3.1.1. Stability regions and amplifications

Once we have solved the boundary-layer equations for a given ${\it\delta}$ and ${\it\epsilon}$ , (2.8)–(2.9), we can solve the PSE (2.21). In figure 4, the profiles of $\partial \text{Re}({\it\phi})/\partial y$ and $\partial \text{Im}({\it\phi})/\partial y$ are displayed which give the horizontal velocity component $u^{\prime }$ of the perturbation, cf. (2.20), for the case ${\it\epsilon}=0.4$ , ${\it\delta}=4.75\times 10^{-4}$ and ${\it\omega}=0.22$ at position ${\it\xi}=-0.2375$ . In addition the profiles of $\text{Im}(a)\text{Im}({\it\phi})$ and $\text{Im}(a)\text{Re}({\it\phi})$ are shown, needed to compute the vertical velocity component $v^{\prime }$ of the perturbation. As can be observed from figure 4, the perturbation velocity $(u^{\prime },v^{\prime })$ decays towards infinity and has its maximum magnitude close to the wall. This can also be observed in figure 5, where contour plots of the modulus and argument of ${\it\phi}$ in (2.20) are plotted as a function of ${\it\xi}$ and $y$ for this case. We observe that the shape function ${\it\phi}$ displays a slow change in ${\it\xi}$ . As such the width of ${\it\phi}$ seems to increase with ${\it\xi}$ . It needs to be noted that the physical significance of ${\it\phi}$ is somewhat limited as it depends on the constraint chosen in order to restrict growth to the amplitude of the Tollmien–Schlichting wave (cf. (2.26)). Whereas figure 5 displays the slow variation of ${\it\phi}$ with respect to ${\it\xi}$ , we give in figure 6 an example of the rapid change with respect to ${\it\xi}$ of the streamfunction ${\it\psi}^{\prime }$ of the perturbation. Figure 6 shows the streamfunction ${\it\psi}^{\prime }$ at time $t=0$ , normalized by the amplitude $A({\it\xi})$ (cf. (2.27)).

Figure 4. Profiles of the perturbation (2.20) computed by means of the PSE. The parameters are ${\it\epsilon}=0.4$ , ${\it\delta}=4.75\times 10^{-4}$ and ${\it\omega}=0.22$ and the profiles were taken at position ${\it\xi}=-0.2375$ . The profiles are only shown up to a value of the ordinate of $y=10$ . However, for the present case the domain extends until $y=45.5$ , a value at which the profiles have decayed sufficiently.

Figure 5. (a) Contours of $|{\it\phi}|$ from (2.20) for the case in figure 4. (b) Contours of $\arg ({\it\phi})$ from (2.20) for the case in figure 4.

Figure 6. Surface plot of $\text{Re}({\it\psi})/A({\it\xi})$ from (2.20) for the case in figure 4.

Computing the amplification of the horizontal velocity component $u^{\prime }$ by means of (2.27) we obtain the curves of zero growth by the criterion of maximum or minimum amplitude $A$ defined in (2.27). A series of neutral curves for ${\it\delta}=8\times 10^{-4}$ is depicted in figure 7 for different values of ${\it\epsilon}$ . These curves separate regions of growth and decay for Tollmien–Schlichting waves in the $({\it\xi},{\it\omega})$ plane. The position ${\it\xi}_{c}$ leftmost on the neutral curve is called the critical position. For ${\it\xi}>{\it\xi}_{c}$ perturbations are expected to grow, while they decay for ${\it\xi}<{\it\xi}_{c}$ . In figure 7 we observe an increase in the size of the unstable regions with ${\it\epsilon}$ , but that the unstable regions remain confined within the decelerating part ( ${\it\xi}>0$ ) of the flow for the cases shown. Blondeaux et al. (Reference Blondeaux, Pralits and Vittori2012), using their method of linear stability, found regions of temporal instability in the $({\it\xi},k)$ plane, where $k$ is the chosen wavenumber. These regions increased with ${\it\epsilon}$ and were also entirely situated in the deceleration region, in accordance with the above result. As mentioned by Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010), instability can be expected in the deceleration part of the flow, since the profile $U_{\mathit{base}}$ displays an inflection point here. Rayleigh’s inflection point theorem is, however, not entirely applicable for the present case, since non-parallel effects are not negligible and viscosity is important in the boundary layer.

Figure 7. Stability domain for ${\it\delta}=8\times 10^{-4}$ . The region bounded by the curves is the unstable region. Here and in the subsequent figures, ${\it\xi}$ and ${\it\omega}$ are scaled by $h_{0}$ and $c_{0}/({\it\delta}h_{0})$ , respectively.

In figure 8 we show the dependence of the unstable region on ${\it\delta}$ , with a fixed amplitude ${\it\epsilon}=0.4$ . Overall, the neutral curves move upstream (decreasing ${\it\xi}$ ) and cover a wider frequency span with decreasing ${\it\delta}$ . For the smaller values of ${\it\delta}$ the unstable domain also forms a ‘thumb’ for lower values of ${\it\omega}$ intruding into the accelerating region. This thumb is probably of viscous nature, since there are no inflection points in the velocity profiles for ( ${\it\xi}<0$ ). As such the form of this thumb is reminiscent of the unstable region of the Blasius boundary layer (Drazin & Reid Reference Drazin and Reid1981). This suggests that the instability mechanism in this case is similar to that of the Blasius boundary layer (Baines, Mujumdar & Mitsudera Reference Baines, Mujumdar and Mitsudera1996). The smallest ${\it\delta}$ , from the figure, which gives instabilities in front of the crest is $0.0001$ and corresponds to a water depth of 16 m. Hence, it may not be observable in laboratory wave tanks.

Figure 8. Stability domain for ${\it\epsilon}=0.4$ . The region bounded by the curves is the unstable region.

The above stability domains in figures 7 and 8 have been computed by means of the PSE method. This method accounts approximately for non-parallel effects of the flow. The OSE (2.16) on the other hand neglects these effects and assumes the flow to be parallel. In figure 9 neutral curves computed by the PSE method and the OSE method are displayed together. They are similar, but we observe that the unstable regions tend to be larger in extent when computed by the OSE. In addition they are shifted downwards on the ${\it\xi}$ axis. The difference in size of the unstable regions and the shift between the critical positions increase for increasing ${\it\delta}$ , which is not surprising, since the normal velocity component scales like ${\it\delta}$ (cf. (2.7)) and so do the non-parallel effects. This can be observed when plotting the difference between the critical positions computed by means of the PSE and the OSE as a function of ${\it\delta}$ (cf. figure 10). The difference seems to increase almost linearly with ${\it\delta}$ . A regression line is added to guide the eye. Since wave tank experiments are usually performed for large values of ${\it\delta}$ , such as the ones used in figure 9, a significant influence on the stability properties due to non-parallel effects needs to be accounted for. For large Reynolds numbers (small ${\it\delta}$ ), on the other hand, the OSE is accurate.

