3.1 Critical suction parameters

As discussed in § 1, a diagram illustrating the relation between the critical suction ratio $R_{crit}$ and the hole diameter $d^{\ast }/\unicode[STIX]{x1D6FF}_{h}^{\ast }$ for different Reynolds numbers $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}$ , was given by MacManus & Eaton (Reference MacManus and Eaton1998, Reference MacManus and Eaton2000) (see figure 1 a). The figure gathers results from their low-speed wind-tunnel measurements as well as transonic results by Blanchard et al. (Reference Blanchard, Seraudie, Breil and Payry1991). (Note that MacManus and Eaton defined $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}$ based on the boundary layer edge velocity, and $R$ as the ratio between the average suction velocity and the boundary layer edge velocity.) In order to further illustrate the variation of $R_{crit}$ with $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}$ , these data are replotted in figure 1(b). As seen, an increase in $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}$ tends to decrease the value of $R_{crit}$ . However, whereas all the data points in figure 1(a) collapse onto a single curve, figure 1(b) shows an appreciable amount of scatter. This suggests that the boundary layer Reynolds number might have a minor influence on the critical suction ratio $R_{crit}$ compared to the hole diameter $d^{\ast }/\unicode[STIX]{x1D6FF}_{h}^{\ast }$ .

Figure 3.

Flow visualisation of streamwise velocity for (a) $R=0.452$ , (b) $R=0.678$ and (c) $R=0.905$ , at $y=0.9$ (viewed from above).

In the thesis by Müller (Reference Müller2012), three cases were studied with the values $d^{\ast }/\unicode[STIX]{x1D6FF}_{h}^{\ast }=2$ , $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}=1924$ and $R=\{0.452,0.678,0.905\}$ (based on the same parameter definitions as in this work). These velocity ratios imply a pipe Reynolds number of $Re_{d}=\{1739.3,2608.9,3482.4\}$ (with $Re_{d}=|V_{h}^{\ast }|d^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ ). As a starting point for the present study, the three cases of Müller (Reference Müller2012) are reproduced. The simulations are started from the synthetic flow field $\boldsymbol{u}_{Bl-p}$ (see § 2.1) and integrated forward in time. The long-time flow responses are visualised in figure 3. For the lowest suction ratio shown in figure 3(a), an apparently stable boundary layer is obtained featuring a central high-speed streak and a counter-rotating vortex pair. With increased levels of suction, the strength of this streak increases, and as seen in figure 3(b), oscillations appear on the vortex legs towards the end of the domain. These oscillations are persistent in time, which is an indication that the flow might have surpassed its critical suction ratio $R_{crit}$ and become unstable. For even higher values of $R$ (figure 3 c), the vortices break down and turbulent flow develops in the wake. Based on these simulations, the critical suction ratio that precipitates boundary layer transition appears to be in the interval $0.452<R_{crit}<0.678$ (see figure 1 for a comparison with the aforementioned data available in the literature). The case with suction ratio $R=0.678$ is selected for further analysis. Before considering a spectral analysis of this configuration, some of its flow characteristics will be further highlighted.

3.2 Flow features

One of the most prominent structures in the flow is the counter-rotating vortex pair shown in figure 4. This pair of vortices originates inside the pipe and brings high-velocity fluid in the outer part of the boundary layer towards the plate, where it creates a high-speed streak behind the suction hole as shown in figure 5. Outside of the primary vortex pair, there is a weak secondary vortex pair with a sense of rotation that is opposite to the principal one, followed by a third slightly stronger vortex pair that co-rotates with the principal one. Similar vortex structures have also been reported by (Meyer Reference Meyer1955; MacManus & Eaton Reference MacManus and Eaton1998, Reference MacManus and Eaton2000). MacManus & Eaton (Reference MacManus and Eaton1998, Reference MacManus and Eaton2000) argued that these secondary vortices may arise due to the strongly inflectional spanwise velocity component (see figure 5).

