2. Governing equations of hypersonic non-equilibrium polyatomic gas flows

The Martian atmosphere is composed mostly of carbon dioxide (96%), nitrogen (1.9%), argon (1.9%), and others. Small admixtures of nitrogen (N2) and argon (Ar) in the Mars atmosphere do not play a significant role in the process of heat transfer to descent vehicles (at least, at moderate velocities of flight till the convective heat transfer prevails). It is possible to restrict ourselves by consideration of model atmosphere as the pure carbon dioxide. The pressure on the planet surface is taken equal to 6.0 mbar. It is 0.6% of Earth's mean sea level pressure. The atmosphere is quite dusty.

The conditions of a flow corresponding to the last stage of flight of space vehicles in an atmosphere of Mars (V<sup>∞</sup> ≤ 6 km/s, r<sup>∞</sup> > 10<sup>5</sup> kg/m<sup>3</sup> , H < 60 km) were studied. Determining process at such velocities is a process of dissociation. Up to 75% of full gas flow energy can be spent on it.

The region where non-equilibrium physical and chemical processes realized is a significant part from all considered regions (Figure 2). Velocity of physical and chemical processes, as a rule, grows together with density of gas. For considered flow conditions, the degree of gas ionization is small and does not bring the appreciable contribution to internal gas energy. The translational degrees freedom becomes equilibrium on distances of several free path lengths of molecules behind front of a shock wave for considered altitude. The distribution of rotation energy also is established slightly later. Therefore, it is usually supposed that translational and vibration degrees of freedom of particles are in equilibrium. At high temperatures that observed in a shock layer, the characteristic times of a vibration energy relaxation of molecules and characteristic times of dissociation become one order. Thus the account of non-equilibrium excitation of vibration degrees of freedom of carbon dioxide molecules is necessary. The region of relaxation behind the bow shock wave has a specific structure that consists of sequential relaxation zones. The flow in shock and boundary layers are being especially non-equilibrium. It causes the energy redistribution of the internal energy.

Since the Martian atmosphere is strongly rarefied, non-equilibrium processes affect heat transfer at the more significant part of the descent trajectory. The most thermal-loaded part of the typical descent trajectory is the region of frozen chemical reactions and equilibrium-excited vibration degrees of freedom.

We consider the conditions typical for a high-temperature shock layer, while the translational and rotational relaxation are supposed to proceed fast as well as intra-mode VV-transitions in CO2, O2, CO and inter-mode VV-exchange between CO2 symmetric and bending modes. All other vibration energy transitions, dissociation, recombination, and exchange reactions are considered to be slower with relaxation times comparable with the mean time of the gas dynamic parameters variation. Such a relation between the characteristic times makes it possible to introduce vibration

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules

The existing experimental and theoretical data on relaxation times of different processes in mixtures containing CO2 molecules show that in a wide range of conditions the following

Here τtr, τrot are the characteristic times of translational and rotation relaxation; τVVm are the

to inter-mode transitions; τ<sup>r</sup> is the characteristic time of chemical reactions; and θ is the mean

On the basis of the kinetic theory principles, the closed self-consistent three-temperature description of a flow in terms of densities of species, macroscopic velocity, gas temperature, and three vibration temperatures are obtained [7, 8]. The set of governing equations contains the conservation equations of mass, momentum, and total energy coupled to the equations of non-equilibrium three-temperature chemical kinetics as well as the relaxation equations for

<sup>1</sup>�2�<sup>3</sup> <sup>&</sup>lt; <sup>τ</sup>VT<sup>3</sup> << <sup>τ</sup><sup>r</sup> � <sup>θ</sup>, m <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3 (4)

http://dx.doi.org/10.5772/intechopen.71666

dt <sup>þ</sup> <sup>r</sup> <sup>∇</sup> � <sup>v</sup> <sup>¼</sup> <sup>0</sup> (5)

dt <sup>þ</sup> <sup>∇</sup> � <sup>P</sup> <sup>¼</sup> <sup>0</sup> (6)

<sup>i</sup> ð Þ i ¼ CO2; CO; O2; O; C (8)

dt <sup>þ</sup> <sup>∇</sup> � <sup>q</sup> <sup>þ</sup> <sup>P</sup> : <sup>∇</sup><sup>v</sup> <sup>¼</sup> <sup>0</sup> (7)

<sup>k</sup>�<sup>m</sup> correspond

45

temperatures for the coupled (symmetric-bending) and asymmetric CO2 modes.

<sup>1</sup>�<sup>2</sup> << <sup>τ</sup>VT<sup>2</sup> � <sup>τ</sup>VV<sup>0</sup>

times of intra-mode VV exchanges; τVT<sup>2</sup> , τVT<sup>3</sup> are the times of VT transitions; τVV<sup>0</sup>

Under condition (5) the set of equations are obtained in the following form: dr

> r dv

The equations of non-equilibrium chemical kinetics written in the following form

Here ni is the number density of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values r<sup>i</sup> = mini is the species density, mi and ni are the species mass and number density, <sup>r</sup> <sup>¼</sup> <sup>P</sup>r<sup>i</sup> is the mixture density; v is the macroscopic gas velocity, e is the mixture total energy per unit mass;

, R12, R<sup>3</sup> are the production terms due to dissociation, recombination, exchange reactions and slow processes of CO2 vibration relaxation; V<sup>i</sup> is the diffusion velocity; P is the pressure

r de

dt <sup>þ</sup> ni<sup>∇</sup> � <sup>v</sup> <sup>þ</sup> <sup>∇</sup> � ð Þ¼ niV<sup>i</sup> <sup>R</sup><sup>r</sup>

relations are valid:

vibration temperatures.

