3.5.2 Equations for Mean Variables in Turbulent Flow
Equations that characterize turbulent flow result from a generalization of the
Navier–Stokes equations for laminar flow of Newtonian fluids, with addition of
terms for random velocity fluctuations. This generalization involves subjecting the
velocity field u j to the Reynolds decomposition into mean components u j , and
fluctuations u j ’ with a zero average.
The Boussinesq approximation makes it possible to neglect density fluctuations
in the various terms in the linear momentum budget equation generic for turbulent
flow, except for the term for gravity because of gravitational acceleration.
As a rule of thumb (Stull 1994), the practical application of the Boussinesq
approximation in linear momentum equation for turbulent flow involves simply
replacing every occurrence of q with
À
q and, g with g À h
0
v =h v
À
Á
À
Á
, where h v is the
virtual potential temperature.
3.5.2.1 Continuity Equation
The continuity equation expresses the mass conservation principle in differential
form, under conditions of incompressibility, by the following expression:
@ u j
À Á
@x j
¼
@
À
u þ u
0
j
@x j
¼ 0
ð3:49Þ
or
@
À
u
@x j
þ
@u
0
j
@x j
¼ 0
ð3:50Þ
so, calculating the means we have:
@ u
@x j
þ
@ u 0 j
@x j
¼
@u
@x j
þ
@u0 j
@x j
¼
@u
@x j
¼ 0
ð3:51Þ
The continuity equation for the mean flow becomes:
@
À
u
@x j
¼ 0
ð3:52Þ
Thus from Eqs. (3.51) and (3.49), the continuity equation for the instantaneous
velocity fluctuations becomes:
@u
0
j
@x j
¼ 0
ð3:53Þ
44
3 Characterization of Turbulent Flow in the Surface Boundary Layer