# Manningrect: Rectangular cross-section for the Gauckler-Manning-Strickler... In iemisc: Irucka Embry's Miscellaneous Functions

## 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

 1 2 3 4 5 6 7 8 9 Manningrect( Q = NULL, n = NULL, b = NULL, Sf = NULL, y = NULL, T = NULL, units = c("SI", "Eng") )

## 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{ρ RV}{μ}

Re

Reynolds number (dimensionless)

ρ

density (kg/m^3 or slug/ft^3)

R

the hydraulic radius (m or ft)

V

the velocity (m/s or ft/s)

μ

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}{≤ft(√{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)

## 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.

## Note

Assumptions: uniform flow, prismatic channel, and surface water temperature of 20 degrees Celsius (68 degrees Fahrenheit) at atmospheric pressure

Note: Units must be consistent

## Source

r - Better error message for stopifnot? - Stack Overflow answered by Andrie on Dec 1 2011. See https://stackoverflow.com/questions/8343509/better-error-message-for-stopifnot.

## 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.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 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