Ri t τtr < τrot < τVVm � τVV<sup>0</sup>

time of gas dynamic parameters change.

dni

tensor; and q is the heat flux.

Figure 2. (a) The region of different flow regimes of flow for MSRO vehicle (red line) and MARS EXPRESS vehicle (blue line); (b) Mars descent vehicles.

The thermo-chemical model of the processes occurring in the shock layer includes the chemical reactions, dissociation and recombination of CO2 molecules, dissociation and recombination of diatomic molecules, exchange reactions, processes of vibration energy exchange between various levels of molecules, influence of the vibration relaxation on the chemical reactions and vice versa (CVDV-processes), processes of excitation and deactivation of the electronic states of molecules, and spontaneous radiation processes for excited particles.

We consider a high-temperature flow of the mixture taking into account vibration excitation and the following exchange reactions, dissociation, and recombination:

$$\rm{CO}\_2 + M \leftrightarrow \rm{CO} + \rm{O} + M,\\
\rm{o} \rm{CO} + M \leftrightarrow \rm{C} + \rm{O} + M,\tag{1}$$

$$\rm O\_2 + M \leftrightarrow O + O + M,\\
\rm OO\_2 + O \leftrightarrow CO + O\_2.\tag{2}$$

$$\text{CO} + \text{CO} \leftrightarrow \text{CO}\_2 + \text{C}, \mathfrak{v}\text{CO} + \text{O} \leftrightarrow \text{O}\_2 + \text{C},\tag{3}$$

where M is a molecule or an atom [10].

As known molecule of СО<sup>2</sup> have three vibration modes with different characteristic temperatures: symmetric, deformation (twice degenerate), and asymmetric. According to fast exchange of vibration energy between the different modes, it is assumed that molecules reach the Boltzmann distribution with a uniform temperature.

We consider the conditions typical for a high-temperature shock layer, while the translational and rotational relaxation are supposed to proceed fast as well as intra-mode VV-transitions in CO2, O2, CO and inter-mode VV-exchange between CO2 symmetric and bending modes. All other vibration energy transitions, dissociation, recombination, and exchange reactions are considered to be slower with relaxation times comparable with the mean time of the gas dynamic parameters variation. Such a relation between the characteristic times makes it possible to introduce vibration temperatures for the coupled (symmetric-bending) and asymmetric CO2 modes.

The existing experimental and theoretical data on relaxation times of different processes in mixtures containing CO2 molecules show that in a wide range of conditions the following relations are valid:

$$
\tau\_{tr} < \tau\_{nt} < \tau\_{VV\_m} \sim \tau\_{VV\_{1-2}'} < \tau\_{VV\_2} \sim \tau\_{VV\_{1-2-3}'} < \tau\_{VV\_3} < \tau\_r \sim \theta, \quad m = 1, \ 2, \ 3 \tag{4}
$$

Here τtr, τrot are the characteristic times of translational and rotation relaxation; τVVm are the times of intra-mode VV exchanges; τVT<sup>2</sup> , τVT<sup>3</sup> are the times of VT transitions; τVV<sup>0</sup> <sup>k</sup>�<sup>m</sup> correspond to inter-mode transitions; τ<sup>r</sup> is the characteristic time of chemical reactions; and θ is the mean time of gas dynamic parameters change.

On the basis of the kinetic theory principles, the closed self-consistent three-temperature description of a flow in terms of densities of species, macroscopic velocity, gas temperature, and three vibration temperatures are obtained [7, 8]. The set of governing equations contains the conservation equations of mass, momentum, and total energy coupled to the equations of non-equilibrium three-temperature chemical kinetics as well as the relaxation equations for vibration temperatures.

Under condition (5) the set of equations are obtained in the following form:

The thermo-chemical model of the processes occurring in the shock layer includes the chemical reactions, dissociation and recombination of CO2 molecules, dissociation and recombination of diatomic molecules, exchange reactions, processes of vibration energy exchange between various levels of molecules, influence of the vibration relaxation on the chemical reactions and vice versa (CVDV-processes), processes of excitation and deactivation of the electronic

Figure 2. (a) The region of different flow regimes of flow for MSRO vehicle (red line) and MARS EXPRESS vehicle (blue

We consider a high-temperature flow of the mixture taking into account vibration excitation

As known molecule of СО<sup>2</sup> have three vibration modes with different characteristic temperatures: symmetric, deformation (twice degenerate), and asymmetric. According to fast exchange of vibration energy between the different modes, it is assumed that molecules reach

CO2 þ M \$ CO þ O þ M, CO þ M \$ C þ O þ M, (1)

O2 þ M \$ O þ O þ M, CO2 þ O \$ CO þ O2, (2)

CO þ CO \$ CO2 þ C, CO þ O \$ O2 þ C, (3)

states of molecules, and spontaneous radiation processes for excited particles.

and the following exchange reactions, dissociation, and recombination:

where M is a molecule or an atom [10].

line); (b) Mars descent vehicles.

