weirbos.broadcrest() function, the computations are exclusively based on the foot-second unit system and careful adherance by the user is required.The discharge equation for an acceptable tail-water condition $h_t$ is $$Q = k_c k_R k_s C_v C b \frac{2}{3} \sqrt{\frac{2}{3}g} H^{1.5}$$ where $Q$ is discharge in cubic feet per second, $k_c$ is the contraction coefficient that also is a function of the abutment rounding $r$, $k_R$ is the approach rounding coefficient that is a function of the approach crest rounding $R$, $k_s$ is the downstream embankment slope coefficient, $C_v$ is the approach velocity coefficient, $C$ is the discharge coefficient, $b$ is the width in feet of the weir crest, and $H$ is total head in feet on the weir, which is computed by $$H = h + v_o = h + \alpha v^2/2g$$ where $h$ is static head in feet on the weir, $v_o$ is velocity head in feet in the approach section, $v$ is mean velocity in feet per second in the section computed by $v=Q/A$ for cross section area $A$ in square feet, which by default is computed by $A=(h + P)B$, but can be superceded. The quantity $g$ is the acceleration of gravity and is hardwired to 32.2 feet per square second. The dimensionless quantity $\alpha$ permits accommodation of a velocity head correction that is often attributable to cross section subdivision. The $\alpha$ is outside the scope of this documentation, is almost always $\alpha=1$, and is made available as an argument for advanced users.
The discharge equation is solved for two conditions that consider the approach velocity head. First, the equation is solved as shown above for $H$ through iteration and $C_v=1$. Second, the equation is solved using $h$ and $C_v$, which is determined by nomograph lookup.
The weirbos.broadcrest() function is vectorized meaning that optional vectors of $h$ can be specified along with an optional and equal length vector $h_t$. The function assumes rectangular approach conditions to compute approach area $A$ if not superceded by the optional A argument, which also can be a vector.
The weirbos.broadcrest() function also permits optional vectors of $L$ and $b/B$ (by the argument contractratio) so that tuning of the weir-computed discharge to a measured discharge potentially can be made. The crest length $L$ can be used to increase discharge slightly by shortening in say the circumstances of a slightly downward sloping crest. (Such potential “crest” sloping is distinct from the downstream embankment slope---do not confuse the two.) The $b/B$ can be used to decrease discharge by decreasing $k_R$ in say the circumstance of an inlet that is rougher or has asperities that slightly increase the expected contraction and reduce flow efficiency. To clarify, the fact that $L$ and $b/B$ can be vectorized as optional arguments shows a mechanism by which tuning of the computational results to measured $Q$ values can occur without replacing the fundamental $C$ and $C_v$ nomographs of BOS and nomograph and lookup tables for $k_c$, $k_R$, and $k_s$ of TWRI3A5 (Hulsing, 1967). In all cases, these coefficients can be superceded by user-specified scalars or vectors in various combinations.
weirbos.broadcrest(h, ht=NULL, b=NULL, B=NULL, P=NULL, L=NULL, R=0, r=0, A=NULL, alpha=1, slopeus="vertical", slopeds="vertical", kc=NULL, kr=NULL, ks=NULL, C=NULL, contractratio=NULL, hhptest=TRUE, extended=TRUE, header="", flowdigits=2, coedigits=3, verbose=FALSE, eps=0.001, maxit=20)contractratio and A to bypass any computations normally involving $B$. This would be the mechanism to bypass the $B$ as a scalar restriction;tan().) The string “vertical” must be provided as the value for slopeus for vertical slopes;tan().) The string “vertical” must be provided as the value for slopeds for vertical slopes;slopeds. Optionally, this coefficient may be a vector;slopeus. Optionally, this coefficient may be a vector;b.over.B[i] <- contractratio[i];attributes() of the returned data frame under the non-original label name of header;flow of the returned data frame.data.frame() is returned and the extended=TRUE version is described below:
weirbos.broadcrest.extended=FALSE version is restricted to the most salient items including $Q_H$, $Q_h$, $Q_{cv}$, $v_o$, $C$, $C_v$, $k_c$, $k_R$, and $k_s$.
Hulsing, Harry, 1967, Measurement of peak discharge at dams by indirect methods: U.S. Geological Survey Techniques of Water-Resources Investigations, Book 3, Chapter A5, 29 p., http://pubs.usgs.gov/twri/twri3-a5/
weir.broadcrest
# Simple, scalar inputs and results
weir.broadcrest(type="BOS", 0.5, b=8, B=11, P=6, L=3.25);
# Vector of heads
weir.broadcrest(type="BOS", c(0.5,0.4,0.3), b=8, B=11, P=6, L=3.25);
# Now compare two solutions by weirbos.broadcrest() to weir3a5.broadcrest()
h <- seq(0.01,1.5, by=0.001)
Qtank <- weir3a5.broadcrest(h, b=7.97, B=10.97, P=1.125, L=3.76, flowdigits=4)
QtankbosA <- weirbos.broadcrest(h, b=7.97, B=10.97, P=1.125, L=3.76, flowdigits=4)
QtankbosB <- weirbos.broadcrest(h, b=7.97, B=10.97, P=1.125, L=3.76,
hhptest=FALSE, flowdigits=4)
plot(Qtank$flow, h, type="l", col=8, lwd=6,
xlab="FLOW, CFS", ylab="HEAD, FEET")
lines(QtankbosA$flow, h, col=2, lwd=4) # see the trunction in the red line
lines(QtankbosB$flow, h, col=3, lwd=2)
# See examples for weir3a5.broadcrest() for additional examples that
# explore higher complexity of argument settings, which test the
# nomographs for k_c, k_R, and k_s
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