A MULTI-TEMPERATURE TVD ALGORITHM FOR RELAXING HYPERSONIC FLOWS

Jean-Luc Cambiei. Eloret Institute, Sunnyvale, California

& Gene P. ~ e n e e s t NASA-Ames Research Center, Moffett-Field, California

Abstract

In this paper we describe in details the extension of a multi-species TVD algorithm, second-order accurate for real- gas flows to multi-temperature formulation. The convec- tion algorithm is coupled t internal relaxation processes and the features of the coupling are also examined. In a first version, we discuss a 3-temperature model, where translational-rotational, vibrational, and electronic excitation energy modes are separately convected. Although several species are present, there is only one vibrational tempera- ture in this model. The second version generalizes to a vi- brational temperature for each molecular specie, with addi- tional couplings between species. In both cases the formula- tion of the numerical scheme is discussed in detail. We also describe our modeling of the chemical kinetics in the ther- mal non-equilibrium environment. Several tests are being conducted on the accuracy and efficiency of the algorithms. The algorithms are also applied to a generic two-dimensional flow field and results are compared with experimental obser- vations.

Nomenclature

= specific heat at constant volume, translational & rotational modes

Ctib = specific heat at constant volume, vibrational mode

C,* = specific heat at constant volume, electronic mode

c = (frozen) speed of sound ES = mass fraction specie s(= pJp)

*Research Scientist. Previous address: ANALATOM Inc., Sunnyvale, Ca.

t ~ e s e a r c h Scientist, Associate Fellow AIAA

dissociation energy barrier for quantum numbers I j total energy per unit volume transl. & rotational energy per unit volume vibrational energy per unit volume electronic energy per unit volume energy of quantum level (Ivj) vibrational energy per unit mass (=Eu/p) electronic energy per unit mass (=E,/p) dissociation rate averaged over all levels dissociation rate from level ( lv j) recombination rate onto level ( Iv j ) rate of vibrational energy exchange (VV) flux-limiter (defined in eq. 11) diffusive flux in characteristic space enthalpy per unit volume (=E + P) enthalpy per unit mass (=?) formation enthalpy of specie s average molecular weight of mixture molecular weight of specie s momentum density (= pu) static pressure universal gas constant velocity field generalized Riemann invariant total energy of atomic states 1'1" prior to recombination or after dissociation small parameter average vibrational energy per mole of specie s average vib. energy removed by dissociation average vib. energy left by recombination ratio of specific heats for translational -rotational modes ratio of specific heats for all energy modes at equilibrium k-th eigenvalue mass density of specie s mixture mass density collision cross-section between species r , s

This paper is declared a work of the US. Government and is not subject to copyright protection in the United States.

I: Introduction

Future aero-space vehicles are likely to operate at very high speeds in the atmosphere and experience severe loads of convective and/or radiative heating. The same is true of fu- ture space missions, which involve high-speed re-entry in the atmospheres of various planetary bodies. The flight regime includes a region where high velocity, low density and high temperature combine to provide a flow with a plethora of non-equilibrium phenomena. The success of these flight mis- sions depends on a complete, predictive understanding of the physics of the flow. The problem is accentuated by the need to approximately reproduce these flight regimes in laboratory experiments. The non-equilibrium effects may not always be large for actual spacecrafts, due to their large dimensions. The models used in ground experiments are however much smaller, and the free stream conditions from various facilities may already be in thermal non-equilibrium. For these rea- sons, we believe there is a need for an accurate understand- ing of of the physics and dynamics of gases in thermal and chemical non-equilibrium.

Numerical simulations are an excellent tool to comple- ment experimental studies, if they can be accurate enough. This implies a certain level of complexity in the theoreti- cal modeling and in the numerical methods employed. We describe here such a numerical scheme that is currently be- ing developed, and which attempts to include a sufficiently high level of precision in the modeling and in the algorithms. The numerical method is Total Variation Diminishing VVD), based on the famed algorithms by A. Harten [I], and leads to much more accurate solutions than other non-equilibrium al- gorithms [2,3], notably in the resolution of discontinuities. The algorithms that we present here form the basis of fu- ture versions with enhanced capabilities. The test cases and the special features of the solution of these non-equilibrium flows is discussed in detail. The current version is limited to non-ionized flows. Translational and rotational degrees of freedom are assumed in equilibrium. The first algorithm is a 3-temperature model, while the second model assumes a dif- ferent vibrational temperature for each molecular specie. The algorithms are studied also in the frozen and equilibrium lim- its, and the effect of the stiffness of the relaxation process is also examined.

11: TVD Algorithm descri~tion

11-A: Three-Temperature model

Internal relaxation processes are driven by density and temperature dependent rates, which can be very sensitive to fluctuations. It is therefore essential that numerical high- frequency oscillations be removed near the discontinuities; this can be achieved by using a monotone numerical scheme. The simplest one is the upwind (or 'donorcell') scheme; unfortunately this linear method is only first-order accurate and leads to severe numerical diffusion at the discontinuities. Higher order accuracy can be achieved only using a non- linear method, such as the Flux Corrected Transport (FCT) of Boris et al. [4,5], or the TVD method of Harten [I] (see also Yee et al [6-81). After experimentation with both schemes, we found that the TVD method lead to better results for non- linear systems of equations, especially for high Mach num- bers. We have used therefore the latter, extended to multiple chemical species [8,9,10] and to multiple temperatures.

