2.1. Governing equations

We consider a unidirectional base flow velocity in the $x$ -direction which has a constant shear in $y$ and a spatially periodic ‘streak’ shear varying in the spanwise $z$ -direction

(2.1) \begin{equation} {\boldsymbol{U}}_{\!{B}} = U_{\!B}(y,z)\boldsymbol e_x = [y + \beta \cos {(k_zz)}] \boldsymbol e_x, \end{equation}

where $\beta$ is the dimensionless streak strength: see figure 1. The system has been non-dimensionalised by the $y$ -shear rate, $S$ , and the initial $k_y$ wavenumber (i.e. by a length scale $L:=L_y/2\pi$ where $L_y$ is the initial perturbation wavelength in $y$ ), so that the streak wavenumber $k_z$ is a parameter of the problem. This allows straightforward access to the well-studied Kelvin problem of $\beta \rightarrow 0$ while using the spanwise wavenumber as an inverse length scale does not.

The Navier–Stokes equations linearised around ${\boldsymbol{U}}_{\!{B}}$ for a perturbation ${\boldsymbol u}:=(u,v,w)$ and associated pressure perturbation $p$ are then

(2.2) \begin{align} \frac {\partial \boldsymbol u}{\partial t} + [y + \beta \cos {(k_zz)}] \frac {\partial \boldsymbol u}{\partial x} & + [v-\beta w k_z \sin {(k_zz)}] \boldsymbol e_x + \boldsymbol{\nabla }p = \frac {1}{\textit{Re}} \varDelta \boldsymbol u, \end{align}

(2.3) \begin{align} & \boldsymbol{\nabla } \boldsymbol{\cdot }\boldsymbol u = 0. \end{align}

Here, the Reynolds number is $\textit{Re}:=L^2 S/\nu$ , where $\nu$ is the kinematic viscosity. Taking $\boldsymbol e_y \! \boldsymbol{\cdot }\! {} \boldsymbol{\nabla }\times {} \boldsymbol{\nabla }\times$ and $\boldsymbol e_y \!\boldsymbol{\cdot }\! {} \boldsymbol{\nabla }\times$ of (2.2) leads to a pair of equations for $v$ and the $y$ -shearwise vorticity $\eta := \partial u / \partial z - \partial w/\partial x$ , respectively,

(2.4) \begin{align} \left [\frac {\partial }{\partial t} + [y + \beta \cos {(k_zz)}] \frac {\partial }{\partial x} - \frac {1}{\textit{Re}}\varDelta \right ] \varDelta v &+ 2\beta k_z \sin {(k_z z)} \left [\frac {\partial ^2 w}{\partial x \partial y} - \frac {\partial ^2 v}{\partial x \partial z} \right ] \nonumber \\ &- \beta k_z^2 \cos {(k_zz)}\frac {\partial v}{\partial x} =0, \end{align}

(2.5) \begin{align} \left [\frac {\partial }{\partial t} + [y + \beta \cos {(k_zz)}] \frac {\partial }{\partial x} - \frac {1}{\textit{Re}}\varDelta \right ] & \eta + \frac {\partial v}{\partial z} + \beta k_z \sin {(k_zz)}\frac {\partial v}{\partial y} - \beta w k_z^2 \cos {(k_zz)} =0. \end{align}

2.2. Kelvin modes

The well-known trick to handle the $y$ -dependent advection term is to allow the wavenumbers of the perturbation to be time-dependent (Kelvin Reference Kelvin1887) giving a ‘Kelvin’ mode

(2.6) \begin{equation} [u,v,w,p,\eta ](x,y,z,t) = [\hat {u},\hat {v},\hat {w},\hat {p},\hat {\eta }](t){\textrm{e}}^{{\textrm{i}}\left [k_x x+(1-k_x t)y+k_z z\right ]}, \end{equation}

which is actually a full solution to the unlinearised perturbation equations since $\boldsymbol {k} \boldsymbol{\cdot }\boldsymbol u=0$ by incompressibility (e.g. Craik & Criminale Reference Craik and Criminale1986). For $\beta \neq 0$ , the extra streak shear unavoidably couples different spanwise wavenumbers and the perturbation ansatz has to be extended to

(2.7) \begin{equation} [u,v,w,p,\eta ](x,y,z,t) = \sum _{m=-M}^M [\hat {u}_m,\hat {v}_m,\hat {w}_m,\hat {p}_m,\hat {\eta }_m](t){\textrm{e}}^{{\textrm{i}}\left [k_xx+(1-k_xt)y+mk_z z\right ]}, \end{equation}

where, formally, $M$ should be infinite but, practically, is chosen large enough so as not to influence the results. This ansatz could be extended further by shifting the spanwise wavenumbers by a fraction of the base wavenumber – so $mk_z \rightarrow mk_z+\mu$ where $-(1/2)k_z \leqslant \mu \lt (1/2)k_z$ is the modulation parameter in Floquet theory – but only $\mu =0$ is treated here so the perturbation has the same wavelength as the base field (i.e. there is no modulation) consistent with numerical observations e.g. Schoppa & Hussain (Reference Schoppa and Hussain2002). Also, since ${\boldsymbol{U}}_{\!{B}}$ is symmetric about $z=0$ , perturbations which are either symmetric (‘varicose’) or antisymmetric (‘sinuous’) in $u$ about $z=0$ can be pursued separately. However, the computations are sufficiently low-cost that it was easier to just consider both cases together.

The continuity equation and definition of $\eta$ make it possible to express $\hat {u}_m$ and $\hat {w}_m$ in terms of $\hat {v}_m$ and $\hat {\eta }_m$ as follows:

(2.8) \begin{equation} \hat {u}_m = \displaystyle\frac {-k_x(1-k_xt)\hat {v}_m - {\textrm{i}}mk_z\hat {\eta }_m}{k_x^2+m^2 k_z^2} \quad \text{and} \quad \hat {w}_m = \displaystyle\frac {-m k_z(1-k_xt)\hat {v}_m + {\textrm{i}}k_x\hat {\eta }_m}{k_x^2+ m^2 k_z^2}, \end{equation}

and then (2.4) and (2.5) require for each integer $m$

(2.9) \begin{align} \dot {\hat {v}}_m +\left [-\displaystyle\frac {2k_x(1-k_xt)}{k_m^2}+ \displaystyle\frac {k_m^2}{\textit{Re}}\right ]\hat {v}_m &= \displaystyle\frac {{\textrm{i}}\beta k_x}{2k_m^2}\left [ 2(m+1)\displaystyle\frac {k_z^2k_{m+1}^2}{h_{m+1}^2} - k_z^2 - k_{m+1}^2 \right ]\hat {v}_{m+1} & \nonumber \\[8pt] &\quad-\, \displaystyle\frac {{\textrm{i}}\beta k_x}{2k_m^2}\left [ 2(m-1)\displaystyle\frac {k_z^2k_{m-1}^2}{h_{m-1}^2} + k_z^2 + k_{m-1}^2 \right ]\hat {v}_{m-1}& \nonumber \\[8pt] &\quad+\, \displaystyle\frac {\beta k_x^2 k_z (1-k_xt)}{k_m^2h_{m+1}^2}\hat {\eta }_{m+1} - \displaystyle\frac {\beta k_x^2 k_z (1-k_xt)}{k_m^2h_{m-1}^2}\hat {\eta }_{m-1},& \end{align}

and

(2.10) \begin{align} \dot {\hat {\eta }}_m +\frac {k_m^2}{\textit{Re}}\hat {\eta }_m = &- \frac {{\textrm{i}}k_x\beta }{2}\left (1- \frac {k_z^2}{h_{m+1}^2} \right )\hat {\eta }_{m+1} - \frac {{\textrm{i}}k_x\beta }{2}\left (1-\frac {k_z^2}{h_{m-1}^2} \right )\hat {\eta }_{m-1} \nonumber \\[8pt] &-\, {\textrm{i}} m k_z\hat {v}_m - \frac {\beta k_z(1-k_xt)}{2}\left (\frac {(m+1)k_z^2}{h_{m+1}^2}-1\right )\hat {v}_{m+1} \nonumber \\[8pt] &-\, \frac {\beta k_z (1-k_xt)}{2}\left (\frac {(m-1)k_z^2}{h_{m-1}^2}+1\right )\hat {v}_{m-1}, \end{align}

where $h_m^2:=k_x^2+m^2 k_z^2$ and $k_m^2:= k_x^2+(1-k_xt)^2+m^2 k_z^2$ . These $2(2M+1)$ coupled complex ODEs are readily integrated forward in time from given initial conditions using MATLAB’s ode45.

