3. Results

We perform nonlinear optimizations of finite-amplitude three-dimensional perturbations for the plane Poiseuille flow, focusing on subcritical values of the Reynolds number, namely, $\mathit{Re}=2000,3000,4000,5000$ . In order to find optimal perturbations rapidly triggering coherent structures generated by nonlinear effects, we focus on the time scale of the Orr mechanism, preventing the dominant effect of the lift-up mechanism which acts on a much larger time scale. Butler & Farrell (Reference Butler and Farrell1992) found that the lift-up time scale for the Poiseuille flow at $\mathit{Re}=5000$ is about equal to 380. On the other hand, performing several linear two-dimensional optimizations, we have found that the transient energy growth due to the Orr mechanism peaks at $t\approx 10$ for all of the considered Reynolds numbers. Thus, $T_{\mathit{opt}}=10$ has been chosen for the following nonlinear three-dimensional optimizations. Moreover, using this short target time, we need a sufficiently large initial energy density in order to trigger nonlinear mechanisms. Therefore, several nonlinear optimizations have been performed increasing the value of $E_{0}$ until, quite suddenly, the initial optimal perturbation localizes in space and the energy gain increases with respect to a linear optimization. Such an increment of the energy gain with respect to the linear case is between 10 % and 30 % for the values of the Reynolds number considered in the present work. The value of $E_{0}$ has been successively bisected in order to determine with an accuracy of two digits the nonlinearity threshold, $E_{0th}$ , i.e. the smallest initial energy at which the nonlinear optimal localizes and the energy gain increases by more than 10 % with respect to the linear value. The values of this energy threshold are shown in figure 1(a) for all of the considered values of $\mathit{Re}$ ; they appear well fitted by the scaling law $E_{0th}\propto \mathit{Re}^{-2.36}$ . The corresponding energy gain at target time obtained by nonlinear optimization is provided in figure 1(b). For all of the considered Reynolds numbers, the energy increases by two order of magnitude in only 10 time units, increasing with $\mathit{Re}$ following the law $\mathit{Re}^{0.84}$ .

Figure 1.

Scaling law for the initial energy (squares), the transition threshold (circles) (a), and the energy gain (b) for the short-time nonlinear optimal perturbation at the nonlinearity threshold.

It is noteworthy that the value of the nonlinearity threshold is greater than the minimal energy capable of inducing transition, as one would anticipate. For example, we have performed optimizations and bisections for $T_{\mathit{opt}}=50$ and $\mathit{Re}=4000$ , finding a transition energy threshold of about $2.5\times 10^{-7}$ which is one order of magnitude lower than the energy corresponding to the nonlinearity threshold. On the other hand, the energy at the nonlinearity threshold is not very high since we have verified that the linear optimal perturbation rescaled with $E_{0th}$ (see figure 1 a) does not lead to transition. In order to evaluate the distance of the nonlinearity threshold from the laminar–turbulent boundary for each optimal perturbation, figure 1(a) also provides the rescaled energy level necessary for placing each optimal perturbation (computed at the nonlinearity threshold) on the laminar–turbulent boundary. Notice that the scaling law for the transition threshold ( $E(0)\propto \mathit{Re}^{-2.35}$ , corresponding about to $\mathit{Re}^{-1.2}$ for the amplitude) is not far from the value provided by the experimental analysis of Cohen, Philip & Ben-Dov (Reference Cohen, Philip and Ben-Dov2009), who found a scaling law for the transition threshold amplitude of $A\propto \mathit{Re}^{-3/2}$ using wall-normal flow injection for inducing hairpin vortices.

Figure 2.

Isosurfaces of the $Q$ -criterion for the nonlinear optimal perturbation obtained with $\mathit{Re}=4000$ , $T_{\mathit{opt}}=10$ , and $E_{0}=2\times 10^{-6}$ : (a) $t=0$ , $Q=0.01$ ( $Q_{\mathit{max}}=0.086$ ; red/green surfaces for positive/negative streamwise vorticity values) and (b)  $t=10$ , $Q=0.2$ ( $Q_{\mathit{max}}=3.88$ ).

Figure 3.

Isosurfaces of the three velocity components: (a,d) $u$ , (b,e) $v$ , (c,f) $w$ (light grey for positive and black for negative values, $u,v,w=\pm 0.01$ ) of the nonlinear optimal perturbation for $T_{\mathit{opt}}=10$ : (a,b,c) $E_{0}=2\times 10^{-6}$ , $\mathit{Re}=4000$ ; and (d,e,f) $E_{0}=1\times 10^{-5}$ , $\mathit{Re}=2000$ .

Figure 4.

