Statistical transition to turbulence in plane channel flow

Intermittent turbulent-laminar patterns characterize the transition to turbulence in pipe, plane Couette, and plane channel flows. The time evolution of turbulent-laminar bands in plane channel flow is studied via direct numerical simulations using the parallel pseudospectral code ChannelFlow in a narrow computational domain tilted by ${24}^{\circ }$ with respect to the streamwise direction. Mutual interactions between bands are studied through their propagation velocities. Energy profiles show that the flow surrounding isolated turbulent bands returns to the laminar base flow over large distances. Depending on the Reynolds number, a turbulent band can either decay to laminar flow or split into two bands. As with past studies of other wall-bounded shear flows, in most cases survival probabilities are found to be consistent with exponential distributions for both decay and splitting, indicating that the processes are memoryless. Statistically estimated mean lifetimes for decay and splitting are plotted as a function of the Reynolds number and lead to the estimation of a critical Reynolds number ${\mathrm{Re}}_{\text{cross}}\simeq 965$, where decay and splitting lifetimes cross at greater than ${10}^6$ advective time units. The processes of splitting and decay are also examined through analysis of their Fourier spectra. The dynamics of large-scale spectral components seem to statistically follow the same pathway during the splitting of a turbulent band and may be considered as precursors of splitting.

Figure 1

(a) Sketch of the laminar profile. (b) Visualization of turbulent bands in a $240\times 108$ streamwise-spanwise domain at $\mathrm{Re}=1000$. Colors indicate the streamwise velocity in the $y=-0.8$ plane. A superimposed black box illustrates a long-narrow computational domain, tilted with an angle $\theta$ relative to the streamwise direction. (c) and (d) Structure of a turbulent-laminar pattern computed in a tilted domain at $\mathrm{Re}=1200$. Plot (c) shows the $x$ component of the velocity in the $(x,z)$ plane at $y=-0.8$. The streamwise and spanwise directions are indicated in red. Plot (d) shows streamwise vorticity in a $(y,z)$ plane with the vertical $y$ scale stretched by a factor of 2. Only the portion of the computational domain containing the turbulent region is shown in (d). As seen in (c), on the downstream side of the turbulent region the flow exhibits weak straight streaks, oriented in the streamwise direction, that slowly diminish as the flow returns laminar.

Figure 2

Space-time diagrams of turbulent bands in a frame moving at the bulk velocity, $U_{\mathrm{bulk}}$, with (a) $\mathrm{Re}=830,L_z=100$, (b) $\mathrm{Re}=1100,L_z=100$, (c) $\mathrm{Re}=1200,L_z=50$. Colors show the perturbation energy $E=\frac{1}{2}(u^2+v^2+w^2)$ as a function of $z$ and $t$, sampled in the $y=-0.8$ plane at a arbitrary value of $x$ (yellow: $E=0.1$; blue: $E=0$). Average band propagation velocities, relative to $U_{\mathrm{bulk}}$, and the degree of fluctuations can be discerned from diagrams. Case (a) is an example of a band moving downstream relative to $U_{\text{bulk}}$, which occurs for $\mathrm{Re}\lesssim 1000$, and then decaying. In case (b), a single band in a domain with $L_z=100$ splits into two bands, resulting in a pair of bands separated in $z$ by distance 50 = $L_z/2$. The change in velocity resulting from a decrease in interaction distance is evident. Note, however, that the time range covered in the plot is large, which visually accentuates the effect. Case (c) shows band splitting in a domain of size $L_z=50$. The resulting bands are closely spaced and interact strongly.

Figure 3

Dependence of the band propagation velocity on the Reynolds number and on the interband distance $L_z$ (left axis: $z$ velocity; right axis: streamwise velocity). Normal-approximated error bars are shown for $L_z=100$.

Figure 4

Energy averaged over $x$, $y$, and $t$ as a function of $z$, for different $L_z$, in (a) linear and (b) logarithmic scales, for a one-band state at $\mathrm{Re}=1000$.

Figure 5

Band decay at $\mathrm{Re}=830$. Plotted is the $x$ velocity in $(x,y)$ planes at $y=-0.8$. For clarity the color scale changes over time.