We are solving a hyperbolic system of equations ' , writ- ten in conservative form (throughout this paper, we describe the scheme in the 1-dimensional form, for reasons of simplic- ity):

with:

Q =

m = pu is the momentum density, and H = E + P is the enthalpy per unit volume. The electronic excitation energies and temperatures are denoted by the * suffix, to distinguish it from free electronic quantities with a traditional e suffix. The translational and rotational degrees of freedom are combined, since they are at the same temperature 2 . Since the pressure is governed by the translational degrees of freedom only, the equation of state is:

Remember that we consider here a neutral gas. 'his may not be the case for vely light diatomic gases, such as H2, which

has a slow rotational relaxation[ll,l2]. We assume that this extreme case is not considered here. If it was, and if the time scales of interest woukl allow an 0bse~ation of the relaxation, the scheme can easily be extended to include this effect.

where rtr is the ratio of specific heats, but considering the translational and rotational degrees of freedom only. h: is the heat of formation of specie s at T = 0 , and R is the Boltzmann constant. We also use the frozen speed of sound, defined for thermally perfect gases [ l l , p1171 as:

Note that the frozen speed of sound, using rtr, is larger than the equilibrium speed of sound (rtrvt < ytr), as expected: thls is the correct speed to use in a numerical simulation, where the CFL condition must be enforced. It is known for example that the characteristic directions are always defined in terms of the frozen speed of sound [I1 pp 203-2101.

The question of the correct speed of sound to be used in a numerical simulation has been raised for example by C. Park [13]. Besides the stability considerations (CFL condition), there is another argument for using this speed in the computa- tion of the wave propagation. Let us assume a 1 -dimensional relaxation region (vibrational only) behind a shock, described by an array of cells of constant width Ax. If our simulation is valid, the exponentially relaxing zone is well resolved by a large number of cells: the amount of energy exchanged be- tween the combined translational-rotational and vibrational modes is then small. This implies that during our time step At, a wave that propagates from one side of the cell to the other sees little relaxation; only the translational and rota- tional modes contribute to its propagation, and the choice of frozen speed of sound is correct. If instead the relaxation zone is not resolved, and most of the equilibriation occurs within a cell, then the correct ratio of specific heats to be used is between the frozen (rt,) and the equilibrium value (7t,v,). It is clear that in the non-equilibrium case, the speed of propagation of the wave is dependent on the time step: this is precisely an effect of the dispersive nature of the re- laxing medium, where waves with different frequencies will be propagated at different speeds. These considerations lead us to the obse~ation that the numerical simulation is abso- lutely accurate if, and only if the relaxation zones are re- solved, which is the purpose of these computations. This implies a minimal gridding size for correct numerical sim- ulations. We will see however that if the time step is chosen appropriately, the case of very stiff relaxation can be handled correctly, leading to propagation speeds asymptotically con- sistent with the equilibrium case.

The TVD scheme for a system of hyperbolic differential

equations is obtained by rewriting the system in its eigenvec- tor form (no summation implied):

where w k is the kth characteristic field and x is the kth eigen- value. The transformation matrices T , T-' that map the sys- tem (1) onto its characteristic form (4) are described below. Let a:+1l2 = w:+~ - W! be the component of A Q in the kth characteristic direction (= Ti+ 12 Ai+ 12 Q). TO write the transformation matrices T and T-' , we need the following definitions for the partial derivatives of the pressure. The derivatives are computed with respect to the conserved vari- ables, i.e. p,, m, E, E,, E,.

Using the definitions of the mass fractions 2, = pi/p, and of the internal energies per unit mass e, = E,/p, e, = E,/p, the jumps in characteristic variables are then:

The inverse transformation matrix is such that:

where A is the diagonal matrix whose elements are the eigen- values of the hyperbolic system of equations. The transfor- mation matrices are listed in Appendix I.

The rest of the algorithm is standard [I]. We define the following functions:

where $J(X) is an eigenvalue modified by artificial viscosity, i.e.:

and where X = XA t/A s. At this point, it is important to re- mark that the time step used here in the computation of the flux-limiter is the convective time step, i.e. is a dummy time scale that is computed by assuming that the CFL number is exactly 1. If the real time step was used, and was very small (a condition imposed by the stiff internal relaxation time scale), the $J function would always be large, and this would lead to excessive numerical diffusion at the shocks. The entropy- violating parameter 6 (artificial viscosity) is present mostly to prevent over-expansion (i.e. rarefaction shocks). The up- dated solution is now given by:

The modified (interface centered) numerical fluxes being:

gi is the anti-diffusion corrected by a non-linear function (min-max) that ensures that no spurious maxima or minima are created, therefore making the method TVD. The form of the limiter is variable, and it usually includes an artificial compressor (ACM), used to sharpen the discontinuities [I]. However, it was observed that when stiff internal processes are coupled to the hydrodynamics (with strong coupling, such as very endothermic/exothermic reactions), it may be unwise to use a very compressive limiter. Spurious oscillations could be generated because of this strong coupling: it is best there- fore to avoid the use of an ACM, or use extreme caution, when the flow is coupled to the relaxation processes.

