Manningcirc: Circular cross-section using the Gauckler-Manning-Strickler equation

Description

Manningcirc and Manningcircy solve for a missing variable for a circular cross-section. The uniroot function is used to obtain the missing parameter.

Usage

Manningcirc(
  Q = NULL,
  n = NULL,
  Sf = NULL,
  y = NULL,
  d = NULL,
  T = NULL,
  units = c("SI", "Eng")
)
Manningcircy(
  y = NULL,
  d = NULL,
  y_d = NULL,
  theta = NULL,
  Sf = NULL,
  Q = NULL,
  units = c("SI", "Eng")
)

Value

the missing parameter (Q, n, or Sf) & theta, area (A), wetted perimeter (P), velocity (V), top width (B), hydraulic radius (R), Reynolds number (Re), and Froude number (Fr) as a list for the Manningcirc function.

the missing parameter (d or y) & theta, area (A), wetted perimeter (P), top width (B), velocity (V), and hydraulic radius (R) as a

list for the Manningcircy function.

Arguments

Q: numeric vector that contains the discharge value [m^3/s or ft^3/s], if known.
n: numeric vector that contains the Manning's roughness coefficient n, if known.
Sf: numeric vector that contains the bed slope (m/m or ft/ft), if known.
y: numeric vector that contains the flow depth (m or ft), if known.
d: numeric vector that contains the diameter value (m or ft), if known.
T: numeric vector that contains the temperature (degrees C or degrees Fahrenheit), if known.
units: character vector that contains the system of units [options are SI for International System of Units and Eng for English units (United States Customary System in the United States and Imperial Units in the United Kingdom)]
y_d: numeric vector that contains the filling ration (y/d), if known.
theta: numeric vector that contains the angle theta (radians), if known.

Details

The Manningcirc function solves for one missing variable in the Gauckler- Manning equation for a circular cross-section and uniform flow. The possible inputs are Q, n, Sf, y, and d. If y or d are not initially known, then Manningcircy can solve for y or d to use as input in the Manningcirc function.

The Manningcircy function solves for one missing variable in the Gauckler- Manning equation for a circular cross-section and uniform flow. The possible inputs are y, d, y_d (ratio of y/d), and theta.

Gauckler-Manning-Strickler equation is expressed as

$$V = \frac{K_n}{n}R^\frac{2}{3}\sqrt{S}$$

V: the velocity (m/s or ft/s)
n: Manning's roughness coefficient (dimensionless)
R: the hydraulic radius (m or ft)
S: the slope of the channel bed (m/m or ft/ft)
$K_n$: the conversion constant -- 1.0 for SI and 3.2808399 ^ (1 / 3) for English units -- m^(1/3)/s or ft^(1/3)/s

This equation is also expressed as

$$Q = \frac{K_n}{n}\frac{A^\frac{5}{3}}{P^\frac{2}{3}}\sqrt{S}$$

Q: the discharge [m^3/s or ft^3/s (cfs)] is VA
n: Manning's roughness coefficient (dimensionless)
P: the wetted perimeter of the channel (m or ft)
A: the cross-sectional area (m^2 or ft^2)
S: the slope of the channel bed (m/m or ft/ft)
$K_n$: the conversion constant -- 1.0 for SI and 3.2808399 ^ (1 / 3) for English units -- m^(1/3)/s or ft^(1/3)/s

Other important equations regarding the circular cross-section follow: $$R = \frac{A}{P}$$

