2.1 Inviscid flow field

The flared cone is sketched in figure 1. This model consists of a $5^{\circ }$ half-angle axisymmetric sharp cone (with a nose radius of 0.05 mm) and a 160 mm-long flared afterbody located 100 mm downstream from the cone apex. The flared body was generated by a circular curve with a radius of 0.931 m. Thanks to the axisymmetric nature of the flow, the computed region can be reduced to a plane containing the axis. The flow conditions correspond to a free stream Mach number of 6, free stream pressure of 570 Pa, stagnation temperature of 433 K, and Reynolds number of $9.70\times 10^{6}~\text{m}^{-1}$ . In our computation, the nose radius was set to zero since the nose radius was too small to affect the stability analysis. The grid resolution was $1000\times 200$ in the streamwise and wall-normal directions.

Figure 1.

Sketch of the flared-cone model and the body oriented coordinate.

Figure 2.

Inviscid flow field pressure contours.

The inviscid pressure contours are shown in figure 2. A shock wave is emitted from the cone apex with a cluster of compression waves generated over the flared region. The shock wave is approximately 15 mm above the cone wall over the flared region while the boundary layer is approximately 1 mm thick. Therefore the shock wave can be disregarded when studying the boundary layer instabilities. The wall values (which are equal to the boundary layer edge values in the Euler/boundary layer approach) of the selected variables are displayed in figure 3. In the straight region, all the variables remain nearly constant, which is consistent with the contour plot in figure 2. In the flared region, the edge pressure and temperature both increase significantly, while the edge Mach number decreases nearly linearly.

Figure 3.

Inviscid wall values versus the axial length: (a) normalized edge pressure, (b) edge Mach number, (c) normalized edge temperature.

2.2 Boundary layer flow

Consider undisturbed compressible flow past an axisymmetric body of radius $r_{0}(x)$ , where $x$ is the coordinate along the body and $y$ is normal to the surface. Let $\bar{U}$ and $\bar{V}$ represent the undisturbed velocity components along the $x$ - and $y$ -directions, then the boundary layer equations in the presence of a pressure gradient are

where $\bar{\unicode[STIX]{x1D70C}}$ is the density, $\bar{T}$ the temperature and $\bar{c_{p}}$ the specific heat. Note that a variable with an overbar denotes the undisturbed mean-flow quantity. ${\mathcal{R}}$ is the gas constant and $r$ is related to $r_{0}$ as

where the local value of the angle $\unicode[STIX]{x1D703}$ signifies the slope of the body surface. For a cylinder, $\unicode[STIX]{x1D703}=0$ , and for a sharp cone, $\unicode[STIX]{x1D703}$ is the cone half-angle. The viscosity, $\bar{\unicode[STIX]{x1D707}}$ , is assumed to be given by the composite relation (Mack Reference Mack1984)

The thermal conductivity, $\bar{k}$ , is prescribed so that the Prandtl number, $Pr$ , is equal to 0.7.

Following Malik & Spall (Reference Malik and Spall1991), we use the Mangler–Levy–Lees transformation:

Define three parameters $F$ , $V$ and $G$ as

Then, the governing equations in the transformed ( $\unicode[STIX]{x1D709}$ , $\unicode[STIX]{x1D702}$ )-space are

where

and $\unicode[STIX]{x1D712}$ is the transverse curvature parameter

The boundary conditions are

Thus, results presented in this paper are for an adiabatic boundary condition. Since the quiet wind tunnel could operate for as long as 30 seconds (Zhang et al. Reference Zhang, Tang and Lee2013), an adiabatic boundary condition for the wall temperature is appropriate. The flat-plate similarity solution was used as the initial profile. Numerical experiments showed that the profiles downstream are insensitive to the initial profile as the downstream position is far from the initial position. The spatial marching scheme (along $\unicode[STIX]{x1D709}$ -coordinate) was the Euler backward difference scheme. The discretization along the $\unicode[STIX]{x1D702}$ -coordinate was implemented by the Chebyshev collocation method. The resulting algebraic equations were solved with a successive iterative method (Nakamura Reference Nakamura1994), that is, the nonlinear terms were linearized first and the linear equations were then solved. This process was repeated until convergence. For example, the term $F_{j}^{2}$ was approximated by $F_{j-1}F_{j}$ , where index $j$ represents the iteration number. For the first iteration ( $j=1$ ), $F_{0}$ was given by the value of $F$ at last step. After solving the linear equations for $F_{j}$ , $F_{j-1}$ was updated using under-relaxation as

where $s$ is the under-relaxation parameter. The optimal value for $s$ depends on the problem as well as the discretization scheme. In this paper, $s=0.4$ was used.

Figure 4.

Boundary layer distributions: (a) normalized streamwise velocity and temperature at $x=180$  mm; (b) the boundary layer thickness (the locations of 99 %  $\bar{U}_{e}$ ). The mean flow was calculated with 250 points in the wall-normal direction.

