3.2.1. Determination of the disturbance propagation velocity

To support the statement that transition is initiated by a superposition of TS waves of different frequencies, the propagation velocity of the disturbances is determined and compared with LST. The concept of a wavepacket is understood as a localised disturbance consisting of partial waves of different frequencies, propagation angles and phase velocities. An example for a typical signal is shown in figure 8(a). The phase velocity $c$ of a constituent wave mode can be obtained in principle from the measurements of two closely distributed sensors. As the wavelength of some of the TS waves in the present study was less than the streamwise hot-film sensor distance of ${\rm\Delta}x=28~\text{mm}$ , fourth-order Butterworth filters were applied to limit the signal bandwidth to a narrow frequency band around the investigated TS mode (Kenchi et al. Reference Kenchi, Matsubara and Toshihiko2008). The time difference between two prefiltered sensors signals was then deduced from the cross-correlation ${\it\varrho}_{12}$ . Figure 8(b) compares one of the most amplified two-dimensional TS modes at a frequency of $700~\text{Hz}$ with the results obtained from the hot-film signals using a pass band of $680~\text{Hz}\leqslant f\leqslant 720~\text{Hz}$ . Despite the limited temporal and spatial resolution, the theoretical and experimental phase velocities agree very well in the streamwise region $0.32\leqslant X/L_{c}\leqslant 0.5$ .

Figure 8.

Procedure for determining the phase and group velocity of the TS wavepackets (lower side, $\bar{{\it\alpha}}=-0.92^{\circ }$ , $\mathit{Re}=3.215\times 10^{6}$ ). (a) Hot-film signals of two consecutive sensors and their envelope on the lower side of the wing glove. (b) Streamwise distribution of the phase velocity determined from the cross-correlation of band-pass filtered signals and comparison with LST.

In a process without dissipation, the group velocity describes the velocity of the energy transport of a wavepacket in space (Boiko et al. Reference Boiko, Grek, Dovgal and Kozlov2002). It is defined as the partial derivative of the angular frequency ${\it\omega}$ with respect to the modulus of the wavenumber $|\boldsymbol{k}|=k$ :

The sensor distance in the present case is sufficient to detect the propagation velocity of the disturbances, as dispersion needs larger distances in the aerofoil boundary layer to become significant. Yet, the typical shape of the TS wavepacket needs to be known to enable a spatiotemporal correlation of the type ${\it\varrho}_{12}$ . This shape is obtained by applying a discrete Fourier transform to the digitally band-pass filtered (300–2000 Hz, fourth-order Butterworth) sensor signals. Following Gaster & Grant (Reference Gaster and Grant1975), the envelope of a deterministic wavepacket can be obtained by using the real and the imaginary part of an inverse Fourier transform of the previously obtained discrete Fourier coefficients ${\hat{S}}$ . As the imaginary part describes a similar wavepacket shifted by $90^{\circ }$ in phase, the envelope is defined as the absolute value

Figure 8(a) illustrates a result of the procedure for two consecutive sensor positions. Although growing in space and being displaced in time, the general shape of the envelope of the signals is retained and a good correlation between the two envelopes can be expected. In contrast to most laboratory experiments, e.g. Gaster & Grant (Reference Gaster and Grant1975), Cohen, Breuer & Haritonidis (Reference Cohen, Breuer and Haritonidis1991) or Medeiros & Gaster (Reference Medeiros and Gaster1999), the wavepackets observed in flight are not isolated but one wavepacket gradually transitions into another. To retrieve the time difference between two consecutive signals, the cross-correlation ${\it\varrho}_{12}$ is computed. If the envelope shapes of the investigated disturbances are similar, the correlation shows a distinct peak. The displacement of the peak from the abscissa yields the time difference ${\it\tau}$ necessary to compute the propagation velocity with the known sensor distance ${\rm\Delta}x$ . An example for an investigated flight case is included in figure 9(a). For deterministic TS waves the proposed procedure is underlined by considering the coherence between the two signals

where $G_{12}$ , $G_{11}$ and $G_{22}$ are the auto-spectral and the cross-spectral densities of two signals, i.e. the Fourier transforms of the corresponding cross and auto-correlations. As the wavepackets disperse only gradually, a strong coherence is found for the frequency band of the TS wave hump.

Figure 9.

Procedure for determining the group velocity of the TS wavepackets (lower side, $\bar{{\it\alpha}}=-0.92^{\circ }$ , $\mathit{Re}=3.215\times 10^{6}$ ). (a) Coherence ${\it\gamma}_{12}$ between the signals and the cross-correlation coefficient ${\it\varrho}_{12}$ of the two signal envelopes; (b) coherence averaged for $300\leqslant f\leqslant 1000~\text{Hz}$ .