44 Advances in Some Hypersonic Vehicles Technologies

the Boltzmann distribution with a uniform temperature.

$$\frac{d\rho}{dt} + \rho \left| \nabla \cdot \mathbf{v} \right| = 0 \tag{5}$$

$$
\rho \frac{d\mathbf{v}}{dt} + \nabla \cdot \mathbf{P} = 0 \tag{6}
$$

$$
\rho \frac{de}{dt} + \nabla \cdot \mathbf{q} + \mathbf{P} : \nabla \mathbf{v} = 0 \tag{7}
$$

The equations of non-equilibrium chemical kinetics written in the following form

$$\frac{dn\_i}{dt} + n\_i \nabla \cdot \mathbf{v} + \nabla \cdot (n\_i \mathbf{V}\_i) = \mathbf{R}\_i^r \ (i = \mathbf{CO}\_2, \mathbf{CO}, \mathbf{O}\_2, \mathbf{O}, \mathbf{C}) \tag{8}$$

Here ni is the number density of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values r<sup>i</sup> = mini is the species density, mi and ni are the species mass and number density, <sup>r</sup> <sup>¼</sup> <sup>P</sup>r<sup>i</sup> is the mixture density; v is the macroscopic gas velocity, e is the mixture total energy per unit mass; Ri t , R12, R<sup>3</sup> are the production terms due to dissociation, recombination, exchange reactions and slow processes of CO2 vibration relaxation; V<sup>i</sup> is the diffusion velocity; P is the pressure tensor; and q is the heat flux.

We consider the condition that corresponds to rapid translational and rotational relaxation, VV<sup>m</sup> is the vibration energy exchanges within modes and VV<sup>0</sup> <sup>12</sup> is the exchange between symmetric and bending CO2 modes. In this case the vibration CO2 distributions depend on the vibration temperatures T<sup>12</sup> of the combined (symmetric + bending) mode, and T<sup>3</sup> of the asymmetric mode. The vibration distributions of CO and O2 are supposed to be close to the thermal equilibrium; vibration spectra are simulated by the harmonic oscillator model. Values Evibr1(T12,T3) = E12(T12) + E3(T3), Evibr2(T), Evibr3(T) are the specific vibration energies of molecular species CO2, CO, and O2, respectively; thus E12(T12), E3(T3) are the specific vibration energies of non-equilibrium CO2 modes.

The equations of non-equilibrium vibration kinetics written in the following form

$$
\rho\_1 \frac{dE\_{12}}{dt} + \nabla \cdot \mathbf{q}\_{12} = R\_{12} - m\_1 E\_{12} R\_1' + E\_{12} \nabla \cdot (\rho\_1 \mathbf{V}\_1) \tag{9}
$$

derivation leads to the definition of two coefficients of viscosity: shear coefficient of viscosity and

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules

Transport coefficients (bulk and shear viscosity, heat conductivity, diffusion, pressure- and thermo-diffusion of multi-component gas mixture) are calculated according to the basic kinetic theory. The algorithms for thermal conductivity, vibration thermal conductivity, diffusion,

Here [A;B] are the bracket integrals depending on the cross sections of rapid processes (see for

The values of diffusion fluxes can be written down through thermodynamic forces (external

η ¼ kT B½ � ; B =10, ζ ¼ kT F½ � ; F , prel ¼ kT F½ � ; G (12)

∇ln T (13)

http://dx.doi.org/10.5772/intechopen.71666

47

ij, DTi are connected to the

; A<sup>i</sup> � �=3n: (15)

cij, Bcij, Fcij, Fcij are found from the linear integral equa-

� �∇ln <sup>p</sup>: (14)

DTi

ij is the multi-component coefficients of diffusion, DTi is the coefficients of thermo-

nj <sup>n</sup> � cj

Here nj is the number of jth mole components, n is the common mole's number. Let us take the

kinetic theory methods. In order to define the multi-components diffusion coefficients, it is necessary to solve the system of the linear algebraic equations that in case of gas mixture have

; <sup>D</sup><sup>i</sup> � �=3n, DTi <sup>¼</sup> <sup>D</sup><sup>i</sup>

tions for the first-order correction to the distribution function. The relaxation pressure determine by the slow non-equilibrium processes. This quantity is basically supposed to be small

The total heat transfer of a multi-component mixture is defined by effects of heat conductivity

; A<sup>i</sup> � � are the bracket integrals depending on the cross sections of rapid

ij and Dij, where Dij is the binary diffusion coefficients.

thermal diffusion, shear, and bulk viscosity coefficients computation are developed.

