Page 190 - Bird R.B. Transport phenomena

174 Chapter 5 Velocity Distributions in Turbulent Flow

(c) Show that Eq. 5.4-2 and dimensional considerations lead to the following form for the tur-
bulent kinematic viscosity:

,/'> = yPi = \Vj/pWz U2 (5C.1-3)
9
Here Л is a dimensionless constant that has to be determined from experiments.
(d) Rewrite the equation of motion for the jet using the expression for the turbulent kine-
matic viscosity from (c) and the stream function from (b). Show that this leads to the follow-
ing differential equation:
i f ,2 + i Fpn _ A f w = 0 (5C.1-4)

For the sake of convenience, introduce a new variable
77 = f /4Л = x/4Az (5C.1-5)
and rewrite Eq. 5C.1-4.
(e) Next verify that the boundary conditions for Eq. 5C.1-4 are F(0) = 0, F'(0) = 0, and F(°°) = 0.
(f) Show that Eq. 5C.1-4 can be integrated to give
IFF - F" = constant (5C.1-6)

and that the boundary conditions require that the constant be zero.
(g) Show that further integration leads to
2
F -F = С 2 (5С.1-7)
where С is a constant of integration.

(h) Show that another integration leads to
F= -CtanhCry (5C.1-8)
and that the axial velocity can be found from this to be
Vj/oWC 2
2
v z = J * V sech C-q (5C.1-9)
(i) Next show that putting the axial velocity into the expression for the total momentum of
the jet leads to the value С = лКЗЛ for the integration constant. Rewrite Eq. 5C.1-9 in terms of
A rather than C. The value of A = 0.0102 gives good agreement with the experimental data. 2
The agreement is believed to be slightly better than that for the Prandtl mixing length
empiricism.
(j) Show that the mass flow rate across any line z = constant is given by

w = 2^3A / ^ (5C.1-10)

5C.2 Axial turbulent flow in an annulus. An annulus is bounded by cylindrical walls at r = aR
and r = R (where a < 1). Obtain expressions for the turbulent velocity profiles and the mass
flow rate. Apply the logarithmic profile of Eq. 5.3-3 for the flow in the neighborhood of each
wall. Assume that the location of the maximum in the velocity occurs on the same cylindrical
surface r = bR found for laminar annular flow:

2
H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 4th edition (1960), p. 607 and
Fig. 23.7.