2. Theoretical analysis

For the Poiseuille flow in the 2-D plane driven by a constant pressure gradient with the base flow $U(y) = (1-y^2) \boldsymbol{e_x}$ , the Reynolds number $ \textit{Re} = U_{\textit{max}} h / \nu$ where $h$ is the half-depth of the channel. Let the disturbances be $\boldsymbol{u} =\boldsymbol{u}_{\textit{tot}} - \boldsymbol{U}$ , $p = p_{\textit{tot}} - P$ , where $\boldsymbol{u}_{\textit{tot}}$ and $ p_{\textit{tot}}$ are the total velocity vector and pressure, and $U$ and $P$ are the velocity vector and pressure of the steady state base flow. Therefore, the NS equations are

(2.1) \begin{align} \frac {\partial \boldsymbol{u} } {\partial 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}{\textit{Re}} {\nabla} ^2 \boldsymbol{u}, \quad \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} = 0. \end{align}

Here $\boldsymbol{u}$ is the velocity disturbance vector $(u, v)$ , and the imposed conditions are $\boldsymbol{u}(x, \pm 1) = 0$ on the walls. According to (2.1), we can obtain the governing equation of wall-normal velocity $v$ without the pressure gradient

(2.2) \begin{align} \frac {\partial {\nabla} ^2 v} {\partial t } + U {\nabla} ^2 v_x - \frac { {\rm d}^2 U}{{\rm d} y^2} v_x - \frac {{\nabla} ^4 v}{\textit{Re}} = \boldsymbol{Non} (v), \quad \ u_x + v_y = 0, \end{align}

where the nonlinear terms $\boldsymbol{Non} (v)$ are

(2.3) \begin{align} \boldsymbol{Non} (v) = v_x(u_{\textit{yy}} - v_{\textit{xy}}) + v_y(v_{\textit{xx}} + v_{\textit{yy}}) - u(v_{\textit{xxx}} + v_{\textit{xyy}}) - v(v_{\textit{xxy}} + v_{\textit{yyy}}). \end{align}

The nonlinear interactions are evoked after the Orr amplification, and the nonlinear edge state could be reached (Huang et al. Reference Huang, Gao, Gao and Xi2024). For the periodic boundary condition in the streamwise direction, we can decompose the velocity disturbance into Fourier modes. For example,

(2.4) \begin{align} v(x,y,t) = \sum _{n=0}^{N} \sum _{k_t = 0}^{N_t} \sum _{k_x=0}^{N_x} \tilde {v}_n(y) \exp ^{(-i \omega k_t t + i \alpha k_x x)}, \end{align}

where $\tilde {v}_n(y)$ is the $n$ th continuous function in the wall-normal direction $y$ , $\omega$ is the frequency and $\alpha$ is the streamwise wavenumber. Here $n, k_t$ and $k_x$ are the summation indices with the truncation order of $N, N_k$ and $N_x$ , respectively. However, wave-like solutions are popular in stability analysis according to previous studies (Schmid & Henningson Reference Schmid and Henningson2001; Chapman Reference Chapman2002). Therefore, the wave-like solutions are employed in this study,

(2.5) \begin{align} v(x,y,t) = \sum \tilde {v}(y) \exp ^{(-i \omega t + i \alpha x)}, \end{align}

where $\tilde {v}(y)$ is the continuous function in the wall-normal direction. Then, we can obtain

(2.6) \begin{align} (i \alpha U - i \omega ) \left ( \frac {\mathrm{d}^2 } {\mathrm{d} y^2} - \alpha ^2 \right ) \tilde {v} - i \alpha \frac {\mathrm{d}^2 U } {\mathrm{d} y^2} \tilde {v} - \frac {1}{\textit{Re}} \left (\frac {\mathrm{d}^2 } {\mathrm{d} y^2} - \alpha ^2 \right )^2 \tilde {v} &= \boldsymbol{Non} (\tilde {v}), \end{align}

(2.7) \begin{align} \ i \alpha \tilde {u} &= - \mathrm{d} \tilde {v}/ \mathrm{d}y. \end{align}

According to the continuity (2.7), the nonlinear terms are approximated as follows:

