Relation between IAS and Mach number

[Rivet gun]

This quote comes from the FAA's "Airplane Upset Recovery Training Aid" referring to Mach stability:

"This stability can be independent of airspeed if, for example, the airplane crosses a cold front. When the outside air temperature changes, the Mach number changes, even though the indicated airspeed may not change."

Now I thought that at a constant flight level, the relationship between IAS and Mach number is independant of temperature. So if you maintain a constant flight level and constant IAS, you also fly a constant Mach number even if the air temperature changes.

But then the source document was written by senior test pilots from Boeing and Airbus, so maybe I have got it wrong??

[bubbers44]

Mach speed is affected by temperature. IAS is affected by altitude with constant mach. If you were above the trop and climbed 4,000 feet your IAS would decrease significantly but mach would be constant because temp would be constant. Say you cruise at .80.

[rhovsquared]

as a (bio)chemist I can't resist this,
Mach number depend on TAS which in turn depends on EAS (IAS corrected for compressibility and position error)

TAS=EAS/(sigma)^(1/2) and sigma is the ratio of densities this is the HTBJ part.

Now the partial pressures of the atmosphere add to give total pressure P (it's value can be determined from the the altitude, hypsometric formula of altimetry:8) from the ideal gas law PV=nRT; n=moles of gas, R=gas constant 0.0831L*atm/mole*K.

n=mass/molar mass(MW) and rho or density =mass/volume

so I write PV=(m/Mm)RT; rearanging that I have PMm=rhoRT=dRT

so TAS depends on rho which depend on T and the two vary directly; actually with that equation you can relate mach no. to T or P or d=rho

[bubbers44]

What? My 727 onboard computer doesn't understand.

[Start4&3]

Say that in English Mr. Biochemist.:eek:

[hawk37]

Rivet Gun, I believe you are right, since you did say constant flight level. I don't believe these test pilots would be wrong, perhaps a misquote. Perhaps Bubbers and Rhov can specifically respond respond to your query, and state whether they agree with you or not. Of course, I have little formal qualifications, and have known to be completely wrong.

Hawk

[bubbers44]

It was difficult to know exactly what IAS would do flying a FL and flying into colder air because the mach would increase because of the colder air and the speed of sound would of course decrease but how much, if any, would this change the IAS? Also would the true altitude change? That is why I did the constant mach example. I don't want to get in to too much physics because then it is too hard to understand. Previous example proves that.

[30/30 Green Light]

Bubbers,I can't remember all the physics behind it but back in my 707 days cruising @ M.81 the formula used to X check or in case of Machmeter failure was:SAT+TAS=521(?) at any altitude.I'm pretty sure it was 521 but it was a while ago and it worked spot on every time.

[specialrider987]

Mach is the ratio of an aircraft's True Airspeed (TAS) to the local speed of sound.

The local speed of sound is only dependent on temperature. Nothing else.

Therefore, it is conceivable that at a constant Mach one could cross a front and have a change to the TAS in order to maintain constanct Mach, which would require a change in IAS.

The colder the air mass, the higher the Mach number for a constant TAS

The warmer the air mass, the lower the Mach number for a constant TAS

I hope this helps... it's been a while!

[chornedsnorkack]

It was difficult to know exactly what IAS would do flying a FL and flying into colder air because the mach would increase because of the colder air and the speed of sound would of course decrease but how much, if any, would this change the IAS? Also would the true altitude change?

Well, suppose you have a constant true altitude - say, sea level - and the temperature changes, for example from +40 to -40 while the pressure remains unchanged...

If pressure is not changed, flight level is not changed. But the density of air changes. Therefore the IAS changes at a constant TAS.

Right?

So, the same ought to apply at any higher FL... as the FL follows pressure, flying into colder air should increase IAS if the TAS is unchanged...

[Rivet gun]

, which would require a change in IAS.

Not so, at least as I understood it.

You are flying a constant flight level and the autothrottle holds a constant Mach number. You fly from a colder to a warmer air mass:

Going to the warmer air the local speed of sound increases, therefore TAS increases.

The warmer air is less dense, which affects the IAS / TAS ratio.

