2.1. Numerical experiments

To investigate the role of different linear mechanisms, we perform direct numerical simulations of incompressible turbulent channel flows driven by a constant mean pressure gradient. Hereafter, the streamwise, wall-normal and spanwise directions of the channel are denoted by $x$, $y$ and $z$, respectively, the corresponding flow velocity components by $u$, $v$, $w$, and pressure by $p$. The density of the fluid is $\rho$, the kinematic viscosity of the fluid is $\nu$, and the channel height is $h$. The wall is located at $y=0$, where no-slip boundary conditions apply, whereas free stress and no penetration conditions are imposed at $y=h$. The streamwise and spanwise directions are periodic.

The simulations are characterised by the friction Reynolds number, ${Re}_{\tau }$, defined as the ratio of the channel height to the viscous length scale $\delta _v = \nu /u_{\tau }$, where $u_{\tau }$ is the characteristic velocity based on the mean skin friction at the wall $u_{\tau }^2 \equiv \nu \langle \partial u(x,0,z,t)/ \partial y\rangle _{xzt}$. Here, the Reynolds number is ${Re}_{\tau } = h/\delta _v \approx 180$. The streamwise, wall-normal and spanwise sizes of the computational domain are $L_x^+ \approx 337$, $L_y^+ \approx 186$ and $L_z^+ \approx 168$, respectively, where the superscript $+$ denotes quantities scaled by $\nu$ and $u_{\tau }$. Jiménez & Moin (Reference Jiménez and Moin1991) showed these ‘minimal channels’ contain an elementary turbulent flow unit comprised of a single streamwise streak and a pair of staggered quasi-streamwise vortices, that reproduce the dynamics of the flow in larger domains. Hence, the current numerical experiment isolates the few, most relevant, coherent structures involved in self-sustaining turbulence in the buffer layer. It also provides an ideal testbed for studying linear mechanisms, as it enables the identification of a meaningful base flow for these elementary coherent structures. In appendix A we assess the sensitivity of the key results presented in this study to changes in the domain extent ($L_x$ and $L_z$). We find that our conclusions still hold when the size of the computational domain is doubled in each direction.

We integrate the incompressible Navier–Stokes equations

(2.1a)\begin{gather} \frac{\partial\boldsymbol{u}}{\partial t} = - \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} - \frac{1}{\rho}\boldsymbol{\nabla} p + \nu \nabla^2 \boldsymbol{u} + \boldsymbol{f}, \end{gather}

(2.1b)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}

with $\boldsymbol {u} {\stackrel {\textrm {def}}{=}} (u,v,w)$ and $\boldsymbol {f} = (u_{\tau }^2/h,0,0)$.

The simulations are performed with a staggered, second-order, finite differences scheme (Orlandi Reference Orlandi2000) and a fractional-step method (Kim & Moin Reference Kim and Moin1985) with a third-order Runge–Kutta time-advancing scheme (Wray Reference Wray1990). The solution is advanced in time using a constant time step chosen appropriately so that the Courant–Friedrichs–Lewy condition is below 0.5. The code has been presented in previous studies on turbulent channel flows (Lozano-Durán & Bae Reference Lozano-Durán and Bae2016; Bae et al. Reference Bae, Lozano-Durán, Bose and Moin2018b, Reference Bae, Lozano-Durán, Bose and Moin2019). In addition, we performed various numerical experiments (summarised in the second column of table 2) in which we time advance two sets of equations: one for the base flow $\boldsymbol {U}$ and one for the fluctuations $\boldsymbol {u}'$. In this manner, we are able to independently control the dynamics of $\boldsymbol {U}$ and $\boldsymbol {u}'$. We discuss in detail these additional experiments in § 6.

Table 2. List of cases of turbulent channel flows with and without constrained linear mechanisms. The friction Reynolds number is ${Re}_{\tau } \approx 180$ for all cases. The cases are labelled following the nomenclature: R, regular wall turbulence with feedback $\boldsymbol {U}\rightarrow \boldsymbol {u}'$ allowed; NF, no-feedback from $\boldsymbol {U}\rightarrow \boldsymbol {u}'$ allowed; SEI, suppressed exponential instabilities; TG, only transient growth without exponential nor parametric instabilities; NLU, no linear lift-up of the streak; NPO, no linear push-over of the streak; NO, no linear Orr of the streak.

The streamwise and spanwise grid resolutions are $\Delta x^+\approx 6.5$ and $\Delta z^+\approx 3.3$, respectively, and the minimum and maximum wall-normal resolutions are $\Delta y_{{min}}^+\approx 0.2$ and $\Delta y_{{max}}^+\approx 6.1$. The corresponding number of grid points in $x$, $y$ and $z$ are $64 \times 90 \times 64$, respectively. All the simulations presented here were run for at least $300h/u_{\tau }$ units of time after transients. This time period is orders of magnitude longer than the typical lifetime of individual energy-containing eddies (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014), and allows us to collect meaningful statistics of the self-sustaining cycle.

2.2. Base flow

We partition the flow into fluctuating velocities $\boldsymbol {u}' {\stackrel {\textrm {def}}{=}} (u', v', w')$ and base flow $\boldsymbol {U}$, defined as the time-varying mean streamwise velocity $\boldsymbol {U} {\stackrel {\textrm {def}}{=}} (U, 0, 0)$, where

(2.2)\begin{equation} U(y, z, t) {\stackrel{\textrm{def}}{=}} \langle u \rangle_x = \frac{1}{L_x} \int_{0}^{L_x} u(x,y,z,t) \,\mathrm{d}\kern0.05em x, \end{equation}

such that $u' {\stackrel {\textrm {def}}{=}} u - U$, $v' {\stackrel {\textrm {def}}{=}} v$ and $w' {\stackrel {\textrm {def}}{=}} w$. Hereafter, $\langle \boldsymbol {\cdot } \rangle _{i,j,k,\ldots }$ denotes averaging over the directions (or time) $i$, $j$, $k$,…, for example,

(2.3)\begin{equation} \langle u \rangle_{xzt} = \frac{1}{L_x L_z T_s} \int_{0}^{L_x}\int_{0}^{L_z}\int_{0}^{T_s} u(x,y,z,t) \, \mathrm{d}t\,\mathrm{d}z\,\mathrm{d}\kern0.05em x, \end{equation}

where $T_s$ is a time horizon long enough to remove any time fluctuations. Figure 2 illustrates this flow decomposition and figure 3 depicts three typical snapshots of the base flow defined in (2.2). Note that because we are using a minimal box for the channel, only a single energy-containing eddy fits in the domain. Hence, $U$ computed in minimal boxes is a meaningful base flow ‘felt’ by individual flow structures. This would not be the case in larger domains in which the effect of the multiple structures present in the flow cancels out and does not contribute to $U$.

Figure 2. Decomposition of the instantaneous flow into a streamwise mean base flow and fluctuations. Instantaneous isosurface of streamwise velocity for (a) the total flow $u$, (b) the streak base flow $U$ and (c) the absolute value of the fluctuations $|u'|$. The values of the isosurfaces are 0.6 (a,b) and 0.1 (c) of the maximum streamwise velocity. Shading represents the distance to the wall from dark ($y=0$) to light ($y=h$). The arrow in panel (a) indicates the mean flow direction. Results for case R180.

Figure 3. Examples of base flow, defined as $U(y,z,t) {\stackrel {\textrm {def}}{=}} \langle u \rangle _x$, for a turbulent channel flow at ${Re}_{\tau }\approx 180$ (case R180 from § 3). The examples are representative instances with (a,b) strong streak activity and (c) quiescent times with weak streak activity. The shading represents the value of the streamwise velocity in wall units.

We have not included in the base flow (2.2) the contributions from the streamwise averages of $v$ and $w$ components, $V {\stackrel {\textrm {def}}{=}} \langle v \rangle _x$ and $W {\stackrel {\textrm {def}}{=}} \langle w \rangle _x$, as these are not traditionally included in the study of stability of the streaky flow. Indeed, the vast majority of studies reported in table 1 do not account for $V$ and $W$ in their analysis. The results obtained using $(U,0,0)$ as a base flow were repeated for a base flow consisting of $(U,V,W)$, and a concise overview of the findings can be found in appendix B. In summary, the conclusions drawn for base flows $(U,0,0)$ or $(U,V,W)$ are similar and, thus, we focus on the former for simplicity.

The equation of motion for the base flow $\boldsymbol {U}=(U,0,0)$ is obtained by averaging the Navier–Stokes equations (2.1) in the streamwise direction,

(2.4a)\begin{gather} \frac{\partial \boldsymbol{U}}{\partial t} + \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{U} = -\mathcal{D}\langle \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}' \rangle_x - \frac{\mathcal{D}}{\rho}\boldsymbol{\nabla} \langle p \rangle_x + \nu \nabla^2 \boldsymbol{U} + \boldsymbol{f}, \end{gather}

(2.4b)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} = 0, \end{gather}

where the operator $\mathcal {D}$ sets the $y$- and $z$-components of the nonlinear terms and pressure to zero for consistency with $\boldsymbol {U} = (U,0,0)$ (see appendix B). Subtracting (2.4) from (2.1) we get that the fluctuating flow $\boldsymbol {u}'$ is governed by

(2.5)\begin{equation} \frac{\partial\boldsymbol{u}'}{\partial t} = \underbrace{\mathcal{L}(U)\boldsymbol{u}'}_{\substack{\textit{linear}\\ \textit{processes}}} + \underbrace{\boldsymbol{N}(\boldsymbol{u}')}_{\substack{\textit{nonlinear}\\ \textit{processes}}}, \end{equation}

where $\mathcal {L}(U)$ is the linearised Navier–Stokes operator for the fluctuating state vector about the instantaneous $\boldsymbol {U}$ (see figure 2b) such that

(2.6)\begin{equation} \mathcal{L}(U)\boldsymbol{u}' = \mathcal{P}\left[ -\boldsymbol{U}\boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}' - \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{U} + \nu \nabla^2 \boldsymbol{u}' \right]. \end{equation}

The operator $\mathcal {P}$ accounts for the kinematic divergence-free condition, $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {u}' = 0$. Conversely, $\boldsymbol {N}(\boldsymbol {u}')$ collectively denotes the nonlinear terms, which are quadratic with respect to fluctuating flow fields,

(2.7)\begin{equation} \boldsymbol{N}(\boldsymbol{u}') = \mathcal{P}\left[ -\boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}' +\mathcal{D}\langle \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}' \rangle_x \right]. \end{equation}

We are interested in the dynamics of $\boldsymbol {u}'$ governed by (2.5). Note that the flow partition $\boldsymbol {U} + \boldsymbol {u}'$ as defined in (2.2) implies that the energy injection into the velocity fluctuations is ascribed to linear processes from $\mathcal {L}(U)$, since the term $\boldsymbol {N}(\boldsymbol {u}')$ is only responsible for redistributing the energy in space and scales among the fluctuations, i.e. the domain integral of $\boldsymbol {u}'\boldsymbol {\cdot }\boldsymbol {N}$ vanishes identically and, thus,

(2.8)\begin{equation} \frac{\partial }{\partial t} \left\langle E \right\rangle_{xyz} = \left\langle \boldsymbol{u}' \boldsymbol{\cdot} \mathcal{L}(U)\boldsymbol{u}'\right\rangle_{xyz}, \end{equation}

where $E {\stackrel {\textrm {def}}{=}} |\boldsymbol {u}'|^2/2$ is the fluctuating turbulent kinetic energy.

