Bifurcations to turbulence in transitional channel flow

In wall-bounded parallel flows, sustained turbulence can occur even while laminar flow is still stable. Channel flow is one of such flows and displays spatiotemporal fluctuating patterns of localized turbulence along its way from and to featureless turbulence. By direct numerical simulation, we study the observed inconsistency between turbulence decay according to a two-dimensional directed-percolation $\left(2D\text{−}\mathrm{DP}\right)$ scenario and the presence of sustained oblique localized turbulent bands (LTBs) below the DP critical point. Above Reynolds number ${\mathrm{Re}}_{\mathrm{g}}\approx 700$, sustained LTBs are observed; most LTBs have the same orientation so that the spanwise symmetry of the LTB pattern is broken below ${\mathrm{Re}}_2\approx 1000$. The frequency of transversal splitting, by which an LTB generates another one with opposite obliqueness, so that turbulence spreading becomes intrinsically two dimensional, increases in the range ${\mathrm{Re}}_{\mathrm{g}}<\mathrm{Re}<{\mathrm{Re}}_2$. It reaches a critical rate at ${\mathrm{Re}}_2$, beyond which symmetry is restored. The $2D\text{−}\mathrm{DP}$ behavior is retrieved only above ${\mathrm{Re}}_2$. A mean-field model is proposed which qualitatively accounts for the above symmetry-restoring bifurcation by considering interactions between space-averaged densities of LTBs propagating in either direction.

Figure 1

(a) $\mathrm{Re}=850$ (${\mathrm{Re}}_{\mathrm{b}}\approx 789$): one-sided LTB regime. (b) $\mathrm{Re}=1050$ (${\mathrm{Re}}_{\mathrm{b}}\approx 947$): two-sided LTB regime. (c) $\mathrm{Re}=1200$ (${\mathrm{Re}}_{\mathrm{b}}\approx 1012$): strongly intermittent loose continuous network of LTBs. (d) $\mathrm{Re}=1800$ (${\mathrm{Re}}_{\mathrm{b}}\approx 1237$): weakly intermittent loose banded pattern. (e) $\mathrm{Re}=3000$ (${\mathrm{Re}}_{\mathrm{b}}\approx 1604$): tight banded pattern. (f) $\mathrm{Re}=4000$ (${\mathrm{Re}}_{\mathrm{b}}\approx 1843$): nearly featureless state. Flow direction is from the left to right. The wall-normal velocity field on the center plane $(y=0)$ is displayed. Domain size is $500\times 250$. See video in the Supplemental Material [43].

Figure 2

Bifurcation diagram for plane channel flow as obtained in our simulations (corresponding values of ${\mathrm{Re}}_{\mathrm{b}}$ between parentheses). Laminar flow is always recovered in the long-time limit for $\mathrm{Re}<{\mathrm{Re}}_{\mathrm{g}}\approx 700$ ($\mathrm{Re}\equiv {\mathrm{Re}}_{\mathrm{b}}$ up to ${\mathrm{Re}}_{\mathrm{g}}$). Event A corresponds to the onset of transversal splitting at $\mathrm{Re}\approx 800$ (${\mathrm{Re}}_{\mathrm{b}}\approx 795$). The extrapolated $2D$-DP threshold is found at ${\mathrm{Re}}_{\mathrm{DP}}\simeq 984$ (${\mathrm{Re}}_{\mathrm{b}}\simeq 905$). The transition from one-sided to two-sided propagation of LTBs takes place at ${\mathrm{Re}}_2\simeq 1011$ (${\mathrm{Re}}_{\mathrm{b}}\simeq 924$). Event B marks the opening of laminar gaps in the turbulent branches, sufficiently long-lived to allow the development of DAHs. It takes place at $\mathrm{Re}\approx 1200$ (${\mathrm{Re}}_{\mathrm{b}}\approx 1012$). Finally, “featureless” turbulence is present for $R>{\mathrm{Re}}_{\mathrm{t}}\approx 3900$ (${\mathrm{Re}}_{\mathrm{b}}\approx 1820$). The Tollmien-Schlichting instability threshold is at ${\mathrm{Re}}_{\mathrm{TS}}=5772$ (analysis refers to laminar base flow, hence identical ${\mathrm{Re}}_{\mathrm{b}}$).