Figure 9. Comparison of the unstable region for the cases ${\it\epsilon}=0.4$ and (a) ${\it\delta}=1.34\times 10^{-3}$ , (b) ${\it\delta}=2.67\times 10^{-3}$ and (c) ${\it\delta}=4.49\times 10^{-3}$ computed by means of the OSE and the PSE.

Figure 10. Distance ${\it\xi}_{c}^{PSE}-{\it\xi}_{c}^{OSE}$ between the critical position ${\it\xi}_{c}$ computed by means of the PSE and the OSE as a function of ${\it\delta}$ . The amplitude was set to ${\it\epsilon}=0.4$ .

All of the solitary waves display some region of instability in the deceleration region. However, for a given disturbance to become significant, or observable, the crucial quantity is the total, accumulated amplification as obtained by (2.27) for the PSE method. For ${\it\delta}=8\times 10^{-4}$ ( $h_{0}=1\ \text{m}$ ) amplifications are depicted in figure 11. We observe a striking increase with ${\it\epsilon}$ . Disturbances for ${\it\epsilon}=0.1$ may only be amplified $100$ times, say, whereas the factor for ${\it\epsilon}=0.4$ is above $10^{7}$ . The variation of the accumulated amplification with ${\it\delta}$ is shown in figure 12. With decreasing ${\it\delta}$ , we observe a strong increase in amplification. From the figure we may infer that rather strong disturbances are required for instabilities to become visible for typical small-scale wave tank experiments with $h_{0}\sim 0.1$ – $0.2\ \text{m}$ . For large ${\it\delta}$ , we observe that after the critical position the amplification grows to a maximum before declining again. For the case $({\it\epsilon}=0.4,\ {\it\delta}=4.5\times 10^{-3})$ , the growth of the amplitude of the Tollmien–Schlichting wave is weak such that at ${\it\xi}=19.5$ the resulting amplitude is lower than at the critical position. Maximum amplification of approximately 1.5 for this case is at ${\it\xi}=7.44$ . Although in principle unstable, any growth of Tollmien–Schlichting waves will hardly be observable in experiments for this case. The characteristic quantities for each case in figures 11 and 12 are tabled in tables 1 and 2, respectively. In order to quantify the amplification for each case ( ${\it\delta},{\it\epsilon}$ ), we might relate the maximum amplitude to the minimum amplitude of the critical Tollmien–Schlichting wave at the neutral curve. This would give us a function $A_{max}/A_{min}({\it\delta},{\it\epsilon})$ for each ${\it\delta}$ and ${\it\epsilon}$ . However, the computation of $A_{max}$ is difficult, since it is far downstream outside of the computational domain for many cases. A more straightforward criterion would be to compute the amplitude at ${\it\xi}=19.5$ and to relate this value to the minimum amplitude. We used the value at ${\it\xi}=19.5$ at the rightmost end of the computational domain. The value ${\it\xi}=19.5$ seemed reasonable to us, since the maximum extension in time behind the crest used by Vittori & Blondeaux (Reference Vittori and Blondeaux2011), for example in figure 1 in Vittori & Blondeaux (Reference Vittori and Blondeaux2011) corresponds approximately to a spatial location between ${\it\xi}=21$ and ${\it\xi}=23.6$ . The solitary wave itself has passed well before this point in time even for an amplitude of ${\it\epsilon}=0.05$ . In figure 13, we plotted the isolines of the function

(3.1) $$\begin{eqnarray}\log _{10}\frac{A({\it\xi}=19.5)}{A_{min}}({\it\delta},{\it\epsilon}).\end{eqnarray}$$

The isolines exhibit that for each ${\it\epsilon}$ , the amplification increases strongly when decreasing ${\it\delta}$ . However, when increasing ${\it\epsilon}$ , the increase in amplification is large in the beginning but slows down for larger ${\it\epsilon}$ .

Figure 11. Amplification of Tollmien–Schlichting waves for the cases listed in table 1 using the PSE solver.

Figure 12. Amplification of Tollmien–Schlichting waves for the cases listed in table 2 using the PSE solver.

Figure 13. Isolines of the function $\log _{10}(A({\it\xi}=19.5)/A_{min})({\it\delta},{\it\epsilon})$ and lines of $Re_{\mathit{Sumer}}=2\times 10^{5}$ and $Re_{\mathit{Sumer}}=5\times 10^{5}$ .

Table 1. Critical parameters for the case ${\it\delta}=8\times 10^{-4}$ , for different values of ${\it\epsilon}$ .

Table 2. Critical parameters for the case ${\it\epsilon}=0.4$ for different values of ${\it\delta}$ .

A quantity of interest, also investigated in Blondeaux et al. (Reference Blondeaux, Pralits and Vittori2012), is the phase speed of the critical Tollmien–Schlichting wave. For a given frequency ${\it\omega}$ , the wavenumber of the Tollmien–Schlichting wave can approximately be given by $\text{Im}(a)$ , where $a$ is defined in (2.20), allowing us to compute the phase speed $c_{p}={\it\omega}/\text{Im}(a)$ as a function of ${\it\xi}$ . The phase speed $c_{p}$ gives us the celerity of the wave in the moving frame of reference. In the absolute frame of reference the phase speed is given by $c_{p}-c$ , which is plotted in figures 14 and 15. The parameters for the plotted cases are given in tables 1 and 2, respectively. The Tollmien–Schlichting waves seem to first propagate in the direction of the solitary wave and then reverse their direction of propagation. This result has also been obtained by Blondeaux et al. (Reference Blondeaux, Pralits and Vittori2012) for their perturbations. They proposed that the flow reversal in the boundary layer is causing the Tollmien–Schlichting waves to reverse their direction of propagation too. From figure 14, we observe in addition that for increasing amplitudes ${\it\epsilon}$ , the Tollmien–Schlichting waves travel with an increasing phase speed. The reason may be that the magnitude of the particle velocities in the base flow becomes higher for increasing ${\it\epsilon}$ . For increasing ${\it\delta}$ , we also observe an increase in the magnitude of the phase speed, although only slightly (cf. figure 15). We suspect, however, that this is not primarily due to the increase in ${\it\delta}$ , but as the critical frequency ${\it\omega}_{c}$ is higher for increasing ${\it\delta}$ (cf. table 2), the critical Tollmien–Schlichting wave travels with a higher phase speed.