In the rest of the paper, in addition to solutions of the Navier–Stokes equations (2.1), we modify (2.6) to preclude the energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$ for targeted linear mechanisms. The simulations carried out are summarised in table 2, which includes the active linear mechanisms for energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$ and whether the cases are capable of sustaining turbulent fluctuations. The details on how the equations of motion are modified for each case are discussed in the remainder of the paper.

4.1. Energy transfer via exponential instability

The first mechanism considered is modal instability of the instantaneous base flow. At any given time, the exponential instabilities are obtained by eigendecomposition of the matrix representation of the linear operator $\mathcal {L}(U)$ in (2.5),

(4.2)\begin{equation} \mathcal{L}(U) = \mathcal{Q} \varLambda \mathcal{Q}^{-1}, \end{equation}

where $\mathcal {Q}$ consists of the eigenvectors organised in columns, $\mathcal {Q}^{-1}$ is the inverse of $\mathcal {Q}$, and $\varLambda$ is the diagonal matrix of associated eigenvalues, $\lambda _j + i\omega _j$, with $\lambda _j, \omega _j \in \mathbb {R}$. Equation (4.1) admits solutions of the form $\boldsymbol {u}'_{{linear}} \sim \boldsymbol {c} \exp {[(\lambda _j + \textrm {i}\omega _j)t]}$, with $\boldsymbol {c}$ a constant. Hence, we say that the base flow is unstable if any of the growth rates $\lambda _j$ is positive. More details on the stability analysis are provided in appendix C along with the validation of our implementation in appendix D. Figure 7 shows a representative example of the streamwise velocity for an unstable eigenmode. The predominant eigenmode has typically a sinuous structure of positive and negative patches of velocity flanking the velocity streak side by side, which may lead to its subsequent meandering and eventual breakdown. Varicose-type modes are also observed but they are less frequent.

Figure 7. Representative exponential instability of the streak. (a) Instantaneous isosurface of the base flow $U$. The value of the isosurface is $0.6$ of the maximum and the shading represents the distance to the wall. (b) Isosurface of the instantaneous streamwise velocity for the eigenmode associated with the most unstable eigenvalue $\lambda _{max} h/u_{\tau } \approx 3$. The flow structure of the eigenmode is consistent with a sinuous instability. The values of the isosurface are $-0.5$ (dark) and $0.5$ (light) of the maximum streamwise velocity.

Figure 8(a) shows the probability density functions of the growth rate of the four least stable eigenvalues of $\mathcal {L}(U)$. On average, the operator $\mathcal {L}(U)$ contains two to three unstable eigenmodes at any given instant. Denoting the Fourier streamwise wavenumber as $k_x$, the most unstable eigenmode usually corresponds to $k_x=2{\rm \pi} /L_x$, although occasionally modes with $k_x=2{\rm \pi} /(2L_x)$ become the most unstable. The sensitivity of our results to $L_x$ is further discussed in appendix A. The history of the maximum growth rate supported by $\mathcal {L}(U)$, denoted by $\lambda _1 = \lambda _{{max}}$, is shown in figure 8(b). The flow is exponentially unstable ($\lambda _{{max}}>0$) more than 90 % of the time. The average e-folding time for an exponentially unstable perturbation is roughly $h/u_{\tau }$, which is comparable to the bursting duration $T_b$.

Figure 8. (a) Probability density functions of the growth rate of the four least stable eigenvalues of $\mathcal {L}(U)$, $\lambda _1>\lambda _2>\lambda _3>\lambda _4$. (b) The time series of the most unstable eigenvalue $\lambda _{max}=\lambda _1$ of $\mathcal {L}(U)$. (c) The time series of the ratio of $\lambda _{max}/ \lambda _U$, where $\lambda _U$ is the growth rate of the base flow given by (4.3). The horizontal dashed and dotted lines are $\lambda _{max}/ \lambda _U=1$ and $\lambda _{max}/ \lambda _U=10$, respectively. Results for regular channel R180.

The ansatz underlying modal instability is that the spatial structure of the base flow remains constant in time. Therefore, we expect the above linear instability to manifest in the flow only when $\lambda _{{max}}$ is much larger than the time rate of change of the base flow $U$, defined as

(4.3)\begin{equation} \lambda_U {\stackrel{\textrm{def}}{=}} \left\langle \frac{1}{2} \frac{|\mathrm{d} E_U /\mathrm{d}t|}{E_U} \right\rangle_{yz}, \end{equation}

where $E_U {\stackrel {\textrm {def}}{=}} U^2/2$ is the energy of the base flow. The ratio $\lambda _{{max}}/\lambda _U$ for $\lambda _{{max}}>0$, shown in figure 8(c), is approximately 5 on average and occasionally achieves values of 20, i.e. the time changes of $U$ might be 5 to 20 times slower than the e-folding time of the most unstable eigenmode. The growth of the modal instabilities is not overwhelmingly faster than the changes on the base flow. However, considering that the exponential growth of disturbances is supported for a non-negligible fraction of the flow history (roughly 90 % of the time as shown before), modal instability of $\mathcal {L}(U)$ still stands as a potential mechanism sustaining the fluctuations. Note that the argument above does not imply that exponential instabilities are necessarily relevant for the flow when $\lambda _{{max}}/\lambda _U$ is large, but only that they could manifest based on their characteristic time scales. In fact, we show in § 6.2 that exponential instabilities are not a requisite to sustain turbulence.

4.2. Energy transfer via transient growth

The second linear mechanism considered is the non-modal transient growth of the fluctuations. The linear dynamics of (4.1) can be formally written as

(4.4)\begin{equation} \boldsymbol{u}'_{{linear}}(t+T) = \varPhi_{t \rightarrow t+T}\, \boldsymbol{u}'_{{linear}}(t). \end{equation}

The propagator $\varPhi _{t \rightarrow t+T}$ maps the fluctuating flow from time $t$ to time $t+T$ and represents the cumulative effect of the linear operator $\mathcal {L}(U)$ during the period from $t$ to $t+T$. If the base flow remains constant for $t\in [t_0, t_0+T]$, then the propagator simplifies to

(4.5)\begin{equation} \varPhi_{t_0 \rightarrow t_0+T} = \exp \left( \mathcal{L}_0\,T \right), \end{equation}

where $\mathcal {L}_0$ denotes $\mathcal {L}(U(y, z, t_0))$.

Equation (4.4) accounts for both the modal and non-modal growth of $\boldsymbol {u}'$ for $t\in [t_0, t_0+T]$. The exponential growth of the fluctuating velocities was already quantified in § 4.1. Here we are concerned with the transient growth of $\boldsymbol {u}'$ supported by $\mathcal {L}_0$. To that end, we exclude from the analysis any growth of fluctuations due to the modal instabilities of $\mathcal {L}_0$. This is achieved by the modified operator $\tilde {\mathcal {L}}_0$,

(4.6)\begin{equation} \tilde{\mathcal{L}}_0 {\stackrel{\textrm{def}}{=}} \mathcal{Q} \tilde{\varLambda} \mathcal{Q}^{-1}, \end{equation}

where $\tilde {\varLambda }$ is the stabilised counterpart of $\varLambda$ in (4.2) obtained by setting the real part ($\lambda _j$) of all unstable eigenvalues of $\varLambda$ equal to $-\lambda _j$, while their phase speed and eigenmode structure are left unchanged. We assessed that the transient growth properties of $\tilde {\mathcal {L}}_0$ are mostly insensitive to the amount of stabilisation introduced in $\varLambda$ when $\lambda _j>0$ are replaced by $-a\lambda _j$ with $a\in [1/10,10]$. The potential effectiveness of transient growth due to a base flow $U(y,z,t_0)$ is then characterised by the energy gain $G$ over some time period $T$, defined as

(4.7)\begin{equation} G(t_0, T, \boldsymbol{u}'_{0}) {\stackrel{\textrm{def}}{=}} \frac{\left\langle \boldsymbol{u}'_T \boldsymbol{\cdot}\boldsymbol{u}_T' \right\rangle_{xyz}}{\left\langle \boldsymbol{u}'_0\boldsymbol{\cdot}\boldsymbol{u}_0' \right\rangle_{xyz}}, \end{equation}

where $\boldsymbol {u}'_{T} {\stackrel {\textrm {def}}{=}} \boldsymbol {u}'_{{linear}}(x,y,z,t_0+T)$, $\boldsymbol {u}'_0 {\stackrel {\textrm {def}}{=}} \boldsymbol {u}'_{{linear}}(x,y,z,t_0)$ and $T$ is the time horizon for which the gain $G$ is computed.

The energy, being a bilinear form, can be expressed as an inner product, e.g.

(4.8)\begin{equation} \left(\boldsymbol{u}', \boldsymbol{u}'\right) {\stackrel{\textrm{def}}{=}} \left\langle \boldsymbol{u}'\boldsymbol{\cdot}\boldsymbol{u}' \right\rangle_{xyz}. \end{equation}

Using the definition (4.8) and the form of the propagator (4.5) for the frozen linear operator $\tilde {\mathcal {L}}_0$, the energy gain is rewritten as

(4.9)\begin{equation} G\left(t_0, T, \boldsymbol{u}'_0\right) = \frac{\left(\boldsymbol{u}'_T , \boldsymbol{u}'_T \right)}{\left( \boldsymbol{u}'_0, \boldsymbol{u}'_0 \right)} = \frac{\left( {e}^{\tilde{\mathcal{L}}_0 T} \boldsymbol{u}'_0 , {e}^{\tilde{\mathcal{L}}_0 T}\boldsymbol{u}'_0 \right)} {\left(\boldsymbol{u}'_0 , \boldsymbol{u}'_0 \right)} = \frac{\left( \boldsymbol{u}'_0 , {e}^{\tilde{\mathcal{L}}_0^{{\dagger}ger} T} {e}^{\tilde{\mathcal{L}}_0 T}\boldsymbol{u}'_0 \right)}{\left( \boldsymbol{u}'_0 , \boldsymbol{u}'_0 \right)}. \end{equation}

In the last equality, dagger ${\dagger}ger$ denotes the adjoint operator. Note that, for $T \rightarrow \infty$, the energy gain (4.9) tends to $0$, since the operator $\tilde {\mathcal {L}}_0$ is exponentially stable. The maximum gain over all initial conditions $\boldsymbol {u}'_0$, denoted by $G_{max}(t_0,T) = \mathrm {sup}_{\boldsymbol {u}'_0}(G)$, is given by the square of the largest singular value of the stabilised linear propagator $\tilde {\varPhi }_0$ (Butler & Farrell Reference Butler and Farrell1993; Farrell & Ioannou Reference Farrell and Ioannou1996),

(4.10)\begin{align} \tilde{\varPhi}_{t_0 \rightarrow t_0+T} & = \exp(\tilde{\mathcal{L}}_0 T), \end{align}

(4.11)\begin{align} & = \mathcal{M} \varSigma \mathcal{N}^{{\dagger}ger}, \end{align}

where $\varSigma$ is a diagonal matrix, whose positive entries $\sigma _j$ are the singular values of $\exp (\tilde {\mathcal {L}}_0 T)$ and the columns of $\mathcal {M}$ and of $\mathcal {N}$ are the output modes (or left-singular vectors) and input modes (or right-singular vectors) of $\exp (\tilde {\mathcal {L}}_0 T)$, respectively.

