WATER JET CONFERENCE - Waterjet Technology Association

In the limit as h and k tend to zero, the finite-difference scheme yields the exact
solution. As a check on the uncertainty associated with practical finitedifference
approximations, solutions were obtained using successively finer meshes. The maximum
percentage change between successive solutions was determined and recorded together
with other relevant information in table 1. Nozzle design (2) of figure 13 was used with
U i=5 m/s.
TABLE 1
N K ΔX / L (Δu /U ) max No. iterations approx. cpu time
______________________________________________________________
12 12 0.016 69 2 mins
20 20 0.010 +0.39% 69 8 mins
32 32 0.00625 +0.12% 130 15 mins
40 40 0.005 +0.06% 210 28 mins
As an adequate compromise between time and uncertainty for the purpose of this
study, A x= 0.0025 and h/k = 4 were selected. This selection provided sufficient wall
points for the boundary layer analysis of section 2.5. Resultant potential flow solutions
were within 0.5% of the exact solution. The value of the relaxation parameter was set to
1.75 throughout and is suggested as sufficiently close to optimum.
Figure 5 illustrates typical examples of the comparison of the wall velocity distribution
with that for simple one-dimensional flow. Figure 6 indicates typical examples of the
radial velocity distribution at the nozzle exit. The figures (5,6) highlight the differences in
velocity distribution between nozzle contractions terminating in parallel cylindrical
sections and those terminating in straight taper sections. The well documented regions of
velocity undershoot which occur in both cases and the region of wall velocity overshoot
in the vicinity of the joint between the taper section and the parallel section may be
observed. A notable difference between the two examples is the velocity defect near the
wall associated with the nonuniform exit velocity in the case of the straight taper exit
section nozzle as opposed to the nearly uniform exit velocity distribution associated with
the cylindrical exit section nozzle. It must be stated however that for contractions
terminating in non-parallel exit sections the flow contracts on leaving the exit plane. The
fluid accelerates towards uniform parallel flow associated with a minimum in jet flow
diameter, dj min., before subsequent downstream expansion.
Thus in the case of the non-parallel exit section nozzle exit velocity profile
non-uniformity is subsequently relaxed and perfectly uniform flow is achieved
downstream of the nozzle exit. A jet contraction coefficient may be defined as C c = (d j
min./d o). The associated hydraulic advantages are that additional viscous drag due to fluid
contact with the contraction boundary is not incurred and finite, favourable pressure
gradients exist at the nozzle exit. In addition it is well known that in the absence of
strongly favourable pressure gradients e.g. in flat plate or developing pipe flow,
boundary-layer growth is rapid.
Strictly the potential flow solution of the contracting jet influences the upstream
nozzle flow solution to some degree. The exit boundary condition of uniform flow an
arbitrary distance downstream of the nozzle exit is somewhat artificial but provides an
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