The code validation is presented in appendix B. The wall-normal boundary layer profiles for the normalized streamwise velocity and temperature at $x=180$  mm are shown in figure 4(a). The velocity distribution is nearly linear across the boundary layer, which is characteristic of high-speed boundary layers. Also the frictional heating increases the wall temperature for this high-speed flow to more than five times that at the boundary layer edge. Figure 4(b) illustrates the downstream variation of the boundary layer thickness. The boundary layer thickness grows continuously along the straight region but decreases in the flared region. This was also observed by Pruett & Chang (Reference Pruett and Chang1998) and Laible & Fasel (Reference Laible and Fasel2011).

5.1 Parametric study

In these calculations, mode ( $f_{2nd},0$ ) ( $f_{2nd}=350$  kHz, $n=0$ ) was chosen to represent the second-mode waves, since it is one of the most amplified components and has a frequency close to the measured peak in the PKU experiments, as shown in § 6. The low-frequency disturbances, however, were more difficult to identify in the PSE simulations because the unstable regions in the $n$ – $f$ plane for low-frequency disturbances (including the Görtler mode and the first mode) are much larger than the second-mode instability region, and the azimuthal wavenumber could not be measured in the PKU experiments. Thus, a parametric study was performed first to clarify the effects of the frequency and azimuthal wavenumber. This subsection considers the case I interactions with the case II interactions discussed in § 5.4.

Figure 9.

Spectra of the initially seeded oblique waves at $x=180$  mm: (a) contours of the linear amplitudes $\ln A_{lin}/A_{0}$ with a level step of 0.5; (b) contours of the nonlinear amplitudes $\ln A_{non}/A_{0}$ with a level step of 1; (c) contours of the ratios of the nonlinear amplitudes to the linear amplitudes $\ln A_{non}/A_{lin}$ with a level step of 0.5.

The second-mode wave ( $f_{2nd}$ , 0) and a low-frequency wave $(f_{low},n)$ were initially seeded in each PSE simulation. The main interactions can be summarized as

The interactions with the MFD are not shown here but were considered in the PSE simulations. Only six modes, i.e. the second mode, its first harmonic, MFD, the low-frequency mode, the difference mode and the sum mode, were considered in each PSE simulation. Such a simple model turns out to suffice for the parametric study.

Figure 10.

Typical amplitude evolution in the four regions: (a) mode $(140,50)$ in region $A$ , (b) mode $(10,50)$ in region $B$ , (c) mode $(60,10)$ in region $C$ and (d) mode $(30,50)$ in region $D$ . The dashed lines correspond to linear evolutions.

A total of 1258 cases, covering 34 frequencies and 37 azimuthal wavenumbers, were calculated. Two of these cases, $(0,10)$ and $(5,10)$ , failed to converge before the last breakdown and are therefore disregarded here. The second and low-frequency modes were seeded at $x=100$  mm (the beginning of the flare region) with initial amplitudes equivalent to the r.m.s. temperature fluctuations of 5 % and 0.01 %. The initial second-mode amplitude was chosen such that the second mode became saturated at approximately $x=170$  mm as observed in the experiments (see figure 27). Different initial oblique-wave amplitudes may change the breakdown positions of the PSE simulations but do not seem to alter the basic evolution process. The contours of the low-frequency wave amplitudes in the $n$ – $f$ plane are shown in figure 9 for a streamwise location of $x=180$  mm. The linear distribution is shown in figure 9(a). As expected, the amplitudes of the unstable first and Görtler modes increase while those of the stable modes decrease. The amplitude spectrum in the nonlinear case is illustrated in figure 9(b). The low-frequency modes are all amplified from the initial amplitude. The most amplified ones are the low-frequency Görtler modes ( $f<20$  kHz and $20<n<80$ at $x=180$  mm) for which breakdowns of the PSE simulations occur first.

Figure 11.

Amplitude and phase-velocity evolution of low-frequency waves with a fixed azimuthal wavenumber and different frequencies (a,c) and a fixed frequency and different azimuthal wavenumbers (b,d).

The contours of the amplitude ratios defined as $\ln A_{non}/A_{lin}$ are shown in figure 9(c) to further clarify the effects of the second mode on the evolution of the oblique waves. If the amplitude ratio is negative (the white space in the figure), then the second mode suppresses the low-frequency mode. Otherwise, the second mode plays a positive role. The results show that most of the modes are amplified compared to the linear case, of which two regions, $A$ and $B$ , are most prominent. Region $A$ essentially consists of linear stable modes. The large amplitude ratios in that region are mainly due to the small linear amplitudes. The attenuated modes lie in regions, $C$ and $D$ . Figure 10 shows the typical amplitude evolutions of the modes in these four regions. The selected modes are $(140,50)$ , $(10,50)$ , $(60,10)$ and $(30,50)$ from region $A$ , $B$ , $C$ and $D$ , respectively. It shows that all these four modes evolve linearly at first, and then interact with their difference/sum modes as well as the second mode. In the latter stage, these four modes behave quite differently which results in different characteristics of these regions. Mode $(140,50)$ exhibits large oscillations, although it still grows moderately. Mode $(10,50)$ is strongly amplified at a nearly constant rate. Mode $(60,10)$ remains saturated. Mode $(30,50)$ is slightly attenuated for a long distance and then amplified like mode $(10,50)$ , which implies that region $D$ may eventually be incorporated into region $B$ . Moreover, mode $(10,50)$ and mode $(30,50)$ have the same amplification rates as the difference/sum modes, which suggests that these oblique waves are in resonance. This resonance is explored in detail based on Floquet theory in the next subsection.

