3.1 Formation of boundary-layer modes

Local stability analysis, based on the temporal approach, has been carried out at different streamwise locations, corresponding to the frequencies $f_{6}$ and $f_{10}$ shown in figure 6, for two-dimensional disturbances (i.e. $\unicode[STIX]{x1D6FD}=0$). The streamwise wavenumbers for the LST analysis were computed from $\unicode[STIX]{x1D6FC}_{n}=2\unicode[STIX]{x03C0}f_{n}/(u_{t,e}+a_{e})$, where $u_{t,e}$ and $a_{e}$ are respectively the local wall-tangential velocity component and speed of sound at the boundary-layer edge at each considered $x$-location. The considered phase speed for the LST analysis corresponds to the local fast-mode phase speed, thus resulting in the local streamwise wavenumber of the fast acoustic waves outside the boundary layer (i.e. the local forcing) being used as input for the LST analysis at all the streamwise locations at both the frequencies $f_{6}$ and $f_{10}$. Figure 7 shows the spectrum of the complex eigenvalues at different positions along the wedge for the frequency $f_{10}$. Moving from low to high values of $\unicode[STIX]{x1D714}_{r}$ in the spectrum corresponds to moving from low to high phase speeds(i.e. the output phase speeds of the LST analysis). The two horizontal branches of modes at $\unicode[STIX]{x1D714}_{i}\approx 0$ positioned at the left and right extremes of the graph represent the slow and fast acoustic-wave continuous spectra respectively. The branch of modes positioned in the middle of the graph, at approximately $\unicode[STIX]{x1D714}_{r}=0.26$, represents the entropy/vorticity wave continuous spectrum.

Performing the LST analysis at several $x$ positions is equivalent to tracking the movement of some discrete modes of interest. The evolution of the discrete modes helps, in turn, to understand the corresponding trend of the numerical pressure distribution shown in figure 6. In particular, at the streamwise location corresponding to the first peak ($x\approx 150$) in the curve of the pressure response, figure 7 shows the presence of a discrete stable mode close to the continuous spectrum of the fast acoustic waves (the blue asterisk symbol, indicating the position $x=143$, with $\unicode[STIX]{x1D714}_{r}=0.294$). This mode corresponds to mode F, which is synchronised with the forced fast acoustic waves in the leading-edge region, thus experiencing a resonance-induced amplification (Fedorov & Khokhlov Reference Fedorov and Khokhlov2001), as shown by the numerical results. However, as is evident in figure 7, mode F gradually moves away from the fast acoustic-wave continuous spectrum, corresponding to a de-synchronisation process. This results in a downstream decay of the wall response, due to the decay of mode F, which decreases its phase speed (as it moves towards lower values of $\unicode[STIX]{x1D714}_{r}$) and becomes more stable (moving towards higher negative values of $\unicode[STIX]{x1D714}_{i}$).

The region $x\approx 250{-}500$ is characterised by an oscillatory wall response in figure 6 (for frequency $f_{10}$) that suggests a modulation process as mode F traverses the entropy/vorticity wave continuous spectrum. Such a behaviour was previously observed in Ma & Zhong (Reference Ma and Zhong2003b), Ma & Zhong (Reference Ma and Zhong2005), Zhong & Ma (Reference Zhong and Ma2006) and Zhong & Wang (Reference Zhong and Wang2012), and described as a modulation between different competing modes, such as the internal decayed mode F and the external mode of the forcing fast acoustic waves. At the same time, another discrete mode, close to the slow acoustic-wave continuous spectrum, starts moving rapidly from very low values of $\unicode[STIX]{x1D714}_{i}$ to higher values, becoming less stable with increasing phase speed. This mode corresponds to mode S, which is excited due to a synchronisation with the decaying mode F, after the latter has crossed the convected wave continuous spectrum.

Upstream of $x=641$ mode S corresponds still to the first (lower-frequency) instability mode, then for $x>641$ mode S develops in a second (higher-frequency) instability mode (or Mack mode) (Mack Reference Mack1984), which emerges as a consequence of the wavelength exchange (or synchronisation) mechanism between mode F and mode S (Fedorov & Khokhlov Reference Fedorov and Khokhlov2001; Fedorov Reference Fedorov2003, Reference Fedorov2011; Ma & Zhong Reference Ma and Zhong2003a,Reference Ma and Zhongb, Reference Ma and Zhong2005; Zhong & Ma Reference Zhong and Ma2006; Fedorov & Tumin Reference Fedorov and Tumin2011; Zhong & Wang Reference Zhong and Wang2012). Further downstream, the Mack mode shows a decreasing frequency (and phase speed), and becomes progressively less stable. Downstream of $x=700$ in figure 6, the $f_{10}$ curve appears now much smoother than in the previous modulation region, which indicates that there is a dominant mode in this region. This mode grows as a consequence of a resonance, due to a (second) synchronisation with the fast acoustic waves, consistent with the numerical results of Ma & Zhong (Reference Ma and Zhong2003b), Ma & Zhong (Reference Ma and Zhong2005), Zhong & Ma (Reference Zhong and Ma2006) and Zhong & Wang (Reference Zhong and Wang2012). For this reason, we refer to this mode as the second fast mode, or mode F II, combining both Fedorov’s and Zhong’s notation (Fedorov Reference Fedorov2011; Zhong & Wang Reference Zhong and Wang2012). The downstream evolution of the second fast mode is represented by a short branch very close to the fast acoustic-wave continuous spectrum in the top right corner of figure 7.

Similar characteristics have been observed in the corresponding spectrum at the frequency $f_{6}$, with the difference that the evolution of the modes is delayed and extends only up to approximately the position where mode F traverses the convected wave continuous spectrum, as can be observed in figure 6. Thus, the evolution of the boundary-layer modes at the higher frequencies is quicker than at the lower frequencies, and results in the mode F/mode S wavelength exchange mechanism taking place closer to the leading edge.

