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form of the thermodynamic equation to understand its behavior more precisely.
Unlike the preceding analysis where we found similarity solutions, we will now
make use explicitly of governing equations. We make the additional simplifying
assumptions:
a. The flow is steady state (
t
¼
0). (It has forever been thus. So, in the words of
the former rock group The Talking Heads, ''
@=@
'').
b. The plume is symmetrical about its center and can therefore be described in
cylindrical coordinates, where u and w are the radial and vertical components of
the wind, respectively. There is therefore no azimuthal variation in any variable.
The azimuthal wind component,
v
, is set to zero, so there is no vertical vorticity.
Radial variations in variables are similar at all heights (i.e., for all values of z).
c. The
...
same as it ever was
...
radial
component of
radial
inflow represents
the
entrainment of
environmental air and can be parameterized as
u
¼
w
ð
2
:
179
Þ
is a positive constant (fractional) entrainment rate.
d. The radial profiles of vertical velocity and buoyancy are such that each is zero
beyond the mean radius of the thermal (R) and constant within the mean radius
of the thermal. Such a variation is termed a ''top hat'' profile (
Figure 2.13
), not
necessarily in honor of Fred Astaire and Ginger Rogers whose 1935 movie of the
same name involved a top hat (worn by the former). The top hat profile is the
simplest possible profile, though a Gaussian profile may be more realistic.
where
We will end up with four equations in four unknowns (u, w, B, and R). First,
the Boussinesq continuity equation in cylindrical coordinates is
1
=
r
@=@
r
ð
ru
Þþ@
w
=@
z
¼
0
ð
2
:
180
Þ
We eliminate u as a variable by substituting the entrainment relation (2.179) into
(2.180); so we now have three equations in three unknowns. After integrating the
Figure 2.13. Example of a top hat profile for vertical velocity (w) and buoyancy (B).
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