A Simple Iteration for Shock Wave Calculations with the Enthalpy Coefficient

BOYE AHLBORN Department ofPhysics, University ofBritish Coluntbia, Vnncsuver V6T I W5

Received January 16,1975

The enthalpy coefficient g -- klu can be used to separate the thermodynamical and the hydrodynamical aspects of shock and other step waves in partially ionized gases, providing a simple recursion formalism for the exact calculation of the equilibrium end parameters of such waves.

Le coefficient d'enthalpie g .= hlu peut 6tre utilisC pour skparer les aspects thermodynamique et hydrdynamique des chocs ou des autres ondes en Cchelon dans des gaz partiellement ionisks, ce qui fournit un formalisme de r6currence simple pour le calcul des parametres d'Cquilibre de telles ondes. [Traduit par le journal] Can. J. Phys., 53,976(1975)

Introduction The equilibrium calculations of shock front

and other gas dynamical discontinuities in par- tially and fully ionized gases require the use of the hydrodynamical conservation equations and thermodynamical equations of state including the Saha ionization formula. Due to the ex- ponential temperature dependence in the Saha equation there is no analytical solution for this set. With the help of modem computers it is, of course, not difficult to calculate tables for the parameters behind the discontinuity such as the pressure, p, the density, p, the temperature, T, and the enthalpy, h, as function of the Mach number, M. However, it is often required to calculate just a single value, and for the under- standing of the experiment it would be better if one could clearly recognize how dissociation and ionization changes the flow pattern.

In this paper it is shown how to separate the hydrodynamical and the thermodynamical ef- fects in step waves with the enthalpy coefficient g and obtain the end conditions with a simple iteration in a 'back of the envelope' type calcu- lation. [I1

FIG. 1. Comparison of g, y,, and c,/c, as functions of temperature for argon at p = 1 atm. Numerical values for cp and c, from Drellishak (1963) and for g and 7, from Bosnjakovic et al. (1962).

ionized gas one has to distinguish the sound speed coefficient y, defined by

Thermodynamical Coefficients the ratio of the specific heats cp/c, In a cold gas the same adiabatic exponent, y,

serves in different thermodynamical relations to 123 cp - @hlaT), -- give the sound speed and the enthalpy as func- CU (aulaT), tions of pressure and density and to describe and the enthalpy coefficient g (Ahlborn and adiabatic processes. In a partially dissociated or Salvat 1961, 1967)

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AHLBORN: A SIMPLE ITERATION FOR SHOCK WAVE CALCULATIONS 977

which is a measure for the energies of all internal degrees of freedom, but includes in first approxi- mation only the dissociation and ionization energy, D and E,; n, and n, being the number densities of atoms and molecules respectively and p, the partial pressure of the molecules. The definition [3] holds even in the absence of thermal equilibrium. While the three coefficients y,, c,/c,, and g are identical and equal to y at room temperatures, they differ markedly at higher temperatures, as may be seen in Fig. 1 for argon at p = 1 atm. All fall in the range between 1 and 513 but rather large errors can be encountered when using the wrong coefficient in factors of the form (y - 1).

Gas Dynamical Formalism

For gas dynamical calculations, most important is the enthalpy coefficient g. Firstly because it can be used to express the enthalpy as function of pressure and density

and secondly because g relates the escape speed of a generalized Chapman Jouguet wave in a simple way with pressure and density, as will be shown later.

Relation [el] is extremely useful in step wave calculations where one wants to eliminate the enthalpy from the conservation of energy, +v12 + h1 + Wlp,vl = *vZ2 + h,, so that it may be combined with the conservation of mass, plvl = p2v2, and the conservation of momentum, plv12 + p1 = p2u2, + p2, to obtain the ratio of the densities (Ahlborn and Salvat 1961).

and pressures

across the step wave if the initial conditions, index 1, the Mach number MI, and the energy source W (erg/cm2 s) are known. v1 and v, are the 'intake' and 'exhaust' velocities, respectively. In the case of a shock wave, where W = 0 and M I > 1, the positive root has to be taken so that the standard Rankine-Hugoniot relations are found :

These equations are exact, but the enthalpy coefficient g, is initially unknown. However, for an end state in equilibrium g is a known function of pressure and enthalpy. Figure 2 shows this relation in graphical form, which is quite suffi- cient for the present use. In order to find the end conditions (Index 2) behind the shock for

a given value of the shock velocity v,, one applies a simple iteration. First a somewhat reasonable value of the enthalpy coefficient g, is assumed and p2 and h2 are calculated from [a] and [9]. While these pressure and enthalpy values are not yet very accurate, they are good enough to get a better value for g2 by graphical

FIG. 2. Enthalpy coefficient for argon. Data from Bosnjakovic et al. (1962).

