Numerical prediction of nonequilibrium hypersonic flow around Brazilian satellite SARA

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Numerical prediction of nonequilibrium hypersonic flow around Brazilian satellite SARA

Ghislain Tchuen^I; Yves Burtschell^II; David E. Zeitoun^II

^IInstitut Universitaire de Technologie Fotso Victor Université de Dschang, BP. 134 Bandjoun - Cameroun

^IIUniversité de provence, Polytech'Marseille-DME, 5 rue Enrico Fermi, Technopole de Chateau Gombert, 13453 Marseille Cedex 13, France

ABSTRACT

Hypersonic flows past Brazilian satellite SARA at zero angle of attack in chemical and thermal nonequilibrium are investigated using an axisymmetric Navier-Stokes solver. The numerical solutions were carried out for freestream conditions equivalent to a typically re-entry trajectory with a range of Mach numbers from 10 to 25. The gas was chemically composed by seven air species O;N;NO;O₂, N₂;NO+; e^– with 24 steps chemical reactions scheme and thermically characterized by a multi-temperature model. Comparisons have been made between the present computation and the distribution of pressure coefficient and the heat transfer obtained recently with Direct Simulation Monte Carlo Method[1]. The study also points out the influence of nonequilibrium phenomena like ionization, vibrational and electronic excitation on aerothermodynamic flow parameters.

1 Introduction

Recently, Sharipov predicted by the DSMC method and for a frozen flow (without chemical reactions), the aerothermodynamic parameters which are necessary for the calculation of the ballistic trajectory of re-entry and for the creation of an adequate aerothermic protection of a small reusable ballistic Brazilian satellite SARA[1]. The atmospheric conditions used goes from the free molecular regime up to the hydrodynamic medium.

The development of the satellite re-entry into atmosphere requires accurate prediction of the thermal protection system for extremely high temperature. The vehicle flying with hypersonic velocity through the Earth's upper atmosphere. This creates a detached shock wave around the vehicle and the kinetic energy is transformed into the internal energy. Therefore, the shock layer is the site of intensive physico-chemical nonequilibrium processes such as vibrational excitation, dissociation- recombination, electronic excitation, significant ionization and radiative heating[2]. These are commonly referred as high-temperature effects which causes considerable difficulties for accurate numerical and experimental simulations of the flowfield. Successful conception of such high technology would be obtained after some knowledge of the thermochemical nonequilibrium phenomena and how they affect the performance of the vehicle.

In this study, an effort is being spent to the numerical prediction of the aerothermodynamic characteristics of satellite for very high altitude with a Knudsen number relatively suitable to the application of Navier-Stokes equations. According to the Mach number and heating effects observed, nonequilibrium chemical, vibrational and electronic modes are taken into account. The series of calculations are carried out taking into account, in a progressive way the assumptions which allow as well as possible to approach the reality of physics within the flow. The numerical simulations begin with a frozen air flow, followed by the taking into account of a chemical kinetics with 5 species in which the various couplings are included. The last step is made by an extension for conditions being able to cause significant ionization of gas (chemical kinetics with 7 species) and the electronic excitation of the species. An efficient and robust thermochemical nonequilibrium Navier-Stokes code based on Upwing technology with Riemann's solver has been developed.

The various upstream flow conditions are considered with a wide range of Mach numbers(10 up to 25). SARA is a sphere-cone and its shape is shown in Fig. 1. The nose raduis is equal to 0.27544 m. The numerical results obtained under the same conditions of simulation have been well compared with available DSMC computations[1].

2 Governing equations

The governing equations for a real gas are reported following the simplifications discussed by Lee[3]. The full laminar Navier-Stokes equations for two-dimensional conservation equations are written as:

The mass conservation equation for each species, s,

The momentum conservation equation in x and y directions,

The total energy equation,

The conservation equation of vibrational energy for each nonequilibrium molecule,

The electron-electronic energy conservation equation

In these equations, the electric field due to the presence of electrons in flow is expressed as:

The shear stresses are modelled using the hypothesis of a newtonian fluid as:

The dynamic vicosity m_s of each species is given by Blottner et al.[4] and the thermal conductivity of each species is derived from Eucken's[5]. The total viscosity and conductivity of the gas are calculated using Wilke's semi-empirical mixing rule[6].

