1. Introduction
Oscillatory flows are ubiquitous in nature (Spalart & Baldwin Reference Spalart and Baldwin1987; Jensen, Sumer & Fredsøe Reference Jensen, Sumer and Fredsøe1989), biomechanical systems (Ku Reference Ku1997; Taylor & Draney Reference Taylor and Draney2004) and engineering applications (Gatti & Quadrio Reference Gatti and Quadrio2016; Adcock et al. Reference Adcock, Draper, Willden and Vogel2021), with ongoing research including a particular focus on transition to turbulence in oscillatory boundary layers (Blennerhassett & Bassom Reference Blennerhassett and Bassom2008; Biau Reference Biau2016; Xiong et al. Reference Xiong, Qi, Gao, Xu, Ren and Cheng2020; Blondeaux, Pralits & Vittori Reference Blondeaux, Pralits and Vittori2021; Gong et al. Reference Gong, Xiong, Mao, Cheng, Wang and Zhang2022). The canonical oscillating flow is the Stokes boundary layer (SBL), the flow above a sinusoidally oscillating flat plate. The transition to turbulence in this flow is complex (von Kerczek & Davis Reference von Kerczek and Davis1974; Blennerhassett & Bassom Reference Blennerhassett and Bassom2008; Mujal-Colilles et al. Reference Mujal-Colilles, Christensen, Bateman and Garcia2016; Blondeaux & Vittori Reference Blondeaux and Vittori2021); the SBL is a subcritical flow that exhibits turbulent dynamics at Reynolds numbers
$\textit{Re} = U_0 \sqrt {T_0 / \nu }$
(where
$U_0$
is the velocity oscillation amplitude,
$T_0$
is the oscillation period and
$\nu$
is the fluid’s kinematic viscosity) below the critical Reynolds number
$\textit{Re}_c \approx 2511$
at which the laminar solution becomes linearly unstable to normal mode perturbations (Blennerhassett & Bassom Reference Blennerhassett and Bassom2002), although convective linear instability occurs below this critical value in domains with a large horizontal extent (Ramage et al. Reference Ramage, Davies, Thomas and Togneri2020; Pretty, Davies & Thomas Reference Pretty, Davies and Thomas2021). The critical Reynolds number found by Blennerhassett & Bassom (Reference Blennerhassett and Bassom2002) is based upon a Floquet analysis of perturbations growing over the whole oscillation cycle.
Ozdemir, Hsu & Balachandar (Reference Ozdemir, Hsu and Balachandar2014) summarised several theoretical, experimental and numerical works on the SBL and categorised the flow into four regimes: (i) laminar (
$\textit{Re} \lt \textit{Re}_{c_1}$
); (ii) disturbed laminar (
$\textit{Re}_{c_1}\lt \textit{Re} \lt \textit{Re}_{c_2}$
), where some disturbances such as spanwise vortices are observed, but without enough growth to trigger transition; (iii) intermittently turbulent (
$\textit{Re}_{c_2}\lt \textit{Re} \lt \textit{Re}_{c_3}$
), where some turbulent features such as bursts of energy during the mid and late deceleration phases are observed, which laminarise during the early acceleration phase; (iv) turbulent (
$\textit{Re} \gt \textit{Re}_{c3}$
). Around the onset of the intermittently turbulent regime, Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2014) observed a self-sustaining transitional behaviour, and this has been described in terms of a ‘bypass’ transition in which streamwise streaks triggered by free stream turbulence grow, form hairpin vortices and then eject a spanwise vortex back into the free stream, which subsequently breaks down to turbulence and reinitiates the cycle (Xiong et al. Reference Xiong, Qi, Gao, Xu, Ren and Cheng2020; Gong et al. Reference Gong, Xiong, Mao, Cheng, Wang and Zhang2022). Mier, Fytanidis & García (Reference Mier, Fytanidis and García2021) and Fytanidis, García & Fischer (Reference Fytanidis, García and Fischer2021) reported that this self-sustaining transitional behaviour is associated with a phase-lag between the wall-shear stress and the laminar velocity, with disturbances growing during the deceleration phase and decaying during the acceleration phase (see also Luo & Wu Reference Luo and Wu2010), an observation in agreement with the transitional cycles observed in other, wall-bounded oscillating flows (Pier & Schmid Reference Pier and Schmid2017; Ebadi et al. Reference Ebadi, White, Pond and Dubief2019; Pier & Schmid Reference Pier and Schmid2021; Linot, Schmid & Taira Reference Linot, Schmid and Taira2024).
Crucially, however, such a ‘self-sustaining’ bypass transition relies upon turbulent motions and strong spanwise vortices. This sets it apart from the coherent motions of the vortex–wave interaction states (Hall & Smith Reference Hall and Smith1991; Hall & Sherwin Reference Hall and Sherwin2010), often realised as ‘edge states’ controlling the transition to turbulence in steady wall–bounded shear flows (Schneider & Eckhardt Reference Schneider and Eckhardt2006; Skufca, Yorke & Eckhardt Reference Skufca, Yorke and Eckhardt2006; Kim & Moehlis Reference Kim and Moehlis2008; Duguet, Schlatter & Henningson Reference Duguet, Schlatter and Henningson2009; Schneider, Marinc & Eckhardt Reference Schneider, Marinc and Eckhardt2010; Eaves & Caulfield Reference Eaves and Caulfield2015) that follow the self-sustaining process (SSP) (Waleffe Reference Waleffe1997) of streamwise vortices created via the lift-up mechanism (Landahl Reference Landahl1980) and sustained by weak three-dimensional waves. From a dynamical systems point of view, laminar and turbulent flows can be seen as attractors in a high-dimensional state-space, whose basins of attraction are separated by a manifold termed the ‘edge of chaos’ (or simply the ‘edge’) (Skufca et al. Reference Skufca, Yorke and Eckhardt2006), containing local attractors known as ‘edge states’. Characterisation of these edge states offers an alternative view of the physics associated with transition to turbulence which is self-contained, omitting the need to invoke pre-existing free stream turbulence to initiate transition, and, in the case of steady shear flows, requiring only streamwise vortices. Furthermore, the mechanisms underlying such states are also known to control important processes in the fully turbulent flow (see e.g. Kawahara, Uhlmann & van Veen Reference Kawahara, Uhlmann and van Veen2012; Budanur et al. Reference Budanur, Short, Farazmand, Willis and Cvitanović2017).
In steady wall-bounded shear flows, these edge states rely on the lift-up mechanism, a linear transient growth mechanism by which parcels of fluid are advected (‘lifted’) through a shear flow, for example taking high-speed fluid and moving it to a low-speed region, creating streamwise momentum defects. Continued advection leads to large transient growth over a long (
${O}(\textit{Re})$
) time scale, and is responsible for producing large-amplitude (
${O}(1)$
) streamwise streaks when realised at finite amplitude. A steady background shear allows the lift-up mechanism to accumulate momentum defects in fixed locations over a long period of time, and so it is often the linear mechanism with the largest, and longest sustained, growth (Butler & Farrell Reference Butler and Farrell1992) and hence is integral to the SSP (Waleffe Reference Waleffe1997). However, in the SBL, the lift-up mechanism is not an optimal linear transient growth mechanism (Biau Reference Biau2016) since the laminar shear flow is periodically oscillating, preventing momentum defects from accumulating in a single location, and reversing sign every half-period. Instead, the Orr mechanism (Orr Reference Orr1907) is dominant, which allows for moderate growth as spanwise vortices are tilted by the background shear over a short time scale. It is known that even small disruptions to the lift-up mechanism have the potential to significantly affect edge state dynamics (Eaves & Caulfield Reference Eaves and Caulfield2015).
These observations raise the interesting question as to what self-sustaining mechanism(s) edge states in the SBL utilise to balance dissipation (relaminarisation) against instability (transition to turbulence). On the one hand, a traditional interpretation of the SSP as using the lift-up mechanism due to its optimal linear growth would potentially point to new, Orr-based edge states for the SBL. On the other hand, transitional SBL flows are observed to consist of streamwise streaks (Sarpkaya Reference Sarpkaya1993; Costamagna, Vittori & Blondeaux Reference Costamagna, Vittori and Blondeaux2003) and so perhaps some nonlinearity associated with the edge state dynamics allows the lift-up mechanism enough ‘time’ to create large-amplitude streaks. The most well-studied edge states have been steady, and so the different components of the SSP act all together at the same time, however, in this second, lift-up based scenario, any realisation of the SSP in the SBL would evolve periodically in order to accommodate the alternating shear direction of the background flow, opening up the potential for components of the SSP to act sequentially. In order to make some connection with the bypass transition scenario, whatever mechanisms arise within the edge state dynamics should shed light upon how boundary layer ejections could be initiated without resorting to pre-existing turbulence.
In this work, we compute edge trajectories in the transitional regime of the SBL and analyse their dynamics in order to determine the underlying physical mechanisms which sustain them. We demonstrate that the edge states are periodically evolving versions of the SSP, utilising the lift-up mechanism, and that a temporal interplay between the laminar shear and streak nonlinearity results in streak migration away from the wall (non-turbulent ejections) and sequential activation of different components of the flow. In § 2 we present the equations of motion, the numerical model and the edge tracking algorithm. In § 3 we provide an overview of the edge trajectory dynamics, which are examined in closer detail in section § 4 which presents a complete analysis of the periodic self-sustaining process (PSSP). Conclusions are drawn in § 5.
2. Methodology
2.1. Equations of motion
The SBL is the oscillatory flow above a flat plate, in which an oscillation of period
$T_0$
is driven either by a sinusoidal oscillation in the velocity of the plate itself (in the
$x$
-direction) of magnitude
$U_0$
or a sinusoidal oscillation of the pressure gradient above a stationary plate; in this work, we study the former, oscillating plate flow. These oscillations set up a boundary layer against the plate whose thickness is determined diffusively, leading to the characteristic length scale
$\delta = \sqrt {\nu T_0}$
in which
$\nu$
is the kinematic viscosity of the fluid. We decompose the instantaneous velocity into the sum of a laminar component
$\boldsymbol{U}=(U,0,0)$
and a perturbation velocity field
$\boldsymbol{u}=(u,v,w)$
, such that
$\boldsymbol{u}^{\textit{tot}} = \boldsymbol{U} + \boldsymbol{u}$
. Using index notation, the dimensionless incompressible Navier–Stokes equations for this flow read
\begin{align} \dfrac {\partial u_i}{\partial x_i} = 0, \\[-28pt] \nonumber \end{align}
\begin{align} \dfrac {\partial u_i}{\partial t} + \textit{Re} \left [ U_{\!j} \dfrac {\partial u_i}{\partial x_{\!j}} + u_{\!j} \dfrac {\partial U_i}{\partial x_{\!j}} + u_{\!j} \dfrac {\partial u_i}{\partial x_{\!j}} \right ] = - \dfrac {\partial p}{\partial x_i} + \dfrac {\partial ^2 u_i}{\partial x_{\!j} \partial x_{\!j}}, \\[6pt] \nonumber \end{align}
where the Reynolds number is given by
\begin{equation} \textit{Re} = \frac {U_0 \sqrt {\nu T_0}}{\nu } = U_0 \sqrt {\frac {T_0}{\nu }}, \end{equation}
which follows the definition adopted by Biau (Reference Biau2016). We note that this definition of the Reynolds number is a factor of
$\sqrt {\pi }$
larger than that of von Kerczek & Davis (Reference von Kerczek and Davis1974), Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2014) and others, and is a factor of
$2\sqrt {\pi }$
larger than that of Blennerhassett & Bassom (Reference Blennerhassett and Bassom2002), Ramage et al. (Reference Ramage, Davies, Thomas and Togneri2020) and others; all values quoted here, including the critical linear stability threshold
$\textit{Re}_c\approx 2511$
have been converted to match (2.3). The boundary conditions are
\begin{align} u_i^{\textit{tot}} = \delta _{\textit{1i}} U(0,t) \quad &\textrm {at} \quad y = 0, \end{align}
\begin{align} u_i^{\textit{tot}} \rightarrow 0 \quad &\textrm {as} \quad y\rightarrow \infty , \end{align}
where
$\delta _{\textit{ij}}$
is the Kronecker delta.
