Oscillating Hydrofoil Frequency

[LiamH1]

Homework Statement

Evening,

As part of a project we are building a pico oscillating hydroelectricity generation system. Our system is based on the flow of a river providing lift onto a hydrofoil, which is then connected to a flywheel via a mechanism, the energy and rotational speed in the flywheel is the rated up using gears, connected to a generator and electricity is produced.

We currently have all the theoretical equations derived and are using the following equations of motion:

h = h0*sin(wt)
a = a0*sin(wt+p)

Where h = heaving motion(m), a = pitching motion(rad), w = the frequency of oscillation, t = time and p = the phase angle between the heaving motion and pitching motion. a0 and h0 are the maximum and minimum pitch.

We use these equations to derive a power curve for 1 oscilaltion cycle, the area under this curve is then found to determine the total energy being put into the flywheel which is then converted into electricity. In order for our generator to operate correctly it must turn at a set RPM which we intend to rate down using the gear system. This gives us an RPM or rotational speed the flywheel the must turn at.

However the issue we are having is that we are unable to determine an equation that links the flow velocity of the river (U) and the maximum heave (h0) to the oscillation frequency of the hydrofoil, and therefore the the oscillation frequency of the flywheel. This will enable us to determine the required dimensions of the hydrofoil for set flow conditions which will produce the desired RPM to operate the generator. Is anyone aware of a relation between flow velocity (U), maximum heave (h0) and the oscillation frequency (w)?

Homework Equations

w = 2*pi*f
f = frequency (hz)
h = h0*sin(wt)
a = a0*sin(wt+p)

The Attempt at a Solution

We have attempted multiple solutions, however we have our doubts over these. If anyone has further solutions we can attempt to verify these through experiment.

Solution 1:
Sqrt(Mav/J)
Where Mav is the average Torque applied to the fin and J is the moment of inertia.

Solution 2:

Strouhal Number St = (k*a)/(pi*c)
Where k is the reduced frequency of the foil = w*c/2*U, a = the amplitude of the motion = 2*h0 and c = the chord length of the hydrofoil.

Thanks in advance.

 

[struess]

Hi there,
I'm not 100% sure what your system looks like, but I'd start by calculating the lifting force on the hydrofoil and convert that into some kind of torque applied to the flywheel. The force on the foil is

Lift = 0.5*ρ*V²*A*CL

where ρ is the fluid density, V is fluid velocity, A is the planform area (chord*span), and CL is a characteristic of the hydrofoil shape (usually ~1 for moderate angles of attack).

How is the heaving motion achieved? Does the hydrofoil change angle of attack at the top/bottom of the cycle? If so, that lifting force will be periodic with the motion. A picture is worth 1000 words..