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978 CAN. J. PWYS. VOL. 53, 1975

interpolation from Fig. 2. With this improved value of the enthalpy coefficient the equilibrium end values, p2 and h,, are obtained in quite a good approximation, because both do not de- pend sensitively on g,. A further iteration is of course easily done, if required. The best g, value is used to calculate the compression ratio from [7], which depends sensitively on g,. The tem- perature is found by interpolation in thermo- dynamical tables or graphs which give the enthalpy as a function of pressure and tempera- ture; see references under the figures. Other step waves where the pressure and enthalpy can be given as functions of p,, g,, v,, and W can be treated in the same manner. For instance, for generalized Chapman Jouguet (C.J.) waves the solution is single valued, i.e. the square root in [5] and 161 cancels, so that for M,' >> 1 the conditions

are found. Again the end conditions are func- tions of the initially unknown g2 but an esti- mated value for the enthalpy coefficient and one or two iterations with i l l ] and [12] and Fig. 2 quickly leads to the end conditions behind the C.3. wave. It is worthwhile to notice that the escape velocity v2 which is found from [4], [10], and [12] is exactly

Since g2 f y, for partially ionized gas, the escape velocity for these generalized C.J. waves is not exactly sonic. However, the difference between a,, the sound speed at this point, and the escape velocity v, is small so that one can often use the approximation a2 z gp/p (Ahlborn 1966).

The enthalpy coefficients for helium, air, and hydrogen are given in Figs. 3, 4, and 5. Some temperature values are indicated in Fig. 5 to show that g approaches the limiting value 513 only very slowly, namely when kT has become large compared to E,. A rather large error can therefore be made if the enthalpy coefficient for

FIG. 3. Enthalpy coefficient for helium. Data from Lick and Emmons ((1962).

FIG. 4. Enthalpy coefficient for air. Data from Bur- horn and Wienecke (1 960b).

fully ionized gas, say hydrogen at 40 000 K is taken to be 513.

In conclusion the enthalpy coefficient g = h/(h - pip) is easily available for equilibrium conditions. It includes the thermodynamical properties of the medium; g w 1.15 for barely ionized gas and approaching 513 very slowly at temperatures above 106 K.

The enthalpy coefficient appears in all hydro- dynamical calculations where the enthalpy has been eliminated so that the end conditions behind step waves are obtained as function of the initially unknown g,. A quite good value for g, is then quickly found with one or two simple iterations making use of otherwise calcu- lated thermodynamical data or tables so that, in fact, the hydrodynamical and thermodynami- cal properties of the experiment can be separately evaluated.

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AHLBORN: A SIMPLE ITERATION FOR SHOCK WAVE CALCULATIONS

FIG. 5. Enthalpy coefficient for hydrogen. Data from Burhorn and Wienecke (1960~).

AHLBORN, B. 1966. Phys. Fluids, 9,1873. BURHORN, F. and WIENECKE, R. 1%0a,b. Z. Phys. Chem. AHLBORN, B. and SALVAT, M. 1%1. Max Planck Institut 215,285,269.

f i r Plasmaphysik (IPP), Garching, Germany, Report DRELLISHAK, K. S. 1%3. Phys. Fluids, 6, 1280. 312. LICK, W. J. and EMMONS, H. W. 1962. Thermodynamical

1967. Z. Naturforsch. 220,260. properties of helium (Harvard University Press, Cam- BOSNJAKOVIC, F., SPRINGE, W., and KNOCKE, K. F. 1962. bridge, Massachusetts).

Z. Flugwiss. 10,413.

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