Acoustics - Large-amplitude Wave Motion in Air

Numerical methods and mathematical models of Elmer
raback
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Re: Acoustics - Large-amplitude Wave Motion in Air

Post by raback »

Hi

Usually I trust Mika's formulas more than my own ones. I guess

R = C_p - C_v = (C_p/C_v - 1)C_v = (γ − 1)C_v

where γ is the ratio of heat capacities. So I would think it is formally ok.

-Peter
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Re: Acoustics - Large-amplitude Wave Motion in Air

Post by cohor »

Hello Peter,

Thank you for the explanation, your equations are true for ideal thermally perfect gases. I have missed the most important point, that this model works only for gases that obey the ideal gas law (and that my gases didn't obey this law). I have run few simulations for compressed refrigerants like ammonia and r-134a using this model and the results were so much off from reality that the correctness of the model was suspicious. Then comparing the measured specific gas constant Rs of the gases with the value calculated from the equation R=(γ − 1)C_v they were very different. This is why I thought maybe this equation is wrong.

Now I realize that the gases I want to model don't obey the ideal gas law, especially pressurized. Another model for real gases will be necessary for my simulations. Run some more simulations also for Nitrogen and Carbon Dioxide that are nearly ideal gases at 1bar. Here is some data that illustrates which gases can be reasonably simulated with this CompressibleNS solver up till what densities:

Tetrafluorethan – R134a
p=1bar; T=0C; rho=4.4926291871; rho_error=2.9%; Cp=0.8154; Cv=0.7203; Rs=0.081488856; Rs_error=-16.7%; c=154.2; c_simulated=177.57; c_error=-15.16%

p=10bar; T=40C; rho=39.1876628596; rho_error=25%; Cp=1.134; Cv=0.8824; Rs=0.081488856; Rs_error=-208.8%; c=140.9; c_simulated=332.5; c_error=-135%

p=100bar; T=110C; rho=910.2; rho_error=320%; Cp=1.722; Cv=1.026; Rs=0.081488856; Rs_error=-754%; c=283.4; c_simulated=21.3; c_error=92.5%

Ammonia – NH3
p=1bar; T=0C; rho=0.7498690909; rho_error=1.5%; Cp=2.178; Cv=1.641; Rs=0.4882175; Rs_error=-10%; c=414.3; c_simulated=459.6; c_error=-11%

p=10bar; T=25C; rho=8.31365; rho_error=21%; Cp=3.1295; Cv=2.1245; Rs=0.4882175; Rs_error=-106%; c=407.8; c_simulated=710.3; c_error=-74%

p=100bar; T=130C; rho=47.895; rho_error=-5.7%; Cp=3.646; Cv=1.4125; Rs=0.4882175; Rs_error=-357%; c=193.35; c_simulated=1644.9; c_error=-750.7%

CO2
p=1bar; T=0C; rho=1.951; rho_error=0.68%; Cp=0.8267; Cv=0.6319; Rs=0.1889241; Rs_error=-3%; c=258.1; c_simulated=274.15; c_error=-6.2%

p=10bar; T=0C; rho=20.84; rho_error=7.5%; Cp=0.9291; Cv=0.6677; Rs=0.1889241; Rs_error=-38%; c=248.7; c_simulated=325.6; c_error=-31%

p=100bar; T=0C; rho=974.1; rho_error=402.7%; Cp=2.178; Cv=0.9282; Rs=0.1889241; Rs_error=-562%; c=639.8; c_simulated=1041.8; c_error=62.8%

Nitrogen
p=1bar; T=0C; rho=1.234; rho_error=0.04%; Cp=1.041; Cv=0.7429; Rs=0.2968039; Rs_error=-0.4%; c=337; c_simulated=355.15; c_error=-5.4%

p=10bar; T=0C; rho=12.39; rho_error=0.4%; Cp=1.06; Cv=0.7457; Rs=0.2968039; Rs_error=-5.9%; c=338; c_simulated=363.4; c_error=-7.5%

p=100bar; T=0C; rho=125.2; rho_error=1.5%; Cp=1.241; Cv=0.7701; Rs=0.2968039; Rs_error=-58.7%; c=362; c_simulated=473.5; c_error=-30.8%

The trouble of finding a model valid for real gases remains. Regards,
-cohor
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Re: Acoustics - Large-amplitude Wave Motion in Air

Post by raback »

Hi Cohor

What are the features that are relevant to you? Do you need the effect of the thermal boundary layers. If you neglect that you could use the standard FlowSolve and write your own material model.

-Peter
cohor
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Re: Acoustics - Large-amplitude Wave Motion in Air

Post by cohor »

Hello Peter,

I have just taken a cursory glance at FlowSolve in Models Manual. It looks like using ρ=p/c^2 and a polynomial function c = c(p, T, . . . ) the real compressible gas behaviour could be well modelled. Unfortunately, the modelling of viscous-thermal boundary layers is also required, like it is done in this CompressibleNS model. Basically I would need a model just like CompressibleNS but with the ability to use real gas models (not ideal gas). The thermal effects are very important for high amplitude acoustics in pressurized gases because the temperature can vary even several 10C between high pressure and low pressure points of the wave (due to the compression).

Can the FlowSolve not model the viscous-thermal boundary layers?
-cohor
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Re: Acoustics - Large-amplitude Wave Motion in Air

Post by raback »

Hi cohor,

You could try whether FlowSolve works for you. I know that you can obtain pressure waves with the "pressure dependent" model. However, that is not applicable directly as it lacks some loss terms. If it works then solving it weakly coupled with the heat equation could do the trick for you.

-Peter
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