The laminar flow
$\boldsymbol{U} = (U(y,t),0,0)$
is directed along the
$x$
-axis and varies in the vertical
$y$
-direction. The laminar flow is the well-known solution to the Stokes second problem, and is given by
\begin{equation} U(y,t) = \cos ( 2 \pi t -\sqrt {\pi } y ) e^{- \sqrt {\pi } y}. \end{equation}
The laminar flow is periodic with period
$T=1$
, and it has a (time-)shift–reflect symmetry
\begin{align} U(y,t+T/2) = -U(y,t). \end{align}
Local features of the laminar flow (such as maxima, minima, inflection points, etc.) move upward away from the wall at a constant speed of
$2\sqrt {\pi }$
.
With this laminar flow, the boundary conditions for the perturbation velocity are
\begin{align} u_i = 0 \quad &\textrm {at} \quad y = 0, \end{align}
\begin{align} u_i \rightarrow 0 \quad &\textrm {as} \quad y\rightarrow \infty . \end{align}
However, for numerical purposes a bounded domain is used, in which the streamwise
$x$
- and spanwise
$z$
-directions are periodic, and a large wall-normal extent
$L_y$
is chosen, at which impenetrable and stress-free boundary conditions are applied,
\begin{equation} v = 0 \quad \textrm {and} \quad \frac {\partial u}{\partial y} = \frac {\partial w}{\partial y} = 0 \quad \textrm {at} \quad y = L_y. \end{equation}
A schematic representation of the oscillating wall problem is presented in figure 1.
Figure 1. Problem diagram for flow above an oscillating wall. The lateral boundaries (
$x$
and
$z$
) are periodic. In the wall-normal direction, the perturbation velocity satisfies the no-slip condition, while the upper boundary is impenetrable and stress free.
2.2. Numerical implementation and set-up
We solve (2.1–2.2) for the perturbation velocity
$u_i$
with boundary conditions (2.8) and (2.10) using the direct numerical simulation solver Diablo (Taylor Reference Taylor2008). This code employs pseudospectral methods to compute spatial derivatives and uses a third-order Runge–Kutta scheme for time integration, and a 2/3 dealiasing rule is applied to deal with the nonlinear terms. The simulations are carried out in parallel using the message passing interface library.
The equations are solved in a rectangular domain of size
$L_x=L_z=8.2$
and
$L_y=10.0$
. The size of
$L_x$
(and also
$L_z$
) is chosen to match the wavenumber of the largest transiently growing linear disturbance at
$\textit{Re}=1000$
(Biau Reference Biau2016). The periodic
$x$
-and
$z$
-directions are discretised using a uniform grid with
$N_x = 64$
and
$N_z = 32$
Fourier modes, respectively. Initial tests used the same resolution as Biau (Reference Biau2016),
$N_x=N_z=128$
, but this was scaled back for efficiency since turbulent scales do not need to be resolved in order to accurately compute the much simpler edge trajectories, and there was no significant loss in accuracy found when doing so. The
$y$
-direction is discretised using
$N_y=241$
grid points, which are stretched away from the wall to ensure an efficient and accurate representation of the structures near the wall. The results are insensitive to other values of
$N_y$
around this choice. A narrower geometry with
$L_z=4.1$
and
$N_z=16$
is also briefly considered, and the resulting dynamics are discussed in the next section.
2.3. Edge tracking
Trajectories along the edge manifold may be found by ‘edge tracking’ (Toh & Itano Reference Toh and Itano2003; Skufca et al. Reference Skufca, Yorke and Eckhardt2006; Schneider, Eckhardt & Yorke Reference Schneider, Eckhardt and Yorke2007; Kreilos et al. Reference Kreilos, Veble, Schneider and Eckhardt2013), an iterative procedure in which trajectories of initial conditions either side of the edge are computed and bisected depending on how they evolve in time. To classify trajectories as leading to laminar or turbulent flow, we use the
$L_2$
-norm of the perturbation velocity field as a proxy (
$E=0$
is the laminar state by construction),
\begin{equation} E(\boldsymbol{u}) = \dfrac {1}{V_{\varOmega }}\int \limits _{\varOmega } \dfrac {1}{2} \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{u} \,\, {\textrm d}\varOmega =\dfrac {1}{V_{\varOmega }}\int \limits _{\varOmega } \dfrac {1}{2} u_i u_i \,\, {\textrm d}\varOmega , \end{equation}
where
$\varOmega = [0,L_x) \times [0,L_y) \times [0,L_z)$
and
$V_{\varOmega } = L_x L_y L_z$
. Starting with a pair of initial conditions that lead to the laminar and turbulent states, and defining suitable upper and lower thresholds on
$E(\boldsymbol{u})$
for transition to turbulence (
$E_T^*$
) and decay to the laminar state (
$E_L^*$
), we can iteratively define new initial conditions that remain near the edge for long times. As the flow is oscillatory in nature, the threshold conditions are met if the average energy of the flow remains above or below these thresholds for a defined time window
$t_{\textit{a}v\textit{g}}$
.
If at time
$t_0^n$
we have an initial condition that evolves towards turbulence,
$\boldsymbol{u}^{n,0}_T$
, and an initial condition that evolves towards the laminar state,
$\boldsymbol{u}^{n,0}_L$
, then a new initial condition at this time,
$\boldsymbol{u}^m_{\lambda }$
, is defined by linear interpolation:
\begin{equation} \boldsymbol{u}^m_\lambda = \boldsymbol{u}^{n,0}_L + \lambda _m \big(\boldsymbol{u}^{n,0}_T - \boldsymbol{u}^{n,0}_L \big). \end{equation}
Iterative bisection, enumerated by
$m$
, is performed on the parameter
$\lambda$
over the interval [0,1], which generates a set of initial conditions on the laminar and turbulent sides of the edge (
$\lambda$
is increased if the trajectory from
$\boldsymbol{u}^m_{\lambda }$
decays to the laminar state, and decreased if it transitions to turbulence). As the bisection proceeds, the trajectories on either side of the edge spend longer periods of time close to the edge and close to each other. The first pair of initial conditions used to start the bisection process consisted of
$\boldsymbol{u}^{0,0}_T$
taken from a minimal seed trajectory (for the definition of a minimal seed, see Kerswell (Reference Kerswell2018)) as it approached the edge state on the turbulent side of the edge manifold (minimal seeds in this problem will be reported at a later date), along with
$\boldsymbol{u}^{0,0}_L = \boldsymbol{0}$
, the laminar flow state. Results using an alternative initialisation with
$\boldsymbol{u}^{0,0}_T$
taken as a random sample of the turbulent flow did not show any qualitative difference; the former option for
$\boldsymbol{u}^{0,0}_T$
was chosen when generating a long edge trajectory, owing to efficiencies associated with initialising the algorithm already nearby to the edge.
Once the change in
$\lambda$
between two consecutive iterations is less than a threshold
$\Delta \lambda ^*$
, it is no longer efficient to continue bisecting between the original two initial conditions in (2.12) to track the edge. Instead, a new pair of laminar and turbulent initial conditions for use in (2.12) are generated from the trajectories on either side of the edge. Letting
$m_L$
and
$m_T$
be the number of initial conditions found on the laminar and turbulent sides of the edge, respectively,
$\boldsymbol{u}_{L}^{n,m_L}$
is the most recently found initial condition on the laminar side of the edge, with trajectory
$\boldsymbol{u}_{\textit{lam}}(t)$
, and
$\boldsymbol{u}_{T}^{n,m_T}$
is the most recently found initial condition on the turbulent side of the edge, with trajectory
$\boldsymbol{u}_{\textit{turb}}(t)$
. We then set a new initial time
$t_0^{n+1}=t^*$
and new laminar and turbulent initial conditions for use in (2.12) as
$\boldsymbol{u}_L^{n+1,0} = \boldsymbol{u}_{\textit{lam}}(t^*)$
and
$ \boldsymbol{u}_T^{n+1,0} =\boldsymbol{u}_{\textit{turb}}(t^*)$
, where
$t^*$
is the last time for which
$\vert E(\boldsymbol{u}_{\textit{lam}}(t)) - E(\boldsymbol{u}_{\textit{turb}}(t))\vert \lt \Delta E^*$
. Different averaging times and threshold values were tested and
$t_{\textit{a}v\textit{g}}=0.5$
,
$\Delta \lambda ^*=10^{-4}$
,
$\Delta E^*=10^{-6}$
,
$E_L^*=10^{-4}$
and
$E_T^*=10^{-3}$
showed to be suitable for achieving an accurate representation of flow dynamics in the vicinity of the edge. These values were determined during the initial stages of investigating the edge trajectory;
$E_T^*$
was initially set a little below the turbulent average value (which is readily estimated from a single turbulent simulation), and
$E_L^*$
was set very low, at
$10^{-8}$
. These initial values allowed for a short section of edge trajectory to be computed, after which the thresholds were adjusted for efficiency to more closely sandwich the observed edge properties. The values were regularly reviewed to ensure that they were not interfering with the computation of the edge trajectory itself. A schematic representation of the bisection and start time shifting procedure is presented in figure 2.
3. Edge tracking results
Following the procedure described in § 2.3, a set of edge trajectories was generated for the baseline simulation, whose energies
$E(\boldsymbol{u})$
are shown in figure 3. The results show that, near the edge, the energy is oscillatory with a dominant frequency of
$T/2$
caused by the back-and-forth forcing of the wall (see figure 3
b). However, the energy on the edge is not periodic (see figure 3
a), indicating that the edge state is not a simple periodic orbit, but rather a chaotic saddle, albeit of a strong oscillatory nature. A total simulated time of approximately 50 periods allows for a detailed description of the dynamics on the edge. Here we focus on
$t\gt 16$
, after the edge trajectory has settled onto the edge state.
Figure 2. Bisection algorithm scheme. Block 1 indicates that when the energy, averaged over a time window of size
$t_{\textit{a}v\textit{g}}$
, is larger than
$E_T^*$
or smaller than
$E_L^*$
, then
$\lambda$
is scaled down or up, respectively. Block 2 shows that when two consecutive values of
$\lambda$
are closer than
$\Delta \lambda ^*$
, a new starting point is chosen to reinitialise the bisection process. Block 3 shows how the new starting point is chosen. The latest turbulent and laminar trajectories remain nearby for an extended period, until the energy difference between them is larger than
$\Delta E ^*$
. The last time for which the energy difference is below
$\Delta E ^*$
is the new starting point.
Figure 3. (a) Kinetic energy of the different trajectories bisected to follow the edge. The first (transient) 16 time units of edge tracking are omitted. The red trajectories lead to turbulence and the blue ones relaminarise. The inferred edge trajectory is shown with a dashed black line. The thresholds for classifying trajectories as laminar (
$E_L^*$
) and turbulent (
$E_T^*$
) are marked with horizontal dashed lines. The time window over which the energy is averaged is shown near
$t=22$
. (b) A detailed version of (a) for
$23.5 \lt t \lt 28$
.
