3.1 Robustness of the minimal seed

3.1.1 Changes to the base flow

Inspired by Kühnen et al.’s (Reference Kühnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b ) experiments, we first study the minimal seed for transition with a modified base flow. By adding a suitable body force, the laminar base profile becomes more ‘plug-like’, i.e. flatter in the centre of the pipe, than the parabolic profile of HPF. Kühnen et al. (Reference Kühnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b ) considered an initially fully turbulent state and showed that, by flattening the streamwise velocity profile, they were able to completely relaminarise the flow. Here, instead, our aim is to investigate how the basin of attraction of the laminar state is affected by flattening of the base flow and whether this modification affects the minimal seed, i.e. the closest point of approach of the edge to the laminar state shown schematically in figure 1.

Following Kühnen et al. (Reference Kühnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b ) we use the following family of profiles for the base flow ${\mathcal{W}}(r;\unicode[STIX]{x1D6FF},c)\hat{\boldsymbol{z}}$ (see their equation (19)), where

(3.1) $$\begin{eqnarray}{\mathcal{W}}(r;\unicode[STIX]{x1D6FF},c)=(1-\unicode[STIX]{x1D6FF})\left[1-\frac{\cosh (cr)-1}{\cosh (c)-1}\right].\end{eqnarray}$$

The parameter $\unicode[STIX]{x1D6FF}$ is the centreline difference between the laminar profile and the target profile and $c$ is set by the constant mass-flux condition. The force $F=F(r)\hat{\boldsymbol{z}}$ required to generate such a target velocity profile is obtained by substituting ${\mathcal{W}}(r;\unicode[STIX]{x1D6FF},c)$ in the Navier Stokes equations, i.e. $F(r):=\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{N}\boldsymbol{S}$ . For example, the case with $\unicode[STIX]{x1D6FF}=0.2$ and $c=3.5935$ is shown in figure 2. The forcing decelerates the flow in the central part and accelerates it near the wall while the mass flux is kept fixed.

Figure 2. (a) Comparison of the parabolic profile (red solid line) and the forced profile (green dashed line) of the laminar flow. (b) The shape of the force chosen to decrease the centreline velocity of the laminar flow by 20 % at $Re=1750$ (green dashed line). The force is negative (pointing upstream) in the centre and positive (pointing downstream) near the wall. For comparison, the pressure gradient in a laminar flow $|\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z|=4/Re$ is also shown (red solid line). Similar forced profiles and forcings are reported by Hof et al. (Reference Hof, De Lozar, Avila, Tu and Schneider2010) (figure S2 of their supplementary material) and Kühnen et al. (Reference Kühnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b ) (figure 7 of their extended data).

The effect of the forcing is studied for the parameters corresponding to the works by PK10 and PWK12 as summarised in table 1. As discussed in PWK12, the choice of parameters in PK10 was not ideal: the Reynolds number is close to the first appearance of turbulent state, the target time is short and as a result the algorithm struggles to discern between conditions that relaminarise and conditions that trigger turbulence, and in a tightly constrained geometry the basin boundary is highly fractal. Nevertheless it is useful here to show the effect of the global forcing.

Table 1. Parameters and resolution for PK10 and PWK12 cases.

Figure 3 shows the maximum growth (at the target time $T_{opt}$ ) as a function of $E_{0}$ for the cases with forced and parabolic base profile. The initial energy is gradually increased until $E_{c}$ is reached.

Figure 3. Comparison of $G$ versus $E_{0}$ with parabolic and forced base profile for (a) PK10 and (b) PWK12 cases (refer to table 1). For each case, the thick black cross indicates the initial energy $E_{fail}$ at which the optimisation algorithm first fails to converge, as we gradually increase $E_{0}$ . Up to the last data point before $E_{fail}$ we were able to converge the algorithm. The critical energy of the minimal seed is thus bracketed between these two values of $E_{0}$ . Due to the convergence issues explained in the text and in PWK12, the search for $E_{c}$ in the cases with a flattened base profile was not refined as much as in the unforced cases.

Most of the data points have been verified by feeding the algorithm with at least two or three different initial conditions (for example, a snapshot from a turbulent run, another NLOP at a lower $E_{0}$ or a turbulence-inducing initial conditions at a higher $E_{0}$ ). The initial energies $E_{fail}$ at which the optimisation algorithm first fails to converge are marked with a black cross in figure 3. According to the conjecture 1 of PWK12, these correspond to the critical initial energies $E_{c}$ , where the edge touches the energy hypersurface at one velocity state. Due to the reasons mentioned above, in the PK10 case, especially for $\unicode[STIX]{x1D6FF}=0.2$ , convergence is sometimes not clear and deteriorates (becomes slower and slower) as $E_{0}$ is increased and approaches $E_{fail}$ . The last data points before $E_{fail}$ , for both values of $\unicode[STIX]{x1D6FF}$ , appear to show convergence, but there still remains some doubt even after running the algorithm for more than 1000 iterations. Furthermore, the perturbations corresponding to $E_{0}=E_{fail}$ decay immediately after the time $T_{opt}$ is reached, because at this low Reynolds number turbulence is intermittent and appears only in the form of decaying puffs.