(2.8) \begin{align} \boldsymbol{Non} (v) & = v_x (u_{\textit{yy}} - v_{\textit{xy}}) + v_y (v_{\textit{xx}} + v_{\textit{yy}}) - u(v_{\textit{xxx}} + v_{\textit{xyy}}) - v(v_{\textit{xxy}} + v_{\textit{yyy}}) \nonumber \\& \quad \sim i\alpha \tilde {v} \left ( - \frac {1}{i \alpha } \frac {{\rm d}^3 \tilde {v} }{{\rm d}y^3 } - {i \alpha } \frac {{\rm d} \tilde {v} }{{\rm d}y } \right ) + \frac {{\rm d} \tilde {v} }{{\rm d}y } \left ( - \alpha ^2 \tilde {v} + \frac {{\rm d}^2 \tilde {v} }{{\rm d}y^2 } \right ) \nonumber \\& \quad - \frac {1}{i \alpha } \frac {{\rm d} \tilde {v} }{{\rm d}y } \left ( - i \alpha ^3 \tilde {v} + i \alpha \frac {{\rm d}^2 \tilde {v} }{{\rm d}y^2 } \right ) - \tilde {v} \left ( i \alpha ^2 \frac {{\rm d} \tilde {v} }{{\rm d}y } + \frac {{\rm d}^3 \tilde {v} }{{\rm d}y^3 } \right ) \nonumber \\& \quad \sim \tilde {v} \left ( - \frac {{\rm d}^3 \tilde {v} }{{\rm d}y^3 } + \alpha ^2 \frac {{\rm d} \tilde {v} }{{\rm d}y } \right ) + \frac {{\rm d} \tilde {v} }{{\rm d}y } \left ( - \alpha ^2 \tilde {v} + \frac {{\rm d}^2 \tilde {v} }{{\rm d}y^2 } \right ) \nonumber \\& \quad - \frac {{\rm d} \tilde {v} }{{\rm d}y } \left ( - \alpha ^2 \tilde {v} + \frac {{\rm d}^2 \tilde {v} }{{\rm d}y^2 } \right ) - \tilde {v} \left ( i \alpha ^2 \frac { {\rm d}\tilde {v} }{{\rm d}y } + \frac {{\rm d}^3 \tilde {v} }{{\rm d}y^3 } \right )\!. \end{align}

The wall-normal derivative would be more significant near the channel walls for the disturbance of the wall mode (Chapman Reference Chapman2002; Huang et al. Reference Huang, Gao, Gao and Xi2024). Then, the leading terms are $v_x u_{\textit{yy}}$ and $v v_{\textit{yyy}}$ , obviously with the same magnitude as $\tilde {v} {\rm d}^3 \tilde {v}/ {\rm d}y^3$ . Therefore, $\boldsymbol{Non} (\tilde {v}) \sim \tilde {v} {\rm d}^3\tilde {v}/{\rm d}y^3$ is chosen for the following analysis for convenience.

Lin (Reference Lin1944) introduced $(\alpha \textit{Re})^{1/3}$ into the linearized NS equations (Orr–Sommerfeld (OS) equations) to study the linear instability analysis; asymptotic analysis was performed and great success was achieved since the critical Reynolds number $ \textit{Re}_c$ is predicted to be approximately $8000$ ( $ \textit{Re}^{1/3}$ is approximately 20) for the 2-D plane Poiseuille flow. Here $ \textit{Re}_{c} = 5772.2$ is well known now for the disturbance of wall mode in the 2-D plane Poiseuille flow (Orszag Reference Orszag1971; Schmid & Henningson Reference Schmid and Henningson2001). Chapman (Reference Chapman2002) performed the asymptotic analysis to investigate the subcritical transition in channel flows, mainly focused on the wall and centre modes. Therefore, inspired by the asymptotic analysis (Lin Reference Lin1944; Chapman Reference Chapman2002), nonlinear disturbances of the wall mode (Huang et al. Reference Huang, Gao, Gao and Xi2024) are considered, and the amplitude required for the nonlinear interaction could be achieved when it is sufficient to induce instability and trigger the subcritical transition. For the plane Poiseuille flow, the Orr amplification of the disturbance occurs near the channel wall due to the notable mean shear, and the nonlinear optimal disturbances are found to be wall modes (Huang et al. Reference Huang, Gao, Gao and Xi2024). Before the nonlinear terms are notable, let us consider the OS equation

