2.1. Model formulation

Incompressible plane Couette flow is considered, where two infinitely wide and long parallel plates, separated by a wall-normal distance $2h$, are set to move in opposite directions with velocity $\pm U_0$. All variables are made dimensionless with length and velocity scales $h$ and $U_0$, respectively. The dimensionless flow domain of interest is considered to be $\boldsymbol {x}(=(x,y,z)) \in [0,2{\rm \pi} ] \times [-1,1] \times [0,{\rm \pi} ]$, equipped with periodic boundary conditions in the streamwise $(x)$ and spanwise $(z)$ directions and the velocity $\tilde {\boldsymbol {u}}|_{y=\pm 1} = (\pm 1,0,0)$ specified on the upper and lower wall-normal $(y)$ boundaries. The velocity field is first decomposed into the laminar state, $\boldsymbol {u}_0=(y,0,0)$, and the fluctuations about this state, $\boldsymbol {u}'$

(2.1)\begin{equation} \tilde{\boldsymbol{u}}(\boldsymbol{x},t) = \boldsymbol{u}_0(y) + \boldsymbol{u}'(\boldsymbol{x},t). \end{equation}

The evolution of this fluctuating velocity field is described by the following equations:

(2.2)\begin{gather} \partial_t \boldsymbol{u}' + (\boldsymbol{u}_0\boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}' + (\boldsymbol{u}'\boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}_0 + (\boldsymbol{u}'\boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}'={-}\boldsymbol{\nabla} p + \frac{1}{Re}\nabla^2\boldsymbol{u}', \end{gather}

(2.3)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}' = 0, \end{gather}

with boundary conditions given by $\boldsymbol {u}'|_{y=\pm 1} = (0,0,0)$, where $Re=U_0 h/ \nu$ is the Reynolds number, and $\nu$ is the kinematic viscosity.

Following the formulation of Cavalieri & Nogueira (Reference Cavalieri and Nogueira2022), we define an inner product

(2.4)\begin{equation} \langle \,\boldsymbol{f},\boldsymbol{g} \rangle = \frac{1}{4{\rm \pi}^2} \int_0^{\rm \pi} \int_{{-}1}^1 \int_0^{2{\rm \pi}} \boldsymbol{f}(\boldsymbol{x}) \boldsymbol{\cdot} \boldsymbol{g}(\boldsymbol{x}) \,\mathrm{d}\kern0.06em x \, \mathrm{d}\kern0.05em y \, \mathrm{d}z, \end{equation}

and introduce a modal decomposition of the fluctuating velocity field

(2.5)\begin{equation} \boldsymbol{u}'(\boldsymbol{x},t) = \sum_{j=1}^N a_j(t) \boldsymbol{\varPhi}_j(\boldsymbol{x}), \end{equation}

where the modes $\boldsymbol {\varPhi }_j$ are defined to be divergence free $\boldsymbol {\nabla } \boldsymbol {{\cdot }}\boldsymbol {\varPhi }_j =0$, and orthonormal with respect to the defined inner product, i.e. $\langle \boldsymbol {\varPhi }_i,\boldsymbol {\varPhi }_j \rangle = \delta _{ij}$, where $\delta _{ij}$ is the Kronecker delta. Hence, by construction, any linear superposition of modes satisfies both the incompressibility condition and the boundary conditions of the system.

Substituting the decomposition (2.5) into (2.2) and taking an inner product with the mode $\boldsymbol {\varPhi }_i$ leads to a system of ordinary differential equations describing the evolution of the mode coefficients $a_i(t)$

(2.6)\begin{equation} \frac{{\rm d}a_i}{{\rm d}t} = \frac{1}{Re} \sum_{j=1}^N L_{ij} a_j + \sum_{j=1}^N \tilde{L}_{ij} a_j + \sum_{j=1}^N \sum_{k=1}^N Q_{ijk} a_j a_k, \end{equation}

where

(2.7a)\begin{gather} L_{ij} = \langle \nabla^2 \boldsymbol{\varPhi}_j,\boldsymbol{\varPhi}_i\rangle, \end{gather}

(2.7b)\begin{gather}\tilde{L}_{ij} ={-}\langle (\boldsymbol{\varPhi}_j \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}_0 + \boldsymbol{u}_0\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{\varPhi}_j), \boldsymbol{\varPhi}_i \rangle, \end{gather}

(2.7c)\begin{gather}Q_{ijk} ={-}\langle (\boldsymbol{\varPhi}_j \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{\varPhi}_k), \boldsymbol{\varPhi}_i \rangle. \end{gather}

Here, $L_{ij}$ and $\tilde {L}_{ij}$ represent the viscous term and the linear interaction with the laminar state, respectively, whereas $Q_{ijk}$ represents the quadratic nonlinear interaction between the different modes. The pressure dependence is seen to vanish due to the divergence-free property of the modes.

This system of ordinary differential equations still describes the exact dynamics in the limit as $N\rightarrow \infty$ by the orthogonality of the modes, although here we attempt to choose the modes in an optimal way such that we can model the dynamics of the system using a greatly reduced number of degrees of freedom.

Many choices exist for the modes $\boldsymbol {\varPhi }_j(\boldsymbol {x})$. In this study, POD modes of the stochastic response of the linearised Navier–Stokes equations are chosen, and the detailed computational procedure is described in Cavalieri & Nogueira (Reference Cavalieri and Nogueira2022), who followed the method of Jovanović & Bamieh (Reference Jovanović and Bamieh2005). The velocity covariance resulting from the stochastic response is obtained by formulating the controllability Gramian of the Navier–Stokes equations linearised about the laminar basic state, $\boldsymbol {u}_0$, and is computed by solving the associated Lyapunov equation. The POD modes are obtained by computing the eigenfunctions of the velocity covariance that are ranked in terms of their energy (i.e. eigenvalue).

Compared with a typical POD approach, the use of POD modes from the stochastic response of the linearised Navier–Stokes equations removes the necessity to form these modes using large numerical or experimental datasets, as they can be directly computed from the forced linear system. The results of Cavalieri & Nogueira (Reference Cavalieri and Nogueira2022) suggest that these modes provide an efficient set of basis functions for the construction of the ROM, as the resulting ROM provides equal or more accurate results than a Galerkin–POD approach constructed from DNS data, at least in terms of the second-order velocity statistics.

