This article presents a calculation of a reactive flow around an axisymmetric blunt body in the shape of a hemispherecylinder. Experiments for the hypersonic shock tunnel remain a valuable source of information on the physicochemical phenomena that occur in the flow of air surrounding the model. Numerical simulations were used to supplement the theoretical and experimental studies. The simulations were performed by powerful calculators and powerful mathematical tools. In the present work, we implemented a numerical technique to simulate the hypersonic flow around an axisymmetric blunt body. The gas considered is the air in a standard state composed of 21% O₂ and 79% N₂. The infinite conditions of the flow are those of the atmosphere at a given altitude. In addition, it is necessary to take into account the air under a non-equilibrium state at the exit of a nozzle (Haoui at al., 2001).

Upstream the body, an intense detached shock wave occurred resulting in a very large temperature increase located at the nose of the body just after the shock. For this case, the air starts to dissociate and transforms into a gas composed of five species, O₂, N₂, NO, O, and N. There are several models that treat chemical kinetics that differ according to the number of chemical reactions. A realistic study must consider all the probable chemical reactions occurring in the flow. For this study, the model with seventeen chemical reactions is selected, 15 dissociation reactions and 2 exchange reactions (Haoui et al., 2001).

Speeds of the reactions employed are based on Arrhenius law. This is in addition to the expression of Landau and Teller (1936) for non-equilibrium of vibration. The time characteristic of vibration is given by the Blackman model (Blackman, 1955). The time characteristic of translation and rotation of molecules are very short in comparison to the time characteristic transition of the flow; however, the time it takes to return to the equilibrium state for these modes is very fast. Note that equilibrium of translation and rotation is carried out in any point of the flow and at any moment. By considering the range of temperatures in which the present study is placed, which is approximately 6,000 K, the phenomena of ionization of the species can be neglected. Similarly, effect of radiation is ignored. The chemical kinetics and the non-equilibrium of vibration are taken into account in the computational domain. The system of nonlinear partial derivative equations which governs this flow are solved by an explicit, unsteady numerical scheme (Goudjo and Desideri, 1989).

The Euler equations for the mixture in non-equilibrium contain mass conservation, momentum and energy equations of the evolution of chemical species (N₂, O₂, NO, O, N) and the energy of vibration of the molecules. It should be noted that only O₂ and N₂ species are considered in the non-equilibrium state. The NO molecule is studied under vibrational equilibrium. The time characteristic of vibration of this molecule is lower than that of O₂ by two orders of magnitude between the temperatures 3,000 K and 7,000 K (Wray, 1962).

In its vectorial form, the system of equations is written in 3D as follows:

Where:

The flux W, F, G, H and Ω are:

The energy per unit of mass e is such as:

h⁰_f is the enthalpy of formation of the species s in J/kg:

The pressure of the mixture is obtained by the equation of state:

Where

The temperature of the mixture is calculated based on the energy Eq. (3). The source term of the chemical equation of evolution of species s is given by:

Where:

v'_s and v'_s are the stochiometric mole numbers of the reactants and products of species s, respectively, for each chemical reaction (r) such as:

Both forward and backward reaction rates are represented by K_f and K_b. An empirical expression for the forward reaction rate K_f may be written as

The backward reaction rate K_b is a function of the equilibrium constant K_ep

The constants A, n and the temperature characteristic of dissociation T_d are given in Table 1 by Gardiner model (Gardiner, 1984). The equilibrium constant of the chemical reaction is given as a polynomial of the 4th degree:

Where z = 10000/T , and the coefficients c₀ through c₄ are provided for each reaction (Table 2). The term of energy production of vibration ω_vs is:

where e_v(T) is the equilibrium energy of vibration at the temperature of translation-rotation, which is expressed as:

where r is the constant of a particular gas and θ_v is the temperature characteristic of vibration. It is specified for each molecule. In the present work:

e_v(T_v) is the energy at the temperature of vibration T_v. The time characteristic of vibration of a species s in the mixture τ_s is a function of the temperature, the pressure and the molar fractions X_i . It can be calculated as follows (Lee, 1986):

Where s = O₂, N₂ and i = O₂, N₂, NO, O, N ^. τ_{s, j} is the time characteristic of vibration of the species s in a mixture containing species i. We have:

Experiments for binary exchanges, made it possible to evaluate vibrational relaxation time of O₂ in monatomic oxygen (Kiefer and Lutz, 1967) and of N₂ in O (Breshears and Bird, 1968), both correlated by (Thivet, 1989). The pressure p is in atmospheres and τ in seconds.

