Page 50 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 33

In some of the work to be presented later (e.g. in Chapter 7, Volume 2), particular forms
of the foregoing for pipe flows - containing considerable empirical input - is utilized.
Thus, for pipe flow, a friction factor, f, is sometimes defined via

(2.97)

where U is the mean flow velocity; f is given empirically, for instance by the Colebrook
equation,
-- - -2 log,o { = -} , (2.98)
+
2.51
a
1
3.7
Rea
where Re is the Reynolds number based on the diameter, D, and k,/D is the relative
roughness.
Reverting now to equation (2.85) for a more general analysis of turbulent flow, it is
noted that -w is often not measured, but modelled mathematically. For example, by
means of Boussinesq’s eddy viscosity concept, one may write

(2.99)

where K is as given by (2.93); v, is the eddy viscosity which, unlike v,, (or v = v,,
in laminar flow), it is not a constant but is dependent on the flow field. The chosen
form depends on the turbulence model adopted - see, for instance, Launder & Spalding
(1972), Jones & Launder (1972), Launder & Sharma (1974), Rodi (1980), Lesieur (1990),
So et al. (1991), Wilcox (1993).
Perhaps the simplest model is based on Prandtl’s mixing-length hypothesis for 2-D or
axisymmetric flows, in which

v, = l2 (2.100)

where 1 is Prandtl’s mixing length, y = x2 is the coordinate measured away from the
wall, and U = U1 is the mean flow velocity.+ In the case of smooth pipes, for instance,
Nikuradse’s measurements yield the following empirical expression (Schlichting 1960):
1
- = 0.14-0.08 (2.101)
R
R being the pipe radius.
There are many other models, including so-called two-equation models, for turbulent
flow (Wilcox 1993). One of the first and most popular was pioneered by Launder and
Spalding. It is based on two scalar functions, already defined: K = i-, the average
turbulence (kinetic) energy per unit mass; and E, the rate of decay of turbulence energy
per unit mass, which is also the rate of transfer of energy from the large eddies to smaller
ones, and hence, in this latter capacity, it is independent of viscosity. In this so-called,

tIncidentally, this is the equation, with 1 c( y. originally used by Prandtl to prove the law of the wall.