HEAT FLUX AND DIFFUSION VELOCITIES BEHIND SHOCK WAVE : STATE-TO-STATE APPROACH

In the present paper, nonequilibrium vibrationdissociation kinetics of N2/N and O2/O mixtures and its in§uence on di ̈usion and heat transfer in relaxation zone behind shock waves are studied. For the description of nonequilibrium reacting gas §ows, the most accurate model [1] based on detailed state-to-state molecular distributions over internal energy levels is used. This approach receives much attention for investigation of di ̈erent gas §ows; in shock heated gases, the stateto-state kinetics has been studied in a number of papers [26]. Unfortunately, the practical implementation of the state-to-state approach for viscous reacting mixtures occurs extremely computationally consuming since a large number of equations for vibrational distributions should be solved; moreover, at each time and space step of numerical simulation, the state-dependent transport coe©cients have to be computed solving the high-rank transport linear systems containing equations for each chemical component and vibration level. A simpli¦ed algorithm for the transport coe©cients calculation which reduces noticeably the number of equations has been proposed in [7]. Nevertheless, direct implementation of state-to-state transport algorithms is still complicated. In the present paper, a simpli¦ed approach applying a postprocessing procedure is used: ¦rst, Progress in Flight Physics 9 (2017) 327-334 DOI: 10.1051/eucass/201709327


INTRODUCTION
In the present paper, nonequilibrium vibrationdissociation kinetics of N 2 /N and O 2 /O mixtures and its in §uence on di¨usion and heat transfer in relaxation zone behind shock waves are studied.For the description of nonequilibrium reacting gas §ows, the most accurate model [1] based on detailed state-to-state molecular distributions over internal energy levels is used.This approach receives much attention for investigation of di¨erent gas §ows; in shock heated gases, the stateto-state kinetics has been studied in a number of papers [26].Unfortunately, the practical implementation of the state-to-state approach for viscous reacting mixtures occurs extremely computationally consuming since a large number of equations for vibrational distributions should be solved; moreover, at each time and space step of numerical simulation, the state-dependent transport co-e©cients have to be computed solving the high-rank transport linear systems containing equations for each chemical component and vibration level.A simpli-¦ed algorithm for the transport coe©cients calculation which reduces noticeably the number of equations has been proposed in [7].Nevertheless, direct implementation of state-to-state transport algorithms is still complicated.In the present paper, a simpli¦ed approach applying a postprocessing procedure is used: ¦rst, the gasdynamic parameters and their derivatives are calculated for an inviscid §ow and then, the transport terms are evaluated on the basis of this solution.In this way, the impact of nonequilibrium kinetics on the heat transfer has been studied in shock heated mixtures.The contribution of various processes and the in §uence of di¨erent initial conditions on di¨usion velocities and total energy §ux in the §ows of air components have been estimated.

KINETICS BEHIND THE SHOCK WAVE
In the present paper, inviscid §ows of N 2 /N and O 2 /O mixtures in the relaxation zone have been considered behind a plane shock wave.The kinetic scheme for the binary mixtures includes vibrational energy exchanges at the collisions of molecules (VV), exchanges of translational and vibrational energies (VT), and dissociation and recombination reactions.A nonequilibrium §ow under the considered conditions is described by the equations for the vibrational level populations n i of molecular species and number densities of atoms n a coupled to the conservation equations for the momentum and total energy [1]: Here, x is the distance from the shock front; p = nk B T is the pressure; n is the mixture number density; T is the gas temperature; k B is the Boltzmann constant; v is the macroscopic velocity; and the subscript ¤0¥ denotes the parameters in the free stream.In Eq. ( 4), where ρ is the mixture density and ρ m , ρ a , h m , and h a are the mass densities and speci¦c enthalpy of molecular and atomic components: with R m and R a being the speci¦c gas constants of molecules N 2 and O 2 and atoms N and O, m a being the atom mass, ε i being the vibrational energy of a molecule at the ith vibrational level, and ε a being the formation energy of atoms.In calculations, the vibrational energy is simulated by the anharmonic oscillator model with total numbers of vibrational levels l = 46 for N 2 and l = 35 for O 2 .
The right-hand sides of Eqs. ( 1) and ( 2) contain the state-dependent rate coef-¦cients of considered VV and VT energy transitions and dissociation recombination reactions.The rate coe©cients for vibrational energy transitions in the source terms R VT i and R VV i are calculated using the generalized Schwartz, Slawsky, and Herzfeld formulas [8,9], rate coe©cients of dissociation are described by the TreanorMarrone model [10] modi¦ed for state-to-state approach [1].Rate coe©cients of forward and backward kinetic processes are connected by the detailed balance principle.