To provide a better understanding of the energy dynamics, an energy evolution equation is derived by projecting (2.2) onto the perturbation velocity field
$u_i$
to obtain the local kinetic energy density defined as
$e = ({1}/{2}) u_i u_i$
. The energy transport equation reads
\begin{equation} \dfrac {\partial e}{\partial t} = - u_{\!j}\dfrac {\partial p}{\partial x_{\!j}} -\textit{Re} \left [ u_i \dfrac {\partial }{\partial x_{\!j}} \left ( u_i U_{\!j} + u_{\!j} U_i + u_{\!j} u_i \right ) \right ] + \dfrac {\partial ^2 e}{\partial x_{\!j} \partial x_{\!j}} - \dfrac {\partial u_i }{\partial x_{\!j}} \dfrac {\partial u_i }{\partial x_{\!j}}. \end{equation}
Defining the average operator
$\langle \boldsymbol{\cdot }\rangle _{x_i}$
as
$\langle a \rangle _{x_i} = ({1}/{L_{x_i}}) \int _{0}^{L_{x_i}} a \, {\textrm d}x_i$
, where
$i=1,2,3$
for
$x,y$
and
$z$
, respectively, the total energy
$E$
is given by
$E = \langle e \rangle _{x,y,z}\equiv \langle e \rangle _{\varOmega }$
. Integrating (3.1) over the whole domain, taking into account the boundary conditions and the incompressibility condition (2.1) leads to the global energy balance equation
\begin{equation} \dfrac {{\textrm d}E}{{\textrm d}t} =\left \langle -\textit{Re} \, u v \dfrac {\partial U }{\partial y} \right \rangle _{\varOmega } - \left \langle \dfrac {\partial u_i }{\partial x_{\!j}} \dfrac {\partial u_i }{\partial x_{\!j}} \right \rangle _{\varOmega } \equiv \mathcal{P}(t) - \mathcal{D}(t), \end{equation}
where
$\mathcal{P}(t)$
is the instantaneous production rate, and
$\mathcal{D}(t)$
is the instantaneous dissipation rate. The production term has the general form of a stress (
$-\textit{Re} \, uv$
) acting on a strain rate (
$\partial U / \partial y$
), from which the energy transfer process from the laminar velocity gradient (induced by the oscillation) to the perturbation velocity field is clear. Furthermore, the oscillation period of
$T/2$
observed in the energy is in agreement with the period
$T$
of
$\partial U / \partial y$
provided that
$uv$
is also periodic with period
$T$
(and zero mean).
Figure 4. (a) Time-series of production rate (red) and dissipation rate (blue) for
$20\lt t\lt 30$
. (b) A detailed plot for
$21 \lt t \lt 22$
showing that within half a period, there is a production-dominated (P–D, red) stage and a dissipation-dominated (D–D, blue) stage. The right-hand axis plots the magnitude of the wall velocity
$U_b=U(0,t)$
(grey dashed). The sign of the wall acceleration,
$d|U_b(t)|/{\textrm d}t$
, is shown to correspond with the P–D and D–D stages. The times of the six snapshots in figure 5 are indicated with dotted lines.
Figure 4 plots time series of the production and dissipation, and shows that they are in phase, but that there are stages within the
$T/2$
cycle when production is dominant and stages when the dissipation is dominant. This observation is in broad agreement with previous work on transitional behaviour in the Stokes layer, which indicates a growth of flow disturbances during the deceleration phase (which is, accordingly, a production-dominated phase) and their decay during the acceleration phase (Luo & Wu Reference Luo and Wu2010; Ozdemir et al. Reference Ozdemir, Hsu and Balachandar2014). A closer comparison between these observations and the transitional regime of Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2014) is made in § 4.4. The behaviour of the edge trajectory suggests the presence of an internal self-sustained dynamics that balances the energy transfer among structures in the flow. To further understand the dynamics along the edge, figure 5 plots snapshots of
$\langle e \rangle _{x}$
on the
$y$
–
$z$
plane at times
$t_1=21.11$
,
$t_2=21.25$
,
$t_3=21.35$
,
$t_4=21.50$
,
$t_5=21.58$
and
$t_6=21.67$
, which cover a total time just over
$T/2$
.
Figure 5. (a–f) Snapshots of
$\langle e \rangle _x$
in the
$y$
–
$z$
plane for six times
$t_1=21.11$
,
$t_2=21.25$
,
$t_3=21.35$
,
$t_4=21.50$
,
$t_5=21.58$
and
$t_6=21.67$
. To the right of each snapshot is the corresponding instantaneous laminar velocity
$U$
(teal) and normalised shear
$(2\pi )^{-1/2} (\partial U /\partial y)$
(red) with black dots at its maximum and minimum. (g) Time-evolution of the average vertical distribution of energy
$\langle e \rangle _{x,z}$
and its instantaneous maximum (red dashed). (h) Average spanwise distribution of energy
$\langle e \rangle _{x,y}$
.
Figure 5(a–d) (times
$t_1$
to
$t_4$
) show a well-defined region in the
$y$
–
$z$
plane that concentrates most of the kinetic energy of the flow, which modifies its shape and migrates upwards over time. This region is located around (and just above) the instantaneous location of a local maximum in the absolute value of laminar shear,
$|\tau _{\ell }|=| \partial U / \partial y |$
, and rises with it. We label this location
$y_{\tau _{\ell }}^{\textit{min}}$
(given that
$\partial U/\partial y\lt 0$
). In figure 5(d) the region of concentrated kinetic energy reaches a height
$y\approx 2$
and begins to spread out. After
$t=t_4$
a new region of concentrated energy is formed near the wall, seen in figure 5(e,f) (times
$t_5$
and
$t_6$
), and the process repeats, although this new region aligns with the location of maximal positive laminar shear, labelled
$y_{\tau _{\ell }}^{\textit{max}}$
. Figure 5(g) plots the vertical distribution of the
$x$
–
$z$
averaged energy
$\langle e \rangle _{x,z}$
over an extended period of time, demonstrating that this basic
$T/2$
cycle repeats indefinitely, with flow structures periodically forming a little above the plate, rising, and dissipating around
$y\lesssim 3$
.
The location of the flow structures in the spanwise direction varies sporadically in time, as can be seen in figure 5(h), which shows the spanwise distribution of the
$x$
–
$y$
averaged energy
$\langle e \rangle _{x,y}$
. The time window studied in figure 5(a–f) contains flow structures located in a region around the centre of the spanwise domain, while the time window between
$t=23$
and
$t=26$
contains the same structures located around the (periodic) spanwise boundaries. This ‘jumping’ of the structure by an amount
$L_z/2$
is characteristic of a spatially localised structure which nevertheless feels the influence of its periodically located neighbours, owing to the domain size
$L_z$
being too small for the structure to evolve entirely freely, but large enough for it to appear essentially isolated for extended periods (Khapko et al. Reference Khapko, Kreilos, Schlatter, Duguet, Eckhardt and Henningson2016). These jumps are associated with elevated energy
$E$
during a single half-period
$T/2$
in figure 3, as there are essentially two structures side by side, and are distributed randomly in time. If the domain size
$L_z$
were large enough, then the jumps would presumably cease since the structure can evolve entirely independently, although this may instead induce a spanwise drift (Khapko et al. Reference Khapko, Kreilos, Schlatter, Duguet, Eckhardt and Henningson2013). Nevertheless, we shall treat the structure as an essentially isolated object in the following analysis, as this is a reasonable approximation for much of the flow evolution between jumps.
Figure 6. (a–d) Snapshots of high-speed (
$u_{S\textit{max}} = 0.5 \mbox{max} \{ u\}$
, purple) and low-speed (
$u_{S\textit{min}} = 0.5 \mbox{min} \{ u \}$
, orange) streamwise velocity isosurfaces, at times
$t=t_2$
,
$t_3$
,
$t_4$
and
$t_5$
. Streamwise-averaged perturbation energy
$\langle e \rangle _x$
is plotted on the plane
$x=0$
. The instantaneous height of maximum perturbation energy (
$y_{\textit{max}\{ \langle e \rangle _x \} }$
) is plotted on the planes
$x=0$
and
$z=L_z$
with a dotted line. The instantaneous laminar flow is plotted on the planes
$z=0$
and
$z=L_z$
(centred at
$x=L_x/2$
). (e) Time evolution of the
$x,z$
-averaged perturbation energy
$\langle e \rangle _{x,z}$
, its instantaneous global maximum (continuous line) and oblique lines (dashed) with slope
$2\sqrt {\pi }$
. Dotted blue and red lines indicate locations of zero laminar shear, labelled
$y_-$
and
$y_+$
, defined in § 4.1.
To provide a more detailed characterisation of the three-dimensional nature of these structures, figure 6(a–d) plot isosurfaces of high-speed streaks (
$u_{S\textit{max}} = 0.5 \mbox{max} \{ u \}$
) and low-speed streaks (
$u_{S\textit{min}} = 0.5 \mbox{min} \{ u \}$
) at times
$t_2$
to
$t_5$
. The
$x$
-averaged energy
$\langle e \rangle _x$
is shown on the plane
$x=0$
, and the height of maximal energy (
$y_{\textit{max}\left \{ \langle e \rangle _x \right \} }$
) is plotted on the planes
$x=0$
and
$z=L_z$
. The instantaneous laminar flow profile is shown on the planes
$z=0$
and
$z=L_z$
(shifted to be centred at
$x=L_x/2$
) for reference. These figures show that the streamwise structures are streak-like, and instantaneously (at least visually) carry the majority of the energy of the perturbed flow, as in the SSP (Hall & Smith Reference Hall and Smith1991; Waleffe Reference Waleffe1997). Indeed, the streamwise velocity is
${O}(1)$
within these streaks whilst typical cross-stream velocities are
${O}(10^{-3})$
. Within a single half-period
$T/2$
(times
$t_2$
to
$t_4$
), a single streak with
$u\gt 0$
dominates the perturbation energy. In the following half-period (
$t_5$
) the streak is replaced by one of the opposite sign (
$u\lt 0$
) and the dynamics of the new streak are essentially the same as the first. This goes some way to explaining the apparent
$T/2$
period within the energy budget; the flow itself has a period of
$T$
, but the reflection symmetry after a time
$T/2$
results in preperiodic motions that are not apparent in positive-definite quantities such as the energy.
Figure 6(e) shows the instantaneous vertical distribution of the
$x$
–
$z$
-averaged kinetic energy,
$\langle e \rangle _{x,z}$
, along with the locations of its global maximum and oblique lines of slope
$2\sqrt {\pi }$
, showing that the laminar flow propagation speed controls the spatial location of these nonlinear structures on the edge. The no-slip boundary condition (
$\boldsymbol{u}=0$
at
$y=0$
) prevents the streaks from forming at the wall, and instead they begin to form around
$y\approx 1/4$
and reach substantial amplitude at
$y \approx 1$
. As the streaks migrate upwards, they begin to lose a substantial amount of energy around
$y\approx 2$
and essentially do not propagate into
$y\gtrsim 3$
(this will be demonstrated explicitly in the following section). Although the laminar flow, and hence also the energy production rate
$\mathcal{P}$
, decays exponentially away from the wall, there is nothing inherently preventing these streaks from continuing to propagate to
$y\rightarrow \infty$
while viscously decaying through
$\mathcal{D}$
; therefore, there must be some further dynamics contained within the other flow components that essentially cut off the streaks beyond a maximum height. The exact physical mechanisms driving these observations will be explained in the following section. A supplementary movie accompanying figure 6 is available, which shows the streak dynamics and includes a jump of the streaks from the centre of the domain to the spanwise boundaries.
Figure 7. (a) Snapshot at
$t=20.35$
of high and low speed streaks, as in figure 6 for the narrower domain. (b) The corresponding perturbation energy from edge tracking.