Figure 4.

Vortex structures for the suction ratio of $R=0.678$ visualised by the $\unicode[STIX]{x1D706}_{2}$ -criterion (Jeong & Hussain Reference Jeong and Hussain1995). The view is from above, and the sense of rotation of the different vortices are indicated by their colour and the arrows.

Figure 5.

Velocity in the $yz$ -plane at $x=10$ ( $x^{\ast }/d^{\ast }=5$ ) for the suction ratio $R=0.678$ (viewed from upstream). Isocontours of the streamwise velocity component are shown with black lines (ranging from $0.0$ to $1.0$ and separated by $0.04$ units) and the spanwise velocity is shaded.

Figure 6.

Visualisation of the instantaneous flow inside the pipe at various distance from the pipe inflow. The out-of-plane $y$ -velocity is shaded, and positive and negative $y$ -vorticity is shown with yellow and blue contours, respectively, and ranges from $-29.0$ to $29.0$ with $1$ unit separation.

Regarding the flow within the pipe, a separated zone develops at the upstream side of the pipe wall close to the pipe junction, as plotted in figure 6. Further down in the pipe, the counter-rotating vortices create two streaks with fluid moving in the upward direction that are symmetrically positioned about the $xy$ -plane. These streaks move away from the wall with distance from the inlet and grow into very intricate structures that eventually destabilise and trip the flow inside the pipe. This breakdown is further illustrated in figure 7 where typical power spectra (PS) from two probes within the pipe are shown. These power spectra are computed from the $x$ -velocity component using Welch’s method of averaged modified periodograms (see e.g. Heinzel, Rüdiger & Schilling Reference Heinzel, Rüdiger and Schilling2002), where each segment has been windowed with a Hann function using an overlap of 50 %. (The length of each segment was approximately 123 time units, and the effective noise bandwidth of the window was 0.0122.) Near the inlet of the pipe ( $5\unicode[STIX]{x1D6FF}_{h}^{\ast }$ from the pipe inflow), large energetic structures appear with several amplified frequencies. Further down into the pipe ( $20\unicode[STIX]{x1D6FF}_{h}^{\ast }$ from the pipe inflow), these large structures have broken down and consequently their corresponding energies have decreased.

Figure 7.

Power spectra (PS) of the $x$ -velocity (cross-stream component) measured in the pipe at the positions $(0.0,-5.0,-0.5)$ (——) and $(0.0,-20.0,-0.5)$ (——).

Figure 8.

Eigenvalue spectrum for $R=0.678$ (the superscripts $r$ and $i$ refer to the real and the imaginary parts of $\unicode[STIX]{x1D714}$ , respectively). The eigenvalues whose corresponding eigenvectors have their support only in the pipe are marked with filled symbols, whereas eigenvalues whose eigenvectors have their support in the pipe and in the boundary layer are marked with open symbols. The eigenvectors have different symmetries and are either varicose (○) or sinuous (▵).

A separation bubble and a pair of counter-rotating vortices have also been observed in the simulations presented by MacManus & Eaton (Reference MacManus and Eaton1996). However, for their comparably low values of $R$ , no unsteadiness of the separation bubble was seen. The inhomogeneity of the vertical velocity component near the pipe inlet (figure 6 a) was also observed in the compressible simulations by Müller (Reference Müller2012) using flow parameters comparable to the present ones. However, in contrast to figures 6 and 7, that author did not report any unsteadiness inside the pipe, but rather argued for the opposite. The reason for this difference in results is unknown. In fact, a preliminary DNS study on a coarser grid investigating different parameter combinations of $R$ and $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}$ , suggests that for the present values of $d$ , $s$ and $L$ , the pipe always transitions for a lower $R$ or $Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}$ than the boundary layer (Brynjell-Rahkola et al. Reference Brynjell-Rahkola, Barman, Peplinski, Hanifi and Henningson2015). Consequently, the separation bubble and flow conditions in the pipe seem to play a crucial role in the transition process.