The comparison of the experimentally obtained disturbance velocity with LST is not straightforward. A ‘steepest descent’ approach would be formally correct (Schmid & Henningson Reference Schmid and Henningson2001), however, a simpler approach is chosen here for convenience. To determine the partial derivative $\partial {\it\omega}/\partial k$ in (3.1), the two-dimensional TS eigenvalues ( $k_{x}=k_{x,r}+\text{i}k_{x,i}$ , $k_{z}=0$ ) are computed for frequencies ranging from 100 to 2000 Hz in steps of 50 Hz. Hence, only the propagation in the streamwise direction is considered. The frequency distribution enables a rather smooth determination of the derivative $\partial {\it\omega}/\partial k_{x,r}$ for all streamwise positions, which is depicted in a contour plot in figure 10(a). Yet the desired group velocity of the TS wavepackets is given by only one value for each streamwise position. Therefore, it is assumed that the bulk of the kinetic energy is transported with the frequency producing the maximum $N$ -factor at each streamwise position. These positions are marked with a solid curve in the position–frequency plane in figure 10(a). Another characteristic position corresponds to the maximum disturbance growth rate $k_{x,i}$ , which is represented by the dashed line.

Figure 10.

Comparison of the disturbance propagation velocity with LST results (lower side, ${\it\alpha}=-0.92^{\circ }$ , $\mathit{Re}=3.215\times 10^{6}$ ). Solid line: $c_{g}(\max (N))$ . Dashed line: $c_{g}(\min (k_{x,i}))$ . (a) LST results; (b) disturbance propagation velocity.

To compare the LST results with the experimental findings, the procedure described above is used to compute the propagation velocity along the entire streamwise hot-film row on the lower side of the aerofoil. Figure 9(b) depicts the streamwise evolution of the mean coherence, which has been averaged in the frequency band 700–1000 Hz. Only in the region $0.38\leqslant X/L_{c}\leqslant 0.52$ is a significant coherence of $\bar{{\it\gamma}}_{12}>0.5$ observed. This is exactly the region where the TS hump protrudes in the spectra after overcoming the sensor noise. The coherence in figure 9(b) declines downstream when the flow randomises in the breakdown process of the boundary layer. Moreover, the comparison of the experimentally determined propagation velocities and the LST results show very good agreement in figure 10(b). The solid line corresponding to the $\max (N)$ criterion is closely approached by the experimental values within the coherent region. The curve corresponding to maximum modal growth describes slightly lower velocities. The characteristic propagation velocity on the lower side of the aerofoil is found at $c_{g}/U_{\infty }\approx 0.47$ for the non-self-similar boundary layer.

Under calm conditions, the coincidence of the frequency range of unstable modes as well as the good agreement of the propagation velocity compared with LST substantiate the assumption that transition on the wing glove is initiated by the TS type of instability.

3.2.2. Three-dimensional and nonlinear aspects of the transition process

The question of which TS-wave frequencies and propagation angles are involved in the transition process under calm conditions can be assessed by using the experimentally obtained frequency–wavenumber spectra. The hot-film data from the spanwise sensor row ${e^{\prime }}_{mn}=e^{\prime }(t_{m},z_{n})$ can be represented in the spectral domain by applying the discrete Fourier transform twice, first in the spanwise direction $z$ and then in the time domain $t$ . The transformation results in a two-dimensional matrix of Fourier coefficients in the discrete frequency–wavenumber domain. The resolution of the spatial wavenumber detection is limited. The maximum detectable wavenumber is determined by the Nyquist–Shannon theorem to $k_{z}=2{\rm\pi}/(2{\rm\Delta}z)=628.3~\text{m}^{-1}$ , where ${\rm\Delta}z=5~\text{mm}$ is the spanwise sensor distance. The minimum detectable wavenumber $k_{z}=62.8~\text{m}^{-1}$ , i.e. the largest wavelength, is determined by the length of the spanwise sensor row with 21 equidistantly distributed sensors.

The spanwise sensor row is locally fixed and different boundary-layer states can thus only be observed by carefully changing the angle of attack. Slight changes in the angle of attack result in a deceleration or an acceleration of the lower side boundary layer and lead to the streamwise displacement of the transition process observed in figure 6. Figure 11 shows frequency–wavenumber spectra measured in flight for four different angles of attack. While the frequency axis in the contour plots scales the temporal periodicity of the signals, the spanwise wavenumber accounts for the spanwise periodicity contained in the boundary-layer disturbances. The displayed results correspond to the four different angles of attack highlighted with plus markers in figure 6(b). The superimposed lines of constant wave angles ${\it\Psi}$ in the first three plots were calculated using the boundary-layer method and the linear stability method described above.

Figure 11.

Frequency–wavenumber spectra obtained with the spanwise hot-film row on the lower side: (a) $X/L_{c}=0.482$ , ${\it\alpha}=-0.16^{\circ }$ , $\mathit{Re}=3.032\times 10^{6}$ ; (b) $X/L_{c}=0.482$ , ${\it\alpha}=-0.60^{\circ }$ , $\mathit{Re}=3.142\times 10^{6}$ ; (c) $X/L_{c}=0.482$ , ${\it\alpha}=-0.92^{\circ }$ , $\mathit{Re}=3.215\times 10^{6}$ ; (d) $X/L_{c}=0.482$ , ${\it\alpha}=-1.29^{\circ }$ , $\mathit{Re}=3.325\times 10^{6}$ .

Figure 12.

