# How to handle the formula for computing net thrust of closely coupled engine systems for an engine with a nacelle that has non-circular dimensions?

I am working on a paper and I wish to use this equation for calculating the thrust of an engine in a nacelle: $$Fn= \displaystyle \oint V_x\rho\vec V\cdot\vec n \ dA + \displaystyle \oint(P-P_{\infty})\vec n_x \ dA - \displaystyle \oint \vec \tau_x \ dA$$

This equation was given for a simplified case: the common circular nacelle. However, I would like to apply it to the flattened bottom nacelle as on the 737. I have attempted to fit an equation to this by adjusting the parameters of a cardiod in polar coordinates. I'm having some trouble changing the above to polar integrals, however I'm not sure as to how this conversion would affect the rest of the equation. I could apply the rectangular form instead and keep the above integrals, but I don't think there is a away to express y explicitly in terms of x; it must be done implicitly and so that integral wouldn't really be possible.

• Can we get some @PeterKampf love on this question? – Jihyun May 14 '18 at 15:47
• Did you mean to include an image under "here:" – Jamiec May 14 '18 at 15:50
• No, that was an error on my part, thanks for pointing that out. Unless you think I should place an image there for help. – Jihyun May 14 '18 at 19:32
• I really don't see any aspect of the formula that limits it being applied to only a circular exhaust nozzle. But it all seems a very pointless formula anyway, because to apply it you need to know the pressure and velocity in a piecewise fashion, across the whole face of the nozzle exit plane. Engine performance programs don't give you that data. You would need a 3D CFD model of the nozzle exit. And after all that effort, are you really going to end up with a more accurate answer than the standard 1D thrust equation: FN = m x (Ve - Vi) + A8 x (Ps8 - Pamb) ? – Penguin May 15 '18 at 9:45
• ... You raise a valid point @Penguin. I will take a further look at this from your point of view, but for the time being, if you'd like to just post this as an answer I will mark it as accepted if I don't get anything more detailed by today. Thanks for your second perspective. – Jihyun May 15 '18 at 14:09

In review of my earlier question, @Penguin and my math professor's commentaries agree on the point that the equation $$Fn= \displaystyle \oint V_x\rho\vec V\cdot\vec n \ dA + \displaystyle \oint(P-P_{\infty})\vec n_x \ dA$$ is the meat of the problem and that when you get the piecewise pressure and velocity, the third term $$- \displaystyle \oint \vec \tau_x \ dA$$ was inconsequential, as it (as far as I can mathematically deduce) describes the core cowl and the external nacelle. As it is an independent characteristic, I'm confident I can just adjust the modelling for a different shape. Thanks again to @Pengiun for providing an outside perspective to my dilemma.