Edge tracking was also performed at a higher Reynolds number of
$\textit{Re}=1200$
using the same computational set-up, and the dynamics were qualitatively similar to those at
$\textit{Re}=1000$
. Due to the moderate Reynolds numbers used and the small gap between them, no clear scaling of flow structures with
$\textit{Re}$
was observed. Another set of edge tracking results were computed in a narrower domain with half the spanwise extent (
$L_z = 4.1$
, using the same spatial and temporal resolution) for
$\textit{Re} = 1000$
, to see whether or not a truly periodic edge state could be identified. However, the edge trajectory in this case is significantly more chaotic than in the wider domain, and no period of nearly periodic motion can be identified for further analysis of the flow structures. Figure 7 shows a snapshot of the flow in the narrow domain, from which it is seen that the structures are similar to those of the wider domain discussed above. However, the flow is too confined, and alternating signed streaks ‘compete’ for space as they form and migrate upwards, and a consistent pattern of dynamics does not clearly emerge. The next section will focus entirely on the wide domain with
$\textit{Re}=1000$
and demonstrate that rolls in the
$y$
–
$z$
plane, which sustain the streamwise streaks during their migration, span the entire domain, and it is likely that these rolls do not have enough room in the narrow domain to establish a periodic sustained flow.
4. The periodic self-sustained process
To fully understand the streak dynamics, including their creation, migration, growth and sudden decay, we analyse the effect of wall-oscillation on the SSP, by decomposing the full perturbation velocity into its streak, roll and wave (SRW) components. The average of the velocity field over the
$x$
-direction,
$\boldsymbol{\mathcal{U}} = \langle \boldsymbol{u} \rangle _x$
, can be decomposed into streak and roll components as
$\boldsymbol{\mathcal{U}} = \boldsymbol{\mathcal{U}}_s + \boldsymbol{\mathcal{U}}_r \equiv (\mathcal{U},0,0)_{s} + (0,\mathcal{V},\mathcal{W})_{r}$
. The full perturbation velocity field can be expressed as
\begin{equation} \boldsymbol{u} = \boldsymbol{\mathcal{U}}_s + \boldsymbol{\mathcal{U}}_r + \boldsymbol{\hat {u}} = (\mathcal{U},0,0)_{s} + (0,\mathcal{V},\mathcal{W})_{r} + (\hat {u} , \hat {v} , \hat {w})_{w} \end{equation}
where the subscript
$s$
denotes the streak velocity, the
$r$
subscript denotes the roll velocity and the
$w$
subscript and hat decoration denotes the remaining part of the fully three-dimensional velocity field. Introducing this decomposition into (2.2), we obtain momentum equations for
$\mathcal{U}$
,
$\mathcal{V}$
and
$\mathcal{W}$
,
\begin{align} \dfrac {\partial \mathcal{U}}{\partial t} + \textit{Re} \,\, \mathcal{U}_{\!j}\dfrac {\partial \mathcal{U}}{\partial x_{\!j}} &= -\textit{Re} \left \{ \mathcal{V}\dfrac {\partial U}{\partial y} + \dfrac {\partial \left \langle \hat {u} \hat {u}_{\!j} \right \rangle _x}{\partial x_{\!j}} \right \}+ \dfrac {\partial ^2 \mathcal{U}}{\partial x_{\!j} \partial x_{\!j}}, \end{align}
\begin{align} \dfrac {\partial \mathcal{V}}{\partial t} + \textit{Re} \,\,\mathcal{U}_{\!j} \dfrac {\partial \mathcal{V}}{\partial x_{\!j}} &= - \dfrac {\partial \mathscr{P}}{\partial y} - \textit{Re} \dfrac {\partial \left \langle \hat {v}\hat {u}_{\!j} \, \right \rangle _x}{\partial x_{\!j}} + \dfrac {\partial ^2 \mathcal{V}}{\partial x_{\!j} \partial x_{\!j}}, \end{align}
\begin{align} \dfrac {\partial \mathcal{W}}{\partial t} + \textit{Re} \,\, \mathcal{U}_{\!j} \dfrac {\partial \mathcal{W}}{\partial x_{\!j}} &= -\dfrac {\partial \mathscr{P}}{\partial z} -\textit{Re} \dfrac {\partial \left \langle \hat {w} \hat {u}_{\!j} \right \rangle _x}{\partial x_{\!j}} + \dfrac {\partial ^2 \mathcal{W}}{\partial x_{\!j} \partial x_{\!j}}, \end{align}
where
$\mathscr{P} \equiv \langle p \rangle _x$
is the the
$x$
-averaged pressure. Mass conservation within the SRW decomposition reads
\begin{equation} \frac {\partial \mathcal{V}}{\partial y} + \frac {\partial \mathcal{W}}{\partial z} =0 \quad \textrm {and} \quad \frac {\partial \hat {u}_i}{\partial x_i}=0, \end{equation}
and momentum equations for the wave components
$\boldsymbol{\hat {u}}$
can be obtained by subtracting (4.2–4.4) from (2.2), though the resulting equations are not needed here.
4.1. Cycle description and energy transport dynamics
To unveil the dynamics among streaks, rolls and waves, we analyse the evolution of the energy within the SRW-decomposition. Defining the streak energy density as
$\mathcal{E}_s \equiv ({1}/{2}) \mathcal{U}^2$
, the roll energy density as
$\mathcal{E}_r \equiv ({1}/{2})(\mathcal{V}^2+\mathcal{W}^2)$
, and the
$x$
-averaged wave energy density as
$\hat {\mathcal{E}}\equiv \langle ({1}/{2}) |\boldsymbol{\hat {u}}|^2 \rangle _x$
, we have by construction that the
$x$
-averaged total perturbation energy density
$\mathcal{E}\equiv \langle e \rangle _x$
is given by
$\mathcal{E} = \mathcal{E}_s + \mathcal{E}_r + \hat {\mathcal{E}}$
, given that the cross-terms vanish. The SRW momentum equations (4.2–4.4) may be converted into evolution equations for the streak energy density and roll energy density, and an equation for the
$x$
-averaged wave energy density can be constructed by observing that
$({\partial \hat {\mathcal{E}}}/{\partial t}) = ({\partial \mathcal{E}}/{\partial t}) - ({\partial \mathcal{E}_s}/{\partial t}) -({\partial \mathcal{E}_r}/{\partial t})$
. These equations are presented in the Appendix.
The energy density evolution equations contain flux terms which move energy within the domain, in addition to production from the laminar flow, transfer between rolls, streaks and waves, and dissipation terms. To better elucidate the key mechanisms sustaining the motion, we integrate the equations over the whole domain to give evolution equations for the total streak
$E_s = \langle \mathcal{E}_s \rangle _{\varOmega }$
, roll
$E_r = \langle \mathcal{E}_r \rangle _{\varOmega }$
and wave
$\hat {E} = \langle \hat {\mathcal{E}} \rangle _{\varOmega }$
energies,
\begin{align} \dfrac {\text{d}E_s}{\text{d}t} & = \langle \mathcal{P}_{\ell \rightarrow s} \rangle _{\varOmega } - \langle \mathcal{T}_{s\rightarrow w} \rangle _{\varOmega } - \langle \mathcal{D}_{s} \rangle _{\varOmega }, \end{align}
\begin{align} \dfrac {\text{d}E_r}{\text{d}t} & = \langle \mathcal{T}_{w \rightarrow r} \rangle _{\varOmega } - \langle \mathcal{D}_{r} \rangle _{\varOmega }, \end{align}
\begin{align} \dfrac {\text{d} \hat {E} }{\text{d}t} & = \langle \mathcal{P}_{\ell \rightarrow w} \rangle _{\varOmega } + \langle \mathcal{T}_{s\rightarrow w} \rangle _{\varOmega } - \langle \mathcal{T}_{w\rightarrow r} \rangle _{\varOmega } - \langle \mathcal{D}_w \rangle _{\varOmega }. \end{align}
The production (
$\mathcal{P}$
), transfer (
$\mathcal{T}$
) and dissipation (
$\mathcal{D}$
) terms are also given in the Appendix. Their indices indicate the source and the destination of the transferred energy (in the case of
$\mathcal{P}$
and
$\mathcal{T}$
) or the component which is dissipating energy (in the case of
$\mathcal{D}$
), where the laminar flow is denoted by
$\ell$
. For example, the term
\begin{equation} \mathcal{P}_{\ell \rightarrow s}\equiv -\textit{Re} \mathcal{U}\mathcal{V} \frac {\partial U}{\partial y} \end{equation}
corresponds to the production of perturbation energy by the laminar flow, creating streaks
$\mathcal{U}$
(and mediated by the vertical roll component
$\mathcal{V}$
). This is the process that creates streaks via the so-called lift-up mechanism, which forms part of the SSP (Waleffe Reference Waleffe1997). The volume-average of each energy component in (4.6–4.8) are plotted in figure 8 along with the total energy
$E$
whose evolution in terms of these components is
\begin{equation} \frac {\text{d}E}{\text{d}t} = \mathcal{P}(t)-\mathcal{D}(t) \equiv (\langle \mathcal{P}_{\ell \rightarrow s} \rangle _\varOmega +\langle \mathcal{P}_{\ell \rightarrow w} \rangle _\varOmega ) - (\langle \mathcal{D}_{\ell } \rangle _\varOmega + \langle \mathcal{D}_{r} \rangle _\varOmega + \langle \mathcal{D}_{w} \rangle _\varOmega ). \end{equation}
Figure 8. (a) Time series of energy components
$E_{\alpha }$
:
$E$
(black);
$E_s$
(red);
$E_r$
(green);
$\hat {E}$
(blue). (b) Time series of individual production, transfer and dissipation terms (labelled
$\dot {E}_{\alpha \rightarrow \beta }$
):
$\langle \mathcal{P}_{\ell \rightarrow s} \rangle _{\varOmega }$
(light red circles);
$\langle \mathcal{P}_{\ell \rightarrow w} \rangle _{\varOmega }$
(dark red squares);
$\langle \mathcal{T}_{s \rightarrow w} \rangle _{\varOmega }$
(light green upwards triangles);
$\langle \mathcal{T}_{w \rightarrow r} \rangle _{\varOmega }$
(dark green diamonds);
$\langle \mathcal{D}_{s} \rangle _{\varOmega }$
(dark blue crosses);
$\langle \mathcal{D}_{r} \rangle _{\varOmega }$
(mid bue filled circles);
$\langle \mathcal{D}_{w} \rangle _{\varOmega }$
(light blue downwards triangles). (c,d) The same as (a) and (b) for
$20.75\leqslant t \leqslant 21.9$
. Dashed sections of
$\langle \mathcal{P}_{\ell \rightarrow w} \rangle _{\varOmega }$
are negative values.
Figure 8 clearly shows a scale difference in energy and energy transfer rate terms between rolls, streaks and waves. Figure 8(a) shows that the most energetic component is the streaks, with energy that is almost indistinguishable from the total energy. The second most energetic structures are the waves, which have an average energy around one and a half orders of magnitude smaller than the streaks, and the rolls are the least energetic structures with average energy around two orders of magnitude smaller than the streaks. This hierarchy is as expected from vortex-wave interaction theory (Hall & Smith Reference Hall and Smith1991), albeit with relative sizes that do not match the high Reynolds number asymptotic theory, owing to the relatively modest Reynolds number used here (see Hall & Sherwin Reference Hall and Sherwin2010).