3.3 Linear stability analysis

The eigenvalue spectrum containing the most unstable/least stable eigenvalues for the case with the intermediate suction ratio $R=0.678$ are plotted in figure 8. (Since the spectrum is symmetric about the imaginary axis, only half of it is shown.) Comparison of these eigenvalues with the spectra plotted in figure 7 shows a quite good agreement between the linear analysis and the DNS. All of the eigenvectors have their support in the pipe and a few modes also extend into the boundary layer. As can be seen, many of the modes inside the pipe are highly unstable, whereas two modes in the boundary layer (and the pipe) are slightly unstable, and another two are lightly dampened. In contrast, a preliminary stability analysis (Brynjell-Rahkola et al. Reference Brynjell-Rahkola, Barman, Peplinski, Hanifi and Henningson2015) performed on a coarser mesh suggests that all the modes for $R=0.452$ that contain both the boundary layer and the pipe are stable (in agreement with the nonlinear DNS results of § 3.1).

Two kinds of symmetry are observed among the eigenvectors. One where the $x$ - and $y$ -components are anti-symmetric and the $z$ -component is symmetric with respect to the $xy$ -plane, and another where this symmetry is reversed (i.e. the $x$ - and $y$ -components are symmetric and the $z$ -component is anti-symmetric). Throughout the rest of this article, modes with these different symmetries will be referred to as sinuous and varicose, respectively.

In figure 9, the most unstable eigenvector (a sinuous pipe mode) is shown. The mode resides close to the pipe inlet in the same region where the flow was seen to destabilise and break down in the DNS simulation (figure 6). In fact, the five most unstable (pipe) modes have their maximum amplitude in the interval $-6<y<-3$ . This shows that the transition to turbulence observed in the pipe section to a large extent can be explained by an instability near the pipe inlet. Instabilities that are restricted to the pipe would, however, not be able to explain the origin of oversuction since amplification of such modes does not imply disturbance growth within the boundary layer. This can be realised by considering the lower suction ratio $R=0.452$ , for which the pipe flow transitions to turbulence while the boundary layer remains laminar. Therefore, given the aim of understanding the origin of oversuction, the boundary layer modes will be studied in greater detail throughout the remainder of this article.

The eigenvalues whose eigenvectors span both the pipe and the boundary layer are labelled I–IV in figure 8. Their corresponding eigenvectors are shown in figure 10. Like the other modes, the modes I–IV are strongest in the pipe, but in contrast to the other ones, the eigenvectors also have a weak tail that extends into the boundary layer. Although the amplitudes of these eigenvectors are between $O(10^{-4})$ and $O(10^{-2})$ lower in the boundary layer than in the pipe, our analysis will show that it is precisely these motions that are responsible for the transition in the boundary layer.

As seen in figure 10, the unstable modes I and II have different symmetries. Since the growth rate (and the frequency) of these modes are similar to each other, they can be expected to have similar amplitudes in the flow. Therefore the transition process is unlikely to exhibit a particular symmetry, as also indicated in figure 3.

Figure 9.

Vertical velocity component of the eigenvector corresponding to the most unstable eigenvalue $\unicode[STIX]{x1D714}=\pm 0.4997+\text{i}0.2014$ . Contours corresponding to $\pm 1\,\%$ of the maximum velocity are plotted in white and black colours, respectively, together with base flow vortex structures visualised by the $\unicode[STIX]{x1D706}_{2}$ -criterion (Jeong & Hussain Reference Jeong and Hussain1995) (dark grey).

Figure 10.

Streamwise velocity component of the eigenvectors corresponding to the eigenvalues labelled (a) I, (b) II, (c) III and (d) IV in figure 8. Contours corresponding to $\pm 0.1\unicode[STIX]{x2030}$ of the maximum velocity are plotted in grey and black colours, respectively.