Figure 14. Absolute phase speed $c_{p}-c$ of Tollmien–Schlichting waves for ${\it\delta}=8\times 10^{-4}$ . The parameters of the Tollmien–Schlichting waves are given in table 1.

Figure 15. Absolute phase speed $c_{p}-c$ of Tollmien–Schlichting waves for ${\it\epsilon}=0.4$ . The parameters of the Tollmien–Schlichting waves are given in table 2.

3.1.2. Relation to experiments and previously published results

As mentioned in the introduction, research on the boundary layer under a solitary wave and in particular its stability properties, was initiated when Liu et al. (Reference Liu, Park and Cowen2007) derived a theoretical approximation and performed experiments on this flow. The experiments were performed in an equilibrium depth of $h_{0}=0.1\ \text{m}$ , which corresponds to ${\it\delta}=4.5\times 10^{-3}$ , and with amplitudes ${\it\epsilon}=0.3$ , or less. Figure 12 indicates a maximum amplification as small as 1.5 even for a wave with ${\it\epsilon}=0.4$ . This explains why they did not observe any instabilities in their experiments.

Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) performed direct numerical simulations for a range of ${\it\delta}$ of $2.8\times 10^{-4}\leqslant {\it\delta}\leqslant 1.34\times 10^{-3}$ and found the flow to be stable or not depending on the amplitude ${\it\epsilon}$ . In order to be precise, Vittori & Blondeaux (Reference Vittori and Blondeaux2008) wrote $2.8\times 10^{-5}\leqslant {\it\delta}\leqslant 1.34\times 10^{-4}$ , but presented data only for the above range. We therefore assume that a typo happened in Vittori & Blondeaux (Reference Vittori and Blondeaux2008) concerning the range of investigated ${\it\delta}$ . Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) performed experiments for a range of $2.8\times 10^{4}\leqslant Re_{\mathit{Sumer}}\leqslant 2\times 10^{6}$ and found that the flow turns unstable for Reynolds numbers larger than $Re_{\mathit{Sumer}}>2\times 10^{5}$ . The critical set of parameters $({\it\epsilon}_{c},{\it\delta}_{c})$ by Vittori & Blondeaux (Reference Vittori and Blondeaux2008), was mapped to a critical Reynolds number in Vittori & Blondeaux (Reference Vittori and Blondeaux2011). The value of which was found to be $Re_{\mathit{Sumer}}\approx 5\times 10^{5}$ . Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) found a different regime appearing after the laminar regime which they called, disturbed laminar, in the sense that perturbations can be observed to destabilize the flow. They put the Reynolds number of this transition at $Re_{\mathit{Sumer}}=8\times 10^{4}$ . The disturbed laminar changes to the transitional at $Re_{\mathit{Sumer}}=1.1\times 10^{5}$ .

Apart from this quantitative discrepancy, there were also qualitative differences concerning the appearance of the instabilities. Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) observed irregularities in the boundary layer in front of the crest for higher Reynolds numbers $Re_{\mathit{Sumer}}$ . In particular, they presented the case $Re_{\mathit{Sumer}}=2\times 10^{6}$ (cf. figure 10(d) in Sumer et al. Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010, in which instabilities are observable for ${\it\xi}<0$ ). Vittori & Blondeaux (Reference Vittori and Blondeaux2011) applied $({\it\delta}=2.8\times 10^{-4},{\it\epsilon}=0.2)$ , corresponding to $Re_{\mathit{Sumer}}=2.6\times 10^{6}$ and $({\it\delta}=4.75\times 10^{-4},{\it\epsilon}=0.5)$ , corresponding to $Re_{\mathit{Sumer}}=3.6\times 10^{6}$ and did not observe any sign of instability in the acceleration region. Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) conjectured that for sufficiently strong perturbations a nonlinear instability can develop in front of the crest for Reynolds numbers larger than $Re_{\mathit{Sumer}}=1.1\times 10^{6}$ , but did not observe any flow transition in front of the crest.

We shall relate these findings to the present results. There are of course differences in the present work to the works by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011), Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) and Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010). As stated in § 2.1 we start with a more accurate description of the outer flow than what was used in the previous studies. In addition, we do not neglect the nonlinear and non-parallel effects in the boundary layer. Moreover, the references considered temporal growth of instabilities, whereas the present study focuses on spatial growth. A more crucial difference is that Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) in their simulations and Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) in their experiments did not directly control the amplitude of the perturbation which might lead to a retarded appearance of the instability in their simulations and experiments, respectively. The importance of this point should not be underestimated and we shall discuss it in more detail below.

Concerning the appearance of instabilities in front of the crest, the question of whether there exists a nonlinear instability mechanism in front of the crest for this boundary layer as conjectured by Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) cannot be answered by the present method. Linear instability is, however, possible in front of the crest and we observe that the neutral curve for the case $({\it\delta}=10^{-4},{\it\epsilon}=0.4)$ in figure 8 is the first to cross the line ${\it\xi}=0$ . This case corresponds to a Reynolds number $Re_{\mathit{Sumer}}=6\times 10^{7}$ , which is an order of magnitude larger than what Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) found and what Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) reported to be the lower bound in order to observe nonlinear growth in front of the crest. If the observed irregularities in front of the crest in Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) have their origin in a nonlinear growth mechanism and whether this mechanism is strong enough or not (as Ozdemir et al. Reference Ozdemir, Hsu and Balachandar2013 suggested) to induce a bypass transition is impossible to say without further knowledge of the experimental conditions. Concerning the quantitative discrepancy between the critical Reynolds numbers in the works by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011), Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) and Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010), we discussed the cases ${\it\delta}=8\times 10^{-4}$ and ${\it\epsilon}=0.1,0.2,0.3,0.4$ above (cf. figures 7 and 11 and table 1). These cases have also been investigated by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011). In figure 1 in Vittori & Blondeaux (Reference Vittori and Blondeaux2011), time profiles of the horizontal velocity component at a point in space are plotted. In addition, the case ${\it\epsilon}=0.1$ has been considered stable by both Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) and Vittori & Blondeaux (Reference Vittori and Blondeaux2011) (see figure 5 in Vittori & Blondeaux Reference Vittori and Blondeaux2011). The cases ${\it\epsilon}=0.3$ and ${\it\epsilon}=0.4$ , on the other hand, were found to be unstable by both references. However, the case ${\it\epsilon}=0.2$ was classified unstable by Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010), whereas Vittori & Blondeaux (Reference Vittori and Blondeaux2011) deemed it stable. Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013), on the other hand, would consider this case as disturbed laminar with the possible appearance of roller pairs.

