Page 186 - Bird R.B. Transport phenomena

170 Chapter 5 Velocity Distributions in Turbulent Flow

To avoid working with two dependent variables, we introduce the stream function as
discussed in §4.2. For axially symmetric flow, the stream function is defined as follows:
дф
i
i дф
-''
V =
=
** -H ~ ' M (5 6910)
This definition ensures that the equation of continuity in Eq. 5.6-4 is satisfied. Since we know
that v is z" 1 X some function of f, we deduce from Eq. 5.6-9 that ф must be proportional to z.
z
2
Furthermore ф must have dimensions of (velocity) X (length) , hence the stream function
must have the form
U)
ф(г, z) = v zF(0 (5.6-11)
in which F is a dimensionless function of f = r/z. From Eqs. 5.6-9 and 10 we then get

The first two boundary conditions may now be rewritten as

B.C. 1: at f = 0, | - F = 0 (5.6-14)

B.C. 2: atf = O, ^ - ^ = 0 (5.6-15)

If we expand F in a Taylor series about f = 0,

F(O = a + bt + c£ + de + et 4 + -- (5.6-16)
2
then the first boundary condition gives a = 0, and the second gives b = d = 0. We will use this
result presently.
Substitution of the velocity expressions of Eqs. 5.6-12 and 13 into the equation of motion
in Eq. 5.6-5 then gives a third-order differential equation for F,

This may be integrated to give
F'
= F" - j + Q (5.6-18)

in which the constant of integration must be zero; this can be seen by using the Taylor series
in Eq. 5.6-16 along with the fact that a, b, and d are all zero.
5
Equation 5.6-18 was first solved by Schlichting. First one changes the independent vari-
able by setting f = In p. The resulting second-order differential equation contains only the de-
pendent variable and its first two derivatives. Equations of this type can be solved by
elementary methods. The first integration gives
2
ff' = IF + \F + C 2 (5.6-19)
Once again, knowing the behavior of F near f = 0, we conclude that the second constant of in-
tegration is zero. Equation 5.6-19 is then a first-order separable equation, and it may be solved
to give

^ , (5.6-20)
2

5
H. Schlichting, Zeits. f. angew. Math. u. Mech., 13, 260-263 (1933).