Table 2.

Relative contributions from the production mechanisms to the growth rate in (3.2) for the selected modes. The tuples $(j,k)$ denote the different terms $-E_{\unicode[STIX]{x1D6FA}}^{-1}\int _{\unicode[STIX]{x1D6FA}}(\hat{u} _{j}^{r}\hat{u} _{k}^{r}+\hat{u} _{j}^{i}\hat{u} _{k}^{i})\unicode[STIX]{x2202}U_{j}/\unicode[STIX]{x2202}x_{k}\,\text{d}\unicode[STIX]{x1D6FA}$ normalised by $|\unicode[STIX]{x1D714}^{i}|$ . The numbering in the first column refers to the mode labelling introduced in figure 8.

Table 3.

Relative contributions from the dissipation mechanisms to the growth rate in (3.2) for the selected modes. The tuples $(j,k)$ denote the different terms $E_{\unicode[STIX]{x1D6FA}}^{-1}Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}^{-1}\int _{\unicode[STIX]{x1D6FA}}[(\unicode[STIX]{x2202}\hat{u} _{j}^{r}/\unicode[STIX]{x2202}x_{k})^{2}+(\unicode[STIX]{x2202}\hat{u} _{j}^{i}/\unicode[STIX]{x2202}x_{k})^{2}]\,\text{d}\unicode[STIX]{x1D6FA}$ normalised by $|\unicode[STIX]{x1D714}^{i}|$ . The numbering in the first column refers to the mode labelling introduced in figure 8.

Valuable information regarding the mechanisms promoting the growth of these modes can be obtained by considering the temporal evolution of the perturbation energy $E_{\unicode[STIX]{x1D6FA}}$ as described by the Reynolds–Orr equation. This equation is derived from (2.5) (see Schmid & Henningson Reference Schmid and Henningson2001) and reads

(3.1) $$\begin{eqnarray}\frac{\text{d}E_{\unicode[STIX]{x1D6FA}}}{\text{d}t}=-\int _{\unicode[STIX]{x1D6FA}}u_{j}^{\prime }u_{k}^{\prime }\frac{\unicode[STIX]{x2202}U_{j}}{\unicode[STIX]{x2202}x_{k}}\,\text{d}\unicode[STIX]{x1D6FA}-\frac{1}{Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}}\int _{\unicode[STIX]{x1D6FA}}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\,\text{d}\unicode[STIX]{x1D6FA}-\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FE}u_{j}^{\prime }u_{j}^{\prime }\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

wherein two similar indices imply summation, and $j,k\in \{1,2,3\}$ . Indices $\{1,2,3\}$ hence refer to the three spatial directions $\{x,y,z\}$ , respectively. The first term on the right-hand side of (3.1) represents the exchange of energy between the perturbation and the base flow (i.e. energy production), the second term represents dissipation of energy by viscosity and the third term corresponds to a volume force (here taken to be the sponge force that makes the disturbance vanish on the inflow and outflow boundaries). By substituting the mode ansatz of § 2.2.2 into (3.1), an equation for the growth rate of a mode may be derived

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{i} & = & \displaystyle -\frac{1}{E_{\unicode[STIX]{x1D6FA}}}\int _{\unicode[STIX]{x1D6FA}}(\hat{u} _{j}^{r}\hat{u} _{k}^{r}+\hat{u} _{j}^{i}\hat{u} _{k}^{i})\frac{\unicode[STIX]{x2202}U_{j}}{\unicode[STIX]{x2202}x_{k}}\,\text{d}\unicode[STIX]{x1D6FA}-\frac{1}{E_{\unicode[STIX]{x1D6FA}}}\frac{1}{Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}}\int _{\unicode[STIX]{x1D6FA}}\left[\left(\frac{\unicode[STIX]{x2202}\hat{u} _{j}^{r}}{\unicode[STIX]{x2202}x_{k}}\right)^{2}+\left(\frac{\unicode[STIX]{x2202}\hat{u} _{j}^{i}}{\unicode[STIX]{x2202}x_{k}}\right)^{2}\right]\,\text{d}\unicode[STIX]{x1D6FA}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{E_{\unicode[STIX]{x1D6FA}}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FE}[(\hat{u} _{j}^{r})^{2}+(\hat{u} _{j}^{i})^{2}]\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