2.3. Kinetic energy and optimal gain

The volume-averaged kinetic energy is defined as

(2.11) \begin{equation} E(t) := \frac {1}{V_\varOmega }\int _\varOmega \frac {1}{2} |{\boldsymbol u(\textbf{x},t})|^2 {\textrm d}\varOmega =\frac {1}{2} \sum _{m=-M}^M \frac {1}{h_m^2} \big[k^2_m|{\hat {v}_m}|^2 +|{\hat {\eta }_m}|^2\big], \end{equation}

where $\varOmega :=[0,2\pi /k_x] \times [0,2 \pi ] \times [0,2\pi /k_z]$ and $V_\varOmega$ is the volume of $\varOmega$ . In this paper we define the ‘optimal gain’ as the largest value of $E(T)/E(0)$ over all possible initial conditions

(2.12) \begin{equation} G(T,k_x,k_z;\beta ,\textit{Re}) := \max _{\boldsymbol {u}(0)}\frac {E(T,k_x,k_z;\beta ,\textit{Re})}{E(0,k_x,k_z;\beta ,\textit{Re})}. \end{equation}

The ‘global optimal gain’ will be the result of optimising this quantity over the wavenumbers

(2.13) \begin{equation} {\mathscr{G}}(T;\beta ,\textit{Re}) := \max _{\{k_x,k_z\}}G(T,k_x,k_z;\beta ,\textit{Re}), \end{equation}

and the ‘overall optimal gain’ will be the result of optimising this over the time $T$

(2.14) \begin{equation} {\mathbb{G}}(\beta ,\textit{Re}) := \max _{T} {\mathscr{G}} (T;\beta ,\textit{Re}). \end{equation}

2.4. Parameter choices: buffer layer estimates

The focus in this study is to consider values for the five physical parameters – $T,k_x,k_x,\beta$ and ${\textit{Re}}$ – appropriate for the buffer layer (e.g. Jimenez Reference Jimenez2018; Lozano-Durán et al. Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021). Given there is no simple characterisation of the mean flow profile here as it adjusts from the viscous sublayer to the log layer and only order of magnitude estimates are sought, we adopt the empirical log layer representation

(2.15) \begin{equation} U^+(y^+) = U/u_\tau \approx \frac {1}{\kappa } \ln y^+ +B, \end{equation}

(where $y^+=yu_\tau /\nu$ is the distance from the wall in so-called viscous units, $u_\tau$ is the friction velocity at the wall, $\kappa \approx 0.4$ and $B \approx 5$ ) even in the buffer region. Then at a typical buffer scale of $y^+=20$ , this gives the local Reynolds number

(2.16) \begin{equation} \textit{Re}_{\textit{local}} \approx y^+ \left ( \frac {1}{\kappa }\ln y^+ +B \right) \approx 250, \end{equation}

which is large but not very large.

A representative $T$ can be found using the time quoted by Lozano-Durán et al. (Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021) for the maximum gain seen in numerical computations, which is $t=0.35h/u_\tau$ ( $h$ being the half-channel width) at $\textit{Re}_{\tau }:=hu_\tau /\nu =180$ and comparable to the eddy turnover time at the appropriate distance from the wall (Markeviciute & Kerswell Reference Markeviciute and Kerswell2024). The local shear rate at $y^+$ using the log layer approximation is $S={\textrm d}U/{\textrm d}y=\textit{Re}_\tau u_\tau /h\, {\textrm d}U^+/{\textrm d}y^+=\textit{Re}_\tau u_\tau /(h \kappa y^+)$ which gives a time in our inverse shear rate units of

(2.17) \begin{equation} T \sim 0.35 \frac {h}{u_\tau } S= 0.35 \frac {\textit{Re}_\tau }{\kappa y^+} \approx 8, \end{equation}

for $y^+=20$ .

In terms of wavenumbers, we take the initial $k_y$ (cross-shear) wavenumber used to non-dimensionalise the model as $2\pi /y^+$ (i.e. the wavelength is the distance to the wall), so the other non-dimensionalised wavenumbers are $k_z \sim O( y^+/(\lambda ^+_z \approx 100) )$ and $k_x \sim O( y^+/(\lambda ^+_x \approx 300) )$ . Therefore both wavenumbers are $O(0.1$ – $1)$ .

The magnitude of $\beta$ can be estimated by comparing shear rates. Typical root-mean-square velocities near the wall are $u_{\textit{rms}} \approx 1$ (in units of $u_\tau$ ) which can be used to estimate typical streak amplitudes. Comparing the ratio of streak shear to wall-normal shear with the model (2.1) gives

(2.18) \begin{equation} \frac {k_z^+ u_{\textit{rms}}}{1/(\kappa y^+)} \approx \frac {k_z\beta /\sqrt {2}}{1}, \end{equation}

or $\beta \approx 3.6$ at $y^+=20$ (the $\sqrt {2}$ factor compensates for the cosine dependence in (2.1)). Given this, we consider values of $\beta$ from 1 to 5 in the below (going to higher $\beta$ causes problems with numerical overflow in our model: see figure 9).

3.1. Orr mechanism

With $k_z=0$ , the shearwise vorticity can only exponentially decay leaving the shrinking of $k^2(t):=k_x^2+(1-k_xt)^2$ with time in the expression for $\hat {v}(t)$ – see (3.2) – the sole growth mechanism. Growth occurs for $t\leqslant 1/k_x$ with the viscous (exponential) damping term moderating this. The fact that the growth time does not scale with viscosity means the Orr mechanism is inertial and so considered ‘fast’.

For large ${\textit{Re}}$ and $1\lesssim \! T \!= o(\textit{Re})$ , the viscous damping can be ignored and then $k_x=(1/2)( \sqrt {T^2+4}-T)$ maximises the global optimal growth at

(3.4) \begin{equation} {\mathscr{G}}^{\textit{Orr}} =1/k_x^2 \sim T^2, \end{equation}

for $T^2 \gg 4$ . For $1 \ll T=O(\textit{Re})$ , the optimal wavenumber is still related to the time via $k_x=1/T$ to leading order and it is simple to show that the overall growth maximum over $T$ (the overall global optimal) is

(3.5) \begin{equation} {\mathbb{G}}^{\textit{Orr}}= \frac {9}{{\textrm{e}}^2}{\textit{Re}}^2, \quad T^{\textit{Orr}}= 3{\textit{Re}}, \quad k_x^{\textit{Orr}}= \frac {1}{3{\textit{Re}}}, \end{equation}

where ${\textrm{e}}=\exp (1)=2.7183$ . So, somewhat paradoxically, the maximum growth over all times produced by the ‘fast’ Orr mechanism actually occurs on a ‘slow’ viscous time scale.

3.2. Lift-up mechanism

With $k_x=0$ , there is no growth possible in $\hat {v}$ but this velocity component forces the shearwise vorticity $\hat {\eta }$ which grows. This growth ultimately is turned off by the viscous decay of $\hat {v}$ and so lift-up is viewed as a ‘slow’ viscous process. The overall optimal lift-up growth ( $k_x=0$ ) is

(3.6) \begin{equation} {\mathbb{G}}^{\textit{lift-up}}= \frac {4}{27e^2}{\textit{Re}}^2, \quad T^{\textit{lift-up}}= \frac {2}{3}{\textit{Re}}, \quad k_z^{\textit{lift-up}}= \frac {1}{\sqrt {2}}, \end{equation}

which is consistent with this. Here, since $1\ll T=O(\textit{Re})$ , the initial shearwise vorticity can be ignored facilitating the derivation of (3.6) while for $1 \lesssim T=o(\textit{Re})$ , this is no longer valid.

