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Gradually Varied Flow | Dynamic Equation of the Flow

If the flow depth is changing gradually over a long distance, then flow is Gradually Varied Flow (GVF) and if the flow depth is rapid over a short distance, then the flow is Rapidly Varied Flow (RVF). Rapidly Varied Flow and Gradually Varied Flow are steady and non-uniform flows.

Gradually Varied Flow

The definition of these flow types go something like this-


Rapidly Varied Flow (RVF) is a significant change in water depth or velocity of flow over a short distance. This distance can be a few times the water depth. Gradually Varied Flow (GVF) is the flow where the water depth changes gradually over a large distance.


In a more formal and technical way, Gradually varied flow is the nonuniform, steady open-channel flows where the changes in velocity and depth are slow enough in the downstream direction that the vertical distribution of fluid pressure from the free surface to the bottom is not much different from hydrostatic.


I have already covered the types of fluid flow such as steady and unsteady flow, uniform and non-uniform flow here.


Assumptions of Gradually Varied Flow


Some of the important assumptions under which the gradually varied flow (GVF) holds ideal are:

  • The slope of the channel/canal is very small. The vertical depth is considered from the bottom.

  • The pressure distribution is hydrostatic because the curvature of streamlines is small.

  • Manning's and Chezy's formula is valid for the computation of velocity.

  • Channel bed slope is small, HGL will lie at the free surface.

  • Velocity distribution is invariant. KE factor ( or Energy corrections factor) is 1.

  • The flow channel is prismatic. The channel has constant shape and alignment.

  • The velocity distribution coefficients are constant.

This is all about the basics of Gradually Varied Flow.


Not in a mood to read all these? Sit back and relax, I have a video ready for you!


A simple and fundamental example of Gradually Varied Flow(GVF) is the transition of flow over a gentle step. It is finished in a very short distance that the flow energy loss due to friction can be ignored, but the accelerations of fluid are still very small that the vertical fluid pressure distribution is close to being hydrostatic.


In a maximum of the gradually varied flows, however, the change takes place over a distance sufficiently great that we can not assume zero energy loss due to the friction at the bottom.


Dynamic Equation of GVF


The general form of gradually varied flow(GVF) equation is:


(dy/dx)=(So-Sf)/(1-Fr^2)


In this equation, So = Bottom slope, positive in the downward direction

Sf = Friction slope. It is positive in the downward direction.

y = Water depth, measured from culvert bottom to water surface

x = Longitudinal distance, measured along the culvert bottom

Fr = Froude number


The friction slope is approximated as the below equation(Manning's equation),


Sf = (nv)^2/R^(4/3)


Where: Sf = Friction slope, positive in the downward direction

n = Manning's roughness coefficient

v = Average cross section velocity

R = Hydraulic radius, (Wetted Area / Wetted Perimeter)


Bottom Slope and Friction Slope Correlation


From the above mentioned Dynamic Equation, the following inferences can be drawn.

  1. If Sf = So, then dy/dx = 0, the surface profile of flow is parallel to the bottom of the channel.

  2. If Sf < So, then dy/dx is positive which means the flow profile is rising gradually.

  3. If Sf > So, then dy/dx is negative which means the flow profile is lowering gradually.

A simple question on Slope calculation. Give it a shot ;-)

In the next article, I will write about the different types of water surface profile. See you then! Let me know by commenting on what all you want for the next one.


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