(2.9) \begin{align} (i \alpha U - i \omega ) \left ( \frac {\mathrm{d}^2 } {\mathrm{d} y^2} - \alpha ^2 \right ) \tilde {v} - i \alpha \frac {\mathrm{d}^2 U } {\mathrm{d} y^2} \tilde {v} - \frac {1}{\textit{Re}} \left (\frac {\mathrm{d}^2 } {\mathrm{d} y^2} - \alpha ^2 \right )^2 \tilde {v} = 0. \end{align}

Since the wall modes near the channel wall obey the following scalings (Chapman Reference Chapman2002):

(2.10) \begin{align} y = -1 + (\textit{Re} \alpha )^{-1/3}{y}^{\prime}, \quad \omega / \alpha = \gamma = (\textit{Re} \alpha )^{-1/3}\gamma^{\prime}. \end{align}

Substituting (2.10) into (2.9), the inner flow near the channel should obey the following equation:

(2.11) \begin{align} \big [ i \alpha (1&- ((\textit{Re} \alpha )^{-1/3} {y}^{\prime} -1)^2 ) - i \alpha (\textit{Re} \alpha )^{-1/3} \gamma^{\prime} \big ] \left [(\textit{Re} \alpha )^{2/3} \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} - \alpha ^2 \tilde {v}_i \right ] \nonumber \\ &+ 2 i \alpha \tilde {v}_i - \frac {1}{\textit{Re}} \left [ (\textit{Re} \alpha )^{2/3} \frac {\mathrm{d}^2 } {\mathrm{d} {y}^{\prime2}} - \alpha ^2 \right ]^2 \tilde {v}_i = 0, \end{align}

which is

(2.12) \begin{align} & (\textit{Re} \alpha )^{1/3} \left [ i ( 2y^{\prime} - \gamma^{\prime}) \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} - \frac {\mathrm{d}^4 \tilde {v}_i } {\mathrm{d} {y}^{\prime 4}} \right ] + 2i \tilde {v}_i - i {y}^{\prime2} \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} \nonumber\\ & \quad + (\textit{Re} \alpha )^{-1/3} \left [ 2 \alpha ^2 \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} - i ( 2y^{\prime} - \gamma^{\prime}) \alpha ^2 \tilde {v}_i \right ] + (\textit{Re} \alpha )^{-2/3} {y}^{\prime2} \alpha ^2 \tilde {v}_i - \frac {\alpha ^4 \tilde {v}_i}{\textit{Re} \alpha } = 0. \end{align}

Therefore, the leading equation for the inner flow near the channel wall is

(2.13) \begin{align} (\textit{Re} \alpha )^{1/3} \left [ i ( 2y^{\prime} - \gamma^{\prime}) \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} - \frac {\mathrm{d}^4 \tilde {v}_i } {\mathrm{d} {y}^{\prime 4}} \right ] = 0, \end{align}

with boundary conditions $ \tilde {v}_i = \mathrm{d} \tilde {v}_i /{\rm d}y = 0$ on the wall $ y = -1$ or ${y}^{\prime} = 0$ . If the coefficient $(\textit{Re} \alpha )^{1/3}$ is not zero, the following equation is obtained:

(2.14) \begin{align} i ( \gamma^{\prime}- 2y^{\prime}) \frac {\mathrm{d}^2 \tilde {v}_i} {\mathrm{d} {y}^{\prime2}} + \frac {\mathrm{d}^4 \tilde {v}_i } {\mathrm{d} {y}^{\prime 4}} = 0, \end{align}

which is the same as the (5.35) in Chapman (Reference Chapman2002). Consequently, according to the rigorous asymptotic analysis in §§ 4 and 5 of Chapman (Reference Chapman2002), the left-hand terms of (2.6) should be approximately $(\alpha \textit{Re})^{1/3} \tilde {v}_i$ near the channel wall. Next, we should focus on the amplitude of the leading nonlinear terms on the right-hand side of (2.6).

