2. Governing equations and numerical simulations

The 2-D plane shear flow is governed by the incompressible NS equations

(2.1a,b)\begin{equation} \boldsymbol{U}_t + \boldsymbol{U} \boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{U} ={-} \boldsymbol{\nabla} P + \frac{1}{Re}\nabla^2 \boldsymbol{U}, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{U} = 0, \end{equation}

where $\boldsymbol {U}$ is the velocity vector $(U, V)$, and $P$ is the pressure. Let the laminar state of plane channel flow $\boldsymbol {U} = (U(y), 0)$, and for the plane Poiseuille flow $U(y) = 1-y^2$. The equations are non-dimensionalized by the maximum velocity $U_{max}$, half-height of the channel $h$, time $h/U_{max}$, and pressure $\rho U_{max}^2$. The Reynolds number $Re = U_{max}h/\nu$, where $\nu$ is the viscosity. The perturbation velocity vector is $\boldsymbol {u} = \boldsymbol {u}_{tot} - \boldsymbol {U}$. Therefore, the governing equations of the perturbation velocities are

(2.2a,b)\begin{equation} \boldsymbol{u}_t + \boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{U} + \boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}+ \boldsymbol{U} \boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}={-} \boldsymbol{\nabla} p + \frac{1}{Re}\nabla^2 \boldsymbol{u}, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} = 0 . \end{equation}

Falkovich & Vladimirova (Reference Falkovich and Vladimirova2018) found the pressure-driven laminar flow induces travelling waves for sufficiently small viscosity and friction. By adding uniform friction to real fluid layers, the turbulent flow was observed at $Re \sim 30\,000$ much larger than $Re_c = 7696$, which is based on the flow rate ($4/3 U_{max}$). The subcritical transition to turbulence in quasi-2-D flows has been studied in the duct flows with moving walls driven by the linear friction under a strong transverse magnetic field (Camobreco et al. Reference Camobreco, Pothérat and Sheard2023). Due to the significant effect of linear friction, the critical Reynolds number $Re_c = 79123.2$, which is based on the half-duct-height and the maximum velocity $U_{max}$ in Camobreco, Pothérat & Sheard (Reference Camobreco, Pothérat and Sheard2021). They calculated the linear optimizations for the quasi-2-D shear flow in a rectangular duct with significant linear friction under a strong magnetic field at $Re = 71\,211$, and found the linear and nonlinear optimizations offer identical results to approach the edge state. However, the nonlinear optimal disturbances with the minimal energy, or the so-called minimal seeds (Pringle et al. Reference Pringle, Willis and Kerswell2012; Eaves & Caulfield Reference Eaves and Caulfield2015), are the perfect choice to trigger the subcritical transition along the edge state between the nonlinear laminar and turbulent attractor basin boundary. Therefore, besides the linear optimizations, we employed fully nonlinear optimizations to investigate the subcritical transition via the nonlinear edge state for 2-D plane Poiseuille flow. The nonlinear optimal disturbance and the related evolution in the channel could be identified by the variational method (see Pringle & Kerswell (Reference Pringle and Kerswell2010), Cherubini et al. (Reference Cherubini, Palma, Robinet and Bottaro2011) and Monokrousos et al. (Reference Monokrousos, Bottaro, Brandt, Di Vita and Henningson2011)). The disturbance energy density is defined as

(2.3)\begin{equation} E(t) = \frac{1}{2V} \int_{V} [u^2(t) + v^2(t)] \,{\rm d}V, \end{equation}

and energy density of streamwise and vertical velocity disturbances

(2.4a,b)\begin{equation} E_u(t) = \frac{1}{2V} \int_{V} u^2(t) \,{\rm d}V,\quad E_v(t) = \frac{1}{2V} \int_{V} v^2(t) \,{\rm d}V, \end{equation}

where $V$ is the volume of the channel, and for 2-D flow $V = 2hLx$ is the area of the channel. By defining the objective functions and introducing the Lagrangian multipliers, the adjoint governing equations are formulated as

(2.5a,b)\begin{equation} \boldsymbol{u^{{\star}}}_t + \boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{ u^{{\star}} } - \boldsymbol{u^{{\star}}} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}={-} \boldsymbol{\nabla} p^{{\star}} - \frac{1}{Re}\nabla^2 \boldsymbol{u^{{\star}}}, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u^{{\star}}} = 0 , \end{equation}

where $\boldsymbol {u^{\star }}$ is the adjoint velocity of disturbance velocity $\boldsymbol {u}$, and $p^{\star }$ is the adjoint variable of pressure disturbance $p$.