In order to determine the relaxation times, which are not given by the experimental correlations, it is supposed that the relaxation time of a species s is the same as the relaxation time of species i, provided that their masses are identical. Therefore, the following approximations can be employed:

The model with 17 reactions is:

O₂+M ←→ 2O+M r = 1 to 5

N₂+M ←→ 2N+M r = 6 to 10

NO+M ←→ N+O+M r = 11 to 15

N₂+O ←→ NO+N r =16

NO+O ←→ O₂+N r = 17

M can be one of the five chemical species O₂, N₂, NO, O or N (in the order).

General information is not lost by seeking the solution at the points of an infinitely small domain Fig. 1. A method developed within the Sinus project of the National Institute of Research in Informatics and Automatic (NIRIA) Sophia- Antipolis (Goudjo and Desideri, 1989) makes it possible to pass from 3D to 2D axisymmetric model by using a technique of the disturbance of domain. By taking advantage of this finding, the problem is thus considered as axisymmetric. The system of Eq. (1) can be written as:

Where mes(C_{i, j}) is the measurement (in m³) of an infinitely small volume of center (i, j). aire (C_{i, j}) is the surface of the symmetry plane passing by the center of an elementary volume. η_a is the integrated normal. For a detailed calculation of η_a, aire (C_{i, j}) and mes(C_{i, j}), we refer to the work of (Goudjo and Desideri, 1989). The third term of the equation expresses the axisymmetric flow condition. Flows, W, F, G and H are given by:

The present numerical method is based on an explicit approach in time and space. The step of time △t is as follows:

The Courant, Friedrich, Lewis (CFL) is a stability factor. V is the velocity of the flow and a is the speed of sound. △x is the small length of the mesh at the same point (i, j). At each time step and for each point (i, j), the system of Eq. (20) can be written as

The choice of the grid plays an important role in determining the convergence of the calculations. Therefore, it is indeed advisable to have sufficiently refined meshes at the places where the gradients of the flow parameters are significantly large, as observed in front of the nose of the obstacle.

In this study, the decomposition of Van-Leer (1982) is selected, namely a decomposition of flow in two parts, F_VL and F_VL.This decomposition applies to the present twodimensional problem by calculating the flow within each interface between two cells. Moreover, through this interface, the normal direction is paramount; thus, a reference mark is placed in the interface and the normal orientation is measured by an intermediate rotation, R, Fig. 2.

The vector W_E (Euler variable) is written W^R_E in the new reference mark

where is obtained from , via the rotation R, in the following way:

where:

The overall transformation R is

Moreover, at each interface (i + 1/2), two neighbor states (i ) and (i + 1) are known. Thus, one can calculate the onedimensional flow, F, through the interface. The total flow, f(W, η), may be deduced from F by applying the opposite rotation, as:

This property makes it possible to use only one component (F for example) of flow f to define the decomposition of flow in two dimensions. Moreover, this method is much easier and simpler to implement than the decomposition of flow in two dimensions f = F^.η_x + G η_y

The expressions of F⁺_VL and F^-_VL in 1-D, which are those of F⁺_VL(W^R) and F^-_VL(W^R) by rotation R, where W^R is defined like the transform of W, can be written in the following from:

Where , u_n and v_n are the velocity in the reference mark of the interface.

On the line of entrance, the parameters M, P, T and Y_S are fixed. Furthermore, as the flow reaches the entrance in a supersonic state, these parameters are unchanged during the iterations.

Because flow is not viscous, a slip condition is applied to the wall. At any point, M(x, y), on the wall the following condition must be checked:

where is the normal to the wall.

6.3 Axis of Symmetry

In any point M of the axis of symmetry, the following condition should be satisfied:

At the exit of the computational domain, which is downstream of the body, the flow is supersonic; the exit values of the flow filed parameters are extrapolated from the interior values.

The body employed for the numerical simulations is a hemisphere followed by a cylinder with a 5^o conical angle, as depicted in Fig. 3. The grid used in calculations is composed of 40 × 180 nodes. For illustration purposes, an example of grid with the dimensions 20 × 90 is presented in Fig. 4. The composition of the air of the upstream the flow is taken equal to 21% O₂ and 79% N₂. The pressure and the temperature related to the air at an altitude of 45 km, are 170 Pa and 295 K, respectively. The Mach number is set to 10. These values of the free stream correspond to the condition of spacecraft reentry into the Earth's atmosphere. Computations based on the 20 × 90 grid begin with the aerodynamic part, which converged after 1,000 iterations with a residue of 10^-3. This is followed by a complete scheme of simulations where the residue reaches 10^-6 and number of iterations equals 2,600 iterations. Note that for the 40 × 180 grid, the iteration count increases to 5,400. The CFL is always taken equal to 0.4. For high precision we are used the 40 × 180 grid. The solution converges to the exact solution and the truncation error converges to zero, which confirms the consistency and the accuracy of our computer code, as shown in Fig. 5. For more details, see the Condition of convergence applied to an axisymmetric reactive flow (Haoui et al., 2003).

