Sources for Splash height of a ball

Question

I was reading on this post on Height of Water 'Splashing'. One of the answers given by user Martin Gales derived a simple correlation which I could logically understand:

From the law of momentum conservation we get $$m_{s}v_{0}=(m_{s}+m_{w})v$$ or

$$v=\frac{\rho_{s}v_{0}}{\rho_{s}+\rho_{w}}=\frac{v_{0}}{1+\frac{\rho_{w}}{\rho_{s}}}$$

So $v$ above is velocity with which water (with volume of $V$) bursts up. To what height of $H$ ?

$$H=\frac{v^{2}}{2g}=\frac{v_{0}^{2}}{2g}\frac{1}{\left (1+\frac{\rho_{w}}{\rho_{s}}\right )^{2} }=\frac{H_{0}}{\left (1+\frac{\rho_{w}}{\rho_{s}}\right )^{2} }$$

or $$\frac{H}{H_{0}}=\frac{1}{\left (1+\frac{\rho_{w}}{\rho_{s}}\right )^{2} }$$

Because $\rho_{w}=1\frac{g}{cm^{3}}$ and $\rho_{s}=3\frac{g}{cm^{3}}$ (roughly) we get an estimation:

$$\frac{H}{H_{0}}=\left ( \frac{3}{4} \right )^{2}\approx 0.6$$

I wanted to use this formula in my experiment to test if that equation really holds. However my professor asked me - although the derivation seems ok as it removes complex details - to find an actual report that supports this derivation and tests if it actually works. Now the problem is that I am not sure if there are any sources which back this derivation so I was wondering if anyone in this forum is aware of any studies conducted to verify this relationship?

I would appreciate any sources and pdf file links.

Answer 1

The use of conservation of momentum in the calculation which you quote does not make any sense to me. So I would not use it. A better idea which is closer to the description below is found in the unanswered question How does a ball cause a splash? (With the relevant math).

The splash phenomenon is called a Worthington Jet.

The object falling into water pulls down a column of air behind it. This column collapses as the water at the sides rushes inwards, causing a "pinch" effect at the middle of the air column. Two columns or jets of water then rush upwards and downwards from the pinch. The upper jet jumps out of the water with a speed which can be several times that of the object at impact. In some circumstances this jet can break up, leaving a droplet at the tip which continues rising as the remainder collapses back into the water.

The above explanation can be found in Worthington jets explanation: fluid phenomenon. This contains a link to useful videos on YouTube which show that spectacular jets can also be achieved by dropping heavy objects into fine sand or powder. Thus surface tension does not have much effect in liquid jets.

Also included is pre-print of the article Generation and Breakup of Worthington Jets After Cavity Collapse by S Gekle and J M Gordillo (2009), from which the following diagrams are taken. These show simulations of the formation of the jets, which are the spikes in the right-hand diagrams.

Gekle and Gordillo's paper develops a numerical model to identify how the formation and break up of the jet depend on the dimensionless Froude Number $Fr=V^2/gR$ and Weber Number $We=\rho R V^2/\sigma$ which characterise the experiment. They not give a complete analytical derivation of the relation between the jet ejection velocity $U$ and the radius $R$ and velocity of impact $V$ of the object (which is modelled as a circular disk) and the density $\rho$ and surface tension $\sigma$ of the liquid. When $U$ is known the maximum height of the splash is $U^2/2g$.

Another factor which has a significant effect on jet velocity $U$ is the wettability of the surface material of the object, which is measured by contact angle. If the surface is hydrophilic, meaning that contact angle is small, then the jet is almost eliminated. See the video Dynamics of Water Entry by TT Truscott et al, MIT. A review article for the whole phenomenon was published by the same team in 2014; it can be accessed from ResearchGate.

Also useful is the Masters Thesis An Experimental Study of Worthington Jet Formation After Impact of Solid Spheres by Jenna Marie McKown, MIT (2011).