Time evolution of the energy gain and velocity (a) and vorticity (b) r.m.s. values for the nonlinear optimal perturbation with $\mathit{Re}=4000$ , $E_{0}=2\times 10^{-6}$ , and $T_{\mathit{opt}}=10$ .

Figure 5.

Time evolution of the nonlinear optimal perturbation for $\mathit{Re}=4000$ , $E_{0}=2\times 10^{-6}$ , and $T_{\mathit{opt}}=10$ . Isosurfaces of $Q$ -criterion ((a–e)  $Q=0.02$ ; ( f)  $Q=0.1$ , green) and streamwise velocity disturbance ( $u=-0.03$ , black). $Q_{\mathit{max}}=0.086,1.03,1.99,7.5,21.5,35.4$ for (a) to ( f), respectively. (a) $t=0$ , (b) $t=5$ , (c) $t=8$ , (d) $t=12$ , (e) $t=16$ , (f) $t=40$ .

Figure 2(a) shows the nonlinear optimal perturbation obtained for $(\mathit{Re},E_{0})=(4000,2\times 10^{-6})$ , composed at $t=0$ of several thin tubes of counter-rotating vorticity alternated in the spanwise direction and fully localized in space. The vortices are characterized by a large streamwise vorticity component and are tilted against the base flow, confirming the presence of linear energy growth mechanisms such as the Orr (Reference Orr1907) and the lift-up (Landahl Reference Landahl1980) mechanisms. However, this optimal solution is strongly different from the linear one computed for the same flow due to: (i) its strong localization; (ii) its smaller wavelength in the spanwise direction; (iii) the presence of an inclination of the vortical structures with respect to the streamwise direction. In particular, figure 2(a) shows three pairs of vortices, all of them having different streamwise inclination and length, characterized by a streamwise vorticity of alternating sign. One can observe that the two inner pairs of vortices at the centre of the structure are very close to each other. When such a structure evolves in time, a strong interaction of the vortices occurs immediately after the tilting in the streamwise direction. Therefore, a ‘hot spot’ is created, leading to the formation of the hairpin vortex at target time $t=T_{\mathit{opt}}=10$ , as shown in figure 2(b). It is also noteworthy that, unlike other parallel shear flows, such as Couette flow (Cherubini & De Palma Reference Cherubini and De Palma2013), this nonlinear optimal structure is characterized by a symmetry with respect to a $z=\text{const.}$ plane, which is maintained when the initial energy $E_{0}$ is increased. We have verified that the optimal perturbation is characterized by this symmetry only for small target times; for target times typical of the lift-up mechanism, the optimal perturbation strongly resembles the one found in Couette flow (Cherubini & De Palma Reference Cherubini and De Palma2013). Figure 3(a–c)  clearly shows the symmetry of the streamwise and wall-normal velocity components, whereas the spanwise component is antisymmetric with respect to a $z$ -constant axis, recalling the structure of varicose perturbations (Schmid & Henningson Reference Schmid and Henningson2001). Moreover, it is worth noting that the streamwise and wall-normal components of the velocity clearly show a ${\it\Lambda}$ structure oriented against the flow. This structure, once tilted by the Orr mechanism, becomes a precursor of the hairpin vortex. Very similar structures are found for different Reynolds numbers, as shown in figure 3(d–f) for $\mathit{Re}=2000$ , although the latter perturbations are characterized by larger amplitudes (see the scaling in figure 1 a).

Direct numerical simulation (DNS) has been employed to study the time evolution of the initial optimal perturbation into a hairpin vortex and its subsequent transition to turbulence. The nonlinear optimal perturbation computed for $\mathit{Re}=4000$ with $E_{0}=2\times 10^{-6}$ has been used to initialize the computation. The results are shown in figure 4 providing the r.m.s. values of velocity (a) and vorticity (b) versus time. It appears that all of the velocity and vorticity components grow together until reaching transition in a relatively short time (at $t\approx 50$ ). It is noteworthy that in other transition scenarios, such as oblique and streaks instability (see Schmid & Henningson Reference Schmid and Henningson2001), the streamwise velocity experiences a much larger growth than the other components at short times. Snapshots of the time evolution of the vorticity (green) and of the negative streamwise velocity (black) are provided in figure 5. The initial vortex tubes are alternated in $z$ and have opposite inclination with respect to the streamwise direction (see figure 5 a), so that the downstream tilting due to the Orr mechanism has already induced at $t=5$ the merging of two of these vortices in a tube of spanwise vorticity resembling a hairpin head (see figure 5 b,c). Patches of positive and negative streamwise velocity can be observed in the flow; for $t\leqslant T_{\mathit{opt}}$ they appear rather localized in the streamwise direction (figure 5 c) and they elongate in the streamwise direction, creating bent streaks only at times larger than the target time (figure 5 d,e). It appears that the hairpin head is originated by a strong localized streamwise velocity defect (Adrian Reference Adrian2007), which is already rather large at $t=0$ . This defect further increases its amplitude due to the Orr mechanism and to a modified lift-up mechanism (see Cherubini et al. Reference Cherubini, De Palma, Robinet and Bottaro2011) driven by the initial vortex tubes, until inducing local inflection points in the instantaneous velocity profile.