Figure 8.

Phase speed of the boundary-layer modes along the wall for the frequencies $f_{6}$ and $f_{10}$. Comparison between DNS and LST results. The dotted horizontal blue lines indicate the local phase-speed values of the fast and slow acoustic waves, and the entropy/vorticity waves.

An alternative perspective on these processes is provided by figure 8, which shows a comparison between the DNS and the LST phase speeds at both the frequencies $f_{6}$ and $f_{10}$. Similar plots have been presented by Ma & Zhong (Reference Ma and Zhong2003b), Ma & Zhong (Reference Ma and Zhong2005), Zhong & Ma (Reference Zhong and Ma2006) and Zhong & Wang (Reference Zhong and Wang2012). The phase speed from the DNS results has been computed through the fast Fourier transformed wall-pressure fluctuations, using the relation for the streamwise wavenumber

(3.1)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(x,f)=\unicode[STIX]{x1D6FC}_{r}+\text{i}\unicode[STIX]{x1D6FC}_{i}=\frac{1}{\text{i}p_{w}^{\prime }(x,f)}\frac{\unicode[STIX]{x2202}p_{w}^{\prime }(x,f)}{\unicode[STIX]{x2202}x},\end{eqnarray}$$

and then applying the relation $c_{ph}=2\unicode[STIX]{x03C0}f/\unicode[STIX]{x1D6FC}_{r}$. For both the frequencies, the numerical phase speed (represented by blue and red solid lines for the frequencies $f_{6}$ and $f_{10}$ respectively) starts at a value of approximately 1.12 in the early nose region. This is slightly lower than the local fast acoustic-wave phase speed of approximately 1.16, which is attributed to non-parallel effects in the early leading-edge region. For $x>100$ both LST and DNS follow the same decreasing trend and it can be concluded that the internal mode generated at the leading edge is mode F.

Further downstream (for $x>200$ at $f_{10}$ and $x>400$ at $f_{6}$) an oscillatory behaviour is seen in figure 8 that can be explained by mode F traversing the convected wave continuous spectrum and entering a synchronisation process with the internal mode S. Here, the numerical phase speed shows a rapid increase, and a second synchronisation with the local fast acoustic waves is reached. Downstream of this synchronisation point ($x=370$ at the frequency $f_{10}$, and $x=830$ at the frequency $f_{6}$), the numerical phase speed no longer follows the LST result for mode F, but instead agrees very well with the LST result for mode F II, as can be seen further downstream for the frequency $f_{10}$ (for the lower frequency, $f_{6}$, mode F II would be reached further downstream of the computational domain outer edge). During the decay of mode F, when the phase speed drops below the value of 0.9, strong oscillations of the numerical solution are observed, as a consequence of the mode F phase speed approaching the phase speed of mode S, which results in an initial growth of the latter, as shown by the LST curves representing mode S (blue and red dash-dot curves with square symbols). These oscillations represent the modulation process between the forcing fast acoustic waves and the decaying mode F, in the region where the latter is synchronised with mode S. Downstream of the intersection point between mode F and mode S LST phase speeds (i.e. the mode F–mode S synchronisation point) at the frequency $f_{10}$, the Mack mode is generated and develops downstream with an initial decrease of the phase speed, as a continuation of the mode F branch, but with increasing growth rate. These results are qualitatively in a very good agreement with the results of Ma & Zhong (Reference Ma and Zhong2005), Zhong & Ma (Reference Zhong and Ma2006) and Zhong & Wang (Reference Zhong and Wang2012).

A further comparison between the DNS and the LST results, including a more detailed identification of the internal modes, is made by considering the eigenfunctions of the different modes. Figure 9 shows the temperature $\hat{T}$ and wall-tangential velocity $\hat{u}$ eigenfunctions (normalised with the maximum inside the boundary layer as $A/A_{max}$) for the frequency $f_{6}$ at $x=259$, which is the position from figure 6 where the first peak of the wall-pressure fluctuation amplitude is reached due to the resonance between leading-edge fast-acoustic mode and mode F. At this position mode F is the dominant mode inside the boundary layer, which is proved by the good agreement between the DNS results for $\hat{T}$ and $\hat{u}$ and the LST eigenfunctions for mode F. As can be seen, the shape of mode F is characterised by a high peak near the wall. These results are also in a good qualitative agreement with the numerical and LST results of Zhong (Reference Zhong2001) for the eigenfunctions of mode F. In the modulation regions the comparison between the DNS profiles and the eigenfunctions from the LST is not as good since no dominant mode exists. In conclusion, the present numerical and theoretical results show the main characteristics of Fedorov’s (Fedorov & Khokhlov Reference Fedorov and Khokhlov2001; Fedorov Reference Fedorov2003) receptivity mechanism for hypersonic boundary layers, and are in good qualitative agreement with other theoretical and numerical studies available in the literature (Zhong Reference Zhong2001; Ma & Zhong Reference Ma and Zhong2005; Zhong & Ma Reference Zhong and Ma2006; Fedorov Reference Fedorov2011; Zhong & Wang Reference Zhong and Wang2012), for different flow conditions.

Figure 9.

Temperature and wall-tangential velocity eigenfunctions for the frequencies $f_{6}$ at the position $x=259$.

Figure 10.

Real part of the Fourier-transformed wall-pressure fluctuations ($p_{w}^{\prime }$) at the different forced frequencies. Case 1, $A_{\infty }=5\times 10^{-3}$. The labels of the vertical ($z$) and horizontal ($x$) axes have been purposely omitted.

