Boundary-layer transition over a rotating broad cone

The route to turbulence in the boundary layer on a rotating broad cone is investigated using hot-wire anemometry measuring the azimuthal velocity. The stationary fundamental mode is triggered by 24 deterministic small roughness elements distributed evenly at a specific distance from the cone apex. The stationary vortices, having a wave number of 24, correspond to the fundamental mode and these are initially the dominant disturbance-energy carrying structures. This mode is found to saturate and is followed by rapid growth of the nonstationary primary mode as well as the stationary and nonstationary first harmonics, leading to transition to turbulence. The amplitudes of these are plotted in a way to highlight the continued growth after saturation of the fundamental stationary mode.

Figure 1

The coordinate system $(x,\theta ,z)$ on the rotating cone.

Figure 2

Transition locations $x_{\mathrm{tr}}$ against nondimensional roughness height $h=h^{*}/{\delta }_{\nu }^{*}$. Black, blue, green, and red markers correspond to $900,1200,1500$, and 1800 rpm.

Figure 3

Power-spectrum density $log\left(E\right)$ for the cone with 24 roughness elements: (a) $h^{*}=8\mu \mathrm{m}$, 900 rpm; (b) $h^{*}=17\mu \mathrm{m}$, 900 rpm; and (c) $h^{*}=8\mu \mathrm{m}$, 1800 rpm. The solid line shows the neutral curve for stationary disturbances from linear stability theory. The dashed lines at (a) and (b) $x=267$, and (c) $x=377$ locate the roughness elements. The black arrow on the abscissa indicates the transition location for the respective case, as in Fig. 2.

Figure 4

(a) rms and (b) spatial growth rate ($-{\alpha }_i$) of the azimuthal velocity fluctuation with different roughness heights (900 rpm): $h^{*}=4\mu \mathrm{m}$ (black), $8\mu \mathrm{m}$ (blue), $13\mu \mathrm{m}$ (green), and $17\mu \mathrm{m}$ (red). The rectangle at $x=267$ indicates the roughness elements. The dash-dotted and solid lines show the stationary components of the fundamental and the first harmonic (${\omega }^{*}/{\mathrm{\Omega }}^{*}=24$ and 48), respectively. In (a), the solid marker (upper right) indicates the total rms level for the turbulent flow. The arrows on the abscissa in (a) show the transition locations as in Fig. 2; in (b), the thin line shows the growth rate based on LLSA.

Figure 5

(a) rms and (b) spatial growth rate ($-{\alpha }_i$) of the azimuthal velocity fluctuation for different rotational speed cases ($h^{*}=8\mu \mathrm{m}$): 900 rpm (black), 1200 rpm (blue), 1500 rpm (green), and 1800 rpm (red). The rectangles at the bottom indicate the roughness elements. The dash-dotted and solid lines show the stationary components of the fundamental and the first harmonic ${\omega }^{*}/{\mathrm{\Omega }}^{*}=24$ and 48, respectively. In (a), the solid marker (upper right) indicates the total rms level for the turbulent flow. The arrows on the abscissa show the transition location as in Fig. 2; in (b), the thin solid and dashed lines show the growth rate for the fundamental and first harmonic based on LLSA.

Figure 6

Amplitude of different disturbance components as a function of the amplitude of the fundamental disturbance: (a) ${v^{\prime }}_{\text{rms},24}$, (b) ${\tilde{v}}_{\text{rms},48}$, (c) ${v^{\prime }}_{\text{rms},48}$ and its ratio to the amplitude of the fundamental disturbance: (a) ${v^{\prime }}_{\text{rms},24}/{\tilde{v}}_{\text{rms},24}$, (b) ${\tilde{v}}_{\text{rms},48}/{\tilde{v}}_{\text{rms},24}$, (c) ${v^{\prime }}_{\text{rms},48}/{\tilde{v}}_{\text{rms},24}$ for ${\mathrm{\Omega }}^{*}$: 900 rpm (black), 1200 rpm (blue), 1500 rpm (green), and 1800 rpm (red). The squares indicate the location of roughness elements and the circles the transition position. The anticlockwise arrows show the direction of increasing $x$.