Frequency–wavenumber spectra obtained with LST. Solid lines represent contours of equal phase speed $c$ . From the inner to the outer curve: $c/U_{e}=0.31,0.33,0.35,0.37,0.39$ . Here $\circ$ -markers depict exemplary modal combinations for the generation of difference modes and $\times$ -markers depict exemplary modal combinations for streak generation: (a) $X/L_{c}=0.482$ , ${\it\alpha}=-0.60^{\circ }$ , $\mathit{Re}=3.142\times 10^{6}$ ; (b) $X/L_{c}=0.482$ , ${\it\alpha}=-0.92^{\circ }$ , $\mathit{Re}=3.215\times 10^{6}$ .

For the highest angle of attack, ${\it\alpha}=-0.16^{\circ }$ in figure 11(a), a concentrated region of high amplitude appears for frequencies between $500~\text{Hz}<f<900~\text{Hz}$ and spanwise wavenumbers of $|k_{z}|<200~\text{m}^{-1}$ . The deviation of the maximum from the zero propagation angle is due to a slight side-slip angle of the wing glove that is consistently present, but is insignificant for the transition development. The picture of TS wavepackets with a predominant two-dimensional propagation direction is typical for a natural transition scenario in a low-disturbance environment, as can be seen in the frequency–wavenumber spectra previously obtained by Peltzer & Nitsche (Reference Peltzer and Nitsche2004) and Seitz (Reference Seitz2007) in similar flight test configurations. The corresponding wave propagation angles remain below $20^{\circ }$ at this early transition stage. Nevertheless, the presence of slightly oblique TS waves with different frequencies is the reason for the appearance of the localised wavepackets and the three-dimensional character of laminar–turbulent transition in the later stages. In the second case, ${\it\alpha}=-0.60^{\circ }$ in figure 11(b), the disturbance amplification is increased due to the increased instability of the boundary layer. Growing disturbances have spread toward higher spanwise wavenumbers, signifying an increased amplification of oblique TS waves. For ${\it\alpha}=-0.92^{\circ }$ in figure 11(c) the boundary layer has become even more unstable, especially toward three-dimensional modes. Besides the amplification of highly oblique TS waves at the fundamental frequencies, fluctuation energy at very low frequencies can be observed to magnify symmetrically for wavenumbers around $\pm 200~\text{m}^{-1}$ . Furthermore, a region of higher-harmonic waves can be seen to evolve from the background noise in the frequency band between 1300 and 1700 Hz. This indicates that the nonlinear transition stage has started. In between $-0.92^{\circ }\geqslant {\it\alpha}\geqslant -1.29^{\circ }$ the disturbance energy grows steeply. The spectrum fills up with fluctuation energy in-between the fundamental wave combinations and the growing low-frequency disturbances with a rich spanwise wavenumber content. For the lowest angle of attack ( ${\it\alpha}=-1.29^{\circ }$ , figure 11 d), the low-frequency content for all spanwise wavenumbers has vastly grown. The flow is highly transitional and the disturbances are randomised. Therefore, no wave angles based on LST are included in the figure.

From these observations it can be concluded that a certain portion of three-dimensional receptivity must be involved in generating the three-dimensional modes (Kachanov Reference Kachanov2000). Furthermore, no significant nonlinear activity at the exact subharmonic wave band was detected in figure 11. At a certain amplification level of the fundamental TS waves, the power spectra fill up with fluctuation energy from the low frequency end. Furthermore, a high amplification of oblique disturbances at $|k_{z}|\approx 200~\text{m}^{-1}$ can be identified in the initial nonlinear transition stage. Except for a significantly weaker maximum at the first overtones of the fundamental waves, there are no other wave interactions visible in figure 11(c). The same qualitative appearance of the frequency wavenumber spectra can also be found in an article of Peltzer (Reference Peltzer2004), although the authors do not discuss this issue.

LST is employed to yield the disturbance growth rates $k_{x,i}$ and the phase velocity $c$ of all modes in the frequency–wavenumber plane ( ${\it\omega}$ , $k_{z}$ ) for each streamwise position. The growth rates of the three-dimensional TS modes are integrated in the streamwise direction to obtain an $N$ -factor distribution. In figure 12 the N-factor maximum corresponds fairly well to the experimentally observed TS wave hump in figure 11(b,c). However, disturbance growth is experimentally found even at higher spanwise wavenumbers $k_{z}$ . This may be explained by the fact that LST based on the parallel flow assumption underpredicts the growth of oblique wave modes, as was shown by Bertolotti (Reference Bertolotti1991).

Three-dimensional disturbances are essential for transition to turbulence in a two-dimensional boundary layer under controlled and natural forcing conditions. In the absence of visible signs of H-type or K-type wave interaction (Herbert Reference Herbert1988; Kachanov Reference Kachanov1994), the question arises: which nonlinear mechanism eventually leads to transition of the aerofoil boundary layer in a low-disturbance flight environment?