3.2 Disturbance growth and breakdown mechanism

The DNS allows the transition process to be followed up to final breakdown to turbulence. We start by illustrating the results of the 3-D simulations for the cases at lower Reynolds number with fast acoustic waves as free-stream disturbances (cases 1 and 2 from table 3). Figures 10 and 11 show contours of the real part of the Fourier-transformed wall-pressure fluctuations at all the considered frequencies (from $f_{1}$ at the top to $f_{10}$ at the bottom) for cases at lower (case 1) and higher (case 2) free-stream disturbance amplitude respectively. Moving from top to bottom in both figures, the effect of frequency on the resonance mechanism can be seen, with the peak wall disturbance amplitude moving towards the leading edge as the frequency is increased. In the lower-amplitude case (figure 10) the wall disturbances appear to be dominated by a 2-D mode through the whole length of the computational domain. In contrast, in the higher-amplitude case (figure 11) the 2-D mode seems to be the most amplified mode only in the leading-edge region and, once the peak has been achieved, 3-D modes (with non-zero spanwise wavenumber) emerge and become the dominant structures further downstream. The solution in the nose region ($x=0{-}200$) is similar between the two different amplitude cases at all the frequencies, with the contour levels scaled by the same one order of magnitude as for the forcing. This means that the solution can be reasonably considered as linear in the early region of the computational domain. Further downstream, the solution becomes different between the cases at different amplitude, meaning that nonlinear effects are no longer negligible. The nonlinearity appears first in the form of long streaky structures that reduce the spanwise coherence of the waves and are seen for all the considered frequencies.

Figure 11.

Real part of the Fourier-transformed wall-pressure fluctuations ($p_{w}^{\prime }$) at the different forced frequencies. Case 2, $A_{\infty }=5\times 10^{-2}$. The labels of the vertical ($z$) and horizontal ($x$) axes have been purposely omitted.

Figure 12 shows the trend of the wall-pressure fluctuation amplitudes along the wedge for the modes $\unicode[STIX]{x1D6FD}=0,1,2$ at the frequency $f_{2}$ for both high- and low-amplitude cases, obtained through a fast Fourier transform (FFT) in time and in the spanwise direction. The wall-pressure fluctuation amplitudes for each mode are normalised with the value of the corresponding free-stream pressure fluctuation amplitude (i.e. as $p_{w}^{\prime }/p_{\infty }^{\prime }$). The wall response very close to the leading edge does not show a significant difference between case 1 and case 2, due to the nonlinear effects being relatively small in this region. In the second half of the computational domain, in contrast, the large nonlinear effects in case 2 provide a markedly different behaviour. In particular, the $\unicode[STIX]{x1D6FD}=2$ mode decays to very low values in case 1, whilst in case 2 it is strongly excited in the second half of the domain. Also, it is evident that the rapid growth of the $\unicode[STIX]{x1D6FD}=2$ mode starts at a position very close to the point where the peak of the 2-D mode F is reached, thus suggesting that the resonance mechanism at the leading edge might play an important role in the excitation of 3-D boundary-layer disturbances. In fact, it is evident in figure 12 and also from comparison between figures 10 and 11 that as soon as the disturbance amplitude reaches its maximum value (corresponding to the peak of mode F resulting from the resonance mechanism) 3-D disturbances are generated and the response pattern in the high-amplitude case diverges from that in the low-amplitude case. This suggests that in the high-amplitude case, the resonance mechanism increases the amplitude to values high enough to trigger nonlinear effects that affects the wall response (and the associated breakdown mechanism) further downstream. The associated flow structure is revealed by figure 13, which shows an instantaneous contour of the wall-tangential velocity component ($u_{t}$) for the $j=10$ grid line inside the boundary layer. It is evident that the excitation of the $\unicode[STIX]{x1D6FD}=2$ mode at the different forcing frequencies corresponds to the generation and downstream growth of streaks in the streamwise direction inside the boundary layer. For the low Reynolds number cases considered so far, these streaks do not break down within the computational domain, so we turn next to the higher Reynolds number cases 3–6 from table 1.

Figure 12.

Extracted modes ($\unicode[STIX]{x1D6FD}=0,1,2$) of the wall-pressure fluctuation amplitudes at the frequency $f_{2}$, for cases 1 and 2.

The effect of amplitude and disturbance type is shown in figure 14, where we plot temperature contours for each of cases 3–6, for which $Re=5625$. It can be seen that streak breakdown is reached for the high-amplitude fast acoustic-wave case (case 4) at this higher Reynolds number, with the transition process occurring downstream of $x\approx 550$. At the low disturbance amplitude, in contrast, for both fast (case 3) and slow (case 5) acoustic waves, a 2-D wave pattern can be observed, as discussed for the lower Reynolds number case. For the fast-wave case, however, the 2-D waves appear more amplified than in the slow-wave case, which is a result of the resonance mechanism for mode F at the leading edge. Finally, the case of high-amplitude slow acoustic waves (case 6) in figure 14 shows the appearance of 3-D structures, consistent with the corresponding fast-wave case, but less amplified and with no breakdown.

In summary, these results for the unswept wedge demonstrate that the resonance mechanism occurring at the leading edge in the presence of free-stream fast acoustic waves induces a high amplification of mode F, which can have important implications in cases where nonlinear effects are not negligible. The resonant interaction between mode F and the fast acoustic waves at high free-stream amplitudes leads to significantly earlier boundary-layer transition, compared to the case of slow acoustic waves.

Figure 13.

Instantaneous contour of the streamwise (tangential to the wall) velocity component ($u_{t}$) along the grid line $j=10$ inside the boundary layer. Case 2, $A_{\infty }=5\times 10^{-2}$.

Figure 14.