2.2. Numerical implementation

To implement this model numerically, the POD modes from the controllability Gramian are represented using a Fourier discretisation in the streamwise and spanwise directions, and a Chebyshev discretisation in the wall-normal direction

(2.8)\begin{equation} \boldsymbol{\varPhi}_j(\boldsymbol{x}) = \hat{\boldsymbol{\varPhi}}_j(y)\,{\rm e}^{\mathrm{i}(m k_{x,0}x + n k_{z,0}z)}, \end{equation}

where $\widehat {{\cdot }}$ denotes the Fourier transform, and $k_{x,0} = {2{\rm \pi} }/{L_x} = 1$ and $k_{z,0} = {2{\rm \pi} }/{L_z} = 2$ denote the fundamental wavenumbers in the streamwise and spanwise directions, respectively. In particular, we use the modes constructed in Cavalieri & Nogueira (Reference Cavalieri and Nogueira2022) with the number of modes $N=600$: $m=\{0,1,2\}, n=\{-2,-1,0,1,2\}$ with $N_{y}=24$, where $N_{y}$ is the number of POD modes used in the expansion for a given $m$ and $n$. Here, we note that only positive streamwise wavenumbers need be considered since negative wavenumber modes can be reconstructed by complex conjugation. The number of degrees of freedom here was chosen by running a preliminary test, and $N=600$ was found to be close to the smallest possible dimension for a system retaining self-sustaining processes at two scales. Two wavenumbers must be considered for each of the two wall-parallel directions to retain the ‘two’ scales, and $N_y=24$ was chosen to ensure the linear stability of the laminar solution. A bifurcation analysis of the ROM was subsequently carried out, revealing that its behaviour is remarkably similar to that of plane Couette flow; i.e. saddle-node bifurcation and the resulting bifurcation cascade leading to turbulence (e.g. Kreilos & Eckhardt Reference Kreilos and Eckhardt2012). These modes can then be used to construct the system of ordinary differential equations (2.6) which are integrated in time using a standard fourth- or fifth-order Runge–Kutta scheme at $Re = 1200$ using the ode45 function in MATLAB.

The time evolution of this system starting from a random initial condition displays chaotic behaviour consistent with turbulence, as shown by Cavalieri & Nogueira (Reference Cavalieri and Nogueira2022). The long trajectory was initialised from a random perturbation of the mode coefficients $a_i$ with the amplitude between 0 and 0.1, and was time integrated using ode45 with a relative error tolerance of $10^{-8}$. Temporal averaging of the flow snapshots from the model over a sufficiently long time leads to the mean and root mean square (r.m.s.) velocity profiles shown in figure 1. These profiles have been compared with reference DNS statistics at a grid resolution of $(N_x, N_y, N_z)=(64, 65, 64)$ computed using the open-source code Channelflow (Gibson Reference Gibson2014, http://channelflow.org/), showing reasonable agreement relative to the greatly reduced computational cost. Note that the number of degrees of freedom of the DNS is $N_{DNS} \approx 10^6$. Computation of the velocity gradient at the wall leads to a friction Reynolds number $Re_\tau \approx 80$, giving a domain size of $(L_x^+, L_y^+, L_z^+) = (504,160,250)$ based on inner units. These dimensions are bigger than the minimal flow unit configurations of Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) and Kawahara & Kida (Reference Kawahara and Kida2001), and are of similar size to that of the minimal multi-scale flow unit formulated by Doohan et al. (Reference Doohan, Willis and Hwang2019). In this regard, we expect similar two-scale behaviour as observed by Doohan et al. (Reference Doohan, Willis and Hwang2019) to be captured by the ROM.

Figure 1. Mean and r.m.s. velocity profiles: (a) $U^+$; (b) $u_{rms}^+$; (c) $v_{rms}^+$; (d) $w_{rms}^+$. The superscript $({\cdot })^+$ denotes the viscous inner-scaled variables. Here, the black and blue lines indicate those from the ROM and DNS, respectively.

3.1. Two-scale energy budget

To formulate a relevant energy budget for a statistically stationary turbulent state, the full velocity field is decomposed as

(3.1)\begin{equation} \tilde{\boldsymbol{u}}(\boldsymbol{x},t) = \boldsymbol{U}(y) + \boldsymbol{u}(\boldsymbol{x},t), \end{equation}

where $\boldsymbol {U}$ describes the mean flow, defined as $\boldsymbol {U}(\boldsymbol {x}) = ({1}/{T})\int _0^T \tilde {\boldsymbol {u}}(\boldsymbol {x},t)\,\mathrm {d}t$ for some suitably long time interval $T$, and $\boldsymbol {u}(\boldsymbol {x},t)$ is the velocity fluctuation about the mean flow.

As discussed in Doohan et al. (Reference Doohan, Willis and Hwang2019), the inner-scaled flow domain at $Re_\tau (\simeq 80)$ considered here is approximately $L_z^+\simeq 250$. Given that the smallest spanwise size of energy-containing structures is $\lambda _z^+\simeq 100$, the computational domain would admit energy-containing structures mainly at two different spanwise length scales: $\lambda _z^+\simeq 125, 250$. To understand the interactions of the two different scales within the flow, we further decompose the fluctuating velocity field into a large- and small-scale fluctuating velocity field

(3.2)\begin{equation} \boldsymbol{{u}}(\boldsymbol{x},t) = \boldsymbol{{u}}_l(\boldsymbol{x},t) + \boldsymbol{u}_s(\boldsymbol{x},t), \end{equation}

where the large- and small-scale velocity fields are defined through the large- and small-scale projection operators, $\mathcal {P}_l\{{\cdot }\}$ and $\mathcal {P}_s\{{\cdot }\}$, which split the fields by spanwise wavenumber in the following way:

(3.3a)\begin{gather} \boldsymbol{u}_l(\boldsymbol{x},t) = \mathcal{P}_l\{\boldsymbol{u}(\boldsymbol{x},t)\} = \sum_{|m|\le m_x} \sum_{|n|\le 1} \hat{\boldsymbol{u}}(y)\, {\rm e}^{\mathrm{i}(m k_{x,0}x + n k_{z,0}\,z)}, \end{gather}

(3.3b)\begin{gather}\boldsymbol{u}_s(\boldsymbol{x},t) = \mathcal{P}_s\{\boldsymbol{u}(\boldsymbol{x},t)\} =\sum_{|m|\le m_x} \sum_{2\le|n|\le n_z} \hat{\boldsymbol{u}}(y)\, {\rm e}^{\mathrm{i}(m k_{x,0} x + n k_{z,0}\,z)}. \end{gather}

The split by spanwise wavenumber here is justified by the observation that the statistical structure of eddies in wall-bounded flows is self-similar with respect to the spanwise wavelength and proportional to the distance from the wall (Hwang Reference Hwang2015). Additionally, both large and small scales are expected to contain a range of streamwise wavenumbers due to the presence of both long streaks and short streamwise vortices as part of the self-sustaining process (Hwang Reference Hwang2015).