For the PWK12 case, convergence is clearer than in the PK10 case due to the larger domain, longer integration time and higher Reynolds number. However, in the forced case at $E_{0}=6.28\times 10^{-5}$ (last data point before $E_{fail}$ ) neither a smooth convergence nor a clear increase of the residual was obtained, even after 1000 iterations. Note also that in the unforced case, $G$ sharply increases when the critical initial energy is reached, while with a flattened base profile, the increase of $G$ is much more gradual, as this case behaves similarly to cases where the system is close to the marginal $Re$ (as in PK10 case discussed above).

Despite these convergence issues, figure 3 shows that by flattening the base profile, $E_{c}$ is moved towards higher values of the initial energy and the maximum growth reached at time $T_{opt}$ is decreased, i.e. the unforced curve $G=G(E_{c})$ is shifted ‘down’ and ‘right’ as $\unicode[STIX]{x1D6FF}$ is increased. Therefore, the presence of the forcing expands the basin of attraction of the laminar base profile and reduces transient growth. For example, for the PWK12 case, the critical energy of the minimal seed moves from $E_{0}=3.73\times 10^{-5}$ to $E_{0}=6.5\times 10^{-5}$ and for $E_{0}<E_{c}$ the NLOP of the forced case reaches less than half of the growth of the unforced case. The energy time series of the minimal seeds for the parabolic and forced cases are shown in figure 4(a). These initial conditions clearly lead to a turbulent episode which survives for at least double the optimisation time. The case $E_{0}=6.28\times 10^{-5}$ for which convergence was critical is also shown. This initial state seems turbulent at the optimisation time but decays straight after.

Figure 4. (a) Time series of energy for the minimal seeds of PWK12 case (parameters provided in table 1) with forced (green dashed line) and unforced (red solid line) base profiles. In the forced case, the time evolution of the NLOP just before $E_{fail}$ is also shown (green dash-dotted line). Although in this case the convergence of the optimisation algorithm was not clear, the decay of the energy straight after $T_{opt}$ (indicated by the vertical dotted line) suggests that this disturbance does not lead to a turbulent episode. The cross-sections of the minimal seeds in $(b)$ the forced (green dashed border) and $(c)$ unforced (red solid border) cases are also shown. Contours indicate axial velocity perturbations (with the laminar flow subtracted off): white or light for positive perturbation, red or dark for negative, outside shade corresponds to zero. The arrows indicate cross-sectional velocities.

Despite the critical initial energy being significantly increased with a flattened base profile, the fully localised structure of the minimal seed remains largely unchanged, as shown in the cross-sections of figures 4(b,c). Therefore, the structure of the minimal seed is found to be fairly robust to changes to the base flow. This is different from Rabin et al.’s (Reference Rabin, Caulfield and Kerswell2014) study of oscillated plane Couette flow where, instead, qualitative changes in the structure of the minimal seed are found when compared to the non-oscillated case. In their study, however, the basic fluid response in the presence of spanwise oscillations becomes time dependent through the additional spanwise component (refer to their equation (2.1)), while in our case only the shape of the laminar flow profile is modified, its dimension (one) and dependencies (only radial) remain unchanged.

Figure 5. Critical initial energy as a function of the filtering ratio ${\mathcal{F}}$ for the unforced PWK12 case (parameters provided in table 1). Insets: cross-sections of the minimal seeds at different values of ${\mathcal{F}}=0$ , $0.75$ and $0.93$ . Contours indicate streamwise velocity beyond the laminar flow (white or light for positive, red or dark for negative) and arrows indicate cross-sectional velocities.

3.1.2 Filtering

In order to assess how robust the minimal seed is to smoothening, we perform the full optimisation procedure over perturbations in a lower-dimensional space for the set of parameters corresponding to the (unforced) PWK12 case. Specifically, at each iteration we project the initial condition onto a subspace where only the first $K_{f}\times M_{f}$ wavenumbers are retained. The resolution in the forward and backward steps is, however, unchanged (i.e.  $K=36$ , $M=32$ ). As a measure of how much we are truncating we introduce the filtering ratio ${\mathcal{F}}$ defined as

(3.2) $$\begin{eqnarray}{\mathcal{F}}=1-\sqrt{\frac{K_{f}}{K}\frac{M_{f}}{M}}.\end{eqnarray}$$

Figure 5 shows the critical energy of the minimal seed as a function of the filtering ratio, where ${\mathcal{F}}=0$ means no filtering (i.e. fully resolved minimal seed) and ${\mathcal{F}}=1$ would imply that no perturbation remains. Note that, for a fixed value of ${\mathcal{F}}$ , values for $K_{f}$ and $M_{f}$ are chosen so that the corresponding ratios $K_{f}/K$ and $M_{f}/M$ are as close as possible, i.e. to avoid cases where the filtering in one direction is much higher than in the other direction. The critical initial energy of the minimal seed remains almost unchanged for ${\mathcal{F}}\leqslant 0.75$ , that is, when only retaining 25 % of the modes. Since we are restricting the initial condition to a subset of the energy hypersurface by adding the filtering constraint, one would expect the minimal seed to occur at a higher initial energy. Our results thus suggest that the edge is locally quite flat near the minimal seed. As ${\mathcal{F}}$ is further increased, larger initial energies are needed in order to trigger turbulence. For example, for ${\mathcal{F}}=0.93$ (i.e. only the first $3\times 2$ modes retained) $E_{c}$ is almost an order of magnitude larger than in the fully resolved case. For values of ${\mathcal{F}}$ larger than this (for example we have tested ${\mathcal{F}}=0.96$ where only $2\times 1$ modes are retained) it seems that it is not possible to trigger transition. As shown by the cross-sections in the insets of figure 5, the structure of the minimal seed remains almost unchanged when the filter is applied, even for cases where the minimal seed occurs at larger $E_{c}$ than in the unfiltered case (refer to the last inset to the right).