Figure 2. A contour plot of optimal growth $G$ in wavenumber space for $T=5$ and $\textit{Re}=200$ . This figure shows that the optimal early time gain occurs for a three-dimensional perturbation, at a location in wavenumber space close to the $k_x$ value associated with the maximum gain possible through the Orr mechanism.

3.3. Fully 3-D optimals

Figure 2 shows the growth landscape over $(k_x,k_z)$ plane for $T=5$ and $\textit{Re}=200$ . The optimal $k_x$ wavenumber is close to the (2-D) Orr optimal value ( $k_z \rightarrow 0$ on the plot) but the optimal $k_z$ is an order of magnitude smaller than the corresponding lift-up value ( $k_x \rightarrow 0$ ). This observation is robust over different $T$ as shown in figure 3(b) and becomes more exaggerated as $T$ increases towards the maximum at $T=O(\textit{Re})$ . It is also noticeable in 3(a) that this overall optimal gain ${\mathbb{G}}$ (maximised over all $k_x,k_z$ and $T$ ) occurs at about the same time as the Orr maximum (this is clearly not true for lift-up). This and the $k_x$ observation suggests using the large ${\textit{Re}}$ results in (3.5) for Orr as a starting point for analysing the full 3-D optimal. Assuming that the initial shearwise vorticity is zero, this leads to

(3.7) \begin{equation} {\mathbb{G}}^{{3D}}= \frac {9\pi ^2}{{\textrm{e}}^2}{\textit{Re}}^2, \quad T^{{3D}}= 3\textit{Re} + \alpha {\textit{Re}}^{9/11}, \quad k_x^{{3D}}= \frac {1}{3\textit{Re}}, \quad k_z^{{3D}}= \gamma {\textit{Re}}^{-8/11}, \end{equation}

where $\alpha :=3^{6/11}/\pi ^{2/11}\approx 1.4786$ and $\gamma :=\pi ^{3/11}/3^{9/11}\approx 0.5562$ . These asymptotic predictions match the numerical computations very well even down to $\textit{Re}=O(30)$ – see figure 4, also providing a posteriori justification for assuming the optimal has $\hat {\eta }(0)=0$ . The formulae in (3.7) give optimal wavenumbers at $\textit{Re}=100$ of $(k_x,k_z)=(3.33, 19.53) \times 10^{-3}$ which are consistent with figure 1(a) of Jiao et al. (Reference Jiao, Hwang and Chernyshenko2021).

Figure 3. (a) optimal growth vs $T$ at $\textit{Re}=1000$ . The solid line is the full 3-D optimal ${\mathscr{G}}(\textit{Re})=\max _{(k_x,k_z)}G(k_x,k_z;\textit{Re})$ , the dashed line is $\max _{k_x} G(k_x,0;\textit{Re})$ (Orr) and the dotted line is $\max _{k_z} G(0,k_z;\textit{Re})$ (lift-up). (b) the corresponding optimal wavenumbers: solid brown/blue for $k_x/k_z$ for the full 3-D optimal, dashed brown for $k_x$ in Orr and dotted blue for $k_z$ in lift up. These plots show that the 3-D optimal closely follows the Orr result in terms of both $G$ and $k_x$ with assistance from lift-up except near the maximum over $T$ (the overall optimal gain ${\mathbb{G}}$ ).

Figure 4. This plot shows the overall optimal gain ${\mathbb{G}}(\beta ,\textit{Re})$ and associated optimising parameters ( $k_x, \,k_z$ and $T$ ) as a function of ${\textit{Re}}$ for $\beta =0$ in symbols. The asymptotic predictions of (3.7) are the lines.

4.1. Two dimensional ( $k_x=0$ ) push-over growth

With $k_x=0$ , the wavenumbers are constant so Orr is absent and it is simpler to work with the velocity field directly. The $\hat {v}$ and $\hat {w}$ momentum equations together with continuity imply $\hat {p}_m=0$ for all $m$ , leaving the 3 evolution equations

(4.1) \begin{align} \dot {\hat {u}}_m &= - \frac {k_m^2}{\textit{Re}}\hat {u}_m - \hat {v}_m + \frac {{\textrm{i}}\beta k_z}{2}(\hat {w}_{m+1}-\hat {w}_{m-1}), \end{align}

(4.2) \begin{align} \dot {\hat {v}}_m &= - \frac {k_m^2}{\textit{Re}}\hat {v}_m, \qquad \dot {\hat {w}}_m = - \frac {k_m^2}{\textit{Re}}\hat {w}_m, \\[9pt] \nonumber \end{align}

where $k^2_m=1+m^2 k_z^2$ . These indicate that $\hat {v}_m$ and $\hat {w}_m$ simply viscously decay in time allowing explicit solutions to be written down as follows:

(4.3) \begin{align} {\hat{u}}_{m} = {\textrm{e}}^{-k_{m}^{2} t/\textit{Re}}\left \{ U_{m}-V_{m} t + \frac {{\textrm{i}} \textit{Re} \beta }{2k_z}\right . &\left[ \frac {W_{m-1}}{1-2m} \big({\textrm{e}}^{-(1-2m)k_{z}^{2} t/\textit{Re}}-1 \big) \right. \nonumber\\ & \left. \left. -\, \frac {W_{m+1}}{1+2m} \big({\textrm{e}}^{-(1+2m)k_{z}^{2} t/\textit{Re}}-1 \big) \right ] \right \}\!, \end{align}

(4.4) \begin{align} \hat {v}_m = V_m {\textrm{e}}^{-k_m^2 t/\textit{Re}},& \qquad \hat {w}_m = W_m {\textrm{e}}^{-k_m^2 t/\textit{Re}}, \end{align}

where $U_m:=\hat {u}_m(0)$ , $V_m:=\hat {v}_m(0)$ and $W_m:=\hat {w}_m(0)$ are the initial conditions which, by continuity, satisfy $V_m+mk_zW_m=0$ . Taking $\textit{Re} \rightarrow \infty$ gives the simple $\hat {u}$ equation

(4.5) \begin{equation} \hat {u}_m = U_m-V_mt + \frac {{\textrm{i}}\beta k_z t}{2} \left ( W_{m+1} - W_{m-1}\right )\!. \end{equation}

The push-over effect is produced by the forcing term proportional to the spanwise streak shear, $\beta k_z$ , and is very similar to the $-V_m t$ lift-up forcing (normalised to have unit shear) although push over drives neighbouring spanwise modes while lift-up is direct. Lift-up can be turned off by setting $V_m=0$ for $\forall m$ which, through continuity, forces $W_m=0$ for $m \neq 0$ . Setting $U_m=0$ for all $m$ as well appears an extreme choice for initial conditions but numerical computations indicate that it is close to optimal for $k_x=0$ (e.g. at $\beta =1$ , $T=5$ and $\textit{Re}=200$ , the error using the extreme choice is only $\approx 1\%$ of the global optimal gain value). When only $W_0$ of the initial conditions is non-zero, the gain is

(4.6) \begin{equation} \hat {G}^{\textit{push}} = {\textrm{e}}^{-2T/\textit{Re}}\left [ 1+\frac {\textit{Re}^2\beta ^2}{2k_z^2}\big(1-{\textrm{e}}^{-Tk_z^2/\textit{Re}}\big)^2\right ] \sim 1+\frac {1}{2}\beta ^2 k_z^2 T^2 \quad \textrm {as}\, \textit{Re} \,\rightarrow \, \infty , \end{equation}

assuming $T \ll \textit{Re}$ and $k_z^2 \ll \textit{Re}/T$ for the asymptotic simplification (we use a $\hat {\quad }$ to indicate a quantity not optimised over all initial conditions). This mirrors the result for lift-up: when $\beta =0$ is used, choosing only $(U_1,V_1,W_1)=(0,1, -1/k_z)$ to be non-zero gives a gain of

(4.7) \begin{equation} \hat {G}^{\textit{lift-up}} \sim 1+k_z^2T^2/\big(1+k_z^2\big), \end{equation}

at large ${\textit{Re}}$ . Hence push over seems to behave very much like lift-up in both the growth possible and the time scales, i.e. growth is ultimately limited by the viscous decay of the spanwise velocities so push over is ‘slow’.