In a flat-plate experiment under different weak levels of free-stream turbulence Matsubara et al. (Reference Matsubara, Takaichi and Kenchi2010) reported two different natural transition mechanisms. For slightly higher turbulence levels they observed the expected transformation of wavepackets into ${\rm\Lambda}$ -shaped structures and their subsequent breakdown. For lower free-stream turbulence localised wavepackets were distorted by a short streak before rapidly breaking down. The latter scenario appears to be related to the observations of Breuer, Cohen & Haritonidis (Reference Breuer, Cohen and Haritonidis1997) in late transition stages of an isolated wavepacket in a controlled flat-plate experiment. They describe that substantial subharmonic resonance in the wavepacket is succeeded by a stage of the short but rapid streak generation. This stage was connected to the ‘oblique transition’ mechanism (Henningson, Lundbladh & Johansson Reference Henningson, Lundbladh and Johansson1993). The mechanism requires a nonlinear interaction of oblique waves to transfer energy into streamwise vortices which subsequently generate streaks through the lift-up effect (Schmid & Henningson Reference Schmid and Henningson2001). If non-modal streak growth (a linear mechanism) is sufficient, streaks become susceptible to secondary instability causing breakdown to turbulence. The first set nonlinear interaction between two oblique waves creates the modes $(f,k_{z})=(0,0)$ , $(2f_{0},0)$ , $(2f_{0},\pm 2k_{z,0})$ and $(0,\pm 2k_{z,0})$ . In the experiments of Elofsson & Alfredsson (Reference Elofsson and Alfredsson2000) and King & Breuer (Reference King and Breuer2001) the dominant energy was found for the streaks created by the stationary vortices $(f,k_{z})=(0,2k_{z,0})$ . However, the initial oblique waves need a certain amplitude for the nonlinear initiation of the process (Elofsson & Alfredsson Reference Elofsson and Alfredsson1998). In flight tests under low environmental disturbances such low-frequency disturbances can in principle be self-generated by non-resonant interactions of the observed broad band of TS waves and wavenumbers. One such exemplary modal combination is included in figure 12(b). Indeed, the low-frequency part of the spectrum in figure 11(c) exhibits a certain similarity to spectral representation of streaks given by Westin et al. (Reference Westin, Bakchinov, Kozlov and Alfredsson1998).

Another explanation is given by de Paula et al. (Reference de Paula, Würz, Krämer, Borodulin and Kachanov2013) who conducted a transition investigation on an aerofoil in a low-turbulence wind tunnel using a disturbance source generating fully controlled broadband frequency–wavenumber spectra. The investigation was limited to two weakly nonlinear stages of transition, i.e. no feedback influence on the fundamental TS waves, omitting explicitly the fully nonlinear state. The first stage is the generation of low-frequency waves from a simple (non-resonant) quadratic combination interaction of fundamental TS waves. Between a two-dimensional $(f_{1},0)$ and an oblique mode $(f_{2},k_{z,2})$ the first interaction set creates the modes $(f,k_{z})=(|f_{1}-f_{2}|,\pm k_{z,2})$ and $(f_{1}+f_{2},\pm k_{z,2})$ . It was shown that oblique ‘difference modes’ of the type $(f,k_{z})=(|f_{1}-f_{2}|,\pm k_{z,2})$ were most efficiently generated, especially when they had a phase velocity close to the one of the fundamental waves (Wu, Steward & Cowley Reference Wu, Steward and Cowley2007). Again, such a generation is plausible from the present results when comparing the amplified frequency–wavenumber content in figure 11(b,c) with figure 12(a,b), where the phase velocity $c$ is included by solid lines. The subsequent rapid growth of the difference modes in the non-self-similar boundary layer is described by the second stage of frequency-detuned subharmonic resonance with the fundamental waves (Würz et al. Reference Würz, Sartourius, Kloker, Borodulin, Kachanov and Smorodsky2012). An initial broadband TS wave spectrum enables a favourable selection of the most efficient combination and a frequency cascade of this mechanism can explain the observed filling of the spectra.

Both mechanisms described above rely on a first (weakly) nonlinear stage, in which ‘seeds’ (stationary vortices or oblique low-frequency TS waves) are generated from the fundamental TS waves leading either to the subsequent creation and non-modal growth of streaks or to resonant, frequency-detuned subharmonic wave interactions. An inherent prerequisite of the first interaction mechanism is sufficient growth of the fundamental waves to generate low-frequency disturbances of significant amplitude. In figure 12 remarkably high $N$ -factors are reached for the parameters where the first signs of nonlinearity are observed experimentally. Hence, in a low-disturbance flight environment self-generated disturbances from the fundamental TS wave hump can assume significantly larger amplitudes than low-frequency background disturbances created directly by the receptivity of the aerofoil boundary layer. The only obvious source for the appearance of low-frequency modes is thus the self-generation by broadband interactions of the fundamental TS waves. The measurement setup in these highly challenging in-flight experiments did not have the necessary resolution to clarify completely which of the described mechanisms eventually triggers breakdown to turbulence. It can be conjectured that both mechanisms, in which combination modes of TS waves are crucial, are possible and may become important at different later stages of the transition process.