The laws of physics conspire such that these effects cancel each other and IAS remains constant.

Another way of looking at this is to consider the inputs required to an Air Data Computer (or traditional instruments).

To compute IAS (theoretically CAS if we correct for position error) requires only impact pressure (this is the pitot - static differential measured by the traditional "airspeed" capsule).

To compute Mach number requires both impact pressure and absolute static pressure (traditional "altitude" capsule).

So far we have both IAS and Mach number without any input of air temperature. Thus the relation between IAS and Mach number is independant of air temperature.

Now if we wish to compute Static Air Temperature (SAT) and TAS, we require also the Total Air Temperature (TAT) input.

[hawk37]

Rivet Gun, guess I am the only one then that thinks you are correct. Seems the responses have all avoided your original question though. For the others who replied, here’s Rivet Gun’supposition from his original post,

“Now I thought that at a constant flight level, the relationship between IAS and Mach number is independant of temperature. So if you maintain a constant flight level and constant IAS, you also fly a constant Mach number even if the air temperature changes.”

The thread then departed to address other aspects, such as mach vs tas, EAS, compressibility, formulas…..

If Rivet is incorrect, then, will anyone go out on a limb and say what direction will the IAS (assuming no error correction/position error) or CAS will take if you fly at a constant mach and flight level and the air temperature gets colder?

Hawk

[specialrider987]

rivet gun

IAS corrected for Pressure = Calibrated

Calibrated corrected for Compressibility = Equivalent

EAS corrected for Density = TAS

it is conceivable in your example that the effects of P and C could cancel one another out on a particular aircraft on a particular day, leaving the IAS constant and the Mach number constant. This would have more to do with the compressibility effect of the aircraft being flown than the atmospheric factors.

The relationship above is universal, not aircraft dependent.

Cheers :)

[hawk37]

SpecialRider, reference your quote

"it is conceivable in your example that the effects of P and C could cancel one another out on a particular aircraft on a particular day, leaving the IAS constant and the Mach number constant. This would have more to do with the compressibility effect of the aircraft being flown than the atmospheric factors"

I just don't see how your argument provides any answer to Rivet Gun's question. If by P you mean pressure, and by C you mean Compressibility correction, then I don't see why you think they "could cancel another out" has any bearing at all. Remember that P is a constant, a constant Flight level. And as per Rivet Gun's post, IAS is a constant, and if position/error correction are zero, then so is CAS. If pressure and calibrated airspeed are constants, then so is compressibility correction. So with all these being constants, good for all days and on all aircraft, I don't see why you say in your quote "on a particular aircraft on a particular day".

[Mad (Flt) Scientist]

Ultimately, the speed of sound in air depends on the air temperature only.

In fact, assuming an ideal gas, the speed of sound c depends on temperature only, not on the pressure. Air is almost an ideal gas.

Therefore, you can have the speed of sound be different for the same density and pressure, as long as the temperature is different. Density and pressure determine flight level and TAS-to-IAS, so you'd be at the same FL, the same IAS, the same TAS, but a different Mach number.

In a single volume of gas, of course, I can't change JUST T and not p and V. But we're talking about a case of moving from one section of the atmosphere to another, so the gas law relationship doesn't have to hold.

So the original text is correct - if you pass a front such that there is a marked temperature change, you may find the relationship between Mach and IAS changes, even at a fixed FL, due to temperature changes.

[chornedsnorkack]

Ultimately, the speed of sound in air depends on the air temperature only.
Therefore, you can have the speed of sound be different for the same density and pressure, as long as the temperature is different. Density and pressure determine flight level and TAS-to-IAS, so you'd be at the same FL, the same IAS, the same TAS, but a different Mach number.
In a single volume of gas, of course, I can't change JUST T and not p and V. But we're talking about a case of moving from one section of the atmosphere to another, so the gas law relationship doesn't have to hold.

Er, why does the gas law not hold in atmosphere?

[Mad (Flt) Scientist]

Yep, that was nonsense.:ouch: The relationship between CAS, pressure altitude and Mach number is unique. So as long as you stay at the same pressure altitude CAS and Mach vary uniformly.