from which the contributions of the different production and dissipation terms to the growth or decay of a given eigenmode can be evaluated. The superscripts $r$ and $i$ refer to the real and imaginary parts, respectively, and the energy of the perturbation in the domain is given by $E_{\unicode[STIX]{x1D6FA}}=\int _{\unicode[STIX]{x1D6FA}}(\hat{u} _{j}^{r2}+\hat{u} _{j}^{i2})\,\text{d}\unicode[STIX]{x1D6FA}$ . (In the stability computations the eigenvectors are normalised to have $E_{\unicode[STIX]{x1D6FA}}=1$ .) The results of evaluating the terms in (3.2) are given in tables 2 and 3, which state the individual production and dissipation mechanisms, respectively.

Figure 11.

Density of the sum of all production (yellow) and dissipation (blue) mechanisms for (a,b) mode I and (c,d) mode II. Contours corresponding to 10 % of the maximum production/dissipation of kinetic energy are plotted together with base flow vortex structures visualised by the $\unicode[STIX]{x1D706}_{2}$ -criterion (Jeong & Hussain Reference Jeong and Hussain1995) (dark grey).

By examining table 2, it is noted that the production of energy generally is dominated by either $-E_{\unicode[STIX]{x1D6FA}}^{-1}\int _{\unicode[STIX]{x1D6FA}}(\hat{v}^{r}\hat{u} ^{r}+\hat{v}^{i}\hat{u} ^{i})\unicode[STIX]{x2202}V/\unicode[STIX]{x2202}x\,\text{d}\unicode[STIX]{x1D6FA}$ or $-E_{\unicode[STIX]{x1D6FA}}^{-1}\int _{\unicode[STIX]{x1D6FA}}(\hat{v}^{r}{\hat{w}}^{r}+\hat{v}^{i}{\hat{w}}^{i})\unicode[STIX]{x2202}V/\unicode[STIX]{x2202}z\,\text{d}\unicode[STIX]{x1D6FA}$ . For mode I and IV it is the former whereas for II and III it is the latter. This implies that energy is extracted from the base flow and transferred to the eigenmodes by vertical and horizontal components of the eigenvectors working against the gradients of the vertical base flow component in the horizontal directions. Similarly, by studying table 3, it is observed that the main mechanisms responsible for dissipation of energy are the horizontal gradients of the vertical eigenvector component, i.e.  $E_{\unicode[STIX]{x1D6FA}}^{-1}Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}^{-1}\int _{\unicode[STIX]{x1D6FA}}[(\unicode[STIX]{x2202}\hat{v}^{r}/\unicode[STIX]{x2202}x)^{2}+(\unicode[STIX]{x2202}\hat{v}^{i}/\unicode[STIX]{x2202}x)^{2}]\,\text{d}\unicode[STIX]{x1D6FA}$ for modes I and IV, and $E_{\unicode[STIX]{x1D6FA}}^{-1}Re_{\unicode[STIX]{x1D6FF}_{h}^{\ast }}^{-1}\int _{\unicode[STIX]{x1D6FA}}[(\unicode[STIX]{x2202}\hat{v}^{r}/\unicode[STIX]{x2202}z)^{2}+(\unicode[STIX]{x2202}\hat{v}^{i}/\unicode[STIX]{x2202}z)^{2}]\,\text{d}\unicode[STIX]{x1D6FA}$ for modes II and III.