In figure 11 the growth of the critical Tollmien–Schlichting wave is shown. The amplifications of the Tollmien–Schlichting waves at ${\it\xi}=19.5$ , as compared with the minimum value at ${\it\xi}_{c}$ , are listed in table 1. If we assume that the Tollmien–Schlichting waves start to roll up into vortices (triggering of the secondary instability) once their amplitude has grown to a value of 1 % of the mean flow (which we set to unity for simplicity), the amplification ( $10^{5}$ ) for the unstable case ${\it\epsilon}=0.2$ in table 1, tells us that the background noise in the experiments in Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) was higher than $10^{-7}$ . On the other hand, the case ${\it\epsilon}=0.1$ (amplification: $10^{2.7}$ ) was classified stable. Therefore, the background noise level in the experiments by Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) might have been between $10^{-7}$ and $10^{-4.7}$ . A similar reasoning finds the background noise in the simulations by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) to be between $10^{-8.5}$ and $10^{-7}$ . Since linear stability predicts a strong decay of the perturbations in the acceleration region, the initial $10^{-4}$ amplitude perturbation, which Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) imposed onto the initial condition before the solitary wave arrived, should have decayed by the time the critical position was reached to values much lower than the $10^{-7}$ estimated above. The NS solver used by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) and Blondeaux et al. (Reference Blondeaux, Pralits and Vittori2012) may have produced a certain level of numerical noise, which might have provided a sufficient level at ${\it\xi}_{c}$ for instabilities to become visible. We find support for this presumption in figure 4 in Blondeaux et al. (Reference Blondeaux, Pralits and Vittori2012), where the level of kinetic energy of the perturbation seems to stay on a stable level of around $10^{-8}{-}10^{-9}$ for all cases of ${\it\epsilon}$ , before arriving at the critical position. Several reasons for the numerical source of noise are possible, such as truncation errors or incomplete pressure solutions. Overall, the triggering mechanism of the instability in these references is not well controlled. A better approach has been chosen by Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) where five different intensities of white noise have been introduced at the beginning of a direct numerical simulation. In addition they followed the evolution of the root-mean-square velocity during the course of the simulations. Although being a major improvement over the approach by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011), the results cannot be transferred to the case where a constant level of background noise is present, as for example in laboratories. As predicted by the present linear stability the perturbations in Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) underwent strong decay before growing again. As a consequence of their analysis, Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013) also obtained a new set of critical Reynolds numbers $Re_{\mathit{Sumer}}$ .

The present investigation is different, it solves the governing equations, i.e. the Orr–Sommerfeld and the PSE, for the perturbations, the Tollmien–Schlichting waves themselves. The amplification of the Tollmien–Schlichting waves can therefore be computed independently from any given initial level of noise, as long as the initial level is small. Since the initial amplitude is crucial for the visual appearance of the perturbation or observation of transition (triggering of the secondary instability), a result from the present analysis is that classifications such as that presented in figure 5 of Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010), figure 5 of Vittori & Blondeaux (Reference Vittori and Blondeaux2011) or in § 4.4 of Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013), or the determination of critical parameters $({\it\delta}_{c},{\it\epsilon}_{c})$ in Blondeaux et al. (Reference Blondeaux, Pralits and Vittori2012), need to be taken with care. The flow is convectively unstable and acts as a broadband amplifier, similar to the Blasius boundary layer (Herbert Reference Herbert1988).

As mentioned above, in figure 13 isolines for the amplification of the critical Tollmien–Schlichting waves are plotted. Knowing the initial amplitude at the neutral curve of the critical Tollmien–Schlichting wave for an experiment or a direct numerical simulation, we might by means of the isolines predict whether disturbances become visible or not. In addition to the isolines, we plotted lines given for the critical Reynolds numbers $Re_{\mathit{Sumer}}=2\times 10^{6}$ and $Re_{\mathit{Sumer}}=5\times 10^{6}$ . For small values of ${\it\delta}$ and ${\it\epsilon}$ , we observe that the lines seem to coincide with the isolines for an amplification of $10^{4}$ and $10^{7}$ , respectively. However, as ${\it\epsilon}$ and ${\it\delta}$ increase the lines do not follow these isolines anymore. As mentioned in § 2.1, the deviation for increasing ${\it\epsilon}$ reflects the limited accuracy of the first-order approximation of the outer flow. The deviation for increasing ${\it\delta}$ , on the other hand, is due to non-parallel effects becoming significant. For small values of ${\it\delta}$ and ${\it\epsilon}$ , the amplification of Tollmien–Schlichting waves gives a theoretical explanation for the appearance of transition beyond a critical $Re_{\mathit{Sumer}}$ in the experiments by Sumer et al. (Reference Sumer, Jensen, Sørensen, Fredsøe, Liu and Carstensen2010) or the simulations by Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) and Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013). The stability properties of the flow are, however, more complex, since neutral curves, critical positions, critical wavelengths, critical frequencies, etc., can be different for a given $Re_{\mathit{Sumer}}$ .

Most important, however, is the fact that the flow may appear to be stable (in the sense that transition does not occur) for a specific $Re_{\mathit{Sumer}}$ or $({\it\delta}_{c},{\it\epsilon}_{c})$ during one experimental run, whereas it appears to be unstable during another run. This observation has been made by Pedersen et al. (Reference Pedersen, Lindstrøm, Bertelsen, Jensen, Laskovski and Sælevik2013) for a related problem, namely the boundary layer of a solitary wave running up a sloping beach, where the occurrence of irregularities was not strictly repeatable. In fact, unless the disturbance level is actively controlled we do not believe that experimental repeatability for a flow transition of this type can be obtained.

3.1.3. Direct numerical simulation

In the analysis above (cf. § 3.1), results have been generated by model equations; namely the OSE and the PSE. In the present subsection a direct numerical simulation based on first principles is performed for validation. Our approach is very different from that of Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011) or Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2013), in the sense that the perturbation is carefully controlled and that we, in contrast to Vittori & Blondeaux (Reference Vittori and Blondeaux2008, Reference Vittori and Blondeaux2011), employ a high-accuracy solution method for the NS equations. We note that different types of NS solvers might give different sets of allegedly critical parameters $({\it\epsilon}_{c},{\it\delta}_{c})$ depending on their truncation errors, numerical dissipation rates etc.

As mentioned in § 2, the present direct numerical simulations are performed using a method developed by Fasel (Reference Fasel1976, Reference Fasel2002) to investigate the stability of the Blasius boundary layer. Thereby a Tollmien–Schlichting wave is introduced at the inlet of the simulation domain and its evolution is monitored.

