Finite Volume Analysis of a Supersonic NonEquilibrium Flow Around an Axisymmetric Blunt Body
 Author: Haoui R.
 Organization: Haoui R.
 Publish: International Journal Aeronautical and Space Sciences Volume 11, Issue2, p59~68, 15 June 2010

ABSTRACT
The aim of this work is to analyze high temperature flows around an axisymmetric blunt body taking into account chemical and vibrational nonequilibrium state for air mixture species. For this purpose, a finite volume methodology is employed to determine the supersonic flow parameters around the axisymmetric blunt body. This allows the capture of a shock wave before a blunt body placed in supersonic free stream. The numerical technique uses the flux vector splitting method of Van Leer. Here, adequate time stepping parameters, along with Courant, Friedrich, Lewis coefficient and mesh size level are selected to ensure numerical convergence, sought with an order of 10^{8}.

KEYWORD
Supersonic flow , Nonequilibrium , Finite volume , Blunt body

1. Introduction
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% O_{2} and 79% N_{2}. 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 nonequilibrium 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, O_{2}, N_{2}, 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 nonequilibrium 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 nonequilibrium 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).
2. Equations
The Euler equations for the mixture in nonequilibrium contain mass conservation, momentum and energy equations of the evolution of chemical species (N_{2}, O_{2}, NO, O, N) and the energy of vibration of the molecules. It should be noted that only O_{2} and N_{2} species are considered in the nonequilibrium state. The NO molecule is studied under vibrational equilibrium. The time characteristic of vibration of this molecule is lower than that of O_{2} 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:h ^{0}_{f} is the enthalpy of formation of the speciess 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} andv '_{s} are the stochiometric mole numbers of the reactants and products of speciess , respectively, for each chemical reaction (r ) such as:Both forward and backward reaction rates are represented by
K _{f} andK _{b}. An empirical expression for the forward reaction rateK _{f} may be written asThe backward reaction rate
K _{b} is a function of the equilibrium constantK _{ep}The constants
A, n and the temperature characteristic of dissociationT _{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 coefficientsc _{0} throughc _{4} 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 translationrotation, 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 vibrationT _{v}. The time characteristic of vibration of a speciess in the mixture τ_{s} is a function of the temperature, the pressure and the molar fractionsX _{i} . It can be calculated as follows (Lee, 1986):Where
s =O _{2},N _{2} andi =O _{2},N _{2},NO, O, N ^{.} τ_{s, j} is the time characteristic of vibration of the speciess in a mixture containing speciesi . We have:Experiments for binary exchanges, made it possible to evaluate vibrational relaxation time of
O _{2} in monatomic oxygen (Kiefer and Lutz, 1967) and ofN _{2} inO (Breshears and Bird, 1968), both correlated by (Thivet, 1989). The pressurep 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 speciesi , provided that their masses are identical. Therefore, the following approximations can be employed:The model with 17 reactions is:
O_{2}+M ←→ 2O+M r = 1 to 5
N_{2}+M ←→ 2N+M r = 6 to 10
NO+M ←→ N+O+M r = 11 to 15
N_{2}+O ←→ NO+N r =16
NO+O ←→ O_{2}+N r = 17
M can be one of the five chemical speciesO _{2},N _{2},NO ,O orN (in the order).3. Axisymmetric Formulation
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 m^{3}) 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 andH are given by:4. Discretization in Time
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 anda 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 asThe 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.
5. Decomposition of VanLeer
In this study, the decomposition of VanLeer (1982) is selected, namely a decomposition of flow in two parts,
F _{VL} andF _{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 writtenW ^{R}_{E} in the new reference markwhere is obtained from , via the rotation
R , in the following way:where:
The overall transformation
R isMoreover, 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 fromF by applying the opposite rotation, as:This property makes it possible to use only one component (
F for example) of flowf 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 dimensionsf =F ^{.}η_{x} +G η_{y}The expressions of
F ^{+}_{VL} andF ^{}_{VL} in 1D, which are those ofF ^{+}_{VL}(W ^{R}) andF ^{}_{VL}(W ^{R}) by rotationR , whereW ^{R} is defined like the transform ofW , can be written in the following from:Where ,
u _{n} andv _{n} are the velocity in the reference mark of the interface.6. Boundary Conditions
6.1 Border Upstream
On the line of entrance, the parameters
M, P, T andY _{S} are fixed. Furthermore, as the flow reaches the entrance in a supersonic state, these parameters are unchanged during the iterations.6.2 Body Surface
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:6.4 Border Downstream
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.
7. Results and Interpretations
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%
O _{2} and 79%N _{2}. 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 thermochemistry, 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 (VT) and with coupling vibrationdissociation (CVD) according to the Park model (Park, 1989) while replacing the temperaturet
T in equation 9 byT ^{1q}.T ^{q}_{v} whereq = 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 streamT _{inf} is 17 instead of 21.3. This is reduction of about20.6% 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
O _{2} andN _{2} compared with the temperature of translationrotation of the mixture. After the shock, the vibrational temperatures are strongly excited and after a distance ofx / r = 1 becomes frozen,T _{vO2}/T inf = 12.7,T _{vN2}/T _{inf} = 8.55 whereasT /T _{inf}  7.53Figures 913 illustrate the evolution of the mass fraction of the chemical species constituting the mixture of air. It is noticed that the dissociation of
O _{2} andN _{2} 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 _{O}2 is equal to 0.19 instead of 0.21 andY _{N2} is equal to 0.78 instead of 0.79. ForNO ,O andN 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% andY _{N} = 3.10^{4}%.Concerning the flow far from the wall, we presented the Machcontours 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 temperaturecontours 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 temperaturecontours of vibration of
O _{2} andN _{2} providing a clear insight that the vibration is at nonequilibrium in the beginning and then it becomes frozen.Furthermore, the concentrationcontours of
O andN 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.
8. Conclusions
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 nonequilibrium, and a frozen state is obtained after the stagnation point, from
x /r = 0.6 for the mass fractions and fromx /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.