Figure 8(b) shows that the scale differences persist among the energy transfer terms. The largest two terms in the energy budget are the production term transferring energy from the laminar flow to the streaks through the lift-up mechanism,
$\langle \mathcal{P}_{\ell \rightarrow s}\rangle _{\varOmega }$
, and the streak dissipation,
$\langle \mathcal{D}_{s} \rangle$
, which dissipates most of this energy, and is almost exactly in phase with and of the same magnitude as
$\langle \mathcal{P}_{\ell \rightarrow s}\rangle _{\varOmega }$
. The small amount of streak energy gained from production which is not dissipated is transferred to the waves via
$\langle \mathcal{T}_{s \rightarrow w}\rangle _{\varOmega }$
. This transfer rate is around an order of magnitude smaller than the streak production and dissipation, and is responsible for powering the smaller amplitude waves. In turn, this energy transfer to the waves is balanced almost entirely by the wave dissipation
$\langle \mathcal{D}_w\rangle _{\varOmega }$
, which is of a similar magnitude as
$\langle \mathcal{T}_{s\rightarrow w}\rangle _\varOmega$
and in phase with it. The waves receive little energy via production from the laminar flow,
$ \langle \mathcal{P}_{\ell \rightarrow w}\rangle _{\varOmega }$
. This production term oscillates somewhat randomly between positive and negative values (backscatter onto the laminar flow) and essentially averages to zero over long periods of time; the laminar flow is stable to linear waves at this Reynolds number (and in this geometry). This production term essentially plays no meaningful role in the wave dynamics as its average magnitude is around an order of magnitude smaller than either the transfer to the waves from the streaks,
$\langle \mathcal{T}_{s\rightarrow w}\rangle _\varOmega$
, or the wave dissipation,
$\langle \mathcal{D}_w\rangle _{\varOmega }$
. The waves lose a small amount of energy to the rolls, via the transfer term
$\langle \mathcal{T}_{w \rightarrow r}\rangle _{\varOmega }$
, which is also an order of magnitude smaller than either
$\langle \mathcal{T}_{s \rightarrow w}\rangle _{\varOmega }$
or
$\langle \mathcal{D}_w\rangle _{\varOmega }$
. The transfer from the waves to the rolls,
$\langle \mathcal{T}_{w \rightarrow r}\rangle _{\varOmega }$
, is of the same magnitude and nearly in phase with the roll dissipation,
$\langle \mathcal{D}_r\rangle _{\varOmega }$
.
Figure 9. Schematic representation of the energy transfer cycle for the PSSP. The input of energy comes from the laminar flow formed from the plate oscillation. This is transferred to the streaks via the lift-up mechanism due to the action of the rolls (the role of the roles is purely advective; no energy is transferred). Streaks dissipate most of the energy and what remains is transferred to the waves through linear instability. Waves dissipate most of this energy and what remains is transferred to the rolls through Reynolds stresses and the cycle repeats.
A closer visualisation of the time series is plotted in figure 8(c,d) for
$20.8\leqslant t \leqslant 22$
. In figure 8(c), it can be seen that the reduction in total streak energy leads to an increment in the total wave energy, which in turn leads to an increment in the total roll energy. The ‘cascade’ process is indicated with a tricoloured arrow in the plot, and indicates that the energy transfers first between the streaks and the waves, and second between the waves and the rolls. Furthermore, as the rolls begin to gain energy, there is a subsequent increase in the streak energy, as expected from the lift-up mechanism, and the cycle repeats. In summary, each flow component receives and loses the majority of its energy from a single source, as in the SSP of Waleffe (Reference Waleffe1997), but the periodic nature of the structures presented here more readily reveals the flow of energy through the system than in other, steady, realisations of the SSP. Additionally, it is often not made explicit that most of the energy at every step of the SSP is dissipated, with only a little being transferred to the next part of the cycle; the largest sink of energy is the streaks and the smallest is the rolls. This cycle is presented schematically in figure 9, which echoes the well-known figure of Waleffe (Reference Waleffe1997), but includes energy sinks (dissipation) and indicates the main flow of energy using different pathway thicknesses and circle sizes for each component of the flow.
This view of the global energies and energy transfer rates does not provide any information on the mechanistic flow processes and flow structures involved in this cycle. It also does not explain exactly how the flow manages to organise itself periodically, for example why the nonlinear streaks move upward at a speed dictated by the linear laminar flow, why they stop after a finite distance and how the waves and rolls manage to create further streaks close to the wall. An explanation based upon the spatial evolution of the various flow components and spatial distribution of the energy transfer rates during the cycle is needed.
4.2. Streak and roll dynamics
Figure 10 shows the distribution in the
$y$
–
$z$
plane of the energy transfer components
$\mathcal{P}_{\ell \rightarrow s}$
,
$\mathcal{D}_s$
,
$\mathcal{T}_{w\rightarrow r}$
and
$\mathcal{D}_r$
, which are responsible for supplying and dissipating energy within the streaks and rolls, at the times
$t_1$
,
$t_2$
,
$t_3$
and
$t_4$
indicated in figure 8(c,d). Overlain on each panel are instantaneous roll streamlines and contours of high streak energy (
$\mathcal{E}_s \geqslant 0.25 \max _{\varOmega } \left \{ \mathcal{E}_s \right \}$
). The laminar velocity and (scaled) shear profiles are displayed to the right of each plot to relate the stage of the PSSP with the stage of the wall oscillation. These profiles are plotted alongside a profile of the normalised strain-rate magnitude of the rolls, whose magnitude is defined as
\begin{equation} \vert \mathcal{S} \vert = \textit{Re} \sqrt {2 \left [ \left ( \dfrac {\partial \mathcal{V}}{\partial y} \right )^2 + \dfrac {1}{2} \left ( \dfrac {\partial \mathcal{V}}{\partial z} + \dfrac {\partial \mathcal{W}}{\partial y} \right )^2 + \left ( \dfrac {\partial \mathcal{W}}{\partial z} \right )^2 \right ]}, \end{equation}
which is normalised via
$\vert \mathcal{S}^* \vert = \vert \mathcal{S} \vert / \max _{\varOmega } \left \{ \mathcal{S} \right \}$
.
Figure 10. Distribution in the
$y$
–
$z$
plane of the energy transfer components
$\mathcal{P}_{\ell \rightarrow s}$
,
$\mathcal{T}_{w \rightarrow r}$
,
$\mathcal{D}_s$
and
$\mathcal{D}_r$
at times
$t_1$
to
$t_4$
, normalised by each of their maximum values across the time window
$t_1\leqslant t \leqslant t_4$
. Roll streamlines are coloured by the local roll energy. Red contour lines show high streak energy (
$\mathcal{E}_s \geqslant 0.25 \max _{\varOmega } \{ \mathcal{E}_s \}$
). Shaded production windows (PWs) are shown for negative and positive streaks (PW
$_-$
, blue; PW
$_+$
, pink). To the right of the panels are the laminar velocity (teal) and normalised shear (orange) along with the profile of the normalised roll strain rate
$| \mathcal{S}^* |$
(grey dashed) defined in (4.11) through the roll stagnation point, labeled by
$y_{\textit{SP}}$
and a filled white circle.
The roll streamlines in figure 10 show that the overall shape of the rolls is persistent throughout the time interval
$t_1\leqslant t \leqslant t_4$
, forming recirculation cells of
${O}(1)$
size. These cells only slightly modify their shape over time, remaining located in the same areas throughout the cycle; their magnitude alone oscillates in time. They form a narrow channel around
$z\approx L_z/2$
of upward roll velocity, and the streaks form, grow and migrate upward in this channel. The channel ends at a stagnation point (in the
$y$
–
$z$
plane) around
$y=y_{\textit{SP}}\lesssim 3$
which appears to present a barrier for the upward migrating streak. The roll velocity is horizontal away from the stagnation point at
$(y,z)\approx (y_{\textit{SP}},L_z/2)$
and gradually turns upward, or downward back towards the wall, at the spanwise extremities of the domain.
It is apparent from the roll structure that the streaks reside largely in a region for which
$\mathcal{V}\gt 0$
around
$z=L_z/2$
below the stagnation point at
$y=y_{\textit{SP}}$
. In order for the streak production rate
$\mathcal{P}_{\ell \rightarrow s} = -\textit{Re} \mathcal{U}\mathcal{V} (\partial U / \partial y)$
to be positive in this region, we therefore require that the combination
$\mathcal{U}(\partial U / \partial y) \lt 0$
since the sign of
$\mathcal{V}$
is fixed. As such, a positive streak (
$\mathcal{U}\gt 0$
) can only grow when the laminar shear is negative (
$\partial U/\partial y\lt 0$
) and vice versa. This observation, along with the fact that the laminar flow
$U(y,t)$
is periodic, and that its features propagate upward at a speed
$2\sqrt {\pi }$
, goes some way to explaining the streak migration.
Figure 11. Schematics of four different stages of the streak development and upward migration at the centreline between the roll recirculation cells. Light pink and blue represent PWs, regions of the flow where the production term
$\mathcal{P}_{\ell \rightarrow s}$
is positive. In darker pink and blue, streak development bands (SDBs) are presented. These are subregions of the PWs where
$\mathcal{P}_{\ell \rightarrow s}$
is of substantial size, and therefore regions where streaks develop. The PWs and SDBs migrate upwards at a speed of
$2\sqrt {\pi }$
according to the locations of
$y_{\pm }(\xi )$
defined in the text and can sustain positive (PW
$_{+}$
, SDB
$_{+}$
,
$\mathcal{U}\gt 0$
) and negative (PW
$_{-}$
, SDB
$_{-}$
,
$\mathcal{U}\lt 0$
) streaks. In (a), there is a PW
$_-$
between
$y_+$
and
$y_{\textit{SP}}$
which is intentionally omitted for clarity of the discussion.
To make this notion precise, we detail key features of the laminar profile. Heights with zero laminar shear are denoted
$y_\pm$
such that the laminar shear transitions from negative to positive through
$y_+$
(i.e.
$\partial ^2 U / \partial y^2|_{y=y_+}\gt 0$
) and vice versa. Locations of maximum and minimum laminar shear are denoted
$y^*_{\pm }$
. This notation is illustrated in figure 11. We introduce the cycle time
$0\leqslant \xi \lt 1$
, where the total time is
$t=kT+\xi$
with
$T=1$
and
$k$
an integer, and consider the evolution of
$y_\pm (\xi )$
and
$y_\pm ^*(\xi )$
. A laminar shear minima appears at the wall when
$\xi =0$
and for
$0\leqslant \xi \lt 1$
is located at
$y_{-}^{*}(\xi )=2\sqrt {\pi }\xi$
. After half a cycle, at
$\xi =T/2=1/2$
, a laminar shear maxima appears at the wall and for
$1/2\leqslant \xi \lt 1$
is located at
$y_{+}^*(\xi )=\sqrt {\pi }(2\xi -1)$
. Above the wall at the beginning of the cycle, the laminar shear changes sign at
$y_+=3\sqrt {\pi }/4$
. As time advances, this location of zero shear is given by
$y_+(\xi ) = \sqrt {\pi }(2\xi +3/4)$
. When
$\xi =1/8$
, another location where the laminar shear changes sign appears at the wall, and for
$1/8\leqslant \xi \lt 1$
is located at
$y_-(\xi )=\sqrt {\pi }(2\xi -1/4)$
. When
$\xi = 5/8$
, a final zero shear location appears at the wall and its location is given by
$y_+(\xi ) = \sqrt {\pi }(2\xi - 5/4)$
.
These locations are illustrated at four times within the cycle in figure 11. They divide the region around
$z\approx L_z /2$
below the stagnation point into locations where the production term
$\mathcal{P}_{\ell \rightarrow s}$
can create positive streaks (
$\partial U/\partial y\lt 0$
, coloured pink) and locations where it can create negative streaks (
$\partial U / \partial y\gt 0$
, coloured blue). These regions are also overlain on the energy transfer term distributions and laminar flow profiles in figure 10. The schematics in figure 11 make a further distinction within each region; for a substantial production term we require that both
$\partial U/\partial y$
and
$\mathcal{V}$
are large enough to provide a meaningful energy input to the streaks. We therefore expect to have reduced streak growth around
$y=y_\pm$
(where
$\partial U/\partial y$
is small) and around
$y=0$
(where
$\mathcal{V}$
is small). This defines SDBs which are indicated with a darker shade in figure 11.
To relate this scheme to the dynamics observed in figure 10, at the beginning of the cycle (
$\xi =0$
) the region
$0\lt y\lt y_+(0)=3\sqrt {\pi }/4$
has negative laminar shear
$\partial U / \partial y\lt 0$
and positive vertical roll velocity
$\mathcal{V}\gt 0$
around
$z\approx L_z/2$
, meaning that a positively signed streak (
$\mathcal{U}\gt 0$
) grows in this region. However, due to the no-slip condition at
$y=0$
(the boundary conditions ensure that
$\mathcal{V}\propto y^2$
near the wall),
$\mathcal{V}$
is too weak near the wall to produce large values of
$\mathcal{P}_{\ell \rightarrow s}$
. The streak therefore grows to a spatial extent of
${O}(1)$
within a SDB, with its lower edge a little separated from the wall.
