Flow Around a River Bend(Horseshoe Bend)

Prepared byBrendan Gallahue and Tushar Dange

Guided by Dr. David J. Willis

IndexIntroPictorial ViewBrief History of CanyonAssumptionsEquations Used and Flow PropertiesGraphsFuture WorkReferences

Horseshoe Bend

Phenomenon Being Examined How does water flow around a 180° bend?

The real life example of this phenomenon can be seen as a part of the Colorado River called Horseshoe bend.

The Colorado River flows through the Grand Canyon and is the reason the Canyon is as vast and deep as it is today.

In this segment we will examine some interesting things about the flow around a river bend such as the river surface level and the velocity distribution

Brief History of the Canyon Over millions of years the flow of water through the canyon has caused the

sedimentary rock to break down and erode away leaving a massive canyon in its place.

The erosive action of the Colorado has been severely constrained by the building of the Glen Canyon Dam, which ended the annual spring floods, but there is still a lot of water flowing relatively quickly through a very narrow gorge.

Before building the dam the Colorado River had spring floods that would exceed a flow rate of 100,000 CFS. All of that snow melting in the Colorado Rockies came pouring down through the Grand Canyon causing a lot of rapid erosion.

Today spring floods are only producing 8,000-10,000 CFS with a maximum of about 20,000 CFS in some parts of the river.

Assumptions The flow of the river is at steady state.

The flow is invicid. If the flow were truly invicid (as we know no real flow is) then there

would be no Grand Canyon because no shear stress to break down the sedimentary rock as the water flows over it.

The flow is incompressible.

There is no vertical change in velocity.

The width of the river is an average of 400 ft across and is uniform across the bend.

Horseshoe Bend

Picture: Google Maps

Equations UsedConservation of Mass

Streamline Bernoulli

Normal Bernoulli

Used to verify that mass is conserved (obvious). Also used to find the surface height at the inner wall of the bend.

Used to find an equation relating the velocity at some radial distance along the bend.

Used to determine the surface height at some radial distance along the bend. Taken on top(red line) and bottom (green line) surfaces.

Horseshoe Bend Flow Properties

Average volume flow rate of 1100 over a period of 105 years.

Ref: http://waterdata.usgs.gov/usa/nwis/uv?site_no=13247500

Horseshoe Bend Flow Properties

Average height of water is 3.5 ft.

Ref: http://waterdata.usgs.gov/usa/nwis/uv?site_no=13247500

Water Surface Height

The change in the depth of the water from the inner surface to the outer surface of the bend is about 1.2 ft.

Water Velocity Around Bend

The velocity of the water around the bend goes from 6.25 at the outer radius to a value of 0.728.

Interesting Future Work If the flow was not assumed to be invicid then the velocity distribution and

the water surface height would be different.

The no slip boundary condition would come into effect at the inner and outer walls of the bend as well as the floor of the river.

It would be interesting to calculate the amount of sediment removed by the flow of the river to create this Wonder of the World.

By utilizing the equations learned in this course and with more allotted time we could also calculate the rate at which the river eroding the canyon today by calculating the shear stresses caused by the flow of the river.

ReferencesGoogle Maps2013 Advanced Fluids MidtermUSGS (United States Geological Survey)

http://waterdata.usgs.gov/usa/nwis/uv?site_no=13247500

THANK YOU

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