The maximum gain $G_{max}$ for R180 as a function of the optimisation time $T$ is shown in figure 9(a). The values of $G_{max}$ also depend on $t_0$; figure 9(a) features the mean and the standard deviation of $G_{max}$ for more than 1000 uncorrelated instants $t_0$. Figure 9(a) reveals that non-normality alone is potentially able to produce fluctuation energy growth of the order of $G_{max} \approx 100$. On average, the time horizon for maximum gain is attained at $T_{max} \approx 0.35 h/u_{\tau }$. Thus, the maximum non-normal energy gain is obtained at a similar time scale as the bursting time, $T_p$ (see § 3). For an elapsed time of $T_{max}$, the auto-correlation of the base flow,

(4.12)\begin{equation} C_{UU} {\stackrel{\textrm{def}}{=}} \left\langle \left[U(y,z,t)-\langle U \rangle_{t}\right]\left[U(y,z,t+T)-\langle U \rangle_{t}\right] \right\rangle_{yzt}, \end{equation}

has a value of 0.7, as shown in figure 9(a), which is reasonably high to justify the ‘frozen-base-flow’ assumption underlying the calculation of $G$. The probability density function (p.d.f.) of $G_{max}$ at $T_{max}$ (figure 9b) shows that $U(y,z,t_0)$ at certain times can support gains as high as 300.

Figure 9. Energy transfer via transient growth with frozen-in-time base flow. (a) The ensemble average of the maximum energy gain $G_{max}(t_0,T)$ (black solid line, see (4.13)) over different initial instances $t_0$, as a function of the time horizon $T$. Shaded regions denote $\pm$ half standard deviation of $G_{max}(t_0, T)$ for a given $T$. The vertical dashed line denotes $T_{{max}} = 0.35 h/ u_{\tau }$. The blue dotted line is the auto-correlation of $U$, $C_{UU}$ and its values appear on the right vertical axis. (b) Probability density function of gains $G_{max}(t_0,T_{{max}})$. Results for regular channel R180.

The results here support the hypothesis of transient growth of the ‘frozen’ mean streamwise flow $U(y,z,t_0)$ as a tenable candidate to sustain wall turbulence. It is worth noting that the maximum gain $G_{max}$ obtained with a streaky base flow $U(y,z,t_0)$ is considerably larger than the limited gains of around 10 reported in previous studies focused in the buffer layer (Del Álamo & Jiménez Reference Del Álamo and Jiménez2006; Cossu et al. Reference Cossu, Pujals and Depardon2009; Pujals et al. Reference Pujals, García-Villalba, Cossu and Depardon2009). In latter works, the base flow selected was $\langle u \rangle _{xzt}$, which lacks any spanwise $z$-structure and, hence, supports much lower gains compared to $U(y,z,t_0)$.

Figure 10 provides an example of the input and output modes associated with the maximum optimal gain for one selected instant $t_0$. The flow displays a sinuous backwards-leaning perturbation (input mode) that is being tilted forward by the mean shear over the time $T$ (output mode). The process is reminiscent of the linear Orr/lift-up mechanism driven by continuity and wall-normal transport of momentum characteristic of the bursting process and streak formation (Orr Reference Orr1907; Ellingsen & Palm Reference Ellingsen and Palm1975; Kim & Lim Reference Kim and Lim2000; Jiménez Reference Jiménez2013; Encinar & Jiménez Reference Encinar and Jiménez2020). Unlike the studies that used the base flow $\langle u\rangle _{xzt}$, our choice of a spanwise-varying base flow $U(y,z,t)=\langle u \rangle _{x}$ limits both the spanwise extent and location of the input and output modes, which are controlled by the spanwise location of the streak.

Figure 10. Representative sinuous input and output modes associated with the transient growth of the streak. Isosurfaces of (a,c) the input and (b,d) the output wall-normal velocity mode associated with the largest singular value of $\tilde {\varPhi }_{t_0 \rightarrow t_0+T}$ from (4.9) at $T= 0.35h/u_{\tau }$. The isosurface are $-0.5$ (dark) and $0.5$ (light) of the maximum wall-normal velocity. The gain is $G_{{max}}=136$. The coloured contours at $x=L_x$ are 0.2, 0.4, 0.6 and 0.7 of the maximum velocity of the base flow. The result is for the regular channel R180.

4.3. Energy transfer via transient growth with time-varying base flow

In the previous section we have considered frozen-in-time base flows. We now relax this assumption such that the (stabilised) linear operator is time dependent $\tilde {\mathcal {L}}(U(y,z,t))$. The propagator $\tilde {\varPhi }^t_{t_0 \rightarrow t_0+T}$ (now with superscript $t$), is given by the Peano–Baker series (Rugh Reference Rugh1996),

(4.13)\begin{equation} \tilde{\varPhi}_{t_0 \rightarrow t_0+T}^t =\mathcal{I}+\int_{t_0}^{t_0+T}{\tilde{\mathcal{L}}}(t_{1})\,\mathrm{d}t_{1} +\int_{t_0}^{t_0+T}{\tilde{\mathcal{L}}}(t_{1})\int_{t_0}^{{t_{1}}}{\tilde{\mathcal{L}}}(t_2)\, \mathrm{d} t_{2}\,\mathrm{d} t_{1} + \cdots, \end{equation}

where $\mathcal {I}$ is the identity matrix and we have simplified the notation to $\tilde {\mathcal {L}}(t) = \tilde {\mathcal {L}}(U(y,z,t))$. The propagator $\tilde {\varPhi }^t_{t_0 \rightarrow t_0+T}$ represents the cumulative effect of $U(y,z,t)$ from $t_0$ to $t_0+T$ accounting for time variations in the base flow. The energy gain of (4.13) is

(4.14)\begin{equation} G^t(t_0,T, \boldsymbol{u}'_0) = \frac{( \boldsymbol{u}'_0 , (\tilde{\varPhi}_{t_0 \rightarrow t_0+T}^t)^{{\dagger}ger} (\tilde{\varPhi}_{t_0 \rightarrow t_0+T}^t) \boldsymbol{u}'_0)} {( \boldsymbol{u}'_0 , \boldsymbol{u}'_0)}. \end{equation}

In contrast with the frozen-base-flow propagator $\tilde {\varPhi }_{t_0 \rightarrow t_0+T}$ in (4.9), the time variations of the operator $\tilde {\mathcal {L}}(U)$ can either weaken or enhance the energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$. Another difference is that the $G^t$ now admits a finite value at $T\rightarrow \infty$, despite that $\tilde {\mathcal {L}}(U)$ is modally stable at all instances. One potential route to enhance the gain for short times and/or achieve finite gains for long times is the parametric instability of the streak discussed in the introduction (Farrell & Ioannou Reference Farrell and Ioannou2012). However, it is shown below that none of these effects seem to be at play.

To evaluate the transient growth with time-varying base flows, we reconstruct the propagator without exponential instabilities $\tilde {\varPhi }^t_{t_0 \rightarrow t_0+T}$ for case R180. In virtue of the property $\tilde {\varPhi }_{t_0 \rightarrow t_0+T}^t = \tilde {\varPhi }_{t_1 \rightarrow t_0+T}^t \tilde {\varPhi }_{t_0 \rightarrow t_1}^t$ for $t_0\le t_1 \le t_0+T$, the propagator is numerically computed by the ordered product of exponentials under the assumption of small $\Delta t$ as

(4.15)\begin{equation} \tilde{\varPhi}_{t_0 \rightarrow t_0+T}^t \approx \exp \left[\tilde{\mathcal{L}}(t_0+(n-1)\Delta t)\Delta t \right]\cdots \exp\left[ \tilde{\mathcal{L}}(t_0+\Delta t) \Delta t \right] \exp\left[ \tilde{\mathcal{L}}(t_0) \Delta t \right], \end{equation}

where $T = n\Delta t$, with $n$ a positive integer. We saved the time history of $U(y,z,t)$ from R180 at all time steps and used it to compute $\tilde {\varPhi }_{t_0 \rightarrow t_0+T}^t$ via (4.15). We take $\Delta t^+ \approx 0.05$, which is the time step used to integrate the equations of motion. The maximum gain supported by $\tilde {\varPhi }_{t_0 \rightarrow t_0+T}^t$ is compared with its frozen-base-flow counterpart $\tilde {\varPhi }_{t_0 \rightarrow t_0+T}$ in figure 11. The results reveal that energy growth with time-varying base flows is almost identical to the energy growth under the frozen-base-flow assumption up to $T \approx T_{{max}}=0.35 h/u_{\tau }$, which also corresponds to the time for maximum gain for $\tilde {\varPhi }^t_{t_0 \rightarrow t_0+T}$. For longer times $T > T_{{max}}$, the gain with time-varying base flows is depleted with respect to that of the frozen base flow, and tends to zero for $T \rightarrow \infty$ (not shown). The results show that accounting for time variations of the base flow has a negligible effect on energy transfer for short times, but gains for frozen base flows are over estimated for long times otherwise.

Figure 11. Energy transfer via transient growth with time-varying base flow. (a) The ensemble average of the maximum energy gain $G^t_{max}(t_0,T)$ (black solid line, see (4.13)) and $G_{max}(t_0,T)$ (red dashed line, see (4.11)) over different initial instances $t_0$, as a function of the time horizon $T$. Shaded regions denote $\pm$ half-standard deviation of $G^t_{max}(t_0, T)$ for a given $T$. The vertical dashed line denotes $T_{{max}} = 0.35 h/ u_{\tau }$. (b) Probability density function of gains for $G^t_{max}(t_0,T_{{max}})$ (black solid line) and $G_{max}(t_0,T_{{max}})$ (red dashed line). Results for regular channel R180.

The propagator $\tilde {\varPhi }^t_{t_0 \rightarrow t_0+T}$ can also be analysed in terms of input and output modes. The input and output modes for the time-varying base flow are again a backwards-leaning perturbation (input mode) that is being tilted forward by the mean shear (output mode), very similar to the example shown in figure 10, but not shown here for brevity.

6.1. Wall turbulence without explicit feedback from $ {u}'$ to ${U}$

In previous sections, we have acted as if

(6.1)\begin{equation} \frac{\partial\boldsymbol{u}'}{\partial t} = \mathcal{L}(U)\boldsymbol{u}'+ \boldsymbol{N}(\boldsymbol{u}') \end{equation}

is linear in the term $\mathcal {L}(U)\boldsymbol {u}'$. This is obviously not true because $U(y,z,t)$ depends on $\boldsymbol {u}'$ via the nonlinear feedback term $-\langle \boldsymbol {u}' \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {u}' \rangle _x$ (see the base flow evolution equation (3.1b)).

Prior to investigating the cause-and-effect links of linear mechanisms in $\mathcal {L}(U)$, we derive a surrogate system in which the energy injection is strictly linear by preventing the explicit feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$. To achieve this, we proceed as follows.

1. (i) We perform a simulation of R180 for $600 h/u_{\tau }$ units of time (after transients) with a constant time step.

2. (ii) We store the base flow at all time steps. We denote the time series of this base flow as $U_0 = U(y,z,t)$ from case R180.