In figure 11 the density functions of the sum of all production and dissipation mechanisms for modes I and mode II are visualised. As seen, the plotted structures are symmetric with respect to the $xy$ -plane and have a rather different shape for the two eigenmodes. For the varicose mode I, the production and dissipation of kinetic energy are localised close to the pipe inlet beneath the separation bubble that forms on the upstream portion of the wall. For the sinuous mode II on the other hand, production and dissipation of kinetic energy seem to be associated with both the aforementioned region close to the pipe inlet, and the elongated vortex legs that extend through the pipe.

Since the instability of the eigenmodes I and II is modest, small changes in the numerical set-up may have a large impact on the overall character of the boundary layer. In order to ensure that these modes are converged in terms of resolution, the polynomial order is increased from $N=12$ to $N=15$ (mesh M2). This change doubles the number of grid points, but changes the absolute value of the eigenvalues for the two modes by less than 1 %. (The changes in the absolute value of the eigenvalues corresponding to the pipe modes are $O(0.1\,\%)$ .)

Figure 10 shows that all the encountered modes in the boundary layer extend towards the end of the domain where they are dampened by the sponge function. Recent three-dimensional stability analyses considering the flow around a cylindrical roughness element in a Blasius boundary layer have shown that eigenmodes corresponding to isolated eigenvalues can be rather insensitive to changes in the domain size, given that sufficiently long domains are considered (Loiseau et al. Reference Loiseau, Robinet, Cherubini and Leriche2014). In contrast, eigenvalues corresponding to the flow around a cylindrical roughness element in a Falkner–Skan–Cooke boundary layer have been shown to be impossible to converge with respect to the domain size (Brynjell-Rahkola et al. Reference Brynjell-Rahkola, Shahriari, Schlatter, Hanifi and Henningson2017) due to the presence of acceleration and sweep. In order to verify the sensitivity of this parameter on the present flow case, the streamwise extent of the boundary layer is increased by approximately 25 % (mesh M3). However, this changes the eigenvalues of mode I and II less than $0.1\unicode[STIX]{x2030}$ in absolute value, which suggests that the instability mechanisms of the boundary layer are independent of the box length and localised in the domain. (The changes in the absolute value of the eigenvalues corresponding to the pipe modes are $O(0.01\unicode[STIX]{x2030})$ .)

3.4 Sensitivity

Next, the origin of the instability reported in § 3.3 is investigated. This is done through a structural sensitivity analysis as introduced by Giannetti & Luchini (Reference Giannetti and Luchini2007). These authors considered a spatially localised perturbation to the linearised Navier–Stokes equations (2.5) in the form of a force–velocity coupling, and they derived an upper bound for the resulting eigenvalue drift due to this perturbation. Following their analysis, regions in space where the flow is sensitive and where a localised force feedback thus will create a large change in a given eigenvalue, may be identified by evaluating the expression

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D707}(\boldsymbol{x})=\Vert \hat{\boldsymbol{u}}^{\dagger }(\boldsymbol{x})\Vert \Vert \hat{\boldsymbol{u}}(\boldsymbol{x})\Vert ,\end{eqnarray}$$

for a pair of direct and adjoint eigenvectors subject to the normalisation

(3.4a,b ) $$\begin{eqnarray}\max _{\boldsymbol{x}\in \unicode[STIX]{x1D6FA}}\{|\hat{u} (\boldsymbol{x})|\}=1,\quad \int _{\unicode[STIX]{x1D6FA}}|\hat{\boldsymbol{u}}^{\dagger }\boldsymbol{\cdot }\hat{\boldsymbol{u}}|\,\text{d}\unicode[STIX]{x1D6FA}=1,\end{eqnarray}$$