There are many numerical optimized methods to seek the nonlinear optimal disturbance for shear flows (Pringle & Kerswell Reference Pringle and Kerswell2010; Monokrousos et al. Reference Monokrousos, Bottaro, Brandt, Di Vita and Henningson2011; Cherubini & Palma Reference Cherubini and Palma2015; Huang & Philipp Reference Huang and Philipp2020). In order to capture the edge state of subcritical transition, the disturbance energy density (Pringle & Kerswell Reference Pringle and Kerswell2010) and the total time-averaged dissipation are defined (Monokrousos et al. Reference Monokrousos, Bottaro, Brandt, Di Vita and Henningson2011) as the objective function, respectively. In this study, the nonlinear optimizations are conducted by the well-developed direct-adjoint-looping method based on the direct numerical simulation (DNS) codes Diablo (Taylor Reference Taylor2008). The Fourier truncations are used in the streamwise $x$-direction, and the dealiasing technique which removes the highest $1/3$ portion of the Fourier spectrum is employed. The finite difference method is used for the spatial discretizations in the wall-normal direction, and a combined implicit–explicit Runge–Kutta–Wray Crank–Nicolson scheme for time integration. The numerical energy threshold could be obtained by the bisection technique. Since the simulated turbulence effects are very strong for long 2-D channels (Jiménez Reference Jiménez1987, Reference Jiménez1990), the wavenumber $\alpha _{domain} = 0.1$ is mainly considered to define the computational domain length $Lx = 2{\rm \pi} / \alpha _{domain} = 20 {\rm \pi}$, and $\alpha$ is used for convenience and refers to the channel length if not specified in this study. According to Jiménez (Reference Jiménez1990), the target time $T = 2000$ is sufficient to observe the saturated nonlinear interactions for 2-D plane Poiseuille flow. The edge tracking method is shown in figure 1, the edge state is bracketed by energy trajectories of red solid and dashed lines with initial energy density of $9.24967\times 10^{-6}$ and $9.24965\times 10^{-6}$, respectively, for $Re = 3500$. Therefore, for convenience, we only show the two bounded trajectories in this paper. The edge tracking technique is very similar to that shown in figure 1 in Zammert & Eckhardt (Reference Zammert and Eckhardt2014). We performed the grid sensitivity study with three grid resolutions, i.e. $512 \times 256$, $1024 \times 512$ and $2048 \times 1024$, the corresponding critical energy densities that induce the subcritical transitional flow are $1.025\times 10^{-5}$, $9.2497\times 10^{-6}$ and $9.241\times 10^{-6}$, respectively. The energy relative difference between the grids $1024 \times 512$ and $2048 \times 1024$ is less than $0.1\,\%$. Therefore, the grid resolution $1024 \times 512$ is employed with more than 256 points distributed near the channel walls within $h/4$. The time step 0.01 is chosen for all numerical optimizations and DNS in this study.

Figure 1. The edge tracking in 2-D plane Poiseuille flow at $Re = 3500$: the solid lines are the initial densities of trajectories that trigger subcritical transition; the dashed lines are trajectories of laminar; and the edge state (dotted line) bracketed by the trajectories of red lines.