3.4.1. The role of the instantaneous pressure gradient

To check the validity of the above considerations, the role of the pressure gradient on the lower side is investigated in detail. Figure 16(a) shows the temporal fluctuations of the pressure coefficient $C_{p}^{\prime }$ obtained with the miniature pressure sensors on the lower side of the wing glove in the streamwise region between $0.07\leqslant X/L_{c}\leqslant 0.64$ . Fluctuations of $\pm 0.15$ are observed for short time periods and changes in the sign of the pressure coefficient become apparent. The variations are intensified in the leading-edge region which is underlined by the RMS values of the pressure fluctuations included in figure 16(b). Whereas positive deflections generally lead to a stabilisation of the boundary layer, negative deflections have a destabilising effect on the lower side of the wing section. It can be observed that the upstream fluctuations of boundary-layer disturbance growth in figure 15(b) correspond to negative deflections of the pressure coefficient $C_{p,l}$ in figure 16(a) at approximately the same time. Note that these results qualitatively correlate with the quasi-steady expectations in figure 6.

Figure 16.

Fluctuations of the pressure coefficient on the lower side: (a) temporal fluctuations; (b) RMS of $C_{p}^{\prime }$ on the lower side; (c) instantaneous pressure distributions due to a downward directed gust.

Thus, to some extent the fluctuating transition position in figure 15 can be explained by the quasi-steady considerations in § 3.2. Yet, the characteristic time scales of some of the observed transition excursions are of the order of $0.2~\text{s}\leqslant {\rm\Delta}t\leqslant 1~\text{s}$ in the present experiments, which will be further verified in § 3.4.1. Within such periods the aerofoil travels only a distance corresponding to 6–30 chord lengths. For such dimensionless distances the time derivatives assume an important role in the conservation equations (Bisplinghoff et al. Reference Bisplinghoff, Holt and Halfman1996). The relevance of unsteady inviscid aerofoil effects as well as unsteady boundary-layer effects can be assessed more clearly when considering these time periods as half-cycles of an oscillation and by determining the characteristic frequency of a corresponding harmonic excitation. Table 3 provides an estimation of these relations, including the reduced frequency ${\it\kappa}$ (McCroskey Reference McCroskey1982) and the boundary-layer frequency parameter ${\it\kappa}_{{\it\delta}}$ (Lighthill Reference Lighthill1954; Hill & Stenning Reference Hill and Stenning1960; Patel Reference Patel1975). Both dimensionless numbers express ratios between the fluid convection time scale and the time scale of a harmonic excitation caused by the oncoming flow or the outer flow of the boundary layer, respectively. According to Leishman, Rycroft & Shyy (Reference Leishman, Rycroft and Shyy2006) unsteady inviscid effects become significant for ${\it\kappa}>0.05$ and they consider ${\it\kappa}>0.2$ highly unsteady. Note that unsteady distortions of the boundary-layer profiles compared with the quasi-steady case increase with the boundary-layer length ( ${\it\kappa}_{{\it\delta}}\propto x$ ), which is significant for the investigated aerofoil (Patel Reference Patel1975; Reeh Reference Reeh2014).

Table 3.

Estimation of dimensionless time and frequency scales to characterise the unsteadiness of the aerofoil flow ( $U_{\infty }$ , $L_{c}=1.35$ , $x\approx X$ ).

For an unsteady outer flow, the boundary condition for the boundary-layer development, i.e. the unsteady edge velocity, cannot solely be determined from the surface pressure since Bernoulli’s equation contains the time derivative of the velocity potential. It was already inferred from figure 15 that the variation of the pressure gradient plays an outstanding role for the boundary-layer development and affects transition. In order to demonstrate typical amplitudes observed in these transient events, characteristic instantaneous pressure distributions for one such event at $10.3~\text{s}\leqslant t\leqslant 11.2~\text{s}$ are depicted in figure 16(c). A strong deviation of ${\rm\Delta}C_{p}$ is observed at $t=10.645~\text{s}$ , especially in the leading edge part. The deflection corresponds to the aerofoil response to a downward directed gust. On the lower side of the aerofoil ${\rm\Delta}C_{p,l}\geqslant 0.2$ and the pressure gradient at this particular time instant is completely adverse, promoting a significant deceleration of the boundary layer. The distributions for $t=10.3~\text{s}$ and $11.2~\text{s}$ almost coincide. In the following it will be shown that the duration of the bulk of the deflection occurs in less than $0.4~\text{s}$ .

3.4.2. Investigation of the streamwise excursions of the transition front

The striking rapidness of some of the upstream transition fluctuations with durations of less than 0.3 s in figure 15(a) calls for a more profound investigation in order to determine their nature. Considering the increased level of small-scale turbulence in the oncoming flow (cf. figure 13 c and table 2), the spectral composition of these transition excursions is of particular interest. Studer et al. (Reference Studer, Arnal, Houdeville, Seraudie and Govindarajan2006) showed in a low-disturbance wind tunnel that the unsteady boundary-layer development on an aerofoil can significantly affect the transition mechanism. Their results show the rapid appearance of turbulent bursts which are preceded by nonlinear instabilities. The mechanism was first described by Obremski & Fejer (Reference Obremski and Fejer1967) in an experimental study of an oscillating flat-plate boundary layer. Tani (Reference Tani1969) and Loehrke, Morkovin & Fejer (Reference Loehrke, Morkovin and Fejer1975) explain the occurrence of such transition by the rapid quasi-steady growth of linear instability waves travelling through unsteadily distorted, highly unstable boundary-layer profiles.

