Estimating the force exerted by a water jet on a flat plate - Bernoulli vs Momentum Equation

Hi,

Case 1:

Imagine a steady liquid water jet with unit cross sectional area - incompressible and inviscid. density D and uniform velocity V. Operating atmospheric pressure is 0.0 Pa.

The jet is in air ( no air resistance ) and hits a stationary flat plate perpendicularly.

Applying momentum equation gives - the force on the plate by the water jet as : D*V*V = DVV.

Hence the pressure on the plate = DVV / Area = DVV. (Area = 1)

Case 2.

For the same scenario the Stagnation pressure is = 0.5*D*V*V = 0.5 D V^2
The stagnation pressure is the pressure just before the plate, when the fluid completely stops on impact.

However, the two answers are different.

I have read the followings in the literature.

1. In practice, when the water first hit the plate it creates very high pressure (water hammer), which lasts for few micro seconds.

2. Then the pressure reduces and settles at the stagnation pressure.

My question is:

What quantity ( relevance in practice) is calculated using the momentum equation.

PS. - I have seen that , the initial water hammer on the plate can be calculated by Density*V*Speed of sound in water. Which is different than what is estimated though the momentum equation.

It depends on where the energy goes in the collision and flow. The momentum equation by itself is not the whole story...we need to do energy accounting as well. The Momentum Equation works in concert with the First Law of Thermodynamics. Fundamentally, you "think" you are finding "strange things" because you are applying one without the other.

You also have to be careful about mass conservation. If a flow is hitting a flat plate it’s being deflected. Bringing the flow to a complete halt (as you do in one of your scenarios ) you are unintentionally accelerating the flow from ##v## to 0. Where is the mass going? That is not steady flow, you have to account for momentum accumulation in the control volume as well. In that scenario there must be non-conservative work done on the flow. All the energy it had on the way in must be lost to heat.

We can try to work some basic problems to see the process If you would like.