3. Subcritical transitional flow

According to the linear stability analysis, the critical Reynolds number $Re_{c}$ is approximately 5772.2 for 2-D plane Poiseuille flow (Orszag Reference Orszag1971). Figures 2(a) and 2(b) compare the time variation of kinetic energy of linear and nonlinear optimal disturbances at the subcritical Reynolds number $Re = 3500$ for different channel lengths. The linear optimal disturbances in this study are calculated for the maximal transient energy growth. For example, the maximal energy gain is 34.08 with optimal target time $T_{opt} = 12.55$ with optimal wavenumber of $1.53$ at $Re = 3500$. We calculated the linear optimal disturbances with target time $T_{opt}$ by the numerical optimization (Butler & Farrell Reference Butler and Farrell1992), while the nonlinear optimal disturbances are captured for $T = 2000$. Both of the optimal disturbances could reach the chaotic transitional flow via their edge states of different energy norms with different channel length, as shown in figures 2(a) and 2(b). For linear optimal disturbances, the initial energies are rescaled up to $E_0 = 1.44909 \times 10^{-3}$ and $E_0 = 4.076686 \times 10^{-4}$ for $\alpha = 0.4$ and $\alpha = 0.1$, respectively, to trigger the chaotic transitional flow via the edge state related to the energy plateau; while the flows are finally relaminarized after the experienced edge states for $E_0 = 1.449085 \times 10^{-3}$ and $E_0 = 4.076684 \times 10^{-4}$. The energy norm of the edge state is approximately $2 \times 10^{-3}$, which is larger than the initial energy for $\alpha = 0.4$. While the energy norm of edge state is approximately $2 \times 10^{-4}$, which is much smaller than the initial energy for $\alpha = 0.1$. The linear optimal disturbances experience amplification, decay in an oscillatory way and then monotonically to touch the edge state. For the nonlinear optimal disturbances, or minimal seeds, the critical initial energies are $E_0 = 3.602621 \times 10^{-5}$ and $E_0 = 9.24967 \times 10^{-6}$ for $\alpha = 0.4$ and $\alpha = 0.1$, respectively. The required energies of nonlinear optimal disturbances are approximately $2.5\,\%$ of those of linear optimal disturbances for both $\alpha = 0.4$ and $\alpha = 0.1$. The minimal seeds are amplified rapidly and decay to the edge states, whose energy norms are approximately $7.0 \times 10^{-4}$ and $1.25 \times 10^{-4}$ for $\alpha = 0.4$ and $\alpha = 0.1$, respectively, before reaching the transitional flow.

Figure 2. $(a)$ The time history of perturbation kinetic energy for linear (black lines) and nonlinear (red lines) optimal disturbances at $Re = 3500$ and $\alpha = 0.4$. $(b)$ The time history of perturbation kinetic energy for linear (black lines for $T_{opt}$ and green lines for $10T_{opt}$) and nonlinear (red lines) optimal disturbances at $Re = 3500$ and $\alpha = 0.1$. $(c)$ The time variation of perturbation kinetic energy of nonlinear optimal disturbances for different $Re$; the black, red and green lines are corresponding to $Re = 2400$, 3500 and 5000, respectively. $(d)$ The perturbation kinetic energy varying with time from $t = 0$ to 4000 for the minimal seeds of 2-D plane Poiseuille flow for different $Re$. (The solid lines depict the chaotic transitional flow while the dashed lines depict the relaminarized flow.)