Instantaneous temperature ($T$) contours along the grid line $j=10$ inside the boundary layer. (a) Case 3: fast acoustic waves, $A_{\infty }=5\times 10^{-3}$, (b) case 4: fast acoustic waves, $A_{\infty }=5\times 10^{-2}$, (c) case 5: slow acoustic waves, $A_{\infty }=5\times 10^{-3}$, (d) case 6: slow acoustic waves, $A_{\infty }=5\times 10^{-2}$.

For the specific case of hypersonic wind tunnels, it is known from previous studies, e.g. Duan et al. (Reference Duan, Choudhari and Wu2014), Wagner et al. (Reference Wagner, Schülein, Petervari, Hannemann, Ali, Cerminara and Sandham2018) and Cerminara et al. (Reference Cerminara, Durant, André, Sandham and Taylor2019), that slow acoustic waves are likely to be the most dominant acoustic disturbance radiated from the turbulent boundary layer on the nozzle wall. This does not mean, however, that fast modes are not of interest. Another relevant source of disturbances is represented by entropy spottiness that may come from the wind-tunnel reservoir. As the entropy waves, as well as the vorticity waves, show a receptivity mechanism similar to that of the fast acoustic waves, as described by Fedorov’s mechanism (Fedorov & Khokhlov Reference Fedorov and Khokhlov2001; Fedorov Reference Fedorov2011), and proved numerically by Ma & Zhong (Reference Ma and Zhong2005), an in-depth study of the breakdown mechanism associated with fast acoustic waves is relevant for these types of non-acoustic disturbances. Moreover, in agreement with several previous numerical studies, e.g. Ma & Zhong (Reference Ma and Zhong2003a), Balakumar (Reference Balakumar2009) and Egorov et al. (Reference Egorov, Sudakov and Fedorov2006), the present study shows that the fast mode, although stable, has growth factors higher than the slow mode in the leading-edge region (upstream of the Mack mode excitation). Furthermore, it is important to mention that fast acoustic waves may be important in other scenarios, e.g. in flight, where structure vibrations supplement the noise generated from turbulent boundary layers.

Figure 15.

Fourier-transformed response of the $x$-velocity component at the subharmonic frequency $f_{0}$ inside the boundary layer, along the $j=10$ grid line. Case 9, $A_{\infty }=5\times 10^{-3}$, slow waves.

4.1 DNS results

For the swept cases, pressure gradients normal to the free-stream direction lead to a laminar base flow that contains inflectional cross-flow boundary-layer profiles. Figure 16 shows the variation of the wall pressure and the Mach number at the edge of the boundary layer along the chordwise direction ($x$), considering both the unswept and swept base flow. Due to the weaker shock wave induced by a non-zero sweep angle, the wall pressure is lower in the swept wedge case, while the Mach number at the boundary-layer outer edge is higher over the whole wedge length, compared to the unswept case. Figure 17 shows examples of inflectional cross-flow boundary-layer profiles at different distances from the leading edge. The coordinate directions for the cross-flow were defined in figure 4, with $u_{\unicode[STIX]{x1D702}}$ denoting the velocity in the cross-flow direction. Figure 18 shows a close up of the Mach number field in the leading-edge region, which highlights the shock stand-off distance as well as the very thin subsonic region around the circular leading-edge profile.

Figure 16.

(a) Wall-pressure variation and (b) Mach number variation at the boundary-layer edge in the chordwise ($x$) direction, for the unswept and swept base flows.

Figure 17.

Cross-flow boundary-layer profiles at several distances from the leading edge.

Figure 18.

Mean Mach number field in the leading-edge region.

An overview of the transitional flow is provided by figures 19 and 20, showing contours of temperature inside the boundary layer for cases 7–10. The cases shown in figure 19 all have a similar breakdown mechanism, which appears more rapid in the high-amplitude slow-wave case. All the cases, even at the lower amplitude, show the start of nonlinear breakdown to turbulence. In each case, transition appears to be related to the formation and breakdown of streamwise-oriented streaks. The streaks are visible as elongated structures of alternate high and low temperature, corresponding respectively to low and high values of the streamwise velocity.

The case of fast acoustic waves at higher disturbance amplitude shown in figure 20 is different to the others in several aspects. First, this is the case where transition occurs at the earliest location, thus revealing that nonlinear effects related to the higher amplitudes are prevalent for fast acoustic waves. Secondly, this case shows the presence of spanwise-coherent structures up to $x\approx 450$. This is consistent with the results obtained for the unswept case, in which it was shown that the resonance of the 2-D fast acoustic modes in the leading-edge region is a powerful mechanism of amplification of the internal waves, which can rapidly trigger transition at the higher amplitudes. A third noticeable difference of the fast-wave case at high amplitude is that the streamwise streaks have a much higher wavelength compared to the narrow streaks observed in the other cases. This is consistent with the presence of an unstable cross-flow mode.

Figure 19.

Instantaneous temperature ($T$) contours along the grid line $j=10$ inside the boundary layer. (a) Case 7: fast acoustic waves, $A_{\infty }=5\times 10^{-3}$, (b) case 9: slow acoustic waves, $A_{\infty }=5\times 10^{-3}$, (c) case 10: slow acoustic waves, $A_{\infty }=5\times 10^{-2}$.

Figure 20.

Instantaneous temperature ($T$) contours along the grid line $j=10$ inside the boundary layer. Case 8, fast acoustic waves, $A_{\infty }=5\times 10^{-2}$. (a) Near leading-edge region, (b) downstream region.

Figure 21.

Real part of the Fourier-transformed wall-pressure fluctuations ($p_{w}^{\prime }$) at the frequencies $f_{2}$ and $f_{10}$, at low (case 7) and high (case 8) amplitude. Fast acoustic waves.