The evolution equations for turbulent fluctuations are subsequently projected onto the large- and small-scale velocity subspaces using the definition of $\mathcal {P}_l\{{\cdot }\}$ and $\mathcal {P}_s\{{\cdot }\}$, yielding two fluctuation equations, one equation each for the large and small scales, respectively. Multiplying each of these equations by the corresponding large- or small-scale velocity and averaging in the streamwise and spanwise direction yields energy budget equations for each component of the velocity at large

(3.4a)\begin{gather} \frac{\partial E_{ul}}{\partial t} = {P}_{ul} + T_{ul} + \varPi_{ul} + T_{\nu,ul} + \varepsilon_{ul}, \end{gather}

(3.4b)\begin{gather}\frac{\partial E_{vl}}{\partial t} = T_{vl} + \varPi_{vl} + T_{\nu,vl} + T_{p,vl} + \varepsilon_{vl}, \end{gather}

(3.4c)\begin{gather}\frac{\partial E_{wl}}{\partial t} = T_{wl} + \varPi_{wl} + T_{\nu,wl} + \varepsilon_{wl}, \end{gather}

and small scales

(3.5a)\begin{gather} \frac{\partial E_{us}}{\partial t} = {P}_{us} + T_{us} + \varPi_{us} + T_{\nu,us} + \varepsilon_{us}, \end{gather}

(3.5b)\begin{gather}\frac{\partial E_{vs}}{\partial t} = T_{vs} + \varPi_{vs} + T_{\nu,vs} + T_{p,vs} + \varepsilon_{vs}, \end{gather}

(3.5c)\begin{gather}\frac{\partial E_{ws}}{\partial t} = T_{ws} + \varPi_{ws} + T_{\nu,ws} + \varepsilon_{ws}, \end{gather}

where $P$ denotes the energy production, $T$ the turbulent transport, $\varPi$ the pressure strain, $T_\nu$ the viscous transport, $T_p$ the pressure transport and $\varepsilon$ the dissipation at each scale and component, where the subscripts $l$ and $s$ denote large and small scales, and $u$, $v$ and $w$ the streamwise, wall-normal and spanwise components. Full details of each term are included in Appendix A. Since only the velocity field is computed directly by the ROM, the fluctuating pressure field must be computed using the Poisson equation for pressure projected onto the large and small scales

(3.6a)\begin{gather} \nabla^2 p_l ={-} U\frac{\partial^2 u_l}{\partial x^2} - \frac{\partial v_l}{\partial x} \frac{\partial U}{\partial y} - \mathcal{P}_l\{\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u})\}, \end{gather}

(3.6b)\begin{gather}\nabla^2 p_s ={-} U\frac{\partial^2 u_s}{\partial x^2} - \frac{\partial v_s}{\partial x} \frac{\partial U}{\partial y} - \mathcal{P}_s\{\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u})\}, \end{gather}

with boundary conditions specified as $\partial _y p_l |_{y=\pm 1}= 0$, and $\partial _y p_s |_{y=\pm 1}= 0$ (Kim Reference Kim1989).

The various multi-scale energy budget terms have been computed from the ROM. The temporal averaging has been taken over a dataset which is roughly 18 000 dimensionless time units in length $(T^+ = T u_\tau ^2 /\nu \approx 96\,500)$. They have been plotted over a channel half-height as a function of the wall-normal direction in figure 2. The overall statistical behaviour of the two-scale energy budget terms are qualitatively very similar to those found by Doohan et al. (Reference Doohan, Willis and Hwang2021), who used the DNS results of their two-scale shear stress-driven flow model.

Figure 2. Wall-normal variation of the time-averaged terms of the energy budget equations (3.4) and (3.5) for (a,b) streamwise, (c,d) wall-normal and (e,f) spanwise components: (a,c,e) large scale; (b,d,f) small scale.

A large amount of energy production is seen in both the large and small scales (blue lines in figure 2a,b), featuring peaks in production in the near-wall region at approximately $y^+\approx 10$, the location of which is consistent with that reported by Bech et al. (Reference Bech, Tillmark, Alfredsson and Andersson1995) and Afzal, Seena & Bushra (Reference Afzal, Seena and Bushra2018) as well as many others. The magnitude of the peak of the small-scale production is seen to be larger than that of the large scales, similar to the results reported by Doohan et al. (Reference Doohan, Willis and Hwang2021), although slightly less production is observed overall in the logarithmic region of the flow compared with this work.

The pressure strain terms (yellow lines in figure 2) sum to zero at each scale as a consequence of continuity, i.e. $\varPi _{ul}+\varPi _{vl}+\varPi _{wl}=0$ and $\varPi _{us}+\varPi _{vs}+\varPi _{ws}=0$. The streamwise components of the pressure strain are negative across the whole wall-normal domain, whereas the wall-normal and spanwise pressure strains are seen to be positive, except at $y^+ \lesssim 20$. This is indicative of the pressure strain terms being responsible for the redistribution of the turbulent kinetic energy from the streamwise to the wall-normal and spanwise components, also observed in Doohan et al. (Reference Doohan, Willis and Hwang2021). However, the pressure strain terms have a relative magnitude lower than that of Doohan et al. (Reference Doohan, Willis and Hwang2021), presumably due to the low number of streamwise and spanwise Fourier modes used for the present ROM. For $y^+ \lesssim 20$, large peaks in the spanwise and wall-normal components are observed. The near-wall negative wall-normal pressure strain is attributed to the well-known ‘splat effect’ (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988; Perot & Moin Reference Perot and Moin1995; Lee & Moser Reference Lee and Moser2019), whereby fluid motions moving towards the wall must be turned to move parallel to the wall due to its impermeability.

The streamwise turbulent transport term at the large scales is seen to be positive, especially in the near-wall region (green line in figure 2$a$), implying that the turbulent transport from the small scales injects energy into the large scales. This is the inverse energy transfer (or energy feeding from the small scales), previously observed (Cho et al. Reference Cho, Hwang and Choi2018; Doohan et al. Reference Doohan, Willis and Hwang2021), and it gradually disappears on increasing $y^+$. In general, the turbulent transport terms follow similar trends to those of Doohan et al. (Reference Doohan, Willis and Hwang2021) but are more oscillatory (green lines in figure 2). This lack of smoothness is likely a result of the nonlinear interactions between the truncated modes not being fully captured by the model, although the consequences of this are not immediately clear. For this reason, the turbulent transport terms have been further decomposed and studied in § 3.2 as their role is of significant interest in relation to the accuracy of the multi-scale behaviour in ROM.

The largest amounts of dissipation are seen in the near-wall region through the streamwise component (orange line in figure 2$a$), similar to Doohan et al. (Reference Doohan, Willis and Hwang2021). There is slightly greater large-scale dissipation than small-scale dissipation for all components (orange lines in figure 2), suggesting that the cascade of energy from large to small scales (Kolmogorov Reference Kolmogorov1941) is not fully resolved. This is likely due to the truncated nature of the model, whereby energy cannot be transferred to the smaller scales neglected by the model, as a result of spectral blocking. Nevertheless, the amounts of dissipation at each scale are still comparable due to the low $Re$ considered here, as in Doohan et al. (Reference Doohan, Willis and Hwang2021). However, at higher Reynolds numbers, the energy cascade is expected to be more poorly resolved due to the increasing number of small-scale eddies for turbulence dissipation. As such, the number of degrees of freedom of the ROM would need to be increased further.

