WATER JET CONFERENCE - Waterjet Technology Association

skin friction coefficient but are closed using a third auxiliary relation which may take a
variety of forms and relies entirely or partly on empirical observation.
According to the literature, the only readily adaptable method which takes into
account, empirically, relaminarization phenomena is the method for the prediction of
axisymmetric turbulent boundary layers in conical nozzles due to Au (1972). A version of
the momentum equation for axisymmetric incompressible flow is developed which is
shown to be a more general form of the two dimensional Karman equation utilized by
previous workers. Closure is achieved using an extended version of Heads' entrainment
equation (1958) and resultant entrainment functions are formulated empirically from the
experimental results. Good agreement with experiment for boundary layer flow in
venturi-meters over the inlet Reynolds number range 1 x 10 5 < Re < 5 x 10 5 was obtained.
In figure 10 the flow element is bounded by the wall surface BC and an imagined
inner surface AD of radius r c. It is assumed that flow is axisymmetric and the velocity at
the edge of the boundary layer (r = r c ) is given by the potential flow solution at the wall.
From continuity the mass flow across the annular surface AB is equal to the sum
of the mass flows from the cylindrical surface AD and the annular surface CD. The mass
flows have associated momentum fluxes and the change of momentum flux in the
x-direction in the element equals the sum of all the applied forces on the element. These
applied forces consist of pressure forces on the surfaces AB and CD, the x-direction
pressure force due to the sloping wall and x direction force due to wall friction.
The x-direction momentum equation may be written:
R
R
d
dx [ u2rdr] − U d
1 dp
[ purdr] +
dx ∫ 2 dx (R2 2
∫
− rc ) + R w = 0 (2-31)
rc
rc
Applying the one-dimensional Euler equation and assuming incompressible flow,
equation (2-31) becomes:
d
dx
R
=
R − {[
2
2RU (H2 + 2) − (H + 2)]
U dU
dx − dR
R dx
64
Cf
+ } (2-32)
2
where H = * / (2-33)
Cf = w \ 1
2 U 2 (2-34)
Au uses the Ludwieg-Tillmann equation (1949) for skin friction coefficient:
Cf = 0.246(10) −0.678H Re −0.268 (2-35)
White (1974) reports that (2-35) correlates available data to within +10% and that
equation (2-36) is a more accurate correlation (+3%). Equation (236) is adopted in this
analysis.