( $\Vert \cdot \Vert$ is the Euclidean norm in $\mathbb{C}^{3}$ ). As argued by Chomaz (Reference Chomaz2005) and Giannetti & Luchini (Reference Giannetti and Luchini2007), regions where $\unicode[STIX]{x1D707}(\boldsymbol{x})$ is non-zero may be ascribed the role of a ‘wavemaker’ that indicates where in space the instability mechanism resides. Equation (3.3) and (3.4) evaluated for mode I is visualised in figure 12. As seen, the ‘wavemaker’-region is localised inside the pipe beneath the separation bubble. It is symmetric about the $xy$ -plane and embraces the vertical elongated vortices that form in the pipe and were seen to break down in figure 6. Comparing the isocontours corresponding to 1 % and 10 % of the maximum value, it is noted that the sensitivity increases towards the centre of the pipe. This implies that the ‘core’ of the instability lies on the inner sides of the two vortices, and thus that a small control device will have the largest impact on the instability if placed between the two vortex legs. The corresponding region for mode II has a similar shape, strength and location.

Figure 12.

Structural sensitivity. Contours of $\unicode[STIX]{x1D707}$ corresponding to 10 % (blue) and 1 % (yellow) of the maximum sensitivity for mode I are plotted together with base flow vortex structures visualised by the $\unicode[STIX]{x1D706}_{2}$ -criterion (Jeong & Hussain Reference Jeong and Hussain1995) (dark grey).

Figure 13.

Sensitivity to momentum forcing. Contours corresponding to 5 % (yellow), 15 % (blue) and 50 % (black) of the maximum value of (a) $\Vert \hat{\boldsymbol{u}}_{III}^{\dagger }\Vert$ and (b) $\Vert \hat{\boldsymbol{u}}_{IV}^{\dagger }\Vert$ are plotted ( $\Vert \cdot \Vert$ is the Euclidean norm in $\mathbb{C}^{3}$ ).

Since mode III and mode IV are asymptotically stable, they will ultimately decay in time unless they are excited by a harmonic driving force. It is therefore of interest to also consider the sensitivity of these modes to forcing. As described by Hill (Reference Hill1995), the sensitivity of a given flow to momentum sources may be assessed by studying the adjoint eigenvectors. The adjoint eigenvectors exhibit the same symmetry as their corresponding direct eigenvectors. In figure 13 the norm of the adjoint eigenvectors corresponding to modes III and IV (here denoted $\hat{\boldsymbol{u}}_{III}^{\dagger }$ and $\hat{\boldsymbol{u}}_{IV}^{\dagger }$ , respectively) are visualised. By comparing the spatial distribution of the sensitivity of the two modes, it is seen that both are susceptible to momentum disturbances that enter through the sucked streamtube. However, the low frequency mode appears more sensitive to lateral forcing, whereas the high frequency mode is more sensitive to forcing directly in front of the hole. In addition to the boundary layer, the adjoint eigenvectors are also seen to reach into the pipe and cover the pipe junction and the separation bubble. Due to the presence of the unstable modes, excitation of the lightly dampened modes III and IV can thus be expected to appear through momentum forcing by the unsteady fluctuations in the pipe and around the pipe orifice. As a result, these modes, although asymptotically stable, are likely to be excited and contribute to the transition process in the boundary layer.

Figure 14.

An instantaneous snapshot of the unsteady disturbance component of a nonlinear DNS. Contours of the streamwise velocity component $u-U$ corresponding to $\pm 10^{-4}$ are plotted in grey and black colours, respectively.

3.5 Analysis of the nonlinear flow

With this insight into the stability and receptivity of the flow, it is instructive to study how well the results from the linear analysis apply to the nonlinear flow, and to what extent they can be used to interpret the transition scenario observed. To obtain a first qualitative picture of the disturbance field present in the DNS, the steady base flow solution is subtracted from the instantaneous velocity field shown in figure 3(b). The resulting unsteady component of the flow is plotted in figure 14. This shows the presence of an unsteady perturbation with spatially varying wavelength in the aft of the suction hole that is reminiscent of the modes shown in figure 10. Note that the velocity in the DNS and the base flow is $O(1)$ , and thus that the strength of the perturbation in the boundary layer is comparable to the relative strength of the eigenvectors in the boundary layer.

