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160<br />
Multi-scale exact coherent structures in channel flow<br />
T. Itano ∗ , F. Waleffe † and S. Toh ‡<br />
Recent discoveries of coherent solutions of the Navier-Stokes equations in channel<br />
and pipe flows are opening new avenues for understanding the dynamics of shear<br />
turbulence 1 . These discoveries have been made possible by a bifurcation method that<br />
leads to unstable saddle-node branches of solutions that do not bifurcate from the<br />
simple laminar solution 2 .<br />
Here, the method is applied to find another type of traveling wave solutions in<br />
channel flow, that contains not a single structure but several structures at different<br />
scales. This multi-scale structure is of great interest since it is similar to what is<br />
observed in wall-bounded shear flows. Our new solution contains two distinct types of<br />
structures simultaneously. One is a near-wall structure (NWS) located in the bufferlayer,<br />
consisting of wavy low-speed streaks flanked by staggered quasi-streamwise<br />
vortices. The other is a mean-flow perturbation with a spanwise wavelength that<br />
is twice that of the near-wall structure. The latter is suggestive of the large-scale<br />
structure (LSS) observed over the buffer layer in turbulent channel flow 3 . We call<br />
this new solution the “double-decker structure” (DDS).<br />
The bifurcation diagram of these DDS in the (cx,ReP )spaceisshowninfigure<br />
(a) below, where cx is the streamwise phase speed of the solution normalized<br />
by the laminar centerline velocity. The near-wall structure previously obtained 1 is<br />
also calculated, which is shown as a reference by a curve with triangles in the figure.<br />
The second type of DDS is emerged at ReP =1553 in the diagram. It is clearly<br />
found that the existence of perturbation reduces the phase speed of the second type<br />
of DDS. In figure (b), the DDS is visualized by contours of the first and second Fourier<br />
components of streamwise velocity with respect to the spanwise wavenumber.<br />
cx 0.60<br />
0.55<br />
0.50<br />
0.45<br />
0.40<br />
Lz<br />
double-decker 1<br />
double-decker 2<br />
1400 1600 1800 ReP<br />
(a) The bifurcation diagram at<br />
the streamwise and spanwise wavelengths,<br />
(α, γ) = (1.00, 1.94).<br />
(b) The cross view of the second type<br />
of DDS at ReP = 1553. ûkz=1(y, z) =<br />
−0.01(red), ûkz=2(y, z) = −0.1(cyan).<br />
∗Faculty of Engineering, Kansai University, Osaka, Japan<br />
† Department of Mathematics and Engineering Physics, University of Wisconsin, Wisconsin, USA.<br />
‡ Department of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto,<br />
Japan.<br />
1Waleffe Phys. Fluids 15, 1517 (2003), Wedin and Kerswell, J. Fluid Mech. 508, 333 (2004)<br />
2Nagata, J. Fluid Mech. 217, 519 (1990)<br />
3Toh and Itano, J. Fluid Mech. 524, 249 (2005)
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