Figure 22.

Real part of the Fourier-transformed wall-pressure fluctuations ($p_{w}^{\prime }$) at the frequencies $f_{2}$ and $f_{10}$, at low (case 9) and high (case 10) amplitude. Slow acoustic waves.

A first step towards the identification of the main wave modes involved in the receptivity process at different frequencies is made using a fast Fourier transform of the wall-pressure response, as was done in figures 10 and 11 in the previous section. Figures 21 and 22 show the Fourier-transformed pressure fluctuation amplitudes at the forcing frequencies $f_{2}$ and $f_{10}$ (representative of the low- and high-frequency response) and at the different amplitudes, for fast and slow acoustic waves, respectively. The initial response to fast acoustic waves is characterised by a dominant 2-D wave pattern, for both frequencies and both amplitudes. This is a result of the same resonance mechanism between the forcing fast acoustic waves and the induced fast mode (or mode F) that was discussed in detail in § 3. At the higher frequency, $f_{10}$, the peak of the fast mode shifts closer to the leading edge, within $x=100{-}300$, whereas downstream of this region the amplitude decreases, consistent with the de-synchronisation of mode F. In the high-amplitude case, downstream of $x=400$, the 2-D mode is seen to be competing with another emerging structure that is aligned with the mean flow direction. As a result of this modulation, the wall response shows high amplitudes for a longer distance downstream, compared to the case with low free-stream disturbance amplitude. A similar structure is also present in the leading-edge region, seen more clearly at the higher frequency, but also present at the lower frequency.

In the case of slow acoustic-wave forcing, shown in figure 22, the 2-D mode does not play the same role as in the fast-wave case. At low frequencies a complex three-dimensional pattern appears in the leading-edge region, suggesting the presence of several oblique modes that can be more clearly distinguished further downstream. These low-frequency modes decay downstream, and do not appear to be directly responsible for the sudden breakdown to turbulence seen near $x=800$. At the higher frequencies, the response near the leading edge shows a 2-D wave pattern that is strongly modulated by cross-flow-like modes, whereas further downstream oblique waves with higher spanwise wavenumber are clearly present.

These numerical results of the wall response at the different frequencies reveal the presence of at least three relevant modes that may play a determining role in the transition process. These modes consist of 2-D waves (wave fronts parallel to the leading edge), waves aligned in the mean flow ($\unicode[STIX]{x1D709}$) direction (suggestive of cross-flow instabilities), and waves with wave fronts aligned in the cross-flow ($\unicode[STIX]{x1D702}$) direction. The type of the free-stream acoustic waves, either fast or slow, determines which of the above mentioned modes is dominant in a certain flow region. To further understand the flow behaviour we next use LST to determine the most relevant unstable modes in the boundary layer.

Figure 23.

Temporal growth rate ($\unicode[STIX]{x1D714}_{i}$) from linear stability analysis at different $x$ locations on the wedge.

4.2 Results from linear stability analysis

Profiles for analysis using LST are extracted from the computed solution for steady laminar flow over the swept wedge. Figure 23 shows contours of the temporal growth rate ($\unicode[STIX]{x1D714}_{i}$) taken at different $x$-locations along the wedge, as well as at different $x$-wise and spanwise wavenumbers ($\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FD}$). With reference to figure 4, the wavenumbers $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the wavenumbers in the $xz$-reference system, i.e. with axes oriented in the chordwise and leading-edge directions. Hence, in this section, the wavenumber $\unicode[STIX]{x1D6FC}$ will be defined as the $x$-wise wavenumber (instead of streamwise wavenumber), to distinguish it from the wavenumber in the main flow (i.e. streamwise) direction. The ranges of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ cover the free-stream forcing waves, for example the $x$-wise wavenumbers corresponding to the imposed frequency range are $\unicode[STIX]{x1D6FC}=0.126{-}0.353$ and $\unicode[STIX]{x1D6FC}=0.2{-}0.57$ for fast and slow acoustic waves, respectively. The corresponding forcing spanwise wavenumbers are $\unicode[STIX]{x1D6FD}=\pm 0.11$ (or alternatively $m=\pm 1$) and $\unicode[STIX]{x1D6FD}=\pm 0.22$ ($m=\pm 2$) for both fast and slow waves and at each forcing frequency. At all the considered locations ($x=100$, 200, 300, 500, 700, 800) the region of dominant instability (i.e. positive $\unicode[STIX]{x1D714}_{i}$) can be seen in figure 23 as the narrow elongated red regions with negative $\unicode[STIX]{x1D6FD}$. Within these regions, increasingly negative $\unicode[STIX]{x1D6FD}$ correspond to increasingly positive $\unicode[STIX]{x1D6FC}$ with approximately a constant ratio ($\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC}\approx -0.83$, i.e. $\unicode[STIX]{x1D703}_{xz}\approx -40^{\circ }$), i.e. there is a constant orientation of the wave vector, which coincides with the local direction of the maximum cross-flow inside the boundary layer. The dominant unstable modes are travelling cross-flow waves, which are known to be associated with free-stream disturbances (Li et al. Reference Li, Choudhari, Duan and Chang2014, Reference Li, Choudhari, Paredes and Duan2016; Bartkowicz et al. Reference Bartkowicz, Subbareddy and Candler2010) at low frequencies, such as those imposed in our simulations ($f_{1}^{\ast }=12.6~\text{kHz}$, $f_{10}^{\ast }=35.3~\text{kHz}$). The phase speed of the most unstable cross-flow mode from LST, taken in the direction of the wave vector ($c_{ph}=\unicode[STIX]{x1D714}_{r}/\sqrt{\unicode[STIX]{x1D6FC}^{2}+\unicode[STIX]{x1D6FD}^{2}}$), is $c_{ph}=0.057$ at $x=400$, and $c_{ph}=0.058$ at $x=500$, and the corresponding wavenumbers are $\unicode[STIX]{x1D6FC}=0.138$ and $\unicode[STIX]{x1D6FD}=-0.11$. The spanwise wavenumber coincides with the imposed $m=1$ wavenumber, whereas the $x$-wise wavenumber approximately corresponds to the lowest imposed $\unicode[STIX]{x1D6FC}$ for fast acoustic waves. The most unstable cross-flow waves are travelling waves with a significantly lower phase speed compared to the imposed free-stream fast and slow acoustic waves. The corresponding frequency is $f=0.0016$, which is an order of magnitude lower than our minimum imposed non-dimensional frequency ($f_{1}=0.0175$). In dimensional units, the frequency of the most unstable cross-flow mode, considering a reference nose radius of $R^{\ast }=1.2$  mm, is $f_{mode}^{\ast }=1.15~\text{kHz}$, representing a subharmonic of the free-stream forcing.