[Table 1.] Forward constants (Gardiner models)

[Table 2.] Table 2. Equilibrium constants

[Fig. 1.] Mesh form.

[Fig. 2.] Interface and its normal.

[Fig. 3.] Computational domain.

[Fig. 4.] Grid of (20 × 90).

[Fig. 5.] Grid convergence test, perfect gas.

[Fig. 6.] Variation of residue.

[Fig. 7.] Variation of translationalrotational temperatures. PG: perfect gas, CVD: coupling vibrationdissociation.

[Fig. 8.] Variation of vibrational temperatures.

[Fig. 9.] Evolution of the mass fraction of O2. PG: perfect gas, CVD: coupling vibrationdissociation.

[Fig. 10.] Evolution of the mass fraction of N2. PG: perfect gas, CVD: coupling vibrationdissociation.

[Fig. 11.] Evolution of the mass fraction of NO. PG: perfect gas, CVD: coupling vibrationdissociation.

[Fig. 12.] Evolution of the mass fraction of O. PG: perfect gas, CVD: coupling vibrationdissociation.

[Fig. 13.] Evolution of the mass fraction of N. PG: perfect gas, CVD: coupling vibrationdissociation.

[Fig. 6.] Mach number contours. CVD: coupling vibrationdissociation.

[Fig. 15.] Temperature contours.

[Fig. 16.] Temperature contours of vibration of O2.

[Fig. 17.] Temperature contours of vibration of N2.

[Fig. 18.] Concentration contours of O.

[Fig. 19.] Concentration contours of N.

[Fig. 20.] Comparison computationalexperimental: (a) Computational Mach contours; (b) test with a hemisphere model of a nominal Mach number 10 nozzle (Nagamatsu, 1959).