Figure 6. Energy thresholds $E_{c}$ versus $Re=2400$ , 3500, 5000, 7000 and 10 000 for different types of initial conditions: turbulent snapshots (cyan diamonds), artificially generated global (purple plusses) or localised (blue crosses) random fields and the minimal seed (black plus symbols correspond to the largest tested value of $E_{0}$ below which transition never occurs, while the red squares correspond to the smallest tested value of $E_{0}$ for which transition occurs at least once). The green stars and the yellow circles are the energies of the minimal seed after the Orr and oblique phases, respectively. The lines represent the power-law scalings obtained by fitting the last four data points. Insets: cross-sections of a typical global (purple dash-dotted border) and localised (dashed blue border) random initial condition. Contours indicate streamwise velocity beyond the laminar flow (white or light for positive, red or dark for negative) and arrows indicate cross-sectional velocities.

This study shows that the minimal seed is robust to quite severe spectral filtering, i.e. the small-scale structure of the minimal seed is not important. To confirm this, we also looked to see where the energy of the fully resolved minimal seed is distributed across the Fourier modes in $\unicode[STIX]{x1D703}$ and $z$ . For ${\mathcal{F}}=0.75$ and $0.93$ we retain ${\approx}96\,\%$ and $23\,\%$ of the energy, respectively. It thus appears that, despite the fully localised structure of the minimal seed, most of the energy is carried by the large-scale modes.

It is possible that radial filtering may have a more significant effect on the energy thresholds, but the structure remains essentially similar, thus suggesting that truncation is not a simple route towards a new set of more ‘typical’ initial conditions. Conversely, it shows that the structure does not have to be perfectly formed to be the optimal. From this it seems reasonable that the minimal seed might be realised in a laboratory, but whether it would be ‘naturally’ realised among ‘ambient’ disturbances remains unclear at this stage.

3.2 Statistical study of transition to turbulence

To establish whether the minimal seed may be used to model typical ambient perturbations, we compare the critical initial energy of the minimal seed with that of a random disturbance. The latter is generated by scattering energy randomly over the subset of wavenumbers $K\times M=12\times 9$ . At $Re=2400$ this truncation corresponds to ${\mathcal{F}}\approx 0.7$ , before the rapid increase in $E_{c}$ (figure 5), where the minimal seed could still be captured quite faithfully. An arbitrary complex amplitude $A_{n}$ is generated for each of the spectral modes in the chosen subset and the radial dependence introduced in the form $\sum _{n=1}^{N_{s}}A_{n}r\sin (n\unicode[STIX]{x03C0}r)$ with $N_{s}=5$ (the projection scheme integrated into the time stepping algorithm is used to ensure that the initial condition is solenoidal). These initial conditions are fed into direct numerical simulation (DNS) with time integration $T=125(D/\overline{W})$ . Five Reynolds numbers are considered: $Re=2400$ , 3500, 5000, 7000 and 10 000, with the numerical resolution appropriately enlarged for increasing Reynolds number. The same subset of wavenumbers used at $Re=2400$ is employed for all the other Reynolds number considered. For each $Re$ we consider 10 to 12 distinct random initial conditions and for each of them we gradually increase $E_{0}$ until turbulence is triggered and sustained for a time $T(D/\overline{W})$ . The criterion to check for relaminarisation is $E_{3d}<10^{-8}$ , where $E_{3d}$ is the energy associated with the streamwise-dependent modes only, as this quantity decays very rapidly when the flow relaminarises.

An analogous statistical study is carried out using 10 different snapshots from a turbulent run as initial condition (in a similar fashion to Schneider & Eckhardt (Reference Schneider and Eckhardt2008)). Both the random and snapshot initial conditions are global disturbances, while the minimal seed is fully localised. A third set of random localised initial conditions is obtained by scattering energy randomly over the wavenumbers as above, then by multiplying this global disturbance by a smooth spatial windowing function ${\mathcal{B}}(z)$ and ${\mathcal{B}}(\unicode[STIX]{x1D703})$ so that the disturbance occupies only $1/5$ and $1/3$ of the streamwise and azimuthal domain, respectively. For the range of Reynolds numbers considered here we do not observe a strong localisation of the minimal seed in the radial direction and therefore we do not localise the random disturbance in this direction. The smoothing function is defined using equation 8 of Yudhistira & Skote (Reference Yudhistira and Skote2011). For example, for the localisation in the streamwise direction we use:

(3.3) $$\begin{eqnarray}{\mathcal{B}}(z)=g\left(\frac{z-z_{start}}{\unicode[STIX]{x0394}z_{rise}}\right)-g\left(\frac{z-z_{end}}{\unicode[STIX]{x0394}z_{fall}}+1\right),\end{eqnarray}$$

with

(3.4) $$\begin{eqnarray}g(z^{\ast })=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }z^{\ast }\leqslant 0\\ \{1+\exp [1/(z^{\ast }-1)+1/z^{\ast }]\}^{-1}\quad & \text{if }0<z^{\ast }<1\\ 1\quad & \text{if }z^{\ast }\geqslant 1,\end{array}\right.\end{eqnarray}$$

where $z_{start}=2L/5$ and $z_{end}=3L/5$ indicate the spatial extent over which the disturbance is non-zero, and $\unicode[STIX]{x0394}z_{rise}=\unicode[STIX]{x0394}z_{fall}=L/10$ are the rise and fall distances of the disturbance. A similar expression to (3.3) is used for ${\mathcal{B}}(\unicode[STIX]{x1D703})$ with $\unicode[STIX]{x1D703}_{start}=2\unicode[STIX]{x03C0}/3$ , $\unicode[STIX]{x1D703}_{end}=4\unicode[STIX]{x03C0}/3$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{rise}=\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{fall}=\unicode[STIX]{x03C0}/3$ .

The critical initial energies of the minimal seeds are found with a two-digit accuracy for all the Reynolds numbers considered, except for $Re=10\,000$ for which only one-digit accuracy is reached due to large computational cost of the simulations. The curves $E_{c}=Re^{-\unicode[STIX]{x1D6FE}}$ obtained by fitting critical energies found for the turbulent snapshot, global and localised random sets of data points and for the minimal seed are shown in figure 6. The data points at $Re=2400$ are not used to obtain the fittings as turbulence can still be transient at this relatively low Reynolds number. A power-law exponent $\unicode[STIX]{x1D6FE}\approx 2.8$ for the minimal threshold energy is obtained. The same exponent was obtained experimentally by Peixinho & Mullin (Reference Peixinho and Mullin2007) and later confirmed numerically by Mellibovsky & Meseguer (Reference Mellibovsky and Meseguer2009) using ‘push–pull’ perturbations with constant flow rate. Minimal energies for transition to turbulence were also calculated by Duguet et al. (Reference Duguet, Monokrousos, Brandt and Henningson2013) for plane Couette flow (refer to their figure 3), by Cherubini, De Palma & Robinet (Reference Cherubini, De Palma and Robinet2015) for the asymptotic suction boundary layer (refer to their figure 9) and, very recently, by Z. Huang, C. P. Caulfield and G. Xi (personal communication) for plane channel flow. Our exponent is close to the $\unicode[STIX]{x1D6FE}\approx 2.7$ obtained by Duguet et al. (Reference Duguet, Monokrousos, Brandt and Henningson2013) for plane Couette flow and $\unicode[STIX]{x1D6FE}\approx 3$ found by Huang et al. for plane Poiseuille flow. Furthermore, our study is reminiscent of figure 19 in Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998), where the streamwise-vortex and oblique-wave transition scenarios for plane channel flows are compared to two-dimensional linear optimals and noise, although the power-law exponents are not reported in their study. The minimal seed curve $E_{c}=Re^{-2.8}$ is steeper than those pertaining to all the other forms of disturbances considered. The power-law exponent $\unicode[STIX]{x1D6FE}\approx 2$ for the turbulent snapshot recalls the scaling found by Hof et al. (Reference Hof, Juel and Mullin2003) and Peixinho & Mullin (Reference Peixinho and Mullin2007) in their experiments using small impulsive jets to perturb the flow. For the random global and localised disturbances, larger exponents than for the turbulent snapshot are obtained, i.e.  $\unicode[STIX]{x1D6FE}\approx 2.2$ and $2.3$ , respectively (refer to table 2 for a summary of the different exponents $\unicode[STIX]{x1D6FE}$ discussed above). Most noticeably, the critical energy of the minimal seed is almost three orders of magnitude lower than that of the global disturbances considered here. With localisation of the random disturbances, the critical energy drops by almost an order of magnitude with respect to the global initial conditions, but it is still significantly larger than that of the minimal seed.

Table 2. Power-law scalings $E_{c}=Re^{-\unicode[STIX]{x1D6FE}}$ obtained in the present study and in the literature for different types of disturbances and in different geometries (HPF: Hagen–Poiseuille flow, PCF: plane Couette flow, ASBL: asymptotic suction boundary layer, PPF: plane Poiseuille flow). For asymptotically large Reynolds number we expect to see scaling with wall units, which leads to an exponent $\unicode[STIX]{x1D6FE}=2.5$ .

Although the minimal seed has a very peculiar structure, which might be unlikely to be generated in a laboratory, it evolves to a structure that looks much more familiar to a ‘natural’ disturbance over a relatively short time scale corresponding to the Orr and oblique-wave phases. The early stages of the minimal seed evolution are shown in the three-dimensional visualisations of figure 7 at $Re=2400$ .

Figure 7. Snapshots showing isocontours of streamwise perturbation velocity during the early stages of the time evolution of the minimal seed for $Re=2400$ . (a)  $t=0$ , (b)  $t=1.1D/\overline{W}$ (end of the Orr phase) and (c)  $t=2.5D/\overline{W}$ (end of the oblique phase). The isocontours in each snapshot correspond to 50 % of the maximum (light/yellow) and 50 % of the minimum (dark/red) of the streamwise perturbation velocity in the pipe at that time.

