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

In iemisc: Irucka Embry's Miscellaneous Functions

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 parameters.

Usage

Manningrect(
  Q = NULL,
  n = NULL,
  b = NULL,
  Sf = NULL,
  y = NULL,
  Temp = 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.
Temp	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 or 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 perimeters 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 perimeters 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 perimeters 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)

Z = \frac{\sqrt{2}}{2}my^2.5

Z

the Section factor (m or ft)

y

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

m

the horizontal side slope

E = y + \frac{Q^2}{2gA^2}

E

the Specific Energy (m or ft)

Q

the discharge [m^3/s or ft^3/s (cfs)] is VA

g

gravitational acceleration (m/s^2 or ft/sec^2)

A

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

y

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

VH = \frac{V^2}{2g}

VH

the Velocity Head (m or ft)

V

the velocity (m/s or ft/s)

g

gravitational acceleration (m/s^2 or ft/sec^2)

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)

Value

the missing parameters (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

Please refer to the iemisc: Manning... Examples using iemiscdata [https://www.ecoccs.com/R_Examples/Manning_iemiscdata_Examples.pdf] and iemisc: Open Channel Flow Examples involving Geometric Shapes with the Gauckler-Manning-Strickler Equation [https://www.ecoccs.com/R_Examples/Open-Channel-Flow_Examples_Geometric_Shapes.pdf] for the cross-section examples using iemiscdata

Author(s)

Irucka Embry

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, https://web.archive.org/web/20210416091858/https://www.wcc.nrcs.usda.gov/ftpref/wntsc/H&H/xsec/manningsNaturally.pdf. Retrieved thanks to the Internet Archive: Wayback Machine

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. Ven Te Chow, Ph.D., Open-Channel Hydraulics, McGraw-Hill Classic Textbook Reissue, New York City, New York: McGraw-Hill Book Company, 1988, pages 21, 40-41.

10. Gary P. Merkley, "BIE6300 - Irrigation & Conveyance Control Systems, Spring 2004", 2004, Biological and Irrigation Engineering - OCW. Course 2, https://digitalcommons.usu.edu/ocw_bie/2/.

11. The NIST Reference on Constants, Units, and Uncertainty, Fundamental Constants Data Center of the NIST Physical Measurement Laboratory, "standard acceleration of gravity g_n", https://physics.nist.gov/cgi-bin/cuu/Value?gn.

12. Wikimedia Foundation, Inc. Wikipedia, 15 May 2019, "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.

iemisc documentation built on June 22, 2024, 9:45 a.m.