The amplitude and phase-velocity evolution of the low-frequency waves in region $B$ are shown in figure 11 for various frequencies and azimuthal wavenumbers. The phase velocities slowly decrease during the linear stage, sharply drop from approximately 0.9 to approximately 0.4 before the resonant growth begins with a slow growth section followed by a rapid increase. For a fixed azimuthal wavenumber the length of the linear stage increases with the frequency while the growth rate in the resonance stage decreases with the frequency. This observation implies that region $B$ will broaden at higher frequencies. Modes with shorter linear stages and stronger resonance stages have earlier breakdowns of the simulation. For a fixed frequency, the mode with an azimuthal wavenumber of 50 is the first to induce breakdown of the simulation. Meanwhile, modes with larger azimuthal wavenumbers have lower phase velocities at the resonance stage.

Of all the cases considered in the parametric study, two groups are of particular interest. One is the steady waves. The interaction between this group and the second mode is indeed a fundamental resonance. The present results show that the fundamental resonance is a dominant interaction, which is consistent with the findings of Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2014). They modelled and investigated a natural transition scenario using wavepackets and then investigated the fundamental resonance in detail in Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2015). The other is the subharmonic waves with half the second-mode frequency (175 kHz). The interaction between this group and the second mode corresponds to a subharmonic-resonance type. However, the present work indicates that the subharmonic resonance is weak in a hypersonic boundary layer, as was also observed by Husmeier & Fasel (Reference Husmeier and Fasel2007) and Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2014).

5.2 Floquet analysis

This subsection shows that Floquet theory can well describe the resonance stage and explain the selection mechanism observed in the parametric study in § 5.1. The details of Floquet theory and its application to secondary instability studies can be found in Herbert (Reference Herbert1988) and Ng & Erlebacher (Reference Ng and Erlebacher1992). In the parametric-resonance stage in this case, the base flow is the undisturbed laminar flow plus the second mode, the MFD and the harmonics as

where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D714}$ are the wavenumber and the circular frequency of the second-mode wave.

According to Floquet theory, the periodic base flow given by equation (5.5) can support three-dimensional disturbances for a given azimuthal wavenumber, $n$ , as follows:

where $\unicode[STIX]{x1D6FE}$ is the characteristic exponent and $\unicode[STIX]{x1D70E}_{d}$ is a detuning parameter that distinguishes the various types of secondary instabilities. The fundamental resonance corresponds to $\unicode[STIX]{x1D70E}_{d}=0$ . The subharmonic resonance is associated with the case of $\unicode[STIX]{x1D70E}_{d}=0.5$ . For $0<\unicode[STIX]{x1D70E}_{d}<0.5$ , the secondary waves are called combination resonance modes (or detuned modes). When $\unicode[STIX]{x1D6FE}$ is real, the secondary and primary disturbances travel with the same phase velocity in the downstream direction, i.e. they are phase locked. For complex $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{r}+i\unicode[STIX]{x1D6FE}_{i}$ , the phase velocities of the two disturbances differ. Practical calculations only need to retain three terms in the solution to (5.6) as

where these three terms are analogous to the difference, low-frequency and sum modes in our case, respectively.

Substituting equations (5.5) and (5.7) into the Navier–Stokes equations, neglecting the non-parallel effects and the terms of $O({\mathcal{A}}^{2}),O(\unicode[STIX]{x1D716}^{2})$ and higher orders give a complex eigenvalue problem of the form

where ${\mathcal{M}}$ is a function of the base flow alone and $\unicode[STIX]{x1D701}$ is the vector of unknown variables $\hat{\unicode[STIX]{x1D6F9}}_{j}$ ( $j=-1,0,1$ ). This equation along with the boundary conditions in (3.5) was solved using the same numerical methods as for the LST problem.

Figure 12.

Growth rates (a) and phase velocities (b) of the low-frequency modes at $x=180$  mm predicted by Floquet theory.

Figure 12 shows the contours of the growth rates and phase velocities, predicted by Floquet theory, of the low-frequency waves at $x=180$  mm. This figure was plotted based on the most unstable eigenvalue from each of 7326 cases. The region with the large growth rates ( $\unicode[STIX]{x1D6FE}_{r}>0.01$ ) approximately coincides with regions $B$ and $D$ in figure 9(c). Note that Floquet theory only predicts the growth rates when the modes are in resonance, whereas in PSE simulations, the modes need to evolve to reach the resonant states. As shown by the parametric study in § 5.1, higher-frequency modes will start to grow farther downstream due to resonance. This is why region $D$ in figure 9(c) does not show any amplification.

