Quickstart with rivr
Overview
rivr provides functions for calculating the normal depth and critical depth,
computing gradually-varied flow profiles, and solving the unsteady flow
equations using finite-differencing methods. Outputs are formatted for
manipulation with dplyr and visualization with ggplot2. This vignette
provides a brief introduction to the main functions and example applications.
Computing normal and critical depth
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The normal depth \(y_n\) is defined as the flow depth at which
\[
Q = \frac{C_m}{n} AR^{2/3}S_0^{½}
\]
Where \(Q\) is the flow rate, \(n\) is Manning's coefficient, \(A\) is the
cross-sectional flow area (also a function of flow depth), \(R\) is the
hydraulic radius and \(S_0\) is the bed slope. The critical depth \(y_c\) is
defined as the flow depth at which
\[
\frac{dE}{dy} = 1 - \frac{Q^2}{gA^3}\frac{dA}{dy} = 0.
\]
Both \(y_n\) and \(y_c\) are non-linear functions of \(y\). rivr provides
functions for computing normal and critical depths, as shown below. In both
cases a Newton-Raphson scheme is used to solved the equations. | require(rivr)
flow = 250; mannings = 0.045 ; Cm = 1.486; gravity = 32.2
width = 100; slope = 0.001; sideslope = 0
yn = normal_depth(slope, mannings, flow, yopt = 2, Cm, width, sideslope)
yc = critical_depth(flow, yopt = 2, gravity, width, sideslope)
print(c(normal.depth = yn, critical.depth = yc))
## normal.depth critical.depth
## 1.7113010 0.5789949
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Standard-step method for gradually-varied flow profiles
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The standard step method can be used to solve steady-state water surface
profiles. The solution to gradually-varied flow profiles is based on the
non-linear ordinary differential equation
\[
\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}
\]
and is appropriate for cases where \(\frac{dy}{dx}\) is small. The standard-step
method operates by stepping along the channel by a constant distance interval,
starting from a cross-section where the flow depth is known (the control section).
The flow depth is computed at the adjacent cross-section (target section). The
computed value at the target is then used as the basis for computing flow depth at
the next cross-section, i.e. the previous target section becomes the new control
section for each step. A Newton-Raphson scheme is used each step to compute the
flow depth and friction slope. Technically, the average friction slope of the
control and target section is used to compute the flow depth at the target section. | flow = 250; mannings = 0.045 ; Cm = 1.486; gravity = 32.2
width = 100; slope = 0.001; sideslope = 0
gvf = compute_profile(slope, mannings, flow, y0 = 2.7, Cm, gravity, width,
sideslope, stepdist=50, totaldist=3000)
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The output is formatted for easy visualization with ggplot2.
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| require(ggplot2)
ggplot(gvf, aes(x = x, y = y + z)) + geom_line(color='blue')
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Unsteady flow
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Unsteady flow models solve the shallow water equations (conservation of mass
and conservation of momentum). Kinematic wave models (KWM) use a truncated form
of the momentum equation while dynamic wave models (DWM) solve the mass and
momentum equations simultaneously. A variety of numerical schemes can be used to
solve these equations. rivr provides an interface to multiple explicit finite
differencing schemes (one KWM scheme and two DWM schemes) for computing unsteady
flows. The MacCormack scheme is recommended for most applications. | baseflow = 250; mannings = 0.045 ; Cm = 1.486; gravity = 32.2
width = 100; slope = 0.001; sideslope = 0
numnodes = 301; xresolution = 250; tresolution = 10;
times = seq(0, 30000, by = tresolution)
wave = floodwave = ifelse(times >= 9000, baseflow,
baseflow + (750/pi)*(1 - cos(pi*times/(60*75))))
downstream = rep(-1, length(wave))
mn = c(1, 101, 201)
mt = c(501, 1501, 3001)
uf = route_wave(slope, mannings, Cm, gravity, width, sideslope,
baseflow, wave, downstream, tresolution, xresolution, numnodes,
mn, mt, "Dynamic", "MacCormack", "QQ")
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Accessing the unsteady flow outputs is greatly simplified by using dplyr.
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| require(dplyr)
uf.nodes = filter(uf, monitor.type == "node")
ggplot(uf.nodes, aes(x=time, y=flow, color=factor(distance))) + geom_line()
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| uf.times = filter(uf, monitor.type == "timestep")
ggplot(uf.times, aes(x=distance, y=flow, color=factor(time))) + geom_line()
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