The overall optimal gain in (4.6) can be found by optimising over both $k_z$ and $T$ : $\partial \hat {G}^{\textit{push}}/\partial k_z = \partial \hat {G}^{\textit{push}}/\partial T = 0$ requires

(4.8) \begin{align} 2k_z^2Te^{-k_z^2T/\textit{Re}} & -\textit{Re}\big(1-{\textrm{e}}^{-k_z^2T/\textit{Re}}\big)=0, \end{align}

(4.9) \begin{align} {\textit{Re}}^2 \beta ^2 \big(1-{\textrm{e}}^{-k_z^2T/\textit{Re}}\big) & \big[{\textrm{e}}^{-k_z^2T/\textit{Re}}\big(1+k_z^2\big)-1\big] = 2k_z^2. \end{align}

The first of these is solved by noticing that the equation reduces to $2\zeta + 1 - {\textrm{e}}^{\zeta }=0$ with $\zeta :=k_z^2T/\textit{Re}$ which has the unique positive solution $\zeta \approx 1.256$ . Then, using ${\textrm{e}}^{-\zeta }=(1+2\zeta )^{-1}$ in (4.9) gives the optimal wavenumber as $\hat {k}^{push}_z=\sqrt {2\zeta }$ , and thus the maximising time and maximal gain for only-non-vanishing- $W_0$ initial conditions as

(4.10) \begin{equation} \hat {T}^{\textit{push}}=\tfrac {1}{2}{\textit{Re}} \quad \text{and} \quad \hat {{\mathbb{G}}}^{\,\textit{push}}=\frac {\zeta \beta ^2}{{\textrm{e}}(1+2\zeta )^2} {\textit{Re}}^2. \end{equation}

This again shows that the push-over mechanism, in the absence of Orr ( $k_x=0$ ) and with initial conditions chosen to eliminate lift-up, possesses the same scalings as the lift-up mechanism.

Figure 5. Contour plots of the optimal gain over the $(k_x,k_z)$ wavenumber plane for the parameter set $\beta =1$ , $T=5$ and $\textit{Re}=200$ . The truncation value is set to $M=20$ and the contour levels are the same across all the plots. Panel (a) shows the full problem (all terms included), while panels (b), (c) and (d) have the lift-up, push-over and Orr mechanisms removed, respectively. The black line traces the wavenumber ranges used for figure 7. The point of this figure is to show that the large growth observed is substantially suppressed when either the push-over or Orr mechanism is removed, but is largely independent of the lift-up mechanism.

4.2. Transient growth of three-dimensional perturbations

Larger energy growth occurs for 3-D disturbances, where all three mechanisms (lift-up, push over and Orr) can play a role. Figure 5 displays contours of the optimal gain in wavenumber space for four different scenarios at representative parameter values ( $\beta =1$ , $T=5$ and $\textit{Re}=200$ ). The left-most plot shows the gain at $T$ at a given pair of wavenumbers $(k_x,k_z)$ maximised over all initial conditions for the full equations. The other three plots extending to the right have the three growth mechanisms excluded in turn by artificially removing the terms which drive them: figure 5(b) has lift-up suppressed by removing $\partial {{\boldsymbol{U}}_{\!{B}}}/\partial y$ from the equations; figure 5(c) has push over suppressed by removing $\partial {{\boldsymbol{U}}_{\!{B}}}/\partial z$ ; and figure 5(d) has the streak advection (the $\beta \cos (k_z z)\partial {{\boldsymbol u}}/\partial x$ term) removed to suppress the ‘spanwise’ Orr mechanism (removing all of the advection changes the whole character of the flow). All optimal disturbances here are sinuous with varicose disturbances always producing less growth in this augmented Kelvin model across the parameters considered: e.g. see figure 6.

Figure 6. Contour plots of the optimal gain over the $(k_x,k_z)$ wavenumber plane for $\beta =1$ , $T=5$ and $\textit{Re}=200$ . Both use $M=20$ in the model, but the first plots the optimal ‘sinuous’ disturbances, while the second plots the optimal ‘varicose’ disturbances (recall that this terminology refers to the symmetry or antisymmetry of $u$ about $z=0$ ). This plot demonstrates that all of the large growth occurs for sinuous perturbations.

Figure 7. A plot showing the optimal gain as $k_z$ is varied, with $k_x=k_z/5$ and parameters $\beta =1$ , $T=5$ and $\textit{Re}=200$ . The range of $k_z$ used here is indicated in figure 5(a) using a black line. The blue line represents the gain when all terms are included in the equations (corresponding the the contours in figure 5(a)). The red line shows the gain when the lift-up mechanism is removed, while the yellow line shows gain when the push-over mechanism is removed and the purple line shows gain when the spanwise Orr mechanism is removed (which correspond to the contours in figures 5(b), 5(c) and 5(d), respectively). This plot demonstrates that both the push-over and spanwise Orr mechanisms are crucial for large growth of kinetic energy, even down to some moderate wavenumbers. It also shows that for smaller wavenumbers, a switch occurs and lift-up becomes the dominant growth process.

A key feature of the figure 5(a) is the global optimal gain indicated by the yellow region, which reaches values of ${\mathscr{G}} \approx 10^4$ at $k_x \approx 2.7$ and $k_z\approx 15$ . A comparison of this peak and the surrounding region with the same location in the other plots reveals that removing the lift-up mechanism leaves the optimal gain contours almost completely unaltered, while removing either the push-over or spanwise Orr mechanism reduces the size of gain significantly. This importance of push over and lack of importance of lift-up for a streaky base flow is fully consistent with earlier observations (Lozano-Durán et al. Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021; Markeviciute & Kerswell Reference Markeviciute and Kerswell2024).

The optimal wavenumbers for this global optimal are, however, on the large side for the buffer layer (see § 2.4) but a line drawn along the major axis of the contours – approximately $k_z=5k_x$ – suggests a direction of ‘shallowest’ descent in the wavenumber plane (shown as a black line in figure 5(a) for $1\lt k_z\lt 10$ ). Figure 7 shows how the optimal gain behaves along this line for $1\lt k_z\lt 10$ when the various mechanisms are excluded. At $k_z=1$ , removing lift-up is actually more detrimental to the gain than removing either spanwise Orr or push over, but for $k_z \gtrsim 3$ , the situation is reversed and mimics that for the global optimal gain. The figure also shows that the presence of lift-up now actually hinders growth (the crossing of blue and red lines at $k_z \approx 3$ ). This effect is also present in figure 5(b) and is consistent with the observations of Lozano-Durán et al. (Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021) and figure 7 in Markeviciute & Kerswell (Reference Markeviciute and Kerswell2024).

So, the conclusion is, in this unbounded streaky shear flow at these parameters, wavenumber pairings reaching down to those appropriate for buffer-layer dynamics show the clear symbiosis of Orr and push-over mechanisms and the unimportance of lift-up as highlighted by the global optimal gain pairing.

