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:
This is a sort of survey of analytic models of the jump, with some new work on these models, and the unification of all the different models. Bohr's newest work is examined in some new ways, and we find a prediction for the width of the jump, and the fact that separation and the jump are inherently tied : seeking the radius at which separation occurs may be best formula for finding the radius of the jump.
The focus of this report are a thorough documentation of practical observation of the circular jump with design of measurement apparatus. We built the experiment from the ground up. The analysis and results focus on comparison of the r(Re,Fr) predictions from the analytic models above. We also have nice proof of the separation in the jump, and height contours of the jump itself.
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).
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
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 Send Me Email
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