Mechanics of Fluids

344 Boundary layers, wakes and other shear layers
the idea of mixing length permitted useful progress in the investigation of
turbulent flow. Similar results are, however, obtainable by more rigorous
methods, and the concept is now largely outmoded.
8.12 VELOCITY DISTRIBUTION IN TURBULENT FLOW
It was Prandtl who first deduced an acceptable expression for the variation
of velocity in turbulent flow past a flat plate and in a circular pipe. He
used his mixing length concept (Section 8.11) together with some intuitive
assumptions about its variation with distance from the boundary. Using
somewhat different assumptions, von Kármán and others obtained the same
basic result. The expressions are all semi-empirical in that the values of
the constant terms have to be determined by experiment but the forms of
the expressions are derived theoretically.
However, the same results may be obtained from dimensional analysis
without the hypothesis of mixing length and in fact with less far-reaching
assumptions. We shall here use this more general method based on dimensional
analysis.
For fully developed turbulent flow in a circular pipe we require primarily
to know the way in which the time-average value of the velocity varies
with position in the cross-section. If the flow is steady – in the sense that the
time-average value at a given point does not change with time – then considerations
of symmetry indicate that this velocity u is the same at all points at
the same distance from the pipe axis. The independent variables that affect
the value of u are the density ϱ and dynamic viscosity µ of the fluid, the
radius R of the pipe, the position of the point (distance y from the pipe wall,
or radius r = R − y from the axis), the roughness of the pipe wall – which
may be represented by some characteristic height k of the surface bumps (as
in Section 7.3), and the shear stress τ0 at the wall. (The pressure drop divided
by length could be used as a variable in place of τ0, as it is simply related
to τ0, but τ0 – here taken as positive – is more convenient for our purpose.)
Application of the principles of dimensional analysis suggests the following
relation:
�
u
R
= φ1
(τ0/ϱ) 1/2 ν
� �1/2 τ0
ϱ
, y
�
k
,
R R
(8.41)
where ν = µ/ϱ and φ{}means ‘some function of’. (Suffixes on the φ symbols
will be used to distinguish one function from another, but in general the form
of the function will not be known.)
Noting that (τ0/ϱ) 1/2 has the same dimensions as velocity, it is convenient
to make the substitution
uτ =
� �1/2 τ0
ϱ
where uτ is known as the friction velocity. Its introduction allows velocity
distribution equations to be expressed in simpler and more compact forms.