The end of the Orr phase is identified by analysing the flow topology of the NLOP close to $t=0$ : the streaks that are initially tightly layered and inclined back into the oncoming flow, are tilted away from the wall by the Orr mechanism and slightly separated. In a short pipe, as in the PK10 case, the evolution of the flow structure during the Orr phase is relatively clearer to visualise than for longer pipes (refer to figure 1 of PK10 and our figure 7). Therefore, there is some discretion in our choice of the energy size at the end of the Orr phase, but this has a negligible effect on our ensuing discussion. As in the short-pipe case of PK10, our data suggest that the Orr phase occurs in a very short time scale ( $0<t(D/\overline{W})\lesssim 0.5{-}1$ ) and gives rise to an initial spurt of energy growth. By the end of the Orr phase, the helical modes starts to become dominant, thus signalling that the oblique-wave mechanism has come into play. In PK10 the end of the oblique-wave phase was clearly signalled by the ‘shoulder’ in the time evolution of the total perturbation energy $E$ and the corresponding ‘bump’ in the time evolution of the three-dimensional energy $E_{3d}$ , as reproduced in figure 8(a). For the present choice of $L=5D$ , which is approximately three times longer than that in PK10, the ‘shoulder’ in the energy time series is not clearly distinguishable and $E_{3d}$ does not decay straight after the peak because of the longer length-scale modes (refer to red solid and green dashed lines in figure 8 b). However, by defining a three-dimensional energy $\tilde{E}_{3d}=E_{3d}(k>3)$ that includes only energy from streamwise wavenumbers $k>3$ , we are able to observe the ‘bump’ in $\tilde{E}_{3d}$ (refer to the blue dash-dotted line in figure 8 b), where we identify the end of the oblique phase. Both the time scales of the Orr ( $0<t(D/\overline{W})\lesssim 0.5{-}1$ ) and oblique-wave ( $0.5{-}1\lesssim t(D/\overline{W})\lesssim 2.5{-}3.5$ ) phases are found to be similar to the PK10 case and to remain almost unchanged with Reynolds number.

Figure 8. Time evolution of the total perturbation energy $E$ (red solid line) and the energy associated with the streamwise-dependent modes only $E_{3d}$ (green dashed line) in $(a)$ the case of a short pipe $L=0.5\unicode[STIX]{x03C0}D$ at $Re=1750$ (from PK10) and in $(b)$ the case of a longer pipe ( $L=5D$ ) at $Re=5000$ (present study). In the latter case the time evolution of $\tilde{E}_{3d}=E_{3d}(k>3)$ is also displayed (blue dash-dotted line) to show the characteristic ‘bump’ towards the end of the oblique phase.

If we compare the energies of the minimal seed after the Orr and oblique phases with the random localised initial conditions (refer to figure 6), then the gap is reduced to less than an order of magnitude in the former case and, most noticeably, to practically zero in the latter case.

Comparing the scaling of the critical initial energy of the minimal seed with the scaling of its energy at the end of the Orr phase suggests that the growth produced via the Orr mechanism is independent of the Reynolds number, as expected due to the inviscid nature of the Orr process. The growth produced via the oblique-wave mechanism is almost of $O(Re)$ . While the growth factor of $O(Re^{2})$ for the maximum linear transient growth due to the lift-up mechanism is well documented (e.g. Schmid & Henningson Reference Schmid and Henningson2012), to the best of our knowledge, this is the first time that scaling laws are obtained numerically for the Orr and oblique-wave mechanisms.

As shown by PWK12, the NLOP tracks the laminar–turbulent boundary $\unicode[STIX]{x1D6F4}$ before either relaminarising or triggering turbulence. The two bracketing cases shown in figure 6 as black plusses (relaminarising disturbances) and red squares (turbulence-inducing disturbances) are further refined with bisections until the difference in the initial energies is less than 0.005 %. These refined bracketing trajectories are shown in figure 9(a) for the range of Reynolds numbers considered and provide evidence of an edge tracked by the minimal seed. Our data (figure 9 b) suggest that the energy of an edge state $E_{\unicode[STIX]{x1D6F4}}$ decreases with increasing $Re$ , approximately as $Re^{-1}$ .

Figure 9. (a) Trajectories close to the edge (solid lines: ‘just above’, dashed line: ‘just below’) at different Reynolds numbers in the range $2400{-}10\,000$ . (b) Energy of the edge as a function of $Re$ . At each Reynolds number, $E_{\unicode[STIX]{x1D6F4}}$ is calculated as the point on the laminar trajectory where the normalised difference from the turbulent trajectory becomes greater than 5 %. The values of $E_{\unicode[STIX]{x1D6F4}}$ are also marked with black circles in (a). The dashed curve $Re^{-1}$ is added to show that our scaling assumption is reasonable. This power-law exponent is not calculated by fitting the data but the curve is shifted so that it passes closer to the last three data points (consistent with figure 6).