11-B: Multiple Vibrational Temperatures Model

Since the real gas contains multiple molecular species, each with a given degree of vibrational excitation, the equi- libration of the internal energy proceeds through a variety mechanisms. Vibrationally excited molecules can exchange energy quanta with other molecules during a collision: this process is rapid for resonant or near resonant exchanges,

when the vibrational quanta are of the same frequency. The strong coupling between vibrational and rotational excitation [14] at high temperature makes it such that the energy ex- change can be accompanied with the exchange of multiple ro- tational quanta, making the process very likely to occur. This can be seen in the fast increase of the probability of vibra- tional energy transfer between different molecules as a func- tion of temperature, as described for example in Ref.[lS].

This Vibrational-Vibrational (VV) coupling is responsi- ble for the rapid equilibration of the different vibrational tem- peratures (one for each specie) to a common temperature. When the time scales of the VV coupling are not resolved, the 2 or 3 temperatures model described above can be used. Otherwise the scheme must be generalized, in a trivial way, to the 2+Nu model, where N, is the number of species with vibrational degrees of freedom.

The system of equations (1) to be solved is now for:

where we have not written down the terms for the electronic energy, again for reasons of simplicity. The jumps in charac- teristic variables are now:

The inverse transformation matrix T-' is listed in Appen- dix 11. Similar modifications can be implemented in the case of multiple modes of electronic excitations. Notice the simi- larity in structure between the vibrational degrees of freedom and the chemical (mass fractions) degrees of freedom.

III: Tem~erature Evaluation

The solution of the euler equations (1) needs the correct evaluation of the translational temperature. The vibrational and electronic temperatures may be needed as well for VV relaxation and chemical kinetics. In order to evaluate these temperatures, we will assume that the internal energy modes can be separated. This implies that one can write relations of the form:

If for each specie, the specific heats can be computed as a function of the three temperatures, one can then iteratively solve for Ttr, T,, T, simultaneously, given et,, e,, e,. This separation is however not trivial, since at high temperature there is an increasingly strong coupling between the rota- tional energy and the vibrational energy, for example. Each energy level in a molecule (diatomic) can be characterized by three quantum numbers Eluj. If the ground state is denoted by Em there are 6 ways to partition the energy into different modes, some of them being listed in Table 1 below.

There is no absolute method of separating the thrcc en- ergy modes and potentially any of the above partition cases could be used. We however decide for case I, for the follow- ing (heuristic) reasons: the rotational quanta are the easiest to be exchanged, and involve little energy gaps except at very high temperatures. Therefore the rotational transitions will frequently occur at constant vibrational quantum number V .

When a vibrational transition occurs, it may also involve rota- tional transitions. If the translational-rotational temperature is not too large, the energy gap of a vibrational transition A 8, is larger than A Ej. The same argument applies to electronic levels. Therefore, when a hierarchy of energy levels is clearly established, as is the case for the lower temperature range, the first (I) partition is valid. At very high temperatures the ar- gument does not apply. In that case, an exact solution is not

Table 1: Partitioning of intemal energy modes.

possible unless we intend to solve the master equations. This is presently impossible with the current generation of super computers, due to limitations in both memory and computa- tional speed. However, it is worth estimating the error made by choosing one partition method versus another. For this purpose, we have calculated the equilibrium (Tt, = T, = T,) specific heats of each mode for three different cases (I to 111). The results are plotted in Figure la-b for cases I and HI. There was very litthe difference between cases I and 11. The sum is of course identical, since it is independent of the way the en- ergy is distributed. One can see that the greatest difference between cases I (or II) and I11 is 2 15% for C,* at tempera- tures greater than 10,000"K.

For the non-equilibrium case, one can compute the par- tition functions and their moments, and tabulate the specific heats and internal energies as a function of the three temper- atures. The resulting 3-dimensional tables would be a se- vere memory penalty if a constant step in temperature was used. For example, using AT = 50 OK in each direction, up to 50,000"K would require 2 x lo9 words for each specie! Instead, we use a constant relative step A TIT. The tabula- tion proceeds now via the formula: T, = To bn, where n is the index and To is the temperature scale. For example, us- ing b = 1.05 and To = 10O0K, one needs only 130 points to span the desired temperature range (b130 E 568). We also further reduce the memory needs by using a larger step (To = 20O0K, b = 1.6) for the orthogonal directions (i.e. for T,, T, in the tabulation of Cz, etc..). The total memory requirement is now < 10' words per specie. The resulting profiles for the specific heats are still very smooth. Using lin- ear interpolations during the computation of the internal tem- peratures is sufficient for a very good accuracy. The memory requirements can be reduced in later versions of this numer- ical capability by developing polynomial and/or exponential fits to the data.