These transport coefficients are defined by functions Bcij, Fcij, Gcij:

J<sup>i</sup> ¼ �r<sup>i</sup>

d<sup>j</sup> ¼ ∇

The basic way of finding the independent transfer coefficients D<sup>∗</sup>

Dij <sup>¼</sup> <sup>D</sup><sup>i</sup>

cij , <sup>A</sup>ð Þ<sup>3</sup>

cij , <sup>A</sup>ð Þ <sup>c</sup>;<sup>1</sup>

of various kinds of energy and diffusion. The heat flux is given by the formula

X j D∗ ijd<sup>j</sup> � r<sup>i</sup>

nj n � � <sup>þ</sup>

coefficient of bulk viscosity.

instance [7, 8] for definition).

mass forces it is neglected):

difference between D<sup>∗</sup>

; D<sup>i</sup> � �, D<sup>i</sup>

processes. Functions Acij, Að Þ <sup>12</sup>

compared to p, and by value prel is neglected.

the next form

Here Di

diffusion, d<sup>j</sup> is the diffusion driving forces:

Here D<sup>∗</sup>

$$
\rho\_1 \frac{dE\_3}{dt} + \nabla \cdot \mathbf{q}\_3 = R\_3 - m\_1 E\_3 R\_1' + E\_3 \nabla \cdot (\rho\_1 \mathbf{V}\_1) \tag{10}
$$

xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values q12, q<sup>3</sup> are the fluxes of vibration energy in the combined and asymmetric modes, respectively.

The vibration relaxation of molecules time of СО<sup>2</sup> molecules is calculated under the usual formulas by approximation of theoretical and experimental data for particles of different types.

### 3. Transport and source terms

The transport theory of polyatomic gas mixtures taking into account internal molecular structure, different rates of vibration transitions, and unharmonicity has been developed for a five component CO2/O2/CO/O/C mixture taking into account vibration excitation of diatomic molecules.

The transport properties in the viscous gas approximation are determined by the first-order distribution functions. The zero-order and the first-order distribution functions are known and express by means different gradients, the diffusive driving forces, etc. [7, 8].

Pressure tensor, diffusion velocity, heat flux, and vibration energy fluxes are expressed in terms of macroscopic parameters gradients and transport coefficients.

The pressure tensor is obtained in the form:

$$\mathbf{P} = \begin{pmatrix} p - p\_{rel} \end{pmatrix} \text{ I} - 2\eta \text{ S} - \zeta \nabla \cdot \mathbf{v} \text{I}. \tag{11}$$

Here S is the strain rate tensor, I is the unit tensor, η is the shear viscosity coefficients, and ζ and prel are the bulk viscosity coefficient and relaxation pressure appearing in the diagonal elements of the pressure tensor due to rapid inelastic non-resonant processes. The Navier-Stokes equation derivation leads to the definition of two coefficients of viscosity: shear coefficient of viscosity and coefficient of bulk viscosity.

Transport coefficients (bulk and shear viscosity, heat conductivity, diffusion, pressure- and thermo-diffusion of multi-component gas mixture) are calculated according to the basic kinetic theory. The algorithms for thermal conductivity, vibration thermal conductivity, diffusion, thermal diffusion, shear, and bulk viscosity coefficients computation are developed.

These transport coefficients are defined by functions Bcij, Fcij, Gcij:

We consider the condition that corresponds to rapid translational and rotational relaxation,

symmetric and bending CO2 modes. In this case the vibration CO2 distributions depend on the vibration temperatures T<sup>12</sup> of the combined (symmetric + bending) mode, and T<sup>3</sup> of the asymmetric mode. The vibration distributions of CO and O2 are supposed to be close to the thermal equilibrium; vibration spectra are simulated by the harmonic oscillator model. Values Evibr1(T12,T3) = E12(T12) + E3(T3), Evibr2(T), Evibr3(T) are the specific vibration energies of molecular species CO2, CO, and O2, respectively; thus E12(T12), E3(T3) are the specific vibration

The equations of non-equilibrium vibration kinetics written in the following form

dt <sup>þ</sup> <sup>∇</sup> � <sup>q</sup><sup>12</sup> <sup>¼</sup> <sup>R</sup><sup>12</sup> � <sup>m</sup>1E12R<sup>r</sup>

dt <sup>þ</sup> <sup>∇</sup> � <sup>q</sup><sup>3</sup> <sup>¼</sup> <sup>R</sup><sup>3</sup> � <sup>m</sup>1E3R<sup>r</sup>

fluxes of vibration energy in the combined and asymmetric modes, respectively.

express by means different gradients, the diffusive driving forces, etc. [7, 8].

terms of macroscopic parameters gradients and transport coefficients.

P ¼ p � prel

xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values q12, q<sup>3</sup> are the

The vibration relaxation of molecules time of СО<sup>2</sup> molecules is calculated under the usual formulas by approximation of theoretical and experimental data for particles of different types.

