Water jet strikes a plate and slows down due to frictional forces?

[stanley650586031]

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When a water jet strikes a plate (3m plate) at an angle (like 60 degrees), velocity of water decreases as it travels downstream. I know the major contributor that slows down water is probably wall frictional force but how to quantify this phenomena?

I have been thinking this question for over a week but still couldn’t find the appropriate scientific method to quantify this phenomena.
My first question will be how to correctly calculate the shear stress for a case like water striking a 2m long plate at 60 degrees angle. I think this is a transient analysis that the water jet comes out of the nozzle, strikes the plate and then travels downstream until it reaches the end of the plate and leaves it. However, the shear stress at wall is constantly changing as water moves downstream because velocity gradient is also changing constantly due to growing boundary layer.
Also, I know the viscous force of a liquid is dynamic viscosity x velocity gradient (du/dy) for a 2-D laminar flow, and I did the flow simulation to obtain the velocity gradients at wall for a couple of locations from the origin (0.2m, 0.4m, 0.6m….. etc.) and then calculated the shear stress for these locations. However, for transient case like this, how should I use these velocity gradients to calculate shear stresses? Do I have to obtain all the velocity gradients along the plate and then derive the shear stress for every location accordingly since this is a transient analysis?
My second question will be how to correctly calculate calculate wall frictional wall forces for the same case. I know as water moves downstream the plate, its velocity decreases slowly due to frictional forces between wall and water. However, the velocity gradient is changing all the time which changes frictional forces as water moves downstream. Also, what is correct surface area to use to calculate frictional forces for each location for each time moment?
My third question would be would surface roughness of the plate contribute to slowing down the velocity of water as it travels. Intuitively thinking, I think it would but I couldn’t find the relationship between the surface roughness and water speed. For a solid object, it is a lot easier find the relationship but for a liquid I have no idea on that. Would surface roughness contribute to the other kind of frictional forces for a liquid? or it wouldn’t ?
In a nutshell, I just want to find like correlation or a couple of equations that approximate how water slows down as it travels downstream after coming out the nozzle.

Any feedback, comments and thoughts are really really appreciated
Thank you
Stanley

 

[anorlunda]

A sketch may help. I'm not quite sure how you define downstream.

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[stanley650586031]

Imagine a flat plate (2m long, 1 mm thick) in 2D... like a rectangle if you look at in front view. A water jet strikes at the beginning of the plate (left side of the plate) at 60 degrees. After the water jet strikes the plate, probably 80 percent of water goes to the right and the remaining amount goes to the left. For water going the the right, it continues to travel and velocity of it decreases as it travels to the right side of the plate (this is what downstream means) until it reaches the end of the plate.

 

[anorlunda]

Thanks, that helps. The water will also spread out in a fan shape as it moves downstream correct? If the film thickness of the water layer stays constant (may not be true), and since water is incompressible, can't you just use the conservation of mass and volume to determine velocity?

If the water flowed down a channel, and if it slowed as it went downstream, that would cause the water behind it to pile up. That's why I think film thickness and fan size/shape are important.

 

[jrmichler]

I assume this is what you described:

Some random thoughts:

The analysis of how much flow in each direction should be in any undergrad fluids book. I found one source that specifically assumed that the velocity of each flow along the plate was equal to the velocity of the incoming flow. But we know that the flow immediately spreads out perpendicular to the direction of flow, so it has a transverse velocity component. This is the fanning out, and is not part of the simplified analysis in undergrad fluids books. There may be an analytic solution, but I think you need CFD to fully solve the wall impact problem.

The water slows down as it slides along the plate. It may, or may not, form a hydraulic jump (search the term).

This could be treated as an open channel flow problem. All of the open channel flow equations that I have seen were for culverts and rivers, where velocities are lower, depths are deeper, and the channel is rougher. Look up open channel flow, and study the theory and derivation. Try a civil engineering fluids book, civil engineers do a lot of open channel flow. You might find something useful. CFD might work for this part, but getting the friction right might be a challenge. And the thin, spreading flow will definitely be a challenge.

You are right that roughness is important. You might start with something smooth, such as glass or polished metal.

Have fun, it's an interesting problem.