Glowing Waterfalls
Kevin Hayes Andrew
Dan Beutel
Math 164
May 2, 2003
1
Introduction
Picture a glowing blue waterfall crashing into a lovely glowing lake. In designing an optimally
cool looking waterfall, we, as mathematically inclined types, feel the need to understand
how the water will flow over the waterfall and what is necessary to produce an interesting,
entertaining, waterfall.The lake and waterfall glow due to a blacklight actived dye. Figure 1
is a picture from the last time this party was held, about 6 years ago.
Figure 1: Lake filled with blacklight activated dye, similar to the one our waterfall will fall
into.
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2.1
Questions
What is the velocity field as the fluid flows off the waterfall?
Since the water enters freefall as it leaves the channel, we can model its path in the air.
Modeling particles from a sampling at the edge of the waterfall as falling under the force of
gravity, we can develop a picture of the water as it falls.
2.2
How does the shape of the waterfall edge affect the appearance
of the waterfall?
Currently, we still expect the entire waterfall to come to the same angle to the horizontal.
We’re looking for a variable-shaped lip: straight line, curve, alternating between channels,
etc. Each channel still ends square, and perpendicular to the water flow.
2.3
How can we maximize the party experience?
This question is both qualitative and highly dependent on opinion. To maximize the party
experience, we would want the ”best looking” waterfall possible, preferably with a certain
visual impact. Some sub-questions to be answered first would be: How bright can we
make/do we want the waterfall? Can we see through the waterfall? Do we want to?
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Coordinates
Our system consists of water falling a set distance. We use the 3-dimensional Cartesian
coordinate system, with the x- and y- axes defining the horizontal, and the z-axis the vertical
as shown in Figure 2. The channels are set so that the water flows at some angle between
the x- and z- axes (no velocity in the y-direction). Once the water enters freefall, gravity is
the only external force acting on it, so its acceleration is only in the z-direction.
Figure 2: Coordinate system we are using
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Channels
To ensure an even flow across the width of the waterfall, we divide the water up into a
number of small channels. The entire waterfall is about 1 m wide, with 5 mm wide channels.
This also means that to solve the velocity field, we only need to deal with a simple, already
solved problem.
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Numerical Methods
Modeling and finding numerical solutions consisted of several steps, finding the velocity field,
the trajectory of the falling water, and then making a graphical representation of the results.
5.1
Velocity field in the channel
The velocity field in a square channel, −L ≤ y ≤ L, −L ≤ z ≤ L is given by
∞
4 X
(−1)n cosh(Nn y)
L −z −
cos(Nn z)
L n=0 Nn3 cosh(Nn L)
!
u = u0
2
2
(1)
where the flow is only in the downstream (x̂) direction. This velocity field is scaled by a
constant u0 that is determined by the pressure gradient and other system parameters. This
is shown graphically in Figure 3. This expression is found by solving the Navier-Stokes
equations for laminar flow in a straight pipe of constant cross section and is found in WardSmith [6].
Figure 3: Velocity in a cross-section of a square channel
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5.2
Obtaining the Velocity Field
As a first approximation, one can use the velocity field u = sin(πy/L) sin(πz/L) for the
velocity in the pipe. This isn’t too bad, but we can improve by using the cosh(y) cos(z) expansion shown in Equation 1. The difference between these, using 12 terms of the expansion,
is shown graphically in Figure 4. The maximum difference is .1248. The expansion changes
less than 10−4 for up to 20 terms, so we are confident we have used sufficient terms in the
expansion.
Figure 4: Difference between our velocity field expansion and the approximation u = sin(π ∗
y/L) ∗ sin(π ∗ z/L). Maximum error = .1248
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5.3
Velocity field at the channel end
At the end of the channel, we expect that the velocity field won’t change significantly based
on our understanding of the literature for our parameters. Specifically, we looked at the
work of Clarke [1], Naghdi and Rubin [3], Rubin [4] and Toison [5].
5.4
Trajectory of water
Using the velocity u of a particle (think: water molecule) as it exits the pipe, we can exactly
calculate its trajectory by applying the equations
x(t) = x(t = 0) + ut
y(t) = y(t = 0)
z(t) = z(t = 0) − gt2
until z = −h where h is an input parameter, usually 5 meters.
5.5
Drawing Waterfalls
To generate the surface plots off the falling water as seen in Figures 5 and 6. Combining
trajectories for particles starting at various points throughout the channel gives a series of
surface plots. In the following figures, each surface represents particles starting with the
same z values across the pipe.
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Fall From One Channel
Water free falling from one channel exhibits falls as shown in Figures 5 and 6.
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Fall From Five Channels
Water free falling from five channels exhibits falls is shown in Figures 7 and 8.
Figure 5: One channel waterfall, from above
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Figure 6: One channel waterfall, from below
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Fall Over a Jagged Edge
Water free falling out of ten channels which represent a jagged edge pipe is shown in Figures 9
and 10.
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Conclusions
The velocity field is given above in Equation 1. We concluded that falling off the waterfall
didnt significantly affect velocity.
The channels tend to produce a choppy, disjointed, look when different lengths. We think
that a straight lip looks the best and will maximize the party experience.
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Extensions
Extensions to this model would most likely focus on the qualitative question of how to maximize the party experience.What is the right concentration of dye to get the best color?Is this
Figure 7: Five channel waterfall, from above
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Figure 8: Five channel waterfall, from below
as bright as possible, or do we want to be able to see through the waterfall as well?Another
direction would be to integrate molecular interaction into the model.One could model the
interaction of colliding streams of water, or determine the appearance of the waterfalls surface: smooth and glasslike or rough and choppy.Currently our model does not take these
into account, plotting a smooth surface over our points and would predict colliding streams
of water to pass directly through each other. Could one model how the surface type affects
the appearance of the dye?All of these would take the analysis back to the falling water from
the channel water, to the visible water.
References
[1] N.S. Clarke. On two dimensional inviscid flow in a waterfall. Journal of Fluid Mechanics,
22:359–69, 1965.
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Figure 9: Ten channel jagged waterfall, from above
[2] Frederic Dias and E.O. Tuck. Weir flows and waterfalls. Journal of Fluid Mechanics,
230:525–39, 1991.
[3] P.M. Naghdi and M.B. Rubin. On inviscid flow in a waterfall. Journal of Fluid Mechanics,
103:375–87, 1981.
[4] M.B. Rubin. Relationship of critical flow in waterfall to minimum energy head. ASCE
Journal Hydraulic Engineering, 123:82–4, 1997.
[5] Fabrice Toison and Jacques Hureau. Open-channel flows and waterfalls. European Journal
Mechanics B Fluids, 19:269–83, 2000.
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Figure 10: Ten channel jagged waterfall, from below
[6] A. J. Ward-Smith. Internal Fluid Flow: The Fluid Dynamics of Flow in Pipes and Ducts.
Clarendon Press, Oxford, 1980.
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