Page 46 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS. DEFINITIONS AND METHODS 29

(f) Turbulent flows
Due to a three-dimensional instability of laminar flow, the flow field becomes turbulent:
the flow velocity and pressure are no longer steady but contain randomly fluctuating
components. Two-dimensional disturbances in the laminar flow field eventually become
three-dimensional, and this is soon followed by turbulence. The critical Reynolds number
for the onset of turbulence is best stated in terms of the width of the flow’ and depends on
the shape of the laminar velocity profile; it is typically O( lo2) for profiles with inflection
points and 6(103) or more for profiles of single curvature. Thus, boundary layers in
falling pressures are a good deal more stable than those that are suffering a pressure rise;
similarly, jets and wakes are also very unstable.
When turbulence appears, as originally observed and described by Reynolds for pipe
flow, the flow field may be expressed as V + v and P + p, where the lower-case quantities
represent the fluctuating components about the mean (with zero average) and V and P are
the mean components; V = (VI, U2, U3}T and v = (u1, u2, ~ 3 in )an ~(XI, x2, x3)-frame.
Substitution into the Navier-Stokes equations and averaging yields
au; au; 1 ap v --E ,,),
-+u +- a i,j=l,2,3, (2.85)
at ax, p ax; ax, axj

where the indicia1 notation is utilized, in which repeated indices imply summation; e.g.
is
Uj(aUj/axj) = E&, [Uj(aUj/axj)]. The new term -q the correlation of ui and
ii,, obtained by multiplying the two, integrating over a long time (appropriate to the
flow under investigation), and then dividing by the time interval. The quantity --pW
represents additional normal and shear stresses due to additional momentum transfer
associated with the velocity fluctuations,f the so-called Reynolds stresses. Thus, in a
simple two-dimensional shear flow predominantly in the XI -direction, the viscous shearing
stress p(aUl/axz) is increased by -pm, which has the same sign as aUl/ax2 and is
sometimes written as p,(aUl/axz), where the subscript t is for ‘turbulence’; ut = pr/p
is the so-called kinematic eddy viscosity. To differentiate the quantities associated with
viscous stresses from those related to turbulence, or equivalently the quantities associated
with velocity fluctuations at the molecular (Brownian) scale from the turbulent ones, the
subscript m (for ‘molecular’) is introduced, as in u, in equation (2.85); v, here is the
same as u in equation (2.63).
The Reynolds stresses are generally much larger than the viscous ones, except near
walls, in the viscous sublayer (Hinze 1975); on the wall itself, all turbulent fluctuations
vanish. One of the central problems of turbulent flows is the derivation of satisfactory
relations for Reynolds stresses in terms of the mean flow field (Townsend 1961).
The spatial structure of a turbulent flow may be described statistically by correlation
functions or by spectra. The general space-time correlation function between, say, u; at
point x and uj at point x + r is defined by

‘The width of a jet or a wake, or the thickness of a boundary layer.
*This is an essential characteristic of turbulence. As noted by Townsend (1961). ‘a sharp increase in friction,
or in heat and mass transfer is frequently used to determine the onset of turbulent motion if direct observation
of the fluctuations is inconvenient’.