Figure 17 shows the temporal variations of the AC part of the hot-film raw signals for such a transition excursion ( $10.2~\text{s}<t<11.2~\text{s}$ , from the first to the eighth sensor of the streamwise hot-film row). This time span is marked with dashed lines in figure 15(b). The voltage signals of the considered time sequence have been normalised with the maximum value observed in the considered time sequence. The highly nonlinear transition stage appears as sudden bursts of high amplitudes, which marks the course of the transition excursion. Starting the consideration of the plot at $t\approx 10.4~\text{s}$ , $X/L_{c}=0.42$ and proceeding in time, transition moves upstream until reaching a minimum streamwise position at $t\approx 10.64~\text{s}$ . Continuing along the timeline from this upstream position, the transition front moves downstream again and reaches the last depicted sensor in less than 0.1 s. Before and after the excursion of the transition front, localised waveforms are visible at the downstream positions indicating that the transition front spends more time in this region. Yet, only strong low-frequency, nonlinear modulations can be observed in the depicted time interval. According to figure 7(b), amplified TS waves are expected at significantly higher frequencies (700–1500 Hz). Completed breakdown and homogenised turbulence can only be observed downstream of the sensor position $X/L_{c}=0.353$ within the time interval of the transition excursion.

Figure 17.

Surface plot of the spatiotemporal disturbance evolution of a transition excursion. Lines depict the normalised hot-film signals. Positions for the wavelet analysis are marked with thick lines.

The time period over which the transition excursion appears corresponds to the $C_{p,l}$ fluctuations shown in figure 16, which reach a minimum at approximately the same time when the transition front reaches its most upstream position. A minimum in $C_{p,l}$ leads to a deceleration of the boundary layer in the present flight case. It can be inferred from these considerations that boundary-layer transition is mainly driven by the pressure gradient under these disturbed flow conditions. It is important to note that the coherence between the movement of the transition front and the excursions in the pressure gradient excludes the possibility of a pure bypass transition from a convective disturbance generated by free-stream turbulence.

3.4.3. Wavelet analysis of the temporal transition fluctuations

The high nonlinearity at streamwise positions far upstream of the mean transition region calls for deeper insight into the spectral properties of the process to determine whether it starts linearly or whether it is intrinsically nonlinearly. The appearance of the transition front excursion is highly localised in space and time. Conventional Fourier analysis, which is conceptually delocalised in time, may not shed sufficient light into all of the details involved in the time–frequency domain. Therefore, wavelet analysis is employed to investigate the transient process. The use of this method is inspired by the works of Hudgins, Friehe & Mayer (Reference Hudgins, Friehe and Mayer1993), Studer et al. (Reference Studer, Arnal, Houdeville, Seraudie and Govindarajan2006) and Bernardini et al. (Reference Bernardini, Benton, Chen and Bons2013), who also applied wavelet analysis in transient boundary-layer flows. Since the important basics of wavelets are presented in each of these studies and further information can be found in the review of Farge (Reference Farge1992), only a brief summary of the most important aspects of the continuous wavelet transform applied in the present case is given. The transform can be interpreted as an inner product between an analysing function, i.e. the wavelet, and the signal to be analysed. It is defined by the integral

The function ${\it\psi}^{\ast }$ denotes the complex conjugate of the wavelet which is intrinsically localised in time. It can be compressed or stretched by the scale parameter $a$ and shifted in time by the parameter ${\it\tau}$ to approximate locally the function $e^{\prime }(t)$ to be analysed. In order to be admissible for the transform, the analysing function ${\it\psi}$ needs to fulfill certain mathematical requirements (Farge Reference Farge1992). The strength of wavelet analysis is that a great variety of analysing functions exists and a suitable one can be chosen depending on the particular physical process. Here, a Morlet wavelet is used for its good representation of sine-like signal parts as they are expected due to determinism involved in boundary-layer transition:

The wavelet function constitutes a plane wave modulated by a Gaussian envelope. The scales used in wavelet analysis cannot directly be connected to frequencies in the Fourier sense. The maximisation of a given wavelet with a sine wave yields the centre frequency $f_{c}$ . In (3.5) the scale parameter $a$ is connected to the centre frequency $f_{c}$ of the Morlet wavelet by the relation $f_{p}=f_{c}/a$ , where $f_{p}$ is the so-called pseudo-frequency. In the present application of the Morlet wavelet the pseudo-frequency is a good approximation of the real frequency. The scales $a$ are prescribed as $1/\log (a)$ in order to obtain a favorable distribution of pseudo-frequencies between 1 Hz and 8 kHz important for the present investigation. The resulting wavelet coefficients ${\hat{W}}$ from (3.5) yield the energy spectrum ${\hat{W}}\hat{W^{\ast }}=|{\hat{W}}^{2}|$ , which represents the energy content of the signal in the time-scale domain or the time-pseudo-frequency in the present case.