The inner solution has the form of the Airy function for the leading terms, according to (Lin Reference Lin1944)

(2.15) \begin{align} \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} \sim \textrm {Airy} \big[ (2y^{\prime}- \gamma^{\prime}) i^{1/3} \big], \quad \frac {\mathrm{d}^4 \tilde {v}_i } {\mathrm{d} {y}^{\prime 4}} \sim (2y^{\prime}- \gamma^{\prime}) \textrm {Airy} \big[(2y^{\prime}- \gamma^{\prime}) i^{1/3} \big]. \end{align}

In other words,

(2.16) \begin{align} \frac {\mathrm{d}^4 \tilde {v}_i } {\mathrm{d} {y}^{\prime 4}} \sim (2y^{\prime}- \gamma^{\prime}) \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}}. \end{align}

It is straightforward to obtain

(2.17) \begin{align} \frac {\mathrm{d}^2 \tilde {v}_i } {\mathrm{d} {y}^{\prime2}} \sim (2y^{\prime}- \gamma^{\prime}) \tilde {v}_i , \quad \frac {\mathrm{d}^3 \tilde {v}_i } {\mathrm{d} {{y}^{\prime}}^3} \sim (2y^{\prime}- \gamma^{\prime})^{3/2} \tilde {v}_i, \quad \text{and} \quad \frac {\mathrm{d}^4 \tilde {v}_i } {\mathrm{d} {y}^{\prime 4}} \sim (2y^{\prime}- \gamma^{\prime})^{2} \tilde {v}_i. \end{align}

For the outer solution, the unsteady and diffusive terms are usually omitted (Chapman Reference Chapman2002) to obtain the inviscid flow equation. Then, (2.9) could be simplified as follows:

(2.18) \begin{align} (i \alpha U ) \left ( \frac {\mathrm{d}^2 } {\mathrm{d} y^2} - \alpha ^2 \right ) \tilde {v}_o - i \alpha \frac {\mathrm{d}^2 U } {\mathrm{d} y^2} \tilde {v}_o = 0. \end{align}

Therefore, the outer inviscid flow could be obtained by solving the equation

(2.19) \begin{align} \big( 1-y^2 \big) \left ( \frac {\mathrm{d}^2 } {\mathrm{d} y^2} - \alpha ^2 \right ) \tilde {v}_o + 2 \tilde {v}_o = 0 . \end{align}

The outer flow solution has the following form:

(2.20) \begin{align} \tilde {v}_{o} \sim c_1[ 1 + (1+y) \log (1+y)] + c_2 \big[(1+y) + 0.5(1+y)^2 + \ldots \big], \end{align}

with two certain non-zero coefficients (Chapman Reference Chapman2002). It implies that the inner limit of the outer solution has the following estimations $\tilde {v}_{o} \sim c_1$ and $ \mathrm{d} \tilde {v}_{o} /\mathrm{d} y \sim c_1+ c_2$ as $y$ tends to −1, which offers the boundary condition approximation for the outer limit of the inner solution.

According to nonlinear non-modal analysis (Kerswell Reference Kerswell2018), the nonlinear energy-conserved edge state is approached with a sufficiently long time period. Moreover, the subcritical transition is triggered by the optimal disturbance with an energy slightly larger than the critical energy (Rabin et al. Reference Rabin, Caulfield and Kerswell2012; Kerswell Reference Kerswell2018; Huang et al. Reference Huang, Gao, Gao and Xi2024). Consequently, the subcritical transition processes could be treated as two main stages including the Orr amplification and the evoked nonlinear interactions for 2-D plane Poiseuille flow, because the triggered subcritical transition after the nonlinear edge state is easily achieved by increasing the critical energy slightly. Therefore, if the required magnitude of the nonlinear edge state could be quantified, the transition thresholds might be estimated with the transient growth of $ \textit{Re}^2$ . For the sustained nonlinear edge state, the left-hand leading term of (2.6) should be balanced by the leading nonlinear term to induce instability. In this study, we assumed that the two terms are balanced near the matched boundary, which has not been investigated before. For the inner flow, all the terms on the left-hand side are considered and are of the order of $(\textit{Re} \alpha )^{1/3} \tilde {v}_i$ after the Orr amplification according to (2.14) (Chapman Reference Chapman2002). While for the outer flow, the leading nonlinear term on the right-hand side is dominated by $\tilde {v}_{o} \mathrm{d}^3 \tilde {v}_o / \mathrm{d} y^3$ . Near the matched boundary, i.e. the innermost of the outer solution, the solution is assumed to be sufficiently continuous, which is physically reasonable, then $ \tilde {v}_{o} \sim \tilde {v}_{i}$ and ${\rm d}^3 \tilde {v}_{o}/{\rm d}y^3 \sim {\rm d}^3 \tilde {v}_{i} /{\rm d}y^3$ . Based on the scaling (2.10) in the inner solution region, we can obtain