The computation of the intemal and translational temper- atures proceeds via a standard iterative scheme. The method is however not efficient for some special cases, namely when the specific heats are nearly zero. This happens for example to the vibrational specific heat for T, 5 300°K and for the electronic specific heat for T, 5 600°K. In this case we simply decide to discard the check on convergence for these temperatures. This may result in an error in the intemal tem- perature, but since it is very low, it has very little effect on the dynamics of the system, whether the hydrodynamics or the re- laxation. We have carefully tested our iterative algorithm by specifying a random (but known) set of values for the inter- nal energies, and a random set of initial temperatures. In all cases the correct temperatures were computed, and with the

modification prescribed above, the efficiency was very satis- factory.

IV: Internal Energy Relaxation

The Translational-Vibrational (TV) coupling is modeled after the Landau-Teller 1161 formulation. This model is valid for not too-high temperatures (adiabatic collision approxima- tion): it has also been revised by several authors [11,12], but the e x p ( ~ - ' / ~ ) dependence remains the same (it is a conse- quence of the nature of the approximations made). The rates are given in the form given by Millikan-White [17]:

The relaxation mechanism is exponential in terms of the total vibrational energy [ll,pp. 294-3001:

(note: the relaxation equation for the vibrational tempera- ture is not in this simple form). As discussed in [12], this time-scale becomes too small at high temperature, and should be modified by the collision time-scale (it always takes at least one collision in order to transfer energy): T + T + 7,.

When several vibrational species need to be considered separately, there is an additional energy transfer, the V-V cou- pling: the vibrational energies of each active specie satisfy a system of ODE, similar to the chemical kinetics, i.e.:

where k;, is the number of collisions per unit time where en- ergy transfer takes place (from r to s) and ., = Eu,/n, is the average vibrational energy of a molecule of type s. These rates can be obtained from the collision rates and the proba- bilities of vibrational energy transfer [3,11,12,18].

where a,, is the cross-section for the collision of molecules 7 ,9 , %,, = d- is the relative velocity, and P,!: is the probability of de-excitation of molecule r during a colli- sion with molecule s, with subsequent excitation of molecule s . The probabilities of excitationlde-excitation are related -

A single quantum is exchanged, as part of the Landau-TeUer approxi- mation. 'Ihis implies a high-temperature limit on the validity of this model.

through the principle of detailed balance. We can therefore re-write the transfer equation (20) as:

where the equilibrium energies E,";, are such that all vibra- tional temperatures are equal.

The coupling of the relaxation and the hydrodynamics is done by the Operatar-Splitting method. This scheme has been repeatedly used in reactive flow simulations 151 with great success. The critical 0bse~ation is that the time-step must be limited by the smallest of the time scales; if the relax- ation process is very stiff, it will induce large changes of the eigenvalues of the system (either through temperature or 7 variations) during the convective time scale. This cannot be allowed, since the numerical scheme assumes constant eigen- values during the time step. Furthermore, the relaxation equa- tion eq. (19) requires the knowledge of E,"q(Tt,). However, for stiff systems, the exchange of energy will be relatively large, and so is the change in Tt, and E,"Q. One requires then to re-evaluate the translational temperature after significant exchange of energy. The problem is solved then by limiting the time step such that:

where c is a small parameter (E 0 .I). It can be convinc- ingly argued that for this time step, a second-order accurate numerical method is unnecessary. In the case a particular re- laxation process is not resolved by our spatial accuracy, it seems unnecessary to be limited by this very small time step since the energy exchange would be equilibriated within one cell. In order to achieve faster convergence to a steady state, we loosen the time step restriction, for that process only, by using for example c -- 5 - 10. This of course is possible if the algorithm used for the relaxation is inherently stable for large time steps.

The relaxation of the internal electronic energy proceeds through similar T-E and V-E couplings. Of course, when elec- trons are significantly abundant, there will be additional cou- plings with the fkee electrons: the latter may be the dominant mechanism of energy exchange, due to their extreme mobility and their large cross-sections of collisons with other free and bound electrons (i.e. internal electronic modes). The elec- tron gas component will be implemented in a later phase, and these couplings are not considered here. The total frequency of inelastic collisions can be estimated by averaging the prod- uct u,,v,, over the velocity distribution, restricted for veloc- ities larger than v, such that the excitation energy of an elec- tronic transition 1 + 1' is:

The frequency of relaxation then is obtained after averag- ing over all transitions, weighted by the distribution of initial states. It can be easily seen that the final result has a leading order term of the form

and therefore is exponentially suppressed except for very large temperatures. The excitation of internal electronic en- ergy through heavy particle collisions is therefore not a rapid process. For free electrons, the pre-exponential terms are much larger than for heavy particles and therefore, if the gas is substantially ionized, there will be a rapid equilibrium estab- lished between the free electrons and the internal electronic modes.

