Evolution of conditionally averaged second-order structure functions in a transitional boundary layer

We consider the bypass transition in a flat-plate boundary layer subject to free-stream turbulence and compute the evolution of the second-order structure function of the streamwise velocity, $\langle d{u}^2\rangle (\vec{x},\vec{r})$, from the laminar to the fully turbulent region using DNS. In order to separate the contributions from laminar and turbulent events at the two points used to define $du(\vec{x},\vec{r})$, we apply conditional sampling based on the local instantaneous intermittency, $\tau$ (1 for turbulent and 0 for laminar events). Using $\tau (\vec{x},t)$, we define two-point intermittencies, ${\gamma }^{\left(TT\right)}, {\gamma }^{\left(LL\right)}$, and ${\gamma }^{\left(TL\right)}$, which physically represent the probabilities that both points are in turbulent patches, both are in laminar patches, or one is in a turbulent and the other in a laminar patch, respectively. Similarly, we also define the conditionally averaged structure functions, ${\langle d{u}^2\rangle }^{\left(TT\right)}, {\langle d{u}^2\rangle }^{\left(LL\right)}$, and ${\langle d{u}^2\rangle }^{\left(TL\right)}$, and decompose $\langle d{u}^2\rangle (\vec{x},\vec{r})$ in terms of these conditional averages. The derived expressions generalize existing decompositions of single-point statistics to two-point statistics. It is found that in the transition region, laminar streaky structures maintain their geometrical characteristics in the physical and scale space well inside the transition region, even after the initial breakdown to form turbulent spots. Analysis of the ${\langle d{u}^2\rangle }^{\left(TT\right)}$ fields reveals that the outer mode is the dominant secondary instability mechanism. Further analysis reveals how turbulent spots penetrate the boundary layer and approach the wall. The peaks of ${\langle d{u}^2\rangle }^{\left(TT\right)}$ in scale space appear in larger streamwise separations as the transition progresses and this is explained by the strong growth of turbulent spots in this direction. On the other hand, the spanwise separation where the peak occurs remains relatively constant and is determined by the initial inception process. We also analyze the evolution of the two-point intermittency field, ${\gamma }^{\left(TT\right)}$, at different locations. In particular, we study the growth of the volume enclosed within an isosurface of ${\gamma }^{\left(TT\right)}$ and note that it increases in both directions, with growth in the streamwise direction being especially large. The evolution of these conditional two-point statistics sheds light on the transition process from a different perspective and complements existing analyses using single-point statistics.

Figure 1

(a) Free-stream turbulent intensity Tu vs ${\mathrm{Re}}_x$; (b) skin friction coefficient $C_f$ vs ${\mathrm{Re}}_{\theta }$; $\left(c\right)$ mean streamwise velocity $U^{+}$ vs $y^{+}$; (d) $\mathrm{rms}\left(u\right)/\mathrm{rms}{\left(u\right)}_{\max }$ vs the Blasius variable, $\eta$.

Figure 2

Flow snapshots tracking a spot from its incipient formation (top row) to its fully developed form (bottom row); wall-normal velocity fluctuations at $Y=5L_0$ (left panels); streamwise velocity fluctuations in the plane at $Z=25L_0$ (indicated as horizontal dashed lines in the left panels) that pass through the center of the spot (right panels). Note the translation of the domain visualized in the streamwise direction in the right panels. The time interval between each snapshot is $190L_0/U_{\infty }$.

Figure 3

Contour plot of the intermittency field $\gamma$. The dashed red line represents the boundary layer thickness. The solid green, red, and blue vertical lines represent the streamwise locations where the second-order structure functions are calculated. The green line is in the laminar region $((X-X_0)/L_0=540)$ and the blue line in the turbulent region $((X-X_0)/L_0=2415)$. The eight red lines, $TRM1$ and $TR0-TR6$, are in the transition region and their coordinates are 940, 1065, 1215, 1365, 1515, 1665, 1815, and 1965, respectively.