For $x>200$ the LST results show the presence of other instabilities. The first to emerge is located at the low wavenumbers ($\unicode[STIX]{x1D6FC}<0.2$), peaking at a small positive value of $\unicode[STIX]{x1D6FD}$, for example peaking at $\unicode[STIX]{x1D6FD}=0.1$, $\unicode[STIX]{x1D6FC}=0.1$ at $x=300$. This corresponds to a first mode instability (Reshotko Reference Reshotko1976), with a wave orientation perpendicular to the cross-flow mode. The region of first mode instability is present at all the other locations downstream of $x=200$ along the wedge, but the associated growth rates are very low.

At the location $x=700$ another unstable mode appears, with a growth rate lower than the dominant cross-flow mode, but higher than the first mode growth rate. This mode is a higher-frequency mode (i.e. higher wavenumber $\unicode[STIX]{x1D6FC}$), and corresponds to higher positive $\unicode[STIX]{x1D6FD}$ values. The values of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are approximately equal within this high-frequency instability region, which means that these modes are inclined of an angle of about $45^{\circ }$ with respect to the $x$-axis, i.e. they are effectively 2-D modes in the mean flow reference system (rotated by the sweep angle with respect to the geometry reference system), and correspond to 2-D second modes (Mack Reference Mack1984). The phase speed of the modes in this region is about $c_{ph}\approx 0.85{-}0.87$, which is consistent with the phase speed associated with the Mack mode (Reshotko Reference Reshotko1976; Ma & Zhong Reference Ma and Zhong2005; Zhong & Wang Reference Zhong and Wang2012). The corresponding non-dimensional frequency associated with these modes is about $f=0.06$, which in for a reference nose radius of 1.2 mm would give a dimensional frequency of 43 kHz. This would explain the generally low growth rates ($\unicode[STIX]{x1D714}_{i}$ values) of the second mode seen here. The experiments about the second instability mode are usually performed over smaller nose radii, e.g. 0.1 mm (Estorf et al. Reference Estorf, Radespiel, Schneider, Johnson and Hein2008; Wagner et al. Reference Wagner, Schülein, Petervari, Hannemann, Ali, Cerminara and Sandham2018), as it is known that a sharper leading edge provides higher growth rates for the Mack mode, as was shown for example from Malik & Balakumar (Reference Malik and Balakumar2007) for a flat plate with different blunt nose radii. For a nose radius lower than 0.5 mm, i.e. a sharp leading edge, this non-dimensional frequency of 0.06 would correspond to a dimensional frequencies $f^{\ast }>100~\text{kHz}$, which then would include the usual frequency range (100–300 kHz) associated with second mode experiments (Estorf et al. Reference Estorf, Radespiel, Schneider, Johnson and Hein2008) in hypersonic wind tunnels.

The overall growth in amplitude of an instability wave can be quantified using the $N$-factor (Crouch & Ng Reference Crouch and Ng2000), defined by

(4.1)$$\begin{eqnarray}N=\log \left(\frac{A(x)}{A(x_{0})}\right)=\int _{x_{0}}^{x}-\unicode[STIX]{x1D6FC}_{i}(x)\,\text{d}x,\end{eqnarray}$$

in which the spatial growth rate $-\unicode[STIX]{x1D6FC}_{i}$ is approximated from $\unicode[STIX]{x1D714}_{i}$ using Gaster’s relation (Gaster Reference Gaster1962), which can be expressed as

(4.2)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}_{i}^{T}}{\unicode[STIX]{x1D6FC}_{i}^{S}}=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{r}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}_{r}}=-c_{g}.\end{eqnarray}$$

In the above equation the superscripts $T$ and $S$ stand for temporal and spatial, respectively, whereas the term $c_{g}$ is known as the group velocity. Figure 24 shows the computed $N$-factors for the most unstable cross-flow mode from LST compared with the $N$-factors of different Mack modes taken in their unstable region. A logarithmic scale is used along the $N$ axis in order to show the difference between the $N$-factor levels of the cross-flow and Mack modes. As can be seen, the amplification ($N$) factors of the Mack modes resulting from the LST, shown in a form to outline part of the envelope of maximum amplification factors in figure 24, are very small compared to those of the cross-flow mode, thus suggesting that the unstable Mack modes are not responsible for transition. Nor is the unstable first mode expected to be important, since its growth rate is an order of magnitude lower than the cross-flow mode.

Figure 24.

The $N$-factors of the most unstable cross-flow mode and Mack modes at different $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ wavenumbers.

Table 4.

Comparison of temporal and spatial approaches. Assessment of Gaster’s transform.

