Abstracts - KTH Mechanics

36
Equilibrium boundary layers revisited
Y. Maciel a
The purpose of this presentation is to clarify some aspects of the similarity theory
of the outer region of turbulent boundary layers (TBL), as well as concepts that are
related to it, and to bring new ideas on the subject. The similarity analysis of the outer
region of TBL is presented in a general form and in a manner which is consistent with
the asymptotic theory for most pressure gradient conditions (assuming a two-region
structure of the TBL). In this general form, the outer scales are not determined a priori
and it is not assumed that the mean velocity defect and the three Reynolds stresses
share a common velocity scale. This similarity analysis therefore encompasses all the
various similarity analyses found in the literature. It is shown that in the limit of an
infinite Reynolds number, the use of different scales for mean velocity defect and
Reynolds stresses, as was done by Castillo and George1, is unnecessary. In the
asymptotic limit, the Reynolds normal stresses do not intervene, even in strong
adverse-pressure-gradient (APG) flows, and self-similarity leads to a common velocity
scale for the velocity defect and the Reynolds shear stress. The only necessary
condition for self-similarity is, not surprisingly, a constant ratio of the turbulent and
streamwise time scales.
At a finite Reynolds number, if multiple scales are assumed, then there exist five
conditions for self-similarity. If a single turbulent velocity scale is assumed instead,
then there exist three conditions for self-similarity at finite Reynolds number. These
are two conditions on any two of the characteristic turbulent scales (length, time and
velocity) and a corollary condition on the streamwise evolution of the freestream
velocity. As a consequence of the multiple conditions necessary for self-similarity at a
finite Reynolds number, turbulent boundary layers found in the real world are almost
never in a state of equilibrium, although they might closely approach it, contrarily to
what has been claimed in some recent papers.
It is also argued that the most appropriate turbulent (outer) velocity scale for all
TBL is the Zagarola-Smits velocity and not the friction velocity or the freestream
velocity for ZPG and mild PG and not the scales proposed by Mellor and Gibson2 or
Perry and Schofield3 for strong APG. The ZS velocity scales correctly the mean
velocity defect and all the Reynolds stresses in all pressure gradient and roughness
conditions while all the other proposed scales do not. For ZPG and mild PG flow
cases, if one accepts the ZS velocity as the turbulent velocity scale, then it should also
be the inner velocity scale. Experimental evidence suggests that it is indeed a proper
inner velocity scale for ZPG and mild PG flow cases, like the friction velocity. For
strong APG flows, the ZS velocity cannot be used in the inner region and has to be
1/3
replaced by the viscous/pressure-gradient velocity, u = ( −νUdU
/ dx ) .
p e e
a Mechanical Engineering Dept., Laval University, Quebec city, G1K 7P4, Canada.
1Castillo and George, AIAA Journal, 39, 1 (2001).
2Mellor and Gibson, J. Fluid Mech., 24, 2 (1966).
3Perry and Schofield, Phys. Fluids, 16, 12 (1973)