In Camobreco et al. (Reference Camobreco, Pothérat and Sheard2023), the linear optimal disturbances with larger target time $8 T_{opt}$ resemble the nonlinear optimal disturbances, and the leading adjoint mode represent the optimal initial disturbance in short channels. The linear optimal disturbances of $10 T_{opt}$ are calculated with the energy gain 3.67. The linear optimal disturbances are still uniformly distributed in the channel and very similar to those of $T_{opt}$ (see figure 3a). Figure 2(b) shows the comparison of energy time variation, the initial energy of $10 T_{opt}$ is approximately $2.3831\times 10^{-4}$, which is much smaller than $4.076684 \times 10^{-4}$ for $T_{opt}$. Moreover, the uniform linear optimal disturbances reach the same edge state with the same energy, which is larger than that of the minimal seed. The linear optimal disturbances with larger target time are more efficient in triggering transitional flow than those of maximal energy growth; it is consistent with Camobreco et al. (Reference Camobreco, Pothérat and Sheard2023) since the leading adjoint eigenmode is a more efficient initial condition and plays a very important role for larger target time. Moreover, the initial energy is still larger than the energy near the edge state, which means that the energy is not amplified compared with the initial energy. Figure 2(c) shows the energy time variation of minimal seeds for different $Re$, in this study, the subcritical transitional flow could be observed as $Re \geqslant 2400$ for $\alpha = 0.1$. The critical initial energies are $E_0 = 4.922 \times 10^{-4}$ and $E_0 = 2.6186 \times 10^{-6}$ for $Re = 2400$ and 5000, respectively. The linear optimal disturbance could not induce the subcritical transitional flow for $Re = 2400$; while it requires $E_0 = 1.63153 \times 10^{-4}$ of linear optimal disturbance at $Re = 5000$ to trigger the transitional flow. It can be concluded that the critical initial energy of nonlinear optimal disturbance is less than 2.5 % of that of linear optimal disturbance with target time $T_{opt}$. To check if the minimal seeds could sustain the transitional flow after the target time $T = 2000$, figure 2(d) shows the variation of disturbance energy with respect to time from 0 to 4000 for different $Re$. The sustained transitional flow induced by the minimal seeds could be observed after the target time $T = 2000$.

Figure 3. The comparison of vorticity snapshots in the subcritical transition triggered by the (a–i) linear and (a $'$–i $'$) nonlinear optimal disturbances for 2-D plane Poiseuille flow at $Re = 3500$. The time instants for snapshots, (a–i) and (a $'$–i $'$), are $t = 0$, 20, 50, 100, 200, 400, 1000, 1500 and 2000, respectively.

At $Re = 5000$, the initial perturbation energy increases rapidly to the peak by the Orr mechanism, then it decreases gradually to the energy plateau related to the nonlinear edge state, the nonlinear evolution sustains along the basin boundary with a sufficiently long period before it drives the flow into chaotic transitional flow, in figures 2(c) and 2(d). The energy time variation is almost the same as that of $E_0 > E_D$ in figure 2(a) of Camobreco et al. (Reference Camobreco, Pothérat and Sheard2023), which demonstrates that the subcritical transition mechanism (nonlinear TS waves) in 2-D plane Poiseuille flow is captured. Fully developed turbulence in 2-D plane Poiseuille flow has been observed when the Reynolds number is far beyond the critical value $Re_c$ (Falkovich & Vladimirova Reference Falkovich and Vladimirova2018; Markeviciute & Kerswell Reference Markeviciute and Kerswell2021), which indicates that it might not be achieved if the Reynolds number is not greater than 5772.2 for 2-D plane Poiseuille flow. Therefore, it is not surprising that the obvious turbulent kinetic energy oscillation is absent in 2-D flow transitional flow, and the fully developed turbulence is absent for $Re < Re_c$. As $Re$ decreases, the nonlinear optimal disturbances with increased initial kinetic energy could still trigger 2-D subcritical transitional flow, and the perturbation energy variation is the same as $Re = 5000$. By employing nonlinear optimizations, the subcritical transition of 2-D plane Poiseuille flow could be captured at $Re = 2400$. It is interesting that the energy growth at the peak for the nonlinear optimal disturbances is much smaller than the linear optimal disturbances, which might be attributed to the long target time and nonlinear interactions. For example, the maximum energy growth of linear optimal disturbance is approximately 34.08 for $Re = 3500$, while the maximum energy growth at the peak in only approximately 15.1. Moreover, the perturbation energy of the nonlinear edge state is smaller than the energy peak by Orr amplification for larger $Re$; while the energy of the edge state is slightly larger than the energy peak for $Re = 2400$. The kinetic energy of the transitional flow increases with $Re$.