DIFFUSION AND HEAT FLUX
Expressions for di¨usion velocities and heat §ux can be written on the basis of state-to-state approach as follows [1]: Here, V MD i , V MD a , V TD i , V TD a , and V DVE i are the contributions of the mass di¨usion, thermal di¨usion, and di¨usion of vibrational energy: q HC , q MD , q TD , and q DVE are the energy §uxes associated with the heat conductivity of translational and rotational degrees of freedom (Fourier §ux), mass di¨usion, thermal di¨usion, and the transfer of vibrational energy carried by excited molecules: Here, D mm , D ma , D aa , D T m , and D T a are the multicomponent di¨usion and thermal di¨usion coe©cients for each molecule and atom; d m and d a are the di¨usive driving forces depending on gradients of the level populations and atom densities; and λ ′ is the thermal conductivity coe©cient.
First, let us present the §ow parameters obtained as a solution of Eqs.(1)(4) in the relaxation zone behind the shock wave under the following conditions in the free stream: T 0 = 271 K; p 0 = 100 Pa; M 0 = 15 and 18; n m = p 0 /(k B T 0 ); and n a = 0. Initial distributions in the free stream are assumed to be the Boltzmann ones with the temperature T 0 .The gas parameters just behind the shock are found with the use of the RankineHugoniot relations under the assumption of frozen vibrational distributions and mixture composition within the shock front.
Figure 1a shows the variation of the gas temperature in N 2 /N and O 2 /O mixtures with respect to the distance from the shock front.One can notice that the relaxation process in O 2 /O mixture proceeds much faster and more actively than in the mixture of nitrogen molecules and atoms.In particular, one can see slower decrease in the gas temperature for N 2 /N mixture than for O 2 /O in the beginning of the relaxation zone.Then, the value of temperature in N 2 /N mixture occurs about twice higher than in O 2 /O mixture for both Mach numbers.
Figure 1b depicts number densities of N and O atoms as functions of x and shows intensive dissociation of O 2 molecules for both cases.The di¨erence between n N and n O is explained, ¦rstly, by the high rates of the vibrational energy transfer of O 2 molecules, in consequence of which the excitation of the oxygen molecules occurs in a narrow zone behind the shock front compared to the nitrogen molecules.Secondly, dissociation of oxygen is considerably more active than dissociation of nitrogen.These facts are illustrated also in Fig. 2 which show the level populations of nitrogen and oxygen molecules in the relaxation zone for di¨erent M 0 .One can see that level populations (for i > 0) rise in the beginning  of relaxation zone as a consequence of TV excitation and then slightly decrease due to VT deactivation and dissociation.
Self-consistent calculation of the transport terms is extremely timeconsuming, it requires the solution of transport linear systems at each step of computations.Since we are interested in preliminary estimates of qualitative behavior of the §uxes, let us use a simpli¦ed approach.For the investigation of transport processes, let us substitute the macroparameters and their gradients obtained as solutions of the inviscid §ow equations into the expressions of di¨usion velocities and total heat §ux.
As one can see (Fig. 3), di¨usion §uxes of high vibrational states are weak; the most important contribution is given by i = 0 and 1.It is seen that the mass di¨usion §ux for nitrogen atoms is negligible since dissociation of N 2 molecules is rather slow; for oxygen atoms, the mass di¨usion reaches its maximum, when di¨usion §uxes for all molecular vibrational states n i V i /n become zero.For both mixtures, n a V a /n is small near the shock front due to the existence of ¦nite incubation time for the dissociation reaction.Since vibrational excitation proceeds faster in oxygen, the values of n i V i /n are greater in O 2 /O mixture.
Figure 4 presents the variation of the total heat §ux in the relaxation zone for di¨erent Mach numbers.Let us discuss ¦rst the nitrogen mixture.One can see that for M 0 = 15, the heat §ux varies strongly near the shock front, then (for x > 0.2 cm), it is almost constant.One can notice nonmonotonic behavior of the total heat §ux with x for M 0 = 18 which is explained by the strong competition of di¨erent dissipative processes near the shock front.For oxygen mixture, one can see qualitatively similar results.But competition of di¨erent processes and, as a consequence, nonmonotonic heat §ux behavior occurs already for M 0 = 15.The absolute value of total heat §ux is signi¦cantly larger for oxygen, because all the processes proceed faster and the gradients of gasdynamic variables are larger than for nitrogen.
To understand the reason for di¨erent behavior of the total heat §ux for various initial conditions, let us consider the contributions of multiple dissipative processes to the total energy §ux (Fig. 5).The Fourier §ux due to heat conductivity q HC and the §ux caused by the di¨usion of vibrational energy q DVE give the contribution of the same order near the shock front and their absolute values are rather high near the shock front.The signs of these terms are opposite which causes a strong compensation e¨ect leading to much lower values of the total §ux q.The contribution of the vibrational energy di¨usion to the heat §ux in shock heated §ows occurs dominating and it leads to the negative sign of q.The in §uence of mass di¨usion becomes noticeable only at x > 0.05 cm for nitrogen and at x > 0.02 cm for oxygen mixture due to the dissociation incubation time.At such distance from the shock front, the in §uence of q HC and q DVE (a) N2/N; (b) O2/O; 1 ¡ q HC ; 2 ¡ q DVE ; 3 ¡ q TD ; and 4 ¡ q MD becomes smaller than of q MD .Thus, the total heat §ux behavior in the state-tostate approach is nonmonotonic with x for N 2 /N mixture at M 0 = 18 and O 2 /O mixture at M 0 = 15 and 18.For lower Mach numbers, dissociation of molecules is not enough e©cient to provide a considerable e¨ect of mass di¨usion; thus, the heat §ux decreases monotonically with the distance.The thermal di¨usion §ux is practically zero behind the shock wave.

CONCLUDING REMARKS
Di¨usion and heat §ux in the relaxation zone behind the shock wave in stateto-state approximation have been considered for nitrogen and oxygen mixtures for Mach numbers 15 and 18.It is seen that for higher Mach numbers, kinetic and dissipative processes proceed more e©ciently.The contribution of heat conductivity, thermal di¨usion, mass di¨usion, and di¨usion of vibrationally excited molecules to the total energy §ux is analyzed.Comparison of oxygen and nitrogen shows that for O 2 /O, fast vibrational relaxation and dissociation lead to the sharp variation of gasdynamic parameters near the shock front and, as a consequence, to greater values of the total heat §ux.

Figure 5
Figure 5The total heat §ux behind the shock as a function of x for M0 = 18;