4.3. A two-variable model of the push-over and Orr interaction

Remarkably, it is possible to strip the model down to just 2 fields and still retain the symbiotic interaction between push over and Orr. There are 3 distinct steps which achieve this. Firstly, the spanwise truncation $M$ is reduced down to $1$ decreasing the number of (complex-valued) fields from $2(2M+1)$ (with $M=20$ or so) down to 6. Secondly, the symmetry: $\hat {v}_0=0$ , $\hat {v}_1=-\hat {v}_{-1}$ and $\hat {\eta }_1=\hat {\eta }_{-1}$ appropriate for the sinuous mode (discussed in Schoppa & Hussain Reference Schoppa and Hussain2002) is imposed so 6 becomes 3 complex fields. These evolve as follows:

(4.11) \begin{align} \frac {{\textrm d}\hat {v}_1}{{\textrm d}t} &= -\frac {k_1^2}{\textit{Re}}\hat {v}_1 +\frac {2k_x(1-k_xt)}{k_1^2}\hat {v}_1 -\frac {\beta k_z (1-k_xt)}{k_1^2}\hat {\eta }_0, \end{align}

(4.12)

(4.13)

where, recall, $k_0^2:=k_x^2+(1-k_xt)^2$ , $k_1^2:=k_0^2+k_z^2$ and $h_1^2:=k_x^2+k_z^2$ . Here the Orr, lift-up and push-over terms are labelled in the last two equations and the Orr terms only rely on the streak flow (indicated by being premultiplied by $\beta$ ). The fact that the term involving $\hat {v}_1$ in (4.12) is only an Orr term, and that (4.12) contains no push-over contributions is not obvious; see the Appendix for details.

Figure 8. A sequence of contour plots showing the optimal gain in wavenumber space, close to the optimal pair, for the parameter set $\beta =1$ , $T=5$ and $\textit{Re}=200$ . The colour bar is shown on the right-hand side and is the same for all three plots. Panel (a) has truncation value $M=20$ , (b) has $M=1$ with the sinuous symmetry used and (c) uses the further simplification of having $\hat {v}_1$ removed. This figure demonstrates that the underlying growth process is present in all three cases with the reduced model harbouring significantly enhanced growth.

Finally, $\hat {v}_1$ can be jettisoned meaning (4.11) is ignored and $\hat {v}_1$ set to zero in the $\hat {\eta }$ equations, leaving simply

(4.14)

(4.15)

The a posteriori justification for such a drastic reduction is provided by figure 8 which shows the gain in the full ( $M=20$ ) system (the same data as figure 5(a) but replotted over a smaller wavenumber area and with different contour levels), the gain produced by the three-variable model (4.11)–(4.13), and the reduced model (4.14)–(4.15). This comparison clearly shows that: (i) the $M=1$ reduction is surprisingly effective in retaining the gain behaviour and (ii) the reduced model also retains the global optimal gain feature and in fact, as also observed, experiences significantly enhanced growth with lift-up removed. Figure 9 shows this picture is repeated for smaller $\textit{Re}=100$ and target time of $T=2$ but a larger $\beta =5$ value (note the increase in contour scale for $\beta =5$ compared with $\beta =1$ which is explained by the scaling in (4.27)).

Figure 9. Contour plots in wavenumber space of the optimal gain for $\beta =1$ (a,b) and $\beta =5$ (c,d), with other parameters set to $T=2$ , $\textit{Re}=100$ . The first column shows the full model with truncation $M=20$ while the second column shows the reduced two-variable model given by (4.14) and (4.15).

For $Tk_1^2 \ll \textit{Re}$ , diffusion can be dropped in (4.14)–(4.15) and then setting $\hat {\eta }_0=\eta _0$ and $\hat {\eta }_1=i \eta _1$ for convenience gives just

(4.16) \begin{equation} \frac {\textrm d}{{\textrm d}t}\left (\!\! \begin{array}{c} \eta _0\\ \eta _1 \end{array} \!\!\right ) = \left ( \!\!\begin{array}{cc} 0 & \beta k_x^3/\left(k_x^2+k_z^2\right) \\ \tfrac {1}{2}\beta \left(k_z^2-k_x^2\right)/k_x & 0 \\ \end{array} \!\!\right ) \left (\!\! \begin{array}{c} \eta _0\\ \eta _1 \end{array} \!\!\right )\!. \end{equation}

To arrange for the Euclidean norm to correspond to the energy, the vorticities need to be rescaled – $\tilde {\eta }_0:=\eta _0/h_0$ and $\tilde {\eta }_1:=\sqrt {2} \eta _1/h_1$ – so that

(4.17) \begin{equation} E:=\dfrac {1}{2}\big(|\hat {\eta }_0/h_0|^2+2|\hat {\eta }_1/h_1|^2\big)=\dfrac {1}{2}\big(|\tilde {\eta }_0|^2+|\tilde {\eta }_1|^2\big), \end{equation}

and the system becomes

(4.18) \begin{equation} \frac {\textrm{d}}{\textrm{dt}}\left (\!\! \begin{array}{c} \tilde {\eta }_0\\ \tilde {\eta }_1 \end{array} \!\!\right ) = {\mathbb{L}} \left (\!\! \begin{array}{c} \tilde {\eta }_0\\ \tilde {\eta }_1 \end{array} \!\!\right ) := \left ( \!\!\begin{array}{cc} 0 & a \\ b & 0 \\ \end{array} \!\!\right ) \left (\!\! \begin{array}{c} \tilde {\eta }_0\\ \tilde {\eta }_1 \end{array} \!\!\right )\!, \end{equation}

where

(4.19) \begin{equation} a:= \frac {\beta k_x^3}{k_x^2+k_z^2}\frac {h_1}{h_0\sqrt {2}}=\frac {\beta k_x^2}{\sqrt {2\left(k_x^2+k_z^2\right)}} , \qquad b:= \frac {\beta \left(k_z^2-k_x^2\right)}{2 k_x}\frac {h_0\sqrt {2}}{h_1}=\frac { \beta \left(k_z^2-k_x^2\right)}{ \sqrt {2\left(k_x^2+k_z^2\right)}} .\end{equation}

This system can produce energy growth in 2 distinct and independent ways.

1. (i) Algebraic growth due to the non-normality of ${\mathbb{L}}$ . This occurs when the magnitudes of the off diagonal elements of ${\mathbb{L}}$ do not match i.e.

   (4.20) \begin{equation} \frac {a^2}{b^2} =\left (\frac {k_x^2}{k_z^2-k_x^2} \right )^2\neq 1 \, \Rightarrow \, k_z \neq \sqrt {2} k_x, \end{equation}
   which ensures ${\mathbb{L}} {\mathbb{L}}^T \neq {\mathbb{L}}^T {\mathbb{L}}$ - the defining property that ${\mathbb{L}}$ is a non-normal matrix.

2. (ii) Exponential growth due to a linear instability with growth rate $\sigma$ where

   (4.21) \begin{equation} \sigma ^2:= ab = \frac {1}{2} \beta ^2 k_x^2 \left ( \frac {k_z^2-k_x^2}{k_z^2+k_x^2} \right )\!. \end{equation}
   This needs $k_z \gt k_x$ and so push over dominates streak Orr in (4.15). Since only streak Orr operates in (4.14), this clearly shows that push over and streak Orr are needed for this linear instability.

