Manningtrap: Trapezoidal cross-section for the Gauckler-Manning-Strickler equation

Description

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

Usage

Manningtrap(
  Q = NULL,
  n = NULL,
  m = NULL,
  m1 = NULL,
  m2 = NULL,
  Sf = NULL,
  y = NULL,
  b = NULL,
  T = NULL,
  units = c("SI", "Eng"),
  type = c("symmetrical", "non-symmetrical")
)

Value

the missing parameter (Q, n, b, m, Sf, or y) & area (A), wetted perimeter (P), velocity (V), top width (B), hydraulic depth (D), 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.
m1: numeric vector that contains the "cross-sectional side slope of m1:1 (horizontal:vertical)", if known.
m2: numeric vector that contains the "cross-sectional side slope of m2:1 (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.
b: numeric vector that contains the bottom width, 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)]
type: character vector that contains the type of trapezoid (symmetrical or non-symmetrical). The symmetrical trapezoid uses m while the non- symmetrical trapezoid uses m1 and m2.

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 trapezoidal 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)

$$A = y\left(b + my\right)$$

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
b: the bottom width (m or ft)

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

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: the bottom width (m or ft)

$$B = b + 2my$$

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
b: the bottom width (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, 153.
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.
Andrew Chadwick, John Morfett, and Martin Borthwick, Hydraulics in Civil and Environmental Engineering, Fourth Edition, New York City, New York: Spon Press, 2004, pages 132-133.
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.
Robert L. Mott and Joseph A. Untener, Applied Fluid Mechanics, Seventh Edition, New York City, New York: Pearson, 2015, page 376, 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)
# Exercise 4.1 from Sturm (page 153)

Manningtrap(Q = 3000, b = 40, m = 3, Sf = 0.002, n = 0.025, units = "Eng")
# Q = 3000 cfs, b = 40 ft, 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



# Practice Problem 14.19 from Mott (page 392)
# See nchannel in iemiscdata for the Manning's n table that the following
# example uses
# Use the minimum Manning's n value for 1) Natural streams - minor streams
# (top width at floodstage < 100 ft), 2) Lined or Constructed Channels,
# 3) Concrete and 4) float finish.

data(nchannel)

nlocation <- grep("float finish",
nchannel$"Type of Channel and Description")

n <- nchannel[nlocation, 3][1] # 3 for column 3 - Normal n

Manningtrap(y = 1.5, b = 3, m = 3/2, Sf = 0.1/100, n = n, units = "SI")
# y = 1.5 m, b = 3 m, m = 3/2, Sf = 0.1/100 m/m, n = 0.023, units = SI
# units
# This will solve for Q since it is missing and Q will be in m^3/s

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