Page 185 - Bird R.B. Transport phenomena

§5.6 Turbulent Flow in Jets 169

Circular hole

Fig. 5.6-1. Circular jet
emerging from a plane
wall.

SOLUTION In order to use Eq. 5.4-3 it is necessary to know how b and v z max - v 7 min vary with z for the cir-
cular jet. We know that the total rate of flow of z-momentum 7 will be" the same for all values
of z. We presume that the convective momentum flux is much greater than the viscous mo-
mentum flux. This permits us to postulate that the jet width b depends on /, on the density p
and the kinematic viscosity v of the fluid, and on the downstream distance z from the wall.
2
The only combination of these variables that has the dimensions of length is b °c Jz/pv , so
that the jet width is proportional to z.
We next postulate that the velocity profiles are "similar/' that is,
= № where £ = —^- (5.6-1)
Hz)
which seems like a plausible proposal; here v ZfTnax is the velocity along the centerline. When
this is substituted into the expression for the rate of momentum flow in the jet (neglecting the
>=гг pv\rdrde (5.6-2)
contribution from r vv)

Jo Jo
we find that

2 2
2
2
=
] = 27T b vl max J f £d£ = constant X pb v z>n (5.6-3)
P
Since / does not depend on z and since b is proportional to z, then v~ z.max has to be inversely
proportional to z.
in Eq. 5.4-3 occurs at the outer edge of the jet and is zero. Therefore because b <*
The v zMn
(0
oc z~\ we find from Eq. 5.4-3 that /x is a constant. Thus we can use the equations
z and v z>max
(0
of motion for laminar flow and replace the viscosity /x by the eddy viscosity /x , or v by v {t)
In the jet the main motion is in the z direction; that is \v r\ « \v z\. Hence we can use a
boundary layer approximation (see §4.4) for the time-smoothed equations of change and write
continuity: (5.6-4)
motion: V v r (5.6-5)
> dr z ~ r Tr[
These equations are to be solved with the following boundary conditions:

B.C.I: atr = 0, v r = 0 (5.6-6)
B.C. 2: at r = 0, dvjdr = 0 (5.6-7)
B.C. 3: at z = °o, у = Q (5.6-8)
The last boundary condition is automatically satisfied, inasmuch as we have already found
that u 2/max is inversely proportional to z. We now seek a solution to Eq. 5.6-5 of the form of Eq
5.6-1 with b = z.