Computer Calculations of Relaxation Regions and Equilibrium Conditions for Shock Waves with Tables for CO 2 and N 20

Johannesen 2 has discussed a Rayleigh-line method for the analysis of vibrational relaxation regions in shock waves. In this method the flow of the real gas in the relaxation region is treated as the flow of an ideal gas (termed the ~-gas by Johannesen) with heat extraction. The Rayleigh-line method is readily adapted for use on a computer; programs based on this method have been used to deal with experimental data for CO2-He mixtures and N20, and also to provide tables of equilibrium conditions for pure C02 and N20. Conditions (a) immediately behind the diffusion resisted part of the wave are found using the normal shock relations for the ideal gas and the known conditions (1) in front of the shock wave. The Mach number m, of the a-gas is related to the Mach number Ma of the real gas by rnl = MI (Tx/y~) ~ • (1) The total temperature, To1, which does not vary through the diffusion-resisted part of the shock wave, is given by, To,= TI(1 +~y=-1)mi2). "Fhe flow variables for the a-gas can be expressed in terms of the corresponding quantities at sonic conditions , i.e. at m = 1. This gives the well-known Rayleigh-line relations, T (V~+ 1) 2 m 2 Tb-(1 +v~m2) 2 ' (3) T O 2(y~ + 1)m 2 (1 + ~y~-1)m 2) To-~ = (1 + 7~m2) 2 ' (4) Pb _ v _ (y~+l)m z p Vb 1 + v~m 2 ' Pb 1 + v~m 2 " (6) *Replaces A.R.C. 27 581. where suffix (b) denotes sonic conditions, and the symbols have their usual meanings. The conditions (b) can be found from the known conditions (1) using the relations (3) to (6). The increase in vibrational energy at any point in the relaxation region is equal to the energy that has been extracted from the c~-gas, i.e. where ¢r denotes the vibrational energy and ff denotes its local equilibrium value. For the purpose of the computer program it is necessary to obtain an analytic expression for the equilibrium value of the vibrational energy. The simple harmonic oscillator approximation is suitable for most gases. For example, for C02 and N20 we have 2c _ c2 c3 t cp,-cp, (exp(C1/T)-1 ~ exp(C2/T)-1 ~ exp(Ca/T)-1 (8) where x is the mole fraction of oscillators present and C1, C2, and C3 are the characteristic temperatures of the various modes. For CO2 these are …