Natural frequency

Thanks to a parallel with the oscillating pendulum problem, we suppose that the waves' natural frequency has the following expression : $ f_0= \frac{1}{2\pi} \sqrt{\frac{g}{L}} $.

In a first time, we will check that the fluid's viscosity does not influence the natural frequency and we will verify the theoretical expression. 

Influence of fluid's viscosity

The natural frequency is the same for the four viscosities so the viscosity does not have any influence on the natural frequency.

The theoretical expression gives a natural frequency equal to 0,27 Hz but with Fluent results (plot) we obtain a natural frequency equal to 0,35 Hz. The expression is not right.

The expression of the natural frequency is modified supposing that it depends on all the dimensions of the problem. The expression becomes : $ \displaystyle f_0 = \frac{k}{2 \pi} \sqrt{\frac{g}{L*}} $ with L*=L*(L,h) a characteristic length and k a constant. We will find this characteristic length studying the influence of the initial level (h) and of the tank's length (L) on the natural frequency (f0).

Influence of initial level

The Fast Fourier Transform of initial water level for five various initial water levels (first plot) allows to know that the initial water level has an influence on the natural frequency. Thanks to these Fast Fourier Transforms, the natural frequency can be extracted for all the cases and we obtain the second plot.

 

 

 

 

This plot represents the natural frequency (obtained after a fast fourier transform) for four initial water levels. Blue points are the Fluent results. A curve fitting returned a square root function (red points). So the natural frequency is proportional to the squared root of the initial water level : $f_0 \propto \sqrt{h} $. Therefore the characteristic length is proportional to the inverse of the initial level : $ L^* \propto \frac{1}{h} $. We can suppose that the characteristic length is proportional to the squared of tank's lenght over the intial water level : $ L^* \propto \frac{L^2}{h} $. We will verify this expression studying the influence of tank's lenght on the natural frequency.

Influence of tank's length

The Fast Fourier Transform of water level for five various tank's length (first plot) allows to know that the length have an influence on the natural frequency. Thanks to these Fast Fourier Transforms, the natural frequency can be extracted for all the cases and we obtain the second plot.

 

This plot represents the natural frequency (obtained after a fast fourier transform) for five tank's length. Blue points are the Fluent results. A curve fitting returned an inverse function (red points). So the natural frequency is proportional to the inverse of the tank's length : $ f_0 \propto \frac{1}{L} $. Therefore the characteristic length is proportional to the squared of tank's length : $ L^* \propto  L^2 $. 

Thus the characteristic length is proportional to the squared of tank's lenght over the intial water level : 

$ \Rightarrow \displaystyle L^* \propto \frac{L^2}{h} $  

All cases will be ploted in function of this characteristic length in the next part.

Influence of the characteristic length

Thanks to the previous studies, the expression of the characteristic length is known as $ \displaystyle L^* = \frac{L^2}{h} $. All the studied cases are gathered on the following plot : 

Blue points are Fluent Results. Red points are the mathematical approximation :

$ \displaystyle f_0 = \frac{2.73}{2 \pi} \sqrt{\frac{g}{L*}} $

Results and approximation match for almost all the cases. For the first one, there is a difference which can be explained by the non-validity of the long waves hypothesis for this case (L=1.5m).

The characteristic length and so the expression of the natural frequency were groped for but we find the waves celerity. In fact, with the long waves hypothesis, the waves' celerity is equal to : $ c = \sqrt{gh} $

So the new expression of the natural frequency is :

$ \displaystyle f_0 = 0.53 \frac{c}{L} $

$\Rightarrow$ To conclude this part about the natural frequency, we found an expression function of a characteristic length and it does not depend on the fluid's viscosity.