Pattern of Breakdown of Laminar Flow into Turbulent Spots

The breakdown into turbulent spots is the least understood stage of the laminar-turbulent transition process. With cellular-automaton stochastic simulations and stability analysis, we show that the pattern of breakdown in boundary-layer flow bears a connection to laminar instability and may be reconstructed using macroscopic properties of the transition zone, such as persistence times and transitional intermittency. We propose experimental tests of our ideas.

Figure 1

The laminar-turbulent transition process on a boundary layer in the flow past a semi-infinite flat plate.

Figure 2

Examples of two-dimensional patterns of disturbance vorticity observed in a destabilized laminar boundary layer; $z$ is the spanwise coordinate. (a) T2: a subharmonic secondary instability of Tollmien-Schlichting waves; (b) S2: secondary instability of streamwise streaks [4].

Figure 3

Schematic of the simulation with random spot birth: according to a Poisson distribution in time, and uniformly distributed in $z$.

Figure 4

(a) Intermittency in a decelerating flow, $m=-0.06$. $\eta$ is streamwise distance scaled by transition zone length. Solid line: stochastic simulation, with $N_z=49,N_t=10$, similar to the dominant secondary mode; symbols: experiment [19]. (b) The parameter $F$ for data in (a). Also shown are the simulation results of random breakdown (long dashes), and of a regular breakdown where lateral merger would be dominant, $N_z=10,N_t=40$ (dot-dashed line). Experimental error is likely to be significant at either end of the transition zone, since switches from laminar to turbulent flow are difficult to distinguish. Also, when $\gamma \sim 1$, $F$ depends sensitively on $\gamma$, so errors appear amplified.

Figure 5

Intermittency in flat-plate flow. (a) T2 route: circles, experiment [20]; solid line, regular breakdown; $N_z/N_t=2.5$; long dashes, random breakdown; triangles, random spot birth, distributed over $X=1$ to 10. (b) S2 route: symbols, experiment [21]; line, regular breakdown; $N_z/N_t=5$.

Figure 6

Variation of the burst rate with the intermittency for different scenarios of spot birth. The regular pattern conforms to subharmonic T2.

Figure 7

Probability density function of persistence time of laminar flow at (a) $\gamma =0.1$ and (b) $\gamma =0.5$. Dashed line, random breakdown; solid line, 90% regular (subharmonic T2); long dashes, 50% regular. Thin dot-dashed line, predominant lateral merger as in Fig. 4b.