In figure 18 the result of the wavelet analysis is demonstrated for the four different streamwise positions marked in figure 17, corresponding to the second, fourth, sixth and the eighth streamwise sensor. Maybe the most important insight into the transition mechanism can be inferred from the contour plot in figure 18(a) at the first streamwise station. Only a small portion of the time–frequency domain contains energy. At $t\approx 10.6~\text{s}$ and within the frequency band of 500–1800 Hz disturbances are amplified. The concentration in a particular frequency band suggests that transition is indeed initiated by a linear amplification process of TS waves before the start of the nonlinear evolution. For the second station at $X/L_{c}=0.287$ short-time nonlinear processes can already be observed by the broader frequency distribution approximately at the same time when the transition front reaches the most upstream position. Yet, the energy transfer occurs on a very short time scale. At earlier and later time instants, amplification of the linear fundamental frequencies is observed. Further downstream, the nonlinear stage at $X/L_{c}=0.331$ and $X/L_{c}=0.375$ covers longer time periods. The last streamwise station in figure 18(d) includes the homogenisation of turbulence at $t\approx 10.65~\text{s}$ and another nonlinear amplification can be observed at $t\approx 10.95~\text{s}$ . The nonlinear stage in figure 18(b–d) is preceded and followed by amplification in the TS wave band.

Figure 18.

Contours in logarithmic scale of the normalised wavelet coefficient depicting the disturbance energy in the time–frequency domain at four streamwise positions: (a) $X/L_{c}=0.242$ ; (b) $X/L_{c}=0.287$ ; (c) $X/L_{c}=0.331$ ; (d) $X/L_{c}=0.375$ .

Figure 19.

Comparison of transition measurements with quasi-steady N-factor computations, which are based on the instantaneous pressure distribution at $t=10.65~\text{s}$ in figure 16(c). Legend numbers denote frequencies $f$ in Hz.

From these results it can be concluded that the transient appearance of the ‘transition flashes’ is associated with the transient development of the laminar boundary layer which is driven by rapid changes of the flow field along the boundary-layer edge. These changes are mainly caused by vertical gusts and the resulting aerofoil motion in flight through moderate atmospheric turbulence, as demonstrated in the study of Reeh et al. (Reference Reeh, Weismüller and Tropea2014). A pure bypass transition attributed to an entrainment of free-stream turbulence can be excluded from the present results. Nevertheless, a combined effect of unsteady boundary-layer development and higher small-scale free-stream turbulence forcing on transition cannot be excluded. In their generic wind-tunnel experiments, Studer et al. (Reference Studer, Arnal, Houdeville, Seraudie and Govindarajan2006) investigated two frequencies of periodic pressure gradient changes in an aerofoil boundary layer. Their ‘low-frequency’ case exhibits strong similarities to the present results. In their ‘high-frequency’ case, the unsteady boundary-layer effects were strong enough to cause transition by the convection of a turbulent front after formation of an initial turbulent spot. This type of transition, first discovered by Obremski & Fejer (Reference Obremski and Fejer1967), is not observed in the present flight cases.

Quasi-steady $N$ -factor calculations are conducted for this event to demonstrate differences in the obtained transition $N$ -factors with calm flight conditions. The computations are based on the instantaneous pressure distribution. Note that the most amplified frequencies correspond to the observations of the wavelet spectra in figure 18. For the depicted time instant $t=10.65~\text{s}$ the hot-film RMS curve is based on 1600 samples corresponding to an averaging interval of ${\rm\Delta}t=0.1~\text{s}$ . The $N$ -factor corresponding to the peak RMS distribution is found to be $N\approx 10.5$ . In comparison with the findings under calm conditions in figure 7(b) a reduction of ${\rm\Delta}N\approx 3$ is observed. Despite the observed reduction, the comparably high $N$ -factor of $N=10.5$ still substantiates a dominant role of TS waves in the transition process.

Two systematic errors are introduced by employing quasi-steady boundary-layer computations based on the instantaneous pressure distribution. First, the contribution of the time derivative in the Bernoulli equation to the velocity distribution necessary for the boundary-layer computation cannot be accounted for (Reeh Reference Reeh2014). Second, the role of unsteady distortions of the boundary-layer profiles is neglected as the time history of the boundary-layer development is completely omitted by the quasi-steady procedure (Hill & Stenning Reference Hill and Stenning1960; Patel Reference Patel1975). The impact of both effects on the present results needs to be assessed in future numerical investigations using unsteady computation methods. The quasi-steady application of linear stability methods for transition prediction is still justified since the frequencies of the observed TS waves are about two orders of magnitudes higher than the frequencies associated with the unsteady aerofoil response, a reasoning which is also pointed out by Obremski & Morkovin (Reference Obremski and Morkovin1969).