The mass diffusion fluxes for s-species are given by Fick's law with a single diffusion coefficient D_s as:

Where the expression of diffusion coefficient is obtained by assuming a constant Lewis number (L_e=1.2). When ionized species are included, the diffusion of ions and electrons is modeled with ambipolar diffusion as recommended in ref[7]. The heat flux (conductive, chemical, vibrational and electronic counterparts) are defined as:

with

The total energy of the mixture per unit volume

is splitted between the translational-rotational, kinetic, vibrational, electron-electronic contributions, and the latent chemical energy of the species. The total pressure is given by Dalton's law as the sum of partial pressure of each species regarded as perfect gas.

3 Numerical simulations

The governing equations are integrated using an efficient and robust thermochemical nonequilibrium Navier-Stokes solver. The present code is based on the Upwing technology with exact Riemann's solver algorithm, in conjunction with a second-order MUSCL -TVD type scheme approach[8], coupled with a multi-block finite volume scheme.

The system of equations (1)-(5) can be reduced in only one as follows:

Where U is the conservative vector, F_c the convectif flux, F_v the viscous flux and W the source term.

The explicit formulation which gives the variation of U_i,j during time Dt on each cell (i,j) can be written in two dimensional axisymmetric coordinate as[9]:

where a = 1 for an axisymmetry coordinate system, and a = 0 for planar two-dimensions. This equation allows us to calculate numerically the all unknown variables in all computational domain. The exact Riemann solver and the Mimmod limiter function are used for the convective fluxes. The viscous terms are classically discretized by second-order central difference approximation. The source terms W are treated implicitly to relax the stiffness. The semi-implicit predictor-corrector schema can be written as:

where Rⁿ is the residual calculated explicitly at time step n. The steady state is obtained after convergence of the unsteady formulation of the discretized equations. The algorithm is second order accurate in space and time.

The modified speed of sound which takes into account the nonequilibrium electronic is implemented in the flux splitting procedure as[10]:

where classical frozen speed of sound is obtained when T = T_e.

4 Results and comparisons

The conditions of simulation are defined from the following parameters:

The Mach number is given as:

where c is the speed of sound defined earlier

The Reynolds number is given as:

where R is the largest satellite radius as shown in Fig. 1. These two parameters allow to define the gas rarefaction which is characterized with the Knudsen number as[1]:

The various aerothermodynamic parameters calculated along the wall in this paper are defined as follows:

Skin friction coefficient

Pressure coefficient

Wall heat transfer coefficient

Upstream conditions

The upstream conditions are chosen in order to make comparisons with Sharipov's results given in ref.[1]. In this paper the results are presented for 3 values of the Reynolds number (0,1 ; 10 and 4000) and for Mach number (5, 10, 20). The conditions used in this work are gathered in the table 1, and are extract from the abacus of the standard atmosphere[11]. The altitude of 80 km is retained because of the Reynolds number is close to 4000.

Boundary conditions

The boundary conditions are imposed along the satellite. The wall temperature is selected equal to the temperature of the upstream flow from which the various characteristics are given in the table 1. The walls are supposed chemically noncatalytic, no-slip and no-temperature jump boundary conditions are used. The freestream is hypersonic and all flow variables are known. The outflow is supersonic and therefore the zero gradient exit condition is appropriate. The influence of the temperature of wall on the variables aerothermodynamics was studied by Sharipov[1] and is not included in this study.

Figure 2 shows the geometry and grid used in the calculations. The grid point is densely distributed near the wall and near the shock standoff distance. The minimum grid spacing in x and y directions are respectively carried in the table 1. The time step is classically determined by explicit stability criteria and the Courant-Friedrichs-Lewy (CFL) number used in all computations varying from 0.01 to 0.4.