3. (iii) We time integrate the system

   (6.2)\begin{gather} \frac{\partial\boldsymbol{u}'}{\partial t} = \mathcal{L}(U_0)\boldsymbol{u}'+ \boldsymbol{N}(\boldsymbol{u}'), \end{gather}
   (6.3)\begin{gather}U_0 = U(y,z,t) \ \text{from case R180}. \end{gather}

Equation (6.2) is initialised from a random, incompressible velocity field and it is integrated for $600 h/u_{\tau }$ units of time using the same time step as in R180. Equation (6.2) is akin to the Navier–Stokes equations, in which the equation of motion of $\boldsymbol {U}$ is replaced by $\boldsymbol {U} = (U_0, 0, 0)$. We refer to this case as ‘channel flow with no-feedback’ or NF180 for short. Note that the base flow $U_0$ has no explicit feedback from $\boldsymbol {u}'$ in (6.2), although it has been implicitly ‘shaped’ by the velocity fluctuations of R180 and, as such, it contains dynamic information of actual wall turbulence. The key difference in (6.2) is that the term $\mathcal {L}(U_0)\boldsymbol {u}'$ is now strictly linear while preserving both the modal and non-modal properties of $\mathcal {L}(U)$ in R180.

The flow sustained in NF180 is turbulent as revealed by the history of the turbulent kinetic energy in figure 12(a). Moreover, the footprint of the flow trajectory projected onto the $\langle P\rangle _{xyz}$–$\langle D\rangle _{xyz}$ plane in figure 12(b) also exhibits a similar behaviour to R180: the flow is organised around $\langle P \rangle _{xyz} = -\langle D \rangle _{xyz}$ with excursions into the high/low dissipation and production regions with predominantly counterclockwise motions. This assessment is merely qualitative and some differences are expected between the $\langle P\rangle _{xyz}$–$\langle D\rangle _{xyz}$ trajectories in R180 and NF180.

Figure 12. (a) The history of the domain-averaged turbulent kinetic energy of the fluctuations $\langle E \rangle _{xyz}$. Note that only $30 h/u_{\tau }$ units of time are shown in the panel but the simulation was carried out for $600 h/u_{\tau }$. (b) Projection of the flow trajectory onto the average production rate $\langle P \rangle _{xyz}$ and dissipation rate $\langle D \rangle _{xyz}$ plane. The arrows indicate the time direction of the trajectory, which on average rotates counterclockwise. The red dashed line is $\langle P \rangle _{xyz} = -\langle D \rangle _{xyz}$ and the red circle $\langle P \rangle _{xyzt} = -\langle D \rangle _{xyzt}$. The trajectory projected covers $15 h/u_{\tau }$ units of time. The results are for NF180.

The mean turbulence intensities for NF180 are shown in figure 13. Statistics are collected once the system reaches the statistically steady state. The mean velocity profile is omitted as it is identical to that of R180 in figure 6(b). For comparison, figure 13 includes the one-point statistics for R180 (previously shown in figure 6c–e). The main consequence of precluding the nonlinear feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$ is an increase of the level of the turbulence intensities, i.e. the feedback mechanism counteracts the growth of fluctuating velocities in R180. Despite these differences, we can still argue that the turbulence intensities in NF180 are alike those in R180 by noting that the friction velocity $u_{\tau }$ is no longer the appropriate scaling velocity for NF180. The traditional argument for $u_{\tau }$ as the relevant velocity scale for the energy-containing eddies is that the turbulence intensities equilibrate to comply with the mean integrated momentum balance,

(6.4)\begin{equation} -\!\langle u v \rangle \approx u_{\tau}^2 (1- y/h), \end{equation}

after viscous effects are neglected (Townsend Reference Townsend1976; Tuerke & Jiménez Reference Tuerke and Jiménez2013). As a result, $u_{\tau } \approx \sqrt {-\langle u v \rangle /(1-y/h)}$ stands as the characteristic velocity for all wall-normal distances. However, we have altered the momentum equation for case NF180, which renders the balance in (6.4) invalid. A more general argument was made by Lozano-Durán & Bae (Reference Lozano-Durán and Bae2019) by which the characteristic velocity of the energy-containing eddies, $u_{\star }$, is controlled by the characteristic production rate of turbulent kinetic energy, $P_{\star } \sim u_{\star }^2/t_{\star }$, where $t_{\star }$ is the time scale to extract energy from the mean shear

(6.5)\begin{equation} t_{\star} \sim \frac{1}{\sqrt{ (\partial U/\partial y)^2+ (\partial U/\partial z)^2}}. \end{equation}

Taking as characteristic production rate

(6.6)\begin{equation} P_{\star} \sim \sqrt{ \left(u'v'\frac{\partial U}{\partial y}\right)^2 + \left(u'w'\frac{\partial U}{\partial z}\right)^2 }, \end{equation}

a characteristic velocity scale is constructed as

(6.7)\begin{equation} u_{\star}(y) {\stackrel{\textrm{def}}{=}} \sqrt{ \frac{\left\langle P_{\star} t_{\star} \right\rangle_{xzt}}{1-y/h}}, \end{equation}

which generalizes the concept of friction velocity. The factor $1/\sqrt {1-y/h}$ is introduced for convenience in analogy with $u_{\tau }$ in (6.4) so that $u_{\star }$ reduces to $u_{\tau }$ for the regular wall turbulence.

Figure 13. (a) Streamwise, (b) wall-normal and (c) spanwise mean root-mean-squared fluctuating velocities as a function of the wall-normal distance for case R180 normalised by $u_{\tau }$, case NF180 normalised by $u_{\tau }$ and NF180 normalised by $u_{\star }$.

Figure 13 shows that the turbulence intensities, when scaled with $u_{\star }$, resemble those of R180, at least for $y>0.1h$ where viscous effects are negligible. This suggests that the underlying flow dynamics of NF180 is of a similar nature as the regular channel case R180 under the proper scaling. Thus, hereafter we utilise NF180 as the reference case for comparisons as we have shown that it exhibits similar dynamics to regular wall turbulence while being truly linear in $\tilde {\mathcal {L}}(U)\boldsymbol {u}'$. Occasionally, we allow back the feedback $\boldsymbol {u}'\rightarrow \boldsymbol {U}$.

6.2. Wall turbulence without exponential instability of the streaks

We modify the operator $\mathcal {L}(U_0)$ so that all the unstable eigenmodes are rendered stable at all times. We refer to this case as the ‘non-feedback channel with suppressed exponential instabilities’ (NF-SEI180) and we inquire whether turbulence is sustained under those conditions. The approach is implemented by replacing $\mathcal {L}(U_0)$ at each time instance by its exponentially stable counterpart $\tilde {\mathcal {L}}(U_0)$, introduced in (4.6). The governing equations for the channel with suppressed exponential instabilities are

(6.8)\begin{gather} \frac{\partial\boldsymbol{u}'}{\partial t} = \tilde{\mathcal{L}}(U_0)\boldsymbol{u}'+ \boldsymbol{N}(\boldsymbol{u}'), \end{gather}

(6.9)\begin{gather}U_0 = U(y,z,t) \ \text{from case R180}. \end{gather}

The stable counterpart of $\mathcal {L}(U_0)$ given by $\tilde {\mathcal {L}}(U_0)$ guarantees an exponentially stable wall turbulence with respect to the base flow at all times, while leaving other linear mechanisms almost intact. Note that the analysis § 4.1 was performed a priori using data from R180, while in the present case the nonlinear dynamical system (6.8) is actually integrated in time. The simulation was initialised using a flow field from R180, from which the unstable and neutral modes are projected out, and integrated in time for $300h/u_{\tau }$ after transients. It was assessed that initialising the equation with a random velocity field yields exactly the same conclusions.

It is useful to note that the stabilisation of $\mathcal {L}(U_0)$ in (6.8) can be interpreted as the addition of a forcing term to the right-hand side of (6.8) by considering the approximation to $\tilde {\mathcal {L}}(U_0)$,

(6.10)\begin{equation} \hat{\mathcal{L}}(U_0) = \mathcal{L}(U_0) - \sum_{j=1}^n 2 \lambda_j \boldsymbol{\mathcal{U}}_j \boldsymbol{\mathcal{U}}^{{\dagger}ger}_j \approx \tilde{\mathcal{L}}(U_0), \end{equation}

where $\boldsymbol {\mathcal {U}}_j$ is the eigenmode of $\mathcal {L}(U_0)$ associated with eigenvalue $\lambda _j>0$, and $n$ is the total number of unstable eigenvalues. The factor 2 on the right-hand side of (6.10) transforms $\lambda _j>0$ for $\mathcal {L}(U_0)$ into $-\lambda _j$ for $\hat {\mathcal {L}}(U_0)$ in analogy with the stabilised operator $\tilde {\mathcal {L}}$; see (4.6). Equation (6.10) is approximate, as $\mathcal {L}(U_0)$ is highly non-normal. However, we confirmed that the largest eigenvalues and eigenmodes of $\hat {\mathcal {L}}(U_0)$ and $\tilde {\mathcal {L}}(U_0)$ are almost identical most of the time (see discussion in appendix E). In virtue of (6.10), the modification of $\mathcal {L}(U_0)$ in (6.10) is easily interpretable: stabilising $\mathcal {L}(U_0)$ is equivalent to introducing a linear drag term, $-\mathcal {F}\boldsymbol {u}'$, in which the drag coefficient depends on the base flow $U(y,z,t)$, i.e.

(6.11)\begin{equation} \mathcal{F}(U) =\sum_{j=1}^{n} 2 \lambda_j \boldsymbol{\mathcal{U}}_j \boldsymbol{\mathcal{U}}^{{\dagger}ger}_j, \end{equation}

that counteracts the growth of the unstable modes at a rate proportional to the growth rate of the mode itself.

The results of integrating (6.8) are presented in figures 14 and 15. The probability density functions of $\lambda _j$ and a segment of the time series of the maximum modal growth rate of $\tilde {\mathcal {L}}(U_0)$ are shown in figure 14, which confirms that the system is successfully stabilised.

Figure 14. (a) Probability density functions of the growth rate of the four least stable eigenvalues of $\tilde {\mathcal {L}}(U_0)$, $\lambda _1>\lambda _2>\lambda _3>\lambda _4$. (b) The history of the most unstable eigenvalue $\lambda _{max}$ of $\tilde {\mathcal {L}}(U_0)$. Results are for the channel with suppressed modal instabilities NF-SEI180.

Figure 15. (a) The history of the domain-averaged turbulent kinetic energy of the fluctuations $\langle E \rangle _{xyz}$. Note that only $30 h/u_{\tau }$ units of time are shown in the panel, but the simulation was carried out for more than $300 h/u_{\tau }$. (b) Projection of the flow trajectory onto the average production rate $\langle P \rangle _{xyz}$ and dissipation rate $\langle D \rangle _{xyz}$ plane. The arrows indicate the time direction of the trajectory, which on average rotates counterclockwise. The red dashed line is $\langle P \rangle _{xyz} = -\langle D \rangle _{xyz}$ and the red circle $\langle P \rangle _{xyzt} = -\langle D \rangle _{xyzt}$. The trajectory projected covers $15 h/u_{\tau }$ units of time. Results are for the channel with suppressed modal instabilities NF-SEI180.

Figure 15(a) shows the history of the turbulent kinetic energy for NF-SEI180 after initial transients. The result verifies that turbulence persists when $\mathcal {L}(U_0)$ is replaced by its modally stable counterpart $\tilde {\mathcal {L}}(U_0)$. The patterns of the flow trajectories projected onto the production–dissipation plane (figure 15b) exhibits features similar to those discussed above for R180 and NF180.

