# Impact Force/Pressure of a wave on a vertical wall given PIV velocity field

I have PIV (Particle Image Velocimetry) data of liquid sloshing on a rectangular tank which produces a wave impacting one of the vertical walls. In the picture below you can see the wave about to hit the wall on the left

I'm trying to find a way to correlate the velocity fields with the data from the pressure sensors. To do that I'd like to obtain the impact force and then the pressure from the velocity field.

I started with a simple approach to find the impact force of a particle of mass $m = \rho \mathcal{V}$ hitting a vertical wall with a velocity $\overrightarrow{V}$ with $\alpha = (\overrightarrow{x},\overrightarrow{V})$ and $d$ the distance of the particle from the wall.

The force $\overrightarrow{F}$ that drives this particle is the impact force.

Using the kinetic inergy theorem I have :

$\frac{1}{2}mV^2 - \frac{1}{2}m0^2 = \overrightarrow{F}.\overrightarrow{r}$ where $\overrightarrow{r} = -d \overrightarrow{x} + (y_{\text{current}} - y_{\text{impact}}) \overrightarrow{y}$ is the displacement vector between the current position and the impact point (assuming the origin coincides with the impact wall and the particles moves from the right to the left where it will meet the wall of coordinate $y = 0$ hence $\alpha \in [\frac{\pi}{2} \frac{3\pi}{2}]$).

Making use of the trigonometry I eventually find that the magnitude of the impact force would be :

$F = -\frac{\rho \mathcal{V} \ cos(\alpha) \ V^2}{2d}$

I'd then just need to divide that force with the sensor surface area $S$ to have a pressure to be compared.

My questions :

Am I using the right approach here ?

How can I use my PIV measurements ? I have a regularly spaced grid of vectors just before the impact (with some space between the wave and the wall) so at each point of the grid I have $V, d$ and $\alpha$ but what's more tricky is what to choose for $\mathcal{V}$ ? Should it be proportional to the cube of each interrogation window ? Should I average my vectors on a certain slice ?

That's where I would need some advice !

Hope I was clear enough, don't hesitate to ask for precisions or suggest something different.

• From your data you can find the velocity potential at (or close to) the free surface, plus its temporal derivatives. This is sufficient information to prescribe the evolution of the fluid, plus any of its other properties (like pressure) up until the flow is no longer irrotational. See, eg, the paper by Cooker and Peregrine (sciencedirect.com/science/article/pii/037838399290020U) or Chan and Melville (rspa.royalsocietypublishing.org/content/royprsa/417/1852/…). – Nick P Jun 13 '15 at 20:14
• Thank you for pointing me to those papers, I'll look into the free surface tracking, but as I have the independant measurements of velocity as well, couldn't there still be a way to use those ? – Luc Rebillout Jun 13 '15 at 20:27