iemisc (version 0.9.9)

Manningrect: Rectangular cross-section for the Gauckler-Manning-Strickler equation

Description

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

Usage

Manningrect(
  Q = NULL,
  n = NULL,
  b = NULL,
  Sf = NULL,
  y = NULL,
  T = NULL,
  units = c("SI", "Eng")
)

Value

the missing parameter (Q, n, b, 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.

b

numeric vector that contains the bottom width, 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 rectangular 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)

$$A = by$$

A

the cross-sectional area (m^2 or ft^2)

y

the flow depth (normal depth in this function) [m or ft]

b

the bottom width (m or ft)

$$P = b + 2y$$

P

the wetted perimeter of the channel (m or ft)

y

the flow depth (normal depth in this function) [m or ft]

b

the bottom width (m or ft)

$$B = b$$

B

the top width of the channel (m or ft)

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

1. 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.

2. 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.

3. 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.

4. 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.

5. Wikimedia Foundation, Inc. Wikipedia, 26 November 2015, “Manning formula”, https://en.wikipedia.org/wiki/Manning_formula.

6. 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.

7. 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.

8. Robert L. Mott and Joseph A. Untener, Applied Fluid Mechanics, Seventh Edition, New York City, New York: Pearson, 2015, page 376, 379-380.

9. Wikimedia Foundation, Inc. Wikipedia, 17 March 2017, “Gravitational acceleration”, https://en.wikipedia.org/wiki/Gravitational_acceleration.

10. Wikimedia Foundation, Inc. Wikipedia, 29 May 2016, “Conversion of units”, https://en.wikipedia.org/wiki/Conversion_of_units.

See Also

Manningtrap for a trapezoidal cross-section, Manningtri for a triangular cross-section, Manningpara for a parabolic cross-section, and Manningcirc for a circular cross-section.

Examples

library("iemisc")
library(iemiscdata)
# Example Problem 14.4 from Mott (page 379)
# 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) Lined or Constructed Channels,
# 3) Concrete, and 4) unfinished.

data(nchannel)

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

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

Manningrect(Q = 5.75, b = (4.50) ^ (3 / 8), Sf = 1.2/100, n = n, units =
"SI")
# Q = 5.75 m^3/s, b = (4.50) ^ (3 / 8) m, Sf = 1.2 percent m/m, n = 0.017,
# units = SI units
# This will solve for y since it is missing and y will be in m



# Example Problem 14.5 from Mott (page 379-380)
# 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) Lined or Constructed Channels,
# 3) Concrete, and 4) unfinished.

data(nchannel)

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

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

Manningrect(Q = 12, b = 2, Sf = 1.2/100, n = n, units = "SI")
# Q = 12 m^3/s, b = 2 m, Sf = 1.2 percent m/m, n = 0.017, units = SI
# units
# This will solve for y since it is missing and y will be in m