The turbulence intensities for NF-SEI180 are presented in figure 16 and compared with those for NF180. The mean profile is the same as R180 (not shown). Notably, the channel flow without exponential instabilities is capable of sustaining turbulence. The new flow equilibrates at a state with fluctuations depleted by roughly 10–20 %. The outcome demonstrates that, even if the linear instabilities of the streak manifest in the flow, they are not required for maintaining wall turbulence.

Figure 16. (a) Streamwise, (b) wall-normal and (c) spanwise root-mean-squared fluctuating velocities as a function of the wall-normal distance for the non-feedback channel NF180 and the non-feedback channel with suppressed exponential instabilities NF-SEI180.

It is worth emphasising that, according to the post-processing analysis in § 4, modal instabilities stand as a viable mechanism to explain the energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$. Yet, we have demonstrated here that turbulence remains almost unchanged in their absence. This illustrates very vividly the risks of evaluating linear (and other) theories a priori without accounting for the cause-and-effect relations in the actual flow.

6.2.1. Case with explicit feedback from ${u}'$ to ${U}$ allowed

It was shown above that turbulence is sustained despite the absence of exponential instabilities. This was demonstrated for NF-SEI180, in which the nonlinear feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$ was excluded. We have seen in § 6.1 that inhibiting the feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$ actually enhances the turbulence intensities with respect to $u_{\tau }$. This may cast doubts on whether the ‘weaker’ fluctuations from R180 can be sustained when modal instabilities are also cancelled out. To clarify this point, we resolve a channel with suppressed exponential instabilities in which the feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$ is allowed. The equations of motion in this case are

(6.12a)\begin{gather} \frac{\partial\boldsymbol{u}'}{\partial t} = \tilde{\mathcal{L}}(U)\boldsymbol{u}'+ \boldsymbol{N}(\boldsymbol{u}'), \end{gather}

(6.12b)\begin{gather}\frac{\partial \boldsymbol{U}}{\partial t} = -\boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{U} -\mathcal{D}\langle \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}' \rangle_x - \frac{\mathcal{D}}{\rho}\boldsymbol{\nabla} \langle p \rangle_x + \nu \nabla^2 \boldsymbol{U} + \boldsymbol{f}, \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} = 0. \end{gather}

We refer to this case as ‘regular channel with suppressed exponential instabilities’ or R-SEI180. Note that the only difference of (6.12) from the original Navier–Stokes equations (2.1) is the modally stable $\tilde {\mathcal {L}}(U)$ instead of $\mathcal {L}(U)$. The base flow $\boldsymbol {U}$ is now dynamically coupled to $\boldsymbol {u}'$ via the nonlinear term $-\mathcal {D}\langle \boldsymbol {u}' \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {u}' \rangle _x$ in (6.12b). A similar experiment was done by Farrell & Ioannou (Reference Farrell and Ioannou2012) for Couette flow at low Reynolds numbers. We initialise simulations of (6.12) from a flow field of R180 after projecting out the unstable and neutral modes from this initial condition. It was checked that using random velocities as the initial condition yields the same results.

The history of $\lambda _{max}$ for $\tilde {\mathcal {L}}(U)$, shown in figure 17(a), confirms that modal instabilities are successfully removed. Figure 17(b) contains the evolution of the turbulent kinetic energy and shows that turbulence persists under the stabilised linear dynamics of (6.12a). The flow trajectories projected onto the production–dissipation plane (figure 17c) also exhibit similar features to those discussed above for R180 and NF-SEI180.

Figure 17. (a) The history of the most unstable eigenvalue $\lambda _{max}$ of $\tilde {\mathcal {L}}(U)$. (b) The history of the turbulent kinetic energy of the fluctuation energy $E = \tfrac1{2}|\boldsymbol{u}'|^2$ averaged over the channel domain. Note that only $30 h/u_{\tau }$ units of time are shown in panels (a,b), but the simulation was carried out for more than $300 h/u_{\tau }$. (c) Projection of the flow trajectory onto the average production rate $\langle P \rangle _{xyz}$ and dissipation rate $\langle D \rangle _{xyz}$ plane. The arrows indicate the time direction of the trajectory, which on average rotates counterclockwise. The red dashed line is $\langle P \rangle _{xyz} = -\langle D \rangle _{xyz}$ and the red circle $\langle P \rangle _{xyzt} = -\langle D \rangle _{xyzt}$. The trajectory projected covers $15 h/u_{\tau }$ units of time. Results for channel with suppressed modal instabilities but with feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$ allowed (R-SEI180).

The mean velocity profiles and turbulence intensities for R-SEI180 and R180 are shown in figure 18. The results are consistent with the trends reported in figure 16 for NF-SEI180 and NF180: turbulence without modal instabilities is sustained despite allowing the feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$. As in NF-SEI180, the resulting velocity fluctuations are depleted by roughly 10 %. Figure 19 portrays snapshots of the streamwise velocity at three different instants for the R-SEI180 simulation. As in R180 (cf. figure 5), the spatial organisation of the streak cycles through different stages of elongated straight motion, meandering and breakdown, although the first two states (figure 19a,b) occur more frequently than in R180. Indeed, if we consider the common definition for the streamwise velocity fluctuation $u''=u - \langle u \rangle _{xzt}$, which contains part of the streaky flow, the new flow in R-SEI180 attains an augmented streak intensity as clearly depicted in figure 18(b). The outcome is consistent with the occasional inhibition of the streak meandering or breakdown via exponential instability, which enhances $u''$, whereas wall-normal ($v''=v'$) and spanwise ($w''=v'$) turbulence intensities are diminished due to a lack of vortices succeeding the collapse of the streak (namely, mechanism (i) discussed in the introduction). This behaviour was also observed in many drag reduction investigations (Jung, Mangiavacchi & Akhavan Reference Jung, Mangiavacchi and Akhavan1992; Laadhari, Skandaji & Morel Reference Laadhari, Skandaji and Morel1994; Choi & Clayton Reference Choi and Clayton2001; Ricco & Quadrio Reference Ricco and Quadrio2008).

Figure 18. (a) Streamwise mean velocity profile, and (b,c) streamwise, (d) wall-normal and (e) spanwise mean root-mean-squared fluctuating velocities as a function of the wall-normal distance for the regular channel (R180) and the channel with suppressed exponential instabilities but with the feedback from $\boldsymbol {u}'$ to $\boldsymbol {U}$ allowed (R-SEI180). Note that the streamwise fluctuating velocity in panel (b) is defined as $u'' = u - \langle u \rangle _{xzt}$, while in panel (c) is defined as $u'=u-U$.

Figure 19. Instantaneous isosurface of the streamwise velocity at different times for R-SEI180. The value of the isosurface is 0.65 of the maximum streamwise velocity. Shading represents the distance to the wall from dark ($y=0$) to light ($y=h$). The arrow indicates the mean flow direction.

As a final comment, Lozano-Durán et al. (Reference Lozano-Durán, Karp and Constantinou2018) showed in a preliminary work that turbulence was not sustained when $\mathcal {L}(U)$ was stabilised by $\mathcal {L}(U)-\mu \mathcal {I}$, where $\mu >0$ is a damping parameter and $\mathcal {I}$ is the identity operator. However, it can be shown that introducing the linear drag $-\mu \boldsymbol {u}'$ reduces the gains supported by $\mathcal {L}(U)$ by a factor of $\exp (- 2 \mu T)$, with $T$ the optimisation time. Hence, stabilising $\mathcal {L}(U)$ via a linear drag term $-\mu \boldsymbol {u}'$ also disrupts the transient growth mechanism severely and this was the cause for the lack of sustained turbulence in Lozano-Durán et al. (Reference Lozano-Durán, Karp and Constantinou2018). Conversely, we have shown that $\tilde {\mathcal {L}}(U)$ is physically interpretable as the stabilisation of $\mathcal {L}(U)$ via a linear forcing directed toward modal instabilities ($\tilde {\mathcal {L}}(U)\boldsymbol {u}' \approx \mathcal {L}(U)\boldsymbol {u}'-\mathcal {F}\boldsymbol {u}'$, see (6.11)). This entails a much gentler modification which leaves almost intact the transient growth mechanisms of $\mathcal {L}(U)$, as opposed to $\mathcal {L}(U)-\mu \mathcal {I}$.

6.3. Wall turbulence exclusively supported by transient growth

The effect of non-modal transient growth as the main source for energy injection from $\boldsymbol {U}$ into $\boldsymbol {u}'$ is now assessed by ‘freezing’ the base flow $U(y,z,t_i)$ at the instant $t_i$. In order to steer clear of the potential effect of exponential instabilities, the numerical experiment here is performed using the stabilised linear operator $\tilde {\mathcal {L}}(U(y,z,t_i))$. For a given $U(y,z,t_i)$, we refer to this case as ‘channel flow with modally stable, frozen-in-time base flow’, or NF-TG180$_i$, with $i$ an index indicating the case number. Let us denote the flow for NF-TG180$_i$ as $\boldsymbol {u}_{\{i\}}$ (and similarly for other flow quantities). The governing equations for NF-TG180$_i$ are

(6.13a)\begin{gather} \frac{\partial\boldsymbol{u}'_{\{i\}}}{\partial t} = \tilde{\mathcal{L}}_{\{i\}}\,\boldsymbol{u}'_{\{i\}}+ \boldsymbol{N}(\boldsymbol{u}'_{\{i\}}), \end{gather}

(6.13b)\begin{gather}U_{\{i\}} = U(y,z,t_i) \ \text{from case R180}, \end{gather}

where $\tilde {\mathcal {L}}_{\{i\}} = \tilde {\mathcal {L}}(U(y,z,t_i))$. The set-up in (6.13) disposes of energy transfers that are due to both modal and parametric instabilities, while allowing the transient growth of fluctuations. For a given $t_i$, the simulation is initialised from NF-SEI180 at $t=t_i$ (projecting out neutral and unstable modes), and continued for $t>t_i$. We performed more than 500 simulations using different frozen base flows $U(y,z,t_i)$ extracted from R180.

The evolution of the turbulent kinetic energy is shown in figure 20(a) for ten cases NF-TG180$_i$, $i=1,\ldots ,10$. After freezing the base flow at $t_i$, most of the cases remain turbulent, while some others decay before $40 h/u_{\tau }$. Turbulence was sustained in 80 % of the NF-TG180$_i$ simulations. In the cases for which turbulence persists, the projection of the flow trajectory onto the $\langle P \rangle _{xyz}$–$\langle D \rangle _{xyz}$ plane is reminiscent of the self-sustaining cycle for R180; an example is shown in figure 20(b). Since $\tilde {\mathcal {L}}_{\{i\}}$ is modally stable, a necessary ingredient to sustain turbulence in NF-TG180$_i$ is the scattering and generation of new disturbances by $\boldsymbol {N}(\boldsymbol {u}'_{\{i\}})$. Indeed, we verify in appendix F that the system (6.13) decays when the nonlinear term $\boldsymbol {N}(\boldsymbol {u}'_{\{i\}})$ is discarded.

