Steady and Unsteady Open-Channel Flow Computation

Description

This package is designed as an educational tool for students and instructors of undergraduate courses in open channel hydraulics. Functions are provided for computing normal and critical depths, steady (e.g. backwater curves) and unsteady (flood wave routing) flow computations for prismatic trapezoidal channels. See the vignettes to get started.

Channel geometry

Description

Compute geometry relations for trapezoidal channels.

Usage

channel_geom(y, B, SS)

Arguments

Details

Channel geometry relations are routinely calculated in numerical solutions of steady, gradually-varied and unsteady flows. This function is used extensively by internal procedures and is made accessible to the user for convenience.

Value

Named vector:

Examples

channel_geom(1.71, 100, 0) # rectangular channel
channel_geom(5.79, 6.1, 1.5) # trapezoidal channel with sideslope 3H:2V

Gradually-varied flow profiles

Description

Compute the gradually-varied flow profile of a prismatic channel.

Usage

compute_profile(
  So,
  n,
  Q,
  y0,
  Cm,
  g,
  B,
  SS,
  z0 = 0,
  x0 = 0,
  stepdist,
  totaldist
)

Arguments

Details

Computes the longitudinal water surface profile of a prismatic channel using the standard step method by solving the non-linear ODE

\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}

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.

Value

data.frame with columns:

Examples

# example M1 profile
compute_profile(0.001, 0.045, 250, 2.7, 1.486, 32.2, 100, 0, stepdist = 10, totaldist = 3000)
# example M2 profile
compute_profile(0.001, 0.045, 250, 0.64, 1.486, 32.2, 100, 0, stepdist = 10, totaldist = 3000)
# example S2 profile
compute_profile(0.005, 0.01, 250, 2.65, 1.486, 32.2, 10, 0, stepdist = 10, totaldist = 2000)
# example S3 profile
compute_profile(0.005, 0.01, 250, 0.5, 1.486, 32.2, 10, 0, stepdist = 10, totaldist = 2000)

Channel conveyance

Description

Calculate the channel conveyance.

Usage

conveyance(n, A, R, Cm)

Arguments

Details

Channel conveyance is routinely calculated in numerical solutions of steady, gradually-varied and unsteady flows. This function is used extensively by internal procedures and is made accessible to the user for convenience.

Value

The channel conveyance.

Critical depth

Description

Calculate the critical depth.

Usage

critical_depth(Q, yopt, g, B, SS)

Arguments

Details

The critical depth is the water depth at which a channel flow regime will transition from supercritical to subcritical (or vice versa). Calculation of the critical depth is based on a specific energy formulation, i.e.

E = y + z + \frac{Q^2}{2gB^2y^2}

where y is the flow depth, z is the elevation relative to some datum (assumed to be 0), and the last term represents kinetic energy. More specifically, the function operates by finding the point where the derivative of specific energy w.r.t. y is zero, i.e. y = y_c when

\frac{dE}{dy} = 1 - \frac{Q^2}{gA^3}\frac{dA}{dy} = 0

.

Value

The critical depth y_c [L].

Examples

critical_depth(250, 2, 32.2, 100, 0) # rectangular channel
critical_depth(126, 1, 9.81, 6.1, 1.5) # trapezoidal channel with sideslope 3H:2V

Shiny Demonstrations

Description

Demonstrate package functionality via Shiny apps

Usage

demo_shiny(ex)

Arguments

Details

Demonstrations available: "gvf" Gradually-varied flow.

Examples

## Not run: 
# get list of available demos
demo_shiny()
# run the gradually-varied flow demo
demo_shiny("gvf")

## End(Not run)

Froude Number

Description

Calculate the Froude Number.

Usage

froude(Q, g, A, DH)

Arguments

Details

The Froude number is a dimensionless measure of bulk flow characteristics that represents the relative importance of inertial forces and gravitational forces. For open channel flow, the Froude number of open channel flow is defined as

Fr = \frac{v}{\sqrt{gD_H}}

where v = \frac{Q}{A} is the flow velocity, g is the gravitational acceleration and D_H is the hydraulic depth. The Froude number is related to the energy state of the flow and can be used to identify flows as either supercritical (Fr < 1) or subcritical (Fr > 1).

Value

The Froude Number (dimensionless).

Examples

froude(250, 32.2, 171, 1.71) # subcritical flow
froude(250, 32.2, 57.9, 0.579) # critical flow
froude(250, 32.2, 45, 0.45) # supercritical flow

Normal depth

Description

Calculate the normal (equilibrium) depth using Manning's equation.