A little after the beginning of the cycle, at
$\xi _1=0.11$
, the laminar shear maxima
$y_{-}^{*}(\xi )$
has moved upwards to the lower edge of the streak, as shown in figure 10(a). At this time, the streak has essentially not moved from its initial position at
$\xi =0$
, but has become stronger and larger (c.f. figures 8
c and 6
e, respectively) and the production
$\mathcal{P}_{\ell \rightarrow s}$
is centred in the middle of the streak. There is substantial dissipation
$\mathcal{D}_s$
in a ring around the streak, owing to increased streak velocity gradients.
The total streak dissipation is lower than the total production (
$\langle \mathcal{D}_s \rangle _\varOmega \lt \langle \mathcal{P}_{\ell \rightarrow s} \rangle _\varOmega$
) for
$\xi _1\leqslant \xi \leqslant \xi _2 = 0.25$
and so the streak continues to grow in magnitude during this period. However, it starts to migrate upwards, as do the concentrations of
$\mathcal{D}_s$
and
$\mathcal{P}_{\ell \rightarrow s}$
, as can be seen in figure 10(b). After
$\xi =1/8$
, the region
$0\leqslant y \lt y_-(\xi )$
has positive laminar shear
$\partial U/\partial y\gt 0$
and positive vertical roll velocity
$\mathcal{V}\gt 0$
, so that any positively signed streak velocity
$\mathcal{U}\gt 0$
in this region experiences negative production
$\mathcal{P}_{\ell \rightarrow s}$
, i.e. the lift-up mechanism acts to destroy the streak in
$0\leqslant y \lt y_-(\xi )$
. This ultimately means that the streak cannot exist (with a significant amplitude for any significant period of time) outside of
$y_-(\xi )\lt y\lt y_+(\xi )$
. This region with positive production migrates upward at speed
$\text{d} y_- / \text{d} \xi = \text{d} y_+ / \text{d} \xi = 2\sqrt {\pi }$
, and hence so does the streak. Interestingly, the streak advection term
$\textit{Re} \mathcal{V}(\partial \mathcal{U} / \partial y)$
in (4.2) has an advection velocity
$\textit{Re} \mathcal{V}$
around two to three times larger than
$2\sqrt {\pi }$
within and around the streak (depending on the time during the cycle), so that the streak is effectively hindered from migrating upward with the roll advection since any part of the streak crossing the upper edge of the SDB at
$y_+$
is subject to destruction from
$\mathcal{P}_{\ell \rightarrow s}\lt 0$
.
As this upward migration proceeds, the magnitude of the laminar shear decreases exponentially in time; within
$y_-(\xi )\lt y\lt y_+(\xi )$
, we have
$|\partial U/\partial y| \leqslant \sqrt {\pi }e^{-2\xi }$
. The streak energy saturates at a time
$\xi _2\lt \xi \lt \xi _3 = 0.35$
(see figure 8
c) and at
$\xi _3$
the total streak dissipation is larger than the production (
$\langle \mathcal{D}_s\rangle _\varOmega \gt \langle \mathcal{P}_{\ell \rightarrow s} \rangle _{\varOmega }$
). Throughout the period of growth and uplift,
$\xi _1\lt \xi \lt \xi _3$
, the centre of the streak and its production remain above the location of the minimum laminar shear,
$y_{-}^{*}(\xi )$
. Instead, the streak centres itself in the middle of the production region
$y_-(\xi )\lt y\lt y_+(\xi )$
. This can be seen in figure 10(b,c) in which
$y_{-}^{*}(\xi )$
is located around a quarter of the way up the streak (it is straightforward to show that
$y_+-y_- = 4(y^*-y_-)$
).
However, the ultimate fate of the streak is not to be gradually lost to dissipation while continuing to move upwards as the production rate continues to decrease exponentially. Instead, the upward migration of the PW
$y_-(\xi )\lt y\lt y_+(\xi )$
eventually moves the centre of the streak on top of the roll velocity stagnation point, see figure 10(d) at
$\xi _4=0.5$
. The streak is essentially ripped apart in the stagnation point due to the high strain rate in this region (a peak of global strain-rate magnitude is observed around the stagnation point in figure 10
a–d and its magnitude is around 15 when the streak crosses it), substantial wave activity is created (see figure 8
c,d), and the local streak dissipation rapidly destroys what remains. Meanwhile, a new streak is formed close to the wall, below
$y_-(\xi _4)=3\sqrt {\pi }/4$
, in which the laminar shear is positive (
$\partial U / \partial y \gt 0$
) and the vertical roll velocity is (still) positive (
$\mathcal{V}\gt 0$
), and so the production term
$\mathcal{P}_{\ell \rightarrow s}$
produces a negatively signed streak in this region (
$\mathcal{U}\lt 0$
). The cycle then repeats itself during
$0.5\leqslant \xi \lt 1$
, with the sign of the streak reversed from the description above.
This description of the streak cycle requires that the roll velocity magnitude is sufficient for the lift-up mechanism to act against the laminar shear in the correct way. The rolls are sustained by energy transfer from the waves (
$\mathcal{T}_{w \rightarrow r}$
), and dissipate energy in regions where vertical and spanwise velocity gradients are concentrated (
$\mathcal{D}_r$
). Figure 10(a–c) show that the roll dissipation is concentrated around the stagnation point in a ‘figure-eight’ pattern throughout most of the cycle. This is the location where the streamlines change direction from vertical to horizontal most rapidly, and therefore where the largest gradients are expected to be located. The magnitude of the dissipation decreases from
$\xi _1$
to
$\xi _4$
. At the end of the (single) streak cycle,
$\xi =\xi _4=0.5$
, there is also dissipation in the area between the old streak and the new streak, within which
$\mathcal{W}$
changes sign. This dissipation region is also present at time
$\xi _1$
, between the previous streak and the new streak that has just developed.
Just after the beginning of the cycle, at
$\xi _1$
, there is a peak in the transfer from the waves to the rolls,
$\mathcal{T}_{w \rightarrow r}$
(figure 10
a). This is from wave activity associated with the final destruction of the previous streak at the stagnation point. As such, the energy transfer from waves to rolls is concentrated around the stagnation point, and acts primarily to accelerate the rolls horizontally (
$\mathcal{W}$
) away from the stagnation point. This energy transfer takes the form of a wave Reynolds stress (see the Appendix), and a detailed description of which terms contribute to this Reynolds stress is given in the next section. Mass conservation ensures that this local forcing enhances the rotation of the entirety of the cells, and concentrates the streamlines within the stagnation point. At the intermediate times
$\xi _2$
and
$\xi _3$
, the energy transfer from the waves decays (figure 10
b,c) which leads to the streamlines receding from the stagnation point, and expanding the dissipation pattern away from it.
By the end of the cycle at
$\xi =\xi _4$
, the total energy transfer from the waves to the rolls,
$\langle \mathcal{T}_{w \rightarrow r} \rangle _\varOmega$
, is halfway to its maximum value (see figure 8
d), and its distribution starts resembling that observed at time
$\xi _1$
(see figure 10
d). This reflects the fact that the destruction of the old streak is underway by time
$\xi _4$
, and waves are already being produced.
4.3. Waves
To complete the description of the PSSP, the distribution of the wave energy density,
$\hat {\mathcal{E}}$
, the transfer from the streaks to the waves,
$\mathcal{T}_{s\rightarrow w}$
, and the wave dissipation,
$\mathcal{D}_w$
, must be explored. At high Reynolds number, wave energy, and its transfer terms, are expected to concentrate along lines of constant total streamwise velocity,
$U+\mathcal{U} = c$
, where
$c$
is the (real) phase speed of the waves (so-called critical layers). Figure 12 shows the wave energy components
$\hat {\mathcal{E}}$
,
$\mathcal{T}_{s\rightarrow w}$
,
$\mathcal{T}_{w\rightarrow r}$
and
$\mathcal{D}_w$
at the same times
$t_1$
to
$t_4$
along with contour lines of constant total streamwise velocity (critical layer candidates). The timing and spatial distribution of the wave energy and its transfer terms agree with the dynamics deduced when analysing the rolls. In particular, the wave energy
$\hat {\mathcal{E}}$
is high during the streak destruction period of the cycle (figure 12
a,d), and the energy concentrates around the stagnation point where this destruction occurs. As expected, the wave dissipation
$\mathcal{D}_w$
is also high during these periods and overlaps spatially with the wave energy. A closer inspection of figure 12(c,d) indicates that, in the lead-up to the streak destruction, the wave energy concentrates in the region between the upper part of the streak and the stagnation point. This indicates that the streak begins to lose stability (to waves) before it completely overlaps with the region of largest strain rate.
Figure 12. Distribution in the
$y$
–
$z$
plane of the energy transfer components
$\mathcal{T}_{s\rightarrow w}$
,
$\mathcal{T}_{w \rightarrow r}$
,
$\hat {\mathcal{E}}$
and
$\mathcal{D}_w$
at times
$t_1$
to
$t_4$
, normalised by each of their maximum values across the time window
$t_1 \leqslant t \leqslant t_4$
. Red contour lines show high streak energy (
$\mathcal{E}_s\geqslant 0.25 \max _\varOmega \{ \mathcal{E}_s \}$
). Dotted lines ranging from orange to brown are contour lines of total streamwise velocity
$U+\mathcal{U}=c$
for
$-0.1\leqslant c \leqslant 0.1$
. The stagnation point is indicated with a filled white circle.
Figure 13. (a i, b i) A reproduction of figure 12(a) at
$t=t_1$
of (a)
$\mathcal{T}_{s\rightarrow w}$
and (b)
$\mathcal{T}_{w \rightarrow r}$
in more detail around the stagnation point. The thicker dotted line corresponds
$c=-0.075$
. (a ii, b ii) Vertical profiles of the SRW Reynolds stress terms that contribute to the energy transfer terms at the three spanwise locations
$z_L$
,
$z_C$
and
$z_R$
shown in (a i) and (b i).
Since the laminar flow decays exponentially away from the wall (
$|U|\leqslant 0.1$
for
$y\gtrsim 1.3$
), but the streaks are
${O}(1)$
before they dissipate, contours of total streamwise velocity
$U+\mathcal{U}$
in the vicinity of the streak are strongly shaped by the streak velocity
$\mathcal{U}$
and are similar to contours of the streak velocity alone for
$y\gtrsim 2$
. During the middle part of the cycle, when the streak is rising from its initial position towards the stagnation point (figure 12
b,c) this results in open contours of
$U+\mathcal{U}$
bending up and around the streak. However, at the end of the cycle (and beginning of the next cycle), a region of closed contours forms around the stagnation point (figures 12
d and 12
a). The waves are most active during this time, and so they become ‘trapped’ in this region by the closed total streamwise velocity contours. This trapping causes the transfer to the rolls,
$\mathcal{T}_{w\rightarrow r}$
, to be focused around the stagnation point.
Figure 13 shows the two transfer terms,
$\mathcal{T}_{s\rightarrow w}$
and
$\mathcal{T}_{w\rightarrow r}$
, in more detail around the stagnation point when the waves are most active at time
$t_1$
. The transfer from the waves to the rolls,
$\mathcal{T}_{w\rightarrow r}$
, is focused around the contour
$c=-0.075$
, and so the waves are essentially stationary during the relatively short time that they are active, since they would otherwise take a time
$L_x/c \approx 110$
to traverse the streamwise length of the domain. Figure 8(d) shows that
$\mathcal{T}_{s \rightarrow w}$
and
$\mathcal{T}_{w \rightarrow r}$
are in phase, so that the energy transfer from streak to waves and then from waves to rolls takes place almost simultaneously within this closed contour region. However, figure 13 shows that the spatial distribution of the two terms is different; transfer to the waves (
$\mathcal{T}_{s\rightarrow w}$
) is centred in a region just below the stagnation point, but transfer to the rolls (
$\mathcal{T}_{w\rightarrow r}$
) ejects fluid horizontally away from the stagnation point, as discussed above.
