The Circular Hydraulic Jump

by Charles Bloom

The Hydraulic Jump is an intriguing theoretical problem in fluid mechanics, partly because it is so well known, but still not well handled by theory (a paper by Tomas Bohr in April, 1997 offers a promising new approach, but this has not yet been fully explored). I studied the problem in my last semester as a physics undergraduate at UT (Spring 1997). My work on the problem is presented here:

Check out some photos

My papers on the hydraulic jump

links

this is the only other site I know of about the jump:

While working on the computational simulation of the jump, I wrote plot (24k,zip) an interactive plotter for Win95 (zoom,scroll,etc. on 2-d data or 2-d vector field data).

Circular Hydraulic Height Models

Blackford.hyd gnuplot data file for Blackford model

Bloom.hyd gnuplot data file for Bloom model

Tani.hyd gnuplot data file for Tani model

hydr_H.plt gnuplot instructions to plot the three .hyd files

hydr_h.c Hydraulic Height profile model solver

hydr_h.gif image of hydr_h.plt plot

The Tani model (the same model used by Tomas Bohr in his recent paper) should be correct for r -> 0 and r -> large (not infinity!) but we realize it is no good in the region of the jump (dh/dr goes to infinity at r = (5/12)(Fr^-2)a). Also, Tani's model is many-valued in the area of the jump, h(r) and v(r) spiral around a critical attractor. I used by hydr_h solver to run Tani's model from rmin -> rcritical and then from rmax -> rcritical , so you can see the : spiraling h(r) . The Bohr-predict jump radius is somewhere (about half way) between the two critical points shown here. You find this radius by finding the radius at which it is "legal" to jump between the two solutions. This jumping from the r->0 good solution to the r->large good solution means that you avoid any dh/dr -> infinity points, and this is the hydraulic jump. "legal" here means that you must make the jump such that flux and momentum flux are conserved. This means that:

r^2*[H(r)*h(r)]*[H(r)+h(r)] = Q^2 / 2*pi^2*g

Solve this for r; note that this is symmetric in H <-> h

Circular Hydraulic Jump Exact Time-Evolution Method

Hydraulic source, 4/3/97 , not yet working. I never got this to work, but there is interesting stuff in it. These four models are different levels of approximation:

hydr1.c exact

hydr2.c Presssure = g(h - z) fixed.

hydr3.c Like 2, but on a re-scaled domain

hydr4.c exact, but using 3 as a base

First we got some wavy solutions: hydr_waves.jpg

Then we got what appears to be a stable "kink" - stationary after 380000 time-steps; the rest of the flow is uniform. hydr_stable.jpg

Work continues.

Here's a picture of (du/dt) which looks like separation


Charles Bloom / cb at my domain
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