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Fluid Mechanics

12.3 Uniform and Gradually Varied Flow

3 min readLast Updated on July 19, 2024

Open channels are crucial in hydraulic engineering, carrying water in rivers, canals, and drainage systems. Uniform flow maintains constant depth and velocity along the channel, balancing gravity and resistance forces. This equilibrium shapes the flow's behavior and energy distribution.

Gradually varied flow (GVF) introduces complexity, with depth and velocity changing gradually along the channel. GVF profiles are classified based on depth relationships and channel characteristics. Understanding these concepts is vital for designing and analyzing open channel systems effectively.

Uniform Flow in Open Channels

Characteristics of uniform channel flow

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  • Uniform flow maintains constant depth, velocity, and cross-sectional area along the channel length
    • Steady flow exhibits no change in flow characteristics over time at a given location
    • Prismatic channel has a consistent cross-sectional shape and slope (rectangular, trapezoidal)
  • Balance between gravitational forces and resistance forces drives the flow
    • Gravitational force determined by the channel slope and fluid weight
    • Resistance force caused by friction between the fluid and channel boundary (roughness of concrete, natural streambed)
  • Energy grade line (EGL) and hydraulic grade line (HGL) run parallel to the channel bottom in uniform flow
    • EGL represents total energy head as the sum of elevation head, pressure head, and velocity head
    • HGL represents the sum of elevation head and pressure head (piezometric head)

Chezy and Manning equations application

  • Chezy equation calculates average flow velocity (VV) using Chezy coefficient (CC), hydraulic radius (RR), and channel slope (SS): V=CRSV = C \sqrt{RS}
    • Chezy coefficient (CC) depends on channel roughness and flow characteristics (m^(1/2)/s)
    • Hydraulic radius (RR) is the ratio of cross-sectional area (AA) to wetted perimeter (PP) (m)
    • Channel slope (SS) is dimensionless
  • Manning equation expresses average flow velocity (VV) using Manning's roughness coefficient (nn), hydraulic radius (RR), and channel slope (SS): V=1nR2/3S1/2V = \frac{1}{n} R^{2/3} S^{1/2}
    • Manning's roughness coefficient (nn) depends on channel material and surface characteristics (s/m^(1/3))
    • Tabulated nn values available for various channel types (concrete, natural streams)
  • Continuity equation calculates discharge (QQ) as the product of average flow velocity (VV) and cross-sectional area (AA): Q=VAQ = VA

Gradually Varied Flow in Open Channels

Analysis of gradually varied flow

  • Gradually varied flow (GVF) exhibits gradual changes in depth and velocity along the channel length
    • Flow is steady and non-uniform
    • Streamlines are nearly parallel with hydrostatic pressure distribution
  • GVF profiles classified based on the relationship between actual depth (yy), normal depth (yny_n), and critical depth (ycy_c)
    • Normal depth (yny_n) occurs in uniform flow for given discharge and channel characteristics
    • Critical depth (ycy_c) occurs when Froude number (FrFr) equals 1
      • Fr=VgyFr = \frac{V}{\sqrt{gy}}, where gg is acceleration due to gravity and yy is flow depth
  • GVF profiles categorized as mild (MM), critical (CC), steep (SS), horizontal (HH), or adverse (AA)
    • MM profiles: y>yn>ycy > y_n > y_c
    • CC profiles: y=ycy = y_c
    • SS profiles: yn>yc>yy_n > y_c > y
    • HH profiles: zero channel slope
    • AA profiles: negative channel slope

Water surface profiles in prismatic channels

  • Gradually varied flow equation (GVFE) analyzes water surface profiles: dydx=S0Sf1Fr2\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}
    • dydx\frac{dy}{dx} is the slope of the water surface
    • S0S_0 is the channel bottom slope
    • SfS_f is the friction slope (energy grade line slope)
    • FrFr is the Froude number
  • GVFE is a first-order differential equation solved numerically using methods like:
    1. Direct step method
    2. Standard step method
  • Boundary conditions required to solve GVFE:
    • Known water depth at a specific location (downstream control, sluice gate, overfall)
    • Known discharge and flow regime (subcritical or supercritical)
  • Computed water surface profile should match expected GVF profile type based on boundary conditions and channel characteristics
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.