Lastly, the pressure and viscous transport terms appropriately balance the other terms, and their behaviours are also consistent with those found in Doohan et al. (Reference Doohan, Willis and Hwang2019). The integrated energy is seen to be balanced over all terms to within 2 % relative to the magnitude of the turbulent production.

3.2. Inter- and intra-scale energy transport

To access the role of the turbulent transport in greater detail, we decompose these terms further, following the methods of Doohan et al. (Reference Doohan, Willis and Hwang2021) and Kawata & Alfredsson (Reference Kawata and Alfredsson2019)

(3.7a)\begin{gather} T_{ul} = T_{ul,-} +T_{ul,\#} -T_{u,\updownarrow}, \end{gather}

(3.7b)\begin{gather}T_{vl} = T_{vl,-} +T_{vl,\#} -T_{v,\updownarrow}, \end{gather}

(3.7c)\begin{gather}T_{wl} = T_{wl,-} +T_{wl,\#} -T_{w,\updownarrow}, \end{gather}

(3.7d)\begin{gather}T_{us} = T_{us,-} +T_{us,\#} +T_{u,\updownarrow}, \end{gather}

(3.7e)\begin{gather}T_{vs} = T_{vs,-} +T_{vs,\#} +T_{v,\updownarrow}, \end{gather}

(3.7f)\begin{gather}T_{ws} = T_{ws,-} +T_{ws,\#} +T_{w,\updownarrow}. \end{gather}

Here, $T_-$ denotes the intra-scale spatial turbulent transport, $T_\#$ the inter-scale spatial turbulent transport and $T_\updownarrow$ the inter-scale turbulent transport. Here, the use of intra refers to the involvement of velocity fields of only the same scale, whereas inter refers to terms that include the nonlinear coupling of velocity fields of different scales. Spatial transport refers to terms that can be rewritten as the divergence of a vector field, meaning that they do not contribute to the loss or gain of energy when integrated over the spatial domain. On the contrary, the inter-scale turbulent transport cannot be written as the divergence of a vector field and is thus directly responsible for energy exchanges between the large and small scales. Importantly, these inter-scale turbulent transport terms appear in both the large- and small-scale energy transport equations with opposing signs, thus cancelling each other out when considering the total energy budget of both the large and small scales. The full details of these equations are included in Appendix A.

The time-averaged energy transport terms in (3.7) are plotted as a function of the wall-normal direction in figure 3. The inter-scale turbulent transport terms, $T_{\updownarrow }$, are responsible for the direct transport of energy between scales. In the streamwise component, only inter-scale turbulent transport, $T_{\updownarrow }$, from the small to large scales is observed across the entire domain in the mean (figure 3a,b). This is contrary to the DNS results of Doohan et al. (Reference Doohan, Willis and Hwang2021), which show a contribution to the energy cascade in the outer region of the flow in the streamwise direction. With that being said, the spanwise and wall-normal components are seen to contribute to the energy cascade from large to small scale in a similar way to that of Doohan et al. (Reference Doohan, Willis and Hwang2021), suggesting that the cascade is still being captured, at least to some degree, in the outer region of the flow (figure 3c–f). The streamwise and spanwise components of the inter-scale turbulent transport, $T_{\updownarrow }$, show a large amount of energy transport from the small to large scales in the near-wall region (figure 3). This is representative of the near-wall energy feeding process from small to large scale, consistent with both Cho et al. (Reference Cho, Hwang and Choi2018) and Doohan et al. (Reference Doohan, Willis and Hwang2021). Note that this inverse energy transfer from small to large scale (e.g. Cho et al. Reference Cho, Hwang and Choi2018; Andreolli, Quadrio & Gatti Reference Andreolli, Quadrio and Gatti2021) has been known to break the self-similarity of the energy-containing motions in the region close to the wall, where the attached eddy hypothesis is not supposed to be valid (for a further discussion, see Hwang Reference Hwang2016).

Figure 3. Wall-normal variation of the time-averaged intra- and inter-scale energy transport terms in (3.7) for (a,b) streamwise, (c,d) wall-normal and (e,f) spanwise components: (a,c,e) large scale; (b,d,f) small scale.

The inter-scale spatial transport, $T_{\#}$, describes an energy transport that results from interactions between the different scales. The streamwise inter-scale spatial transport terms show local maxima at a wall-normal location similar to that of the peaks of the inter-scale turbulent transport and production, suggesting that these transport terms also play a role in transporting the turbulent kinetic energy produced by the flow between the scales. However, all of these inter-scale spatial transport terms exhibit similar trends at both the large and small scales (at least with respect to their signs) to those in Doohan et al. (Reference Doohan, Willis and Hwang2021), suggesting that these terms act in a similar manner at both scales.

The intra-scale spatial transport, $T_{-}$, represents energy transport within each scale. At the large scales, these terms follow reasonably similar trends to Doohan et al. (Reference Doohan, Willis and Hwang2021). However, no intra-scale transport is exhibited by the ROM at the small scales, a feature that is not in line with the DNS results of Doohan et al. (Reference Doohan, Willis and Hwang2021). This is a consequence of the spanwise averaging over the small-scale intra-scale transport terms equalling zero. For example, the streamwise intra-scale spatial turbulent transport is given by

(3.8)\begin{equation} T_{us,-} ={-}\langle u_s (\boldsymbol{u}_s\boldsymbol{\cdot} \boldsymbol{\nabla} u_s) \rangle_{x,z}. \end{equation}

We note that $T_{us,-}$ takes the form $\langle\, f(x,y,n,t) \,{\rm e}^{\mathrm {i}2(n_1+n_2+n_3)z}\rangle _{x,z}$, where $n_i$ refers to the spanwise wavenumber related to the $i$th term in the triple correlation of (3.8). Since $n_1+n_2+n_3 \neq 0$ at the small scales on account of the model's truncation (i.e. $n_i=\pm 2$ from (3.3)), this spanwise average will always be zero at the small scales unless additional wavenumbers are considered in the present ROM.

The immediate effect of this on the results of the model is not apparent, but with that being said, intra-scale processes that are responsible for the transport of energy from the production to the dissipation are seen to persist, with the pressure strain being largely responsible for this redistribution. Additionally, the viscous transport, $T_\nu$, which itself only involves intra-scale terms, is seen to be active in relation to the production and dissipation terms at both scales.