(2.21) \begin{align} \tilde {v}_{o} \frac { \mathrm{d}^3 \tilde {v}_o } { \mathrm{d} y^3} \sim \tilde {v}_{o} \frac { \mathrm{d}^3 \tilde {v}_i } { \mathrm{d} y^3} = \tilde {v}_{o} \big [(\textit{Re} \alpha )^{1/3} \big ]^3 \frac { \mathrm{d}^3 \tilde {v}_i } { \mathrm{d} {{y}^{\prime}}^3} = (\textit{Re} \alpha ) \tilde {v}_{o} \frac {\mathrm{d}^3 \tilde {v}_i }{ \mathrm{d} {{y}^{\prime}}^3 }. \end{align}

Since the inner solution has the form of the Airy function as shown in (2.17), we can obtain

(2.22) \begin{align} \frac {\mathrm{d}^3 \tilde {v}_i} { \mathrm{d} {{y}^{\prime}}^3} \sim (2y^{\prime}-\gamma^{\prime})^{3/2} \tilde {v}_i = \big[(2+2y-\gamma )(\textit{Re} \alpha )^{1/3}\big]^{3/2}\tilde {v}_i. \end{align}

Hence, the magnitude of the term could be estimated as

(2.23) \begin{align} \frac {\mathrm{d}^3 \tilde {v}_i} { \mathrm{d} {{y}^{\prime}}^3} \sim (\textit{Re} \alpha )^{1/2}\tilde {v}_i , \end{align}

if only the leading terms are considered. Therefore, we can obtain the magnitude of the leading nonlinear term near the matched boundary

(2.24) \begin{align} \tilde {v}_{o} \frac { \mathrm{d}^3 \tilde {v}_o} { \mathrm{d} y^3} \sim (\textit{Re} \alpha ) \tilde {v}_{o} \frac {\mathrm{d}^3 \tilde {v}_i } {\mathrm{d} {{y}^{\prime}}^3 } \sim (\textit{Re} \alpha )^{3/2} \tilde {v}_{o} \tilde {v}_i . \end{align}

Consequently, $ (\textit{Re} \alpha )^{3/2} \tilde {v}_{o} \tilde {v_i}$ should be balanced by the left-hand term $(\textit{Re} \alpha )^{1/3} \tilde {v_i}$ to maintain the nonlinear edge state and induce instability to the inner solution near the outer limit of the inner solution. That is to say,

(2.25) \begin{align} (\textit{Re} \alpha )^{1/3} \tilde {v_i} \sim (\textit{Re} \alpha )^{3/2} \tilde {v}_{o} \tilde {v_i}, \ \tilde {v}_{o} \sim (\textit{Re} \alpha )^{-7/6}. \end{align}

It is noteworthy that the leading nonlinear term is caused by the mutual nonlinear interactions between $u$ and $v$ . This implies that the perturbation magnitudes of both $u$ and $v$ are approximate $(\textit{Re}\alpha )^{-7/6}$ along the edge state to trigger the subcritical transitional flow. Recall that the $O(\textit{Re} \alpha )$ amplification of velocity $v$ by the Orr mechanism according to (2.9) before nonlinear interactions are evoked. Consequently, the optimal initial perturbation of wall mode for velocity $v$ is required to be $(\textit{Re}\alpha )^{-13/6}$ in order to facilitate the onset of subcritical transitional flow. As concluded in the previous study (Huang et al. Reference Huang, Gao, Gao and Xi2024), the nonlinear optimal disturbances manifest as wall modes, with the initial kinetic energy of disturbance $u$ being approximately $(\textit{Re} \alpha )^{2/3}$ times greater than that of $v$ , i.e. $E_{u0} \sim (\textit{Re} \alpha )^{2/3} E_{v0}$ . That is to say, the scaling law of the transition threshold is dominated by the velocity perturbation $u$ and should be around $(\textit{Re} \alpha )^{-11/3}$ for the subcritical transitional flow in the 2-D plane Poiseuille flow.