The validity of Gaster’s relation for the considered flow is assessed by direct comparison between the approximated spatial growth rates ($\unicode[STIX]{x1D6FC}_{i}$) obtained from Gaster’s relation and those obtained by a spatial linear stability analysis. The comparison has been made at two $x$ locations, namely $x=400$ and $x=500$, in which the temporal LST results indicate the presence of cross-flow instabilities. At both locations, the most unstable mode has been considered, as Gaster’s relation is known to be valid only for modes with growth rate close to zero. The results of this comparison are tabulated in table 4. Results for both temporal and spatial approaches are shown at both the locations. The most unstable mode found at these locations is a cross-flow mode with $x$-wise and $z$-wise wavenumbers $\unicode[STIX]{x1D6FC}=0.138$ and $\unicode[STIX]{x1D6FD}=-0.111$. The corresponding wavenumbers in the reference system aligned with the free-stream flow direction (streamwise, $\unicode[STIX]{x1D709}$) and the cross-flow ($\unicode[STIX]{x1D702}$) direction are $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D709}}=0.019$ and $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D702}}=-0.1761$. In table 4 the streamwise wavenumber in the flow reference system is given, whose real part is indicated as $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D709},r}$. The term $\unicode[STIX]{x1D714}_{r}$ represents the real part of the frequency obtained from the temporal approach, which is used as input in the spatial approach. The streamwise phase speed $c_{ph}$ (i.e. the phase speed of the mode in the $\unicode[STIX]{x1D709}$ direction) represents the ratio between $\unicode[STIX]{x1D714}_{r}$ and $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D709},r}$, and is an output in both the temporal and spatial approaches. The spatial growth rate $\unicode[STIX]{x1D6FC}_{i}^{S}$ is the output of the spatial analysis and is approximated by (4.2) using the temporal approach. As can be seen, the spatial growth rates from the spatial LST and Gaster’s transform are very similar at both the locations. The relative error between the two values is 1.9 % at $x=400$ and 1.6 % at $x=500$. These results, obtained for the most unstable mode, confirm the validity of Gaster’s transform as well as the suitability of the temporal LST for the present case.

No other modes have been found in LST. In particular no linear instability modes were found for the chordwise and spanwise wavenumbers of the narrow streaks that are seen in the flow visualisations in figure 19. Thus these features are not associated with a primary flow instability. Nor are they obviously a secondary instability of a finite-amplitude cross-flow mode. The results of the cross-flow mode are streamwise vortices and lifted streaks that would be expected to support well-known secondary streak instabilities, for example of the sinuous or varicose type (Kohama Reference Kohama1987; Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001; Li et al. Reference Li, Choudhari, Duan and Chang2014). In order to understand the final stages of transition we need find the origin and growth mechanism of the narrow streaky flow features. This will be the subject of the following section.

4.3 Streak-induced breakdown mechanism

The wavenumbers associated with the narrow streaks are approximately one order of magnitude higher than the forced wavenumbers of the free-stream noise. They are not the result of a local primary flow instability and we are not aware of any published secondary instability mechanisms that lead to streaks with much higher spanwise wavenumber, but they are oriented in the same direction as the primary instability. Understanding the origin and growth of the mode associated with the narrow streaks requires a more in-depth analysis of the DNS results in the upstream region. Figure 25 shows 3-D views of the $u^{\prime }$ and $v^{\prime }$ velocity components in the flow region where the narrow streaky structures appear and rapidly grow. Figure 25(b) gives a good impression of the underlying wave system in the region $x=480{-}620$. In this region there is a change taking place from structures parallel to the leading edge towards more oblique structures. The corresponding $u$-component of the fluctuations (figure 25a) shows very pronounced narrow streaks that are present even at the start of this image and hence do not appear to be connected to the change in orientation of the underlying waves. Nor is a numerical source likely. A lack of flow resolution with a high-order code of the type used here gives oscillations with wavelength twice the grid spacing when the flow is under-resolved. However, with approximately seven wavelengths across the span and 100 grid points, we already have better than 12 points per wavelength of the narrow streaks and no evidence of any under-resolution of the local flow features.

Figure 25.

Surface views of the chordwise and wall-normal velocity component fluctuations. Case 9.

To pin down the origin of the narrow streaks, figure 26 shows the temperature contours in the region $x=300{-}750$ inside the boundary layer at different $j$ grid lines. Downstream of $x=500$, narrow streaks ($m\approx 7{-}9$) start to be evident, and downstream of $x=650$ their amplitude reaches very high levels. The streaks form in a limited region within the boundary layer that is close to the maximum cross-flow velocity. For example $j=15$ corresponds to $y_{n}\approx 5$ at $x=600$ in figure 17.

Figure 26.

Temperature fluctuation contours inside the boundary layer, at different $j$-grid lines. Case 9.

The origin of the localised region in $y$ in which the $m=O(10)$ mode develops can be deduced from figure 27, showing cross-sections of the streamwise vorticity fluctuations at two different $x$ stations. At the upstream station, $x=400$, the vorticity shows two narrow layers of positive and negative vorticity, one on top of the other. The location at $y=32$, corresponding to a wall-normal distance of $y_{n}=3$, locates this feature at the inflection point of the cross-flow profiles shown in figure 17. At $x=525$ spanwise oscillations have started to form and it is clearly the roll-up of the streamwise vorticity that leads, via a lift-up mechanism, to the narrow velocity streaks seen in figure 25. If we take the thickness of the layer as approximately 2 and the velocity difference as 0.05 from the cross-flow profiles on figure 17, the local shear layer Reynolds number is approximately 500, which is sufficient for the roll-up of the shear layer to develop via an essentially inviscid mechanism. Figure 28 shows the eigenfunctions of streamwise, vertical and cross-flow velocity components as well as temperature of the most unstable cross-flow mode ($\unicode[STIX]{x1D6FC}=0.138$, $\unicode[STIX]{x1D6FD}=-0.11$), from LST at the location $x=400$. As can be seen, the cross-flow mode fluctuations peak at approximately the same location of the thin vorticity fluctuation layer seen in figure 27 (the location of the inflection point of the cross-flow profile), i.e. $y_{n}\approx 3{-}4$, thus confirming that the thin vorticity layer is a feature associated with the cross-flow instability. A further confirmation is given by figure 28(b), which depicts the profile of the streamwise vorticity fluctuations computed from the eigenfunctions of the velocity components in figure 28(a). The vorticity profile shows a narrow layer of oscillations, with maximum located at the peak location of the eigenfunctions, consistent with the narrow vorticity fluctuation layer seen in figure 27.