The transport theory of polyatomic gas mixtures taking into account internal molecular structure, different rates of vibration transitions, and unharmonicity has been developed for a five component CO2/O2/CO/O/C mixture taking into account vibration excitation of diatomic molecules. The transport properties in the viscous gas approximation are determined by the first-order distribution functions. The zero-order and the first-order distribution functions are known and

Pressure tensor, diffusion velocity, heat flux, and vibration energy fluxes are expressed in

Here S is the strain rate tensor, I is the unit tensor, η is the shear viscosity coefficients, and ζ and prel are the bulk viscosity coefficient and relaxation pressure appearing in the diagonal elements of the pressure tensor due to rapid inelastic non-resonant processes. The Navier-Stokes equation

<sup>I</sup> � <sup>2</sup><sup>η</sup> <sup>S</sup> � <sup>ζ</sup><sup>∇</sup> � vI: (11)

<sup>12</sup> is the exchange between

<sup>1</sup> þ E12∇ � ð Þ r1V<sup>1</sup> (9)

<sup>1</sup> þ E3∇ � ð Þ r1V<sup>1</sup> (10)

VV<sup>m</sup> is the vibration energy exchanges within modes and VV<sup>0</sup>

energies of non-equilibrium CO2 modes.

46 Advances in Some Hypersonic Vehicles Technologies

3. Transport and source terms

The pressure tensor is obtained in the form:

r1 dE<sup>12</sup>

> r1 dE<sup>3</sup>

$$
\eta = kT[B,B]/10,\ \zeta = kT[F,F].\ p\_{rel} = kT[F,G] \tag{12}
$$

Here [A;B] are the bracket integrals depending on the cross sections of rapid processes (see for instance [7, 8] for definition).

The values of diffusion fluxes can be written down through thermodynamic forces (external mass forces it is neglected):

$$\mathbf{J}\_i = -\rho\_i \sum\_{\vec{j}} D\_{i\vec{j}}^\* \mathbf{d}\_{\vec{j}} - \rho\_i D\_{T\_i} \nabla \ln T \tag{13}$$

Here D<sup>∗</sup> ij is the multi-component coefficients of diffusion, DTi is the coefficients of thermodiffusion, d<sup>j</sup> is the diffusion driving forces:

$$\mathbf{d}\_{\circ} = \nabla \left( \frac{n\_{\circ}}{n} \right) + \left( \frac{n\_{\circ}}{n} - c\_{\circ} \right) \nabla \ln p. \tag{14}$$

Here nj is the number of jth mole components, n is the common mole's number. Let us take the difference between D<sup>∗</sup> ij and Dij, where Dij is the binary diffusion coefficients.

The basic way of finding the independent transfer coefficients D<sup>∗</sup> ij, DTi are connected to the kinetic theory methods. In order to define the multi-components diffusion coefficients, it is necessary to solve the system of the linear algebraic equations that in case of gas mixture have the next form

$$D\_{\vec{\eta}} = \left[\mathbf{D}^{\dot{\mathbf{i}}}, \mathbf{D}^{\dot{\mathbf{i}}}\right] / \mathfrak{Im}, \ D\_{\overrightarrow{\mathrm{Ti}}} = \left[\mathbf{D}^{\dot{\mathbf{i}}}, \mathbf{A}^{\dot{\mathbf{i}}}\right] / \mathfrak{Im}. \tag{15}$$

Here Di ; D<sup>i</sup> � �, D<sup>i</sup> ; A<sup>i</sup> � � are the bracket integrals depending on the cross sections of rapid processes. Functions Acij, Að Þ <sup>12</sup> cij , <sup>A</sup>ð Þ<sup>3</sup> cij , <sup>A</sup>ð Þ <sup>c</sup>;<sup>1</sup> cij, Bcij, Fcij, Fcij are found from the linear integral equations for the first-order correction to the distribution function. The relaxation pressure determine by the slow non-equilibrium processes. This quantity is basically supposed to be small compared to p, and by value prel is neglected.

The total heat transfer of a multi-component mixture is defined by effects of heat conductivity of various kinds of energy and diffusion. The heat flux is given by the formula

$$\mathbf{q} = -\lambda^{\prime}\nabla T - \lambda\_{12}\nabla T\_{12} - \lambda\_{3}\nabla T\_{3} - p\sum\_{i=1}^{5}D\_{T\_{i}}\mathbf{d}\_{i} + \sum\_{i=1}^{5}\rho\_{i}h\_{i}\mathbf{V}\_{i} \tag{16}$$

theoretically in the literature [18] for weak and strong non-equilibrium conditions but up to

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules

The experiments and kinetic theory show that bulk viscosity can significantly influence shock wave structure for polyatomic molecules. In polyatomic gases of the deviation from local equilibrium effects itself as bulk viscosity. From the Chapman-Enskog's theory, it can be proved that for any perfect monoatomic gas, the coefficient of bulk viscosity is equal to zero. Bulk viscosity results from contributions from the several internal degrees of freedom of the

The bulk viscosity coefficient is defined by rotational energy transitions of all molecular species and VT vibration energy transfer in CO and O2 molecules and can be written as follows

ζ ¼ �kT x1f <sup>1</sup>, <sup>10</sup> þ x2f <sup>2</sup>,<sup>10</sup> þ x3f <sup>1</sup>,<sup>10</sup> þ x4f <sup>4</sup>,<sup>1</sup> þ x5f <sup>5</sup>,<sup>1</sup>

where k is the Boltzmann constant, xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2,

ij mm0 nn<sup>0</sup> :