I'm not doing very well the last few days :(

[hawk37]

Mad Scientist, are you then saying that if one maintains a given Flight Level, and given CAS, that should the aircraft then fly into an air mass that is now at a different temperature, the mach number does not change?

This was the crux of Rivet Gun's question, which seems to be still debated.

Thanks, Hawk

[Mad (Flt) Scientist]

Yes. But it's essentially impossible to do so.

The relationship between CAS, Mach and pressure altitude (FL) is fixed. Therefore if you do as suggested, and
maintains a given Flight Level, and given CAS,

you MUST maintain a fixed Mach also.

I think the problem is that the original text was presumably talking about flying into a frontal system at a fixed GEOMETRIC altitude - in which case the FL won't stay fixed, and there will be a change in Mach for constant CAS. Since it's talking about a case with a sharply delineated temperature profile, changing quite rapidly over a short distance, unless one flew a sudden climb or dive to maintain fixed FL, one CANNOT actually do what the question posits, i.e. fly at constant CAS and FL across a temperature change.

The original article has chosen a bad example to try to explain the difference between CAS and Mach, IMO. They might have done better to consider a constant CAS descent, where they could have talked about the Mach effects just as easily.

[hawk37]

Sure, ok, if maintaining a constant GEOMETRIC altitude then I can see that the same CAS will not mean the same mach when flying through a temperature change. I don't think anyone was having difficulty with that. thanks for your input
Hawk

[rhovsquared]

Sorry about that:\

I figured I would divide this problem into three categories
1. speed of sound: The value of SoS depends upon the MEDIUM, and is affected by temp. humidity and winds. its transmission rate depends upon the elasticity of the medium [the Bulk and Young's modulus] (5X faster in steel than air). because sound travels as longitudinal waves (compressions and rarefactions) if the media is dense (humidity) or warmer (there more energetic) molecules to push around and SoS increases. As you get colder SoS decreases

2.mach number/TAS/EAS/IAS: as every one has said, increases with TAS which increases with altitude for a constant EAS, and increases as SoS decreases
3. gas laws: to avoid being timed out i'll post now:}
I feel I've not explained well

moles are a measure of the amount of a substance. Carbon 12 weigh 12 grams for every mol (n) and one mole of carbon has 6.023*10^23 C atoms. one mole of H2 gas a molecule weighs 2g but has same number of particles(Avogadro's No [NA]).
for an ideal gas (no molecular interaction)

The number of moles is dependent on 'state variables' P,V, and T. with any of these state variables one can find another unknown varialble, so the gas laws allow one to explain how SoS depends on any thing you want because all of the knowns lets say P,V and n can be used to rewrite the expression any how you'd like the above was a little algebra to relate every thing to density instead.
with out these laws thermodyamic would be speculation-at least till I build that entropy meter :} :}

lastly. I forgot this b4 Sigma = rho/rho(msl)

[bubbers44]

We certainly have had an interesting discussion about this and probably all have learned from it. I admit I haven't studied aerodynamics for 30 years but 30 years ago I wanted all the answers and found most of them in books. Asking fellow pilots usually ended up in multiple answers, most being wrong.

Captains wanting the flight attendants to move to the front galley to save fuel because the tail was dragging enroute, etc. One chief pilot insisted you had to hold downwind rudder when enroute to keep from weather vaning into the wind. In a discussion like this everybody gets their input and usually the truth prevails benefiting everybody. Thanks to everybody for contributing.

[Rivet gun]

Lets get rid of these darn moles! The universal gas constant is 8.314 J/mol/K. If we multiply this by the average moles per kilogram for dry air (34.53) we get the specific gas constant for air which is 287.05 J/Kg/K. Call this R

We can now express the ideal gas law as:

P = R rho T

where P is pressure (Pa), rho is density (Kg/ m^3) and T is temperature (K). Humidity complicates matters because it changes R, so lets us stick to considering dry air.

Mad (Flt) scientist: the ideal gas law certainly DOES apply across air masses provided we are considering dry air. Furthermore aircraft can and do maintain constant FL across air mass boundries, they do not maintain geometric altitude (ATC would complain otherwise). The reason they can do so is because the temperature change is gradual, not a step change. For this discussion we need only consider the end states, i.e at one time the aircraft is flying in air at temperature T1, at a later time it is flying at the same FL and Mach number in air at temperature T2.