As suggested by Kerswell (Reference Kerswell2018) (refer also to appendix B of Kerswell, Pringle & Willis (Reference Kerswell, Pringle and Willis2014)), the minimal seed couples together (via the nonlinear effects) the Orr, oblique-wave and lift-up mechanisms, which occur on different time scales and are uncoupled in the linearised dynamics. By ensuring that the energy of the preceding phase feeds into the following, the minimal seed is thus able to produce a much larger overall growth than any possible in the linearised problem. Our data support this picture and suggest that the oblique-wave process produces a growth of almost $O(Re)$ , which is then further magnified by the lift-up mechanism up to an edge state whose energy scales approximately as ${\sim}Re^{-1}$ . From this, it follows that the lift-up mechanism only produces an energy growth of approximately $O(Re)$ , rather than the usually quoted growth factor of $O(Re^{2})$ . A possible explanation follows from the length scales of the minimal seed becoming finer (and thus the rolls experiencing more dissipation) as the Reynolds number increases. This is evidenced by the cross-sections shown in figure 10 of the time evolution of the minimal seed up to the beginning of the lift-up phase for $Re=2400$ and 10 000. Rolls advect the mean shear to drive high and low-speed streaks. The diffusion term for a roll of spanwise wavelength $\ell$ suggests that such a roll survives a time ${\sim}Re\,\ell ^{2}$ . For a shear of $O(1)$ , the growth in amplitude of a streak is then ${\sim}Re\,\ell ^{2}$ . The usual argument with $\ell =O(1)$ then implies an energy growth ${\sim}Re^{2}$ . Here, an energy growth ${\sim}Re$ suggests a length scale $\ell \sim Re^{-1/4}$ .

It appears that we do not yet see scaling with wall units, $\ell ^{+}=Re_{\unicode[STIX]{x1D70F}}\ell$ and $u^{+}=Re/Re_{\unicode[STIX]{x1D70F}}u$ , where $Re_{\unicode[STIX]{x1D70F}}\equiv u_{\unicode[STIX]{x1D70F}}R/\unicode[STIX]{x1D708}$ is the wall Reynolds number ( $R$ is the radius of the pipe) and $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D708}\unicode[STIX]{x2202}\tilde{w}/\unicode[STIX]{x2202}r}_{|wall}$ is the friction velocity. For a localised perturbation ( $l^{+}$ constant) we have $Re_{\unicode[STIX]{x1D70F}}\approx \sqrt{2Re}$ . This would suggest a scaling $\ell \sim Re^{-1/2}$ . In the early stages of the minimal seed dynamics, however, assuming that the energy in wall units $E^{+}$ of a localised perturbation is constant leads to a energy scaling $E\sim Re^{-2.5}$ . At finite Reynolds number we observe a scaling of $E\sim Re^{-2.8}$ .

The energy of the minimal seed at the end of the oblique phase and the critical initial energy of the localised random disturbances are similar over the range studied. Furthermore, their flow topology is also broadly similar, as can be seen by comparing the cross-section shown in the bottom inset of figure 6 for a typical random localised initial condition with those shown in the rightmost panel of figure 10 for the minimal seed at the end of the oblique phase. In this sense, we will regard the minimal seed at the end of the oblique phase as a reasonable proxy for the transition threshold for random disturbances.

Figure 10. (a) Time evolution of the minimal seed for $Re=10\,000$ . From left to right: $t=0$ , $t=1.25D/\overline{W}$ (end of the Orr phase) and $t=3.75D/\overline{W}$ (end of the oblique phase). (b) Time evolution of the minimal seed (rotated by $\unicode[STIX]{x03C0}$ ) for $Re=2400$ . From left to right: $t=0$ , $t=1.1D/\overline{W}$ (end of the Orr phase) and $t=2.5D/\overline{W}$ (end of the oblique phase). The cross-sections are taken at the streamwise location of maximum enstrophy. In the left half-sections ( $t=0$ ) and right half-sections (end of oblique phase), ten contours are used between the extremes of the corresponding streamwise velocity perturbations. The central half-sections (end of Orr phase) are scaled with the extremes of the corresponding streamwise velocity perturbations at the end of the oblique phase (right half-sections). The cross-sectional velocities (indicated with arrows) are scaled differently for visualisation reasons.

3.3 Control via a body forcing

Motivated by the recent experiments performed by Hof et al. (Reference Hof, De Lozar, Avila, Tu and Schneider2010), Kühnen et al. (Reference Kühnen, Scarselli, Schaner and Hof2018a ,Reference Kühnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof b ), as discussed in § 1.2, we study the effect of adding a localised body force that mimics the presence of a baffle in the core of the flow. To study the influence of the baffle on the transition threshold, following the results of the previous section, we use the minimal seed at the end of the oblique phase as an initial condition, and verify that it measures transition similarly to random localised disturbances.

Consider a mesh of stationary point objects in the flow and assume that each point experiences a drag proportional to the total velocity. The baffle is then approximated by the following forcing