Figure 16. (a) Stability domain for the case ${\it\epsilon}=0.4$ and ${\it\delta}=8\times 10^{-4}$ . (b) Horizontal perturbation $u^{\prime }$ of the horizontal velocity component recorded at a distance $y=0.4943$ from the wall for a Tollmien–Schlichting wave of ${\it\omega}=0.24$ computed by means of the NS solver. (c) Amplification of the Tollmien–Schlichting wave with ${\it\omega}=0.24$ computed by means of the OSE solver, the PSE solver and the NS solver.

Figure 16 shows three plots for the case ${\it\epsilon}=0.4$ and ${\it\delta}=8\times 10^{-4}$ . In figure 16(a), the stability domain of the flow, computed using the PSE solver, has been plotted by the criterion of maximum or minimum amplitude $A$ in (2.27). We now pick a particular frequency ${\it\omega}$ and follow the evolution of a Tollmien–Schlichting wave with this frequency. In figure 16(b), we have chosen the Tollmien–Schlichting wave with the critical frequency ${\it\omega}_{c}=0.24$ . The velocity field $u({\it\xi},y,t)$ at a specific point in time $t$ computed by the NS solver, allows us to compute the perturbation velocity $u^{\prime }({\it\xi},y,t)$ by subtracting the base flow $U_{\mathit{base}}({\it\xi},y)$ from the velocity field:

(3.2) $$\begin{eqnarray}u^{\prime }=u-U_{\mathit{base}}.\end{eqnarray}$$

In figure 16(b), a transect of $u^{\prime }$ at $y=0.4943$ is plotted. We clearly observe a sinusoidal wave which decays until it reaches the critical position ${\it\xi}_{c}=0.82$ and then starts to grow, albeit slowly at the beginning. This qualitative picture can be analysed further. Using the solution $u(x,y,t)$ by the NS solver, we compute the amplitude of the Tollmien–Schlichting wave by means of the envelope of $u^{\prime }$ at its maximum, i.e.

(3.3) $$\begin{eqnarray}A({\it\xi})=\max _{t,y}|u^{\prime }({\it\xi},y,t)|.\end{eqnarray}$$

This envelope can then be compared with the resulting amplifications using the PSE and the OSE, computed by means of (2.27) and (2.19), respectively. The result of this comparison is depicted in figure 16(c). All three methods predict first a decay of the Tollmien–Schlichting wave followed by growth. The results by the NS solver and by the PSE solver agree remarkably well. The results by the Orr–Sommerfeld solver on the other hand display an earlier growth of the Tollmien–Schlichting wave compared to the other two solvers, a feature which was also observed by looking at the stability domains in the subsection above. This indicates that non-parallel effects do lead to quantitative differences.

A closer look at figure 16(c) reveals that for ${\it\xi}\gtrsim 2.5$ the amplification curves computed by the parabolized stability method and the NS solver start to mildly deviate. This indicates that the effect of nonlinearity becomes increasingly significant. We can study this effect further. By introducing a superposition of Tollmien–Schlichting waves at the inlet of our computational domain, we can monitor the perturbation velocity $u^{\prime }$ in horizontal direction at different locations further downstream. The perturbation velocity $u^{\prime }$ of the Tollmien–Schlichting waves introduced at the inlet is thus given by

(3.4) $$\begin{eqnarray}u^{\prime }({\it\xi}_{inlet},y,t)=\mathop{\sum }_{k=1}^{N_{{\it\omega}}}\text{Re}\{A_{k}u_{k}(y)\exp \text{i}\left({\it\alpha}_{k}{\it\xi}_{inlet}-{\it\omega}_{k}t\right)\}\end{eqnarray}$$

where $N_{{\it\omega}}$ is the number of Tollmien–Schlichting waves used, $u_{k}$ is the shape function at the inlet, computed by the PSE solver, and ${\it\alpha}_{k}=\text{Im}(a_{k})$ is the wavenumber corresponding to the frequency ${\it\omega}_{k}$ . The frequencies ${\it\omega}_{k}$ have been chosen such that ${\it\omega}_{1}$ equals $0.15$ and ${\it\omega}_{N_{{\it\omega}}}$ equals $0.35$ with $N_{{\it\omega}}=61$ . They are distributed such that we have an equidistant spacing of Fourier modes:

(3.5) $$\begin{eqnarray}{\it\omega}_{k}={\rm\Delta}{\it\omega}(k-1)+{\it\omega}_{1},\end{eqnarray}$$

where ${\rm\Delta}{\it\omega}=({\it\omega}_{N_{{\it\omega}}}-{\it\omega}_{1})/(N_{{\it\omega}}-1)$ . The initial amplitude $A_{k}$ of each Fourier mode is set uniformly to $A_{k}=5\times 10^{-4}/N_{{\it\omega}}$ . The positions at which we monitor the perturbation velocity $u^{\prime }$ are given by ${\it\xi}=2.40,2.96$ and $3.12$ . At these locations we record the time series of $u^{\prime }$ at a distance of $1.64$ from the wall. A decomposition of the signal into its Fourier modes allows us to find the amplitudes $A_{k}$ at these downstream locations. Figure 17 presents the amplitudes $A_{k}$ as a function of ${\it\omega}_{k}$ . In addition, we plot the amplitudes $A_{k}$ predicted by the present linear PSE solver for comparison. We observe that the signal received further downstream, consists of discrete spikes separated by ${\rm\Delta}{\it\omega}$ , indicating that the perturbation $u^{\prime }$ at these locations has its origin in the perturbation introduced at the inlet. When repeating the computation at lower spatial and temporal resolutions, the simulation displays the emergence of a continuous spectrum with modes filling the space between the spikes for the locations further downstream. These modes which are absent for finer resolutions have probably their origin in the truncation error of the simulation growing in parallel to the Tollmien–Schlichting waves. Returning to figure 17, for the direct numerical simulation and the PSE we observe a strong growth of the individual modes for increasing ${\it\xi}$ . Growth is strongest for modes around ${\it\omega}=0.26$ which is somewhat higher than the critical frequency ( ${\it\omega}_{c}=0.24$ ). We remark that we measure the amplitude here not by criterion (2.27) but at a specific location in $y$ . For ${\it\xi}=2.40$ , we see that the amplitudes computed by the NS solver and the linear parabolized equation solver are agreeing well. However, further downwards, ${\it\xi}=2.96$ , we observe that the amplitudes computed by the NS solver tend to be somewhat lower than predicted by the PSE solver. This discrepancy is even more emphasized for ${\it\xi}=3.12$ , where the modes computed by direct numerical simulation are significantly lower than the modes computed by the PSE solver. The spectrum reflects the discontinuity of the initial distribution, displaying a sudden jump at ${\it\omega}=0.15$ and ${\it\omega}=3.5$ . However, we observe for ${\it\xi}=2.96$ and ${\it\xi}=3.12$ , the appearance of (significant) modes outside of the interval $[{\it\omega}_{1},{\it\omega}_{N_{{\it\omega}}}]$ , but still with a spacing given by ${\rm\Delta}{\it\omega}$ . In particular for ${\it\xi}=3.12$ , we observe two additional peaks at ${\it\omega}=0$ and ${\it\omega}=5.3$ , located respectively at zero or two times the frequency of the maximum amplitude. As can be seen from the neutral curve in figure 16(a), this is in contrast to the linear analysis which does not predict any growth for frequencies around ${\it\omega}=0$ . As such the primary effect of nonlinearity is to distribute energy away from modes of maximum amplification.

