Bursting and critical layer frequencies in minimal turbulent dynamics and connections to exact coherent states

The dynamics of the turbulent near-wall region is known to be dominated by coherent structures. These near-wall coherent structures are observed to burst in a very intermittent fashion, exporting turbulent kinetic energy to the rest of the flow. In addition, they are closely related to invariant solutions known as exact coherent states (ECS), some of which display nonlinear critical layer dynamics (motions that are highly localized around the surface on which the streamwise velocity matches the wave speed of ECS). The present work aims to investigate temporal coherence in minimal channel flow relevant to turbulent bursting and critical layer dynamics and its connection to the instability of ECS. It is seen that the minimal channel turbulence displays frequencies very close to those displayed by an ECS family recently identified in the channel flow geometry. The frequencies of these ECS are determined by critical layer structures and thus might be described as “critical layer frequencies.” While the bursting frequency is predominant near the wall, the ECS frequencies (critical layer frequencies) become predominant over the bursting frequency at larger distances from the wall, and increasingly so as Reynolds number increases. Turbulent bursts are classified into strong and relatively weak classes with respect to an intermittent approach to a lower branch ECS. This temporally intermittent approach is closely related to an intermittent low drag event, called hibernating turbulence, found in minimal and large domains. The relationship between the strong burst and the instability of the lower branch ECS is further discussed in state space. The state-space dynamics of strong bursts is very similar to that of the unstable manifolds of the lower branch ECS. In particular, strong bursting processes are always preceded by hibernation events. This precursor dynamics to strong turbulence may aid in development of more effective control schemes by a way of anticipating dynamics such as intermittent hibernating dynamics.

Figure 1

(a) Bifurcation diagram for P4 traveling wave solutions along with curves for Newtonian turbulence and laminar flow [38], where the bulk velocities $U_b^{+}$ are plotted as a function of the friction Reynolds number. The number of unstable eigenvalues is indicated on each branch. Dots on each branch correspond to P4 TW solutions shown in panel (b). (b) State-space visualization at ${\text{Re}}_c=1800$, projected onto disturbance kinetic energy (KE), energy dissipation rate ($D$) and normalized instantaneous wall shear stress (${\tau }_w/{\overline{\tau }}_w$) [38]. The gray line indicates a turbulent trajectory. A joint probability function of KE and $D$ for the trajectory is shown at the bottom of the figure. The labeled symbols ($◃$) are P4 solutions at ${\text{Re}}_c=1800$. Based on the $L_2$ distance between the trajectories and P4 traveling waves [38], the closest visits to P4-LB, P4-LB2, and P4-UB are at points (i), (ii), and (iii), respectively. All quantities are calculated only for the bottom half of the channel. (c)–(f) Vortical structures illustrated by the swirling strength ${\lambda }_{ci}$. (c) P4-LB, (d) P4-UB, (e) instant (i), and (f) instant (iii). For panels (c) and (d), the tubes are isosurfaces at $2/3$ of the maximum swirling strength. The maximum swirling strengths are (c) 0.30 and (d) 0.79. The isosurfaces are colored based on the local value of the streamwise velocity, ranging from low (blue) to high (red). The transparent blue isosurfaces indicate critical layer surfaces where the local streamwise velocity matches the wave speed of the solution. For the flow structures in panels (e) and (f), we use the same vortex strength and critical isosurfaces as in panels (c) and (d), respectively.

Figure 2

(a) Time series of the energy dissipation rate ($\mathit{D}$) at ${\text{Re}}_c=1800({\text{Re}}_{\tau }=85)$ in a minimal channel flow. The black dashed and red dotted lines are the mean and the mean + 1.5 times its standard deviation, respectively. (b) Power spectral density of a long data set of $\mathit{D}$ (more than $2\times {10}^{4}h/U_c$). (c) Reynolds number dependence of characteristic burst periods computed from the power spectrum (blue circles) and by the definition of a turbulent burst (red squares). The red line in panel (b) is ${\omega }_b=2\pi /T_b$, where $T_b$ is the mean burst interval (red square) in panel (c).