The dynamics of the nonlinear optimal perturbation shares some important features with the dynamics of the finite-amplitude perturbations analysed by Suponitsky et al. (Reference Suponitsky, Cohen and Bar-Yoseph2005). In particular, they observed the formation of a single hairpin vortex from initial Gaussian vortices of maximum vorticity magnitude ${\it\Omega}_{\mathit{max}}>1$ and streamwise characteristic length $L<5$ . Such constraints are satisfied by the nonlinear optimal perturbations obtained here which are characterized by ${\it\Omega}_{\mathit{max}}\approx 1.8$ and $L\approx 2.5$ . Thus, for better analysing the dynamics of the nonlinear optimal perturbation, we have studied the evolution of the main vortical structures in time following the approach of Suponitsky et al. (Reference Suponitsky, Cohen and Bar-Yoseph2005). Let us consider the centre of the vortical structure (CVS), whose coordinates are defined by the following equation:

where ${\it\Omega}$ is the vorticity magnitude, the denominator is the enstrophy integral, and the index $i=1,2,3$ represents Cartesian components. To identify the spatial orientation of the vortical structure centred at $X_{i}$ , we employ the tensor of enstrophy distribution (TED), defined as follows:

Since $\unicode[STIX]{x1D61B}_{ij}$ is a symmetric tensor, all its eigenvalues are real. The eigenvector associated with the largest eigenvalue identifies the direction of the principal axis of the vortex, along which the vortex has the largest extension. The insets in figure 6(a,b) show by red lines the principal axis of the vortical structures (represented in green by the $Q$ criterion surfaces) extracted from the DNS at $t=5$ and $t=10$ , respectively. As the flow structures are mainly aligned with the principal axis, we evaluate along this axis the distribution of the streamwise and wall-normal velocity components. The black lines in figure 6(a,b) show such distributions versus the abscissa $s$ along the principal axis ( $s=0$ at the CVS) at two different times. At $t=5$ (figure 6 a) we can observe that the velocity components $u$ and $v$ mostly have opposite sign and are equal to zero for $s=0$ . This behaviour recalls a fundamental feature of eddy motion in wall turbulence; namely, fluctuations have higher probability of spending time in the second (Q2) and fourth (Q4) quadrants of the $u{-}v$ plane (Adrian Reference Adrian2007), inducing two kind of events: (i) ejections, characterized by negative streamwise fluctuations lifted away from the wall by positive wall-normal fluctuations; (ii) sweeps, characterized by positive streamwise fluctuations being transported toward the wall.

Figure 6.

Velocity profile along the principal axis of the TED extracted at two different times from the DNS (black) and the linearized DNS (red), initialized by the nonlinear optimal at $(\mathit{Re},E_{0})=(4000,2\times 10^{-6})$ . The insets show the $Q$ -criterion surfaces ((a) $Q=0.2$ with $Q_{\mathit{max}}=1.03$ ; (b) $Q=0.3$ with $Q_{\mathit{max}}=3.9$ ; green) and the principal axis (red lines) at two different times. The dot indicates the CVS. (a) $t=5$ , (b) $t=10$ .

Figure 7.

Contours of the logarithm of the probability density function of the wall-normal and streamwise velocity disturbance in a $u{-}v$ plane ( $w=0$ ) for $0\leqslant y\leqslant 1$ , extracted at $t=0$ (a), 1 (b), 6 (c), 10 (d) from a DNS initialized with the nonlinear optimal with $T_{\mathit{opt}}=10$ and $E_{0}=2\times 10^{-6}$ . The values of the logarithm of the PDF have been normalized with respect to the total number of points of the computational domain.

Figure 8.

Wall-normal and streamwise components of the velocity vector disturbance along the principal axis of the TED for the nonlinear optimal with $E_{0}=2\times 10^{-6}$ : (a)  $t=0$ , (b)  $t=14$ . Isosurface of the $Q$ -criterion ((a)  $Q=0.01$ with $Q_{\mathit{max}}=0.086$ , (b)  $Q=4$ with $Q_{\mathit{max}}=12.05$ ).

