You have three questions here
1. "How to explain this process through fluid dynamics?"
There is a study from "Hasson and Peck", 1964, which explains "Thickness distribution in a sheet formed by impinging jets." It's all basically Bernoulli's equation; "Velocity -> pressure -> velocity" and then simply continuity.
Shortly, With ideal fluid the Kinetic energy can be completly maintained, though there is obviously more or less 90-degree change in direction. Anouther good source is the Book "Zhang, Zhengji, Freistrahlturbinen" You can luckily preview these particular pages (53-54) here;
It's german, but the equations and a picture is there.
2. Is there an equation to explain this fluid dynamical process?
Yes. After answering this question; Height of Water 'Splashing' I realized that it's only Bernoulli's principle.
Velocity + Elevation + Pressure = Constant
At the impact, you have a point where;
- Elevation = 0
- Velocity = 0
- Pressure = all Energy
Which then has no difference compared to any explosion. The pressure is simply released to the easiest direction, and the matter influenced by this pressure will gain velocity. the direction of this velocity is defined similarily as the hydraulic pressure gradient; allways perpendicular to surface.
3. What is the Physics of a Splash?
Velocity - > Pressure -> Velocity to the least pressure direction.
-Bernoulli; that's all.