4.4. Two-variable energy growth

Either mechanism can act in isolation or together depending on the wavenumber pair $(k_x,k_z)$ . In the model studied here, the wavenumbers $k_x$ and $k_z$ are naturally chosen so that both mechanisms are operative as this creates the largest growth. In particular, in the example plotted in figure 8 – $(\textit{Re},T,\beta )=(200,5,1)$ – the optimal wavenumbers are $(k_x,k_z) \approx (2.7, 15)$ for the M = 20 and M = 1 models so that $a \approx 0.34$ and $b \approx 10.1$ . Hence, $k_z \gt k_x$ and there is actually a significant difference in magnitude between the off-diagonal elements. To characterise how much energy growth this produces, the 2 differential equations in (4.18) can be straightforwardly integrated to give

(4.22) \begin{equation} \left (\!\! \begin{array}{c} \tilde {\eta }_0(t)\\ \tilde {\eta }_1(t) \end{array} \!\!\right ) = {\mathbb{A}} \left (\!\! \begin{array}{c} \tilde {\eta }_0(0)\\ \tilde {\eta }_1(0) \end{array} \!\!\right ) := \left ( \!\!\begin{array}{cc} \cosh \sigma t & \sqrt {a/b} \sinh \sigma t \\ \sqrt {b/a} \sinh \sigma t & \cosh \sigma t \\ \end{array} \!\!\right ) \left (\!\! \begin{array}{c} \tilde {\eta }_0(0)\\ \tilde {\eta }_1(0) \end{array} \!\!\right )\!, \end{equation}

where ${\mathbb{A}}=e^{{\mathbb{L}} t}$ . The maximum growth possible, $G(T)$ , at $t=T$ for given wavenumbers $(k_x,k_z)$ is then the largest eigenvalue of the real symmetric matrix ${\mathbb{A}}^T(T) {\mathbb{A}}(T)$ (here, ${\mathbb{A}}^T$ indicates the transpose of ${\mathbb{A}}$ rather than anything to do with the target time $T$ ). This is just the larger of the two real positive eigenvalues of the quadratic

(4.23) \begin{equation} \lambda ^2-\left [2 \cosh ^2 \sigma T+ \left ( \frac {a}{b}+\frac {b}{a}\right ) \sinh ^2 \sigma T \right ]\lambda +1 =0. \end{equation}

If $\lambda _{\textit{max}}\gt 1$ then $\lambda _{\textit{min}}=1/\lambda _{\textit{max}}\lt 1$ , that is, there is only one growth mechanism for a given $(k_x,k_z)$ pairing and no growth implies $\lambda _{\textit{max}}=\lambda _{\textit{min}}=1$ . Assuming either large $\sigma T$ when $\sigma ^2=ab\gt 0$ or $|a/b| \neq O(1)$ for $0 \gt \sigma ^2=-\omega ^2$

(4.24) \begin{equation} G(T,k_x,k_z;\beta ):=\lambda _{{max}} \approx 2 \cosh ^2 \sigma T+ \left ( \frac {a}{b}+\frac {b}{a}\right ) \sinh ^2 \sigma T. \end{equation}

This shows either linear instability enhanced by initial transient growth as $a/b+b/a \geqslant 2$ or only transient growth in a linearly stable system (ignoring the singular case of $\sin \omega T=0$ ). The corresponding optimal condition, which is the associated left eigenvector of ${\mathbb{A}}^T(T) {\mathbb{A}}(T)$ , depends crucially on whether $a/b \ll 1$ or $\gg 1$ . Taking the former and setting $\varepsilon := \sqrt {a/b} \ll 1$ ( $k_z/k_x \gg 1$ ), the optimal initial condition is

(4.25) \begin{equation} \left (\!\! \begin{array}{c} \tilde {\eta }_0(0)\\ \tilde {\eta }_1(0) \end{array} \!\!\right )= \left (\!\! \begin{array}{c} 1\\ \varepsilon \,\textrm {coth} \,\sigma T \end{array} \!\!\right ) \, \Rightarrow \, \left (\!\! \begin{array}{c} \tilde {\eta }_0(T)\\ \tilde {\eta }_1(T) \end{array} \!\!\right )= \left (\!\! \begin{array}{c} \cosh \sigma T\\ \frac {1}{\varepsilon }\sinh \sigma T \end{array} \!\!\right )\!, \end{equation}

along with what it evolves into a time $T$ later (all to leading order in $\varepsilon$ ). This indicates that energy growth through the non-normality of ${\mathbb{L}}$ is produced by initial $\hat {\eta }_0$ generating $\hat {\eta }_1$ as described by (4.15). This is consistent with what Schoppa & Hussain (Reference Schoppa and Hussain2002) found – the initial condition they chose was purely $\hat {\eta }_0$ – but they focussed on the streamwise vorticity as a measure of the growth rather than the shearwise vorticity.

If instead, $a/b \gg 1$ so now $\varepsilon :=\sqrt {b/a} \ll 1$ (or $k_z^2/k_x^2-1 \ll 1$ ), the optimal initial condition is

(4.26) \begin{equation} \left (\!\! \begin{array}{c} \tilde {\eta }_0(0)\\ \tilde {\eta }_1(0) \end{array} \!\!\right )= \left (\!\! \begin{array}{c} \varepsilon \,\textrm {coth} \,\sigma T\\ 1 \end{array} \!\!\right ) \, \Rightarrow \, \left (\!\! \begin{array}{c} \tilde {\eta }_0(T)\\ \tilde {\eta }_1(T) \end{array} \!\!\right )= \left (\!\! \begin{array}{c} \frac {1}{\varepsilon }\sinh \sigma T\\ \cosh \sigma T \end{array} \!\!\right )\!. \end{equation}

This indicates that energy growth through the non-normality of ${\mathbb{L}}$ is now produced by initial $\hat {\eta }_1$ generating $\hat {\eta }_0$ as described by equation (4.14).

Figure 10. The various transient growth (TG) and stability regimes over $0\leqslant k_z/k_x$ . They are (a) $0 \leqslant k_z/k_x \lt 1$ – linearly stable and $\hat {\eta }_1 \rightarrow \hat {\eta }_0$ transient growth; (b) $k_z/k_x = 1$ – linearly stable but unlimited algebraic growth in $\hat {\eta }_0$ ; (c) $1\lt k_z/k_x \lt \sqrt {2}$ – linearly unstable and $\hat {\eta }_1 \rightarrow \hat {\eta }_0$ transient growth; (d) $k_z/k_x=\sqrt {2}$ – linearly unstable with no transient growth; and (e) $\sqrt {2}\lt k_z/k_x$ – linearly unstable and $\hat {\eta }_0\rightarrow \hat {\eta }_1$ transient growth. Sample evolutions are shown for $k_z/k_x=5.6$ in figure 13 and $k_z/k_x=0.95$ in figure 14 for $(\textit{Re},T,\beta )=(200,5,1)$ .

Given instability exists for $ab\gt 0$ or $k_z \gt k_x$ , there are 5 different scenarios as $k_z/k_x$ varies: see figure 10. Optimising energy growth over all wavenumbers should select either $1\lt k_z/k_x \lt \sqrt {2}$ or $\sqrt {2} \lt k_z/k_x$ since both are linearly unstable but have different transient growth mechanisms. Computations here suggest it is the latter which contains the optimum which is consistent with where the instability growth rate is larger for fixed $k_x$ . In this case, the transient growth which occurs is described by (4.15) and push over is the dominant process. This transient growth situation is illustrated in figure 11(a). Here, the spanwise velocity perturbation $\hat {w}_0:={\textrm{i}} \hat {\eta }_0/k_x$ ‘pushes over’ (advects) the base streak velocity $\beta \cos (k_z z) \, \boldsymbol{\hat{x}}$ to create wavy streaks $\hat {u}_1$ (and so $\hat {\eta }_1$ ) via the projection of the term $\hat {w}_0{\partial }/{\partial z} U_{\!B}$ onto the $\hat {u}_1$ equation. The (streak) Orr achieves the same effect of generating streamwise- and spanwise-dependent flow from only streamwise-dependent flow – albeit with the opposite and wrong phase for subsequent instability.

Figure 11. (a) $\hat{\eta}_0$ corresponds to a spanwise velocity $w_{0} \propto \sin k_{x} x$ which ‘pushes over’ the streak velocity field $\beta \cos k_{z} z \, \boldsymbol{\hat{x}}$ to produce wavy streaks as shown created by the streamwise velocity anomalies $u_1 \propto \sin k_x x \sin k_z z$ (black arrows). (b) the spatial gradients in the $u_1$ field imply a doubly periodic pressure field which drives a concomitant spanwise velocity $w_1 \propto \cos k_x x \cos k_z z$ field (green arrows). The advection of this $w_1$ field by the streak velocity – the streak-Orr effect – generates further spanwise velocity (purple arrows) via the term $-\beta \cos k_z z \partial w_1/\partial x$ which feeds back positively on $\hat {\eta }_0$ completing the loop.