To confirm such a similarity, we have computed the spatial probability density function (PDF) of the perturbation velocity components at different times. Figure 7 shows the PDF values (in a logarithmic scale) of the streamwise and wall-normal components of the velocity perturbation, confirming the presence of ejections and sweeps already at $t=0$ and up to the time of formation of the hairpin vortex. For instance, in figure 6(a) a sweep is observed upstream of the CVS ( $s<0$ ) followed by a strong ejection and a weak sweep downstream of the CVS ( $s>0$ ). At $t\approx 10$ , the hairpin head is formed downstream of the CVS ( $s>0$ ), in correspondence with an ejection, as shown in figure 6(b). This is not surprising since, as found in wall turbulence for a channel flow, the eddy associated with an ejection has a hairpin vortex shape (Adrian Reference Adrian2007). Comparing figure 6(a) and figure 6(b), one can observe that when the hairpin head is formed, the zone of ejection increases its intensity, while the sweep region enlarges in the streamwise direction, reducing its intensity. When the evolution of the same initial perturbation is computed by a DNS based on the solution of the linearized NS equations, the velocity distribution for $t>5$ is characterized by increasing values of $u$ and much lower values of $v$ , as shown by the red thin lines in figure 6. Therefore, in the linearized case, streaks with larger intensity will be formed due to the lift-up effect but they will not be associated with wall-normal velocities of opposite signs, rapidly damping the ejection event that is responsible for the lifting of the hairpin head, thus preventing the creation of the hairpin structure.

Figure 8 shows that the alternation of strong sweeps and ejections found at $t=0$ is maintained at larger times, thus representing the basic element for the rapid hairpin formation. Moreover, in figure 8(b) one can observe that the hairpin head is placed right in the zone of interaction of the stronger ejection with the stronger sweep, indicating that the formation of the spanwise vortex connecting the initial quasi-streamwise vortices might be a consequence of an inflectional instability taking place in this interaction zone. This observation is in agreement with the minimal flow-element model proposed by Cohen, Karp & Mehta (Reference Cohen, Karp and Mehta2014), in which a wavy spanwise vortex sheet was necessary to provide the inflection points for creating hairpin vortices from streamwise counter-rotating vortex pairs.

Figure 9(a,b) provides the instantaneous velocity and vorticity profiles at $t=10$ , computed solving the nonlinear and the linearized NS equations, respectively, extracted along a vertical axis passing through the hairpin head obtained in the nonlinear case. In figure 9(a) one can observe an inflection point, located at the ordinate corresponding to the peak of vorticity coinciding with the hairpin head (at $y\approx 0.7$ ), in the outer zone of the velocity profile; whereas, in the near-wall region, at $y\approx 0.15$ , one can observe the deformation of the velocity profile induced by the formation of a negative streak. Inflection points are observed in the linearized case as well (figure 9 b), but are placed closer to the wall ( $y\approx 0.4$ ), consistent with a shear layer instability induced by a streak. In the nonlinear case, the vorticity peak moves in time towards the centre of the channel, reaching in 10 time units an amplitude almost twice that of the linearized case. When the head of the hairpin is pushed by the wall-normal perturbation toward the centre of the channel, it is advected by the base flow at a higher velocity with respect to the part near the wall, stretching the whole vortical structure in the streamwise direction. Then, due to conservation of circulation, stretching provides a further growth of the vorticity, sustaining the hairpin vortex and allowing secondary hairpin structures to be created (Adrian Reference Adrian2007).

Figure 9.

Streamwise velocity (solid lines) and vorticity magnitude (dashed lines) profiles at $t=5$ along a wall-normal line passing through the hairpin head computed by the DNS (a) and by the linearized DNS (b), initialized with the nonlinear optimal for $E_{0}=2\times 10^{-6}$ . Circles indicate inflection points.

Figure 10.

Time evolution of the nonlinear optimal perturbation obtained for $T_{\mathit{opt}}=10$ and $E_{0}=2\times 10^{-6}$ on the plane $z=2$ . Isosurfaces of the $Q$ -criterion ( $Q=0.1$ , green), isolines of the streamwise velocity disturbance (red positive, black negative), and contours of the wall-normal velocity disturbance (white positive, black negative). $Q_{\mathit{max}}=0.61,3.88,21.52$ for (a) to (c), respectively. (a) $t=4$ , (b) $t=10$ , (c) $t=16$ .

The time evolution of the hairpin structure is shown in figure 10, providing DNS snapshots showing isosurfaces of the $Q$ -criterion (green), isolines of the streamwise velocity disturbance (red for positive and black for negative), and contours of the wall-normal velocity (white for positive and black for negative). One can notice the alternation of patches of $u$ and $v$ with different signs, spread all over the domain, and their effect on the lifting and stretching of the hairpin structure. Nonlinear effects are crucial to sustain the alternation of the $u$ and $v$ perturbation components as well as the vortical structures. This can be clearly inferred by comparing figure 10 with figure 11, showing the time evolution of the nonlinear optimal perturbation obtained by a linearized DNS.