We remark that the present nonlinear study is limited as we only present results for a particular spectrum, namely a uniform initial distribution of Tollmien–Schlichting waves with frequency spacing ${\rm\Delta}{\it\omega}$ . It needs to be noted that for the simulations, each of these modes undergoes a phase of decay first, before reaching the neutral curve. The amount by which each mode has decayed by the time it reaches the neutral curve is varying. This is different to the experimental situation where it is assumed that a certain level of noise is present everywhere in the set up, the frequency distribution of this noise being unknown. We emphasize that the present PSE solver is linear. A nonlinear version, as derived by Bertolotti et al. (Reference Bertolotti, Herbert and Spalart1992), would be able to account for the interaction between Tollmien–Schlichting modes. In addition, we only consider two dimensional disturbances, leaving aside any possible interaction with spanwise modes. Nevertheless, despite its limitation, this nonlinear study reveals that the present linear stability analysis describes accurately the early growth of the perturbations destabilizing the flow.

Figure 17. Fourier modes of the perturbation velocity $u^{\prime }$ computed by direct numerical simulation (NS) and linear PSE at $y=1.64$ and different values of ${\it\xi}$ for the case ${\it\epsilon}=0.4$ and ${\it\delta}=8\times 10^{-4}$ .

3.2.1. Stability regions and amplifications

The boundary layer under surface solitary waves is characterized by two parameters, the amplitude ${\it\epsilon}$ and the parameter ${\it\delta}$ . For internal solitary waves, as sketched in figure 2, the system contains two more parameters accounting for the density ratio and the depth ratio between the two layers. In the following we use the lower layer density ${\it\rho}_{1}$ and the lower layer depth $h_{1}$ as parameters. The scaling is such that the upper layer density ${\it\rho}_{2}$ is unity, whereas the upper layer depth is given by $h_{2}=1-h_{1}$ . In addition, we consider waves of elevation ( ${\it\epsilon}>0$ , $h_{1}<h_{c}$ ) and waves of depression ( ${\it\epsilon}<0$ , $h_{1}>h_{c}$ ), where $h_{c}$ is the height of the critical level (Funakoshi & Oikawa Reference Funakoshi and Oikawa1986):

(3.6) $$\begin{eqnarray}h_{c}=\frac{\sqrt{{\it\rho}_{2}/{\it\rho}_{1}}}{1+\sqrt{{\it\rho}_{2}/{\it\rho}_{1}}}h_{0}.\end{eqnarray}$$

We compute the base flow $(U_{\mathit{base}},V_{\mathit{base}})$ using the present boundary-layer solver in combination with the method by Funakoshi & Oikawa (Reference Funakoshi and Oikawa1986). This flow is then the subject of the present linear stability analysis by means of the PSE. Profiles of the horizontal velocity component for a wave of elevation are displayed in figure 18. Qualitatively, the profiles are similar to those of a surface solitary wave. An inversion of the flow is observed in the deceleration region ${\it\xi}>0$ of the wave. Although we consider only two-layered fluids, fluids with a pycnocline with non-zero thickness can relatively well be approximated by a two-layered system, as long as the thickness of the pycnocline is small (cf. Carr et al. Reference Carr, Davies and Shivaram2008). This can also be seen in figure 19, where we plotted the same time histories for the horizontal velocity as in Thiem et al. (Reference Thiem, Carr, Berntsen and Davies2011) for a wave of depression. The pycnocline of the present two-layered system lies in the centre of the middle layer of experiment ‘20538’ in Carr & Davies (Reference Carr and Davies2006). The scaling corresponds to that of Thiem et al. (Reference Thiem, Carr, Berntsen and Davies2011). The data has been taken by a digitization tool from figure 4(a) of Thiem et al. (Reference Thiem, Carr, Berntsen and Davies2011). Although the agreement between experimental data and numerical result is not excellent it is still reasonable in view of the uncertainties involved in such measurements. The amplitude is predicted somewhat lower than for the data. However, the agreement is still closer to the experimental data than for the numerical method of Thiem et al. (Reference Thiem, Carr, Berntsen and Davies2011).

Figure 18. Horizontal velocity profiles for the boundary-layer flow under an internal solitary wave of elevation with ${\it\epsilon}=0.1$ , ${\it\delta}=0.0023$ , ${\it\rho}_{1}=1.25$ and $h_{1}={\textstyle \frac{1}{3}}$ .

Figure 19. Time histories for the horizontal velocity in the boundary layer of a two-layered fluid corresponding to the case ‘20538’ of Carr & Davies (Reference Carr and Davies2006) at $z/h_{2}=0.1,0.2,0.4$ (wave of depression). The experimental data by Carr & Davies (Reference Carr and Davies2006) was scanned from Thiem et al. (Reference Thiem, Carr, Berntsen and Davies2011).

Figure 20. Stability domains for the wave of elevation case ${\it\epsilon}=0.1$ , ${\it\delta}=10^{-3}$ and $h_{1}={\textstyle \frac{1}{3}}$ for different values of ${\it\rho}_{1}$ .

Figure 21. Amplification curves for the critical Tollmien–Schlichting waves for the cases in figure 20.