Figure 3 compares the subcritical transitional flow field triggered by linear optimal disturbances and minimal seeds, represented by disturbance vorticity. The linear optimal disturbances distribute near the channel walls uniformly and symmetrically against the mean shear and filled in the whole channel, that are the wall mode. While the streamwise clustered nonlinear optimal disturbances locate one third of the channel, and the magnitude of nonlinear optimal disturbances is much smaller. The linear optimal disturbances are amplified by a tilting effect (see figure 3a,b); the linear travelling waves are formulated and the disturbance magnitude decreases (see figure 3c,d) uniformly; the disturbance magnitude decreases further non-uniformly and nonlinearly (see figure 3e,f); the decrease of magnitude ceases and the nonlinear TS waves are sustained (see figure 3f–h) before the chaotic transitional flow is reached.

Unlike the linear optimal disturbances, the nonlinear optimal disturbances are streamwise localized wave packets and not symmetric due to the nonlinear interaction in the optimization. The localized optimal disturbances are amplified by the Orr mechanism to obey the mean shear (figure 3b $'$), and the nonlinear interactions are evoked immediately to form the nonlinear TS waves (figure 3c $'$). The nonlinear TS waves and the nonlinear edge state sustains for a sufficiently long time, before the fluid-structure deviates into the 2-D chaotic transitional state (figure 3i $'$). From the vorticity distributions, we can see that the flow induces unsteadiness of the sharp vortex sheets ejected from the wall by the nonlinear travelling waves, and is mediated by the secondary updrafts induced at the walls by vorticity inhomogeneities in the core of the channel, which is consistent with the observations (Jiménez Reference Jiménez1990). Interestingly, the fluid structures are very similar after $t = 400$ for both the linear (figure 3f–i) and nonlinear (figure 3f $'$–i $'$) disturbances. It indicates that the linear optimal disturbances could be treated as candidate to investigate the subcritical transition of shear flows (Camobreco et al. Reference Camobreco, Pothérat and Sheard2023), since the nonlinear TS waves along the linear edge state are very similar to the nonlinear edge state. It should be noted that most of the uniform distributed linear optimal disturbances are decayed. If only parts of disturbances in the channel are considered, the required initial energy density might be reduced. For example, if the first half of the channel is filled with disturbances and then zeroed everywhere outside of the first half, e.g. filtered with a step function, then the initial energy density of the linear optimal is reduced to $1.11882\times 10^{-4}$, compared with $4.076684 \times 10^{-4}$ for the linear optimal disturbances without applying a step function and $9.24967 \times 10^{-6}$ for the nonlinear optimal disturbances. Therefore, quantitatively, the minimal seeds capture the edge state more efficiently for long channels.

4. The wall mode optimal disturbance for subcritical transition

The optimal disturbances are amplified by the Orr mechanism in 2-D plane Poiseuille flow. However, the magnitude of the streamwise velocity perturbation is much larger than that of vertical velocity perturbation. In this section, the quantitative relation between the two perturbation components is investigated. Figure 4 shows the time variation of the perturbation energy of velocity components and their ratio for the optimal disturbances. The time variation of perturbation energy of streamwise velocity $E_u$ shows the same trend as the total kinetic energy (see figure 2a) for different $Re$, while the variations of $E_v$ are quite different from $E_u$. It is obvious that the perturbation energy of streamwise velocity $u$ is much larger than the vertical velocity $v$, which means the streamwise velocity is predominant in the 2-D flow. Correspondingly, the amplification of $E_v$ is much more remarkable than that of $E_u$, since the initial energy norm $E_v$ is much smaller. From figure 3, the initial optimal disturbances are located near the channel wall. According to the well-established linear instability and transient growth theories (Lin Reference Lin1944; Drazin & Reid Reference Drazin and Reid1981; Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993; Schmid & Henningson Reference Schmid and Henningson2001; Chapman Reference Chapman2002; Schmid Reference Schmid2007), the wall mode disturbances of the linearized NS equations (Orr–Sommerfeld equations) of 2-D plane Poiseuille flow locate near the channel walls. Here, the uniformly disturbed linear optimal disturbances are the wall mode as shown in figure 3(a). For the wall mode near the channel wall $y = -1$, there exists the scaling law (Lin Reference Lin1944; Chapman Reference Chapman2002)