Results

Case Mach 10

Figure 3 represents the translational temperature isocontours in the flow at R_e = 4000. The maximum value of the temperature ratio T/T_¥ behind the shock wave is around 20 which is 30% weaker than the value predicted by Sharipov. The obtained temperature can cause only a very weak dissociation of the oxygen molecules. Consequently, chemical nonequilibrium is not taken into account in this case. The skin friction coefficient C_f, the pressure coefficient C_p and the wall heat transfer coefficient C_h are represented in Figs. 4, 5 and 6 as a function of the angle q for two Reynolds numbers (2161.8 and 4000). The Reynolds number has a significant influence on the peak of C_f. For identical Mach numbers, the evolution of C_f strongly depends on two parameters : the radius of the satellite and the Reynolds number. One can notice that the passage of the altitude 80 to 75 km, involves a decrease of the peak of about 41.93%. The results obtained for C_p and C_h are compared in Figs. 5 and 6 with those obtained with DSMC[1]. The influence of the Reynolds number is also pointed out with an increase of 35.4% of C_h at the stagnation point. For the same values of the Reynolds number (4000), one notes a rather good agreement with Sharipov's results[1]. The numerical simulation continues with a Mach number equal to 20 in which the thermochemical nonequilibrium becomes significant.

Case Mach 20

When the velocity of the vehicle increases, more kinetic energy is absorbed by the initial air molecules and the nonequilibrium processes such as vibrational excitation, rapid dissociation- recombinaison take place in the shock layer. Consequently, the physico-chemical phenomena are simultaneously set up and are included here. To simulate this case, the upstream air flow is considered with 2 species (23.30% of O₂ and 76.69% of N₂). Behind the shock wave, the Park's chemical kinetics model[12] with 5 species O, N, NO, O₂, N₂ and 17 chemical reactions is used. O₂ and N₂ are each characterized by their proper vibrational temperature while NO is assumed to be in thermodynamic equilibrium with the translational temperature[13]. The Coupling Vibration-Dissociation (CVD) is taken into account according to the Park's model[12] with an average two temperature assumption. The coupling Vibration-Vibration(V-V) is given by Candler[14]. The influence of coupling V-V on the thermodynamic parameters is presented. A comparison with the DSMC results is also made.

The translational temperature isocontours are plotted in Fig. 7 for frozen flow. The maximum value of the ratio T/T_¥ behind the shock wave is 30% weaker than that predicted by the DSMC as in the case Mach 10. The pressure isolines are drawn in Fig. 8. The upper part represents the solution in which only CVD coupling is taken into account. The lower part is the solution obtained with simultaneous considerations of the CVD and V-V couplings. The thickness of the shock layer is visible and allows to appreciate the position and the intensity of the shock.

The influence of the nonequilibrium phenomena on the detachment of shock on the stagnation line is presented in table 2. Frozen case corresponds to a calculation without chemical reactions. The temperatures along the stagnation line are drawn in Fig. 9. The nonequilibrium vibrational of the molecules O₂ and N₂ leads to a distinct distribution of temperatures (,). This justifies the use of the model with three temperatures in spite of the strong Mach number, contrary to the use of a single temperature of vibration proposed by certain authors [15,16]. Near the wall where temperature is low, one can note the influence of the V-V coupling. The difference between vibrational temperatures is more important without the V-V coupling, which leads to a transfer of a part of O₂ energy of vibration towards N₂, and confirms the tendency to bring closer the two temperatures of vibration. One notices an increase of the temperature behind the shock wave. The trans-rotational temperature T is about 13 000K behind the shock wave. This maximum is not sufficient to cause a significant ionization of gas under the upstream conditions used. Therefore, the ionization and electronic nonequilibrium are neglected.