The corresponding phase velocities, however, exhibit a peculiar distribution. One can observe that a polygonal line divides the contours into two regions. On the right side, the phase velocities are nearly the same as that of the second mode (around 0.9). On the left side, except for the lower-left corner, however, the phase velocities decrease with the azimuthal wavenumber, which is consistent with the observations in figure 11(d).

The reason for this distribution of the phase velocities can be found by a close examination of the eigenvalue spectrum. Figure 13(a,b) shows the eigenvalue spectra for various azimuthal wavenumbers and for various frequencies. For a given set of parameters, there are in general four unstable modes as labelled in the figure. Mode I lies approximately on the real axis with modes II and III almost symmetrical around mode I. Mode IV is in the upper-right quadrant.

Figure 13.

Eigenvalue spectra at $x=190$ mm predicted by the secondary instability calculation for a fixed frequency (20 kHz) with the azimuthal wavenumber varying from 10 to 120 by steps of 5 (a), and for the cases when the azimuthal wavenumber is fixed (50) with the frequency varying from 0 to 80 by steps of 4 (b). The arrows indicate the trajectory directions.

For a fixed frequency, mode IV moves along a nearly quadratic curve, reaching a peak at approximately $n=60$ and then going down until becoming part of the continuous spectrum. Mode II (III) first moves upwards to the left (right), reaches a peak, and then moves downwards. The motion of mode I is quite complex as it first moves upwards, then downwards, then upwards again, and finally downwards. Note that, as with mode IV, the first three modes will all become stable for very large azimuthal wavenumbers. For a fixed azimuthal wavenumber, modes I, II and III change little with the frequency, which indicates that these three modes depend essentially on the azimuthal wavenumber rather than the frequency. Mode IV, however, initially lies on the real axis (representing a fundamental resonance), then moves towards the lower-right side and eventually merges with the continuous spectrum.

These mode shifts change the imaginary part of the dominant eigenvalue, and, thus, the phase velocities of the secondary waves. Therefore, the peculiar distribution of the phase velocity can be explained as follows. For a fixed frequency and quite small azimuthal wavenumbers, mode I prevails and the phase velocity is close to that of the second-mode wave, changing slightly with the azimuthal wavenumber. As the azimuthal wavenumber increases so that mode IV becomes the most unstable mode, the phase velocity starts to decrease due to the increasing imaginary part of the eigenvalue. Finally, mode I again becomes the most unstable mode, causing a jump in the phase velocity. For a fixed azimuthal wavenumber and low frequencies, mode IV dominates. Although the imaginary part increases as mode IV moves, the streamwise wavenumber also increases, with the combined effects resulting in a slightly varying phase velocity. However, the phase velocity still jumps when the dominant mode changes from mode IV to mode I at a critical frequency. Since the phase velocity may jump with both increasing azimuthal wavenumber and increasing frequency, the polygonal line edge between the high and low phase velocity is reasonable and may not turn to a smooth curve by increasing the number of cases calculated. Nevertheless, in spectra close to the edges these four modes usually have comparable growth rates and, hence, have nearly equivalent contributions to the disturbances in the PSE simulations. Therefore, the phase velocities predicted by the PSE simulations will change gradually across the edge (not shown here).

5.3 Interaction between a second-mode wave and an oblique low-frequency wave

The $(20,50)$ mode was used here as a representative single low-frequency mode to study in detail. The evolution of the $(20,50)$ mode captures all the essential features of the most amplified low-frequency modes as shown in figure 11. In addition, this mode frequency is above the cutoff frequency of the PCB sensors, so it is likely detected in experiments. Finally, this mode lies in the Görtler instability region and is, thus, an unsteady Görtler mode. Thus, we refer to this mode as the Görtler mode in this subsection, although the streamwise curvature plays a secondary role in such interactions as will be shown later.

Here, the number of grid points along the wall-normal direction was increased to 151. The initial settlings were the same as those in the parametric study with the initial amplitude of the Görtler mode increased to 0.1 % to induce earlier breakdown. Four modes, the MFD, the difference mode $(f_{2nd}-f_{low},-n)$ , the sum mode $(f_{2nd}+f_{low},n)$ and the harmonic $(2f_{2nd},0)$ were directly triggered via nonlinear interactions. These four modes along with the two seeded modes will generate higher-order modes, in which nine modes were included here: $(3f_{2nd},0)$ , $(4f_{2nd},0)$ , $(2f_{low},2n)$ , $(2f_{2nd}+f_{low},n)$ , $(2f_{2nd}-f_{low},-n)$ , $(2f_{2nd}+2f_{low},2n)$ , $(2f_{2nd}-2f_{low},-2n)$ , $(f_{2nd}-2f_{low},-2n)$ and $(f_{2nd}+2f_{low},2n)$ .

Figure 14.

