Mastering nonlinear flow dynamics for laminar flow control

A laminar flow control technique based on spanwise mean velocity gradients (SVGs) has recently proven successful in delaying transition in boundary layers. Here we take advantage of a well-known nonlinear effect, namely, the interaction of two oblique waves at high amplitude, to produce spanwise mean velocity variations. Against common belief we are able to fully master the first stage of this nonlinear interaction to generate steady and stable streamwise streaks, which in turn trigger the SVG method. Our experimental results show that the region of laminar flow can be extended by up to 230%.

Figure 1

Schematic of the flat plate in the wind tunnel test section. The measurement planes consist of a horizontal and a series of cross-sectional planes. Smoke injection slot is located at $x=180\mathrm{mm}$. Locations of disturbance wave generation slots are depicted in the figure.

Figure 2

(a) Graph of the dominating modes excited in oblique transition. (b) Amplitude of POW $\left(A_{{\omega }_1}^{\mathrm{POW}}\right)$, its subharmonic $\left(A_{{\omega }_1/2}^{\mathrm{POW}}\right)$ and the generated streamwise streaks $\left(A_{\mathrm{ST}}\right)$ for C1 (black) and C2 (gray) configurations. $F_{\mathrm{POW}}=130$. (c) and (d) Reconstructed unsteady $u_u$ (instantaneous) and steady $u_s$ disturbances, respectively, for C1 configuration and $F_{\mathrm{POW}}=130$. In (d) high-speed streaks are located at $z{\beta }_1^{\mathrm{POW}}=-\pi /2,\pi /2$, and $3\pi /2$.

Figure 3

(a) and (b) Disturbance amplitude growth curves for uncontrolled TSW $\left(A_{2\mathrm{D}}^{\mathrm{TSW}}\right)$, controlled TSW $\left(A_{3\mathrm{D}}^{\mathrm{TSW}}\right)$, and POW $\left(A_{{\omega }_1}^{\mathrm{POW}}\right)$ for C1 (a) and C2 (b) configurations. $F_{\mathrm{TSW}}=100$ and $F_{\mathrm{POW}}=130$. (c) and (d) Reconstructed instantaneous unsteady disturbances $u_u$ for uncontrolled (c) and controlled (d) cases. $F_{\mathrm{TSW}}=130$ and $F_{\mathrm{POW}}=340$.

Figure 4

Disturbance energy evolution in the $F\text{−}{\text{Re}}_x$ plane (measured at $y/\delta =0.86$ and $z{\beta }_1^{\mathrm{POW}}=0)$. Measured frequencies, to produce the contour plots, are indicated by white dashed-dotted lines and tick marks on the vertical axes. Branches I and II of the neutral stability curve for the reference BL are plotted as dark dashed-dotted and dashed lines, respectively. Transition locations $\left(\gamma =0.5\right)$ are plotted as cross symbols. Top row: C1 configuration; (a), (b), and (c) correspond to $F_{\mathrm{POW}}=0$ (uncontrolled reference BL), $F_{\mathrm{POW}}=340$, and $F_{\mathrm{POW}}=450$, respectively. (d) Varying $F_{\mathrm{POW}}$ for fixed $F_{\mathrm{TSW}}=115$. Bottom row: C2 configuration; (e), (f), and (g) correspond to $F_{\mathrm{POW}}=0,F_{\mathrm{POW}}=130$, and $F_{\mathrm{POW}}=160$, respectively. (h) Varying $F_{\mathrm{POW}}$ for fixed $F_{\mathrm{TSW}}=100$. Vertical solid lines indicate transition locations for uncontrolled cases in (d) and (h).

Figure 5

Smoke flow visualization for C2 with $F_{\mathrm{TSW}}=145$ and $F_{\mathrm{POW}}=160$. (a) Control off and (b) control on.