Usage

normal_depth(So, n, Q, yopt, Cm, B, SS)

Arguments

Details

The normal depth is the equilibrium depth of a channel for a given flow rate, channel slope, geometry and roughness. Manning's equation is used to calculate the equilibrium depth. Manning's equation for normal flow is defined as

Q = \frac{C_m}{n} AR^{2/3}S_0^{1/2}

where Q is the channel flow, S_0 is the channel slope, A is the cross-sectional flow area, R is the hydraulic depth and C_m is a conversion factor based on the unit system used. This function uses a Newton-Raphson root-finding approach to calculate the normal depth, i.e. y = y_n when

f(y) = \frac{A^{5/3}}{P^{2/3}} - \frac{nQ}{C_mS_0^{1/2}} = 0

.

Value

The normal depth y_n [L].

Examples

normal_depth(0.001, 0.045, 250, 3, 1.486, 100, 0) # rectangular channel
normal_depth(0.0008, 0.013, 126, 5, 1, 6.1, 1.5) # trapezoidal channel with sideslope 3H:2V

Flood wave routing

Description

Route a flood wave down a prismatic channel.

Usage

route_wave(
  So,
  n,
  Cm,
  g,
  B,
  SS,
  initial.condition,
  boundary.condition,
  downstream.condition,
  timestep,
  spacestep,
  numnodes,
  monitor.nodes,
  monitor.times,
  engine = c("Dynamic", "Kinematic"),
  scheme = c("MacCormack", "Lax"),
  boundary.type = c("QQ", "Qy", "yQ", "yy")
)

Arguments

Details

Provides implementations of a Kinematic Wave Model (KWM) and a Dynamic Wave Model (DWM) with the choice of two numerical schemes. The MacCormack scheme is a second-order accurate predictor-corrector scheme that provides efficient flood wave routing. The Lax diffusive scheme can be used to obtain smooth solutions for problems with discontinuities in the boundary conditions, e.g. sudden gate closures. The DWM implementation uses the Method of Characteristics (MOC) to compute the flow regime at the model boundaries, and allows the user to specify boundaries in terms of depths and/or flows. the KWM implementation assumes the normal depth at the upstream boundary and is only first-order accurate.

Value

data.frame with columns:

Examples

## Not run:  
# kinematic wave routing
times = seq(0, 30000, by = 25)
floodwave = ifelse(times >= 9000, 250,
  250 + (750/pi)*(1 - cos(pi*times/(60*75))))
route_wave(0.001, 0.045, 1.486, 32.2, 100, 0, initial.condition = 250, 
  boundary.condition = floodwave, timestep = 25, spacestep = 50, 
  numnodes=301, monitor.nodes = c(1, 101, 201, 301), 
  monitor.times = seq(1, length(times), by = 10), engine = "Kinematic")
# dynamic wave routing with zero-gradient downstream condition using MacCormack scheme
route_wave(0.001, 0.045, 1.486, 32.2, 100, 0, initial.condition = 250, 
  boundary.condition = floodwave, downstream.condition = rep(-1, length(times)), 
  timestep = 25, spacestep = 500, numnodes = 31, engine = "Dynamic", 
  scheme = "MacCormack", monitor.nodes = c(1, 11, 21, 31), 
  monitor.times = seq(1, length(times), by = 10))
# mixed boundary conditions (sudden gate closure) using Lax scheme
lax = route_wave(0.00008, 0.013, 1, 9.81, 6.1, 1.5, 
  initial.condition = 126, boundary.condition = rep(5.79, 2001), 
  downstream.condition = rep(0, 2001), timestep = 1, spacestep = 10, 
  numnodes = 501, monitor.nodes = c(1, 151, 251, 301, 501), 
  monitor.times = c(1, 501, 1001, 1501, 2001), 
  engine="Dynamic", scheme="Lax", boundary.type="yQ")
# extract data for a monitored point
require(dplyr)
filter(lax, monitor.type == "node", node == 151)

## End(Not run)

California Water Olympics

Description

Digitized results from the California Water Olympics. The variables are as follows:

• t The time (in seconds) since the start of the model run.

• Q The flow rate [ft^3 s^{-1}].

• x The distance downstream [ft] at which the hydrograph was recorded.

The data can be used to validate numerical solutions to flood wave routing for a channel under the following conditions:

• Channel width is 100 feet.

• Channel slope is 0.001.

• Channel extent is 150,000 feet.

• Channel roughness (Manning's n) is 0.045.

• Channel sideslope is 0 (rectangular channel).

• Initial flow rate is 250 cfs.

• Upstream boundary condition is defined as

  Q(t < 9000) = 250 + \frac{750}{\pi}(1 - \cos{\frac{\pi t}{4500}})

  Q(t >= 9000) = 250

Usage

data(waterolympics)

Format

A data frame with 40 rows and 3 variables

References

Sobey, Rodney. "H11: Hydrograph Routing." Review of One-Dimensional Hydrodynamic and Transport Models. Bay-Delta Modeling Forum, 15 June 2001. Web. 13 Mar. 2015. <http://www.cwemf.org/1-DReview/>.