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8.4.1 The reformed model

In this section the momentum conservation principle is applied on the control volume in Figure [*]. For large () the wall shear stress can be neglected compared to the inertial terms (the wave is assumed to have a negligible length). The momentum balance reads:

    (8.18)

where
    (8.19)

Given the velocity profile , the shape factor can be obtained in terms of . The expressions for for laminar and turbulent velocity profiles at section 1 easily can be calculated. Based on the assumptions used in the previous section, equation ([*]) reads:
    (8.20)

Rearranging equation ([*]) into a dimensionless form yields:
    (8.21)

Combining equations ([*]) and ([*]) yields
    (8.22)

where is the number which evolves from the momentum conservation equation. Equation ([*]) is the analogue of equation ([*]) and will be referred herein as the ``Bar-Meir's solution''.

= 125 true mm

Figure: The Froude number as a function of the relative height

It has been found that the solutions of the ``Bar-Meir's solution'' 21and the ``energy solution'' can be presented in a simple form. Moreover, these solutions can be applied to any cross section for the transition of the free surface flow to pressurized flow. The discussion here focuses on the circular cross section, since it is the only one used by diecasters. Solutions for other velocity profiles, such as laminar flow (Poiseuille paraboloid), are discussed in the Appendix 22. Note that the Froude number is based on the plunger velocity and not on the upstream velocity commonly used in the two-dimensional hydraulic jump.

The experimental data obtained by Garber poro:garber, and Karni poro:karnit and the transition from the free surface flow to pressurized flow represented by equations ([*]) and ([*]) for a circular cross section are presented in Figure [*] for a plug flow. The Miller's model (two dimensional) of the hydraulic jump is also presented in Figure [*]. This Figure shows clearly that the ``Bar-Meir's solution'' is in agreement with Karni's experimental results. The agreement between Garber's experimental results and the ``Bar-Meir's solution,'' with the exception of one point (at ), is good.

The experimental results obtained by Karni were taken when the critical velocity was obtained (liquid reached the pipe crown) while the experimental results from Garber are interpretation (kind of average) of subcritical velocities and supercritical velocities with the exception of the one point at (which is very closed to the ``Bar-Meir's solution''). Hence, it is reasonable to assume that the accuracy of Karni's results is better than Garber's results. However, these data points have to be taken with some caution23. Non of the experimental data sets were checked if a steady state was achieved and it is not clear how the measurements carried out.

It is widely accepted that in the two dimensional hydraulic jump small and large eddies are created which are responsible for the large energy dissipation [#!fluid:henderson!#]. Therefore, energy conservation cannot be used to describe the hydraulic jump heights. The same can be said for the hydraulic jump in different geometries. Of course, the same has to be said for the circular cross section. Thus, the plunger velocity has to be greater than the one obtained by Garber's model, which can be observed in Figure [*]. The Froude number for the Garber's model is larger than the Froude number obtained in the experimental results. Froude number inversely proportional to square of the plunger velocity, and hence the velocity is smaller. The Garber's model therefore underestimates the plunger velocity.


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