(4.1)\begin{equation} y ={-}1 + y'/(\alpha Re)^{1/3}, \end{equation}

where $y'$ is the scaled coordinate variable near the channel wall. Instead of the rigorous but time-consuming asymptotic analysis in Chapman (Reference Chapman2002), we just focus the continuity equation $u_x + v_y = 0$ for estimation since there are no disturbances near the channel centre. According to the periodic boundary condition in the streamwise direction, the disturbances could be decomposed into Fourier modes, such as

(4.2)\begin{equation} v(x,y) = \sum \tilde{v}(y) \exp^{{\rm i} \alpha x}, \end{equation}

where $\tilde {v}(y)$ are continuous functions in the wall-normal direction. Therefore,

(4.3)\begin{equation} {\rm i} \alpha \tilde{u}(y) + \mathrm{d} \tilde{v}(y) /\mathrm{d}y = 0 \end{equation}

is obtained according to the continuity equation. Considering the wall modes and the scaling (4.1), ${\rm i}\alpha \tilde {u}( y')$ should be balanced by $\mathrm {d} \tilde {v}(y') /\mathrm {d}y' (\alpha Re)^{1/3}$ near the channel wall. Consequently, it can be concluded that $E_u$ is approximately $(Re)^{2/3} E_v$ for the wall mode optimal disturbances if one focuses on the fixed streamwise wavenumber. Figure 4(b) presents the values of $(Re)^{-2/3} E_u/E_v$ varying with $Re$ for the linear optimal disturbances in 2-D plane Poiseuille flow. The values are almost constant, i.e. $0.567 \pm 0.01$, which demonstrates the linear optimal disturbances are the wall modes, quantitatively. However, for the minimal seeds that induce the subcritical transitional flow, the values of $(Re)^{-2/3} E_u/E_v$ are much smaller, which might be the reason why the peak energy amplification is much smaller. The value keeps constant at approximately $0.33$ for large Reynolds numbers, while it is $0.41$ at $Re = 2400$. It might be concluded that minimal seeds are also exact wall modes for large Re, although they are streamwise localized in the channel; the nonlinear optimal disturbances deviate the wall mode to trigger subcritical transitional flow.

Figure 4. The time variation of perturbation energy of velocity components $(a)$ and the ratios between the optimal disturbances $(b)$ in 2-D plane Poiseuille flow at different $Re$.

Figure 5 compares the initial optimal disturbances represented by disturbance streamwise velocity and the disturbance vorticity. Generally, the value of the vorticity is approximately one order larger than the value of the streamwise velocity. According to the definition of vorticity $\omega = u_y - v_x$ and the asymptotic scaling of the wall mode, it is very reasonable to obtain these results. For the nonlinear optimal disturbances, the initial value of streamwise velocity and vorticity decreases with $Re$. Moreover, the disturbance distributions are very similar between $Re = 3500$ and $Re = 5000$, while the disturbances are more distorted for $Re = 2400$. It should be noted that the values of streamwise velocity and vorticity for the linear optimal disturbances $E_0 = 4.076686 \times 10^{-4}$ (see figure 5a,a $'$) are only approximately twice of those of the minimal seeds $E_0 = 9.24967 \times 10^{-6}$ (see figure 5c,c $'$) at $Re = 3500$. The streamwise localization makes the minimal seeds much more efficient to trigger the subcritical transitional flow, since the strong nonlinear interactions are fully considered including the Orr amplification and edge states.

Figure 5. The comparison of streamwise velocity of (a–d) and vorticity (a $'$–d $'$) optimal disturbances in 2-D plane Poiseuille flow at different $Re$. Panels (a,a $'$), (b,b $'$), (c,c $'$) and (d,d $'$) show the results for $Re = 3500$, 2400, 3500 and $5000$, respectively. Panels (a,a $'$) show the linear initial optimal disturbances, while (b,b $'$), (c,c $'$) and (d,d $'$) show the nonlinear initial optimal disturbances.