Contours of the streamwise velocity disturbances normalized by the boundary layer edge velocity at $x=100$  mm: (a) in the $x$ – $\unicode[STIX]{x1D719}$ planes with $y=0.22$  mm ( ${\approx}0.2\unicode[STIX]{x1D6FF}^{\ast }$ ) and $y=0.54$  mm ( ${\approx}0.5\unicode[STIX]{x1D6FF}^{\ast }$ ); (b) in the $y$ – $\unicode[STIX]{x1D719}$ plane with $x=200$  mm. Part (b) also shows the vertical velocity contours with the dashed lines representing negative values. The level step is 0.023 (20  $\text{m}~\text{s}^{-1}$ ).

Figure 15.

(a) Streamwise development of the maximum temperature disturbance amplitude. (b) Comparison of the linear case (dashed lines) with the nonlinear evolution of the second mode with only self-interaction (dotted line). (c) Time- and azimuthal-averaged skin friction coefficient.

Figure 14(a) shows contours of the instantaneous streamwise velocity disturbances (normalized by the boundary layer edge velocity at $x=100$  mm) in planes at constant distances from the wall. In the upstream region, the flow structures are essentially two-dimensional, which corresponds to the large-amplitude second mode initially forced onto the flow. Farther downstream, the two-dimensional structures become more prominent due to the growth of the second mode. Later, three-dimensional ripples appear and eventually evolve into oblique high- and low-speed streak-like structures. These streak-like structures have the same spanwise wavelength as the Görtler mode regardless of whatever smaller structures are inside, which shows that the Görtler wave is dominant. Streamwise (temperature) streaks were also recently observed for a similar flared-cone model in experiments by Berridge et al. (Reference Berridge, Chou, Ward, Steen, Gilbert, Juliano, Schneider and Gronvall2010). Direct numerical simulations by Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2015, Reference Sivasubramanian and Fasel2016) showed that both the second-mode fundamental breakdown and the oblique breakdown may be responsible for these streaks. Their DNS results showed only steady streamwise streaks. The present work, however, shows that unsteady streamwise streaks are also likely to arise. The details of the streak-like structures are illustrated in the slice at $x=200$  mm shown in figure 14(b). Vertical velocity disturbance contours are also shown with the dashed lines representing negative values. The low- and high-speed regions can be seen with the low-speed peaks at slightly higher positions than the high-speed peaks. Their positions coincide with the edges between the positive and negative contour lines of the vertical velocities, implying that the low and high-speed structures are associated with the vertical velocities, that is, positive vertical velocities (upwash) transfer low momentum fluid to high momentum regions resulting in velocity defects in the high momentum regions, and vice versa.

Of the five primary variables, the temperature disturbances are dominant and are found to be the most suitable for tracking the mode evolutions. Figure 15(a) illustrates the evolution of $|\hat{T}_{rms}|_{max}$ versus axis position for the six primary modes and higher-order modes (grey lines). The PSE calculation breaks down at approximately $x=210$  mm, indicating the occurrence of transition. The main feature of the interactions between the second mode and the Görtler mode can be clearly seen by comparing the nonlinear evolution with the linear as well as the self-interaction cases in figure 15(b). This figure shows that the second mode grows fast to high amplitudes, and then becomes saturated due to self-interaction. During the saturation stage of the second mode, the Görtler mode is strongly amplified. Finally, the second mode begins to decay and the rapid growth of the Görtler mode ends. Such rapid decay of the second mode was also observed by Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2015) for fundamental breakdown transition of a boundary layer over a sharp cone and by Fasel et al. (Reference Fasel, Sivasubramanian and Laible2015) over a flared cone. Li et al. (Reference Li, Choudhari, Chang and White2010a ) considered only the self-interactions of the second mode along a compression cone and also observed the rapid decay of the second mode. However, in the present investigation the rapid decay of the second mode is caused by the interactions with other modes, as shown in figure 15(b), instead of self-interactions. This difference may result from the different flow conditions and geometric models. The work of Li et al. (Reference Li, Choudhari, Chang and White2010a ) implied that if the PKU model were much longer, the rapid decay of the second mode could also be observed in the PSE simulations including only self-interactions.

The downstream development of the temporally and azimuthally averaged skin friction coefficient, $C_{f}$ , is shown in figure 15(c). The skin friction coefficient was calculated as

The skin friction coefficient curve obtained from the simulations is compared with a turbulent boundary layer correlation from White (Reference White2006, p. 558). The skin friction coefficient plotted in figure 15(c) initially follows the laminar curve up to 140 mm, then slightly increases until it slowly decays again. Beyond 190 mm, the curve rises steeply, reaches a peak at $x=200$  mm and then drops sharply. The initial deviation of the skin friction coefficient from its laminar value, well before the final transition, is due to the large amplitude of the MFD which is driven by the growing second-mode wave. Similar phenomenon has also been reported by Franko & Lele (Reference Franko and Lele2013) and Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2015). The skin friction coefficient starts to decay again as the second-mode wave becomes saturated. The second sharp rise is induced by the highly amplified Görtler mode which in turn causes the MFD to increase again. The reason for the final drop is yet unclear, and may be caused by either strong nonlinear interactions or the PSE becoming invalid at this stage.