4.5. Two-variable linear instability

The instability looks to be exactly the 2-D linear instability of a streak field as modelled by 2-D Kolmogorov flow (Arnold & Meshalkin Reference Arnold and Meshalkin1960; Meshalkin & Sinai Reference Meshalkin and Sinai1961). In its simplest form, Kolmogorov flow consists of a steady forcing, $\sin \ell y \, \boldsymbol{\hat{x}}$ for some integer $\ell$ , applied to a flow over a two-torus $[0,2\pi /{\hat \alpha }] \times [0,2\pi ]$ . The corresponding one-dimensional base flow is exactly the streak flow studied here with the base shear removed and is linearly unstable at high enough ${\textit{Re}}$ provided $\hat {\alpha }\lt \ell$ which is exactly the condition $k_x \lt k_z$ for instability found in (4.16) (e.g. Marchioro (Reference Marchioro1986) and figure 2 in Chandler & Kerswell (Reference Chandler and Kerswell2013)).

The instability is produced by a further mechanism represented by (4.14) which generates $\hat {\eta }_0$ from $\hat {\eta }_1$ to complement the transient growth mechanism. As discussed above, the latter produces wavy streaks which drive a concomitant spanwise field $\hat {w}_1$ through continuity ( $k_x \neq 0$ ): see figure 11(b). This spanwise field $\hat {w}_1$ then drives $\hat {w}_0$ to close the loop by the streak-Orr mechanism. that is, streak advection of $\hat {w}_1$ given by the projection of the $\beta \cos (k_z z){\textrm{i}}k_x\hat {w}_1$ term onto the $\hat {w}_0$ equation: again see figure 11(b). In this process, the streamwise flow component of the instability, $\hat {u}_1$ , is largest at the inflexion points of the streak field where the spanwise shear is maximal, while the spanwise flow component is largest when the streak is largest.

4.6. Two-variable transient growth with dissipation present

Before studying the nonlinear consequences of the linear instability, we estimate the energy growth possible in the two-variable system with dissipation present. Because of the unlimited growth of the cross-stream wavenumber in the model, dissipation always eventually overpowers the instability to formally give only transient growth. For a general $k_x$ , the time for the cessation of linear instability growth is given by the balance $\beta k_x \sim k_x^2T^2/\textit{Re}$ so $T=O(\sqrt {\beta \textit{Re}/k_x})$ and then a prediction for the optimal growth for a given $k_x \lesssim O(\textit{Re})$ is

(4.27) \begin{equation} G \sim {\textrm{e}}^{\sqrt {\alpha \beta ^3 k_x \textit{Re}} }, \end{equation}

where $\alpha ={1}/{2}(k_z^2-k_x^2)/(k_z^2+k_x^2)$ is $O(1)$ . Formally, if $T\gtrsim O(1)$ , this is maximised for $k_x =O(\textit{Re})$ beyond which diffusion dominates giving a massive gain of $O({\textrm{e}}^{\alpha _1 \textit{Re}})$ ( $\alpha _1$ being a ${\textit{Re}}$ -independent number). This prediction is confirmed in the two-variable model and for the full system: see figure 12. This also shows good correspondence between the optimal wavenumbers in each system which all scale linearly with ${\textit{Re}}$ . Figure 12 also confirms that a push-over-less system has no such exponential growth. However, this overall optimal gain is not the one of practical interest as $k_z=O(\textit{Re})$ (figure 12 actually has $k_z \sim 0.075\textit{Re}$ and $k_x \sim 0.012\textit{Re}$ ) represents unrealistically large streak shear. If, instead $k_z=O(1)$ (and hence $k_x=O(1)$ ), the gain is reduced but still substantial at $O({\textrm{e}}^{\alpha _2 \sqrt {\textit{Re}}})$ generated over $O(\sqrt {\textit{Re}})$ ‘intermediate’ times ( $\alpha _2$ being another constant). This exponential dependence of gain on $\sqrt {\textit{Re}}$ is surprising given that the Orr and push-over mechanisms in isolation only give gains which scale like $\textit{Re}^2$ and then only over ‘slow’ times of $O(\textit{Re})$ (e.g. see §§ 3 and 4.1).

Figure 12. (a) plot shows the global optimal gain $\mathscr G$ as a function of ${\textit{Re}}$ for $\beta =1$ , $T=5$ using the full ( $M=20$ ) system (purple triangles), the reduced two-variable model (dark blue squares) and the full system with push over removed (dull blue circles). The lines drawn through the data are the straight lines between the extreme points of each data set indicating that the full system has the same ${\mathscr{G}} \sim {\textrm{e}}^{\alpha \textit{Re}}$ behaviour as the two-variable model albeit with a smaller $\alpha$ . The push-over-less system does not have this exponential dependence. (b) plot compares the optimal wavenumbers between the full system and the reduced two-variable model (the symbols). Lines through the data (drawn as in the Left plot) indicate that all wavenumbers scale linearly with ${\textit{Re}}$ .

4.7. Nonlinear feedback of the two-variable system

The linear instability in 2-D Kolmogorov flow is known to be supercritical (Sivashinsky Reference Sivashinsky1985) and hence the first nonlinear feedback of the instability on the streak is to reduce its amplitude. As confirmation, the instability here is

(4.28) \begin{equation} \boldsymbol {u}= \left [\!\! \begin{array}{c} \tfrac {2k_z}{h_1^2}\hat {\eta }_1(t) \sin k_z z \\ 0\\ \tfrac {2ik_x}{h_1^2} \hat {\eta }_1(t) \cos k_z z+\tfrac {i }{k_x}\hat {\eta }_0(t) \end{array}\!\! \right ] e^{ i( k_x x+(1-k_x t)y) }+ \textrm{c.c.}; \end{equation}

( $\textrm{c.c.}$ indicating complex conjugate) which, indeed, has the negative feedback on the streak

(4.29) \begin{equation} \frac {k_{x} k_{z}}{2\pi ^{2}} \int ^{2\pi /k_{z}}_{0} \!\! \int ^{2\pi /k_{x}}_{0} (-\boldsymbol {u} \boldsymbol{\cdot}\boldsymbol{\nabla}u) \cos k_{z} z\, {\textrm{d}}x {\textrm{d}}z \, = \, -\frac{4 k_{z}^{2}}{k_{x} h_{1}^{2}}{\textit{Re}} \big(i \hat {\eta }_0 \hat {\eta }_{1}^{*}\big) \lt 0, \end{equation}

as $\hat {\eta }_1 = i\hat {\eta }_0 \sqrt {k_{z}^{4}/k_{x}^{4-1}}/\sqrt {2}$ from (4.15) ignoring diffusion ( $^*$ indicates complex conjugate and ${\textit{Re}}( \, )$ is the real part). So this two-variable system does not re-energise the imposed streaks and moreover has no feedback on any streamwise rolls. To generate the latter, $\hat {v}_1$ needs to be reinstated – i.e. we need to examine the full three-variable $M=1$ system – but at a cost of expecting less growth.

Figure 13. (a) modal energy $E_m:= ({1}/{2h_m^2}) (k_m^2 |\hat{v}_m|^2+|\hat{\eta}_m|^2)$ plotted over $m \geqslant 0$ ( $E_{-m}=E_m$ due to symmetry) for the initial optimal (blue circled crosses) and final state (red filled circles) for $(\textit{Re},T,\beta )=(200,5,1)$ and $M=20$ . This indicates why the $M=1$ truncation works so well: the initial optimal has its energy dominantly in the $m=0$ mode which shifts to $m=1$ by the final state. (b), the time evolution of the optimal for $M=1$ system and $(\textit{Re},T,\beta )=(200,5,1)$ : solid dark blue uppermost line is $\eta _1/10$ ; dashed purple line is $\eta _0$ and the dash-dot pale blue line is $v_1$ . The green solid line indicating negative values is $F/100$ where $F$ is defined in (4.31). The dotted vertical line at $t \approx 0.37$ indicates where $1-k_x t=0$ for $k_x\approx 2.7$ and $k_z\approx 15$ .

