How Is the Damping Coefficient Calculated in a Fluid-Dynamic Dashpot System?

[a.mlw.walker]

So have a loose fitting piston attached to a conrod, attached to a flywheel. The chamber the piston is in is sealed reasonably well around the controd, and is full of a fluid with a viscosity.
Let's say the flywheel is spinning.

I want to work out from the fluid dynamics occurring, what the damping coefficient is around the piston due to fluid moving over it.

What I think is true.

1. There is shear stress occurring, due to the shearing layers of the fluid. I have included that.

2. The model can be assumed to be a pipe, and therefore there is friction occurring due to surface roughness, and the head loss (pressure) can be calculated using a moody chart. I am including this.

3. A pressure gradient builds up across the piston, due to the piston accelerating towards the middle of the stroke, fluid in front of it increases in pressure, and fluid behind it decreases in pressure.
This pressure gradient forces fluid over and around the piston.
This is acting against the motion, and is another source of damping.

If the first two are agreed with, it is the third one I think has the most effect, because my MATLAB code runs nicely but far too fast, i.e for a specified torque applied to the flywheel, the real model runs slower than the simulation (in matlab) - like a tenth of the speed.

I think I need to consider fluid compressibility (currently the fluid is air, but could be changed - so compressibility would change...?)

But I also don't know how to calculate the coefficient that is mulitplied by the velocity of the piston to give the damping term. Does anyone have any (prefferably justified) equation that can help me calculate this coefficient. Assume other variables are obtainable. I believe it could be NOT a constant, although I may be wrong.

I also am not sure whether (using the same equation) I can calculate any leakage past the conrod of the piston, as it moves linearly in and out of the piston chamber - I think there would be some around it

If anyone has any commments about my approach to this they would be very welcome.

Alex

 

[AlephZero]

Item 3 is important, because if you assume the fluid is incompressible, the mean velocity of the fluid through the gap is not the same as the velocity of the piston. In fact the fluid velocity is scaled up in the ratio of (area of the piston)/(area of gap).

Depending on the clearance around the piston, the "flow in a pipe" model may not be appropriate for this. Remember the "pipe" is effectlvely a thin rectangle, with width = 2 pi r and depth = the clearance gap, so any formulas that assume an approximately circular pipe should be treated with a lot of caution. The fact that the "rectangle" is bent round into a circle won't affect the situation much.

If you consider the flow to be similar to lubrication of a sliding bearing, the standard equation is Reynolds' equation (the same Reynolds who invented "the number"), which is the basically Navier-Stokes equation reduced to 1-D flow with the appropriate boundary conditions.

You should be able to find examples of Reynolds' equation used to calculate the viscous forces in sliding and journal bearings etc, but I don't know a reference for your exact application.

Edit: I just noticed your fluid is air. It would stiill be reasonable to assume it was incompressible up to a mach number of about 0.25. Otherwise, things will get MUCH more complicated.

 

[a.mlw.walker]

So the gap is actually a rectangle. and the speed of fluid is scaled up by the area of the piston face/area of rectangle.

When you say the reynolds equation you mean the equation to calculate the Reynolds number? How do you get viscous force from that?

OK I will have a look at Journal Bearings.

If anyone else has any help, would be welcome!
Alex

 

[AlephZero]

When you say the reynolds equation you mean the equation to calculate the Reynolds number?

No, I mean this equation: http://cambridge.org/us/engineering...ended/Section 5.4.2 The Reynolds Equation.pdf