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An “ideal equivalent gas method” for the study of shock waves in supersonic real gas flows

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Sommario

Nella presente nota viene esposto un metodo sviluppato dall'autore per lo studio sistematico dello stato termodinamico e dinamico del gas a valle di un'onda d'urto in regime supersonico, allorchè cioè gli effetti dell'eccitazione dei gradi di libertà vibrazionali delle molecole e della loro dissociazione e successiva ionizzazione invalidano l'ipotesi di gas ideale generalmente adottata.

Viene definito un “gas ideale equivalente” avente rapporto dei calori specifici γs funzione del numero di Mach e della quota di volo ed in base alle equazioni di conservazione della massa, della quantità di moto e dell'energia, valide attraverso all'onda d'urto, vengono derivate delle relazioni definenti lo stato termodinamico e dinamico del gas a valle dell'onda d'urto. Tali relazioni costituiscono una estensione delle classiche relazioni dell'urto valide per il gas ideale alle quali si riducono per γs=γ∞.

La dipendenza del rapporto dei calori specifici γs del gas ideale equivalente, dal numero di Mach e dalla quota è stata stabilita sulla base delle soluzioni ottenute da Huber per il gas reale.

Summary

A method developed by the author for the systematic study of the thermodynamic and dynamic properties of the gas behind a shock wave is reported.

The method is applicable to supersonic flow regimes for which the excitation, dissociation and ionization effects invalidate the usually adopted hypothesis of ideal gas.

An “Ideal Equivalent Gas”, having the ratio of the specific heats “γs” dependent on Mach number and altitude of flight is postulated.

On the basis of the mass, momentum and energy conservation equations, valid through the shock wave, the relations defining the thermodynamic and dynamic state of the gas behind the shock wave are derived. These relations establish an extension of the classic relations valid for the ideal gas and reduce to them identically for γs=γ∞.

The dependence of the ratio of specific heats γs of the “Ideal Equivalent Gas” on Mach number and altitude has been established, over a wide range, on the basis of the real gas solutions derived by Huber.

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Abbreviations

a :

Speed of sound

C p :

Specific heat constant pressure

d :

Contribution of the dissociation energy to the gas enthalpy normalized with respect to the kinetic free stream energy

D :

Dissociation energy

e :

Energy

f :

Degree of freedom of the gas molecules

H :

Enthalpy

(K):

Ratio of ideal equivalent gas value to corresponding ideal gas value

M ∞ :

Free stream Mach number

M 0 :

Mach number relative to zero altitude

P :

Pressure

Q :

Flight altitude

R :

Gas constant

S :

Entropy

T :

Temperature

V :

Velocity

Z :

Compressibility factor

a :

Dissociated atom fraction

β :

Energy content of the gas

γ :

Ratio of specific heats

δ :

Contribution of the dissociation energy to the gas enthalpy

ε :

Ratio of the inverse of the densities across the shock wave

π :

Ratio of the pressures across the shock wave

ϑ h :

Ratio of the enthalpies across the shock wave

ϑ a :

Ratio of the speeds of sound squared across the shock wave

ϑ T :

Ratio of the temperatures across the shock wave

σ :

Density

σ D :

Characteristic dissociation density.

∞ :

Free stream, initial conditions ahead of the shock wave

s :

Final conditions behind the shock wave

0:

Reference condition relative to zero altitude

id :

Ideal gas conditions

di :

Ideal dissociating gas conditions

ie :

Ideal equivalent gas conditions

∼:

Hypersonic flow values

—:

Mean values.

References

  1. Hansen,Approximations for the thermodynamic and transport properties of high temperature air, NASA, TR-R-50.

  2. Hilsenrath-Beckett,Tables of thermodynamic properties of argon free air to 15000° K, A.E.D.C. TN. 56-12.

  3. Huber,Hypersonic shock-heated flow parameters for velocities to 46K feet/sec and altitudes to 323K feet, NASA, TR-R-163.

  4. Courant-Friedrics,Supersonic Flow and Shock Waves, Interscience.

  5. Hayes-Probstein,Hypersonic Flow Theory, Academic Press.

  6. Ames staff,Equations, tables, charts for compressible flow, NACA, Report 1135.

  7. Liepmann-Roshko,Elements of Gasdynamics, John Wiley.

  8. Bethe-Teller,Deviation from thermal equilibrium in shock waves, Bulletin Research Lab., Aberdeen, Rep. X 117.

  9. Roming,The normal shock properties for air in dissociation equilibrium, J.A.S., 2.56.

  10. Lightill,Dynamics of a dissociating gas, J. Fluid Mech. 2.57.

  11. Feldmann,Hypersonic gas dynamics charts for equilibrium air, AVCO rep., 1957.

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Authors and Affiliations

  1. Istituto di Progetto di Aeromobili, Politecnico di Torino, Italy

    Ernesto Vallerani

Authors
  1. Ernesto Vallerani
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Vallerani, E. An “ideal equivalent gas method” for the study of shock waves in supersonic real gas flows. Meccanica 4, 234–249 (1969). https://doi.org/10.1007/BF02133438

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  • DOI: https://doi.org/10.1007/BF02133438

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