In summary, the subcritical transitional thresholds in the 2-D plane Poiseuille flow are as follows. For minimal seeds, $ E_{v\textit{ms}} \sim (\textit{Re} \alpha )^{-13/3}$ and $E_{\textit{ums}} \sim (\textit{Re} \alpha )^{-11/3}$ , this consequently leads to $E_{0ms} = (E_{\textit{ums}}+E_{v\textit{ms}}) \sim (\textit{Re} \alpha )^{-11/3}$ . The disturbance energy required along the saturated nonlinear interacted edge state is $ E_{\textit{ues}} \sim E_{v\textit{es}} \sim (\textit{Re} \alpha )^{-7/3}$ . The theoretical predictions are consistent with the conjecture (see Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993) since $\kappa =11/3 \gt 2$ .

3. Numerical experiments and results

The 2-D turbulent flow is strictly limited in laboratory or nature (Boffetta & Ecke Reference Boffetta and Ecke2012; Sulpizio et al. Reference Sulpizio, Ella, Rozen, Birkbeck, Perell, Dutta, Ben-Shalom, Taniguchi, Watanabe and Holder2019), therefore, numerical experiments based on DNS are the reasonable choice. To capture the subcritical transition experiencing the edge state, the disturbance energy density is defined as

(3.1) \begin{align} E(t) = \frac {1}{2A} \int _{A} \big[u^2(t) + v^2(t) \big]\, {\rm d}A \end{align}

where $A$ is the area of the 2-D channel. The energy densities associated with disturbances in the streamwise and vertical velocity components are

(3.2) \begin{align} E_u(t) = \frac {1}{2A} \int _{A} u^2(t)\, {\rm d}A, \quad E_v(t) = \frac {1}{2A} \int _{A} v^2(t)\, {\rm d}A .\end{align}

Nonlinear optimizations are performed using the well-developed direct-adjoint looping method (Pringle & Kerswell Reference Pringle and Kerswell2010) based on Diablo DNS codes (Taylor Reference Taylor2008). The energy gain of the disturbance is chosen as the objective function (Rabin et al. Reference Rabin, Caulfield and Kerswell2012), and convergence is obtained when the variation of the energy gain between two successive iterations is smaller than $1\times 10^{-6}$ . To identify the minimal seed, the disturbance energy bisection technique is used (Rabin et al. Reference Rabin, Caulfield and Kerswell2012; Huang et al. Reference Huang, Gao, Gao and Xi2024). Firstly, the optimization is performed with the random initial disturbance of a sufficiently large initial disturbance energy to achieve subcritical transitional flow with target time $T$ . Then, the computed disturbance rescaled with a much smaller initial energy is used as the initial guess for the optimization to obtain the relaminarization. Consequently, the disturbance energy is bisected to repeat the optimization to obtain the nonlinear edge state and the numerical transition threshold. To observe saturated nonlinear interactions of the edge state, the target times are $T = 1000$ for $Lx = 8 \pi$ and $T = 2000$ for $Lx = 20 \pi$ , respectively, based on the previous study (Huang et al. Reference Huang, Gao, Gao and Xi2024).

According to previous studies (Jiménez Reference Jiménez1990; Huang et al. Reference Huang, Gao, Gao and Xi2024), long channels with lengths $Lx = 8 \pi$ and $20 \pi$ with grid resolutions $512\times 512$ and $1024\times 512$ , respectively, are considered in the present study based on grid independence tests in Huang et al. (Reference Huang, Gao, Gao and Xi2024). Furthermore, to avoid the effects of numerical resolutions on the transition thresholds at high Reynolds numbers (Duguet et al. Reference Duguet, Brandt and Larsson2010), double grid resolutions $1024\times 1024$ and $2048\times 1024$ are considered for $Lx = 8 \pi$ and $20 \pi$ , respectively, and the difference is less than one per cent at $ \textit{Re} = 5400$ . Due to 2-D nonlinear travelling wave effects on the plane Poiseuille flow, subcritical transition states could be identified at $ \textit{Re} = 2939$ (Hagan & Priede Reference Hagan and Priede2015). Therefore, the range from 2950 to 5400 is focused on verifying the theory of asymptotic analysis.