To complete the picture of the wave dynamics, figure 13 also plots vertical profiles at three spanwise (
$z$
) locations (through the stagnation point,
$z_C$
, to its left,
$z_L$
, and to its right,
$z_R$
) of the streak and roll velocities, along with wave Reynolds stress divergence terms. Figure 13(a) shows that, in the centre of the streak, large streak velocity combines with the spanwise gradient of the spanwise–streamwise wave Reynolds stress (
$\partial \langle \hat {u} \hat {w} \rangle / \partial z$
) to control the energy transfer from streaks to waves,
$\mathcal{T}_{s \rightarrow w}$
(see the profiles for
$z=z_C$
). Physically, this represents waves extracting streamwise momentum from the streaks and redistributing it into spanwise wave momentum. The profiles observed at
$z=z_L$
and
$z=z_R$
in figure 13(b) show that the spanwise roll velocity
$\mathcal{W}$
combines with the spanwise gradient of the spanwise–spanwise wave Reynolds stress (
$\partial \langle \hat {w} \hat {w} \rangle / \partial z$
) to control
$\mathcal{T}_{w \rightarrow r}$
. This represents waves depositing spanwise momentum back into the mean flow (in this case, the rolls). In particular, although the streak velocity
$\mathcal{U}$
changes sign every half-period and the symmetries of the (linearised) wave equations ensure that
$ \langle \hat {u} \hat {w} \rangle$
changes sign also, the fact that the transfer from waves to rolls,
$\mathcal{T}_{w \rightarrow r}$
, is dominated by the sign-definite term
$\langle \hat {w} \hat {w} \rangle$
means that the roll velocity
$\mathcal{W}$
does not change sign, and by extension neither does
$\mathcal{V}$
due to continuity, resulting in the rolls maintaining their flow direction instead of reversing every half-period. This description helps to clarify how the waves mediate the transfer of energy from the streaks to the rolls, allowing the latter to persist and drive the cycle via the lift-up mechanism.
Figure 14. (a) Average shear stress
$\langle \tau \rangle _{x,z}$
at the wall due to the laminar (orange), streak (red) and roll (teal) flows for
$20 \lt t \lt 30$
. Dashed lines indicate negative values. (b) Power of energy transfer from the plate to the fluid due to the laminar (orange) and streak (red) flows. Dashed, negative values indicate that the plate is transferring energy to the fluid.
4.4. Wall stresses
The PSSP dynamics described above have an effect on the shear stresses
$\langle \tau \rangle _{x,z}$
exerted on the wall by the flow. Figure 14(a) plots time series of the average laminar, streak and roll shear stress at
$y=0$
, showing that these three components of the total shear stress are separated by four orders of magnitude, with the laminar shear stress being larger than the streak shear stress, which is in turn larger than the roll shear stress. The streak shear stress is almost periodic, alternating between positive and negative values with a period of
$T$
and a phase-shift of approximately
$T/2$
with respect to the laminar shear stress. The roll shear stress is irregular and remains negative for most of the time window.
A closer inspection of the phase-shift between the laminar and streak-induced shear stresses in figure 14(a) reveals two types of motion within the edge dynamics. During
$20\lesssim t \lesssim 22$
and
$23\lesssim t \lesssim 25$
, the phase-shift between the two shear stresses is close to
$T/2$
. At the other times in the figure, the streak-induced shear stress shifts slightly earlier by around
$T/8$
and changes shape; rather than being nearly symmetric around each local peak value, the peaks become skewed to earlier times. Two such features, at
$t\approx 22.5$
and
$t\approx 25.5$
correspond to jumps of the structure by
$L_z/2$
. However, the period
$26\lesssim t \lesssim 30$
contains no such jumps and yet contains the same shear stress pattern. This period is associated with slightly elevated total perturbation energy (c.f. figures 3
a and 5
g,h) and streak and wave energies (c.f. figure 8
a and the supplementary movie). The effect of this
$T/8$
shift during this more energetic period, though small, is slightly delayed total shear stress peaks, along with slightly elevated shear stresses leading up to and during the laminar shear stress reversals. Within transitional turbulence, which is significantly more energetic than these edge dynamics, Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2014) observe more exaggerated versions of these two features when
$\textit{Re} \approx 1063$
. This suggests that the dynamics presented herein may relate directly to some key processes of transitional turbulence, though future detailed investigation of this hypothesis is necessary.
Figure 15. Power of energy transfer from the plate to the fluid for
$20 \lt t \lt 21$
(indicated by
$0\lt \xi \lt 1$
with
$k=20$
) for the laminar (yellow) and streak (red) flows. Negative values indicate energy transfer from the plate to the fluid. At the bottom of the plot, the sketches show the direction of the plate velocity and the laminar shear force exerted on the plate.
To analyse the influence of the PSSP on the energy input needed to maintain the plate oscillation, figure 14(b) shows the non-dimensional power per unit area of the laminar and perturbation velocity fields at
$y=0$
. These powers are given by
\begin{align} \dfrac {\boldsymbol{P}_{\!U}}{\rho \nu U_0} &\equiv \boldsymbol{P}^*_U = \left \langle \textit{Re} \,U \left . \dfrac {\partial U}{ \partial y} \right |_{y=0} \right \rangle _{x,z} \equiv \left. Re U \frac{\partial U}{\partial y} \right|_{y=0}, \end{align}
\begin{align} \dfrac {\boldsymbol{P}_{\mathcal{U}}}{\rho \nu U_0} &\equiv \boldsymbol{P}^*_{\mathcal{U}} = \left \langle \textit{Re} \, U \left . \dfrac {\partial u}{ \partial y} \right |_{y=0} \right \rangle _{x,z} \equiv \left \langle \textit{Re} \, U \left . \dfrac {\partial \mathcal{U}}{ \partial y} \right |_{y=0}\right \rangle _{z}, \end{align}
where
$\boldsymbol{P}_{U}$
and
$\boldsymbol{P}_{\mathcal{U}}$
are the dimensional power associated with the laminar and the perturbation streak flows, respectively. The figure shows that both the laminar and the streak powers are periodic with a period of
$T/2$
. Similarly to their shear stresses, the laminar and streak power are out of phase with each other.
Figure 14(b) also shows that positive and negative power values are different in magnitude; the laminar power has larger negative peaks than positive peaks and so is a net consumer of energy, whereas the streak power has larger positive peaks than negative peaks and so is a net contributor of energy to the plate motion. The back-and-forth motion of the plate must overcome the inertia of the fluid in order to repeatedly reverse its direction, and so the average energy transfer over each cycle must be from the plate to the fluid. However, there are periods within each cycle during which energy is instantaneously transferred back to the plate, as the inertia of the fluid drives the plate in the same direction as it travels. These windows are short; the laminar flow shear force
$F_{\tau _\ell }$
is in the same direction as the plate velocity
$U$
for
$1/8\lt \xi \lt 1/4$
and
$5/8\lt \xi \lt 3/4$
(for a total of
$1/4$
of the whole cycle), as shown in figure 15. Furthermore, the power of energy transfer from the laminar flow back to the plate is substantially smaller than during the rest of the cycle when the laminar flow gains energy from the plate. On the other hand, the perturbation streak flow is a net contributor to the plate energy (albeit at a substantially lower power), reducing the total energy consumption. The relative timing during the cycle means that the streak inertia contributes most to reducing the power transfer from the plate when the laminar shear force is most strongly opposed to the plate motion.
5. Conclusions
In this work, we characterised and analysed the flow structures of an edge state in the oscillatory SBL. We used direct numerical simulations to perform edge tracking of the manifold that separates flow trajectories which either relaminarise or become turbulent. The edge was tracked for approximately 40 diffusive time units for
$\textit{Re} = 1000$
. The edge dynamics are organised into streak-like structures of size
${O}(1)$
, which originate a little above the oscillating wall and migrate upward at a speed of
$2\sqrt {\pi }$
(the phase speed of travelling features of the laminar flow), dissipating around
$y \lesssim 3$
. This behaviour is repeated cyclically (twice within each wall-oscillation cycle), and results in alternating production- and dissipation-dominated phases within each period. The formation and upward motion of the structures occurs predominantly during the wall deceleration phase, and their loss of coherence and eventual dissipation near
$y \lesssim 3$
occurs during the acceleration phase. These structures are isolated in the spanwise direction for much of the edge state trajectory, sporadically duplicating and occupying the entire spanwise extent of the flow domain before a new spanwise localised state emerges. We focussed on the localised flow periods in order to understand and describe the dynamics of the edge.
We performed a SRW decomposition of the velocity field in order to interpret the cycle using flow components which comprise the well-known SSP (Hall & Smith Reference Hall and Smith1991; Waleffe Reference Waleffe1997), and in doing so detailed a periodic, spatially evolving version of the typically steady or confined SSP. The rolls are composed of four counter-rotating recirculation cells of diameter
$\approx L_z/2$
that converge into a stagnation point at a height
$y_{\textit{SP}}\lesssim 3$
. The rolls are persistent, only slightly modifying their shape over the cycle, and this relatively unchanging flow component allows for the so-called PSSP to be generated.
The stages of the cycle can be summarised as follows: (i) at the beginning of the cycle, the rolls lift high-velocity fluid near the oscillating wall upward to create a large velocity defect, creating and sustaining the streaks; (ii) once a streak has formed, it can only continue to grow and exist within a region having the same sign of laminar shear as when it was created, and regions of constant laminar shear sign migrate upward at a speed of
$2\sqrt {\pi }$
, thus controlling the upward migration of the streaks; (iii) the upward moving streaks eventually reach the stagnation point at
$y_{\textit{SP}}\lesssim 3$
and cannot be transported any further owing to a reversal in the sign of the streak production rate, and are instead torn apart by the stagnation point, dissipating while transferring a small amount of energy to the waves; (iv) lines of constant total streamwise velocity (potential critical layers) trap the waves in the vicinity of the stagnation point, and the waves transfer a small amount of energy to the rolls via a Reynolds stress directed in the spanwise direction away from the stagnation point; (v) mass conservation within the roll system ensures that the upwards roll velocity where the streaks are created maintains its energy against dissipation, thus sustaining the cycle by creating a new (oppositely signed) streak. The dynamics of the edge at
$\textit{Re} = 1200$
are qualitatively similar to those at
$\textit{Re} = 1000$
, although no clear scaling between the two Reynolds numbers was observed, due to their modest size and small separation. Additionally, edge dynamics in a narrower domain at
$\textit{Re} = 1000$
were somewhat chaotic due to their confinement, although qualitatively similar structures were observed.
Finally, a brief analysis of the effect of the edge dynamics on the wall shear stress and power consumption was performed. The shear stress exerted by the streaks contributes (slightly) to reducing the energy needed to move the plate, and this effect occurs when the laminar flow is extracting energy from the plate with the most power.