3.4.4. Some remarks on the effects of small-scale turbulence

The presence of TS waves in all investigated cases, which is also observed by Zanin (Reference Zanin1985) in flights at even higher turbulence intensities, is explained by the relatively weak vortex receptivity processes found on a typical laminar wing section. Considering the elevated turbulence intensity levels in flight through moderately turbulent conditions ( $\mathit{Tu}\leqslant 0.4\,\%$ ), the question whether this forcing is able to create stronger TS waves in the receptivity process is of importance. When drawing comparisons with wind-tunnel experiments, it should be recalled that the turbulence encountered in the in-flight experiments is intrinsically different to wind-tunnel turbulence of nominally similar turbulence intensity levels. For the considered aerofoil boundary layer, the waveband relevant for TS waves amplification is $300~\text{Hz}\leqslant f\leqslant 2~\text{kHz}$ . Figure 20 shows flight-measured kinetic energy spectra for low turbulence intensities. The dashed lines represent the one-dimensional form Kolmogorov’s power law for the inertial subrange used in conjunction with Taylor’s hypothesis

where $C_{T,X}=0.66$ is a universal turbulence constant for the one-dimensional energy spectrum (Sheih, Tennekes & Lumley Reference Sheih, Tennekes and Lumley1971). The energy containing eddies of atmospheric turbulence, i.e. those motions involved in the turbulence generation, are generally produced on length scales of the order of a hundred metres. Therefore, the bulk of the kinetic energy of the turbulence is contained at length scales which are irrelevant for boundary-layer receptivity. In the cascade process of atmospheric turbulence, kinetic energy decays over at least two decades before reaching the TS wave frequencies. Most of the directional information is lost and the statistics of the motion becomes universal. This is a major difference to most wind-tunnel investigations, where a certain degree of anisotropy is inevitable (Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994) and may influence the type of boundary-layer disturbances (Kendall Reference Kendall1998). Furthermore, the wind-tunnel turbulence spectrum may be entirely different (Zanin Reference Zanin1985; Weismüller Reference Weismüller2011). In the TS waveband ordinary wind tunnels may create stronger fluctuations. It is therefore deemed plausible that the TS wave dominated transition scenario can persist at higher nominal $\mathit{Tu}$ values in flight than under common wind-tunnel conditions. Note further that most comparable wind-tunnel studies are conducted for lower flow speeds, where the instability of the boundary layer is also found at lower frequencies. Nevertheless, the comparison in figure 20 shows that the energy density available for the generation of boundary-layer disturbances differs over all relevant frequencies for different turbulence conditions.

Figure 20.

Comparison of one-dimensional turbulence spectra measured by means of hotwire anemometry in gliding flight at low turbulence levels.

Along the boundary layer, the inherently three-dimensional and localised disturbances imposed by free-stream turbulence can continuously feed energy for the generation of TS wavepackets (Kachanov Reference Kachanov2000). However, the convection velocity of free-stream turbulence components is higher than the phase velocity of the TS modes and at given TS frequencies a wavelength reduction is necessary to transfer energy. An energy transfer can be promoted by interactions of external vorticity with localised or distributed surface roughness creating disturbances matching TS waves in their vital aspects, i.e. amplitude, wavenumbers and frequency (Crouch Reference Crouch1992a ,Reference Crouch b ; Dietz Reference Dietz1999; Saric et al. Reference Saric, Reed and Kerschen2002). On a polished wing surface, however, such receptivity sites are largely avoided and Bertolotti (Reference Bertolotti1997) shows that free-stream turbulence components corresponding to the TS wave band do not penetrate efficiently into lower part of the boundary layer, where TS waves exhibit an amplitude maximum. Furthermore, any TS waves created by a leading edge receptivity mechanism are dampened significantly before reaching the point of neutral stability. The previous results and considerations infer a relatively weak influence of atmospheric turbulence on TS wave amplitudes.

Bertolotti (Reference Bertolotti1999) considers a direct coupling of lower-frequency turbulence components to streaky disturbance structures very efficient. The evolution of these Klebanoff modes is directly connected to the transient growth mechanism (Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Schmid & Henningson Reference Schmid and Henningson2001) and various experiments (Kendall Reference Kendall1985; Boiko et al. Reference Boiko, Westin, Klingmann, Kozlov and Alfredsson1994; Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Fransson et al. Reference Fransson, Matsubara and Alfredsson2005; Kenchi et al. Reference Kenchi, Matsubara and Toshihiko2008) have shown that this coupling mechanism has a crucial importance for higher turbulence levels than those encountered in this investigation. The present experimental set-up did not allow a detailed investigation into the possible generation of low-frequency Klebanoff modes. Nevertheless, the typical features of a TS wave initiated transition scenario have been demonstrated under all investigated flight conditions. As explained in §§ 3.4.1 and 3.4.3, no indication of bypass transition was found. On both sides of the aerofoil, streamwise excursions of the transition front were triggered by fluctuations of the pressure gradient. Disturbance amplification is clearly confined to the region, where TS modes experience the highest growth rates according to LST. Hence the boundary-layer response to the natural disturbances supports the dominant role of TS waves in the transition process. The question whether streaky disturbance structures are also present and modify the transition process in gliding flight through atmospheric turbulence needs to be assessed in future studies.