Figure 1.

Time evolution of energy densities of minimal seed at different Re for (a) $Lx = 8 \pi$ and (b) $Lx = 20 \pi$ ; time variation of energy density parts at $ \textit{Re} = 4500$ for (c) $Lx = 8 \pi$ and (d) $Lx = 20 \pi$ .

Figure 2.

The evolution of minimal seed is represented by vorticity at different time instants for $Lx = 8 \pi$ and $ \textit{Re} =4500$ . The time instants for snapshots $(a) {-} (f)$ are $t = 0$ , 20, 30, 400, 700 and 1000, respectively.

3.2. Thresholds for subcritical transition

Theoretical analysis shows that the magnitudes of disturbances of $u$ and $v$ are approximately $O(\textit{Re}^{-7/6})$ around the edge state for the minimal seeds. Recall that the amplification of $O(\textit{Re})$ for velocity $v$ by the Orr mechanism, the optimal wall mode disturbance for $v$ should be $O(\textit{Re}^{-13/6})$ . It should be noted that the optimal disturbance obeys the wall mode scaling rules, the disturbance energy of $E_u$ is approximately $O(\textit{Re}^{2/3})$ times $E_v$ . Therefore, the thresholds that reveal the full processes of the most dangerous and efficient subcritical transitional flow in energy form are listed in table 1, $ E_{v\textit{ms}} \sim \textit{Re}^{-13/3}$ , $ E_{\textit{ums}} \sim \textit{Re}^{-11/3}$ and $E_{0ms} = (E_{\textit{ums}}+E_{v\textit{ms}}) \sim \textit{Re}^{-11/3}$ , since the energy along the edge state is $E_{\textit{es}} \sim \textit{Re}^{-7/3}$ .

Table 1.

Comparisons of theoretical and numerical threshold $\kappa$ for the energy parts. The subscript ms denotes minimal seed and es denotes edge state.

Huang et al. (Reference Huang, Gao, Gao and Xi2024) found that the energy of the initial linear optimal disturbances would be larger than the energy of their edge state for 2-D plane Poiseuille flow. It means that the linear optimal disturbances are not good candidates for theoretical validations. The minimal seed represents the most dangerous disturbance that triggers the subcritical transition (Kerswell Reference Kerswell2018). Therefore, the numerical scaling law of the threshold obtained directly from the minimal seed is perfect to verify these predictions. The energy density and the energy parts of the minimal seed are shown in figures 3(a) and 3(b). The energy threshold decays with the Reynolds number as $ \textit{Re}^{-3.71}$ and $ \textit{Re}^{-3.62}$ for $Lx = 8 \pi$ and $Lx = 20 \pi$ , respectively. The numerical scaling laws agree well with the theoretical predicted value −11/3. Interestingly, the initial energy of the streamwise velocity $E_{\textit{ums}}$ is dominant and close to the total perturbation energy $E_{0ms}$ , while the initial energy of the vertical velocity $E_{v\textit{ms}}$ is much smaller. More precisely, $E_{\textit{ums}}$ scales as $ \textit{Re}^{-3.70}$ and $ \textit{Re}^{-3.61}$ for $Lx = 8 \pi$ and $Lx = 20\pi$ , respectively; while $E_{v\textit{ms}}$ scales as $ \textit{Re}^{-4.37}$ and $ \textit{Re}^{-4.30}$ for $Lx = 8 \pi$ and $Lx = 20 \pi$ , respectively. Furthermore, it should be noted that $E_{\textit{ums}}$ is approximately $ \textit{Re}^{0.67}$ and $ \textit{Re}^{0.69}$ times $E_{v\textit{ms}}$ for $Lx = 8 \pi$ and $Lx = 20 \pi$ , respectively. The numerical results are in perfect agreement with the theoretical analysis of wall mode disturbances.

Figure 3.

The scalings of disturbance energy threshold of minimal seed with respect to $ \textit{Re}$ in log–log scale for (a) $Lx = 8 \pi$ and (b) $Lx = 20 \pi$ ; the scalings of energy of overshoot and edge state with respect to $ \textit{Re}$ in log–log scale for (c) $Lx = 8 \pi$ and (d) $Lx = 20 \pi$ .