Figure 3

Wave speed for the P4 traveling wave solutions as a function of the Reynolds number in (a) outer units (i.e., scaled with $U_c$ and plotted against ${\text{Re}}_c$) and (b) inner units (i.e., scaled with $u_{\tau }$ and plotted against ${\text{Re}}_{\tau }$). The laminar bulk velocity, $U_b$, is plotted with a dashed line.

Figure 4

Power spectral density of a pointwise-sampled velocity at ${\text{Re}}_c=1800({\text{Re}}_{\tau }=85)$ in the channel domain. [(a)–(c)] Power spectra of the streamwise velocity at $y^{+}=5,49,65$ and [(d)–(f)] power spectra of the wall-normal velocity at $y^{+}=5,49,65$, along with the critical layer frequencies and bursting frequency. The relevant critical layer frequency (${\omega }_c=2\pi c_x/L_x$) is calculated using the wave speed ($c_x$) of P4 traveling wave solutions, and the bursting frequency (${\omega }_b=2\pi /T_b$) is calculated using the burst interval $T_b$ (red square) in Fig. 2.

Figure 5

Reynolds number dependence of the power spectra of wall-normal velocities at $y=0.23$ (note that $y=0$ corresponds to the channel center). [(a)–(c)] Power spectra at ${\text{Re}}_{\tau }=101(y^{+}=77)$, at ${\text{Re}}_{\tau }=113(y^{+}=87)$, and at ${\text{Re}}_{\tau }=133(y^{+}=102)$, respectively. [(d)–(f)] Enlargement of power spectra for the frequency regime of $1.2<\omega <1.7$ at ${\text{Re}}_{\tau }=101(y^{+}=77)$, at ${\text{Re}}_{\tau }=113(y^{+}=87)$, and at ${\text{Re}}_{\tau }=133(y^{+}=102)$, respectively. An additional line for the frequency ${\omega }_m$ corresponding to the laminar bulk velocity $U_b$ is shown.

Figure 6

Time series of the area-averaged wall shear stress normalized by its time average value (black dashed line) as a function of time $t$. Shown are hibernating turbulence (blue intervals), bursting events after hibernation (green intervals), and other bursting events not associated with hibernation (red intervals).

Figure 7

Conditionally sampled instantaneous wall shear stress events (gray lines) during intervals of hibernating turbulence at ${\text{Re}}_c=1800({\text{Re}}_{\tau }=85)$. The ensemble-averaged profile and the unconditional mean wall shear stress are shown by the red line and black dashed line, respectively. The time origin of each event is shifted so that its end corresponds to $t=0$ (vertical blue line). The vertical blue dotted line denotes $t=-65$, which is the minimum time duration of a hibernation event based on our criteria.

Figure 8

Classification of turbulent bursts with respect to hibernation: strong and weak bursts. (a) Probability density function for the bursting magnitudes $\mathrm{\Delta }D$ (defined as the increase of the energy dissipation rate) of two burst classes and (b) Reynolds number dependence of the ensemble-averaged bursting magnitudes normalized by the standard deviation of the energy dissipation rate.

Figure 9

[(a), (b)] State-space visualization of DNS trajectories and P4 solutions at ${\text{Re}}_c=1800({\text{Re}}_{\tau }=85)$ projected onto three dimensions. The gray line indicates the turbulent trajectory. (a) The green and red lines represent trajectories of strong and weak bursts, respectively. (b) Trajectories of the unstable manifolds of the P4-LB solution are shown with blue and magenta lines, while those of the P4-LB2 solution are shown with light green, dark green, and pink lines. The solid circles correspond to the snapshots on the right. (c) Time evolution of flow structures along the most unstable manifold of the P4-LB indicated by dots in (b). Time marches downward. The tubes are isosurfaces at the swirling strength of 0.2, colored by the streamwise velocity $u$, 0 (blue) $\le u\le$ 1 (red). The transparent blue isosurfaces indicate critical layer surfaces where the local streamwise velocity matches the wave speed of the P4-LB solution.