A brief comparison of the frequency content in the DNS and in the linear analysis was presented in figures 7 and 8. However, in order to make any strong conclusions regarding the transition of the boundary layer flow, it is important to verify that the disturbance in the boundary layer has the characteristics of the linear eigenmodes I–IV. To this end, a Koopman analysis as described in § 2.3 is carried out. Given the large amplitude ratio of the disturbance between the pipe and the boundary layer, this analysis will be limited to a smaller subset of the domain, namely $[-25,40]\times [0,15]\times [-15,15]$ . This choice will prevent the strong fluctuations in the pipe as well as in the downstream portions of the plate from contaminating and overshadowing the relevant linear mechanisms that initiate the transition process in the boundary layer. Around 300 snapshots are sampled consecutively in time from the nonlinear DNS with a separation of $\unicode[STIX]{x0394}\tilde{t}=0.48$ . (The columns of $\unicode[STIX]{x1D730}_{0}^{m}$ span an approximate subspace with a residual whose Euclidean norm is below $10^{-5}$ .) The time integration is started from the flow field visualised in figure 3(b), which corresponds to an arbitrary time instance after transients have washed away.

The spectral content of the flow in the specified subdomain is plotted in figure 15. As seen, the flow features several amplified frequencies. Among the frequencies, the three most amplified ones are seen to be (in order) $\tilde{\unicode[STIX]{x1D714}}=0.2258$ , $0.1112$ and $0.1933$ , which are in reasonable agreement with the frequencies of the modes I, III and II that were obtained in the linear analysis, i.e.  $\unicode[STIX]{x1D714}^{r}=0.2155$ , $0.0913$ and $0.2062$ , respectively.

Figure 15.

Frequency spectrum of the Koopman operator (the mean flow is omitted). The amplitudes $|D_{k}|$ of the different frequencies are computed from (2.17) and normalised by that of the mean flow, $|D_{1}|$ .

In figure 16, the shape of the Koopman modes corresponding to these frequencies are visualised. In figures 16(a) and 16(b), the modes corresponding to the frequencies $\tilde{\unicode[STIX]{x1D714}}=0.1933$ and $\tilde{\unicode[STIX]{x1D714}}=0.2258$ are shown, respectively. As seen, the former closely resemble the sinuous mode visualised in figure 10(b). The latter on the other hand appears quite noisy and is somewhat difficult to interpret. It appears to be a combination of a sinuous and a varicose mode. Since the modes I and II from the linear analysis have similar frequencies and growth rates, it is possible that the DMD algorithm is unable to fully distinguish between the two, which might explain its distorted structure. In figure 16(c) the mode corresponding to frequency $\tilde{\unicode[STIX]{x1D714}}=0.1112$ is shown. The shape of this mode is again in good agreement with the sinuous mode III visualised in figure 10(c). The Koopman analysis is also able to identify the frequency $\tilde{\unicode[STIX]{x1D714}}=0.4464$ , which is very close to that of the linear mode IV with $\unicode[STIX]{x1D714}^{r}=0.4469$ . However, as seen in figure 10(d), this mode is relatively weak in the subdomain considered here ( $-25\leqslant x\leqslant 40$ ) and can therefore not be expected to appear as a peak in the current analysis.

Figure 16.

Streamwise velocity component of the Koopman modes corresponding to the frequencies (a) $\tilde{\unicode[STIX]{x1D714}}=0.1933$ , (b) $\tilde{\unicode[STIX]{x1D714}}=0.2258$ and (c) $\tilde{\unicode[STIX]{x1D714}}=0.1112$ . Contours corresponding to $\pm 1\,\%$ of the maximum velocity are plotted in grey and black colours, respectively.