Bypass transition delay using oscillations of spanwise wall velocity

Large eddy simulations are performed to investigate the possibility of bypass transition delay in spatially developing boundary layers. An open loop wall control mechanism is employed which consists of either spatial or temporal oscillations of the spanwise wall velocity. Both spatial and temporal oscillations show a delay in the sharp rise in skin friction coefficient which is characteristic of laminar-turbulent transition. An insight into the mechanism is offered based on a secondary filtering of the continuous Orr-Sommerfeld-Squire (OSQ) modes provided by the Stokes layer, and it is shown that the control mechanism selectively affects the low-frequency penetrating modes of the OSQ spectrum. This perspective clarifies the limitations of the mechanism's capability to create transition delay. Furthermore, we extend the two-mode model of bypass transition proposed by T. Zaki and P. Durbin [J. Fluid Mech. 531, 85 (2005)] to cases with wall control and illustrate the selective action of the wall oscillations on the penetrating mode in this simplified case.

Figure 1

Friction coefficient $c_f$ showing transition delay for wall oscillations with different wave numbers ($k=0.0125,0.0314,0.0628$) and $W_m=0.9$.

Figure 2

Transition delay for different start points of oscillation ($x_{\mathrm{start}}=0,200,300,400$). The start locations correspond to ${\text{Re}}_x=0.3,0.9,1.2,\text{and}1.5\times {10}^5$, respectively.

Figure 3

Transition delay for varying amplitude for $k=0.0628$.

Figure 4

Transition delay for varying amplitude.

Figure 5

Coefficient of skin friction development for pre- and posttransitional wall control.

Figure 6

Transition delay for different inlet turbulence intensities ($k=0.0314,{\text{Re}}_{x_{\mathrm{start}}}=0.9\times {10}^5$).

Figure 7

Transition delay with a short pretransitional wall control ($k=0.0314$). The control regions correspond to ${\text{Re}}_x=0.9–1.5\times {10}^5$ and ${\text{Re}}_x=0.9–5.7\times {10}^5$.

Figure 8

Transition delay for temporal oscillation cases ($W_m=0.9$).

Figure 9

Transition delay for $\omega =0.0117$ at different oscillation amplitudes.

Figure 10

Transition delay for temporal oscillation with $W_m=0.25$ and $\omega =0.0117$ and different $x_{\mathrm{start}}$ locations (${\text{Re}}_x=0.3$ and $1.2\times {10}^5$).

Figure 11

Peak streamwise fluctuations $u_{\mathrm{rms}}^{max}$ for different wave-number cases. The control region starts at ${\text{Re}}_x=0.9\times {10}^5$.

Figure 12

Wall normal location of peak streamwise velocity fluctuations $u_{\mathrm{rms}}^{max}$ for different wave-number cases.

Figure 13

Wall-normal profile of streamwise velocity fluctuations $u_{\mathrm{rms}}$ for different wave-number cases at $x=300$. Note that the wall-normal coordinate is scaled using the inlet displacement thickness ${\delta }_o^{*}$.

Figure 14

A near-wall $y-z$ plane at $x=300$ of the instantaneous streamwise velocity with the spanwise mean subtracted from the instantaneous values, for (a) the reference case and (b) wall control with $k=0.0314,W_m=0.9$. Dark (light) regions represent velocity deficit (excess).

Figure 15

(a) Stokes layer profile and (b) boundary layer profile at $x=300$. “o” symbols in panel (a) are the solutions from the analytical equation (7), while the lines represent simulation results.

Figure 16

Eigenfunctions for the (a) low-frequency penetrating mode A and (b) high-frequency nonpenetrating mode B at the inlet($x=0$). ($-$) Real part and ($--$) imaginary part.

Figure 17

A comparison of $u_{\mathrm{rms}}$ contours with and without wall control for a penetrating continuous OS mode (mode A). (a) No wall control and (b) Wall control.

Figure 18

A comparison of $u_{\mathrm{rms}}$ contours with and without wall control for a nonpenetrating continuous OS mode (mode B). (a) No wall control and (b) Wall control.

Figure 19

Instantaneous wall-normal velocity profile at $t=2000$ for the penetrating mode. (–) Cases with control and (- -) cases without control. The vertical dotted line represents the approximate height where the oscillating spanwise velocity amplitude decays to $\approx 0.1W_m$. (a) Wall-normal velocity profile in the free stream at $x=503$, (b) $x=503$, and (c) $x=747$.

Figure 20

Instantaneous wall-normal velocity profile at $t=2000$ for a nonpenetrating mode. (–) Cases with control and (- -) cases without control. The vertical dotted line represents the approximate height where the oscillating spanwise velocity amplitude decays to $\approx 0.1W_m$. (a) Wall-normal velocity profile in the free stream at $x=503$, (b) $x=503$, and (c) $x=747$.

Figure 21

$c_f$ variation for different continuous OS modes in the flow.