Figure 20. (a) The history of the domain-averaged turbulent kinetic energy of the fluctuations $\langle E \rangle _{xyz}$. Different colours denote various cases of NF-TG180$_i$ for $i=1,\ldots ,10$. The time $t_i$ is the instant at which the mean flow is frozen-in-time. (b) Projection of the flow trajectory onto the average production rate $\langle P_{\{5\}} \rangle _{xyz}$ and dissipation rate $\langle D_{\{5\}} \rangle _{xyz}$ plane for NF-TG180$_5$. The arrows indicate the time direction of the trajectory, which on average rotates counterclockwise. The dashed line is $\langle P_{\{5\}} \rangle _{xyz} = -\langle D_{\{5\}} \rangle _{xyz}$ and the circle $\langle P_{\{5\}} \rangle _{xyzt} = -\langle D_{\{5\}} \rangle _{xyzt}$. The trajectory projected covers $15 h/u_{\tau }$ units of time. (c) Mean velocity profile, and (d) root-mean-squared streamwise, (e) wall-normal and (f) spanwise fluctuating velocities for ten cases NF-TG180$_i$, $i=1,\ldots ,10$. The black dashed lines show same results for NF-SEI180.

The one-point statistics for each NF-TG180$_i$ vary for different $U(y,z,t_i)$. To illustrate the differences among cases, figure 20(c–f) contain the mean velocity profiles and fluctuating velocities for NF-TG180$_i$, $i=1,\ldots ,10$. Note that this is only a small sample; the total number of cases simulated is above 500. In some occasions, $U(y,z,t_i)$ is such that the system equilibrates in a state of intensified turbulence with respect to NF-SEI180 (i.e. $\langle E \rangle _{xyzt}$ for NF-TG180$_i$ larger than for NF-SEI180), while other base flows result in weakened turbulence. Figure 21 shows instances of the streamwise velocity for representative cases with intensified (a–c) and weakened (d–f) turbulent states. The intensified turbulence features a highly disorganised state akin to a broken streak, whereas the weakened turbulence resembles the quiescent stages of wall turbulence with a well-formed persistent streak.

Figure 21. Examples of base flows (a,d) and instantaneous isosurfaces of the streamwise velocity at different times (b,c,e,f). Panels (a–c) are for NF-TG180$_5$, which is representative of a state with enhanced turbulence intensities. Panels (d,e,f) are for NF-TG180$_{10}$, which is representative of a state with weakened turbulence. In panels (b,c,e,f), the value of the isosurfaces is 0.65 of the maximum streamwise velocity and shading represent the distance to the wall located at $y=0$. The arrow indicates the mean flow direction.

Figure 22 shows the average turbulent kinetic energy of a given case NF-TG180$_i$ as a function of the maximum gain $G_{\{i\},max}$ at $T=T_{max}$. The results reveal that turbulence is not maintained when $G_{\{i\},max} \lessapprox 50$, although this critical gain might be Reynolds number dependent. The trend also suggests that the level of the turbulence intensities for NF-TG180$_i$ increases with the amount of transient growth supported by each $U(y,z,t_i)$ and scales approximately as

(6.14)\begin{equation} \langle E_{\{i\}} \rangle_{xyzt} \sim G_{\{i\},max}. \end{equation}

We attempt to explain this observation by invoking the severe assumption that $\boldsymbol {N}(\boldsymbol {u}'_{\{i\}})$ acts as a time-varying forcing whose net effect is independent of $\boldsymbol {u}'_{\{i\}}$, i.e. $\boldsymbol {N}(\boldsymbol {u}'_{\{i\}}) \approx \boldsymbol {\mathcal {N}}_{\{i\}}(t)$ (see, for instance, Farrell & Ioannou Reference Farrell and Ioannou1993b; Zare et al. Reference Zare, Georgiou and Jovanović2020; Jovanović Reference Jovanović2021). Under those conditions, the solution to (6.13a) is obtained via the Green's function as

(6.15)\begin{equation} \boldsymbol{u}'_{\{i\}}(t) \approx \tilde{\varPhi}_{\{i\}}(t-t_i)\boldsymbol{u}'_{\{i\}}(t_i) + \int_{t_i}^t \tilde{\varPhi}_{\{i\}}(t-\tau) \boldsymbol{\mathcal{N}}_{\{i\}}(\tau)\,\mathrm{d}\tau, \end{equation}

with $\tilde {\varPhi }_{\{i\}}(t) = \exp [ \tilde {\mathcal {L}}_{\{i\}}(t)]$. The turbulent kinetic energy of (6.15), after transients, is

(6.16)\begin{equation} \langle E_{\{i\}} \rangle_{xyzt} = \tfrac{1}{2} \left\langle \left( \int_{t_i}^t \tilde{\varPhi}_{\{i\}}(t-\tau) \boldsymbol{\mathcal{N}}_{\{i\}}(\tau)\, \mathrm{d}\tau,\ \int_{t_i}^t \tilde{\varPhi}_{\{i\}}(t-\tau) \boldsymbol{\mathcal{N}}_{\{i\}}(\tau) \mathrm{d}\tau \right) \right\rangle_t. \end{equation}

After considering the singular-value decomposition on $\tilde {\varPhi }_{\{i\}}= M_{\{i\}} \varSigma _{\{i\}} N_{\{i\}}^{{\dagger}ger }$ then

(6.17)\begin{equation} \langle E_{\{i\}} \rangle_{xyzt} \sim \sigma_{\{i\},max}^2 \sim G_{\{i\},max}, \end{equation}

which establishes a link between the level of turbulent kinetic energy and the non-normal energy gain provided by the linear dynamics as anticipated by figure 22. Nonetheless, the scatter of the data in figure 22 is still large and the relation between $\langle E_{\{i\}} \rangle _{xyzt}$ and $G_{\{i\},max}$ is not perfectly linear. This is not surprising as the actual mechanism regulating the intensity of turbulence does not depend exclusively on $G_{\{i\},max}$ but also on the replenishment of fluctuations given by the projection of $\boldsymbol {N}(\boldsymbol {u}'_{\{i\}})$ onto $\tilde {\varPhi }_{\{i\}}$.

Figure 22. Mean turbulent kinetic energy conditioned to the maximum gain $G_{{\{i\}},{max}}$ at $T=T_{{max}}$ compiled over NF-TG180$_{i}$. The red solid line represents the mean value; the shaded area denotes $\pm$ one standard deviation.

To evaluate the compound result of NF-TG180$_i$, we define the ensemble average of a quantity $\phi _{\{i\}}$ over cases NF-TG180$_i$ as

(6.18)\begin{equation} \langle \phi_{\{i\}} \rangle_e = \sum_{i=1}^N \frac{\phi_{\{i\}}}{N}, \end{equation}

where ${1,\ldots ,N}$ is the collection of cases NF-TG180$_i$ which remain turbulent. The ensemble average of the mean and r.m.s. fluctuating velocities are presented in figure 23. The results are compared with those from NF-SEI180, which is similar to NF-TG180$_i$ but with time varying $U$. The outcome is striking: the ensemble averages over NF-TG180$_i$ cases (black solid lines) coincide almost perfectly with the one-point statistics for NF-SEI180 (dashed red lines). Given that the current set-up is composed of ‘static’ base flows, $\partial U/\partial t$ ($=0$) does not play any role in the flow dynamics of NF-TG180$_i$. Thus, we conclude that energy transfer via parametric instabilities (intimately related to $\partial U/\partial t$) is not required to sustain the flow. Time variations of $U$ are only necessary to sample the phase space of ‘regular’ turbulence with different non-normal gains so that the ensemble of NF-TG180$_i$ results in nominal wall turbulence statistics.

Figure 23. (a) Mean velocity profile, (b) root-mean-squared streamwise, (c) wall-normal and (d) spanwise fluctuating velocities. The black solid line is the ensemble average of turbulent cases NF-TG180$_{i}$, namely, $\langle \langle u_{\{i\}} \rangle _{xzt} \rangle _e$, $\langle \langle u'^2_{\{i\}} \rangle _{xzt}^{1/2} \rangle _e$, $\langle \langle v'^2_{\{i\}} \rangle _{xzt}^{1/2} \rangle _e$ and $\langle \langle w'^2_{\{i\}} \rangle _{xzt}^{1/2} \rangle _e$; the shaded region denotes $\pm$ one standard deviation with respect to the ensemble average operator $\langle \boldsymbol {\cdot } \rangle _e$; the red dashed line is $\langle u'^2 \rangle _{xzt}^{1/2}$, $\langle v'^2 \rangle _{xzt}^{1/2}$ and $\langle w'^2 \rangle _{xzt}^{1/2}$ for NF-SEI180.

The wall-normal behaviour of the turbulence intensities for NF-TG180$_i$ are determined by the fluctuation energy balance

(6.19)\begin{equation} \left\langle \boldsymbol{u}'_{\{i\}} \boldsymbol{\cdot} \tilde{\mathcal{L}}_{\{i\}}\boldsymbol{u}'_{\{i\}} + \boldsymbol{u}'_{\{i\}} \boldsymbol{\cdot} \boldsymbol{N}(\boldsymbol{u}'_{\{i\}}) \right\rangle_{xzt} = 0. \end{equation}

Similarly, the average turbulent kinetic energy for NF-SEI180 is dictated by the balance

(6.20)\begin{equation} \left\langle \boldsymbol{u}' \boldsymbol{\cdot} \tilde{\mathcal{L}}(U) \boldsymbol{u}' + \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{N}(\boldsymbol{u}') \right\rangle_{xzt}= 0, \end{equation}

where $\tilde {\mathcal {L}}(U)$ and $\boldsymbol {u}'$ are now the linear operator and velocity vector, respectively, for case NF-SEI180. The excellent agreement between NF-SEI180 and the ensemble average over NF-TG180$_{i}$ suggests that

(6.21)\begin{equation} \left\langle \boldsymbol{u}'_{\{i\}} \boldsymbol{\cdot} \tilde{\mathcal{L}}_{\{i\}} \boldsymbol{u}'_{\{i\}} \right\rangle_{xzte} \approx \left\langle \boldsymbol{u}' \boldsymbol{\cdot} \tilde{\mathcal{L}} \boldsymbol{u}' \right\rangle_{xzt}. \end{equation}

An interpretation of (6.21) (and of figure 23) is that the collection of linear transient growth events due to frozen $U(y,z,t_i)$ at different instances $t_i$ provides an accurate representation of the actual time-varying energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$ in NF-SEI180. From a dynamical-systems viewpoint: the sampling of the phase-space under the time varying $U$ is statistically equivalent to an ensemble average of solutions in equilibrium with frozen instances of $U$. This is an indication that the nonlinear dynamics supported by $\boldsymbol {N}$ are in quasi-equilibrium with $\mathcal {L}(U)$, i.e. the way the energy is input into the system changes slowly in time. The latter argument can be posed in terms of the time scales of the base flow $T_U$ and turbulent kinetic energy $T_E$. Defining $T_U$ as the time at which the auto-correlation of $U$ (see (4.12)) decays to 0.5 (similarly for $T_E$ from the auto-correlation of $E$), the ratio $T_U/T_E$ is found to be roughly 10. Therefore, changes in the base flow are ten times slower than changes in the turbulent kinetic energy, which is consistent with the discussion above.

As a final note, in a preliminary work Lozano-Durán et al. (Reference Lozano-Durán, Nikolaidis, Constantinou and Karp2020) noticed that turbulent channel flows decayed when freezing the base flow, which may initially seem inconsistent with the present results. However, a main difference is that in the present work we are imposing the base flow from actual wall turbulence (R180), while Lozano-Durán et al. (Reference Lozano-Durán, Nikolaidis, Constantinou and Karp2020) imposed a base flow from modified turbulence. The statistical sample used in the present work is also far larger than that used by Lozano-Durán et al. (Reference Lozano-Durán, Nikolaidis, Constantinou and Karp2020).

