Manningtri: Triangular cross-section for the Gauckler-Manning-Strickler equation

Description

This function solves for one missing variable in the Gauckler-Manning- Strickler equation for a triangular cross-section and uniform flow. The uniroot function is used to obtain the missing parameter.

Usage

Manningtri(
  Q = NULL,
  n = NULL,
  m = NULL,
  Sf = NULL,
  y = NULL,
  T = NULL,
  units = c("SI", "Eng")
)

Value

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

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.
m: numeric vector that contains the "cross-sectional side slope of m:1 (horizontal:vertical)", 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.
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)]

Details

Gauckler-Manning-Strickler equation is expressed as

$$V = \frac{K_n}{n}R^\frac{2}{3}S^\frac{1}{2}$$

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

$$Q = \frac{K_n}{n}\frac{A^\frac{5}{3}}{P^\frac{2}{3}}S^\frac{1}{2}$$

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 triangular 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)

$$A = my^2$$

A: the cross-sectional area (m^2 or ft^2)
y: the flow depth (normal depth in this function) [m or ft]
m: the horizontal side slope

$$P = 2y\sqrt{\left(1 + m^2\right)}$$

P: the wetted perimeter of the channel (m or ft)
y: the flow depth (normal depth in this function) [m or ft]
m: the horizontal side slope

$$B = 2my$$

B: the top width of the channel (m or ft)
y: the flow depth (normal depth in this function) [m or ft]
m: the horizontal side slope

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

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, 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, 393.
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 14.41 from Mott (page 393)
# 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) Excavated or Dredged Channels, 3)
# Earth, straight, and uniform, 4) clean, recently completed.

data(nchannel)

nlocation <- grep("clean, recently completed",
nchannel$"Type of Channel and Description")

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

Manningtri(Q = 0.68, m = 1.5, Sf = 0.0023, n = n, units = "Eng")
# Q = 0.68 cfs, m = 1.5, Sf = 0.002 ft/ft, n = 0.05, units = English units
# This will solve for y since it is missing and y will be in ft



# Modified Exercise 4.1 from Sturm (page 153)
Manningtri(Q = 3000, m = 3, Sf = 0.002, n = 0.025, units = "Eng")
# Q = 3000 cfs, m = 3, Sf = 0.002 ft/ft, n = 0.025, units = English units
# This will solve for y since it is missing and y will be in ft



# Modified Exercise 4.5 from Sturm (page 154)
Manningtri(Q = 950, m = 2, Sf = 0.022, n = 0.023, units = "SI")
# Q = 950 m^3/s, m = 2, Sf = 0.022 m/m, n = 0.023, units = SI units
# This will solve for y since it is missing and y will be in m

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