A landscape architect is planning an artificial waterfall in a city park. Water flowing at 1.40 m/s will leave the end of a horizontal channel at the top of a vertical wall 3.90 m high, and from there the water falls into a pool.
To sell her plan to the city council, the architect wants to build a model to standard scale, one-seventeenth actual size. How fast should the water flow in the channel in the model?
Although perfect simulation will not be possible, I believe it would be most important to match the Froude number, as explained at:
http://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Froude_Number_and_Flow_States.htm
To determine the required speed of water in the model, we need to consider the following:
1. Start with the actual scenario: We are given that the water flows at 1.40 m/s and falls from a height of 3.90 m in the actual waterfall.
2. Use the scale factor: The model is built to a scale of one-seventeenth the actual size. This means that all the dimensions (such as height, length, and width) in the model will be one-seventeenth of the actual dimensions.
3. Apply the scale factor to the height: Multiply the actual height of the waterfall (3.90 m) by the scale factor (1/17) to find the height of the waterfall in the model.
Height in Model = Actual Height × Scale Factor
= 3.90 m × (1/17)
= 0.23 m (rounded to two decimal places)
4. Use the conservation of energy principle: The potential energy at the top of the waterfall is converted into kinetic energy (movement energy) as it falls. Therefore, we can equate the potential energy at the top to the kinetic energy at the bottom.
Potential Energy at the top = Kinetic Energy at the bottom
The potential energy can be calculated using the formula:
Potential Energy = mass × gravitational acceleration × height
The kinetic energy can be calculated using the formula:
Kinetic Energy = (1/2) × mass × velocity^2
Since the mass of water cancels out in the equation, we can equate the two equations to find the relationship between the velocities:
mass × gravitational acceleration × height = (1/2) × mass × velocity^2
Simplifying and rearranging the equation, we get:
velocity^2 = 2 × gravitational acceleration × height
Now, let's calculate the velocity in the model:
velocity in the model = √(2 × gravitational acceleration × height in the model)
Substituting the values:
velocity in the model = √(2 × 9.8 m/s^2 × 0.23 m)
= √(4.5188)
≈ 2.13 m/s (rounded to two decimal places)
Therefore, the water should flow at approximately 2.13 m/s in the channel of the model to recreate the same flow dynamics as the actual waterfall.