(3.5) $$\begin{eqnarray}\boldsymbol{F}(r,\unicode[STIX]{x1D703},z,t)=-A\,{\mathcal{B}}(z)\,\boldsymbol{u}_{tot}(r,\unicode[STIX]{x1D703},z,t),\end{eqnarray}$$

where $A$ is the (scalar constant) amplitude of the forcing and ${\mathcal{B}}(z)$ is a (scalar) smoothed step-like function (refer to equation (3.3)) that introduces a streamwise localisation of the force. The product $A\,{\mathcal{B}}(z)$ is a measure of the blockage by the fine mesh. The form of the forcing in (3.5) represents a primitive implementation of an immersed boundary method (refer, for example, to equation (2) in Fadlun et al. Reference Fadlun, Verzicco, Orlandi and Mohd-Yusof2000). As a first approximation, we assume that the baffle is uniform in the radial direction. The streamwise modulation ${\mathcal{B}}(z)$ is fixed so that the baffle occupies a fifth of the pipe, and the smoothing effect is felt for a tenth of the pipe upstream and downstream of it. The only control parameter is thus $A$ . Simulations are fed with the minimal seed at the end of the oblique phase and the random localised disturbance obtained in the unforced cases (refer to figure 6), with the energy gradually rescaled until transition is triggered. The time horizon is $T=125(D/\overline{W})$ , as in the statistical study presented in § 3.2. For the random localised disturbance we apply a random $z$ -shift to the unforced field before feeding it into the simulations with forcing. For comparison, we have also calculated the effect of the baffle on the transition threshold using the (unforced) minimal seed as initial condition. In this instance, we consider the cases where the initial disturbance is centred in the baffle and half-length away from it.

Our purpose is to investigate whether the flow can be kept laminar in the presence of the baffle and how much net energy can be saved. The presence of the baffle causes a pressure drop downstream, which is measured by $(1+\unicode[STIX]{x1D6FD})_{lam/turb}^{A}={\mathcal{D}}_{lam/turb}^{A}/{\mathcal{D}}_{lam}$ , where ${\mathcal{D}}_{lam/turb}^{A}$ is the observed value of the dissipation in the presence of the forcing in either the laminar or turbulent case and ${\mathcal{D}}_{lam}$ is the corresponding laminar value in the unforced case. Hereinafter, the superscript ‘ $A$ ’ indicates the forced case ( $A>0$ ) and the subscripts ‘ $lam$ ’ or ‘ $turb$ ’ refer to the flow being laminar ( $E_{0}<E_{c}$ ) or turbulent ( $E_{0}>E_{c}$ ) at the current value of $A\geqslant 0$ . We use the turbulent dissipation ${\mathcal{D}}_{turb}$ in the unforced case as a reference value to quantify the effect of the forcing. In the unforced case, $1+\unicode[STIX]{x1D6FD}\equiv Re_{p}/Re$ where $Re_{p}=W_{cl}R/\unicode[STIX]{x1D708}$ is the Reynolds number for fixed pressure ( $W_{cl}$ is the centreline velocity of the laminar flow). From the Blasius formula (Blasius Reference Blasius1913) for the turbulent friction coefficient $C_{f}\equiv 16Re_{p}/Re^{2}=0.0791Re^{-0.25}$ , it follows that

(3.6) $$\begin{eqnarray}(1+\unicode[STIX]{x1D6FD})_{turb}\equiv \frac{{\mathcal{D}}_{turb}}{{\mathcal{D}}_{lam}}=\frac{Re_{p}}{Re}=\frac{0.0791}{16}Re^{0.75}.\end{eqnarray}$$

For the forcing to be beneficial, the dissipation in the presence of the forcing in either the laminar or turbulent case must be lower than the turbulent dissipation in the unforced case, i.e. ${\mathcal{D}}_{lam/turb}^{A}<{\mathcal{D}}_{turb}$ , or, equivalently $(1+\unicode[STIX]{x1D6FD})_{lam/turb}^{A}<(1+\unicode[STIX]{x1D6FD})_{turb}$ . As a measure of how beneficial the forcing is, we define a ‘laminar’ and a ‘turbulent’ drag reduction as:

(3.7) $$\begin{eqnarray}{\mathcal{D}}{\mathcal{R}}_{lam/turb}=\frac{{\mathcal{D}}_{turb}-{\mathcal{D}}_{lam/turb}^{A}}{{\mathcal{D}}_{turb}}=\frac{(1+\unicode[STIX]{x1D6FD})_{turb}-(1+\unicode[STIX]{x1D6FD})_{lam/turb}^{A}}{(1+\unicode[STIX]{x1D6FD})_{turb}}.\end{eqnarray}$$

As $A$ is increased, the critical initial energy $E_{c}$ can be pushed further from the corresponding value in the unforced case, but the pressure drop also increases. For example, in the case of a random localised initial condition, a forcing with $A=0.005$ avoids turbulence being triggered for values of the initial energy where turbulence was first hit in the unforced case (i.e. the initial energies corresponding to the blue crosses in figure 6), and a considerable laminar drag reduction is obtained, as shown in table 3. However, a slight increase of $E_{0}$ would result in the flow to become turbulent again, with almost no drag reduction being achieved. Hence, with this very low choice of $A$ , an almost null raise of $E_{c}$ is achieved.

Table 3. Effect of a forcing of amplitude $A=0.005$ at different Reynolds numbers. The second and fourth columns are calculated using (3.6) and (3.7), respectively.

At $Re=5000$ we perform a parametric study on $A$ to find the optimum value which provides the largest $E_{c}$ at the minimum cost, i.e. with the maximum drag reduction. The value of $Re=5000$ is chosen to ensure that in the case without a control the turbulence is sustained, i.e. the chances of turbulence decaying randomly are practically nil. The results are summarised in figure 11. In (a) the critical energies are shown as a function of $A$ for the initial conditions considered here. The corresponding curves for the turbulent and laminar drag reductions are shown in (b). Figure 11(a) shows that, as $A$ is gradually increased up to $A\approx 0.02$ , the critical initial energy increases only slightly, but a considerable turbulent drag reduction is obtained. For example, at $A=0.02$ , we obtain ${\mathcal{D}}{\mathcal{R}}_{turb}=20.7\,\%$ and ${\mathcal{D}}{\mathcal{R}}_{lam}=37.8\,\%$ .

