It is well known that pipe Poiseuille flow undergoes transition to turbulence for Re on the order of 2,000. How about a plane Poiseuille flow?
For plane Poiseuille flow Lin (1955) predicted a critical stability Reynolds number of 5314.
LIN, C. C. 1955. The Theory of Hydrodynamic Stability. Cambridge University Press.
recent studies have shown that transition strongly dépends on the relative roughness height along the wall. Re=2000 is the classical admitted value but the transition can be postponed to 10000 for smooth walls.
Thank you Sébastien. But how about plane Poiseuille flow of plane Couette flow, is there any typical values of Reynolds number for the transition to turbulence? Thank you!
In this paper (see the link below) by numerical calculation it is shown that the plane Poiseuille flow can be stable up to Re larger than 2800.
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=380262
Thank you Hiqmet , can you please send me the paper (I have not access to JFM!). Here please there is an other question: I would like to know if the plane Poiseuille flow and plane Couette flow are linearly stable or can be unstable for small perturbations with respect to high Re number. Thank you!
For plane Poiseuille flow Lin (1955) predicted a critical stability Reynolds number of 5314.
LIN, C. C. 1955. The Theory of Hydrodynamic Stability. Cambridge University Press.
In this paper, in the link below, the authors discuss the linearized stability of the plane Poiseuille flow.
http://www.sciencedirect.com/science/article/pii/S0020746297000772
I hope this helps.
Sorry, I do not have access to JFM, but you may request a preprint from the authors.
Thank you for your helpful answers! I still have a question please!
Are the plane Poiseuille flow and plane Couette flow linearly stable for all Re numbers or can be unstable for small perturbations with respect to high Re number. Thank you!
Both plane Couette flow and Hagen-Poseuille flow( in a pipe) from a theoretical perspective are linearly stable to infinitesimal perturbations but may be unstable to finite disturbances. I believe Romanov showed in 1973 that all the normal modes of the linear stability problem for Plane Couette flow are damped for all wave numbers and Reynolds numbers. More details are given in the book "Hydrodynamic Stability" by Drazin and Reid.
Plane Poiseuille flow is linearly unstable to infinitesimal perturbations, and the critical Reynolds number Re=5772 ( computed using a spectral method)
Thank you Brian for the answer. Do you have any idea about the structures observed in a pipe flow? Are these structures the traveling wave solutions? Thank you!
For plane Couette flow, the typical critical reynolds number can be 325, 350, 370, to name a few.
There is a very interesting paper that addresses the stability of plane Couette Flow that has been slightly disturbed: See Barkley & Tuckerman " Stability analysis of perturbed plane Couette flow", Phys. Fluids, 11, 1187, 1999. You can download it here: http://homepages.warwick.ac.uk/~masax/Research/Papers/pf99.pdf
This paper shows that if you disturbed plane Couette flow by a small ribbon placed in the center of the channel, you can cause the flow to become unstable, by exciting stream-wise vortices which in turn can eventually lead to turbulence. This explains why "plane Couette flow" can be come unstable even though a linear stability analysis suggests it is stable to all infinitesimal disturbances, and as Shiuh-Hwa has pointed out the flow can have a critical Reynolds number as low as 300+.
To summarize, in the real world it is very hard if not impossible to create a true plane Couette flow. There are always unwanted disturbances, which Barkley and Tuckerman and others have shown, can trigger subcritical bifurcations that eventually lead to turbulence.
Plane Poiseuille flow is known to become turbulent at Re~1000:
R. Carlson, E. Widnall, and M. F. Peeters, J. Fluid Mech. 121, 487 (1982).
M. Nishioka and M. Asai, J. Fluid Mech. 150, 441 (1985).
F. Alavyoon, D. S. Henningson, and P. H. Alfredsson, Phys. Fluids 29, 1328 (1986).
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