Identifying the pattern of breakdown in a laminar-turbulent transition via binary sequence statistics and cellular-automaton simulations

The laminar-turbulent transition induced by two-dimensional steplike roughness is investigated focusing on the pattern of breakdown. The statistics of the turbulent burst rate is found to be significantly different from the prediction of the classical theory. A systematic investigation of the pattern of breakdown is motivated by this phenomenon. It is identified with heuristic analysis that a pattern of distributed breakdown is responsible for the deviation, in contrast to the concentrated breakdown hypothesis in the classical theory. The pattern indicates that the steps probably induced a bypass transition in present experimental setup, which is different from the current understanding about the step-induced transition. Cellular-automaton simulations are carried out to validate the heuristic analysis. The influences of quasiconcentration and non-Poisson process in spot generation on the breakdown statistics are also discussed based on the simulation results.

Figure 1

Sketch of the patterns of breakdown (the red triangular areas represent the turbulent spots which are moving from the left side to the right side, the dashed line is the transition starting location, the coordinates $x$ and $z$ represent the streamwise and spanwise directions): (a) self-similar evolution of a single spot at different moments $T_1<T_2<T_3$, the dashed-dotted lines are the envelopes of the spot; (b) concentrated breakdown; (c) regular breakdown (K type); (d) distributed breakdown. (b)–(d) The snapshots of a part of the breakdown region.

Figure 2

Power spectrum (PSD) of the velocity signal measured in free stream at $x=0\mathrm{mm}, y=300\mathrm{mm}$.

Figure 3

Sketch of the wind-tunnel experiments of the step-induced boundary-layer transition, (a) top view, (b) side view.

Figure 4

A snapshot for the cellular-automaton simulation of a stochastic breakdown. The red areas represent the turbulent spots.

Figure 5

Typical exponential, Poisson and Gaussian distribution for the interarrival time of turbulent spots in various patterns of breakdown.

Figure 6

Typical velocity signals throughout the transition at the same wall normal distance ($y\approx 1\text{mm}$) and different streamwise stations (from top to bottom: $x=550$, 650, 750, 850, and 950 mm).

Figure 7

Example of the laminar-turbulent separation procedure for intermittent velocity signal measured at $x=750\mathrm{mm}, y=1.6\mathrm{mm}$. From top to bottom: the original velocity signal ($u$), the high-pass filtered velocity signal ($u_{\mathrm{hp}}$), and the binary intermittency function ($\mathrm{\Gamma }$).

Figure 8

Wall-normal and streamwise distributions of the experimental results of the intermittency factor ($\gamma$) in the breakdown region. The region between $y/{\delta }^{*}=0.5$ and 3 is marked by the vertical dashed lines.

Figure 9

Ensemble statistics of experimental results (black squares) of the intermittency factor ($\gamma$) in comparison with the prediction of classical theory (solid line) and the simulation results of case C-E (red circles) in the breakdown region. The corresponding distribution of $F$ is shown in the inset. Error bars indicate standard deviation from the ensemble average.

Figure 10

Wall-normal and streamwise distributions of the experimental results of the burst rate ($B$) in the breakdown region. The region between $y/{\delta }^{*}=0.5$ and 3 is marked by the vertical dashed lines.

Figure 11

Ensemble statistics of experimental results (black squares) of the normalized burst rate ($B/B_{max}$) in comparison with the prediction of classical theory (solid line) and the simulation results of case C-E (red circles) in the breakdown region. Error bars indicate standard deviation from the ensemble average. The quadratic function (dashed line) fitted from the experimental data is also plotted.

Figure 12

Power spectrum for the binary sequence of the intermittency function (the data are shifted in the vertical direction for clarity) obtained from experimental data. Error bars indicate standard deviation from the ensemble average.

Figure 13

Power spectrum for the binary sequence of the intermittency function (the data are shifted in the vertical direction for clarity) obtained from cellular-automaton simulation of a stochastic concentrated breakdown process (case C-E).