Figure 11. Effect of the forcing for different $A$ at $Re=5000$ . (a) Critical initial energies versus forcing amplitude. The critical initial energies for $A=0$ coincide with the data at $Re=5000$ of figure 6 for the same types of disturbance. The blue crosses and yellow circles pertain to simulations fed with a random localised initial condition and the minimal seed at the end of the oblique phase, respectively. Note that the yellow circles are slightly shifted for visualisation reasons. In addition, calculations were performed with the minimal seed (with two different shifts applied) as initial disturbance (red plusses). The dark-coloured symbols indicate cases where the flow remains turbulent as $E_{0}$ is increased further from the first appearance of turbulence, while the light-coloured symbols denote cases where turbulence is intermittent and characterised by short lifetime. The arrows pointing upwards indicate that $E_{c}\rightarrow \infty$ , i.e. a full collapse of turbulence is obtained. (b) Dissipation and drag reduction versus forcing amplitude. For $A<0.03$ either laminar (Ia: $E_{0}<E_{c}$ ) or turbulent (Ib: $E_{0}>E_{c}$ ) drag reductions are possible, for $A\geqslant 0.03$ , only laminar drag reduction (II) is achieved as turbulence is suppressed.

Figure 12. Time series of energy for simulations fed with a random localised disturbance in the presence of a forcing of amplitude (a)  $A=0.02$ and (b)  $A=0.025$ . In (a) the flow remains turbulent as $E_{0}$ is increased further from the first appearance of turbulence (indicated with red dashed line) and turbulence is sustained, while in (b) turbulence is intermittent and characterised by short lifetime. In the latter case, for $E_{0}>E_{c}$ , the flow is found to either relaminarise or remain turbulent but with a rapid decay towards the end of the observation time window $T=125(D/\overline{W})$ . These scenarios are marked with dark and light symbols in figure 11, respectively.

For $A>0.02$ , transition starts to become intermittent, i.e. as $E_{0}$ is increased further from the first appearance of a turbulent episode, the flow either relaminarise or is characterised by short-lifetime turbulence. In these cases, the ‘critical initial energy’ is indicated by light-coloured symbols in figure 11(a) to distinguish them from the cases, indicated with dark-coloured symbols, where turbulence is sustained. An example of these two situations is shown in figure 12, where in (a) the flow remains turbulent for $E_{0}>E_{c}$ and turbulence is sustained, while in (b) initial disturbances relaminarise and some trigger transition, with the disturbance decaying towards the end of the time window in the latter cases. The scenario where turbulence is short lived and intermittent is analogous to cases where the Reynolds number is close to the first appearance of turbulence ( $Re=1800{-}2000$ ), that is, the effect of increasing $A$ is analogous to that of decreasing the Reynolds number. As $A$ is further increased (e.g. at $A=0.025$ ) this effect becomes more and more pronounced (for example, in the case of the random localised initial condition the ‘light crosses’ become dominant with respect to the ‘dark crosses’) until for $A\geqslant 0.03$ a full collapse of turbulence is obtained (no turbulence episodes are observed). Therefore, the forcing does significantly modify the basin of attraction of the laminar state by expanding it while making its fractal nature more evident until, first, the chaotic attractor transitions back to a chaotic saddle and finally the laminar state remains the only global attractor. The fact that collapse of the turbulent attractor is possible for a forcing of this form can be confirmed using an energy stability type of analysis (refer to appendix A): for large enough $A$ , the laminar state becomes the global attractor.

At $A=0.03$ , where full relaminarisation is first obtained, the laminar drag reduction is still significant (approximately 30 %), as shown in figure 11(b). Therefore we can conclude that this is the optimum choice of forcing amplitude $A_{opt}$ . For $A\geqslant 0.03$ it is not possible to determine ${\mathcal{D}}{\mathcal{R}}_{turb}$ as turbulence is not observed and thus the curve ${\mathcal{D}}{\mathcal{R}}_{turb}(A)$ is connected onto ${\mathcal{D}}{\mathcal{R}}_{lam}(A)$ . For $A<0.03$ we can have either laminar or turbulent drag reduction (i.e. the dynamics either sits on curve Ia or Ib in figure 11), while for $A\geqslant 0.03$ , we only have laminar drag reduction (the dynamics sits on curve II) due to the relaminarisation (destabilisation of turbulent state). The forcing is found to be beneficial up to $A_{c}=0.073$ , where ${\mathcal{D}}{\mathcal{R}}_{lam}$ becomes negative, i.e. for $A>A_{c}$ the cost of the control due to the pressure drop downstream of the baffle becomes greater than the gain due to the relaminarisation.

Figure 11 provides further confirmation that the energy of the minimal seed at the end of the oblique phase is a reasonable proxy to measure transition threshold for random localised disturbances. In this sense, the minimal seed at the end of the oblique phase is a useful tool, potentially enabling us to characterise the critical energy using a single simulation in place of a more expensive statistical study.