4.8. Three-variable feedback onto rolls

The $M=1$ linear system, (4.11)–(4.13) has a symmetry that if $(\hat {\eta }_0, \hat {\eta }_1,\hat {v}_1)$ is a solution then so is $(\hat {\eta }_0^*, -\hat {\eta }_1^*,\hat {v}_1^*)$ . This symmetry is adopted by optimal initial conditions so that the flow subsequently can then be assumed of the form $(\hat {\eta }_0, \hat {\eta }_1,\hat {v}_1)=(\eta _0(t),i\eta _1(t),v_1(t))$ where $\eta _0(t),\eta _1(t)$ and $v_1(t)$ are all real variables subsequently (this observation has already been used to produce equation (4.16)). The evolution of these 3 optimal real variables is shown in figure 13(b) for $(\textit{Re},T,\beta )=(200,5,1)$ . The three-variable flow field takes the form

(4.30) \begin{equation} \boldsymbol {u}= \left [\!\begin{array}{c} 2i \left \{ -k_x(1-k_xt) v_1(t)+k_z \eta _1(t) \right \}\sin k_z z/h_1^2 \hspace {1.7cm}\\ \hspace {1.8cm}2 i v_1(t) \sin k_z z\\ 2 \left \{ -k_z(1-k_xt)v_1(t)-k_x \eta _1(t) \right \}\cos k_z z/h_1^2+i\eta _0(t)/k_x \end{array}\!\! \right ] e^{ i( k_x x+(1-k_x t)y) }+\textrm{c.c.}, \end{equation}

and the nonlinear driving of the cross-stream flow component $V(z,y)\boldsymbol {\hat {y}}$ with the same spanwise structure relevant for lift up is

(4.31) \begin{equation} F:=\frac {k_x k_z}{2\pi ^2} \int ^{2\pi /k_z}_0 \!\! \int ^{2\pi /k_x}_0 (-\boldsymbol {u} \boldsymbol{\cdot }\boldsymbol{\nabla }v) \cos k_z z\, {\textrm d}x {\textrm d}z \, = \, -\frac {4 k_z}{k_x}\eta _0(t) v_1(t). \end{equation}

For the representative parameters used in figure 13(b), $\eta _0\gt 0$ and $v_1\gt 0$ and so the feedback $F$ is large and negative (note that $F/100$ is actually plotted). This drives secondary cross-stream velocities anti-correlated ( $ \propto -\cos k_z z$ ) to the imposed streak field which, through the lift up term $-U_{y} V \, \boldsymbol{\hat{x}}$ reinforces the existing streak field. Specifically, where the instability generates $V\gt 0$ , slower moving fluid from the basic shear is ‘lifted up’ to reinforce the slow streaks (where $\beta \cos k_z z \lt 0$ ) and where $V\lt 0$ , corresponding faster-moving fluid is ‘pushed down’ to reinforce the fast streaks (where $\beta \cos k_z z \gt 0$ ). Since the driven flow $V(y,z) \boldsymbol{\hat{y}}$ is synonymous with streamwise rolls, this demonstrates the potential for this streak instability to produce a sustaining cycle of rolls and streaks (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997; Jimenez & Pinelli Reference Jimenez and Pinelli1999; Farrell & Ioannou Reference Farrell and Ioannou2012).

Figure 14 confirms for non-optimal wavenumbers – $k_x =1$ and $k_z=0.95$ – with no instability present that even the transient growth acting alone provides good nonlinear feedback onto streamwise rolls.

Figure 14. The time evolution of the optimal for $M=1$ system and $(\textit{Re},T,\beta )=(200,5,1)$ with non-optimal $k_x=1$ and $k_z=0.95$ so a linear stable situation: solid dark blue line is $\eta _1$ ; dashed purple line is $\eta _0$ and the dash-dot pale blue line is $v_1$ . The green solid line indicating largely negative values is $F/10$ where $F$ is defined in (4.31). The dotted vertical line indicates when $1-k_x t=0$ .

4.9. Why lift up weakens growth

Figure 8 shows that removing $\hat {v}_1$ from the $M=1$ system considerably enhances the optimal growth. This is because both the $\hat {v}_1$ -Orr term in (4.12) and the lift-up term in (4.13) weaken the instability mechanism which dominates the growth found here. Starting with figure 11(a) or (4.13), the presence of the lift-up term works against the push-over effect of $\hat {\eta }_0$ . The $x$ - and $z$ -phases of $\hat {v}_1$ are the same as that for $\hat {u}_1$ created by $\hat {\eta }_0$ (see (4.30)) so $\hat {v}_1$ is either lifting up or pushing down the base shear flow to reduce the magnitude of $\hat {u}_1$ . By this same reasoning, lift up can enhance the transient growth process if Orr dominates push over – i.e. there is no instability – but this scenario is never selected when maximising growth over wavenumber space.

In figure 11(b), the possibility of motion into or out of the $x$ - $z$ plane of the streaks reduces the spanwise flow field $\hat {w}_1$ generated by the $\hat {u}_1$ field through continuity (the pressure field associated with the spatial gradients of $\hat {u}_1$ now can drive both $\hat {v}_1$ and $\hat {w}_1$ fields rather than just $\hat {w}_1$ ). This reduced $\hat {w}_1$ then has weakened spatial gradients and so diminishes the streak-Orr effect. In terms of (4.12), this manifests itself as the $\hat {v}_1$ -Orr term acting against the $\hat {\eta }_1$ -Orr terms once $(1-k_xt)\lt 0$ which happens early in the evolution (e.g. see figure 13(b)).

4.10. Previous work

The linear instability isolated here in (4.16) is inviscid in nature and, at least in terms of the streamwise and cross-stream velocity components, is centred at the spanwise inflexion points of the streaks where the streak shear is maximal. This suggests it is related to the linear instability originally observed by Swearingen & Blackwelder (Reference Swearingen and Blackwelder1987). Against this, the spanwise flow components are maximal at the streak flow extrema, but there are no cross-stream inflexion points in this model and so no Kelvin–Helmholtz type instability (e.g. see figure 19b of Kline et al. Reference Kline, Reynolds, Schraub and Rundstadler1967).

The model has revealed two transient growth mechanisms that can be in play for different wavenumber pairings. The one which is selected by optimising over the wavenumbers corresponds to that found originally by Schoppa & Hussain (Reference Schoppa and Hussain2002) (e.g. their initial condition is very similar to the optimal conditions found here dominated by $\hat {\eta }_0$ ). There, the authors concentrated on the growth of the streamwise vorticity (e.g. see their figure 11(b) and § 4.2) identifying what they called a ‘shearing’ generation of vorticity due to the right hand side term in their (14a) and illustrated in their figure 16. This is a purely Orr term as there are no lift-up or push-over terms in the streamwise vorticity equation and those authors attributed it predominantly to the base cross-stream shear (see just below (13) in Schoppa & Hussain (Reference Schoppa and Hussain2002)). Here, we instead find that push over is the more important mechanism which has to dominate (streak) Orr to see this growth. When this happens, the two-variable inviscid model is also linear unstable, that is, there are no wavenumber pairs for which this transient growth occurs with the flow stable (see figure 10). Plausibly, introducing diffusion which stays bounded rather than growing steadily as here would introduce a threshold for instability so recreating the possibility of a stable, transient growth scenario studied by Schoppa & Hussain (Reference Schoppa and Hussain2002).

Finally, it is worth remarking that, for reasons of simplicity, we have identified the Orr mechanism only with advection terms and these don’t contribute to the perturbation energy equation. This means that there must be either lift up or push over operating as well to actually get energy growth. The model studied here strongly suggests that push over is the key mechanism which is consistent with the findings of Hoepffner et al. (Reference Hoepffner, Brandt and Henningson2005): their figure 6(a) shows that energy input from the streak field dominates that from the base shear.