Direct energy transfer between electronic and vibrational modes is also possible through the exchange of several vibra- tional quanta, in order to match approximately the electronic excitation energy gap. This process is therefore difficult to evaluate theoretically. We nevertheless model the E-V re- laxation process by a pseudo Landau-Teller equation (see eq. 19), with appropriate estimates of the relaxation time scale. The equilibrium energy in that model is obtained by com- puting the equivalent equilibrium temperature of both vibra- tional and electronic energies. Note that this modeling is not totally accurate: if multiple quanta are exchanged, one should parametrize the energy exchange rate by the average energy exhanged, in addition to the time scale.

It is clear that further work is necessary to correctly model the relaxation of the electronic modes: we believe however that the model equation (19) may be used with relative confi- dence as long as the time scale of relaxation is correctly esti- mated. This must be verified however, and we intend further development in this area.

V: Chemical Reactions

If the gas is simultaneously reacting and vibrationally re- laxing, the activation barriers are different for each vibra- tional level, and the rates take a complicated dependence on the translational and vibrational temperatures (see [Il,pp. 365-3681). A simpler approximation [12] assumed a depen- dence on an average temperature = (Tt,Tv) ' I 2 . Wher- ever electronic excitations are specially relevant, this aver- age would be extended to (Z,TvT,)'/3. There is however no physical argument behind this choice of averaging pro- cedure. This model was compared by R. Jaffe [I41 with a

more exact computation using all the energy levels (the 'sum of states' method) on the case of oxygen dissociation. Sig- nilicant differences between the 'sum of states' and Park's model could be observed in the rates. It is not yet clear how- ever whether the difference in rates from one model to an- other would lead to significant differences in thermal quanti- ties for realistic flow conditions. The most dramatic effects related to the chemistry in thermal non-equilibrium flows can be expected to be found in the induction delay for combustion reactions and the radiative power in the recombining flow at the exit of a hypersonic nozzle. The latter requires an accu- rate modeling of the increase in vibrational energy due to the recombination processes.

It is known for example that recombinations tend to leave the molecule in high vibrational states: therefore one should expect some degree of population inversion, where high lev- els are more populated than low lying ones. If this is truely the case, a correct description of the state of the recombining molecular gas in terms of Boltzmann distributions is not pos- sible. At best, one could separate the energy spectrum into a low levels partition and a high levels band, each with a unique temperature. The temperature in the high level band could be allowed to take a negative value if necessary. Our numer- ical scheme can be easily extended to this 'spectral' repre- sentation, although the coupling terms become slightly more complicated. However, the need for such an extension can be determined only after an extensive study and comparisons with experimental data. The multi-level description would not affect the total vibrational energy but the temperature, and therefore the rates of coupling. We assume for the present that this effect is minimal.

The method used in determining the non-equilibrium rates closely follows the 'sum of states' method described in details in Ref. [14]. For each level (lvj), that dissoci- ates into atomic states ( 1'1" j), one can compute a rate of dis- sociation: from the potential energy curve for a given pair of quantum numbers, Ulj(r), one can obtain the energy bar- rier Dlj and the total energy of the final state Arp (see Fig- ure 2). Note that Alrp is the combined translational and elec- tronic energy of the atomic state. The 'true' activation energy is therefore: E;,. = Dlj - (Elvj - Eloj) and the dissoci- ation rate is of the form: k t j Y exp(-ELj/ kTt,). Only the translational temperature occurs in this expression since the energy fluctuation that allows the jump over the energy barrier must be 'borrowed' from the translational modes dur- ing the collision with the 3rd body. By averaging over

41his is strictly valid when the collision partner is an atom. When a molecule is involved, some energy can be gained from the vibrational or rotational motion of the partner, therefore leading to a more complex tem-

the initial states, and after normalization to the equilibrium forward rate ( I -temperature) the overall rate of dissociation can be computed and tabulated as a function of the temper- atures: k d ( ~ t r , T,,, T,) . Simultaneously, the various energy moments can be computed, e.g. the average energy removed by dissociation: F;(T~, , T,, , T,) . Note that this average en- ergy is larger than the average energy contained in vibra- tional modes at temperature T,,, since the dissociation prefer- ably takes place from highly excited levels. This effect leads to a vibrational cooling by dissociation. In this aspect, our method differs from Candler & Mac Cormack [2], who do not take this effect into account. A similar quantity is com- puted for the electronic energy.

The recombination process is more difficult to consider: the influence of thermal non-equilibrium can be exactly ob- tained by simulations of the reaction trajectory at a micro- scopic level, but these detailed simulations are expensive and therefore there is not enough data available on these mecha- nisms. However, some estimates can be obtained by looking at the energetics of the reaction. Let us consider the time re- versal process of the dissociation from a level (lvj), by which two atoms with electronic states 1'1" coalesce to form a bound state, while the extra energy is being transferred to the 3rd body. When each level is in equilibrium with the dissociated state, there is a detailed balance between the rate of dissocia- tion and that of recombination, i.e.:

where the densities are evaluated at the equilibrium tem- perature Teq. If we consider the distribution of atoms of combined translational and electronic energy App in the ex- pression above, it is easy to find that the recombination rate at equilibrium has the form:

This expression can be reduced to:

The argument of the exponential is nothing but the energy barrier seen by the atomic state approaching the bound state from infinity, i.e. the centrifugal barrier. Only the rota- tional/translational modes are involved in overcoming this barrier: therefore the non-equilibrium rate is obtained by sim- ply replacing Teq by Ttr in the above expression. The recom- bination rate depends only of the translational temperature, and can be simply obtained from the known 1-temperature expression.

perature dependence. We presently ignore this effect.