R: the hydraulic radius (m or ft)
A: the cross-sectional area (m^2 or ft^2)
P: the wetted perimeter of the channel (m or ft)

$$A = \left(\theta - \sin \theta\right) \frac{d^2}{8}$$

A: the cross-sectional area (m^2 or ft^2)
d: the diameter of the cross-section (m or ft)
$\theta$: see the equation defining this parameter

$$\theta = 2 \arcsin\left[1 - 2\left(\frac{y}{d}\right)\right]$$

$\theta$: see the equation defining this parameter
y: the flow depth (normal depth in this function) [m or ft]
d: the diameter of the cross-section (m or ft)

$$d = 1.56 \left[\frac{nQ}{K_n\sqrt{S}}\right]^\frac{3}{8}$$

d: the initial diameter of the cross-section [m or ft]
Q: the discharge [m^3/s or ft^3/s (cfs)] is VA
n: Manning's roughness coefficient (dimensionless)
S: the slope of the channel bed (m/m or ft/ft)
$K_n$: the conversion constant -- 1.0 for SI and 3.2808399 ^ (1 / 3) for English units -- m^(1/3)/s or ft^(1/3)/s

Note: This will only provide the initial conduit diameter, check the design considerations to determine your next steps.

$$P = \frac{\theta d}{2}$$

P: the wetted perimeter of the channel (m or ft)
$\theta$: see the equation defining this parameter
d: the diameter of the cross-section (m or ft)

$$B = d \sin\left(\frac{\theta}{2}\right)$$

B: the top width of the channel (m or ft)
$\theta$: see the equation defining this parameter
d: the diameter of the cross-section (m or ft)

$$D = \frac{A}{B}$$

D: the hydraulic depth (m or ft)
A: the cross-sectional area (m^2 or ft^2)
B: the top width of the channel (m or ft)

A rough turbulent zone check is performed on the water flowing in the channel using the Reynolds number (Re). The Re equation follows:

$$Re = \frac{\rho RV}{\mu}$$

Re: Reynolds number (dimensionless)
$\rho$: density (kg/m^3 or slug/ft^3)
R: the hydraulic radius (m or ft)
V: the velocity (m/s or ft/s)
$\mu$: dynamic viscosity (* 10^-3 kg/m*s or * 10^-5 lb*s/ft^2)

A critical flow check is performed on the water flowing in the channel using the Froude number (Fr). The Fr equation follows:

$$Fr = \frac{V}{\left(\sqrt{g * D}\right)}$$

Fr: the Froude number (dimensionless)
V: the velocity (m/s or ft/s)
g: gravitational acceleration (m/s^2 or ft/sec^2)
D: the hydraulic depth (m or ft)

References

Terry W. Sturm, Open Channel Hydraulics, 2nd Edition, New York City, New York: The McGraw-Hill Companies, Inc., 2010, page 2, 8, 36, 102, 120, 123-125, 153-154.
Dan Moore, P.E., NRCS Water Quality and Quantity Technology Development Team, Portland Oregon, "Using Mannings Equation with Natural Streams", August 2011, http://www.wcc.nrcs.usda.gov/ftpref/wntsc/H&H/xsec/manningsNaturally.pdf.
Gilberto E. Urroz, Utah State University Civil and Environmental Engineering - OCW, CEE6510 - Numerical Methods in Civil Engineering, Spring 2006 (2006). Course 3. "Solving selected equations and systems of equations in hydraulics using Matlab", August/September 2004, https://digitalcommons.usu.edu/ocw_cee/3.
Tyler G. Hicks, P.E., Civil Engineering Formulas: Pocket Guide, 2nd Edition, New York City, New York: The McGraw-Hill Companies, Inc., 2002, page 423, 425.
Wikimedia Foundation, Inc. Wikipedia, 26 November 2015, “Manning formula”, https://en.wikipedia.org/wiki/Manning_formula.
John C. Crittenden, R. Rhodes Trussell, David W. Hand, Kerry J. Howe, George Tchobanoglous, MWH's Water Treatment: Principles and Design, Third Edition, Hoboken, New Jersey: John Wiley & Sons, Inc., 2012, page 1861-1862.
Andrew Chadwick, John Morfett and Martin Borthwick, Hydraulics in Civil and Environmental Engineering, Fourth Edition, New York City, New York: Spon Press, Inc., 2004, page 133.
Robert L. Mott and Joseph A. Untener, Applied Fluid Mechanics, Seventh Edition, New York City, New York: Pearson, 2015, page 376, 377-378, 392.
Wikimedia Foundation, Inc. Wikipedia, 17 March 2017, “Gravitational acceleration”, https://en.wikipedia.org/wiki/Gravitational_acceleration.
Wikimedia Foundation, Inc. Wikipedia, 29 May 2016, “Conversion of units”, https://en.wikipedia.org/wiki/Conversion_of_units.

