Mach-Number to Airspeed Conversion Above 65,000 Feet
 

I have an application that I can use to convert mach-number to KEAS at altitudes up to 65,000 feet. Above that, it doesn't work.

Does anybody here have a mach-number to KEAS conversion table for altitudes above 65,000 feet, or a mathematical formula to help perform the conversion?

Jane, you seem to me to be plenty smart enough to figure it out for yourself without a program, but it does need some mathematical background, and I don't know how much you have.

First, a caveat. The speed of sound is functionally dependent only on the square root of temperature in Kelvin. So you need to know what the temperature is at a given altitude - and of course temperature can vary. So speed of sound is not a function of altitude per se and your program is not giving you that. It is probably assuming you are in an International Standard Atmosphere (ISA). Let's make that assumption below.

The speed of sound is sqrt(gamma x R x T), where gamma is the ratio of the specific heat of air at constant pressure to the specific heat of air at constant volume and is usually taken to be about 1.4; R is the gas constant, whose value for "ideal" air is 287 Joules per kilogramm per Kelvin in SI units (or 1716 ft-lb per slug per degree Rankine in English units); T is temperature in Kelvin (you'll have to apply a degrees-Rankine conversion factor in this formula if you are working in English units).

Now, you just need the distribution of temperature with altitude in the ISA. I'm sure there is a nice graph seomwhere on the WWW (it is a linear spline, which is a name for a number of straight line segments joined together at their ends), but I didn't find it. You can get it pointwise from the standard atmospheric calculator at Standard Atmosphere Calculator, but then you can get the speed of sound from it, too, up to 86,000m (about 180,000 ft). To my mind, understanding the relationships of the quantities and using an arithmetic calculator is more fun than plugging numbers into some special computer program.

There is a very nice explanation of the standard atmosphere in Chapter 3 of John D. Anderson Jr.'s Introduction to Flight (6th Edition, McGraw-Hill 2008). This includes the definition of ISA temperature in terms of altitude, given in a graph in Figure 3.4. (Preceding that is a discussion of what "altitude" means!).

You may be wondering about gamma, the ratio of specific heats at constant pressure and volume. That is a matter of (what is called) elementary thermodynamics. Let me spare you the details here, although I seem to remember Pugilistic Animus was inspired to write it down on some other thread, which I forgot. The issue is dealt with quite nicely in Section 4.5 of Anderson. The speed of sound is discussed in Section 4.9.

I think you'll find Anderson quite helpful. If you don't have easy access to a copy, you might think of buying it. It is comprehensive, and Anderson has gone to a lot of trouble over the editions to make it so (it is over thirty years since the first edition). If you need to follow something from very first principles on, you can do it with Anderson. You'll need some facility in following reasoning using differential calculus - I don't know whether you have that.

PBL

PBL

Okay

I can work with Kelvin okay. Which units are you using for pressure?

Of course -- It's important to understand the concept as well as just the answers

That's a real expensive book...

My calculus skills are kind of rusty...

What on Earth are you intending on flying?? ;)

There's an ISA Chart to 70,000 feet at:-

ISA Chart

It is an adaption of stuff floating around on the web.

What a very odd question!

That depends on how much knowledge is worth to you. I think it's cheap.

Rusty? Would you care to be more precise?

PBL

The International Standard Atmosphere and the 1976 US Standard Atmosphere agree up to 32 kilometres. The latter standard is included in the references to the Wikipedia entry on ISA. Working with altitudes greater than this will require taking account of changes to R (gas constant for air). At sufficiently high Mach numbers account may need to be taken of changes to gamma.

Use the Barre de Saint-Venant equation for Mach numbers below 1, and for Mach 1 and above use the Rayleigh supersonic equation. Both are derived in Anderson referenced by PBL above, and are elsewhere located on the Internet. Build your tables for Mach -v- CAS. While EAS is a physically meaningless quantity the step in determining it from TAS is straight forward.