Now to deal with the speed of sound (SoS). This depends on the square root of the ratio P/rho. But from the gas law P/rho = R T. The exact equation also includes the ratio of specific heats (gamma) which for dry air is 1.4. So:

SoS = sqrt(1.4 R T)
= 20.05 sqrt(T) m/s
=39 sqrt (T) knots.

Now back to the original question.

At constant FL pressure is constant, so from the gas law rho T is constant. This is the important point.

If the aircraft flies from a cold air mass (T1) to a warm air mass (T2) and mach number is held constant the TAS increases by the ratio sqrt(T1/T2). But we have established that rho T is constant, so if T increases rho decreases in exact proportion.

The ratio EAS/TAS depends on sqrt(rho1/rho2). Therefore the increase in TAS is exactly cancelled by the decrease in EAS/TAS ratio and EAS remains constant.

To go from EAS to CAS we adjust for compressibility. However compressibility is a function of Mach number and FL, both of which are constant. Therefore compressibility error is constant and CAS is constant.

To go from CAS to IAS we adjust for position error. This is empirical and aircraft specific, however modern flight instrument systems have built in position error correction and for practical purposes we may assume CAS = IAS. Besides at constant FL and Mach number position error would be constant anyway.

Therefore IAS remains constant

QED

[Old Smokey]

TEMPERATURE HAS NO EFFECT!

One of the basic tenets of manometry is that the relationship between CAS (or EAS) and Mach Number depends only upon Static Pressure (Ps). Temperature has no effect whatsoever upon the CAS/EAS/Mach relationship, only Ps, end of story.

I believe that the original quote that spawned this thread was a mis-quote. As an aircraft passes from a cold mass of air to a warm mass of air, the CAS, EAS, and Mach Number will all initially fall due to reduced thrust. After thrust has been adjusted to regain the original target speed, whether it be a CAS or a Mach Number, both will again be in complete agreement with each other. The TAS and TAT will be higher. An example for an aircraft at 300 CAS at 30,000 feet, and ISA-25°C moving to an ISA+25°C air mass is as follows -

ISA-25°C : CAS = 300 : EAS = 285.00 : Mach = 0.7905 : TAS = 439.75 : TAT = -44.0°C
ISA+25°C : CAS = 300 : EAS = 285.00 : Mach = 0.7905 : TAS = 490.76 : TAT = +12.3°C

Now that was one HUGE longitudinal temperature gradient that I chose to prove the point, but even with this ridiculous temperature gradient, the CAS, EAS, and Mach Number have not budged by a 10,000th of a decimal place. TAS and TAT have risen very significantly.

TAS is a very significant value to aviation, it is of extreme importance in navigation, and in aerodynamics, TAS is 'the' V that we consider in the highly important 1/2 Rho V squared formulae applied to both Lift and Drag calculations. TAS is so important that it's possible to be blinded by it when considering manometry, the measurement and calibration of speed measuring instruments.

Consider this, If we measured our speed as CAS, we need Pressure Height and Temperature to calculate TAS. If we measured our speed as a Mach Number, we need only Temperature to calculate TAS. If we now put it into very simplistic terms, we would have 2 formulae, namely -

TAS = CAS X A function of Ps X A function of Temperature, and

TAS = Mach X A function of Temperature.

Now, putting the two formulae together to calculate Mach from CAS (or vice versa), and as both equal TAS, we have -

Mach X A function of Temperature = CAS X A function of Ps X A function of Temperature

Now, as "A function of Temperature" appears on both sides of the equation, it is SELF CANCELLING, leaving us with -

Mach = CAS X A function of Ps

Wot! No Temperature?:uhoh:

Before I'm accused of over-simplification, take a look at the formulae by which the Airspeed Indicator and the Machmeter are calibrated -

Vc = SQR((Y/(Y-1)) X Po/Qc X ((Qc/Po + 1)^((Y-1)/Y) - 1) X SQR (2 X Qc/Rho0)

M = SQR (5 X (Qc/Ps + 1)^((Y-1)/Y) - 1)

(NOTE - For flight at altitude, substitute Ps for Po in the Vc formula).

Where -

Vc = Calibrated Air Speed, ft/sec
M = Mach Number
Y = A constant, being the ratio of specific heat of air at constant pressure to the specific heat of air at constant volume = 1.4
Po = Sea Level Air Pressure
Ps = Static Pressure
Qc = Impact Pressure
Rho0 = Standard Sea level air density

Where is the temperature for either equation? It simply isn't there. :ok: The only time that Temperature came into play was in establishng the STANDARD Sea Level Air Density (.0023769 Slugs per cubic foot), in other words, the standard temperature at Sea Level was used in establishing a CONSTANT for calibration purposes.

As a parting remark, temperature, again, has absolutely no effect upon the Change-Over height from CAS to Mach for a given speed schedule. In other words, if your Climb speed schedule was, for example, 320 CAS / M 0.84, the Change-Over height will be 30,107 feet every day of the year, Winter, Summer, Spring and Fall, over the Arctic, the Sahara Desert.........Everywhere:)

The FAA guy made a mis-quote..... It's been done before:E

Regards,

Old Smokey

[bubbers44]

--------------------------------------------------------------------------------

This quote comes from the FAA's "Airplane Upset Recovery Training Aid" referring to Mach stability:

"This stability can be independent of airspeed if, for example, the airplane crosses a cold front. When the outside air temperature changes, the Mach number changes, even though the indicated airspeed may not change."

Now I thought that at a constant flight level, the relationship between IAS and Mach number is independant of temperature. So if you maintain a constant flight level and constant IAS, you also fly a constant Mach number even if the air temperature changes.

But then the source document was written by senior test pilots from Boeing and Airbus, so maybe I have got it wrong??

Going back to this original question I say mach would change because it is soley related to temperature. If temp decreased, mach would increase because TAS would be constant and the speed of sound would decrease.

[Rivet gun]

Old Smokey,
Thanks, It is nice to get a definitive answer from somebody who really understands this stuff.


Vc = SQR((Y/(Y-1)) X Po/Qc X ((Qc/Po + 1)^((Y-1)/Y) - 1) X SQR (2 X Qc/Rho0)


Since you give a formula for CAS, I'll give another CAS formula for comparison.

Vc = Ao X SQRT (5 X (Qc/Po + 1)^((Y-1)/Y) - 1)

Where:
Ao = Standard speed of sound at sea level (661.5 kts)
All other terms as defined by Old Smokey

I like this one because it is simpler and it works in any units: i.e if you want CAS in knots, enter Ao in knots, if you want it in m/s enter Ao in m/s etc. Also note the formula is very similar to the formula for Mach number (spot the differnces)

(NOTE - do NOT substitute the Po term for flight at altitude. CAS is a function of Qc and all other terms are constants. CAS is not a function of Ps)

I leave it up to more mathematically inclined readers to prove that this CAS formula and Old Smokey's CAS formula are equivalent.

[Old Smokey]

Oh my Gawd!!!!

Thank you Rivet gun, for pointing out my typo. Text suitably edited and emboldened in the corrected area. That will teach me to not be a lazy typist and not use 'cut and paste' so much.

Just in case anyone was foolish enough to copy my post, the correction relates to the value of Gamma, for which I used the letter Y (Sorry, no Greek key-board). I stated -

Y = A constant, being the ratio of specific heat of air at constant pressure to the specific heat of air at constant pressure = 1.4

It should have read (as it now does) -

Y = A constant, being the ratio of specific heat of air at constant pressure to the specific heat of air at constant volume = 1.4

Thany you again Rivet Gun,

Regards,

Old Smokey

[enicalyth]

G'Day OS!

Isn't cp/cv in dry air a variable with temperature? I know only a teensy bit.

Eg Dry air at 273K cp = 1006 joules/per kg/per K; cv = 719 ditto.

Now I'm moist! cp =? cv = ? cp/cv = ? What happens if I add lots of polyatomic atoms instead of believing air is solely a diatomic gas.

Only splitting hares because I've run out of rabbits but if it's always 1.4 why do I have to look up tables?

Anyway how the devil are you? I shall be in the UK for Farnborough and again for the Gatbash. But not Manchester!

See yuh in the Long Bar!

Rgds

Enny

[Rivet gun]

I think it comes down to the ideal gas assumption. In an ideal gas gamma is constant and for a diatomic ideal gas (two atom molecules) the theoretical value is 1.4.

But of course an ideal gas is just a theoretical model. It may be a close enough description of real air for many purposes, but it is not exact. To the extent that real air is not a true ideal gas gamma may vary a litttle bit with temperature.

Also as you point out air is not entirely diatomic (N2, O2). It also contains H2O, CO2 and traces of monatomic gasses which will affect the gamma value.

Nevertheless flight instrument calibration equations are based on ISA which assumes inter alia that air is a ideal gas and that gamma = 1.4.

[FullWings]

I may be barking up the wrong tree here but are we not confusing a static situation with a dynamic one?

Practically, during a rapid transition from one airmass to another (assuming no change in wind velocity) TAS remains constant (inertia) while Mach and IAS change. If the change in temperature is more gradual, autothrottle/drag will keep Mach or IAS (whatever your reference is) constant while TAS changes... Yes?

[Old Smokey]

FullWings, No you're not barking up the wrong tree, what you say is correct. The discussion here is not so much about a change in Mach and CAS following a temperature change (at a constant Pressure Height), but about the FAA reference originally provided by Rivet gun, which indicated that -

the Mach number changes, even though the indicated airspeed may not change,

which I believe to be incorrect. My assertion is that although both Mach and CAS will change with the temperature change (a thrust and an inertia effect), once stabilised again, BOTH Mach and CAS will have returned to their original values.

enicalyth (G'day Enny) raises valid points, as usual, but I think that he's set one of his cats amongst the pigeons. Gamma (may I refer to it as Y) is certainly variable, but as the radical ((Y-1)/Y) and it's inverse is used for instrument calibration purposes for both the Airspeed indication and the Mach meter, any environmental variation in gamma will affect both indications. In Rivet gun's last post he has quite rightly stated - "flight instrument calibration equations are based on ISA which assumes inter alia that air is a ideal gas and that gamma = 1.4". Thus, although both may be somewhat in error due to environmental variation from the calibration standard, the relationship between IAS and Mach will be the same. One will not change whilst the other remains unaffected.

Sorry to awaken a thread in the 11th day of it's hibernation, work, the curse of the Prooning classes, has taken me to far flung pest holes during the course of an interesting discussion.

Regards,

Old Smokey

[javelin]

This works really well on the A320 and A321.

Get airborne and at accel alt dial up 340kts.

Keep 340kts until the mach number is what you want to cruise at.

Swap to Mach on the FCU

When IAS is the same as in the climb perf page, go back to managed speed.

Reverse on descent

That's my relationship between IAS and Mach and it works just fine :E

[rhovsquared]

ohh I'm glad this is back

RivetGun Are you flying in Vacuo????
you can't take moles out of it the adiabatic gas laws don't allow that: Cp-Cv=nR, Y=Cp/Cp+nR


and I didn't say T dependent I said Rewrite as a function of T

ex: dw = -PdV and du = CvdT and dU = dw+dq adiabatically = du = dw

so I can say Cvdt=-Pdv; REWRITING P in terms of T

Cv dT/ T = -nR dV/V since Cv is a constant I can integrate then i have

<Tf/ (dT/T) / Ti> = -nR <Vf / (dV/V) /Vi > = lnTf/Ti = -nRln (Vf/Vi)

using a ln X =lnX^a

and gas laws,

I have Ti/Tf = (Vf/V1)^y

lastly, Pi Vi^y = Pf Vf^y.

applying this judiciously with the equation fo dynamic pressure you can derive the equations (corrected for adiabatic compressibility) provided by Old Smokey
with about four pages of algebra...I' m not doing it:} :} :}

p.s. please let me know if I made an error if anyone finds one:O


edited: to correct a error Cp DIFFERS from Cv by nR NOT Cp+Cv=nR

I was very wrong there!!!