Examples

Run this code

library("iemisc")
library(iemiscdata)

# Practice Problem 14.12 from Mott (page 392)
y <- Manningcircy(y_d = 0.5, d = 6, units = "Eng")

# See npartfull in iemiscdata for the Manning's n table that the
# following example uses
# Use the normal Manning's n value for 1) Corrugated Metal, 2) Stormdrain.

data(npartfull)

# We are using the culvert as a stormdrain in this problem
nlocation <- grep("Stormdrain",
npartfull$"Type of Conduit and Description")

n <- npartfull[nlocation, 3] # 3 for column 3 - Normal n

Manningcirc(d = 6, Sf = 1 / 500, n = n, y = y$y, units = "Eng")
# d = 6 ft, Sf = 1 / 500 ft/ft, n = 0.024, y = 3 ft, units = "Eng"
# This will solve for Q since it is missing and Q will be in ft^3/s



# Example Problem 14.2 from Mott (page 377-378)
y <- Manningcircy(y_d = 0.5, d = 200/1000, units = "SI")

# See npartfull in iemiscdata for the Manning's n table that the
# following example uses
# Use the normal Manning's n value for 1) Clay, 2) Common drainage tile.

data(npartfull)

nlocation <- grep("Common drainage tile",
npartfull$"Type of Conduit and Description")

n <- npartfull[nlocation, 3] # 3 for column 3 - Normal n

Manningcirc(Sf = 1/1000, n = n, y = y$y, d = 200/1000, units = "SI")
# Sf = 1/1000 m/m, n = 0.013, y = 0.1 m, d = 200/1000 m, units = SI units
# This will solve for Q since it is missing and Q will be in m^3/s



# Example 4.1 from Sturm (page 124-125)
Manningcircy(y_d = 0.8, d = 2, units = "Eng")

y <- Manningcircy(y_d = 0.8, d = 2, units = "Eng")
# defines all list values within the object named y

y$y # gives the value of y



# Modified Exercise 4.1 from Sturm (page 153)
# Note: The Q in Exercise 4.1 is actually found using the Chezy equation,
# this is a modification of that problem
# See nchannel in iemiscdata for the Manning's n table that the
# following example uses
# Use the normal Manning's n value for 1) Natural streams - minor streams
# (top width at floodstage < 100 ft), 2) Mountain streams, no vegetation
# in channel, banks usually steep, trees and brush along banks submerged at
# high stages and 3) bottom: gravels, cobbles, and few boulders.

data(nchannel)

nlocation <- grep("bottom: gravels, cobbles, and few boulders",
nchannel$"Type of Channel and Description")

n <- nchannel[nlocation, 3] # 3 for column 3 - Normal n

Manningcirc(Sf = 0.002, n = n, y = y$y, d = 2, units = "Eng")
# Sf = 0.002 ft/ft, n = 0.04, y = 1.6 ft, d = 2 ft, units = English units
# This will solve for Q since it is missing and Q will be in ft^3/s



# Modified Exercise 4.5 from Sturm (page 154)
library(NISTunits)

ysi <- NISTftTOmeter(y$y)

dsi <- NISTftTOmeter(2)

Manningcirc(Sf = 0.022, n = 0.023, y = ysi, d = dsi, units = "SI")
# Sf = 0.022 m/m, n = 0.023, y = 0.48768 m, d = 0.6096 m, units = SI units
# This will solve for Q since it is missing and Q will be in m^3/s

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