4. Results the calculation of transport coefficients

For calculation of transport coefficient of gas mixture, ones need the information how interaction between themselves and each pair of species of gas mixture is going on. It is shown that under considered conditions, the transport coefficients are defined by the collisions with translational and rotational energy changing whereas the reaction rate coefficients depend on the cross sections of slow energy transitions, dissociation, and exchange reactions.

The algorithm for the calculation of transport coefficients has been realized for the 5-temperature model as a program module in a form of Fortran 90 code. The code calls several independent modules: CONSTANT: common constants and variables definition; SPECIFIC HEAT: calculates vibration energy levels, non-equilibrium vibration partition functions, vibration specific heat capacities; OMEGA INTEGRALS: calculates integrals and their ratios using the Lennard-Jones and the Born-Meyer potentials for moderate and high-temperature ranges; BRACKET INTE-GRALS: calculates bracket integrals in the transport linear systems; INVERS: solves systems of linear algebraic equations using the Gauss method.

influence of account of volume viscosity in the equations of flow leads to increase of a heat flux

Figure 3. Coefficient of shear <sup>μ</sup> and volume <sup>ζ</sup> viscosity along a stagnation line. T1: <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 � <sup>10</sup>�<sup>4</sup> kg/m<sup>3</sup>

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

For conditions of a MSRO vehicle, flow values of multi-component diffusion coefficient D\*

have been obtained with help of diffusion flux definition through thermodynamic forces [13]. In Figure 5, the distribution of self-diffusion coefficients along a stagnation line for the some

the main diagonal in the most part of a shock layer surpass values of non-diagonal elements. It is testifies to legitimacy of application of the Fick's law for calculation of diffusion flux. However, near surface of a body and in the field of a shock wave the values of elements (e.g.

Diffusion flux of everyone components depends on own gradient of concentration components and coefficient of self-diffusion. In Figure 7 confirmation of this fact are presented and the diffusion velocities for component СО<sup>2</sup> and СО obtained by using the "exact" expression

<sup>1</sup>j, j 6¼ 1) can be same order. It means that in these zones to use Fick's law it is

ij

,

51

ij (i 6¼ j) of a matrix diffusion, and in Figure 6

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ij (i 6¼ j). It is evident that values of the elements belonging to

up to 10%. The similar tendency takes place and for other conditions of a flow. In Figure 4, the similar data are presented for coefficient of heat conductivity.

.

component of a gas mixture is presented.

shows non-diagonal elements D\*

T2: <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 � <sup>10</sup>�<sup>5</sup> kg/m<sup>3</sup>

D\*

<sup>11</sup> and D\*

incorrect.

The values corresponding to diagonal elements D\*

Eqs. (6)–(11) with kinetic schemes for transport coefficients described above are solved numerically for a flow in a viscous shock layer near the blunt body imitating the form of the spacecraft MSRO (Mars Sample Return Orbiter) for the conditions typical for the re-entering regime.

In Figure 3, coefficients of shear and volume viscosity along a stagnation line are presented. Calculations are obtained for ideal catalytic wall having the constant temperature T = 1500 K and conditions of a flow of MSRO vehicle <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 � <sup>10</sup>�<sup>4</sup> kg/m3 (curves T1) and <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 � <sup>10</sup>�<sup>5</sup> kg/m3 (curves <sup>Т</sup>2). Near surface of a body, the value of volume viscosity about value of shear, and in a shock layer surpasses it approximately in two times. It is established that the mechanism of non-equilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> does not affect on value of volume viscosity. Estimations of influence of volume viscosity on parameters of flow and a heat transfer to a surface of a space vehicle in an atmosphere of Mars are carried out. For the specified parameters of a flow, the

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β ij mm<sup>0</sup>

regime.

data for particles of different types [10].

50 Advances in Some Hypersonic Vehicles Technologies

4. Results the calculation of transport coefficients

linear algebraic equations using the Gauss method.

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

For calculation of transport coefficient of gas mixture, ones need the information how interaction between themselves and each pair of species of gas mixture is going on. It is shown that under considered conditions, the transport coefficients are defined by the collisions with translational and rotational energy changing whereas the reaction rate coefficients depend on

The algorithm for the calculation of transport coefficients has been realized for the 5-temperature model as a program module in a form of Fortran 90 code. The code calls several independent modules: CONSTANT: common constants and variables definition; SPECIFIC HEAT: calculates vibration energy levels, non-equilibrium vibration partition functions, vibration specific heat capacities; OMEGA INTEGRALS: calculates integrals and their ratios using the Lennard-Jones and the Born-Meyer potentials for moderate and high-temperature ranges; BRACKET INTE-GRALS: calculates bracket integrals in the transport linear systems; INVERS: solves systems of

Eqs. (6)–(11) with kinetic schemes for transport coefficients described above are solved numerically for a flow in a viscous shock layer near the blunt body imitating the form of the spacecraft MSRO (Mars Sample Return Orbiter) for the conditions typical for the re-entering

In Figure 3, coefficients of shear and volume viscosity along a stagnation line are presented. Calculations are obtained for ideal catalytic wall having the constant temperature T = 1500 K and conditions of a flow of MSRO vehicle <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 � <sup>10</sup>�<sup>4</sup> kg/m3 (curves T1) and <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 � <sup>10</sup>�<sup>5</sup> kg/m3 (curves <sup>Т</sup>2). Near surface of a body, the value of volume viscosity about value of shear, and in a shock layer surpasses it approximately in two times. It is established that the mechanism of non-equilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> does not affect on value of volume viscosity. Estimations of influence of volume viscosity on parameters of flow and a heat transfer to a surface of a space vehicle in an atmosphere of Mars are carried out. For the specified parameters of a flow, the

the cross sections of slow energy transitions, dissociation, and exchange reactions.