For a flow neglecting the effects of thermo-chemistry, the mass fractions of the chemical species and the energy of vibration are supposedly fixed. The residue of the relative variation of the density is about 10^-5. There is no change in the findings for lower tolerance; Fig. 6 shows the variation of the residue according to the iterations count for a grid of 40 × 180 nodes. Figure 7 illustrates the variation of the temperature along the axis of symmetry until reaching a stagnation point, and then along the surface of the body. The difference between the cases of a perfect gas (PG), a nonequilibrium flow without coupling (V-T) and with coupling vibration-dissociation (CVD) according to the Park model (Park, 1989) while replacing the temperaturet T in equation 9 by T^1-q. T^q_v where q = 0.3 are presented.

Temperatures obtained in the case of the reactive flow are lower than those of PG. At the stagnation point for example, the ratio of the temperature T of the flow to the temperature of free stream T_inf is 17 instead of 21.3. This is reduction of about 20.6% in comparison to a PG. If we take into account the coupling CVD, the difference is weak between this case and the case without the coupling alternative.

Figure 8 depicts the variation of the temperature of vibration of O₂ and N₂ compared with the temperature of translation-rotation of the mixture. After the shock, the vibrational temperatures are strongly excited and after a distance of x / r = 1 becomes frozen, T_vO2/Tinf = 12.7, T_vN2/T_inf = 8.55 whereas T/T_inf - 7.53

Figures 9-13 illustrate the evolution of the mass fraction of the chemical species constituting the mixture of air. It is noticed that the dissociation of O₂ and N₂ starts just after the shock. Then, the concentrations decrease up to the stagnation point followed by a slow recombination because the velocity of the flow is larger compared to the chemical kinetics.

The mass fraction of the chemical species becomes frozen downstream where Y_O2 is equal to 0.19 instead of 0.21 and Y_N2 is equal to 0.78 instead of 0.79. For NO, O and N a fast formation of these species will occur after the shock up to the stagnation point and then a slow disappearance begins until having a freezing, where, Y_NO = 1.16%, Y_O = 0.9% and Y_N = 3.10^-4%.

Concerning the flow far from the wall, we presented the Mach-contours around the body in Fig. 14 where the shock wave detachment is visible in front of the nose of the body. The position of the detachment shock does not vary too much compared to the case without chemistry. For the nonequilibrium case, the position of the shock is at 0.135 times the radius in contrast to the case without chemistry (0.150 times the radius). Figure 15 shows the temperature-contours around the body. At the stagnation point the temperature is 5,015 K, whereas just after the shock it increases to 5,870 K; the difference is absorbed by the dissociation of the molecules. In addition, Figs. 16 and 17 depict the temperature-contours of vibration of O₂ and N₂ providing a clear insight that the vibration is at non-equilibrium in the beginning and then it becomes frozen.

Furthermore, the concentration-contours of O and N are presented in Figs. 18 and 19. The reactive and frozen areas are clearly observed when one moves away from the stagnation point.

Due to the lack of experimental values in the literature, our results were compared with a real photograph of the same flow (Nagamatsu, 1959), Fig. 20. We can observe the position of the detached shock wave and the thickness of the reactive flow between the shock and the body.

The numerical simulation of flows around blunt bodies at high temperatures provided satisfactory results from a numerical and a physical point of view. With high degree of accuracy requirements, computational convergence is achieved and the physical phenomena considered are visible after the detached shock wave and around the blunt body. The choice of the kinetic model for this type of flows is interesting. The model with 17 reactions proves to be more realistic since it considers all the possible collisions between molecules and atoms of the mixture of air. After the shock, the air is in a state of non-equilibrium, and a frozen state is obtained after the stagnation point, from x/r = 0.6 for the mass fractions and from x/r= 1 for the vibration temperatures. At the node of the body, the temperature of gas is found to be lower in comparison to a model where the dissociations of molecules are not implemented in the simulation.