Oblique Laminar-Turbulent Interfaces in Plane Shear Flows

Localized structures such as turbulent stripes and turbulent spots are typical features of transitional wall-bounded flows in the subcritical regime. Based on an assumption for scale separation between large and small scales, we show analytically that the corresponding laminar-turbulent interfaces are always oblique with respect to the mean direction of the flow. In the case of plane Couette flow, the mismatch between the streamwise flow rates near the boundaries of the turbulence patch generates a large-scale flow with a nonzero spanwise component. Advection of the small-scale turbulent fluctuations (streaks) by the corresponding large-scale flow distorts the shape of the turbulence patch and is responsible for its oblique growth. This mechanism can be easily extended to other subcritical flows such as plane Poiseuille flow or Taylor-Couette flow.

Figure 1

Numerical simulation of turbulent spots in PCF at $R=360$ (left) and $R=500$ (right) (streamwise velocity in the midplane). Simulations in a periodic domain $\Lambda =500$ using $(N_x,N_z)=(1536,2048)$ spectral modes. Only a subdomain of extent $250\times 250$ is displayed here.

Figure 2

Top: Developed stripe pattern at $R=325$: streamwise velocity in the midplane and associated $y$-integrated large-scale flow $({\overline{U}}_x,{\overline{U}}_z)$ computed using cutoff wavelength ${\lambda }_c=50$. Middle: Time-averaged premultiplied spectrum for the streamwise velocity component $u_x$ as a function of the streamwise (${\lambda }_x$) and spanwise (${\lambda }_z$) wavelengths. The two white lines indicate $\lambda =20$ and 50. Bottom: PDF of the angle $\alpha$ for the large-scale flow $(U_x,U_z)(y)$ and the $y$-integrated large-scale flow $({\overline{U}}_x,{\overline{U}}_z)$ corresponding to Fig. 2 (top). Simulation with $N_x=N_z=768$ in a domain $\Lambda =250$

Figure 3

Obliquely growing spot in PCF at $R=360$ (grey: streamwise velocity in the midplane) and associated $y$-integrated large-scale flow $({\overline{U}}_x,{\overline{U}}_z)$$(x,z)$ (arrows) computed with ${\lambda }_c=20$. From left to right: $t=200$, 300, and 400. Simulations in a periodic domain with $\Lambda =500$ and $N_x=1536$, $N_z=2048$ spectral modes. Only the subdomain $[-60:60]\times [-60:60]$ is displayed here.

Figure 4

Spanwise cut of the velocity field for the spot displayed in Fig. 3 (right panel) for $z=5$. $y$-dependent large-scale flow $U_x(x,y)$ (thin blue lines) with the base flow subtracted. $y$-averaged streamwise velocities ${\overline{U}}_x(x)$ and ${\overline{U}}_z(x)$ (respectively, red and black lines). Grey zones indicate the overhang regions using the criterion $|{\overline{U}}_x|>0.5\max |{\overline{U}}_x|$. The scale for the velocities is arbitrary but is the same for ${\overline{U}}_x(x)$ and ${\overline{U}}_z(x)$.