Page 189 - Bird R.B. Transport phenomena

Problems 173

5A.2 Velocity distribution in turbulent pipe flow. Water is flowing through a long, straight, level
run of smooth 6.00 in. i.d. pipe, at a temperature of 68°F. The pressure gradient along the
length of the pipe is 1.0 psi/mi.
(a) Determine the wall shear stress т in psi (lty/in. ) and Pa.
2
0
(b) Assume the flow to be turbulent and determine the radial distances from the pipe wall at
which v /v z>max = 0.0, 0.1, 0.2, 0.4, 0.7, 0.85,1.0.
z
(c) Plot the complete velocity profile, v /v zmax vs.y = R-r.
z
(d) Is the assumption of turbulent flow justified?
(e) What is the mass flow rate?
5B.1 Average flow velocity in turbulent tube flow.
(a) For the turbulent flow in smooth circular tubes, the function 1

3
is sometimes useful for curve-fitting purposes: near Re = 4 X 10 , n = 6; near Re = 1.1 X 10 ,
5
6
n = 7; and near Re = 3.2 X 10 , n = 10. Show that the ratio of average to maximum velocity is
<**> = 2n 2
1) * '
and verify the result in Eq. 5.1-5.
(b) Sketch the logarithmic profile in Eq. 5.3-4 as a function of r when applied to a circular
tube of radius R. Then show how this function may be integrated over the tube cross section
to get Eq. 5.5-1. List all the assumptions that have been made to get this result.
56.2 Mass flow rate in a turbulent circular jet.
(a) Verify that the velocity distributions in Eqs. 5.6-21 and 22 do indeed satisfy the differen-
tial equations and boundary conditions.
(b) Verify that Eq. 5.6-25 follows from Eq. 5.6-21.
5B.3 The eddy viscosity expression in the viscous sublayer. Verify that Eq. 5.4-2 for the eddy vis-
cosity comes directly from the Taylor series expression in Eq. 5.3-13.
5C.1 Two-dimensional turbulent jet. A fluid jet issues forth from a slot perpendicular to the xy-
plane and emerges in the z direction into a semi-infinite medium of the same fluid. The width
of the slot in the у direction is W. Follow the pattern of Example 5.6-1 to find the time-
smoothed velocity profiles in the system.
(a) Assume the similar profiles
v /v w = f(O with £ = x/z (5C.1-1)
z
Show that the momentum conservation statement leads to the fact that the centerline velocity
1/2
must be proportional to z~ .
(b) Introduce a stream function ф such that v = -дф/дх and v = Л-дф/dz. Show that the re-
z x
sult in (a) along with dimensional considerations leads to the following form for ф:
(5C.1-2)
Here F(f) is a dimensionless stream function, which will be determined from the equation of
motion for the fluid.

H. Schlichting, Boundary-layer Theory, McGraw-Hill, New York, 7th edition (1979), pp. 596-600.
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