Figure 14

Probability distribution of the laminar persistence time at three stations $x=700$, 750, and 800 mm corresponding to $\gamma =0.33$, 0.51, and 0.68, and the simulation result of case C-E at $\gamma =0.33$. Error bars indicate standard deviation from the ensemble average. The data are plotted in log-linear coordinates in the inset.

Figure 15

Sketch of the heuristic analysis of the breakdown process. (a) The three-dimensional region projected by the two-dimensional breakdown region considering time ($t$) as the vertical axis. As time goes on, the downstream evolution and propagation of the triangular turbulent spot generated at $(x_0,z_0,t_0)$ forms the three-dimensional influence region. Similarly, any turbulent flow in the upstream dependence region would pass the point $(x_0,z_0,t_0)$. The red triangle is a cross section at some streamwise location ($0<x<x_0$) upstream of $(x_0,z_0,t_0)$. (b), (c) Top and side views of (a), and the straight boundary lines correspond to the linear propagation of the turbulent spots.

Figure 16

Experimental results of the burst rate (black squares) in comparison with the distribution in various breakdown scenarios predicted by the heuristic analysis (normalized by the maximum burst rate) $B\propto {\left[-ln\left(1-\gamma \right)\right]}^{\frac{n-1}{n}}\left(1-\gamma \right)$ with various $n$ (red lines).

Figure 17

Experimental results of the intermittency factor (black squares) fitted with the distribution in various breakdown scenarios predicted by the heuristic analysis: $\gamma =1-exp\{-A_n{[(x-x_s)/(x_{0.75}-x_{0.25})]}^n\}$ with various $n$ (red lines).

Figure 18

Comparison of the growth curves of the intermittency factor ($\gamma$) predicted by the heuristic analysis [red lines, $\gamma =1-exp(-A_n{\xi }^n), n=3$, 4, 4.62] and the statistical results of the numerical simulations of the distributed breakdown (open symbols: D-E-01 square, D-E-02 circle, D-E-03 triangular). The distribution of $\gamma$ in classical theory [$\gamma =1-exp(-A_2{\xi }^2)$, dashed-dotted-dotted line] is also plotted as a reference.

Figure 19

Comparison of the distributions of the normalized burst rate ($B/B_{max}$) predicted by the heuristic analysis (red lines, $B\propto (1-\gamma ){[-ln(1-\gamma )]}^{(n-1)/n}, n=3$, 4, 4.62) and the statistical results of the numerical simulations of the distributed breakdown (open symbols: D-E-01 square, D-E-02 circle, D-E-03 triangular). The distribution of $B/B_{max}$ in classical theory ($B/B_{max}=2.33(1-\gamma ){[-ln(1-\gamma )]}^{1/2}$, dashed-dotted-dotted line) is also plotted as a reference.

Figure 20

Statistical results of the probability distribution of the normalized laminar persistence time ($\text{LPT}/{\text{LPT}}_{\mathrm{mean}}$) at different stages of breakdown ($\gamma =0.33$, 0.51, 0.68) in the distributed stochastic breakdown simulations (D-E-01, D-E-02, D-E-03). An exponential distribution fitted from the simulation results is also plotted as a reference (red line). The results are plotted in log-linear coordinates in the inset.

Figure 21

The statistical results of the intermittency factor ($\gamma$, open symbols) as a function of $\xi$ obtained from the quasiconcentrated breakdown simulations (QC-E-01, QC-E-02, QC-E-03, QC-E-04). The prediction of the classical theory (red line) is also plotted for comparison. The region in the lower left corner is plotted in the inset for clarity.

Figure 22

Statistical results of the normalized burst rate ($B/B_{max}$) obtained in the simulation of non-Poisson process of concentrated (squares) and distributed (circles) breakdown. The predictions of the classical theory (dashed-dotted-dotted line) and the heuristic analysis (solid line) are also plotted for comparison.