The equations you'll need can be found in NACA Report 837. Aiken, William S, Jr. (1946.) NACA Report 837: Standard nomenclature for airspeeds with tables and charts for use in calculation of airspeed. Langley, VA. NACA UK Mirror report description page

PBL

Well, one measurement for pressure is PSI, another is pascals. I'm just wondering which one should be used

True enough

I haven't used them in a long time...


selfin

Okay, so just to make sure I have all these variables right
q = dynamic air pressure
R = gas constant for air
rho = ambient air density
gamma = ratio of specific heat at a constant pressure to specific heat at a constant volume

I'll have to give it a look. I'm not particularly an expert on greek characters, so I'll have to find the greek-alphabet for that purpose.

See Equivalent airspeed - Wikipedia, the free encyclopedia

Used, Jane? I didn't need to use either.

This looks to me very much as if you performed a keyword search on my reply. I grant you, the word "pressure" was in there.

So you think "calculus" is plural. Could be.....

Well, I think that clinches it for me. I thought I'd be able to get you if it were true, but it turns out you got yourself. You have done very, very well so far. You ask great questions, and it is a lot of fun thinking of the answers. Considering you are only a few years old, you are doing pretty well.

Do you by chance have a big date at the end of the month? Do we by any chance have a mutual friend called Eliza, whose Dad died almost three years ago now?

Credit, BTW, must go to mike-wsm, who guessed in mid-January. I couldn't believe anyone would go to the trouble of figuring out their way around PPRuNe syntactic tasks, but obviously one of your friends has done so.

PBL

the SR71:{...used a direct EAS indication for operations and many items such as spike position were in terms of EAS...

the POH has since been declassified...good stuff:ok:

I got one on Ebay for about $50.....

Mutt

Mutt is this you?


:p


we are all just grown up children after all :}

PBL

What units do you use? I'm not just asking this stuff to be difficult

The two units of pressure I could readily think of was PSI and pascals.

No. I said that my calculus skills were rusty -- skills are plural.

My knowledge of the greek alphabet is limited because, I'm not fluent in Greek. Yes, some greek characters are covered in math class, but it's been quite awhile since I've taken a math-course.

I can tell you an alpha looks like an "a" drawn as if you were trying to make it look like a fish; beta as if it were a "B" drawn in some kind of calligraphy; a gamma looks like a "y"; a delta like a triangle; an epsilon like a curvy "E";

I know alpha looks like a fish, beta looks like a really fancy B, I can tell you that gamma looks like a y, Delta looks like a triangle, Epsilon like an "E"; a sigma like a jagged "E"; a Pi looks kind of like two "T's put together with the horizontal line up top kind of curvy; a chi like an "x
; a Mu like a backwards "u" with a curvy tail; an Omega looks kind of like two legs with a loop drawn to connect them.

That does cover a number of the characters, but I don't remember what a "rho" looked like and a number of the other greek characters.

Jane,

I guess the answer is in the very first sentence of PBL's reply:

"The speed of sound is functionally dependent only on the square root of temperature in Kelvin." (my emphasis)

Your task is to determine the temperature (real or assumed, as befits your problem) for where you (hypothetical) aircraft is, then you call that 'T' and plug it into the formula that PBL gives you in his third paragraph.

Simple as that. Your real problem is not in calculating a Match number given an airspeed, your real problem is in finding the value for the local speed of sound. PBL goes on to suggest that you might deduce a nominal figure for that variable from tables available on the internet (and in books), which will give you an LSS (or the elements needed to calculate it) from an atmospheric model. Up until that point, pressure has nothing to do with what you're attempting to solve.

LH2,

To convert TAS into IAS you need to know the atmospheric pressure... that's why I asked what units of pressure you need.

You're in a desert, walking along in the sand, when all of a sudden you look down...

FullWings

What does this have to do with the topic?

Jane-DoH,

I could ask you, as you did to FullWings, who is as aware as I am:

What does this have to do with the topic?

If you want to carry on playing the game, I am willing because it's fun gradually finding out what you know and how, but you need to answer my questions if you want me to answer yours.

I think if you are behind on your competence with the greek alphabet, you are going to have a hard time dealing with airspeed and Mach, because all that is so dependent on the greek alphabet, as you remarked.

PBL

Okay, so you want me to be able to make sure that I can compute the mach-number figures based on air-temperature?