6.4. Distilling the transient growth mechanisms

We have shown above that transient growth is the simplest linear model to explain self-sustaining turbulence. In this section we further dissect the relevance of different transient growth mechanisms and the implications for the streaky structure of the base flow. We turn our attention back to (6.2), which can be written as

(6.22a)\begin{gather} \frac{\partial u'}{\partial t} = -U_0\frac{\partial u'}{\partial x} -v'\frac{\partial U_0}{\partial y} - w'\frac{\partial U_0}{\partial z} - \frac{1}{\rho}\frac{\partial p'}{\partial x} + \nu\nabla^2 u' + N_{u'}, \end{gather}

(6.22b)\begin{gather}\frac{\partial v'}{\partial t} = -U_0\frac{\partial v'}{\partial x} -\frac{1}{\rho}\frac{\partial p'}{\partial y} + \nu\nabla^2 v' + N_{v'}, \end{gather}

(6.22c)\begin{gather}\frac{\partial w'}{\partial t} = -U_0\frac{\partial w'}{\partial x} -\frac{1}{\rho}\frac{\partial p'}{\partial z} + \nu\nabla^2 w' + N_{w'}, \end{gather}

(6.22d)\begin{gather}0 = \frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y}+\frac{\partial w'}{\partial z}, \end{gather}

where we have explicitly expanded the linear components, and $N_{u'}, N_{v'}$ and $N_{w'}$ stand for the remaining nonlinear terms. The base flow is $U_0 = U(y,z,t)$ from case R180 and no feedback from $\boldsymbol {u}'$ to $U$ is allowed. Equations (6.22) allow for exponential and parametric instabilities, but we have shown that these are inconsequential for sustaining the flow. Hence, we admit the possibility of both instabilities for the sake of reducing the computational cost of solving (6.22).

We define the streak flow by $U_{{streak}} = U_0(y,z,t) - \langle u \rangle _{xz}$, and proceed to examine three transient growth mechanisms:

1. (i) linear lift-up of the streak by $-v' \partial U_{{streak}}/\partial y$ in (6.22a);

2. (ii) linear push-over of the streak by $-w'\partial U_{{streak}}/\partial z$ in (6.22a); and

3. (iii) linear Orr of the streak by $-\partial p'/\partial y$ in (6.22b) and continuity in (6.22d).

In mechanisms (i) and (ii), velocities perpendicular to the base shear $(0, \partial U_0/\partial y, \partial U_0/\partial z)$ extract energy from the latter to energise the streamwise perturbations, which persist after transients (Ellingsen & Palm Reference Ellingsen and Palm1975). For perturbations in the form of $v'$, the active linear term for energy transfer is $-v' \partial U_{{streak}}/\partial y$, which is referred to as the lift-up effect. For perturbations in the form of $w'$, the energy is transferred through $-w'\partial U_{{streak}}/\partial z$. We label the latter as ‘push-over effect’ to make a clear distinction from the lift-up mechanism, as one relies on spanwise shear of the base flow and the other on the wall-normal shear. The terminology varicose and sinuous is commonly used to refer to the perturbations from mechanism (i) and mechanism (ii), respectively. In mechanism (iii), velocities perpendicular to the base shear are amplified when backwards-leaning perturbations are tilted forward until they are roughly normal to the base shear, and are damped as they continue to be tilted past that point. Mechanisms (i) and (ii) occur concurrently with mechanism (iii), but the amplification in the latter is guided by continuity. In mechanism (iii), the pressure inhibits the cross-shear velocities when the structures are strongly tilted, and releases the inhibition when they are closer to vertical (Orr Reference Orr1907; Jiménez Reference Jiménez2013; Chagelishvili et al. Reference Chagelishvili, Hau, Khujadze and Oberlack2016).

We perform three additional experiments each aiming to suppress one of the linear mechanisms discussed above. In the first experiment, we modify (6.22a) to suppress the linear lift-up of the streak,

(6.23)

while the equations for $v'$, $w'$ and continuity remain intact. We labelled this case as NF-NFU180 (no feedback and no lift-up). The second experiment consists of a channel without linear push-over mechanism of the streak,

(6.24)

which is referred to as NF-NPO180 (no feedback and no push-over). Again the equations for $v'$, $w'$ and continuity remain the same. We might note here that both modal and non-modal growth share the physical source of the fluctuation amplification: $w \partial U/\partial z$ for sinuous motions and $v\partial U/\partial y$ for varicose motions. Hence, by modifying the latter terms we are also interfering with the modal growth of the system, although, as argued above, we do not worry about this in this section.

In the third experiment, we modify the linear dynamics of $v'$ to constrain mechanism (iii). The equation dictating the linear Orr is given by

(6.25)\begin{equation} \frac{\partial v'_{{linear}}}{\partial t} + U_0 \frac{\partial v'_{{linear}}}{\partial x} = -\frac{1}{\rho}\frac{\partial p'_{{linear}}}{\partial y}, \end{equation}

where we have neglected the viscous effects. As seen in (6.25), the Orr amplification of $v'_{{linear}}$ is controlled by $p'_{{linear}}$, which is the only source term on the right-hand side of the equation. The linear pressure can be easily obtained by solving

(6.26)\begin{equation} \frac{1}{\rho}\nabla^2 p'_{{linear}} = -2 \frac{\partial U_0}{\partial y} \frac{\partial v'}{\partial x} - 2\frac{\partial U_0}{\partial z} \frac{\partial w'}{\partial x}. \end{equation}

If we focus on the linear pressure induced by the streak then

(6.27)\begin{equation} \frac{1}{\rho}\nabla^2 p'^{s}_{{linear}} = -2 \frac{\partial U_{streak}}{\partial y} \frac{\partial v'}{\partial x} - 2\frac{\partial U_{streak}}{\partial z} \frac{\partial w'}{\partial x}. \end{equation}

We inhibit the linear Orr mechanism of the streak by introducing a forcing term $f_{{Orr}}$ on the right-hand side of the $v'$ equation to counteract the gradient of the linear pressure:

(6.28)\begin{equation} f_{{Orr}} = -\frac{1}{\rho}\frac{\partial p'^{s}_{{linear}} }{\partial y}. \end{equation}

Then, the system (6.22a)–(6.22d) is modified by replacing (6.22b) by

(6.29)\begin{equation} \frac{\partial v'}{\partial t} = -U_0\frac{\partial v'}{\partial x} -\frac{1}{\rho}\frac{\partial p'}{\partial y} + \nu\nabla^2 v' + N_{v'} - f_{{Orr}}. \end{equation}

We refer to this case as NF-NO180 (no feedback and no Orr).

The three cases are initialised using flow fields from NF180. Different initial conditions were tested and the subsequent evolution of the flow was similar regardless of the details of the initial velocity. The evolution of the turbulent kinetic energy for one initialisation is shown in figure 24(a). The case without streak lift-up (NF-NLU180) is the only one sustained. The r.m.s. velocity fluctuations after transients for NF-NLU180 are shown in figure 24(b). Interestingly, blocking the streak lift-up enhances $u'$ close to the wall, implying that the wall-normal variations of the streak provide a stabilising effect. Conversely, the cases without streak push-over (NF-NPO180) or Orr mechanisms (NF-NO180) decay in less than $5 u_{\tau }/h$. Therefore, we conclude that both streak push-over and the Orr amplification are essential transient growth mechanisms for sustaining the fluctuations. Three more cases analogous to NF-NLU180, NF-NPO180 and NF-NO180 were conducted by allowing the feedback of the fluctuations back into the base flow. The conclusions drawn for these cases are similar to the ones presented above, and they are not included here for the sake of brevity.

Figure 24. (a) The history of the domain-averaged turbulent kinetic energy of the fluctuations $\langle E \rangle _{xyz}$ for NF-NLU180; NF-NPO180; and NF-NO180. The time $t=0$ is the instant at which the simulations are started. (b) The root-mean-squared streamwise (black), wall-normal (blue) and spanwise (dark orange) fluctuating velocities for NF-NLU180 and NF180.

The necessity of push-over ($-w'\partial U_{streak}/\partial z$) points at the spanwise variations of $U_{streak}$ (equivalent to spanwise variations of $U_0$) as a key structural feature to sustain turbulence. To test this claim, we resort to the cases NF-TG180$_{i}$ presented in § 6.3 and inspect their mean turbulent kinetic energy conditioned to the marker for the spanwise-shear strength

(6.30)\begin{equation} \varGamma_{\{i\}} = \left\langle \left( \frac{\partial U_{streak}(y,z,t_i)}{\partial z} \right)^2 \right\rangle_{yzt}^{1/2}. \end{equation}

The results, shown in figure 25(a), corroborate the hypothesis that the average kinetic energy of the flow depends on the strength of $\partial U_{streak}(y,z,t_i)/\partial z$. Additionally, frozen base flows with $\varGamma _{\{i\}}$ below the critical value of $\varGamma _{\{i\},c}^+ \approx 0.03$ are too feeble to maintain turbulence. This is further supported by figure 25(b), which shows that the maximum gain of the base flow also increases with $\varGamma _{\{i\}}$. A visual impression of base flows that are either able or unable to sustain turbulence can be gained from the examples shown in figure 26. The message conveyed by figure 26 agrees with the discussion above: base flows capable of sustaining turbulence are accompanied by strong spanwise variations, $\partial U_{streak}/\partial z$, while base flows unable to maintain turbulence have milder $\partial U_{streak}/\partial z$.

Figure 25. (a) Mean turbulent kinetic energy and (b) maximum gain $G_{{\{i\}},{max}}$ at $T=T_{{max}}$ conditioned to the marker for the spanwise-shear strength $\varGamma _{\{i\}} = \langle ( \partial U_0(y,z,t_i)/\partial z )^2 \rangle _{yzt}^{1/2}$ compiled over NF-TG180$_{i}$. The red solid line represents the mean value; the shaded area denotes $\pm$ one standard deviation.

Figure 26. Examples of base flows able to support turbulence (a–c) and unable to support turbulence (d–f) from cases NF-TG180$_{i}$ discussed in § 6.3. Visual inspection of the base flows suggests that spanwise variations of $U$ are crucial to sustain turbulence.