Figure 16(a) shows the growth rates against the downstream distance. After rapid changes near the inlet, the nonlinearly generated modes have much higher growth rates than the second mode and the Görtler mode up to $x=160$  mm. For $x>160$  mm, the disturbances gradually form three groups (denoted by $G1$ , $G2$ and $G3$ ). $G1$ consists of the second mode, MFD and the harmonics. $G2$ consists of the Görtler mode, the difference/sum modes and the $(2f_{2nd}+f_{low},n)$ and $(2f_{2nd}-f_{low},-n)$ modes. The other modes form $G3$ . The $G1$ growth rates decay and become negative, whereas the $G2$ and $G3$ growth rates increase and eventually all become equal at their maximums at $x=180$  mm. Since the $G3$ modes are driven by the quadratic interactions of the $G2$ modes, the $G3$ -mode growth rates are nearly twice the $G2$ -mode growth rates. The good agreement with the Floquet analysis results indicates that the $G2$ modes start to resonate with the $G1$ modes. For $190<x<200$  mm, the $G2$ - and $G3$ -mode growth rates rapidly decay whereas the growth rates of the $G1$ harmonics increase again due to interactions with the $G2$ and $G3$ modes. Farther downstream, the growth rates of all the modes become comparable and rapidly oscillate, indicating full interactions between these modes.

The phase velocities for each mode are plotted in figure 16(b). The phase velocities vary by less than 5 % through most of the linear stage (from 110 mm to 150 mm). Thus, phase-locked theory can be readily applied in this region. For $x>150$  mm, however, the $(20,50)$ mode phase velocity drops sharply from approximately 0.9 to approximately 0.4 (which means that the wavelength is cut in half) at 180 mm, then slightly increases to about 0.5 at $x=190$  mm, and finally rapidly increases to 0.7 at the breakdown point. On the other hand, the $(330,-50)/(370,50)$ phase velocity increases/decreases from approximately 0.9 at approximately 160 mm to approximately $0.95/0.85$ at 180 mm and then remains nearly constant downstream before the breakdown point. Consequently, the relative differences between the phase velocities become as large as 100 % for $x>180$ mm, which indicates that phase-locked theory may not be valid there. Comparisons with the Floquet theory predictions in figure 16 demonstrate that such rapid changes in the phase velocities are due to the transition from non-resonance to a resonant state.

Figure 16.

(a) Streamwise growth rates for different modes. The thick lines in groups $G1$ and $G2$ represent the growth rates of the second mode and the Görtler mode. (b) Phase velocities for different modes. The velocities were all normalized by the boundary layer edge velocity at $x=100$  mm. The stars represent the Floquet analysis results.

Figure 17.

Comparison of the mode shapes obtained by Floquet theory (solid lines) and the mode shapes obtained by the nonlinear PSE calculation (dashed lines).

In addition to the growth rates and phase velocities, the mode shapes in the parametric-resonance stage also show very good agreement with the Floquet theory predictions as illustrated in figure 17. The minor differences may be due to non-parallel effects and nonlinear interactions with the higher-order modes. Thus, we can conclude that the parametric-resonance stage is governed by the secondary instability and can be well described by Floquet theory. Moreover, the agreement between the PSE and Floquet results verifies the applicability of these two different methods.

Figure 18.

Downstream evolution of the wavenumber differences for various interactions: 1, $|\unicode[STIX]{x1D6FC}_{(330,-50)}+\unicode[STIX]{x1D6FC}_{(20,50)}-\unicode[STIX]{x1D6FC}_{(350,0)}|/\unicode[STIX]{x1D6FC}_{(20,50)}$ ; 2, $|\unicode[STIX]{x1D6FC}_{(20,50)}+\unicode[STIX]{x1D6FC}_{(350,0)}-\unicode[STIX]{x1D6FC}_{(370,50)}|/\unicode[STIX]{x1D6FC}_{(20,50)}$ ; 3, $|\unicode[STIX]{x1D6FC}_{(370,50)}+\unicode[STIX]{x1D6FC}_{(330,-50)}-\unicode[STIX]{x1D6FC}_{(700,0)}|/\unicode[STIX]{x1D6FC}_{(330,-50)}$ ; 4, $|2\unicode[STIX]{x1D6FC}_{(350,0)}-\unicode[STIX]{x1D6FC}_{(700,0)}|/\unicode[STIX]{x1D6FC}_{(350,0)}$ .

A perfect triadic resonance should satisfy the resonance conditions (Craik Reference Craik1971)

which are the implicit assumptions of Floquet theory. The resonance intensities of the interactions (5.1)–(5.4) in the PSE simulations are shown in figure 18 in terms of the downstream evolution of the wavenumber differences, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}$ , among the resonant triads. The wavenumber difference for a resonant triad was defined as