Carr & Davies (Reference Carr and Davies2010) concluded on the basis of their experiments that the boundary-layer flows of internal solitary waves are qualitatively similar to those of surface solitary waves. A quantitative comparison is, on the other hand, not straightforward, because of the additional parameters ${\it\rho}_{1}$ and $h_{1}$ . However, the leading-order solution for the internal solitary wave in a two-layered fluid (Keulegan Reference Keulegan1953) is in the present scaling independent of the density ratio (we are very grateful to one anonymous referee for pointing this out). This can also be observed when we keep all parameters fixed and vary only ${\it\rho}_{1}$ . In figure 20, we plotted the neutral curves for the wave of elevation case ${\it\epsilon}=0.1$ , ${\it\delta}=10^{-3}$ and $h_{1}={\textstyle \frac{1}{3}}$ for different values of ${\it\rho}_{1}$ . The unstable regions grow somewhat in extent, when increasing ${\it\rho}_{1}$ . However, when looking at the amplification of the critical Tollmien–Schlichting waves of the respective cases, figure 21, we observe that the amplification curves almost coincide except for ${\it\rho}_{1}=1.5$ . The case ${\it\rho}_{1}=1.5$ may be different in this respect because its interface is already quite close to the critical level $h_{c}$ , where inviscid instability of the flow becomes important. In addition, the larger values of ${\it\rho}_{1}$ used to generate the plot might already be somewhat beyond the range the scaling given by $c_{0}$ (2.2) is valid. For depression waves, the same behaviour for varying ${\it\rho}_{1}$ is observed (figures not shown). As opposed to the density ratio the layer depths $h_{1}$ and $h_{2}$ enter the coefficients of the leading-order solution scaled by $c_{0}$ . Given the first-order formula for the wave celerity $c$ (Keulegan Reference Keulegan1953):

(3.7) $$\begin{eqnarray}c=c_{0}\sqrt{1+\frac{h_{2}-h_{1}}{h_{1}h_{2}}{\it\epsilon}h_{0}},\end{eqnarray}$$

we observe that the difference $h_{2}-h_{1}$ between the layer depths has a concurrent role to the amplitude ${\it\epsilon}$ of the wave. Since decreasing $h_{1}$ will increase $c$ , we expect that the flow becomes more unstable with decreasing $h_{1}$ . This can also be seen in figure 22, where the unstable regions increase in size when decreasing $h_{1}$ or in figure 23, where the amplifications of the critical Tollmien–Schlichting waves for the cases in figure 22 are larger the lower $h_{1}$ . The roles of $h_{1}$ and $h_{2}$ are inverted for waves of depression (figures not shown). As for surface solitary waves, increasing ${\it\epsilon}$ increases the size of the unstable regions (cf. figure 24). Surprisingly, for larger values of ${\it\epsilon}$ , we see that the neutral curve is moving away from the crest, i.e. the critical position ${\it\xi}_{c}$ decreases first before growing again. It needs to be said that the case for ${\it\epsilon}=0.3$ is close to the inviscid stability criterion for internal waves of this geometry (Funakoshi & Oikawa Reference Funakoshi and Oikawa1986). Looking at the amplification of the critical Tollmien–Schlichting waves for these cases does not give a clear picture. The amplification curves for ${\it\epsilon}=0.2,0.25,0.3$ in figure 25 do not display a significant increased growth of the critical Tollmien–Schlichting wave compared with the case ${\it\epsilon}=0.15$ . The reason for this behaviour is not obvious. It might be related to the fact that for large amplitudes the shape of the internal solitary wave undergoes considerable change, the amplitude of the solitary waves for ${\it\epsilon}>0.15$ being already larger than the lower layer depth $h_{1}={\textstyle \frac{1}{6}}$ . For increasing ${\it\delta}$ , on the other hand (cf. figure 26), we see a more or less monotone increase in the size of the unstable region, as for the surface solitary waves. In addition, we observe again a thumb reaching out into the acceleration region of the wave for small values of ${\it\delta}$ . This ‘viscous’ instability is, however, observable only for water depths larger than approximately 50 m. As for surface solitary waves the amplification of the critical Tollmien–Schlichting waves increases strongly with decreasing ${\it\delta}$ (figure not shown), from a factor of approximately 100 to a factor of $10^{5}$ for the most extreme case in figure 26.

Figure 22. Stability domains for the case ${\it\epsilon}=0.1$ , ${\it\rho}_{1}=1.25$ and ${\it\delta}=10^{-3}$ for different values of $h_{1}$ .

Figure 23. Amplification curves for the critical Tollmien–Schlichting waves for the cases in figure 22.

Figure 24. Stability domains for the case ${\it\delta}=1.96\times 10^{-3}$ , ${\it\rho}_{1}=1.25$ and $h_{1}={\textstyle \frac{1}{6}}$ for different values of ${\it\epsilon}$ .

Figure 25. Amplification curves for the critical Tollmien–Schlichting waves for the cases in figure 24.

Figure 26. Stability domains for the case ${\it\epsilon}=0.1$ , ${\it\rho}_{1}=1.25$ and $h_{1}={\textstyle \frac{1}{3}}$ for different values of ${\it\delta}$ .

3.2.2. Relation to experiments and previously published results

In this subsection, the above analysis is applied to several cases published in the literature. In particular, we analyse the amplification of the depression waves investigated by direct numerical simulation in Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012) and of the depression waves investigated experimentally by Carr et al. (Reference Carr, Davies and Shivaram2008). Concerning waves of elevation, we turn to the experiments by Carr & Davies (Reference Carr and Davies2010).

Similar to what we did in figure 19, we approximate the depression Dubreil–Jacotin–Long internal solitary wave with a tangent hyperbolic density distribution in Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012) by a two-layered fluid. For each experiment in table 1 of Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012), we compute the unstable regions (not shown) and the amplification for a Tollmien–Schlichting wave with frequency ${\it\omega}=0.11$ (cf. figure 27). The critical Tollmien–Schlichting wave has a frequency somewhat higher than the Tollmien–Schlichting wave with maximum amplification, which lies for all cases approximately at ${\it\omega}=0.11$ . The numbers of the runs from table 1 in Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012) are printed on the amplification curves in figure 27. We marked the curves for those cases for which vortex shedding was observed with red colour and dashed lines and the curves for those cases for which no vortex shedding was observed with blue colour and solid lines. Not surprisingly the amplifications of those cases for which no vortex shedding was observed tend to be smaller than the amplifications for those for which vortex shedding was observed. The passage from blue to red or from solid to dashed lines is, however, not perfect. A reason for this might be a variation in the background noise of the solver used by Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012). In addition, it needs to be said that due to the large number of cases a detailed search for the frequency of the Tollmien–Schlichting wave displaying maximum amplification was out of the scope of this work. Therefore the maximum amplification for some of the cases in figure 27 might be somewhat higher than that for ${\it\omega}=0.11$ . The same holds for figures 28 and 29. This might be an additional reason for seeing an overlap of the red and blue amplification curves. Nevertheless the trend is clearly visible. In line with the above analysis for surface solitary waves, we infer that the level of background noise is around $10^{-4.5}$ in the solver used by Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012). Some reservation needs to be taken with respect to the above results, since they apply to a two-layered system, whereas the Dubreil–Jacotin–Long system in Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012) might display differences in some features.