3.3. Temporal dynamics

Having studied the time-averaged variation of the different terms in the energy equations, it is also of interest to study the temporal relationship between these quantities and to confirm the presence of known dynamic processes found in the similar system of Doohan et al. (Reference Doohan, Willis and Hwang2021). These include the large- and small-scale self-sustaining process, the energy cascade, the driving mechanism which transports energy from large-scale processes and injects it into the small scales through the small-scale production and the feeding mechanism which is responsible for transporting energy from small to large scales. To study the temporal correlation of such quantities, we introduce the temporal cross-correlation of two time-varying functions $f(t)$ and $g(t)$

(3.9)\begin{equation} \{\,f\star g\}(\tau) = \frac{\displaystyle\int_{\Delta T} f(t+\tau)g(t)\,\mathrm{d}t}{\left( \displaystyle\int_{\Delta T} f(t)^2\,\mathrm{d}t \right) ^{1/2} \left( \displaystyle\int_{\Delta T} g(t)^2\,\mathrm{d}t \right) ^{1/2}}, \end{equation}

where $\Delta T$ is a suitable time interval of interest.

Of particular interest is the presence of the large- and small-scale self-sustaining processes (Hwang & Bengana Reference Hwang and Bengana2016; Doohan et al. Reference Doohan, Willis and Hwang2021). With this in mind, the energy related to the different velocity components is separated into streamwise-dependent and -independent parts (Lucas & Kerswell Reference Lucas and Kerswell2017; Doohan et al. Reference Doohan, Willis and Hwang2019, Reference Doohan, Willis and Hwang2021). Specifically, these provide observables based on the kinetic energy for the structural components of the self-sustaining process; i.e. straight streaks $(E_{ss})$, wavy streaks $(E_{ws})$, straight rolls $(E_{sr})$ and wavy rolls $(E_{wr})$ at both the large

(3.10a)\begin{gather} E_{ss,l}(t) = \tfrac{1}{2} \langle \langle u_l \rangle_x^2 \rangle _z \big|_0^{80} , \end{gather}

(3.10b)\begin{gather}E_{ws,l}(t) = \tfrac{1}{2} \langle (u_l - \langle u_l \rangle_x)^2 \rangle _{x,z} |_0^{80}, \end{gather}

(3.10c)\begin{gather}E_{sr,l}(t) = \tfrac{1}{2} \langle \langle v_l \rangle_x^2 + \langle w_l \rangle_x^2 \rangle _z |_0^{80}, \end{gather}

(3.10d)\begin{gather}E_{wr,l}(t) = \tfrac{1}{2} \langle (v_l - \langle v_l \rangle_x)^2 + (w_l - \langle w_l \rangle_x)^2 \rangle _{x,z} |_0^{80}, \end{gather}

and small scales

(3.11a)\begin{gather} E_{ss,s}(t)= \tfrac{1}{2} \langle \langle u_s \rangle_x^2 \rangle _z |_0^{40}, \end{gather}

(3.11b)\begin{gather}E_{ws,s}(t)= \tfrac{1}{2} \langle (u_s - \langle u_s \rangle_x)^2 \rangle _{x,z} |_0^{40}, \end{gather}

(3.11c)\begin{gather}E_{sr,s}(t)= \tfrac{1}{2} \langle \langle v_s \rangle_x^2 + \langle w_s \rangle_x^2 \rangle_z|_0^{40}, \end{gather}

(3.11d)\begin{gather}E_{wr,s}(t)= \tfrac{1}{2} \langle (v_s - \langle v_s \rangle_x)^2 + (w_s - \langle w_s \rangle_x)^2 \rangle _{x,z} |_0^{40}, \end{gather}

where the shorthand $f(t)|_{y_1}^{y_2}$ denotes averaging of quantity $f(t)$ in the wall-normal direction over the interval $[y_1, y_2]$ given in wall units. The choice of $y^+=40$ to separate the boundary of the near-wall region (viscous sublayer and buffer layer) from the logarithmic region is chosen to approximately separate phenomenological differences observed in these regions based on figures 1–3. In practice, small changes in the choice of this separating $y^+$ value was seen to have little impact on the results of the following sections.

The cross-correlation of these energies at both scales has been computed and is shown in figure 4. Indeed, correlation does exist between the different kinetic energies, with the shift of the peak of the correlations revealing the order of the processes: $E_{sr} \rightarrow E_{ss} \rightarrow E_{ws} \rightarrow E_{wr} \rightarrow E_{sr}$, consistent with the results of Hamilton et al. (Reference Hamilton, Kim and Waleffe1995), Hwang & Bengana (Reference Hwang and Bengana2016) and Doohan et al. (Reference Doohan, Willis and Hwang2021), indicating that the self-sustaining process at both scales is captured by the ROM. Further, the time scale of these processes both at the large and small scales is also similar to that found by Doohan et al. (Reference Doohan, Willis and Hwang2021). The small-scale process occurs on a time scale that is significantly shorter than that of its large-scale counterpart, as seen by the width of the cross-correlation functions.

Figure 4. Temporal cross-correlation between the energy observables associated with the structural components of the self-sustaining process at (a) large and (b) small scales. Markers show the peak of each cross-correlation.

Similar correlations have been produced for the other energy transport processes identified by Doohan et al. (Reference Doohan, Willis and Hwang2021), namely for the energy cascade, driving and feeding processes, all showing agreement with those found in the original DNS. The details of these correlations can be found in the supplemental material of this paper. In summary, this system contains an energy cascade which transports energy from the large-scale production to small-scale wall-normal and spanwise dissipations mainly through a particular part of the wall-normal inter-scale turbulent transport

(3.12)\begin{equation} T_{v,\updownarrow c} = \langle v_l(\boldsymbol{u}_l\boldsymbol{\cdot} \boldsymbol{\nabla} v_s) \rangle_{x,z}. \end{equation}

Energy transported from large to small scales also involves the driving mechanism, the activation process of small-scale production via energy cascade. The related energy transport is well characterised by the other part of the wall-normal inter-scale turbulent transport

(3.13)\begin{equation} T_{v,\updownarrow d} ={-}\langle v_s(\boldsymbol{u}_s\boldsymbol{\cdot} \boldsymbol{\nabla} v_l) \rangle_{x,z}. \end{equation}

It results in subsequent activation of the small-scale production via the Orr mechanism (Orr Reference Orr1907), promoting a range of other processes such as an increase in small-scale dissipation in response to the breakdown of small-scale wavy streaks as well as a feeding process which transports energy from the small to large scales. Finally, the feeding process is also seen to transport energy from small-scale production mainly to large-scale dissipation terms through the streamwise and spanwise inter-scale turbulent transport, and it involves the activation of small-scale wavy streaks and small-scale wavy rolls (through the spanwise pressure strain).