1

CCCCCCCCCCCCCCCCCCCA f ¼

<sup>1001</sup> <sup>β</sup><sup>2</sup>�<sup>3</sup> 1001

<sup>1001</sup> <sup>β</sup><sup>3</sup>�<sup>3</sup> 1001

<sup>1001</sup> <sup>β</sup><sup>4</sup>�<sup>3</sup> 1001

<sup>1001</sup> <sup>β</sup><sup>5</sup>�<sup>3</sup> 1001

<sup>0011</sup> 0

0011

ζ ¼ ζ<sup>r</sup> þ ζv,CO þ ζv,O<sup>2</sup> (19)

� �, (20)

B � f ¼ s: (21)

f <sup>1</sup>,<sup>10</sup> f <sup>2</sup>,<sup>10</sup> f <sup>3</sup>,<sup>10</sup> f <sup>4</sup>,<sup>1</sup> f <sup>5</sup>,<sup>1</sup> f <sup>1</sup>,<sup>01</sup> f <sup>2</sup>,<sup>01</sup> f <sup>3</sup>,<sup>01</sup> 1

0

0

BBBBBBBBBBBBBBBBB@

�x2ð Þ cu � ctr =cu �x3ð Þ cu � ctr =cu �x4ð Þ cu � ctr =cu �x5ð Þ cu � ctr =cu x1crot,1=cu x2cint, <sup>2</sup>=cu x3cint, <sup>3</sup>=cu

1

CCCCCCCCCCCCCCCCCA

(22)

CCCCCCCCCCCCCCCCCCA s ¼

0

BBBBBBBBBBBBBBBBBB@

<sup>1</sup>�<sup>2</sup> exchanges occur to be resonant and

http://dx.doi.org/10.5772/intechopen.71666

49

now it was not evaluated in real gas flows.

For harmonic oscillators, rapid inelastic VV and VV<sup>0</sup>

therefore do not give contribution to the coefficient ζ.

Coefficients fi,mn are the solutions of the system

<sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>3</sup>

<sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>3</sup>

<sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>3</sup>

<sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>3</sup>

<sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>3</sup>

<sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>3</sup>

<sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>3</sup>

Here matrix B is composed of the bracket integrals β

<sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>4</sup>

<sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>4</sup>

<sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>4</sup>

<sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>4</sup>

<sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>4</sup>

<sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>4</sup>

<sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>4</sup>

x1ctr x2ctr x3ctr x4ctr x5ctr x1crot, <sup>1</sup> x2cint, <sup>2</sup> x3cint,<sup>3</sup>

<sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>1</sup>

<sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>1</sup>

<sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>1</sup>

<sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>1</sup>

<sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>1</sup>

<sup>0110</sup> <sup>0</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup>

<sup>1001</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup>

<sup>1001</sup> <sup>β</sup><sup>3</sup>�<sup>2</sup>

<sup>1001</sup> <sup>β</sup><sup>4</sup>�<sup>2</sup>

<sup>1001</sup> <sup>β</sup><sup>5</sup>�<sup>2</sup>

<sup>0110</sup> 0 0 <sup>β</sup><sup>3</sup>�<sup>3</sup>

<sup>0011</sup> 0 0

Matrix B, vectors f and s are given above. Here сtr, crot, сint are specific heats of translation, rotational, internal degrees of freedom and cu is the total specific heat. The bracket integrals

<sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>5</sup>

<sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>5</sup>

<sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>5</sup>

<sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>5</sup>

<sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>5</sup>

<sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>5</sup>

<sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>5</sup>

The bulk viscosity coefficient can be obtained in the form [18]:

gas.

4—C, 5—O).

β<sup>2</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup>

0

BBBBBBBBBBBBBBBBBBB@

β<sup>3</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>2</sup>

β<sup>4</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>2</sup>

β<sup>5</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>2</sup>

β<sup>1</sup>�<sup>1</sup> <sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>2</sup>

β<sup>2</sup>�<sup>1</sup> <sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup>

β<sup>3</sup>�<sup>1</sup> <sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>2</sup>

B ¼

Here p is the pressure, hi is the specific enthalpy of species i, λ 0 ¼ λtr þ λ<sup>r</sup> þ λν is the thermal conductivity coefficient of all degrees of freedom which deviate weakly from local thermal equilibrium. They include the translational and rotational modes as well as CO and O2 vibration degrees of freedom. Thus, the coefficient λ<sup>v</sup> = λv,CO + λr,O2. Coefficients λ12, λ<sup>3</sup> correspond to the thermal conductivity of strongly non-equilibrium modes: combined (symmetric + bending) and asymmetric ones.

The fluxes of vibration energy in the combined and asymmetric CO2 modes in the harmonic oscillator approach depend only on the gradient of corresponding vibration temperature:

$$\mathbf{q}\_{12} = -\lambda\_{12}\nabla T\_{12} \quad \mathbf{q}\_3 = -\lambda\_3 \nabla T\_3 \tag{17}$$

The thermal conductivity coefficients are expressed in terms of bracket integrals

$$\boldsymbol{k}\lambda = k[\mathbf{A}, \mathbf{A}]/3, \ \lambda\_{12} = k\left[\mathbf{A}^{(12)}, \mathbf{A}^{(12)}\right]/3, \lambda\_3 = k\left[\mathbf{A}^{(3)}, \mathbf{A}^{(3)}\right]/3 \tag{18}$$

The algorithms of transport coefficients calculation are similar for various multi-temperature models and consist of the following steps:


For example, let us consider the bulk viscosity coefficient. In Navier-Stokes equations, the terms involving bulk viscosity multiplied by divergence of velocity and can play a significant role in flow fields with substantial dilatation. The bulk viscosity coefficient was discussed theoretically in the literature [18] for weak and strong non-equilibrium conditions but up to now it was not evaluated in real gas flows.

