# What is the Saint-Venant formula for calculating equivalent air speed (EAS)?

I'm studying some ground exams (for a lapsed IR). In the course notes it says that EAS (Equivalent Airspeed, i.e. a compressibility-corrected airspeed) is calculated (e.g. by an aicraft's Air Data Computer) using the "Saint-Venant formula". The course notes don't actually give said formula.

I assume they are talking about this guy who derived a forerunner of the Navier-Stokes equations. However, most of his work seems to be to do with waves in shallow water and not aerodynamics. I think I know how EAS is calculated, and it doesn't seem to involve anything as hairy as Navier-Stokes (although it does require isentropic flow relations.) But I can't figure out how Saint Venant has anything to do with it. From another forum: "In the EASA learning objectives I can find no reference to 'Saint-Venant' so it is probably just something somebody thought up to confuse you." Can anyone shed any light on this?

• I just read your question multiples times but I'm not sure I'm understanding it correctly. Is it "Which Saint-Venant the formula for calculating EAS is named after?" If so, you should edit your question, especially the title (there can be other most obvious question with such a title, it would help navigate among them without opening each one) – Manu H May 18 '20 at 15:02
• Good point. It looks like another user (Pondlife) edited it already. This makes it a more consistent question. – Halzephron May 19 '20 at 17:55

Have a look at http://zeteamfirst.free.fr/pilotage/anemo/asi_4.htm

It states the Saint Venant equation as being: $$$$Pt - Ps = \frac{1}{2}\rho V_p^2 \Bigl(1+\frac{M^2}{4}\Bigr)$$$$

where:

$$Pt$$ : total pressure

$$Ps$$: static pressure

$$\rho$$: air density

$$V_p$$: TAS

$$M$$: Mach number calculated as ratio of TAS/speed of sound.

Edited: ROIMaison's reply with the reference to Flight Mechanics of High-Performance Aircraft, Nguyen, looks much better to me, and I am tempted to delete my answer, for 2 reasons:

1) except for the $$(1+M^2/4)$$ part, the equation above is exactly Bernoulli's equation for incompressible flow, and,

2) I am quite sure that a Machmeter can do its job without being able to measure air density directly. In Nguyen's text, air density is not needed in the equation, whereas it is needed in the above equation.

I might have been a little too happy about finding an answer to question its validity thoroughly. Lesson learned!

• Both you and ROIMaison provided very helpful answers - first time I've seen an actual equation associated with the name. The two formulas are different but use different inputs. I might see if I can derive one from the other, to check. I upvoted both your answers and approved yours because it was first, but very grateful to both of you. – Halzephron May 19 '20 at 17:52
• You're right that your formula is very similar to the bernouilli equation. I don't really understand how the Mach factor is derived. The equation I found is derived quite easily using the isentropic relations, that's why I was surprise that our two formulae look so dissimilar from each other. @Halzephron would be interested to see if it's possible to derive one from the other. – ROIMaison May 20 '20 at 7:49

I found this snippet

from Flight Mechanics of High-Performance Aircraft, by Nguyen here.

I'm not sure if this answer is consistent with the answer given Ugo..

• Thanks, this and Ugo's answer are the first time I've seen an actual equation associated with M. saint-venant. Your answers have prompted me to do some searching for his name in association with Mach meters instead of ASI's and this has yielded some more promising google hits than I was getting before. – Halzephron May 19 '20 at 17:48
• You're welcome, it also took some effort for me to find a formula. I found this through books.google.com, and there were also some other books that popped up that might be interesting to you. You could have a look over there. – ROIMaison May 20 '20 at 7:14