Figure 27. Amplification for the Tollmien–Schlichting waves with frequency ${\it\omega}=0.11$ for the cases listed in table 1 of Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012).

Figure 28. Amplification for the Tollmien–Schlichting waves with frequency ${\it\omega}=0.1$ for the cases listed in table 1 of Carr et al. (Reference Carr, Davies and Shivaram2008).

Figure 29. Amplification for the Tollmien–Schlichting waves with frequency ${\it\omega}=0.2$ for the cases listed in table 1 of Carr & Davies (Reference Carr and Davies2010).

The same type of analysis can be done for the experimental runs of a depression wave given in table 1 in Carr et al. (Reference Carr, Davies and Shivaram2008). The Tollmien–Schlichting wave with maximum amplification lies around ${\it\omega}=0.1$ for these cases. The amplifications are plotted in figure 28. Again, the numbers of the runs from table 1 in Carr et al. (Reference Carr, Davies and Shivaram2008) are printed as a label on the amplification curves in figure 28. We marked the curves for those cases for which transition was observed with red colour and dashed lines and the curves for those cases for which no transition was observed with blue colour and solid lines. Apart from case ‘140207’ the colours of the lines switch from blue to red for increasing amplification. The approximate level of background noise in the experimental facility used by Carr et al. (Reference Carr, Davies and Shivaram2008) seems to be at a value of $10^{-2.35}\approx 0.004$ . As a side remark, if the experimental runs by Carr et al. (Reference Carr, Davies and Shivaram2008) had been investigated using the solver in Aghsaee et al. (Reference Aghsaee, Boegman, Diamessis and Lamb2012) no transition would have been observed for all cases, since the level of background noise in this solver is much lower than the maximum amplification for the experiments in Carr et al. (Reference Carr, Davies and Shivaram2008). Somewhat later Carr & Davies (Reference Carr and Davies2010) investigated the boundary-layer flow of waves of elevation, presumably at the very same experimental facility. The amplification curves corresponding to the cases listed in table 1 in Carr & Davies (Reference Carr and Davies2010) are plotted in figure 29. The Tollmien–Schlichting wave with maximum amplification was found to have a frequency around ${\it\omega}=0.2$ . The numbers of the experiment by Carr & Davies (Reference Carr and Davies2010) corresponding to the number labels on the amplification curve can be found in table 3. All curves are plotted in blue colour and solid lines since no vortices were observed by Carr & Davies (Reference Carr and Davies2010). This is also consistent with the above observation that the level of background noise in their experimental facility is at $10^{-2.35}\approx 0.004$ . The maximum amplification in figure 29 can be determined to be $10^{0.28}\approx 2$ , not enough to induce transition given the local background noise. The present result gives an answer to the question raised by Carr & Davies (Reference Carr and Davies2010), of why there is no transition observed for the waves of elevation in Carr & Davies (Reference Carr and Davies2010) but there is for the waves of depression in Carr et al. (Reference Carr, Davies and Shivaram2008), although by the criterion of Diamessis & Redekopp (Reference Diamessis and Redekopp2006), cf. figure 2 in Carr & Davies (Reference Carr and Davies2010), the waves of elevation should be more unstable than the waves of depression. Linear theory is not able to predict the new flow regime after transition, however some ‘structures’ in the new flow regime might display features having their origin in features of the Tollmien–Schlichting wave during primary instability. In figure 9 in Carr et al. (Reference Carr, Davies and Shivaram2008), we can measure the spacing between the vortices for experiment ‘080207’ as being 0.080, 0.109, 0.071, 0.137 from left to right. In order to be precise, in the caption in (Carr et al. Reference Carr, Davies and Shivaram2008) the result is referred to as belonging to experiment ‘080206’. However, such an experiment is not given in table 1. We suspect therefore that a typo has happened in the figure caption.

Table 3. Number of the experiment in Carr & Davies (Reference Carr and Davies2010) corresponding to the label number in figure 29.

The time at which this plot is taken corresponds to ${\it\xi}=7.56$ in the present scaling. For ${\it\xi}=7.56$ , we find that the wavelength of the Tollmien–Schlichting wave is 0.13. The correspondence to the spacing of the vortices in Carr et al. (Reference Carr, Davies and Shivaram2008) is reasonable, in view of the sources of uncertainty. A further feature can be found in the mention by Carr et al. (Reference Carr, Davies and Shivaram2008) that the vortices were only observable long after the wave crest has passed. For the case ‘050207’, they mentioned that the vortices only appeared after ${\it\xi}=6.87$ in the present scaling. This is approximately were we find the maximum amplification for this case in figure 28. The delayed observation may correspond to the fact that the Tollmien–Schlichting wave needs time to grow starting from the neutral curve to a level sufficient to trigger the secondary instability.

As for surface solitary waves, the present results indicate that for the cases investigated in figure 28 the instability in the boundary layer is not of parametric nature. The initial amplitude and spectrum of the perturbation plays a major role whether transition is observable or not. For surface solitary waves, we could give some explanation for the appearance of a critical Reynolds number $Re_{\mathit{Sumer}}$ in terms of isocontours of the amplification of Tollmien–Schlichting waves for small ${\it\epsilon}$ and ${\it\delta}$ . On the basis of the above results we consider the quest for a single parameter controlling the transition in the boundary layer of an internal solitary wave given a certain perturbation as extremely difficult. This is because the amplification curves do not display a monotonic behaviour, for example when varying ${\it\epsilon}$ (cf. figure 25). By means of the amplification of Tollmien–Schlichting waves the observations in the literature (Carr et al. Reference Carr, Davies and Shivaram2008; Carr & Davies Reference Carr and Davies2010; Aghsaee et al. Reference Aghsaee, Boegman, Diamessis and Lamb2012) can be explained. As such the boundary layer under internal solitary waves is in general more stable than the one under surface solitary waves, since the maximum amplifications tend to be lower, at least for the cases considered. This is somewhat counter intuitive to the fact that transition in the boundary layer seems to be observed more frequent for internal solitary waves. The explanation might be that the level of background noise in the experiments for internal solitary waves is compared with the characteristic velocity much higher than for surface solitary waves, even if the levels of noise are the same on an absolute scale. The above analysis is relatively crude. It cannot be excluded that, due to the relatively large amplitude the initial perturbation is required to have in order to induce transition, nonlinear interactions between Tollmien–Schlichting waves may be important. In addition, the method might have to be modified if the effect of an additional boundary layer by an opposing current (Stastna & Lamb Reference Stastna and Lamb2002, Reference Stastna and Lamb2008) has to be accounted for. Yet, it does point to an important ingredient so far not taken into account: the growth of Tollmien–Schlichting waves.