The experiments and kinetic theory show that bulk viscosity can significantly influence shock wave structure for polyatomic molecules. In polyatomic gases of the deviation from local equilibrium effects itself as bulk viscosity. From the Chapman-Enskog's theory, it can be proved that for any perfect monoatomic gas, the coefficient of bulk viscosity is equal to zero. Bulk viscosity results from contributions from the several internal degrees of freedom of the gas.

The bulk viscosity coefficient is defined by rotational energy transitions of all molecular species and VT vibration energy transfer in CO and O2 molecules and can be written as follows

$$
\zeta = \zeta\_r + \zeta\_{v, \text{CO}} + \zeta\_{v, O\_2} \tag{19}
$$

For harmonic oscillators, rapid inelastic VV and VV<sup>0</sup> <sup>1</sup>�<sup>2</sup> exchanges occur to be resonant and therefore do not give contribution to the coefficient ζ.

The bulk viscosity coefficient can be obtained in the form [18]:

$$\mathcal{L} = -kT\Big(\mathbf{x}\_1 f\_{1,10} + \mathbf{x}\_2 f\_{2,10} + \mathbf{x}\_3 f\_{1,10} + \mathbf{x}\_4 f\_{4,1} + \mathbf{x}\_5 f\_{5,1}\Big)\_{\prime} \tag{20}$$

where k is the Boltzmann constant, xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O).

Coefficients fi,mn are the solutions of the system

q ¼ �λ<sup>0</sup>

48 Advances in Some Hypersonic Vehicles Technologies

models and consist of the following steps:

bracket integrals as coefficients.

1. Unknown functions Acij, Að Þ <sup>12</sup>

tion function.

ing) and asymmetric ones.

∇T � λ12∇T<sup>12</sup> � λ3∇T<sup>3</sup> � p

The thermal conductivity coefficients are expressed in terms of bracket integrals

cij , Að Þ <sup>c</sup>;<sup>1</sup>

<sup>λ</sup> <sup>¼</sup> <sup>k</sup>½ � <sup>A</sup>; <sup>A</sup> <sup>=</sup>3, <sup>λ</sup><sup>12</sup> <sup>¼</sup> <sup>k</sup> <sup>A</sup>ð Þ <sup>12</sup> ; <sup>A</sup>ð Þ <sup>12</sup> h i

cij , Að Þ<sup>3</sup>

2. Transport coefficients are expressed in terms of expansion coefficients.

conductivity coefficient of all degrees of freedom which deviate weakly from local thermal equilibrium. They include the translational and rotational modes as well as CO and O2 vibration degrees of freedom. Thus, the coefficient λ<sup>v</sup> = λv,CO + λr,O2. Coefficients λ12, λ<sup>3</sup> correspond to the thermal conductivity of strongly non-equilibrium modes: combined (symmetric + bend-

The fluxes of vibration energy in the combined and asymmetric CO2 modes in the harmonic oscillator approach depend only on the gradient of corresponding vibration temperature:

The algorithms of transport coefficients calculation are similar for various multi-temperature

3. Integral equations are reduced to the linear systems of algebraic equations involving

4. Bracket integrals are simplified on the basis of some assumptions about cross sections of

5. Transport coefficients are found as solutions of transport linear systems using some numerical algorithms (for instance, the Gauss method or new iterative procedures). For example, let us consider the bulk viscosity coefficient. In Navier-Stokes equations, the terms involving bulk viscosity multiplied by divergence of velocity and can play a significant role in flow fields with substantial dilatation. The bulk viscosity coefficient was discussed

calculated for particular models of inter-molecular interaction potentials. In the present study, the Lennard-Jones potential is used for low and moderate temperatures whereas in

rapid processes; finally they are expressed in terms of the standard Ωð Þ <sup>l</sup>;<sup>r</sup>

and relaxation times which can be measured experimentally. The Ωð Þ <sup>l</sup>;<sup>r</sup>

the high-temperature interval the repulsive potential is applied.

and Waldmann-Trubenbacher polynomials; the trial functions are introduced accordingly to the right hand sides of integral equations for the first-order correction to the distribu-

Here p is the pressure, hi is the specific enthalpy of species i, λ

X 5

<sup>d</sup><sup>i</sup> <sup>þ</sup><sup>X</sup> 5

0

q<sup>12</sup> ¼ �λ12∇T12, q<sup>3</sup> ¼ �λ3∇T<sup>3</sup> (17)

<sup>=</sup>3, <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>k</sup> <sup>A</sup>ð Þ<sup>3</sup> ; <sup>A</sup>ð Þ<sup>3</sup> h i

cij, Bcij, Fcij, Fcij are expanded in the series of Sonine

i¼1

rihiV<sup>i</sup> (16)

