions for the second-mode fundamental (-▫-) and first harmonic (-♢-) in the second of these two images. The relative magnitude of the harmonic has been multiplied by a factor of 2.25.

Some wavepackets were notable for the clarity of their harmonic content (this usually corresponded to high fundamental signal strength). In the images of figure 15, we show schlieren visualizations of one such propagating wavepacket from a condition-A smooth-surface sequence, together with processed versions that have been bandpass filtered around each of the fundamental and the first harmonic. For clarity, the amplitude of the first-harmonic-filtered image has been multiplied by a factor of 2.25 in comparison to the fundamental filtered image. In the earlier image the harmonic structures are only weakly visible near the centre of the wavepacket, but they become much more distinct in the later image. The angle formed by the harmonic structures to the cone surface is slightly steeper than that of the fundamental structures. The signal energy of both fundamental and harmonic components is more prominent in the upper half of the boundary layer, though there is also energy present near the surface. As noted in Laurence et al. (Reference Laurence, Wagner and Hannemann2014), the harmonic energy appears to be more concentrated towards the wavepacket centre in the streamwise direction than that of the fundamental.

In the lower left panel of figure 15, we show spectra from this wavepacket at both time instants, measured at $y/{\it\delta}=0.7$ (the height above the cone surface corresponding to maximum strength of the outer peak). Comparing to the spectra in the lower right panel of figure 6, we see the much higher signal strength relative to the noise floor that is obtained from considering a single wavepacket rather than the entire sequence. The fundamental and first harmonic peaks at 280 and 550 kHz are visible in each of the spectra, and we see substantial growth in both components as the wavepacket develops. In the downstream spectrum, there is an additional third peak close to where we would expect the second harmonic (820 kHz). To further elucidate the wall-normal distributions of these spectral components, in the lower right panel we show the peak strength of the fundamental and first harmonic versus the normalized wall-normal distance (note that no windowing has been applied to the spectra from which these are calculated, whereas in (g) a Blackman window has been employed). The harmonic also exhibits the double-peaked wall-normal profile we have noted of the fundamental for rope-like wavepackets. The outer peak of the harmonic here is centred closer to the wall than that of the fundamental; indeed, in all rope-like wavepackets examined, either the peaks were approximately coincident or the harmonic peak lay appreciably closer to the wall than the fundamental peak.

Figure 16.

(a,b) Bandpass-filtered versions of the lower image pair in figure 11, with the markers corresponding to the same locations; filters centred around the second-mode fundamental (upper image in each case) and first harmonic (lower image) have been applied. (c) Power spectra at the $y$ -location of maximum signal intensity for these two time instants: -▫- earlier; - $\triangle$ - later. (d) Peak wall-normal distributions for the second-mode fundamental (-▫-) and first harmonic (-♢-) for the first of these two images. The magnitude of the first harmonic has been multiplied by a factor of 5.

In figure 16 we show equivalent results for the lower image pair of figure 11 (condition B); the markers are shown again for reference. In the bandpass-filtered versions of the first image, the oblique harmonic structures of the plaited wavepacket again coincide with those of the fundamental, both in location and roughly in angle. With the onset of breakdown in the second image, we can still make out the oblique fundamental structures in the centre and downstream (right of centre) part of the wavepacket, but the corresponding harmonic structures are visible over a more extended streamwise region, especially to the left of the wavepacket centre. This would help to explain the relatively strong harmonic component seen in the corresponding wall-normal distribution in figure 14(c).

By examining the structures in the fundamental filtered images of figures 15 and 16, we can also see the origin of the changes in the wall-normal distribution near the boundary-layer edge discussed in the previous subsection. When the wavepacket is in a rope-like state (figure 15), the dominant fundamental structures are simply the oblique features associated with the rope-like waves, which reach maximum strength above the midpoint of the boundary layer and then attenuate towards the outer edge. In a plaited wavepacket, on the other hand, although these oblique features persist near the centre of the wavepacket, the fundamental energy is now concentrated both at the peaks of these structures (which now extend above the boundary-layer height) and at a height where each individual wave structure folds over to intersect with the next in the packet. In between these two concentrations there is a relatively quiet zone, leading to a distribution with a split outer peak as seen in the upper panels of figure 14.

In the lower left-hand panel of figure 16 we show power spectra computed at $y/{\it\delta}=0.7$ for the wavepacket at these two time instants. As would be expected of a disturbance at the onset of breakdown, the strength of the fundamental has decreased substantially at the later instant; the first harmonic peak near 700 kHz, on the other hand, remains essentially unchanged and there is substantial growth in the part of the spectrum corresponding to the second harmonic ( $f\gtrsim 1~\text{ MHz}$ ). In the wall-normal distributions of the peak fundamental and first harmonic strengths in the earlier image of the pair (figure 16 d), we see that the first harmonic profile follows that of the fundamental relatively closely, with additional growth near the boundary-layer edge to form the triple-peaked distribution also characteristic of the fundamental in plaited wavepackets. As with the outer peak in rope-like disturbances, this third peak of the first harmonic was always observed to lie either close to the same $y/{\it\delta}$ as the corresponding fundamental peak, or more commonly, nearer to the wall. In a few cases the third peak was absent in the harmonic, though it was present in the fundamental.

