Fitting Equations to an ARS Dam Breach Flume Model Using Simulated Annealing




Weir data files used:

myallraw.prn - all 2682 measurements, 161 KB
my16rawb.prn - all data with 16 inch breach base widths, 143 KB
raw16hh1.prn - 16 inch data with headcut of 1 foot, most like to have aerated jet, 42 KB
weirdat0.prn - all data with 8 inch and 32 inch breach base widths, 18 KB
weirdat1.prn - weirdat0 plus 16 inch base data for small values of upstream dam slope, mu (H:V), and breach side slope, ms (H:V), 36 KB
weirdat2.prn - all 16 inch base width data, 125 KB, comma-delineated ASCII (CDA)
weirdat3.prn - all 2682 measurements, 8, 16 and 32 inch base widths, 147 KB, CDA
weirdat4.prn - all 8 inch data, 21 KB, CDA
weirdat5.prn - all 32 inch data, 3 KB, CDA
weirdat6.prn - 8 and 16 inch data with steep-sided mu and ms, 34 KB

Program and data files - Page with a table of most of the program and data files used in this study in 2007. UNDER CONSTRUCTION

Fitted equations - Page with most of the trial head-discharge equations used in 2007 to fit the weir data listed above. UNDER CONSTRUCTION


May 24 - July 16, 2001, and Oct 23, 2001 - Jan 30, 2002 -- Work done for USDA-ARS

Learn about bernoulli equation and ideal weir equation. Develop equations from integrating the ideal weir equation over the geometric edges of the flume model (June 7, 2001). Fit equations using simulated annealing to sets of weir data taken from 2682 measurements. The weirs were constructed of plywood as prismatic notches in a flume in about 380 different geometries, including base widths of 8, 16 and 32 inches, drops into headcuts of 0, 1/2 and 1 foot, and side and upstream slopes of from 0 up to as much as H:V = 6:1. The experiment took neither velocity data, nor recorded when the "breach jet" was aerated. Aside from physical measurements of the construction of the "dam breach", the experiment measured only the gross flow and upstream height of the water above the breach base, by deliberate choice of the designer. Equations were fitted to data separated into three flow regimes using simulated annealing. Flow with a presumably aerated jet falling into the head cut, by implication because aeration was never recorded. The upstream slope of the "dam" affected this flow, which was the largest for a given upstream water depth. Flow with the breach jet fully in contact with the bottom of the head cut, or notch floor. This flow obeyed that for flow in a trapezoidal channel. And an obvious transition region between the two. In this analysis, upstream bed depth had little apparent effect. This investigator wrote several reports of the findings in research paper format before leaving that employment, most of which are at the "work done" link above.

May 14 - June 11, 2007 -- Redo previous work, developing new equations.

PDF files of the original research notes, dead ends, mistakes and all, are at the "May" and "June" links above. Fit various polynomial equations using simulated annealing to sets of weir data take from the same 2682 measurements. Use objective function, subrouting fcn, based on mean relative error. Investigate weights based on the relative sizes of the upsteam head, he, and the headcut depth, hh, to account for the transition between and effects of fully-aerated jet flow and trapezoidal channel flow. Base equations on dimensionless flow, usually Q/(b*sqrt(g*he^3)). Use a weighting factor of he/(he+hh). Starting May 23, 2007, consider that the relationship is not simple heights, but the relative cross-sectional flow areas of the jet above the breach base, and the headcut (if any) below it. Settle on a weighting ratio, r = (he*he)/(he*he + b*hh), where b is the base width of the breach.

May 28, based on plot of fitted data in Lotus 1-2-3 file, my2807f.123, pick Intermediate Peak and First Order Sequential Formation equations from SPSS TableCurve 2D V4 as possible fitting equations. Go back to polynomial fits occassionally to see relatively unbiased curves and decide upon other functions. Also switch from objective function based on mean relative error to mean absolute error.

June 4, start plotting log distribution of squared error, and using Chi-Square test to determine best fitting method. Use polynomial fitting functions to look for better method. June 7, choose Logistic Dose Response curve to fit both term with ms only (he^(5/2) term) and term with (ms^2 + mu^2)^(~.4) (approximately he^(3/2), but more like he^4.3).

, with

Q(hat) = calculated breach flow, cfs or ft3/s
b = bottom width of gap in plywood breach, ft
g = acceleration of gravity, 32.174 ft/s2
he = height above the verge of the breach (b) of water upstream, ft
hh = depth of headcut below verge of breach (b), ft
ms = slope of the side of the trapezoidal breach cut, horizontal to vertical, dimensionless
mu = slope of the upstream side of the plywood dam, H:V, dimensionless
D1 = 0.46342, D2 = 0.32459, D3 = -0.16306, D4 = 0.51441, D5 = 8.6838,
D6 = -0.10813, D7 = 0.73077, D8 = 0.50789, D9 = 2.5213, D10 = 0.38428
mean absolute error = 0.08304 CFS
standard deviation of absolute error = 0.13877 CFS
mean relative error = 4.91%, std dev of relative error = 5.94%

Calculated in Fortran simulated annealing program, simanh9.for, with input data file, weirdat3.prn

Plotted results in Lotus 1-2-3 file, jn1107a.123:

Qh = calculated flow, cfs

June 11 to July 1, 2007 -- Developing new equations.


July 1 to 18, 2007 -- Developing new equations.

More Detailed Lab Notes on Equation Fitting - Page with almost daily notes on how the fitting was done. UNDER CONSTRUCTION


This page was designed by Donald L. Baker copyright © 2007