The energy moments however may have a dependence on the electronic temperature, since there is a sum over all initial5 states. We need here to compute three quantities, namely:

1. the average vibrational energy the molecule contains af- ter recombination: F;

2. the average electronic energy the molecule contains after recombination: Z:

3. the average electronic energy removed from the free atomic state after recombination F l

The latter is usually much smaller than Z:, and could be ne- glected or could easily be substracted from : , if only the elec- tronic energy of the mixture (molecules + atoms) is consid- ered. However, since we may consider in the future the case of multiple electronic temperatures, one for each specie, we decided to compute all these averaged transferred energies.

VI: Numerical Results

We now turn to systematic tests of the algorithms on some simple 1-dimensional cases. As mentioned in section IV above, we have first tested the iterative algorithm for tempera- ture evaluation by preparing a random set of internal energies and started the algorithm with a different random set of initial temperatures. The convergence rate and accuracy were mon- itored, and the algorithm tuned for efficiency (see discussion in section IV), and excellent results were obtained: the accu- racy wasalways better than 0.5%. The next numerical tests were conducted by simulating the unsteady propagation of a shock from a reflecting surface. The free stream conditions are listed in Table 2.

The composition was varied from pure Nz to pure 02, including air. In most simulations (Figures 2-8), the internal electronic temperature T, was frozen at 275 K, and no relax- ation of this internal mode was allowed. The shock propaga- tion speeds were monitored by measuring the time of arrival of the shock at a given cell (# loo), situated lcm from the reflecting wall. In some situations the relaxation time scales were artificially set to specified values, instead of computed from the Millikan-White [17] formula. The addition of the collision time scale T,, discussed in section IV was not im- plemented: the one-dimensional numerical experiments per- formed here attempt to demonstrate the features of the algo- rithm and validate it with analytical arguments.

S ~ i s is converted into a sum over final states, where the vibrational modes factor out.

Table 2: Free stream conditions.

We first test the monotonicity of the TVD convective al- gorithm (i.e. no relaxation). The shock that can be seen in Figure 3 is very sharp (2 cells needed for capture), and the monotonicity is extremely well enforced on all energy modes. We allow then for a TV energy exchange, and we enforce the time step condition (23) in the simulations. Because the shock possess a certain numerical width, the relaxation will proceed within the shock and may in some extent affect the results. We can partly account for that by identifying the cells which are within the shock. This is best accomplished by looking at the divergence of the eigenvalues at the cell inter- faces. If the divergence is negative, a shock is present. In order to separate the shock from a contact discontinuity, a cutoff on the divergence of the eigenvalues is applied. A cell a will then be considered within the numerical width of the shock if the condition:

(where E N 5%, ci is the speed of sound) is valid for &l characteristic waves k. Figure 4 shows the vibrational tem- perature for a Damkholer number ~ ~ ~ ~ ~ ~ / r ~ - N 500. The dashed line corresponds to the full relaxation (in all cells), while the solid line corresponds to a calculation with the re- laxation in the shock suppressed. Figure 5 shows these pro- files for a Damkholer number of approximately 112. It shows the very short relaxation length scale, which is notreproduced when the energy exchange is allowed to proceed within the shock. Surprisingly, the difference in dynamics is quite small: the difference in arrival times leads to a relative difference in propagation speeds that is less than 0.3%. This situation may vary with the total amount of energy that is being exchanged, and may be worse for very endothermic/exothermic relax- ation processes, although we consider a drop of 5,000"K in translational temperature (out of 17,000°K) not negligible. The effect of the time-step condition (23) may also play an important role in minimizing the errors. Additional research into these numerical features is currently under way.

If we vary the stiffness of the internal process, we obtain the curve of Figure 6, which shows the variation of the prop- agation speed of the reflected shock (measured by the time of arrival at cell # 100, lcm from the reflecting wall) as a func- tion of the stiffness parameter (here the inverse time-scale of

relaxation). The equilibrium limit was measured by using a different algorithm, where no multi-temperature effects were considered (but with no electronic energy, to comply with our freezing of T*). Both frozen and equilibrium limits are reached asymptotically. This constitutes our final check of the 3-temperature (1-T,) algorithm.