4.1. Equilibria

The mean and r.m.s. velocity profiles of the computed equilibrium solutions as functions of the wall-normal direction are shown in figure 5, compared with a long-time trajectory from the ROM shown in black and the DNS in blue. Four of these solutions have been highlighted in figure 5 which closely capture the wall shear stress of the long-time trajectory. Phase portraits showing similar energy observables to those of Doohan et al. (Reference Doohan, Bengana, Yang, Willis and Hwang2022) are shown in figure 6, also highlighting the same equilibria as in the velocity profiles. Here, we note that many of the computed equilibria have non-negligible amounts of energy and production at both large and small scales (figure 6a–d) and, in this sense, they may be viewed to be multi-scale equilibrium solutions.

Figure 5. Mean and r.m.s. velocity profiles for the equilibrium solutions (grey) compared with the long-time trajectory generated by the ROM (black) and DNS (blue): (a) $U^+$; (b) $u_{rms}^+$; (c) $v_{rms}^+$; (d) $w_{rms}^+$. Highlighted equilibria are EQ8 (orange); EQ28 (cyan); EQ48 (purple); EQ52 (green).

Figure 6. Phase portraits of the large- and small-scale energy observables of the (a–d) self-sustaining process (equations (3.10), (3.11)) and (e–i) the two scale energy budget (equations (3.4), (3.5), (3.7)) for the equilibria (blue), EQ8 (orange), EQ28 (cyan), EQ48 (purple), EQ52 (green), compared with linearly spaced contours of the respective bivariate probability density function (dark–light grey) generated from a long-time trajectory of the ROM.

A large number of the equilibria are found to capture the mean flow reasonably well, especially in the near-wall region (figure 5a). However, many did not accurately capture the velocity fluctuations of the flow and are not seen to be highly representative of the chaotic state with respect to observables other than the mean flow. The equilibria found are typically seen to overestimate the near-wall streamwise velocity fluctuations. This is clearly observed in the maxima of the streamwise r.m.s. profiles (figure 5b), which appear to be related to the large amount of energy associated with straight streaks in figure 6(a). Equilibria overestimating the amount of energy related to straight streaks compared with the long-time trajectory were also observed by Doohan et al. (Reference Doohan, Bengana, Yang, Willis and Hwang2022) with the relative magnitude of the overestimation being similar, especially for the small-scale straight streaks.

Wall-normal fluctuations are seen to be of generally lower magnitude than that of the long-time trajectory, although some of the highlighted equilibria with high friction Reynolds numbers are seen to capture the wall-normal r.m.s. velocity profiles quite well (EQ8, EQ28 and EQ48; figure 5c). With an accurate reproduction of the mean wall shear stress, some of the computed equilibria capture the energy production of the large and small scales reasonably well (figure 6e). The spanwise r.m.s. velocities of the computed equilibria are also typically lower than that of the chaotic trajectory (figure 5d). These observations are consistent with some of the phase portraits, in which straight rolls, wavy streaks and rolls are not well captured by the equilibria (figure 6b–d).

Finally, the phase portraits associated with energy cascade and nonlinear scale interactions show that most of the equilibria computed do not properly capture the dynamics of the chaotic trajectory. The streamwise dissipation of many equilibria does not appear near the chaotic trajectory (figure 6f). Furthermore, the phase portraits of the wall-normal and spanwise dissipation (figure 6g), inter-scale energy transport (figure 6h) and pressure strain (figure 6i) all show that most of the computed equilibria have much smaller amplitudes than those of the instantaneous chaotic state. This is consistent with the finding of the DNS study by Doohan et al. (Reference Doohan, Willis and Hwang2021). However, it should be pointed out that most of their equilibria feature the self-sustaining process either at large or small scale, whereas many equilibria in this study have non-negligible amounts of energy and production at both of the scales (figure 6a,b,e) – note that the equilibria of Doohan et al. (Reference Doohan, Willis and Hwang2021) were obtained by continuing the existing solutions obtained in a smaller computational domain (equivalent to lower Reynolds number), whereas those in our study are obtained by directly looking for the zeros of (4.1). This observation suggests that the relatively low amplitude of the observables associated with energy cascade and nonlinear scale interactions is possibly more associated with the lack of unsteadiness in the equilibria, as we shall indeed see in § 4.2.

4.2. Periodic orbits

Similar data as shown for the equilibria are now shown for the periodic orbits computed using the ROM, with mean and r.m.s. velocity profiles shown in figure 7, and phase portraits of large- and small-scale energy observables in figure 9. Highlighted periodic orbits in figures 7 and 9 are chosen as they show reasonable agreement with the mean and r.m.s. statistics, and their dynamics in the state space will be further explored with respect to figure 9.

Figure 7. Mean and r.m.s. velocity profiles for the periodic orbits (grey) compared with the long-time trajectory generated by the ROM (black) and DNS (blue): (a) $U^+$; (b) $u_{rms}^+$; (c) $v_{rms}^+$; (d) $w_{rms}^+$. Highlighted periodic orbits: PO17 (cyan); PO20 (purple); PO26 (green); PO42 (orange).

In general, the periodic orbits characterise the chaotic state more accurately than the equilibria, showing closer agreement to the long-time trajectory in many cases. The mean profiles displayed by the periodic orbits appear to be similar to those shown by the equilibria, with the wall shear stress being captured well (figure 7a). Large maxima seen in the near-wall region of the streamwise r.m.s. velocity profiles in the equilibria are less apparent in the periodic orbits with a multitude of periodic orbits capturing this profile well (compare figure 7b with 5b). Similar to the equilibria, the wall-normal fluctuations displayed by the periodic orbits appear to be mostly smaller than that of the long-time trajectory or DNS, although wall-normal fluctuations in the near-wall region are seen to be a little closer to the long-time trajectory than those of the equilibria (compare figure 7c with 5c). The spanwise r.m.s. profiles also generally exhibit values less than that of the long-time trajectory, although some periodic orbits replicate the near-wall peaks better than those of the equilibria: PO17 and PO26, in particular (compare figure 7c with 5c).

Some recent studies have suggested that taking simple averages over computed invariant solutions (Chandler & Kerswell Reference Chandler and Kerswell2013; Page et al. Reference Page, Norgaard, Brenner and Kerswell2022) is often found to be as good as other more sophisticated approaches (cyclic expansion; Cvitanovic et al. Reference Cvitanovic, Artuso, Mainieri, Tanner, Vattay, Whelan and Wirzba2005). In figure 8, turbulence statistics obtained from a simple average of equilibria and periodic orbits are further compared with those of the long-time trajectory of DNS and ROM. While there is very little difference in the mean profiles from the simple average of equilibria and periodic orbits (figure 8a), the velocity fluctuations from periodic orbits show better agreements with those from the long-time trajectory of DNS and ROM (figure 8b–d). However, the velocity fluctuations from periodic orbits still show considerable differences from those from the long-time trajectory of ROM, indicating that there are a number of orbits which may not be near the turbulent solution trajectory in the state space.