=3 (18)

cd is the integrals

cd is the integrals are

¼ λtr þ λ<sup>r</sup> þ λν is the thermal

i¼1 DTi

$$
\mathbf{B} \times \mathbf{f} = \mathbf{s}.\tag{21}
$$

Here matrix B is composed of the bracket integrals β ij mm0 nn<sup>0</sup> :

B ¼ x1ctr x2ctr x3ctr x4ctr x5ctr x1crot, <sup>1</sup> x2cint, <sup>2</sup> x3cint,<sup>3</sup> β<sup>2</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup> <sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>3</sup> <sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>4</sup> <sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>5</sup> <sup>11</sup> <sup>β</sup><sup>2</sup>�<sup>1</sup> <sup>1001</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup> <sup>1001</sup> <sup>β</sup><sup>2</sup>�<sup>3</sup> 1001 β<sup>3</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>2</sup> <sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>3</sup> <sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>4</sup> <sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>5</sup> <sup>11</sup> <sup>β</sup><sup>3</sup>�<sup>1</sup> <sup>1001</sup> <sup>β</sup><sup>3</sup>�<sup>2</sup> <sup>1001</sup> <sup>β</sup><sup>3</sup>�<sup>3</sup> 1001 β<sup>4</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>2</sup> <sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>3</sup> <sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>4</sup> <sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>5</sup> <sup>11</sup> <sup>β</sup><sup>4</sup>�<sup>1</sup> <sup>1001</sup> <sup>β</sup><sup>4</sup>�<sup>2</sup> <sup>1001</sup> <sup>β</sup><sup>4</sup>�<sup>3</sup> 1001 β<sup>5</sup>�<sup>1</sup> <sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>2</sup> <sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>3</sup> <sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>4</sup> <sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>5</sup> <sup>11</sup> <sup>β</sup><sup>5</sup>�<sup>1</sup> <sup>1001</sup> <sup>β</sup><sup>5</sup>�<sup>2</sup> <sup>1001</sup> <sup>β</sup><sup>5</sup>�<sup>3</sup> 1001 β<sup>1</sup>�<sup>1</sup> <sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>2</sup> <sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>3</sup> <sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>4</sup> <sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>5</sup> <sup>0110</sup> <sup>β</sup><sup>1</sup>�<sup>1</sup> <sup>0011</sup> 0 0 β<sup>2</sup>�<sup>1</sup> <sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup> <sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>3</sup> <sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>4</sup> <sup>0110</sup> <sup>β</sup><sup>2</sup>�<sup>5</sup> <sup>0110</sup> <sup>0</sup> <sup>β</sup><sup>2</sup>�<sup>2</sup> <sup>0011</sup> 0 β<sup>3</sup>�<sup>1</sup> <sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>2</sup> <sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>3</sup> <sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>4</sup> <sup>0110</sup> <sup>β</sup><sup>3</sup>�<sup>5</sup> <sup>0110</sup> 0 0 <sup>β</sup><sup>3</sup>�<sup>3</sup> 0011 0 BBBBBBBBBBBBBBBBBBB@ 1 CCCCCCCCCCCCCCCCCCCA f ¼ f <sup>1</sup>,<sup>10</sup> f <sup>2</sup>,<sup>10</sup> f <sup>3</sup>,<sup>10</sup> f <sup>4</sup>,<sup>1</sup> f <sup>5</sup>,<sup>1</sup> f <sup>1</sup>,<sup>01</sup> f <sup>2</sup>,<sup>01</sup> f <sup>3</sup>,<sup>01</sup> 0 BBBBBBBBBBBBBBBBBB@ 1 CCCCCCCCCCCCCCCCCCA s ¼ 0 �x2ð Þ cu � ctr =cu �x3ð Þ cu � ctr =cu �x4ð Þ cu � ctr =cu �x5ð Þ cu � ctr =cu x1crot,1=cu x2cint, <sup>2</sup>=cu x3cint, <sup>3</sup>=cu 0 BBBBBBBBBBBBBBBBB@ 1 CCCCCCCCCCCCCCCCCA (22)

Matrix B, vectors f and s are given above. Here сtr, crot, сint are specific heats of translation, rotational, internal degrees of freedom and cu is the total specific heat. The bracket integrals β ij mm<sup>0</sup> nn<sup>0</sup> required for the evaluation of the bulk viscosity coefficient are determined by the energy variation in inelastic processes. The system (22) in this form has a unique solution.

The rates of vibration energy transitions are expressed in terms of corresponding relaxation times. The rate coefficients for non-equilibrium CO2 dissociation were calculated using the expressions proposed in Ref. [8] as an extension of the Treanor-Marrone's model [20] for threeatomic molecules. For the recombination rate coefficients, the detailed balance principle is used. For the rate coefficients of exchange reactions and dissociation of diatomic molecules, the Arrhenius formulas are applied. The vibration relaxation of molecules time of СО<sup>2</sup> molecules is calculated under the usual formulas by approximation of theoretical and experimental data for particles of different types [10].