6.5. Relation with previous studies on secondary linear process

Among the studies in table 1 regarding the secondary linear process, those labelled as TG, TG & EXP, and TG PARA advocate for the algebraic transient growth of the perturbations as a central linear mechanism to energise the velocity fluctuations. The work by Farrell, Ioannou and co-workers complemented the transient growth picture with parametric growth of the perturbations resulting from the time variability of the base flow. In their view, the growth of fluctuations is a concatenation of transient growth events that occur as the base flow varies. We have shown here that time changes in the base flow are needed to recover regular turbulence statistics; however, these time changes do not enhance the energy transfer from the base flow to the fluctuations and, thus, they are not strictly necessary to sustain the flow. The works labelled as TG & EXP deemed both transient growth and modal growth relevant for sustaining wall turbulence, as they are intimately entangled in the equations of motion. On the contrary, Schoppa & Hussain (Reference Schoppa and Hussain2002) argued that modal instability was irrelevant to streak breakdown, and that transient growth driven by the streak profile was the dominant process. The mechanism in Schoppa & Hussain (Reference Schoppa and Hussain2002) was referred to as STG (streak transient growth, i.e. secondary linear process) to make a clear distinction with the more traditional transient growth supported by $\langle u \rangle _{xzt}$ (i.e. primary linear process). In this sense, our conclusions are more aligned with Schoppa & Hussain (Reference Schoppa and Hussain2002), and transient growth is identified here to overcome other linear mechanisms. Schoppa & Hussain (Reference Schoppa and Hussain2002) also analysed the physical process at play during transient growth in terms of vortex dynamics. They formulated the problem in the streak-vortex-line coordinate system, that procures a clear interpretation of the perturbation vorticity generation. They found that $w'$ perturbations of moderately low amplitude lead to the generation of new vortices and sustained near-wall turbulence via the ‘shearing’ mechanism. The latter results are also consistent with our analysis in § 6.4, where we showed that the push-over mechanism (represented by $-w' \partial U_{{streak}} /\partial z$) drives the generation of perturbations. Schoppa & Hussain (Reference Schoppa and Hussain2002) traced back the source of fluctuations to $\partial \langle u \rangle _{xzt} /\partial n$, where $n$ is the direction normal to the base flow vortex lines and includes contributions of both $\partial \langle u \rangle _{x} /\partial y$ and $\partial \langle u \rangle _{x} /\partial z$. Here, we have further demonstrated that the spanwise shear $\partial \langle u \rangle _{x} /\partial z \equiv \partial U_{{streak}} /\partial z$ dominates over $\partial U_{{streak}} /\partial y$.

Despite our main conclusions being consistent with Schoppa & Hussain (Reference Schoppa and Hussain2002), the causal analysis followed in the present study is fundamentally different from that adopted in previous works. This is an important point, as our approach allows us to tackle the criticism raised by the community on transient growth as the prevailing mechanism for the secondary linear process. In the remainder of this section, we survey previous criticisms and discuss how our contributions overcome existing deficiencies. First, it is pertinent to clarify that transient growth due to high non-normality of the linearised Navier–Stokes operator is an intrinsic feature of the equations of motion. Transient growth represents the transport of momentum normal to the base shear, which is a necessary requirement for mixing of the flow and, hence, for turbulence. Therefore, the debate in the community is not about the ubiquity of transient growth in turbulence, but about the necessity of additional sorts of instabilities to sustain the fluctuations (as shown in table 1).

The scepticism on the ability of transient growth alone to fuel the generation of $\boldsymbol {u}'$ has materialised in the following forms.

1. (i) Jiménez (Reference Jiménez2018) pointed out that the absence of unstable streaks reported by Schoppa & Hussain (Reference Schoppa and Hussain2002) can be interpreted as an indication that the instability is important. Quoting Jiménez (Reference Jiménez2018): One may think of the low probability of finding upright pencils on a shaking table. Unstable flow patterns would not be found precisely because instability destroys them.’

2. (ii) Jiménez (Reference Jiménez2018) also argued that transient growth as envisioned by Schoppa & Hussain (Reference Schoppa and Hussain2002) is a property of the streak itself, implying that the energy of $\boldsymbol {u}'$ is drawn from the energy of $U_{{streak}}$. The premise is reasonable close to the wall, where $|\boldsymbol{u}'|^2/2$ is a small fraction of streak energy $U^2_{streak}/2$, but it becomes problematic farther from the wall, where both energies are comparable. If the transverse velocities had to obtain their energy from the streak, one would expect a negative correlation between the two energies, but the opposite seems to be true. Instead, Jiménez (Reference Jiménez2018) suggested that the actual source of energy for $\boldsymbol {u}'$ would come from $\langle u \rangle _{xzt}$, which is also part of the base flow in Schoppa & Hussain (Reference Schoppa and Hussain2002).

3. (iii) Other authors have reasoned, as mentioned above, that it is essentially impossible to distinguish between streak transient growth and streak modal instability, as both processes can be traced back to the same source term in the linearised Navier–Stokes equations, namely, $-w' \partial \langle u \rangle _{x} /\partial z - v' \partial \langle u \rangle _{x} /\partial y$ (Hœpffner et al. Reference Hœpffner, Brandt and Henningdon2005; Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017; de Giovanetti et al. Reference de Giovanetti, Sung and Hwang2017). Consequently, both transient growth and modal instability occur concurrently during streak breakdown and both should be considered responsible for replenishing $\boldsymbol {u}'$.

4. (iv) Another criticism comes from the effect of time-varying base flows. Schoppa & Hussain (Reference Schoppa and Hussain2002) investigated the effect of unfrozen streaks on modal instabilities and transient growth. They showed that unfrozen (freely diffusing) streaks are still able to support transient growth with amplifications of the order of 10, whereas the initially unstable base flow provided only a factor of 2. However, the analysis was performed on freely decaying streaks, while streaks in actual wall turbulence are subjected to periods of both growth and decay. Precluding the growth phase of the streaks might have important consequences for the growth of perturbations. For example, Farrell & Ioannou (Reference Farrell and Ioannou2012) have shown that a potential route to enhance the gain for short times and/or achieve finite gains for long times is the parametric instability of the streak discussed in the introduction (Farrell & Ioannou Reference Farrell and Ioannou1999, Reference Farrell and Ioannou2012; Farrell et al. Reference Farrell, Ioannou, Jiménez, Constantinou, Lozano-Durán and Nikolaidis2016). In contrast with the freely diffusing base flow, alternating periods of growth and decay in the base flow can enhance the energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$ and should be taken into consideration.

5. (v) Finally, some authors have criticised or at least found questionable the use of linear stability theory to analyse time-varying base flows and base flows defined by an average (for example, $\langle u \rangle _x$) rather than by a solution of the Navier–Stokes equations. Most of the successful and well-established results from linear stability theory have been derived in the case of laminar-to-turbulence transition with small perturbations, where the underlying assumptions are rigorously satisfied. This is obviously not the case for turbulent flows (see the discussion in Hussain Reference Hussain1983, Reference Hussain1986). One conceptual objection is the fact that stability analysis of ‘frozen’ turbulent profiles would be appropriate only if the time scales of the actual time-varying mean flow were much smaller than those of the instabilities. However, some authors have pointed out that time changes in the turbulent mean flow could be of the same order as those of the instability wave. Thus, the instabilities do not ‘see’ this mean flow and their evolution departs noticeably from that predicted from linear stability theory. Another recurrent objection is granting the status of ‘perturbations’ (assumed to be small, e.g. ${<}0.1\,\%$), to the turbulent fluctuations, $\boldsymbol {u}'$ (which might reach values above $10\,\%$ of the mean flow, especially close to the wall). Hence, the evolution of these (not-so-small) $\boldsymbol {u}'$ ‘perturbations’ is subjected to non-negligible contributions from nonlinear interactions, which might invalidate the predictions from linear stability theory.

We have addressed the criticism discussed above by formulating the problem within a cause-and-effect framework. Whilst there is no such thing as a methodology free from limitations, we have argued that cause-and-effect analysis entails a substantial leap in the study of turbulence compared to non-causal analysis by post-processing of data. Following the same order used to introduce the criticism above, we address each as follows.

1. (i) We have contributed to settle the debate regarding the role played by modal instabilities by showing that these are not necessary to sustain wall turbulence. This was achieved by completely precluding the possibility of exponential growth from the linear Navier–Stokes operator at all times (see § 6.2).

2. (ii) We have shown that inhibiting the effect of $\partial U_{{streak}}/ \partial z$ interrupts the self-sustaining cycle. Hence, the extraction of energy from the streak $\partial U_{{streak}}/ \partial z$ is a necessary condition to maintain $\boldsymbol {u}'$. Note that we do not imply that $\partial \langle u \rangle _{xzt}/ \partial y$ is inconsequential to sustaining turbulence, but that the spanwise variations of the streak are also an active participant in the self-sustaining cycle of turbulence (see § 6.4).

3. (iii) Despite the fact that modal and non-modal growth of the fluctuations originate from the same physical term in the Navier–Stokes equations, we have established a clear distinction between both mechanisms by manipulating the linear Navier–Stokes operator. We have shown that it is possible to block modal instabilities, while maintaining the transient growth mechanism almost intact (see § 6.3).

4. (iv) Both post-processing analysis § 4 and cause-and-effect analysis in § 6 comprise time-varying base flows which evolve in a realistic manner, as they are extracted from actual direct numerical simulation data. As such, the evolution of the base flow experiences periods of both growth and decay consistent with actual wall turbulence, which allows for an accurate estimation of the linear mechanism assisting the growth of $\boldsymbol {u}'$.

5. (v) The validity of linear theories for fully developed turbulence is more subtle, and we have commented on this topic in § 1. Paraphrasing the argument given in the introduction, expressing the fluctuation equation in the form of (1.1) does not require invoking linearisation about $\boldsymbol {U}$ nor assuming that $\boldsymbol {u}'$ is small. If the volume integral of $\boldsymbol {u}'\boldsymbol {\cdot } \boldsymbol {N}(\boldsymbol {u}')$ is zero then the only way of sustaining $\boldsymbol {u}'$ is through the energy injection from $\boldsymbol {u}' \boldsymbol {\cdot } \mathcal {L}(\boldsymbol {U})\boldsymbol {u}'$. Thus, for a partition of the flow $\boldsymbol {U} + \boldsymbol {u}'$, we can always refer to the linear mechanisms supported by $\mathcal {L}(\boldsymbol {U})$ and assess their relevance in sustaining turbulence regardless of how ‘good’ the linearisation is. This is because our cause-and-effect analysis is conducted for the fully nonlinear equations, instead of only for the linear component. Thus, when we inquire about the validity of a particular linearisation we are indeed asking about the usefulness of the partition $\boldsymbol {U} + \boldsymbol {u}'$ in explaining the dynamics of $\boldsymbol {u}'$ via the linear mechanisms supported by $\boldsymbol {U}$, which circumvents the problem of linearisation.

We close this section by discussing some discrepancies with Schoppa & Hussain (Reference Schoppa and Hussain2002). The stability analysis conducted in § 4.1 reveals that our base flow $U(y,z)$ is modally unstable 90 % of the time. On the contrary, Schoppa & Hussain (Reference Schoppa and Hussain2002) found that most streaks have intensities that are too low to be modally unstable. It is unclear what is the root of such a difference – it might be related to the synthetic base flow used in Schoppa & Hussain (Reference Schoppa and Hussain2002), while here we used instantaneous base flow from direct numerical simulations, which are more corrugated and prone to modal instabilities. Other possible explanations are the criteria used by Schoppa & Hussain (Reference Schoppa and Hussain2002) to quantify unstable streaks via a vorticity-based inclination angle, or the use of a minimal turbulent channel in the present study. A second difference of our work with Schoppa & Hussain (Reference Schoppa and Hussain2002) comes from the value of the gains provided by transient growth. In § 4.2 we have shown that our gains are of the order of 100, while the gains reported in Schoppa & Hussain (Reference Schoppa and Hussain2002) are of the order of 10. The cause for this discrepancy is related to the perturbation chosen by Schoppa & Hussain (Reference Schoppa and Hussain2002), which differs substantially from ours. Schoppa & Hussain (Reference Schoppa and Hussain2002) used a physics-motivated perturbation in $w'$. In our case, we are considering optimal perturbations, which lead to much higher amplifications.