Figure 4

Variation of ${\gamma }_{\max }$ (solid line) in the streamwise direction and comparison with the formula of Narashima (dashed line), where $\xi =(X-X_s)/(X_{\gamma =0.75}-X_{\gamma =0.25})$, and $X_s$ is the location where the transition starts, taken to be $(X_s-X_0)/L_0=1100$.

Figure 5

Sketches of the three states. Red ovals represent turbulent spots. Both points within a turbulent spot (state $\mathit{TT}$; left); one point inside a turbulent spot, the other in the laminar region (state $\mathit{TL}$; middle); both points in the laminar region (state $\mathit{LL}$; right).

Figure 6

$\langle d{u}^2\rangle$ on the $(r_3,Y)$ (left) and $(r_1,Y)$ (right) planes in the laminar region. Note that the left and bottom axes are normalized by the local boundary layer thickness ($\delta$), while the top and right axes are normalized by the inlet Blasius similarity variable, $L_0$.

Figure 7

Second-order streamwise velocity correlation function with respect to the spanwise separation at the wall-normal distance $Y_0\sim 4L_0$.

Figure 8

$\langle d{u}^2\rangle$ on the $(r_3,Y)$ (left) and $(r_1,Y)$ (right) planes in the fully turbulent region. The left and bottom axes are in local wall units, while the top and right axes are in global units of $L_0$.

Figure 9

Evolution of $\langle d{u}^2\rangle$ on the $(r_3,Y)$ plane along the streamwise direction. The locations, denoted $TR0\text{−}TR6$, are defined in Fig. 3. In this and all subsequent figures, the dashed red line represents the local boundary layer thickness.

Figure 10

Variation of the $r_3^{\left(\min \right)}/{\delta }^{*}$ along the streamwise direction. The solid red line refers to the maps of $\langle d{u}^2\rangle$ in Fig. 9, while the dashed blue and dotted black lines refer to the maps of ${\gamma }^{\left(LL\right)}{\langle d{u}^2\rangle }^{\left(LL\right)}$ in Fig. 12 and ${\gamma }^{\left(TT\right)}{\langle d{u}^2\rangle }^{\left(TT\right)}$ in Fig. 14, respectively. Circles mark the streamwise locations of Fig. 3.

Figure 11

Evolution of $\langle d{u}^2\rangle$ on the $(r_1,Y)$ plane in the transitional region.

Figure 12

Evolution of ${\gamma }^{\left(LL\right)}{\langle d{u}^2\rangle }^{\left(LL\right)}$ on the $(r_3,Y)$ plane.

Figure 13

Evolution of ${\gamma }^{\left(LL\right)}{\langle d{u}^2\rangle }^{\left(LL\right)}$ on the $(r_1,Y)$ plane.

Figure 14

Evolution of ${\gamma }^{\left(TT\right)}{\langle d{u}^2\rangle }^{\left(TT\right)}$ on the $(r_3,Y)$ plane; their corresponding locations are $TRM1$ and $TR0\text{−}TR6$, shown in Fig. 3.

Figure 15

Evolution of ${\gamma }^{\left(TT\right)}{\langle d{u}^2\rangle }^{\left(TT\right)}$ on the $(r_1,Y)$ plane.

Figure 16

Evolution of ${\gamma }^{\left(TT\right)}$ on the $(r_3,Y)$ plane; pink isolines are defined as ${\gamma }^{\left(TT\right)}=0.7{\gamma }_{\max }^{\left(TT\right)}$.

Figure 17

Three-dimensional plots of ${\gamma }^{\left(TT\right)}$ at streamwise locations $TR2, TR3$, and $TR4$ (from left to right). Black isolines denote ${\gamma }^{\left(TT\right)}=0.7{\gamma }_{\max }^{\left(TT\right)}(r_3=0)$.

Figure 18

Running average of $\langle d{u}^2\rangle$ versus the number of samples at a streamwise location inside the fully turbulent region, $(X-X_0)/L_0=2415$, at the fixed height $Y/L_0=1.3$ and two different $r_1$ separations.