Figure 8. Mean and r.m.s. velocity profiles from simple average over the equilibria (red) and the periodic orbits (green) compared with those of the long-time trajectory generated by the ROM (black) and DNS (blue): (a) $U^+$; (b) $u_{rms}^+$; (c) $v_{rms}^+$; (d) $w_{rms}^+$.

The periodic orbits generally appear to capture the cross-streamwise velocity fluctuations better than the equilibria, as seen in figure 7. The observables associated with energy transport between different velocity components would therefore be better captured by them, and pressure strain has been understood to be mainly involved in this process (e.g. Tennekes & Lumley Reference Tennekes and Lumley1967; Cho et al. Reference Cho, Hwang and Choi2018). Given the form of pressure strain shown in (A4) and (A11), the state-space projections of the periodic orbits with the observables involving $\partial u/\partial x$ are expected to be improved compared with that of the equilibria. Indeed, an examination of the phase portraits in figure 9 reveals that such observables are more convincingly captured in the periodic orbits at both large and small scales than in the equilibria: for example, wavy streaks (figure 9c), cross-streamwise dissipation (figure 9g) and pressure strain (figure 9i), in particular.

Figure 9. Phase portraits of the large- and small-scale energy observables of the (a–d) self-sustaining process (equations (3.10), (3.11)) and (e–i) the two scale energy budget (equations (3.4), (3.5), (3.7)) for the periodic orbits (blue), PO17 (cyan), PO20 (purple), PO26 (green), PO42 (orange), compared with linearly spaced contours of the respective bivariate probability density function (dark–light grey) generated from a long-time trajectory of the ROM.

Overall, the periodic orbits describe the chaotic trajectory both statistically and dynamically better than the equilibria. The periodic orbits are seen to be more representative with respect to the energy of straight streaks in the flow, with less high-energy small-scale outliers, although these high-energy outliers continue to be seen at the large scales (figure 9a). There is a modest improvement in the description of rolls by the periodic orbits (figure 9b,d). Production and dissipation appear to be more consistently captured by the periodic orbits than the equilibria (figure 9e–g). The inter-scale energy transport terms from the periodic obits are also able to explore more of the state space than those from the equilibria.

Despite the improved state-space dynamics offered by the periodic orbits compared with that by the equilibria, the periodic orbits computed in this study do not sufficiently cover the state space visited by the chaotic trajectory, as is evident from figure 9. Imposition of the reflection symmetry might be a potential reason, but the phase portrait and turbulence statistics from the chaotic trajectory subject to the symmetry did not yield any significant difference to those without the symmetry (see Appendix B). It is, however, important to mention that this does not necessarily imply a ‘theoretical’ failure of the attempt to construct the chaotic dynamics in terms of periodic orbits. As mentioned earlier, the periods of the computed orbits are less than $T^+ \leq 141.57$. However, the time scale of the self-sustaining process at large scale is expected to be $T^+ \sim 400\unicode{x2013}500$ (Hwang & Bengana Reference Hwang and Bengana2016; Doohan et al. Reference Doohan, Willis and Hwang2021), considerably longer than the periods of the computed orbits in this study. Therefore, it is presumable that the lack of state-space coverage by the computed orbits originates from their relatively short periods, and the computation of the periodic orbits with longer periods is expected to improve the state-space description by the periodic orbits. Nevertheless, the computation of orbits with longer periods has been found to be challenging even with the present ROM. In particular, we have observed the vanishing recurrence of the chaotic trajectory on increasing Reynolds numbers, causing trouble in identifying suitable initial conditions for the Newton algorithm. This also suggests that a numerical approach with a fundamentally different concept would probably be required to overcome this challenge. The Newton method employed for the computation of periodic orbits in this study is expected to have limited applicability, when the leading Lyapunov exponent of the given system is too large – the leading Lyapunov exponent of the present ROM computed using the algorithm in Parker & Chua (Reference Parker and Chua1989) was $\lambda _1=0.107$ ($\lambda _1^+=0.58$), comparable to the leading eigenvalues of the periodic orbits in table 1. This is also a common feature of some of the most popular algorithms currently available based on the same type of Newton iteration (e.g. Viswanath Reference Viswanath2007; Willis et al. Reference Willis, Short and Cvitanović2016). There have been a few recent works addressing this challenge, and the proposed algorithms do not appear to depend on the magnitude of the leading Lyapunov exponent (e.g. Azimi et al. Reference Azimi, Ashtari and Schneider2022; Page et al. Reference Page, Norgaard, Brenner and Kerswell2022; Parker & Schneider Reference Parker and Schneider2022). Employing such a technique may be a way to overcome this challenge in the future.

Table 1. Friction Reynolds number $Re_\tau$ (with respect to the friction velocity of each orbit), period $T^+$ (with respect to the friction velocity of the ROM), maximum streamwise r.m.s. velocity $\max \{u_{rms}^+\}$, the dimension of the unstable manifold $\dim (\mathcal {W}^u)$, the largest Floquet multiplier $\mu _1$ and corresponding Lyapunov exponent $\lambda _1$ of selected periodic orbits, compared with the ROM whenever possible.

4.3. Multi-scale dynamics in periodic orbits

Thus far, we have seen that a few periodic orbits computed appear to provide some multi-scale dynamics representative of the chaotic state, consistent with the results of figure 9, which show contributions in energy terms from both the large and small scales. Now, we will consider some periodic orbits, highlighted with distinct colours in figures 7 and 9 (PO17, PO20, PO26, PO42, in particular), to explore the exact dynamics that each orbit offers. We shall see that these orbits describe: (i) the self-sustaining process at small and large scales (Hwang & Bengana Reference Hwang and Bengana2016); (ii) the driving mechanism of small-scale production via energy (Doohan et al. Reference Doohan, Willis and Hwang2021); (iii) energy cascade from large to small scales. An overview of these chosen orbits is tabulated in table 1. In particular, their stability properties show a large number of unstable modes with a high leading Lyapunov exponent, with values much larger than those reported at low Reynolds numbers (Viswanath Reference Viswanath2007). We also note the large dimension of the unstable manifold of these orbits, comparable to or larger than that of the multi-scale equilibrium solution found in Doohan et al. (Reference Doohan, Bengana, Yang, Willis and Hwang2022) (i.e. $L_{L2b}$). Invariant solutions with a high-dimensional unstable manifold have often been observed to more closely resemble the turbulent state (see also Doohan et al. Reference Doohan, Willis and Hwang2019).

Figure 10 shows the large- and small-scale velocity fields at an instant in time in PO42. It is clear that a large pair of streaks that extends from the wall into the centre of the domain dominates the large-scale field, with twice as many small-scale streaks dominating the near wall of the small-scale velocity field, in agreement with the phenomenology of the large- and small-scale self-sustaining processes. This structure in the large- and small-scale velocity fields, common to a number of the computed periodic orbits, combines to exhibit a rich variety of dynamics.