where $\unicode[STIX]{x1D6FC}_{min}$ is the minimum of the real parts of the three wavenumbers and the subscript $r$ denotes the real part of the wavenumber. Choudhari, Chang & Jiang (Reference Choudhari, Chang and Jiang2005) defined a similar factor to represent the intensity of a triadic interaction. As the last two conditions of the resonance conditions in (5.10) are automatically satisfied in our PSE simulations, the wavenumber condition is, therefore, a measure of the resonance intensity with a smaller wavenumber difference indicating higher resonance intensity. If the disturbances have the same wave speed, then they are phase locked and in the exact resonance, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}=0$ . However, exact resonance does not necessarily require matching of the phase speeds except for the fundamental and subharmonic resonances. From figure 18, one can observe that after a short transient behaviour all the wavenumber differences fall to low levels. In the region of $110<x<150$  mm, the wavenumber differences for the first two interactions, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}_{1,2}$ , remain at about 10 % while the others are less than 2 %. The large values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}_{1,2}$ are due to the small value of the denominator $\unicode[STIX]{x1D6FC}_{(20,50)}$ which is about a tenth of the wavenumbers of the other modes. Further downstream, the wavenumber differences of the first three interactions slightly grow to peaks at approximately $x=170$  mm and then fall sharply to rather low levels (especially for the second resonant triad) at around $x=180$  mm. These changes follow from the interactions between the Görtler mode and the difference/sum modes, since these modes have comparable magnitudes there. For $x>190$  mm, the wavenumber differences of the first two resonances explode and develop downstream in a complex manner, implying that these are no longer in resonance.

To conclude, the evolution of the Görtler mode can be divided into four stages. The first stage, 100–160 mm, is a linear stage with the Görtler mode closely following the linear evolution. During this stage, the second mode first evolves linearly, and then gradually departs from the linear growth and becomes saturated, with the nonlinear modes experiencing large amplifications. The second stage, 170–180 mm, is a synchronization stage where the detuned modes gradually synchronize (in the sense that the resonance conditions are approximately satisfied) with the second mode. During this stage, the phase velocity of the Görtler mode is cut in half resulting in large differences between the phase velocities of this mode and the second mode. At the same time, the growth rate of the Görtler mode greatly increases. The third stage, 180–190 mm, is a parametric-resonance stage where the detuned modes are strongly amplified at the same nearly constant rate. The last stage, $x>190$  mm, is a full-interaction stage where all the modes are fully coupled as the flow approaches the final transition.

The linear and parametric-resonance stages in the interaction between a primary planar wave and an oblique wave were also theoretically investigated by Wu & Stewart (Reference Wu and Stewart1996) for incompressible wakes and by Al-Salman (Reference Al-Salman2002) for supersonic boundary layers. The present case differs from these works in two main aspects. First, they assumed that the phase velocities of the disturbances are nearly identical, hence, the resonance conditions were always (approximately) satisfied. However, the present study shows that the phase velocities of the disturbances may become quite different and a synchronization stage is generally needed between the linear stage and the parametric-resonance stage. Second, in those works the sum mode was believed to play a negligible role in the phase-locked interactions. However, the present results indicate that the difference and sum modes are of equal importance when the oblique wave has a very low frequency.

Figure 19.

Amplitudes of the streamwise velocity disturbances normalized by the local wall-normal maximums for four modes. The contour levels range from 0.5 to 0.95 with steps of 0.05. The dashed-dotted line denotes the position of the boundary layer edge. (a) Mode $(350,0)$ ; (b) mode $(20,50)$ with the dashed line denoting the critical layer location; (c) mode $(330,-50)$ ; (d) mode $(370,50)$ .

The amplitudes in figure 15 only show the maximum temperature perturbations. To better visualize the evolution of the disturbances, the contours of the streamwise velocity perturbations for $(350,0)$ , $(20,50)$ , $(330,-50)$ and $(370,50)$ are plotted in figure 19. The main part of the second mode initially lies below $y=0.5$ mm (close to the sonic line) and then slowly expands upwards for $x>160$ mm where the nonlinear effects set in. The peak amplitude of the Görtler mode, however, is slightly below the critical layer (denoted by the dashed line). The critical layer is the position where the local laminar mean velocity is equal to the phase velocity. The excellent match between the peak locations and the critical layer locations strongly suggests that the Görtler mode moves with the local mean-flow velocity, a prominent feature of a vortex. The maximum location moves slowly downwards before $x=170$  mm, which is consistent with the decrease in the boundary layer thickness, and then sharply drops from $y\approx 0.8$  mm at $x=170$ mm to $y\approx 0.4$  mm at $x=180$  mm. This sudden shape change may be caused by the nonlinear forcing by the second mode and the difference/sum modes as will be illustrated below. The difference mode $(330,-50)$ mainly lies on the critical layer and the boundary layer edge until $x=160$  mm where these two humps merge and a new peak appears close to the wall. For $x>180$  mm, the hump near the critical layer has totally disappeared, while the hump near the wall has spread out to cover the lower half of the boundary layer. The shape evolution of sum mode $(370,50)$ is similar to that of the difference mode with the only difference being that only one hump exists in the linear stage. The downwards shift of the difference-/sum-mode shapes may result from a combination of a nonlinear force mainly from the second mode and the linear characteristics of the modes themselves, since they lie close to the second-mode parameter region (see figure 6) and, thus, probably tend to behave like the second mode. Indeed, in a simulation (not shown here) with the nonlinear forces turned off once the difference/sum modes were generated showed that their shapes quickly moved downwards within a distance less than 2 mm to cover the lower half of the boundary layer. The difference-/sum-mode shape changes eventually lead to the sudden change of the Görtler mode distribution in the synchronization stage where the nonlinear forces manifest themselves.