The PSSP unveils a fundamental mechanism by which streaks may be raised away from the laminar boundary layer, growing during the deceleration phase and with a phase-lag to the laminar flow. Were this process to occur at a slightly larger amplitude away from the edge state, then presumably adjacent streaks would readily combine into hairpin vortices and deliver spanwise ejections into the free stream which break down to turbulence, as in the observed bypass transition process. This work therefore provides evidence towards a key component of bypass transition, that of streak growth and regeneration, being fundamentally tied to physics within the boundary layer itself and its self-contained interaction with the background laminar flow. Confirmation of this would require an analysis of the linear stability properties of the edge trajectory, or the computation of its optimally growing disturbances (see, e.g. Andersson, Berggren & Henningson Reference Andersson, Berggren and Henningson1999; Luchini Reference Luchini2000; Cherubini & De Palma Reference Cherubini and De Palma2015; Firano et al. Reference Firano, Cherubini, Robinet and De Palma2015). A careful analysis of such disturbances for this fully three-dimensional, time-varying edge trajectory represents a significant future endeavour. We also note that the streaks presented here bear some resemblance to those found in other oscillating shear flows, particularly large laminar separation bubbles (Gaster Reference Gaster1966; Pauley, Moin & Reynolds Reference Pauley, Moin and Reynolds1990) and structures associated with their slow mode of oscillation (Cherubini et al. Reference Cherubini, Robinet and De Palma2010 a, Reference Cherubini, Robinet, De Palma and Alizard2010 b; Rodríguez et al. Reference Rodríguez, Gennaro and Souza2021; Verdoya et al. Reference Verdoya, Dellacasagrande, Lengani, Simoni and Ubaldi2021; Malmir et al. Reference Malmir, Di Labbio, Le Floc’h, Dufresne, Weiss and Vétel2024). However, a direct comparison between these objects is not easily made within the current work.
It remains to be seen how the structures in the PSSP scale with Reynolds number and to what extent the structures in fully developed Stokes-layer turbulence resemble the edge state PSSP described herein. Repeated, extensive attempts were made to converge dynamics on the edge to a periodic orbit using a Newton–GMRES–hookstep solver but to no avail, potentially suggesting that any coherent edge state structures are quasiperiodic. Though computationally expensive, a fully localised edge trajectory in a large domain should be sought. Not only would such a structure be most physically relevant, it may also represent truly periodic behaviour if the quasiperiodicity of the results presented here owe their origin to interactions between neighbouring streaks through the periodic boundaries. Nevertheless, the current identification and full description of periodic self-sustaining motion within the oscillating SBL sets the groundwork for interpreting its structural dynamics alongside better studied steady or confined shear flows.
Supplementary movie
Supplementary movie is available at https://doi.org/10.1017/jfm.2025.10725
Acknowledgements
Dr D. Lucas is thanked for several insightful conversations about the usage of his Newton–GMRES–hookstep solver. Three anonymous reviewers are thanked for their comments and contributions towards the improvement of the manuscript.
Funding
This research was funded by an EPSRC New Investigator Award (EP/W021099/1).
Author contributions
Both authors contributed to all aspects of the manuscript.
Data availability statement
The initial condition and grid file used for edge tracking are available at https://doi.org/10.15132/10000267. The edge tracking extension of Diablo is available at https://github.com/jsandoval001/Diablo_edge_tracking.
Declaration of interests
The authors report no conflict of interest.
Appendix. Energetics of the SRW decomposition
Projecting the streak momentum equation (4.2) onto the streak velocity field
$\mathcal{U}$
, we obtain
\begin{equation} \dfrac {\partial \mathcal{E}_s}{\partial t} + \textit{Re} \mathcal{U}_{\!j} \dfrac {\partial \mathcal{E}_s}{\partial x_{\!j}} = -\textit{Re} \left ( \mathcal{U} \mathcal{V} \dfrac {\partial U}{\partial y} + \mathcal{U} \dfrac {\partial \! \left \langle \hat {u} \hat {u}_{\!j} \right \rangle _x }{\partial x_{\!j}} \right )+ \dfrac {\partial ^2 \mathcal{E}_s}{\partial x_{\!j} \partial x_{\!j}} - \dfrac {\partial \mathcal{U}}{\partial x_{\!j}} \dfrac {\partial \mathcal{U}}{\partial x_{\!j}}, \end{equation}
where
$\mathcal{E}_s = ({1}/{2})\mathcal{U}^2$
is the streak energy density. Projecting (4.3) onto
$\mathcal{V}$
and (4.4) onto
$\mathcal{W}$
, and summing the resulting equations, we obtain
\begin{align} \dfrac {\partial \mathcal{E}_r}{\partial t} + \textit{Re} \mathcal{U}_{\!j} \dfrac {\partial \mathcal{E}_r}{\partial x_{\!j}} &= -\dfrac {\partial \left ( \mathcal{U}_{\!j} \mathscr{P} \right ) }{\partial x_{\!j}} - \textit{Re} \left ( \mathcal{V} \dfrac {\partial \! \left \langle \hat {v} \hat {u}_{\!j} \right \rangle _x }{\partial x_{\!j}} + \mathcal{W} \dfrac {\partial \! \left \langle \hat {w} \hat {u}_{\!j} \right \rangle _x }{\partial x_{\!j}} \right ) \nonumber \\&\quad + \dfrac {\partial ^2 \mathcal{E}_r}{\partial x_{\!j} \partial x_{\!j}} - \dfrac {\partial \mathcal{V}}{\partial x_{\!j}} \dfrac {\partial \mathcal{V}}{\partial x_{\!j}} - \dfrac {\partial \mathcal{W}}{\partial x_{\!j}} \dfrac {\partial \mathcal{W}}{\partial x_{\!j}}, \end{align}
where
$\mathcal{E}_r = ({1}/{2}) ( \mathcal{V}^2 + \mathcal{W}^2 )$
is the roll energy density.
By construction,
$\mathcal{E} = \mathcal{E}_s + \mathcal{E}_r + \hat {\mathcal{E}}$
, where
$\mathcal{E}=\langle e \rangle _x$
is the
$x$
-averaged total energy density and
$\hat {\mathcal{E}} = \langle ({1}/{2}) |\hat {\boldsymbol{u}} |^2 \rangle _x$
is the
$x$
-averaged wave energy density, given that the cross-term
$\langle \boldsymbol{\mathcal{U}}\boldsymbol{\cdot }\hat {\boldsymbol{u}} \rangle _x$
in
$\mathcal{E}$
vanishes since
$\boldsymbol{\mathcal{U}}$
does not depend on
$x$
and
$\langle \hat {\boldsymbol{u}} \rangle _x=0$
. Therefore, the equation for the waves kinetic energy density
$\hat {\mathcal{E}}$
can be obtained from
$({\partial \hat {\mathcal{E}}}/{\partial t}) = ({\partial \mathcal{E}}/{\partial t}) -({\partial \mathcal{E}_s}/{\partial t}) -({\partial \mathcal{E}_r}/{\partial t})$
, which leads to
\begin{align} \dfrac {\partial \hat {\mathcal{E}}}{\partial t} + \textit{Re} \mathcal{U}_{\!j} \dfrac {\partial \hat {\mathcal{E}}}{\partial x_{\!j}} &= - \dfrac {\partial \langle \hat {u}_{\!j} \hat {p} \rangle _x}{\partial x_{\!j}} -\textit{Re} \left [ \langle \hat {u}\hat {v} \rangle _x \dfrac {\partial U }{\partial y} + \langle \hat {u}_i \hat {u}_{\!j} \rangle _x \frac {\partial \mathcal{U}_i}{\partial x_{\!j}} + \left \langle \hat {u}_i \dfrac {\partial ( \hat {u}_i \hat {u}_{\!j} )}{\partial x_{\!j}} \right \rangle _x \right ] \nonumber \\ &\quad+\dfrac {\partial ^2 \hat {\mathcal{E}}}{\partial x_{\!j} \partial x_{\!j}} - \left \langle \dfrac {\partial \hat {u}_i }{\partial x_{\!j}} \dfrac {\partial \hat {u}_i }{\partial x_{\!j}} \right \rangle _x . \end{align}
Integrating these equations over the whole domain gives evolution equations for the total streak
$E_s \equiv \langle \mathcal{E}_s \rangle _\varOmega$
, roll
$E_r \equiv \langle \mathcal{E}_r \rangle _\varOmega$
and wave
$\hat {E}\equiv \langle \hat {\mathcal{E}} \rangle _\varOmega$
energies, and defines production (
$\mathcal{P}$
), transfer (
$\mathcal{T}$
) and dissipation (
$\mathcal{D}$
) rates associated with the laminar (
$\ell$
), streak (
$s$
), roll (
$r$
) and wave (
$w$
) components of the flow,
\begin{align} \dfrac {\text{d}E_s}{\text{d}t} & = \Bigg \langle \underbrace {- \textit{Re} \,\mathcal{U} \mathcal{V} \dfrac {\partial U }{\partial y}}_{\mathcal{P}_{\ell \rightarrow s}} \Bigg \rangle _\varOmega - \Bigg \langle \underbrace {\textit{Re} \, \mathcal{U} \left ( \dfrac {\partial \langle \hat {u}\hat {v} \rangle }{\partial y} + \dfrac {\partial \langle \hat {u}\hat {w} \rangle }{\partial z} \right )}_{\mathcal{T}_{s\rightarrow w}} \Bigg \rangle _\varOmega \nonumber \\&\quad - \Bigg \langle \underbrace {\left ( \dfrac {\partial \mathcal{U} }{\partial y}\right )^2 + \left ( \dfrac {\partial \mathcal{U} }{\partial z}\right )^2}_{\mathcal{D}_{s}} \Bigg \rangle _\varOmega, \end{align}
\begin{align} \dfrac {\text{d}E_r}{\text{d}t} & = \Bigg \langle \underbrace {-\textit{Re} \left [ \mathcal{V} \left ( \dfrac {\partial \langle \hat {v}\hat {v} \rangle }{\partial y} + \dfrac {\partial \langle \hat {v}\hat {w} \rangle }{\partial z} \right ) + \mathcal{W} \left ( \dfrac {\partial \langle \hat {w}\hat {v} \rangle }{\partial y} + \dfrac {\partial \langle \hat {w} \hat {w} \rangle }{\partial z} \right ) \right ]}_{\mathcal{T}_{w \rightarrow r}} \Bigg \rangle _\varOmega \nonumber \\ &\quad - \Bigg \langle \underbrace {\left ( \dfrac {\partial \mathcal{V} }{\partial y}\right )^2 + \left ( \dfrac {\partial \mathcal{V} }{\partial z}\right )^2 + \left ( \dfrac {\partial \mathcal{W} }{\partial y}\right )^2 + \left ( \dfrac {\partial \mathcal{W} }{\partial z}\right )^2}_{\mathcal{D}_{r}} \Bigg \rangle _\varOmega , \end{align}
\begin{align} \dfrac {\text{d} \hat {E} }{\text{d}t} & = \Bigg \langle \underbrace {- \textit{Re} \,\hat {u}\hat {v} \dfrac {\partial U }{\partial y}}_{\mathcal{P}_{\ell \rightarrow w}} \Bigg \rangle _\varOmega + \Bigg \langle \underbrace {\textit{Re} \,\mathcal{U} \left ( \dfrac {\partial \langle \hat {u}\hat {v} \rangle }{\partial y} + \dfrac {\partial \langle \hat {u}\hat {w} \rangle }{\partial z} \right )}_{\mathcal{T}_{s\rightarrow w}} \Bigg \rangle _\varOmega \nonumber \\ &\quad - \Bigg \langle \underbrace {-\textit{Re} \left [ \mathcal{V} \left ( \dfrac {\partial \langle \hat {v}\hat {v} \rangle }{\partial y} + \dfrac {\partial \langle \hat {v}\hat {w} \rangle }{\partial z} \right ) + \mathcal{W} \left ( \dfrac {\partial \langle \hat {w}\hat {v} \rangle }{\partial y} + \dfrac {\partial \langle \hat {w} \hat {w} \rangle }{\partial z} \right ) \right ]}_{\mathcal{T}_{w \rightarrow r}} \Bigg \rangle _\varOmega - \Bigg \langle \underbrace {\dfrac {\partial \hat {u}_i }{\partial x_{\!j}} \dfrac {\partial \hat {u}_i }{\partial x_{\!j}}}_{\mathcal{D}_w} \Bigg \rangle _\varOmega . \end{align}
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