Figure 3

Means (a) and relative standard deviations (b) of the wall-normal energy $E_y\left(t\right)$ and spanwise energy $E_z\left(t\right)$.

Figure 4

(a) Average of streamwise bulk velocity $\langle U_{\mathrm{m}}\rangle$ as a function of $\mathrm{Re}$; the solid line corresponds to the fit and the dashed line to laminar flow, $U_{\mathrm{m}}^{\mathrm{lam}}=2/3$. Open symbols in the inset are for the different domain sizes. (b) Skin friction coefficient ${\mathrm{C}}_{\mathrm{f}}$ as a function ${\mathrm{Re}}_{\mathrm{b}}$. Filled and open symbols are from DNSs (Tsukahara et al. [16], Xiong et al. [37]) and experiments (Patel and Head [47], Dean [46]), respectively. The dashed line is for laminar flow and the dotted line for the fully turbulent regime.

Figure 5

(a) Longitudinal splitting. (b) Longitudinal collision. (c) Transversal splitting. (d) Transversal collision. The same quantity as in Fig. 1, $u_y(x,0,z;t)$, is displayed in a comoving frame with velocity $(u_x^{\left(\mathrm{f}\right)},u_z^{\left(\mathrm{f}\right)})=(0.8,0.1)$. Circles locate DAHs to be followed and numbers identify bands, “P” and “C” stand for parent and child, respectively. See also videos in the Supplemental Material [43].

Figure 6

Transition from one-sided to two-sided flow. (a) Model. (b) Numerical simulation.

Figure 7

Departure of observed streamwise mean flow $\langle U_{\mathrm{m}}\rangle$ from the value predicted by the law $U_{\mathrm{m}}^{\mathrm{fit}}=19.47/\sqrt{\mathrm{Re}}$, solid line in Fig. 4. Fitting the data against a straight line for $850\le \mathrm{Re}\le 1000$ yields ${\mathrm{Re}}_2\simeq 1011$. Triangles are for the larger domain $1000\times 2\times 500$.

Figure 8

Possible candidates for the determination of the turbulent fraction illustrated by $2D$ images of the same localized turbulent band at $\mathrm{Re}=700$. The flow is from left to right and the downstream active head is in the upper right corner in each panel. Domain size is $500\times 250$.

Figure 9

$F_t$ for different $w$. From the bottom to the top: $w=0$ (no filtering, red), $4\delta$, $8\delta$, $12\delta$, $20\delta$, and $40\delta$ (blue). (a) $F_{\mathrm{t}}$ as a function of $\mathrm{Re}$. (b) $F_t^{1/{\beta }_{\mathrm{DP}}}$ as a function of $1/\mathrm{Re}$ with ${\beta }_{\mathrm{DP}}=0.583$.

Figure 10

[(a)–(c)] Minimum of $\mathrm{Err}$ as a function of the different fit parameters for $F_{\mathrm{t}}$ with $w=12\delta$ and control parameter $1/\mathrm{Re}$. (a) Exponent $\beta$. (b) Amplitude $B$. (c) Threshold ${\mathrm{Re}}_{\mathrm{DP}}$. In these panels, the error is displayed as a function of one fitting parameter, the two other being supposed fixed at their optimal value. (d) Turbulent fraction $F_{\mathrm{t}}$ as a function of $\mathrm{Re}$ and its fit against the theoretical expression given in Sec. 4.

Figure 11

Color-level illustration of the DAH of the LTB shown in Fig. 8 using $|u_y(x,0,z)|$. The size of the domain displayed is $50\times 50\simeq 460\delta \times 460\delta$. Flow direction is from left to right. Raw data (left) and after box filtering with the width $w=12\delta$ (routinely used here) and $w=40\delta$. Light-blue lines mark the boundaries between turbulent and laminar regions as determined by the moment-preserving thresholding method.

Figure 12

Moment-preserving thresholding of snapshots at 1200 and 1800 in Figs. 1 and 1 under box filtering with $w=12\delta$. Flow direction is from left to right. Domain size is $500\times 250$.