# Conventional Integral Solutions of the GVF Equation

Chapter

First Online:

## Abstract

The gradually-varied flow (GVF) equation for flow in open channels is normalized using the normal depth, *hn*, before it can be analytically solved by the direct integration method.

## Keywords

Flow Depth Infinite Series Geometric Element Mathematica Software Gaussian Hypergeometric Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- Allen J, Enever KJ (1968) Water surface profiles in gradually varied open-channel flow. Proc Inst Civil Eng 41:783–811Google Scholar
- Bakhmeteff BA (1932) Hydraulics of Open Channels. McGraw-Hill, New YorkGoogle Scholar
- Bernardin L, Chin P, DeMarco P, Geddes KO, Hare DEG, Heal KM, Labahn G, May JP, McCarron J, Monagan MB, Ohashi D, Vorkoetter SM (2011) Maple programming guide. Maplesoft, Waterloo Maple Inc., CanadaGoogle Scholar
- Bresse J (1860) Cours de mécanique appliqué, 2e partie edition. hydraulique. Mallet-Bachelier, ParisGoogle Scholar
- Chow VT (1957) Closure of discussions on integrating the equation of gradually varied flow. J Hydraul Div ASCE 83(1):9–22Google Scholar
- Chow VT (1959) Open-channel hydraulics. McGraw-Hill, New YorkGoogle Scholar
- Chow VT (1981) Hydraulic exponents. J Hydraul Div ASCE 107(HY11):1489–1499Google Scholar
- Chow VT (1955) Integrating the equation of gradually varied flow. Proceedings Paper No. 838. ASCE 81:1–32Google Scholar
- Dubin JR (1999) On gradually varied flow profiles in rectangular open channels. J Hydraul Res 37(1):99–106Google Scholar
- Gibson AH (1934) Hydraulics and its applications, 4th edn. Constable and Co., LondonGoogle Scholar
- Gill MA (1976) Exact solution of gradually varied flow. J Hydraul Div ASCE 102(HY9):1353–1364Google Scholar
- Gunder DF (1943) Profile curves for open-channel flow. ASCE Trans 108:481–488Google Scholar
- Houcque D (2005) Introduction to Matlab for engineering student. Northwestern University, USAGoogle Scholar
- Jan CD, Chen CL (2012) Use of the Gaussian hypergeometric function to solve the equation of gradually-varied flow. J Hydrol 456–457:139–145CrossRefGoogle Scholar
- Jan CD, Chen CL (2013) Gradually varied open-channel flow profiles normalized by critical depth and analytically solved by using Gaussian hypergeometric functions. HESS 17:973–987Google Scholar
- Keifer CJ, Chu HH (1955) Backwater functions by numerical integration. ASCE Trans 120:429–442Google Scholar
- Kirpich PZ (1948) Dimensionless constants for hydraulic elements of open-channel cross-sections. Civil Eng 18(10):47–49Google Scholar
- Kumar A (1978) Integral solutions of the gradually varied equation for rectangular and triangular channels. Proc Inst Civil Eng 65(2):509–515Google Scholar
- Lee M (1947) Steady gradually varied flow in uniform channels on mild slopes. Ph.D. thesis, Department of Civil Engineering, University of Illinois at Urbana-ChampaignGoogle Scholar
- Mononobe N (1938) Back-water and drop-down curves for uniform channels. ASCE Trans 103:950–989Google Scholar
- Patil RG, Diwanji VN, Khatsuria RM (2001) Integrating equation of gradually varied flow. J Hydraul Eng ASCE 127(7):624–625Google Scholar
- Patil RG, Diwanji VN, Khatsuria RM (2003) Closure to integrating equation of gradually varied flow by R. G. Patil, V. N. Diwanji, R. M. Khatsuria. J Hydraul Eng ASCE 129(1):78–78Google Scholar
- Pickard WF (1963) Solving the equations of uniform flow. J Hydraul Div ASCE 89(HY4):23–37Google Scholar
- Ramamurthy AS, Saghravani SF, Balachandar R (2000) A direct integration method for computation of gradually varied flow profiles. Can J Civil Eng 27(6):1300–1305Google Scholar
- Shivareddy MS, Viswanadh GK (2008) Hydraulic exponents for a D-shaped tunnel section. Int J Appl Eng Res 3(8):1019–1028Google Scholar
- Srivastava R (2003) Discussion of “integrating equation of gradually varied flow" by R. G. Patil, V. N. Diwanji, and R. M. Khatsuria. J Hydraul Eng ASCE 129(1):77–78Google Scholar
- Subramanya K (2009) Flow in open channels, 3rd edn. McGraw-Hill, SingaporeGoogle Scholar
- Subramanya K (2009) Flow in open channels, 3rd edn. McGraw-Hill, SingaporeGoogle Scholar
- Subramanya K, Ramamurthy AS (1974) Flow profile computation by direct integration. J Hydraul Div ASCE 100(HY9):1307–1311Google Scholar
- Vatankhah AR, Easa SM (2011) Direction integration of Manningbased GVF flow equation. Water Manage ICE 164(5):257–264Google Scholar
- Vatankhah AR (2010) Analytical integration of the equation of gradually varied flow for triangular channels. Flow Measur Instrum 21(4):546–549Google Scholar
- Vatankhah AR (2011) Direction integration of gradually varied flow equation in parabolic channels. Flow Measur Instrum 22(3):235–234Google Scholar
- Vatankhah AR (2012) Direction integration of Manning-based GVF equation in trapezodial channels. J Hydrol Eng ASCE 17(3):455–462Google Scholar
- Venutelli M (2004) Direct integration of the equation of gradually varied flow. J Irrig Drainage Eng ASCE 130(1):88–91Google Scholar
- Von Seggern ME (1950) Integrating the equation of nonuniform flow. ASCE Trans 115:71–88Google Scholar
- Wolfram S (1996) The mathematica book, 3rd edn. Wolfram Media and Cambridge University Press, ChampaignGoogle Scholar
- Woodward SM, Posey CJ (1941) Hydraulics of steady flow in open channels. Wiley, New YorkGoogle Scholar
- Zaghloul NA (1990) A computer model to calculate varied flow functions for circular channels. Adv Eng Softw 12(3):106–122Google Scholar
- Zaghloul NA (1992) Gradually varied flow in circular channels with variable roughness. Adv Eng Softw 15(1):33–42Google Scholar
- Zaghloul NA (1998) Hydraulic exponents M and N for gravity flow pipes. Adv Water Resour 21(3):185–191Google Scholar
- Zaghloul NA, Anwar MN (1991) Numerical integration of gradually varied flow in trapezoidal channel. Comput Method Appl Mech Eng 88:259–272ADSCrossRefzbMATHGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2014