# Aerospace Control Volume of single expansion ramp nozzle (SERN)

1. May 7, 2017

Hi guys,

I need to determine the performance of a single expansion ramp nozzle (SERN) from CFD results with different nozzle pressure ratios (NPR). For some NPR the nozzle is overexpanded and for some underexpanded.
Now the impact of the control volume definition on especially the axial thrust is confusing me a lot. I attached a sketch of the nozzle with a control volume. The blue dashed line represents my control volume and the red dashed line just the inflow plane and the nozzle surface.

Now my understanding is, that the force acting on the nozzle surface is fixed by the pressure and shear acting on it. If there is an entrainment of external air into the nozzle (e.g. due to overexpanstion), then this entrainment would change the pressure and shear distribution over the surface. Subsequently the effect of an entrainment should already be included in the surface integrals of pressure and shear. And the difference in stream thrust of an arbitrarily chosen exit control volume and the inflow stream thrust should give exactly this surface force, right?

From my CFD simulations I can easily extract the pressure and shear force that are acting on the nozzle surface and I have the inflowing momentum (stream thrust). My first thought was, to simply add up the inflowing stream thrust to the force acting on the nozzle to get the outflowing streamthrust. Subsequently I would use this exit stream thrust for the performance analysis. That worked fine.

Then I thought that if I use the control volume given in the uploaded figure (blue dashed lines) and calculate the momentum flow over the exit portion of my control volume, the difference between this exit stream thrust and my inflow stream thrust should exactly yield the force that is acting on the body of the nozzle. But the problem is, that it doesn't match...

Cheers,

2. May 8, 2017

### Staff: Mentor

What about the contribution of the pressure difference between inlet and outlet to the momentum balance?

3. May 8, 2017

Hi,

I calculate the stream thrust in x-direction (nozzle axis) as: I = m_dot*u_x + p*A_x

So what I first did is calculating the inflow stream thrust I_in and the surface force F_nozzle. Then I can calculate the stream thrust at the outlet:
I_out = F_nozzle - I_in. In this way I don't need to make any assumptions about the control volume at the exit.

The second method that I tried (just to see if my assumption that the exit control volume can be chosen arbitrarily) was to calculate the stream thrust over my exit control volume (blue dashed) and add it to the infow stream thrust: I_in + I_out =? F_nozzle
And at this point I don't get my F_nozzle. For some NPR the difference to the integrated F_nozzle is only below 5%. For a few cases the difference is between 10-20%.

A difference of below 5% seems reasonable to me, since the calculation of I_out via the exit control volume (blue dashed) involves some interpolation that can definitely introduce some error.

4. May 8, 2017

### Staff: Mentor

Are you making sure you use gauge pressures, and not absolute pressures?

5. May 8, 2017

I'm using static pressure. I guess that's what you mean with gauge pressure.

6. May 8, 2017

No. Static pressure and gage pressure are difference concepts. Static pressure is the pressure you would "feel" in a fluid assuming it doesn't get disturbed by your feeling instrument. It's the pressure associated with a pressure force. What @Chestermiller is talking about is the concept of absolute and gage pressures. Absolute pressure is the true, thermodynamic pressure at some point. Gage pressure is referenced against atmosphere, i.e. it is $p_{gage} = p_{abs} - p_{amb}$.

7. May 8, 2017

### Staff: Mentor

Yes. As I'm sure you know, to get the correct force of the gas on the nozzle, it is preferred (and easier) to use gage pressure. That way, one does not need to account for the force of the atmosphere on the outside (back) of the nozzle.

It still isn't clear whether the OP is using gage pressure or absolute pressure in his model calculations. I'm guessing he is using absolute.

8. May 8, 2017

Of course. It's a lot easier to integrate an entrance and exit that are typically planar than to integrate around the the rest of the complex shape. It turns out it's a lot easier to integrate around that complex shape when the integrand is zero.

9. May 8, 2017

### Staff: Mentor

You and I are "preaching to the choir."

10. May 8, 2017

I generally get that question of absolute vs. gage pressure when working with a control volume for my students. A quick picture and some integrals and I generally have them convinced pretty quickly to abandon using absolute pressures for that application. It does introduce some cognitive dissonance when they go to study compressible flows, though, as those absolutely require absolute pressure in the equations since they are thermodynamic in nature. But that's a topic for another time.

11. May 9, 2017

Alright, thanks for telling me the difference ;)

In fact I calculated the inflow and outflow stream thrust as well as the nozzle surface force both with your definition of absolute pressure (my static pressure) and gauge pressure. However, in terms of error it doesn't really make a difference. Right now I'm just gonna live with the error, since for most cases it's not too big and for the ones where I have a little bit of a larger error I can trace that back to a not properly converged solution.

But I can conclude that the definition of the exit control volume is not of importance for such an analysis.

Cheers!

12. May 9, 2017

### Staff: Mentor

Are you including the momentum that exits through the upper horizontal part of the blue control volume that extends beyond the red physical boundary?

13. May 9, 2017