To check for the VV coupling, we must essentially repro- duce the results above in the limit of a very fast VV relaxation. In this regime, all vibrational temperatures will equilibriate rapidly towards a unique temperature E. This temperature should be identical to the one obtained from the 1-T, model, and in the limit of infinitely fast VV relaxation, the dynamics should also be the same. This is confirmed by Figures 7 and 8, which show the temperatures profiles for the 1-T, and N-T, schemes respectively. These results are for a fixed composi- tion (air), and for which the VV time scale was artificially set to sec. In Figure 8, the T, for N2 and O2 cannot be vi- sually separated, and the overall profile is identical to Figure 7. A close look at the data gives an error less than 0 .l% .

In Figure 9, we show the temperature profiles for the case of a dissociating and relaxing oxygen. The electronic tem- perature is here assumed to relax very rapidly with the vibra- tional temperature. The solid lines represent our full chem- istry model, while the dashed lines are obtained by assum- ing that the chemical rates are a function of the translational temperature only. For this low density case (the free stream conditions are the same as before) we see that there is a sig- nificant difference in the temperature profiles. By assuming that the rates are dependent on the internal temperatures but neglecting the effect of dissociation from upper levels would result in differences of a similar order of magnitude.

A close examination of the temperature profiles shows a slight oscillation of the electronic temperature at the shock: this is not however an oscillation of the electronic energy. The problem resides in the sudden changes in translational and vi- brational temperatures. Due to the corresponding changes in the specific heats, the convergence of the algorithm for the evaluation of the internal temperature is slower. The final electronic temperature obtained after a limiting number of it- erations can then be in error: this can also later affect the

relaxation between electronic and vibrational modes, due to an incorrect estimate of the equilibrium temperature between these two modes. This problem can be solved by allowing for a larger limit on the number of iterations, however at the cost of an increase in computer time. There is therefore a need for further research in improving the algorithm for tem- perature evaluation. We presently emphasize however that this effect is small, and does not affect the overall dynamics of the coupling: as the internal temperatures increase (above 1000°K), the convergence becomes very rapid, and the pro- files becomes smoother.

The final case concerns the reproduction of an experiment [19] on the measurement of the shock layer thickness over a cone of half-anglc 45O, in a pure oxygen hypersonic stream. The free stream conditions are also listed in Table 2. This ex- periment is also used as a bench-mark case by Candler [20]. We show the resulting shock stand-off distance in Figure 10, along with the experimental data points of Ref. [19] and the numerical results of Candler [201. There is good agree- ment between our results and the experimental data. There is some ambiguity in defining the exact shock position when the shock-capturing scheme generates a numerical width of one or two cells; for our grid system, the uncertainty in the shock position is however always less than 0.1 mm; this makes the results of Candler (also subject to this uncertainty) in agree- ment with ours. We also show in Figure 11 the temperature profiles at the station x = 13.25 mm. This figure must be compared with Figure 5 of Ref. [191. The computational method used by Spurk is different than ours. The general shape and position of the shock layer are in agreement, how- ever we find that our peak vibrational temperature is about 1,000 "K larger than Spurk's; this gives near the body a differ- ence between the translational and vibrational temperatures of -- 100 OK, compared to 1 ,00O0K for Spurk [19]. This could be explained by the effect of finite width of the shock discon- tinuity, or more probably by the difference in computational methods. Spurk's numerical results on the shock position are also slightly different from ours or Candler's [20], and are only marginally within the scatter of experimental points for axial distances CY 15 mm or more.

These results constitute a first validation of our code; un- fortunately, it also shows that the experiment is not sensitive enough to distinguish between our chemistry model and oth- ers. We propose that the distinction will be more acute in expanding flows and recombining flow experiments: in these situations, the effect of upper levels will be most apparent, leading to vibrational temperatures higher than translational, if the decoupling occurs sufficiently rapidly (i.e. with high area ratio nozzles). We are planning to investigate these flows

in the future.

VII: Conclusions

We have presented in detail the formulation of a TVD numerical scheme, second-order accurate for the solution of the Euler equations in a multi-temperature environment. The relaxation processes are coupled by an Operator-Splitting method, and the overall scheme has been thorougly tested and validated on simple 1-dimensional examples. We have also discussed the thermo-chemical properties for a non- equilibrium gas, and discussed the features of the numerical method for consistent and accurate evaluation of the internal temperatures. We have focused our attention on the numeri- cal methods, their validity, acuracy and efficiency; the level of detail in the mathematical formulation was considered a strict minimum for a complete understanding and reproduction of the method and results. Further work is presently under way to extend the numerical capability, notably in the modeling of the electronic energy relaxation, and the inclusion of free electrons. Further investigations are also planned for more detailed simulations and experimental validations.

VIII: References

1. A. Harten, 'High Resolution Schemes for Hyperbolic Conservation Laws', J. Comp. Phys., vol49 (1983), pp. 357-393

2. G. Candler & R. MacCormack, "The Computation of Hypersonic Ionized Flows in Chemical and Thermal Non-Equilibrium", AIAA-88-05 11.

3. G. Candler , 'The Computation of Weakly Ionized Hy- personic Flows in thermo-Chemical non-Equilibrium', PhD thesis, Stanford University, June 1988.

4. J.P. Boris & D.L. Book, 'Flux Corrected Transport I: SHASTA- AFluid Transport Algorithm That Works', J. Comp. Phys., vol. 11, pp. 38-69.