5.4 Interaction between a second-mode wave and a pair of first-mode waves

The parametric study in § 5.1 shows that interactions between a second-mode wave and a single first-mode wave can hardly lead to transition in the present investigation. However, we will demonstrate in this subsection that interactions between a second-mode wave and a pair of oblique first-mode waves (case II) can also effectively induce transition. Please note that all the interactions observed in case I can all be found in case II. In this case, a second-mode wave $(350,0)$ and a pair of oblique first-mode waves $(60,\pm 30)$ were initially added at $x=100$ mm. The initial amplitudes are the same as those in case I. Mode $(60,30)$ was chosen because it is the most amplified first mode in the linear case. In addition to the fifteen modes in case I, three more modes were also included for the $(0,60)$ , $(0,120)$ spanwise MFDs and the $(350,60)$ fundamental oblique wave. Mode $(0,60)$ is generated by the pair of oblique first-mode waves and interacts with itself to give birth to $(0,120)$ and with the second mode to generate $(350,60)$ . In case II, each single oblique mode should be understood to be paired with its counterpart unless otherwise stated. For example, the $(60,30)$ mode is in fact the $(60,\pm 30)$ modes.

Figure 23.

Maximum r.m.s of the temperature disturbance for case II (a). Eight selected modes and other higher-order modes (denoted by the grey lines). (b) Time- and azimuthal-averaged skin friction coefficient.

The downstream development of the amplitudes of these eighteen modes are shown in figure 23(a). Similar to case I, these curves were obtained from the wall-normal maximums of the r.m.s. of the temperature disturbances. The amplitudes of $(0,60)$ and $(350,60)$ grow rapidly and lead to breakdown of the PSE simulation at approximately $x=200$  mm. This observation indicates that the dominating interaction mechanism is the fundamental resonance among the second mode, $(0,60)$ and $(350,60)$ , rather than the combination resonance among the detuned modes as in case I. The skin friction plotted in figure 23(b) closely resembles that in case I, shown in figure 15(c), indicating that both cases undergo similar transition processes.

Table 1.

Numerical settings for all four simulations.

To further explore the driving mechanism, three simplified simulations were conducted with the different settings shown in table 1. The first simulation ( $S1$ ) consists of only the second mode, its first harmonic, the MFD, the pair of first-mode waves, $(0,60)$ and $(350,60)$ . The second simulation ( $S2$ ) is the same as $S1$ except that the streamwise curvature is set to zero. In this simulation, the first mode at the inlet is slightly stable and its initial amplitude was set to $10^{-2}$ instead of $10^{-3}$ to achieve nearly the same breakdown positions as in the full model. The third simulation ( $S3$ ) includes the second mode, its first harmonic, the MFD, $(0,60)$ and $(350,60)$ . The initial amplitude of $(0,60)$ was set to $10^{-5}$ so that the breakdown position was close to those in the other simulations.

Figure 24.

Maximum of r.m.s. of temperature disturbance versus downstream distance for the case II: comparison of several selected modes for the full model and the other three simulations.

Figure 25.

Normalized streamwise velocity disturbances of the $(0,60)$ mode for the full model (solid lines) and simulation $S3$ (dashed lines) at three streamwise positions: (a)  $x=140$  mm, (b)  $x=170$  mm and (c)  $x=188$  mm. The Floquet theory results (dotted lines) are displayed in (b) and (c) for comparison.

The results of these three simulations are displayed in figure 24 with comparison to the full model. The evolutions of these three simulations and the full model all enter the parametric-resonance stage at approximately $x=170$  mm after which the amplitudes of the spanwise MFD and the fundamental oblique mode are all amplified at nearly the same rates. This confirms the dominant role played by the fundamental resonance. Figure 25 further shows that the mode shapes eventually collapse to the one predicted by Floquet theory, despite the large differences during the linear stage. Similar trends were observed for other primary disturbances (not shown here). Therefore, this fundamental resonance appears to be an ‘attractor’, that is, once the spanwise MFD is present regardless of whether it was initially seeded (as in $S3$ ) or was nonlinearly generated (as in $S1$ , $S2$ and the full model), resonance will occur downstream, and the growth rates and mode shapes will be independent of the linear stage. In addition, the $S2$ result shows that this type of resonance is also insensitive to the streamwise curvature and, thereby, will be found in boundary layers over straight cones.

Figure 26.

The PSD (a) and bicoherence spectra (b) at $x=180$  mm. The dashed line represents the $N$ factors from the LST results. The bicoherence contour level step is 0.02.

Finally, it should be noted that for the pair of low-frequency modes in region $B$ in figure 9(c), the combination resonance is also strong and the dominant interaction may be either the fundamental resonance or the combination resonance depending on their growth rates.