Figure 10. (a) Instantaneous large-scale streamwise velocity isosurfaces $u_l^+= \pm 1.5$ (red/blue) and (b) instantaneous small-scale streamwise velocity isosurfaces $u_s^+= \pm 0.75$ (red/blue) shown at the same instant in time in PO42.

A number of the spatio-temporal processes statistically identified in the long-time chaotic trajectory have also been found in the periodic orbits. Evidence of the small-scale self-sustaining process has been seen in a large number of the periodic orbits, with an example (PO20) shown in figure 11. The time trace of the observables of the small-scale self-sustaining process shows a consistent interchange of dominance between its structural components: $E_{sr,s} \rightarrow E_{ss,s} \rightarrow E_{ws,s} \rightarrow E_{wr,s} \rightarrow E_{sr,s}$. The flow field reconstructions of figure 11 show the evolution of two pairs of streaks (red/blue $u_s^+$ contours) and vortices (black $\omega _{x,s}$ contours), showing straight streaks undergoing an instability, resulting in a wavy streak. This wavy streak is accompanied by a growth of intensity of the neighbouring streamwise vortices which slowly begin to grow between approximately $t^+=60$ and $t^+=10$ in the absence of streaks. During this time, straight streaks are seen to regenerate and the cycle continues.

Figure 11. (a) The evolution of the small-scale self-sustaining process observables normalised by their maximum values over a single period of PO20. (b–d) Flow field visualisations at selected times show small-scale streamwise velocity isosurfaces $u_s^+=\pm 1.5$ (red/blue), and small-scale streamwise vorticity isosurfaces $\omega _{x,s}=\pm 0.55$ (black).

Due to the increased difficulty and expense of converging longer periodic orbits, evidence of the large-scale self-sustaining process has only been found in a single longer orbit (PO42), shown in figure 12. As in the case of the small-scale self-sustaining process, we see a consistent interchange of dominance between the structural components of the large-scale self-sustaining process: $E_{sr,l} \rightarrow E_{ss,l} \rightarrow E_{ws,l} \rightarrow E_{wr,l} \rightarrow E_{sr,l}$. A single pair of large-scale streaks (red/blue $u_l^+$ contours) are seen to dominate the flow field, starting as straight streaks, slowly undergoing an instability to form a wavy streak, resulting in a breakdown in the streak along its length. From this point, a large intensity of large-scale streamwise vorticity is formed which sits alongside the remnants of the broken streak. This vortex is then seen to stretch in the streamwise direction (Schoppa & Hussain Reference Schoppa and Hussain2002) accompanying the regeneration of the original straight streak occupying the full streamwise length of the domain.

Figure 12. (a) The evolution of the large-scale self-sustaining process observables normalised by their maximum values over a single period of PO42. (b–d) Flow field visualisations at selected times show large-scale streamwise velocity isosurfaces $u_l^+=\pm 1.5$ (red/blue), and large-scale streamwise vorticity isosurfaces $\omega _{x,l}=\pm 1$ (black).

Other processes, such as the energy cascade and the driving mechanism, identified statistically in long-time chaotic trajectories of the ROM (§ 3.3) as well as in the DNS of Doohan et al. (Reference Doohan, Willis and Hwang2021), have also been identified in a number of the periodic orbits. An example of the driving mechanism is observed in PO26, shown in figure 13. The first snapshot ($t^+=10$) shows a point of inactivity in the cycle featuring small amounts of small-scale production accompanying some high $u_s^+$ (streaky) structures (figure 13b). By the second snapshot at $t^+=40$, a large peak in the driving wall-normal inter-scale turbulent transport (see (3.13)) from large to small scale (i.e. positive $T_{v,\updownarrow d}^+$) is observed both in the centre of the channel and around the streaks, transporting energy from large to small scales. This is accompanied by a rise in turbulent production (figure 13c), due to an injection of energy into the wall-normal small-scale velocity $v_s^+$ (see also figure 13a). The result of this is an amplification of the small-scale streaky motions observed at $t^+=60$ (figure 13d).

Figure 13. (a) The evolution of the small-scale streak energy $E_{ss,s}^+$, driving component of wall-normal inter-scale turbulent transport $T_{v,\updownarrow d}^+$ and small-scale production $P_{us}^+$ normalised by their maximum values over a single period of PO26, showing the driving mechanism. (b–d) The flow field visualisations at selected times show small-scale streamwise velocity isosurfaces $u_s^+=\pm 2.25$ (red/blue), driving wall-normal inter-scale turbulent transport isosurfaces $T_{v,\updownarrow d}^+=0.11$ (black) and small-scale production isosurfaces $P_{us}^+=0.67$ (grey).

An example of the energy cascade is observed in PO17, shown in figure 14. In the statistical analysis of the long-time chaotic trajectory, energy transport primarily through a particular part of the wall-normal inter-scale turbulent transport associated with the cascade $T_{v,\updownarrow c}$ (see (3.12)) was seen to transport energy from the large-scale production to the wall-normal small-scale dissipation near $y \pm 0.5$. Thus, figure 14 shows snapshots of the flow in the central region of the channel. Large-scale production isosurfaces reveal long streak-like structures in the centre of the channel at the peak of the large-scale production at around $t^+=12$ (figure 14b). Between $t^+=12$ and $t^+=30$ these structures shorten slightly in length accompanied by a growth in regions of high wall-normal inter-scale turbulent transport (figure 14c), which are seen to move down and intersect with the small-scale dissipation contours at $t^+=35$, continuing until roughly $t^+=50$ (figure 14d,e). At this point, an increase in the wall-normal small-scale dissipation is observed, with a drift of the small-scale dissipation isosurfaces towards the central region of the channel. It is also interesting to note that $T_{v,\updownarrow c}^+<0$ during this time interval, indicating the possible existence of an inverse energy cascade, although the time-averaged value of $T_{v,\updownarrow c}^+$ is positive. From $t^+=50$ to $t^+=60$ (figure 14f), the large-scale production increases again, correlated to the intensification of large-scale straight streaks, completing the cycle.

Figure 14. (a) The evolution of the large-scale production $P_{ul}^+$, cascade component of the wall-normal inter-scale turbulent transport $T_{v,\updownarrow c}^+$ and wall-normal small-scale dissipation $\varepsilon _{vs}^+$ normalised by their maximum values over a single period of PO17, showing the energy cascade. (b–d) The flow field visualisations at selected times show large-scale production isosurfaces $P_{ul}^+=0.22$ (grey), wall-normal inter-scale turbulent transport isosurfaces $T_{v,\updownarrow c}^+=0.22$ (black) and wall-normal and spanwise small-scale dissipation $\varepsilon _{vs}^++\varepsilon _{ws}^+=-0.3$ (cyan).