Show that Froude number of a propeller with diameter D and rotational speed n(1/sec) can be given as Fn=n*sqrt(D/g)
I knew it was coming ! William Froude :)
Fr = v/sqrt(Lg)
v = n R or (1/2) n D
(1/2) n D / sqrt (D g)
= (1/2) n sqrt D/sqrt g
= (1/2) n sqrt (D/g)
the 1/2 does not matter, arbitrary constant
Fr = v/sqrt(Lg)
v = n R or (1/2) n D
(1/2) n D / sqrt (D g) <-----
= (1/2) n sqrt D/sqrt g
= (1/2) n sqrt (D/g)
What's the explanation of replacing L with D there?
L is any old length on your model or ship or dam or whatever
It just has to be the same length on the ship and the model and the speed measured at the bow or at the propeller tip as long as you are consistent from ship to model. The point is that you will get the same wave shapes at the same Froude number.
Got it. I also am studying Naval Architecture by the way. And thanks for answering my other question about sloshing water frequency :)
if you multiply any lengths by 4, you must multiply the speeds by 2 to be at the same Fr.
Great !
To understand how the Froude number of a propeller is given by Fn = n * sqrt(D/g), we need to break it down step by step.
First, let's define the Froude number (Fn). The Froude number is a dimensionless parameter used to characterize the flow regime around a structure (in this case, a propeller) immersed in a fluid. It is defined as the ratio of the flow velocity (V) to the square root of the gravitational acceleration (g) times a characteristic length scale (L).
Fn = V / sqrt(g * L)
In our case, the flow velocity is related to the rotational speed of the propeller (n) and the propeller's blade length, which can be approximated as the propeller diameter (D). So, we can rewrite the flow velocity (V) as:
V = n * D
Substituting this into the Froude number equation, we get:
Fn = (n * D) / sqrt(g * L)
Now, let's substitute the characteristic length scale L with the propeller diameter D, as we can consider this as the characteristic length in this case.
Fn = (n * D) / sqrt(g * D)
Next, we can simplify the equation by canceling out the D term in the denominator:
Fn = n * sqrt(D / g)
And that's how we arrive at the relation Fn = n * sqrt(D / g), which represents the Froude number for a propeller with diameter D and rotational speed n.