The non-equilibrium considerations in the description of the relaxation of the atmospheric molecules depend on the relation of the various molecular relaxation times, to a characteristic flow time which is determined by the velocity of the vehicle. If the lack of equilibrium between vibrational, rotational and translational degrees of freedom is neglected as in the case of most flow field calculations, serious errors may result. When these non-equilibrium effects are explicitly included, the conventional approach to fluid dynamics breaks down.
Attempts to overcome this point have been investigated in the literature by developing essentially two models, the multi-temperature approach and the state-to-state approach. In the multi-temperature approach, a temperature is assigned to any degree of freedom that determines all the thermal, physical and chemical properties. Chemical rate coefficients are then calculated as a function of relevant temperatures. A relaxation equation is written for each temperature to complete the description of the system1,2.
In the state-to-state model, developed by our Institute in these last years, any degree of freedom is decomposed in internal levels (vibrational, rotational, electronic). For each level a continuity equation is written. Known the level distribution, the thermal properties of the gas and the global rate coefficients are calculated as the sum over the distributions3,4.
The main difference between the two models lies in the fact that in the multi-temperature approach equilibrium or quasi-equilibrium distribution functions are assumed at the given temperatures, while in the state-to-state approach the concept of temperature loses any meaning.
Depending on the studied conditions, the state-to-state approach shows strongly non-equilibrium vibrational distributions essentially due to the recombination process. These distributions in turn generate a non-Arrhenius character in the dissociation and exchange rates of typical chemical processes3,4.
More recently these ideas have been extended to include effects due to free electrons. In this case too the multi-temperature approach assigns a temperature (with its own conservation equation) to free electrons, which is then coupled to the internal degrees of freedom of molecules through a phenomenological approach. On the other hand the state-to-state approach solves an appropriate Boltzmann equation for the electron energy distribution function (eedf) trying to underline possible non-equilibrium effects in the eedf. These non-equilibrium effects are smoothed by Coulomb electron-electron collisions becoming however important for ionization degrees less than 10-3. Under these conditions the coupling of free electrons with vibrationally excited states and with electronically metastable species becomes very important.
Multi-temperature and state-to-state models have been incorporated in different fluidynamic codes to describe:
a) high enthalpy flows through nozzle expansion3-4
b) the interaction of high enthalpy flows with catalytic and non catalytic surfaces5.
In general sophisticated 2D-3D fluidynamic codes include multi-temperature models for a) and b) cases, while 1D nozzle expansion flows and boundary layer codes have been widely used for incorporating state-to-state kinetics. Recent attempts by Yoshula6 of Wright Patterson Center after the pioneering work of Giordano et al.7 have incorporated a state to state vibrational kinetics in a 2D fluidynamic code describing the interaction of hypersonic flow with non catalytic body.
Catalytic effects of course strongly determine the heat transfer to the body. The state-to-state approach has been recently proposed to better describe the recombination of atomic species on surfaces of different nature5,8.
In general theoretical aerothermochemistry has enormously improved in the last few years. A parallel effort has been performed to get experimental informations to be used in the validation of the different models. Some hypersonic wind tunnels have been built up all over the word (NASA, CIRA, Stuttgard, VKI, Onera). In particular profiles of macroscopic quantities such as pressure, Mach number, translational and vibrational temperature are now available. These data can be and have been used to validate the different models, even though more detailed information concerning internal distributions and eedf should be developed.
LIF9 (Laser Induced Fluorescence) and CARS10 (Coherent Anti-Stokes Raman Spectroscopy) techniques and second derivative Langmuir probe techniques can be applied to this end. These techniques can monitor the tails of the different distributions being an essential tool to assess the importance of state-to-state approach.
Additional effects are due to the presence of electromagnetic fields. Two effects are possible. The thermal heating of the gas due to the field and the appearance of charge particle beams. In that may affect the collisional dynamic in the flow. For critical situations a strong coupling may exist between the electromagnetic fields and the thermo-chemical characteristics of the flow. As a matter of fact, the electromagnetic field will cause the heating of electrons thus affecting the electron-heavy species collisional processes on one hand, and the behavior of the electromagnetic field in the reactive flow will strongly depends on the ionization degree on the other hand. The interaction between the electromagnetic field and the flow may be described by coupling an electromagnetic numerical module to the plasma flow models. Depending on the frequency of the field, such coupling may need the solution of the full Maxwell curl equations or a single equation for the magnetic potential vector. The later method is used when the flow dimension are smaller or of the same order of magnitude than the wavelength of electromagnetic field while the former method is used otherwise. Both of these methods have been already successfully used for the self consistent simulations of RF parallel plate reactor11 or electrical thrusters12.
**References :**
1) H. Fruhauf, M. Fertig and S. Kanne, "Validation of the enhanced URANUS non-equilibrium Navier Stokes code", J. Spacecraft and Rockets 37, april (2000)
2) S. Kanne, H. Fruhauf and E.W. Messerschmid, "Thermochemical relaxation through collisions and radiation", J. Thermophysics and Heat Transfer 14, vol.4 (2000)
3) G. Colonna and M. Capitelli, "Self-consistent model of chemical, vibrational, electron kinetics in nozzle expansion", J. Thermophysics and Heat Transfer 15,308 (2001)
4) G. Colonna and M. Capitelli, "The influence of atomic and molecular metastable states in high enthalpy nozzle expansion nitrogen flows", J.Phys.D: Appl.Phys.34,1812 (2001)
5) I. Armenise, M. Capitelli, C. Gorse, M. Cacciatore and M. Rutigliano, "Non-equilibrium vibrational kinetics of an O_{2}/O mixture hitting a catalytic surface", J. Spacecraft and Rockets 37,318, (2000); 38,482 (2001)
6) E. Yoshula, W.F. Bailey, "Vibration-dissociation coupling using master equations in non-equilibrium hypersonics blunt-body flow", J. Thermophysics and Heat Transfer 15,157 (2001)
7) D. Giordano, V. Bellucci, G. Colonna, M. Capitelli, I. Armenise, C. Bruno, "Vibrationally relaxing flow of N_{2} past an infinite cylinder", J. Thermophysics and Heat Transfer 11, 27 (1997)
8) M. Cacciatore, M. Rutigliano and G.D. Billing, "Eley-Rideal and Langmuir-Hinshelwood recombination coefficients for oxygen on silica surfaces", J.Thermophysics and Heat Transfer 13,195 (1999)
9) G. Dilecce, M. Simek and S. De Benedictis, "The N_{2}(A^{3}S_{u}^{+}) energy transfer to OH(A^{2}S^{+}) in low pressure pulsed RF discharges", J.Phys.D: Appl.Phys, 34, 1799 (2001)
10) V.A. Shakhatov, O. De Pascale, M. Capitelli, "Experimental investigation of rotational distribution of H_{2}(X^{1}S_{g}^{+}) in a radio frequency capacitive discharge plasma by CARS spectroscopy", AIAA paper 3749 (2003)
11) S. Longo, A. Milella, "A one dimensional self-consistent model of charged particle transport and vibrational kinetics in weakly ionized hydrogen", Chemical Physics 274, 219 (2002)
12) F. Taccogna, S. Longo, M. Capitelli, "Particle in Cell with Monte Carlo simulation of a SPT100 exhaust plumes", Journal of Spacecrafts and Rockets, 39, 409 (2002) |