The rise of temperature observed near the wall involves a greater dissociation of the nitrogen and oxygen molecules in the case of V-V coupling. The vibrational temperatures take part in the calculations of forward reaction rates. Fig. 10 shows a discrepancy of dissociation of approximately 52.38% for O₂ and 3.52% for N₂ at the stagnation point. This high dissociation of O₂ compared to N₂ can be explained by the fact that the O₂ molecules dissociate more quickly than N₂.

The evolution of coefficients C_f, C_p and C_h are given in Figs. 11, 12 and 13 as a function of the angle q. The chemical reactions affect the thickness of the shock layer and one notes a decrease of the friction coefficient peak. The peak difference between calculations with coupling CVD and V-V is about 4.5%. The comparison of the pressure coefficient predicted between DSMC and this calculation is shown in Fig. 12. One notes a good agreement (2.12% of difference). In Fig. 13, the maximum value of the heat flux is located to the position q = 0º. The importance of the dissociation reactions (endothermic) more accentuated in the case with V-V is remarkable on the wall heat flux. One observes a diminution of C_h of about 12.12% between the 2 cases. The difference of about 15.38% is also observed between the DSMC and this calculation at the stagnation point for frozen solution. This divergence may be explained by the initial conditions of simulation, because the conditions of M_a = 20 and R_e = 4000 can be obtained at altitude having different thermodynamic variables. Moreover, results presented are with M_a = 20 and R_e = 4323.6.

Case Mach 25

When the Mach number increases for the same altitude, the enthalpy increases and the ionized particles become important behind the shock wave. The air plasma surrounding re-entry vehicle may perturb the communications with the ground control, because plasma absorbs radio waves[19]. The objective of this section is to numerically study the influence of the ionized particles and the electronic excitation on the aerothermodynamics parameters. For that, an equation of electron-electronic relaxation and two additional species are taken into account (NO⁺ and e^–). The choice of NO⁺ is justified by the fact that the first energy of ionization (energy required to cause ionization) of NO is smallest. The flow are analysed thermally by using a model with four temperatures (T, , , T_e). The chemical kinetic model used for seven air species with 24 elementary reactions is proposed by Park[12].

The distribution of temperatures T, and T_e are represented in Fig. 14. The influence of the nonequilibrium electronic excitation can be appreciated. One notes a decay of 12.25% of maximum of the translational temperature behind the shock wave cause by the translation - electronic coupling (T – E) which is defined here by Appleton and Bray[17]. The electronic - vibration coupling (E – V), according to the Landau - Teller[5] model, concerns only the nitrogen molecule in this work, its influence on the evolution of in the stagnation region is significant with a fall of the temperature (» 23%). The skin friction coefficient decreases with the nonequilibrium effects as shown in Fig. 15. The wall heat flux is represented on Fig. 16. One notes an increase in heat flux of about 4.41% in q = 0º due to the electronic contribution. The taking into account of the thermochemical effects and their couplings brings a diminution of 19.04% of heat flux at the stagnation point between frozen and nonequilibrium calculation.

5 Conclusion

This work is based on a series of numerical and fundamental studies of the flow around reentry satellite SARA in order to better predict the true importance for the thermal protection. The physical model accounted thermochemical nonequilibrium processes with a partially ionized gas. A robust 2D multiblock MUSCL-TVD finite volume scheme is used to solve the viscous flow at high Mach number. The correction of the speed of sound due to the presence of the electron temperature is also included. The various features of this complex flow are observed. The importance of the thermochemical phenomena on the aerothermodynamics parameters is pointed out. In the frozen case, the results show a good agreement for C_p and C_h with the DSMC calculations. The consideration of the thermochemical nonequilibrium effects allows to be more close to the physical reality within the flow and improves the prediction of results.

In a future work, the radiative phenomena will be included in the model.

Received on 29 June, 2004. Revised version received on 15 October, 2004

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Publication Dates

• Publication in this collection
  03 May 2005
• Date of issue
  Mar 2005

History

• Accepted
  15 Oct 2004
• Reviewed
  15 Oct 2004
• Received
  29 June 2004