5. E. Oran & J.P. Boris, 'Numerical Simulation of Reactive Flow', Elsevier, 1987.

6. S. Eberhardt & K. Brown, 'A Shock-Capturing Tech- nique for Hypersonic, Chemically Relaxing Flows', AIAA paper 86-023 1.

7. H.C. Yee, 'Construction of Explicit and Implicit Sym- metric TVD schemes and their applications', J. Comp. Phys., vol. 68 (1986), pp. 151-179.

8. J.L. Shinn, H.C. Yee & K. Uenishi, 'Extension of a Semi-Implicit Shock-Capturing Algorithm for 3-D Fully Coupled, Chemically Reacting Flows in General- ized Coordinates', AIAA paper 87-1577.

9. J.L. Cambier et al., 'Numerical Simulations of Oblique Detonation Waves in Supersonic Combustion Cham- bers', to be published in AIAA J. Prop. & Power.

10. J.L. Cambier et al., 'Numerical Simulations of an Oblique Detonation Wave Engine', AIAA paper 88- 0063.

11. J.F. Clarke & M. McChesney, 'The Dynamics of Real Gases', Butterworths (1964).

12. C. Park, 'Non-Equilibrium in Hypersonic Flows', Lec- ture Notes (1988)

13. R.L. Jaffe, 'Rate Constants for Chemical Reactions in High-Temperature Non-Equilibrium Air', AIAA paper 85-1038.

14. C. Park, 'Convergence of Computation of Chemical Re- acting Flows', Progress in Aeron. & Astron., vol. 103 (1985), pp. 478-513.

15. R.L. Taylor, M. Camac and R.M. Feinberg, 'Measure- ments of Vibration-Vibration Coupling in Gas Mix- tures', l l th Int. Symposium on Combustion, 1967.

16. L.D. Landau & E. Teller, Physikalische Zeitschrift der Sowjetunion, vol. 10 (1966), pp. 34-43.

17. R.C. Millikan & D.R. White, 'Systematics of Vibra- tional Relaxation', J. Chem. Phys, vol. 39 (1963), pp. 3209-3213.

18. E.V. Stupochenko, S.A. Losev & A.I. Osipov, 'Relax- ation in Shock Waves', Springer-Verlag, 1967.

19. J.H. Spurk, 'Experimental and Numerical Nonequilib- rium Flow Studies', AIAA Journal, vol. 8 (1970), pp. 1039-1045.

20. G. Candler, 'On the Computation of Shock Shapes in Nonequilibrium Hypersonic Flows', AIAA paper 89- 03 12.

Aknowledgements:

The authors wish to aknowledge R.L. Jaffe at NASA- Ames Research Center for his help in clarifying some aspects of the molecular physics and for providing the necessary data for the evaluation of the non-equilibrium internal energies and specific heats. We also thank G. Candler at NASA-Ames for pointing out to us the existence of Ref. 15.

Figure 1: Non-dimensionalized Specific heats of internal modes, for various partitions. Plots are for equilibrium (unique) temperature. Partitioning defined in Table I.

Figure 2: Generic Potential curve for a molecular bound state and its dissociation products

80.0 100.0 120.0 UO.0 100.0 180.0 a00

cell #

Figure 3: Shock propagation without relaxation. The mono- tonicity of all temperatures shows that the scheme does not produce spurious couplings.

0.0 50.0 100.0 150.0 200

cell #

Figure 4: Weak relaxing case (T,,/T,- N 500). The dot- ted/solid line is for relaxation allowed/forbidden within shock width.

T(x108)"K

0.0 50.0 100.0 150.0 200

cell #

Figure 5: Stiff relaxing case (T,/ T,, N 0.5 ). The relaxation zone behind the shock is one cell wide.

1.4 equilibrium limit

Figure 6: Shock propagation speed vs. relaxation time scale. The vertical axis is the arrival time at 1 cm from the wall. The horizontal axis is the frequency of relaxation u,. The equilibrium limit is obtained from a separate algorithm.

80.0 100.0 120.0 UO.0 180.0 180.0 200.0

cell #

Figure 7: Temperature profile for 1-T, model, air mixture.

cell #

Figure 8: Temperature profiles for Nu-T, model, same con- ditions. The VV coupling is made artificially large, to check the stiff limit with the 1-T, case.

~ ( x l 0 ' ) ' K

cell #

Figure 9: Temperature profiles for reacting case (02 disso- ciation). Solid lines are for the complete non-equilibrium al- gorithm. Dashed lines are for rates function of translational temperature only.

Candler ..... i!!??! I..! ....... ,.. 0 exp ( U D D ~ T aide)

0 computation

Figure 10: Shock layer position versus experimental points. Dashed curve is computation from Ref. [20]. The upper limit Figure 11: Temperature profile across shock layer at axial

is for an ideal gas. distance x=13.25 mm.

Appendix I

The transformation matrices for the 3-temperatures model are:

Appendix II

The inverse transformation matrix for the 2+Nu model is: