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Analytical Solution of Gradually Varied Flow Equation in Non-prismatic Channels

  • Banafsheh Nematollahi
  • Mohammad Javad AbediniEmail author
Research Paper
  • 5 Downloads

Abstract

Computation of water surface profile in open channel is considered as an essential component of channel design in open-channel hydraulics. In this regard, gradually varied flow (GVF) is enumerated as the most frequently occurring flow regime in artificial open channels. The analytical and/or semi-analytical solutions of GVF have been acquired for some types of open channels with prismatic channel cross section using both Manning’s and Chezy’s resistance equations. In conventional hydraulic textbooks, the interpretation of water surface profile is mainly based on prismatic channels. Indeed, the non-prismatic term in the numerator of the governing equation creates a further obstacle for proper interpretation. In this paper, an explicit closed-form semi-analytical solution is presented for a non-prismatic rectangular channel based on Chezy’s resistance equation using Adomian decomposition method. The developed semi-analytical solution compares well with both analytical and numerical solutions obtained from predictor–corrector method with very fine spatial resolution. The derived semi-analytical solution can be effectively used to conduct sensitivity analysis of pertinent parameters and compute channel discharge for a given water surface profile.

Keywords

Analytical solution Gradually varied flow (GVF) Chezy’s equation Adomian decomposition method (ADM) 

List of Symbols

g

Acceleration of gravity

\(\omega\)

A dummy variable representing non-prismatic channel

\(S_{0}\)

Channel bottom slope

A

Channel cross section (m2)

x

Distance along dominant flow direction (m)

\(\sigma\)

Element width (m)

Q

Flow discharge (m3/s)

T

Flow top width (m)

\(S_{\text{f}}\)

Friction slope

q

Lateral inflow per unit longitudinal direction (m2/s)

\(\beta\)

Momentum coefficient

\(\beta^{'}\)

Momentum coefficient for lateral inflow

\(f\left( x \right)\)

Non-homogeneous term corresponding to forcing function

\(F\left[ {h\left( x \right)} \right]\)

Nonlinear term

j

Number of iteration

i

Section index

\(\alpha\)

Spatial variation of bottom width with respect to longitudinal distance

\(\varepsilon\)

Specific tolerance in reference to the assumed accuracy

\(V_{x}\)

The velocity of lateral inflow in the direction of x (m/s)

h

The water depth (m)

t

Time (s)

\(\overline{h}\)

Vertical distance from the center of cross section to the flow water surface (m)

E

Flow specific energy

b

Channel width

b0

A constant term associated with channel width

xb

The longitudinal distance corresponding to boundary point

hb

The water depth corresponding to boundary point

Mh and Nh

Hydraulic exponents

hc

Critical depth

hn

Normal depth

x*

Dimensionless depth

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Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringSchool of Engineering, Shiraz UniversityShirazIran
  2. 2.Department of Civil and Environmental Engineering, School of EngineeringShiraz UniversityShirazIran

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