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May 4, 2015, 11:37 
Ideal gas law and incompressible flow

#1 
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Nick
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Hi guys,
I've read that when Mach is below 0.3, then the flow is considered incompressible. But in textbooks that look at flow thru a turbo jet, in the compressor or diffuser, nozzle etc although the flow is often subsonic and below Mach 0.3, they use the ideal gas law. My question is: Can the ideal gas law be used when Mach is below 0.3? Why? Isn't the flow supposed to be incompressible? Thanks. 

May 4, 2015, 11:53 

#2 
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Michael Prinkey
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M < 0.3 is a rule of thumb, not a law. Compressibility effects generally scale as M^2. So 0.3^2 makes the error scale as 0.09 if you treat 0.3 Mach flow as incompressible. And, really, this is a statement about local Mach number. You can have flows with an average/freestream Mach number well under 0.3, but have high velocity areas with M>0.3 that end up being important to the overall flow behavior.


May 4, 2015, 12:00 

#3 
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Nick
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Thanks for that. How does one know when to use the ideal gas law then ? How do we know compressibility effects are important?


May 4, 2015, 12:21 

#4 
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Filippo Maria Denaro
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real gas are always compressible in their nature, the "incompressibilty" is only a model used to simplify the set of equations.
But if you assume rho= constant, the continuity equation becomes Div v = 0 and if you consider also homoentalpic condition T= constant that would lead to the ideal law of gas p=const. This conclusion is not correct, pressure gradients in the momentum equation still have sense to ensure a divergencefree velocity field bu you have no law for prescribing a value for the pressure depending on density and temperature. Many researchers use the full compressible model, with the ideal law p=rho*R*T even for low Mach flows. They simply disregard the incompressibility as a simplification and retain the full form of the NS equations. At Mach=0.20.3 such formulations work fine, at lower Mach numbers can be necessary to use some preconditioning technique. 

May 6, 2015, 20:52 

#5 
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Nick
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Supposing we accurately measured the temperature and density fields around an airfoil in an experiment where the flow Mach number is below 0.3, would the ideal gas law provide the pressure field (the same one as that measured by pressure sensors on the foil for instance)?


May 6, 2015, 21:52 

#6 
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Martin Hegedus
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The assumption of incompressibility does not mean that the flow is actually incompressible. However, the assumption allows for the conservation of mass and momentum to be decoupled from the conservation of energy and the ideal gas equation. So for incompressible flow, one first solves for pressure and velocity and then for temperature and density.
Edit: Incompressibility means that the density changes are small enough the the equations can be decoupled. 

May 7, 2015, 00:07 

#7 
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Nick
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I don't think you could use the incompressible continuity, momentum and energy equations and the ideal gas law together. I think either you assume compressibility whereby you solve the fully NS equations and gas law OR incompressible flow equations without ideal gas law.


May 7, 2015, 02:11 

#8 
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Martin Hegedus
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There are six unknowns (p, u, v, w, rho, and T) therefore six equations are required. As the Mach number decreases, the compressible and incompressible equations will give nearly the same answer. What the incompressible equations allow for is the solution of p, u, v, and w independently from rho and T since the conservation of mass and momentum are insensitive to changes in density at low Mach numbers.


May 7, 2015, 03:17 

#9  
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Filippo Maria Denaro
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Quote:


May 7, 2015, 08:58 

#10 
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The Ideal Gas Law as in the Equation of State? Compressibility doesn't dictate its use because density is in the equation itself and therefore compressible effects are taken into account. The IGL relation can be used for any ideal gas regardless of Mach number to relate pressure, volume/density, and temperature.


May 7, 2015, 22:07 

#11  
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Nick
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Sure. What I meant was, if we use an incompressible model, T is found from the energy equation rather than the ideal gas law. Do you agree? 

May 7, 2015, 22:09 

#12  
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Nick
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From what you wrote I understand that if the surface pressure sensors were used to measure the static pressure, it would provide the same values as those found from the ideal gas law. 

May 8, 2015, 03:07 

#13  
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Filippo Maria Denaro
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compressible or incompressible model, T (temperature) is always found from the energy equation... in the incompressible model the energy equation is simply decoupled (apart the buoyancydriven flow) and can be solved after the continuity and momentum equations are solved. 

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