4.4 Structure angle

Finally, we consider the ‘structure angle’ of the disturbances, i.e. the inclination of the line of constant phase in the $s$ – $y$ plane. Kimmel, Demetriades & Donaldson (Reference Kimmel, Demetriades and Donaldson1996) performed measurements of the structure angle in laminar and transitional hypersonic boundary layers using two hot-film probes. For the laminar cases, they found the angle to decrease slightly moving away from the surface, reaching a minimum value of approximately 15° at $y/{\it\delta}=0.7$ , then increasing to 40–60° near the boundary-layer edge. Structure angles from limited schlieren visualizations were also reported by Laurence et al. (Reference Laurence, Wagner, Hannemann, Wartemann, Lüdeke, Tanno and Itoh2012), who observed the angle to decrease from 24° at $y/{\it\delta}=0.4$ to 14–15° at $y/{\it\delta}=0.6$ –0.7; and by Parziale et al. (Reference Parziale, Shepherd and Hornung2015), who measured the angle of the main second mode structures near $y/{\it\delta}=0.6$ to be in the range of 11–15°.

Here the structure angle, ${\it\theta}$ , was calculated by first bandpass filtering the image rows around the wavelength of interest (usually the second-mode fundamental), then calculating the cross-correlation coefficients for vertically separated rows at various relative streamwise displacements; the value of ${\it\theta}$ plotted here is the angle corresponding to the maximum of the resulting cross-correlation curves. In figure 17(a) we show the structure angles for the wavepacket in the two images of figure 15 (condition A, smooth surface); for the later image we show both the fundamental and first harmonic angles. For both images the fundamental angle, ${\it\theta}_{fund}$ , is 70–80°near $y/{\it\delta}=0$ , then decreases away from the surface, at first rapidly but then more gently to reach a minimum of 12–13°; ${\it\theta}_{fund}$ then increases sharply to ${\sim}100^{\circ }$ near the boundary-layer edge. The profile from the later image shows a kink at $y/{\it\delta}$ =0.3 (where the fundamental strength is close to minimum), with ${\it\theta}_{fund}$ increasing to $50^{\circ }$ . The first harmonic structure angle, ${\it\theta}_{harm}$ , shows generally similar behaviour to the fundamental, but with some differences: the angle is larger near the wall and the kink at $y/{\it\delta}$ =0.3 is more pronounced; ${\it\theta}_{harm}$ also remains slightly larger near $y/{\it\delta}=0.7$ (consistent with our observations of figure 15), and the sharp rise in ${\it\theta}$ near the boundary-layer edge occurs at smaller $y/{\it\delta}$ . This latter point is consistent with our previous remarks regarding the tendency of the harmonic wall-normal distribution to be shifted slightly inwards relative to the fundamental distribution. The observations just made regarding the nature of the harmonic structure angle compared to the fundamental were found to generally hold over the disturbances examined in these sequences.

Figure 17.

Measured structure angle for the wavepackets seen in (a) figure 15 and (b) figure 12. (a) The curves correspond to: -▫-, the fundamental in the earlier image; - $\triangle$ -, the fundamental in the later image; and ——, the first harmonic in the later image. (b) The curves correspond to: ▫, the rope-like disturbance; $\triangle$ , the plaited disturbance.

To elucidate changes in the structure angle between rope-like and plaited disturbances, in figure 17(b) we show ${\it\theta}_{fund}$ for the two wavepackets of figure 12 (i.e. again condition A, smooth surface). We see in these disturbances that the kink at $y/{\it\delta}=0.3$ has grown to become a much more substantive feature, with the value of ${\it\theta}_{fund}$ reaching 140° for the second (plaited) wave packet. For this more developed disturbance we also notice a slight rise in ${\it\theta}_{fund}$ near $y/{\it\delta}=0.7$ (relative to the other cases examined thus far).

Figure 18.

Measured structure angle for the wavepackets in their various stages of development seen in (a) figure 13 and (b) figure 14. The temporal order of the curves is -○- (earliest), -▫-, -▵-, and -♢- (latest).

In figure 18 we present the development of ${\it\theta}_{fund}$ for the two wavepackets of figures 13 and 14 (condition B, porous surface). For the first (figure 18 a), in the earlier stages of development the angle increases from 70–80° near the wall (i.e. the lines of constant phase are almost normal to the surface) to form a pronounced hump from $y/{\it\delta}=0.2$ –0.4, peaking at ${\it\theta}=150{-}160^{\circ }$ . The curves thereafter follow a similar pattern to that seen previously. This hump is completely absent in the later two curves, but this is an anomalous case: all other plaited wavepackets examined - and some well-developed rope-like packets – exhibited this feature (see also figure 18 b). Its absence here might be linked to the apparent disappearance of the near-wall peak in the wall-normal distributions of figure 13(c,d). Other than this one atypical case, the general trend for this feature near $y/{\it\delta}=0.3$ is for it to grow in width and maximum value as the wavepacket evolves, reaching an angle as large as 160°. As this would involve the structures tilting forward at 20–30° to the upstream wall, this feature may be associated with the folding over of the wavepacket structures back towards the wall.

In figure 18(b) we show the development of ${\it\theta}_{fund}$ for the wavepacket of figure 14. The rise near $y/{\it\delta}=0.7$ is now very pronounced in the earliest curve, but becomes progressively less so as the wavepacket develops further (the peak of the $y/{\it\delta}=0.3$ feature decreasing simultaneously). A small rise was observed near this location for a number of plaited wavepackets, although none quite so pronounced as that seen here; this rise may also be related to the folding back of the wave structures, as this would tend to increase the apparent structure angle. The minimum value of ${\it\theta}_{fund}$ for these curves is 8–9° which was the smallest value recorded amongst the wavepackets examined. Typically, the minimum angle lay in the range 10–14°.