Figure 6

Example of a $(x,z)$ Fourier spectrum of the $x$ velocity $u$ in the $y=-0.8$ plane, for a turbulent band at $\mathrm{Re}=830$. Colors show the modulus of spectral coefficients, spanning from 0 (blue) to 0.02 (red). The moduli of components $(m_x,-m_z)$ and $(-m_x,m_z)$ are equal since the velocity is real.

Figure 7

Illustrative Fourier spectra ${\widehat{u}}_{0,m_z}$ and ${\widehat{u}}_{1,m_z}$ (a) before band decay and (b) in the final relaxation to laminar flow. $\mathrm{Re}=830$. The black symbols ${\widehat{u}}_{1,m_z}$ with $m_z$ surrounding 35 correspond to streaks while the blue symbols ${\widehat{u}}_{0,m_z}$ at low $m_z$ correspond to large-scale structures. Filled symbols indicate ${\widehat{u}}_{0,1}$ and ${\widehat{u}}_{0,2}$.

Figure 8

Time evolution of spectral quantities ${\widehat{u}}_{0,1}$ and ${\widehat{u}}_{\text{streaks}}$ (left) and of $L_2$ norms ${\left|\right|u\left|\right|}_2$, ${\left|\right|v\left|\right|}_2$, and ${\left|\right|w\left|\right|}_2$ (right) for a decay event at $\mathrm{Re}=830$. Times $t_a$, $t_b$, and $t_f$ refer to slices shown on Fig. 5. The band starts to decay at $t_a$, ${\widehat{u}}_{0,1}={\widehat{u}}_{\text{streaks}}$ at $t_b$, and the relaminarization is considered as complete at $t_f$.

Figure 9

Time evolution of ${\widehat{u}}_{0,1}$ and ${\widehat{u}}_{0,2}$ (left) and of ${\left|\right|u\left|\right|}_2$, ${\left|\right|v\left|\right|}_2$, and ${\left|\right|w\left|\right|}_2$ (right) during ten realizations of decay events at $\mathrm{Re}=830$. Time $t^{*}$ and vertical quantities are respectively translated and scaled to obtain the same final value for each realization. Final decay rates for ${\widehat{u}}_{0,1}$ and ${\widehat{u}}_{0,2}$ (a) are $-3.6\times {10}^{-3}$ and $-5.2\times {10}^{-3}$, respectively.

Figure 10

Band splitting at $\mathrm{Re}=1200$. Plotted is the $x$ velocity in $(x,y)$ planes at $y=-0.8$.

Figure 11

Evolution of a band while it splits at $\mathrm{Re}=1200$. (a) Spatiotemporal diagram of the band. Colors show the turbulent perturbation energy $E$ between 0 (blue) and 0.1 (yellow). (b)–(d) Time evolution of spectral quantities ${\widehat{u}}_{0,1}$ and ${\widehat{u}}_{0,2}$ (b), ${\widehat{u}}_{\text{streaks}}$ (c), and the $L_2$-norm ${\left|\right|w\left|\right|}_2$ (d).

Figure 12

Illustrative Fourier spectra ${\widehat{u}}_{0,m_z}$ and ${\widehat{u}}_{1,m_z}$ (a) before and (b) after band splitting at $\mathrm{Re}=1200$. The black symbols ${\widehat{u}}_{1,m_z}$ correspond to streaks while the blue symbols ${\widehat{u}}_{0,m_z}$ at low $m_z$ correspond to large-scale structures. Filled symbols indicate ${\widehat{u}}_{0,1}$ and ${\widehat{u}}_{0,2}$.

Figure 13

Evolution of spectral quantities during ten splittings at $\mathrm{Re}=1200$, in a domain of length $L_z=50$. Each curve represents one simulation, and is colored by ${\widehat{u}}_{0,1}$ to illustrate the transition between a one-band (1) and a two-band state (2).

Figure 14

Survival probability distributions for the decay of a turbulent band, $\mathrm{Re}\in [730,900]$.

Figure 15

Survival probability distributions for the splitting of a turbulent band, $\mathrm{Re}\in [1100,1350]$.

Figure 16

Variation of mean decay times (red) and splitting times (black) with Reynolds number $\mathrm{Re}$. The error bars correspond to $95\%$ confidence intervals. Inset: $lnln{\tau }^{s/d}$ versus $\mathrm{Re}$ and associated linear fits. The crossing point is at ${\mathrm{Re}}_{\mathrm{cross}}\approx 965,\tau \approx 3\times {10}^6$.