Figure 27.

Cross-sections of the streamwise vorticity fluctuations ($\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D709}}^{\prime }$) at different $x$ locations. Case 9.

Figure 28.

Eigenfunctions of the cross-flow instability mode and corresponding vorticity fluctuations at the $x=400$ location. Case 9.

Having identified the $m=O(10)$, corresponding to $\unicode[STIX]{x1D6FD}=O(1)$, mode as responsible for the roll-up of the thin vorticity layer associated with the cross-flow mode, the final part of the puzzle is to find its origin, since this mode is not present in the free-stream forcing. By performing a fast Fourier transform in space at different $x$ stations close to the leading edge, it is possible to verify which higher wavenumbers are generated, and their relative amplitude. Figure 29 shows the spanwise wavenumber spectrum of the $x$-velocity fluctuation amplitudes inside the boundary layer (along the $j=10$ grid line). The spatial fast Fourier transform has been performed in the spanwise direction for the Fourier-transformed (in time) signal at the fundamental frequency $f_{0}$. There is also a pronounced amplification of a higher (unforced) wavenumber, $m=6$, which decays at higher distances from the leading edge ($x=250$, 500), and then increases again further downstream ($x=600$). Other lower peaks are observed at the higher distances for increasing wavenumbers.

Figure 29.

Spectrum of the Fourier-transformed amplitudes of the $x$-velocity fluctuations along the $j=10$ grid lines, in the domain of the spanwise wavenumbers, at different $x$ positions. The spectrum has been computed at the frequency $f_{0}$. Case 9.

Figure 30.

Streamwise distribution of the Fourier-transformed $x$-velocity fluctuations, along the $j=10$ grid lines, at the frequency $f_{0}$ and at high spanwise wavenumbers. Case 9.

The wavenumber $m=11$ is present at a low-amplitude peak at the early location $x=20$, suggesting that it forms inside the boundary layer near the leading edge. The frequency of the most unstable cross-flow mode ($f=0.0016$) is half the fundamental frequency $f_{0}$ of our simulations. Thus, the closest frequencies to the frequency of the most unstable mode are those most responsible for the growth of the $m=O(10)$ modes. Due to limited computational resources, we are not able to process the $m=O(10)$ modes at the low frequency of the most unstable mode, but only at the lowest frequency we can extract from the numerical results, i.e. the fundamental frequency $f_{0}$, for which $m=11$ is found to be the dominant high wavenumber mode, as seen in figure 29. The spatial variation of the $m=6$ and $m=11$ modes is shown on figure 30. The initial amplification at the leading edge is followed by a decay after which the amplitude of the modes grows in the region $x=350{-}500$, followed by a more pronounced growth downstream of $x=500$, where the streaks are seen to grow. In this latter region, the growth of the mode $\unicode[STIX]{x1D6FD}_{11}$ is more rapid than that of $m=6$, which causes $m=11$ to become the dominant wavenumber associated with the streaks. Thus, the spectrum of figure 29 shows that higher wavenumbers are generated in the early leading-edge region. The growth of the narrow streaks appears to be due to a transient growth mechanism associated with the roll-up of the narrow streamwise vorticity layer (as previously described), whose length scale is comparable with the wavelength of the narrow streaks.

Figure 31.

Sketch of the wave system characterising the shock layer region.

Figure 32.

Sketch of wave scattering at the leading edge on an $xy$-plane.

The generation mechanism of the smaller wavelengths at the leading edge can be discussed with reference to the sketches shown in figures 31 and 32, which illustrate some potentially relevant phenomena involving the acoustic waves in the shock layer. Figure 31 shows a top view of the $xz$-plane at the location of the minimum shock stand-off distance, indicated by $\unicode[STIX]{x1D6E5}=0.55$, whereas figure 32 shows the scattering of the incident acoustic waves from the rounded leading edge (in a $xy$-plane). Due to the presence of the shock, there is a complex wave system in the shock layer, characterised by wave refraction, reflection and scattering from the leading edge. The incident free-stream waves are refracted downstream of the shock, with mechanisms described in McKenzie & Westphal (Reference McKenzie and Westphal1968), and propagate towards the wall with different orientations. The leading edge, in turn, can reflect the waves back to the shock (Cerminara & Sandham Reference Cerminara and Sandham2017) and scatter the waves in all directions. If we consider the multiple reflection process, the shock layer can be seen as a wave guide of thickness equal to the stand-off distance, which then promotes the formation of waves with wavelength comparable to the wave guide thickness. The scattering mechanism is particularly sensitive to the surface radius of curvature (in this case the wedge nose radius), and is more effective when a large difference between the wavelength of the incident waves and the body radius of curvature exists, which leads to the generation of scattered waves with wavelength comparable with the nose radius. This (speculative, but in our view plausible) wavelength and frequency conversion process would result in the generation of waves with significantly smaller wavelength compared to the wavelengths imposed in the free stream, as observed for example in figure 15(a) and warrants further investigation.