The numerical results show that the target times are long enough to capture the long-sustained nonlinear edge state and transition thresholds for different channel lengths. This indicates that the theoretical predictions of the disturbance energy of the edge state should be correct. Since the energy varies very slowly along the saturated nonlinear edge state, the naïve and robust mean energy densities are adopted to observe the scaling laws. To measure the disturbance energy along the edge state, the disturbance energy is averaged from $t = 400$ to 700 for $Lx = 8 \pi$ , while the energy is averaged from $t = 400$ to 1400 for $Lx = 20 \pi$ . Therefore, the energy and energy parts of the edge state with saturated nonlinear interactions near the transition point are also summarized and shown in figures 3(c) and 3(d). More interestingly and importantly, $E_{0ms}$ and $E_{\textit{ums}}$ along the nonlinear edge state scale with $ \textit{Re}$ as $ \textit{Re}^{-2.35}$ and $ \textit{Re}^{-2.29}$ , respectively, for $Lx = 8 \pi$ ; while $E_{0ms}$ and $E_{\textit{ums}}$ scale with $ \textit{Re}$ as $ \textit{Re}^{-2.36}$ and $ \textit{Re}^{-2.32}$ , respectively, for $Lx = 20 \pi$ . All the scaling laws are very close to the theoretical prediction $ \textit{Re}^{-2.33}$ along the edge state, since $E_{v\textit{ms}}$ scales with $ \textit{Re}$ as $ \textit{Re}^{-2.46}$ for $Lx = 8 \pi$ and $ \textit{Re}^{-2.43}$ for $Lx = 20 \pi$ , which are slightly smaller than the prediction.

The disturbance energy of the overshoot $E_{os}$ with respect to $ \textit{Re}$ is also presented in figures 3(c) and 3(d). We can see that the disturbance energy overshoots $E_{os}$ are very close to the $E_{0ms}$ along the edge state. More specifically, $E_{os}$ is slightly higher than $E_{0ms}$ for small $ \textit{Re}$ , while $E_{os}$ is slightly lower than $E_{0ms}$ for large $ \textit{Re}$ . Therefore, the scale of $E_{os}$ is sharper than that of $E_{0ms}$ , and $E_{os}$ scales $ \textit{Re}^{-2.58}$ and $ \textit{Re}^{-2.47}$ for $ Lx = 8 \pi$ and $ Lx = 20 \pi$ , respectively. In this study, we focus on saturated nonlinear interactions along the nonlinear edge state with a sufficiently long time period both theoretically and numerically. Therefore, the time-averaged disturbance energy along the nonlinear edge state could be better to show the nonlinear interactions, while the overshoots might not reflect the full information of the nonlinear interactions.

The disturbance energy should be approximately $ \textit{Re}^{-7/3}$ to sustain the nonlinear edge state in 2-D plane Poiseuille flow, which is similar to the results of the 3-D plane Couette flow since the streamwise rolls of approximately $ \textit{Re}^{-2}$ are needed to sustain the edge state with streaks for the coherent states of the lower branch (Wang, Gibson & Waleffe Reference Wang, Gibson and Waleffe2007). It is interesting that the required disturbance energies are close for different shear flows, which indicates that the present asymptotic analysis in 2-D plane Poiseuille flow might be helpful for 3-D shear flows. However, the total disturbance energies to sustain the edge state before triggering the subcritical transition are quite different. For the 2-D plane Poiseuille flow, the streamwise and wall-normal disturbance energies should be approximately $ \textit{Re}^{-7/3}$ , and the transitional threshold is approximately $ \textit{Re}^{-11/3}$ based on the wall mode disturbance (Huang et al. Reference Huang, Gao, Gao and Xi2024). However, for the 3-D plane Couette flow, the streamwise rolls of $ \textit{Re}^{-2}$ are needed to sustain the streamwise streaks of $ \textit{Re}^{0}$ , and the transitional threshold would be $ \textit{Re}^{-2}$ , which is consistent with the results of the linear optimal disturbance (Duguet et al. Reference Duguet, Brandt and Larsson2010). Nevertheless, the results show that the proposed mechanism and theoretical framework to quantify the subcritical transition thresholds are correct in 2-D plane Poiseuille flow, and it might be helpful to estimate the nonlinear transition thresholds for the 3-D plane Couette flow.