Figure 3. Coefficient of shear <sup>μ</sup> and volume <sup>ζ</sup> viscosity along a stagnation line. T1: <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 � <sup>10</sup>�<sup>4</sup> kg/m<sup>3</sup> , T2: <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 � <sup>10</sup>�<sup>5</sup> kg/m<sup>3</sup> .

influence of account of volume viscosity in the equations of flow leads to increase of a heat flux up to 10%. The similar tendency takes place and for other conditions of a flow.

In Figure 4, the similar data are presented for coefficient of heat conductivity.

For conditions of a MSRO vehicle, flow values of multi-component diffusion coefficient D\* ij have been obtained with help of diffusion flux definition through thermodynamic forces [13].

In Figure 5, the distribution of self-diffusion coefficients along a stagnation line for the some component of a gas mixture is presented.

The values corresponding to diagonal elements D\* ij (i 6¼ j) of a matrix diffusion, and in Figure 6 shows non-diagonal elements D\* ij (i 6¼ j). It is evident that values of the elements belonging to the main diagonal in the most part of a shock layer surpass values of non-diagonal elements. It is testifies to legitimacy of application of the Fick's law for calculation of diffusion flux. However, near surface of a body and in the field of a shock wave the values of elements (e.g. D\* <sup>11</sup> and D\* <sup>1</sup>j, j 6¼ 1) can be same order. It means that in these zones to use Fick's law it is incorrect.

Diffusion flux of everyone components depends on own gradient of concentration components and coefficient of self-diffusion. In Figure 7 confirmation of this fact are presented and the diffusion velocities for component СО<sup>2</sup> and СО obtained by using the "exact" expression

Figure 4. Coefficient of heat conductivity along a stagnation line for MSRO vehicle. 1: model of [8], 2: model of [21, 22], 3: model of [23, 24]. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 � <sup>10</sup>�<sup>5</sup> kg/m<sup>3</sup> .

$$V\_i = -\sum\_j D\_{ii}^\* \nabla \mathbf{x}\_j \tag{23}$$

and relation

$$V\_i = -D\_{\vec{\mu}}^\* \nabla \mathbf{x}\_i. \tag{24}$$

In the second case, without taking into account the second term in the right part of above expression. The data resulted in Figure 8 confirm that influence of thermo-diffusion effect is

ii along stagnation line. 1: D\*

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

<sup>11</sup> (CO2–CO2), 2: D\*

<sup>22</sup> (CO–CO), 3: D\*

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53

<sup>55</sup> (O–O).

In Figure 9, the obtained values of diffusion velocities for various component of a mixture with taking in account and without of thermo-diffusion are presented. It is evident that these values basically are much lower than corresponding parameters of mass diffusion. It allows suppose that thermo-diffusion influences are negligible. However for full clearness, it is necessary to

In Figure 10, comparison of effective diffusion coefficients Di for a component of mixture СО<sup>2</sup> and СО is determined in two ways—with the help of binary diffusion coefficients Dij and multi-component coefficients Dij is shown. The data in Figure 10 are presented along a stagnation line across a shock layer for conditions of a flow of the vehicle: V<sup>∞</sup> = 5292 m/s, <sup>r</sup><sup>∞</sup> = 2.5 <sup>10</sup><sup>4</sup> kg/m3 in a case of ideal catalytic surfaces. It is shown that the effective diffusion

small.

take into account change of temperature.

Figure 5. The coefficient of self-diffusion D\*

<sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

coefficient determined with the help of two methods is very close.

.

From the data of Figure 7, it is obtained that for considered flow conditions, it is necessary to take into account and non-diagonal elements of diffusion matrix. Influence of thermo-diffusion and pressure diffusion on parameters of a flow was considered also. As pressure across a shock layer is equal practically constant then process of pressure diffusion can be not taken into account. The temperature in a shock layer changes essentially.

The temperature gradients are observed near a body surface and near a shock wave. In Figure 8, distribution along a stagnation line of sizes of thermo-diffusion coefficient DT for separate component of a gas mixture is shown. In the first case, diffusion velocity speed was calculated under the formula

$$V\_i = -\sum\_j D\_{ij}^\* \nabla \mathbf{x}\_j - D\_{T\_i} \nabla \ln T \tag{25}$$

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Figure 5. The coefficient of self-diffusion D\* ii along stagnation line. 1: D\* <sup>11</sup> (CO2–CO2), 2: D\* <sup>22</sup> (CO–CO), 3: D\* <sup>55</sup> (O–O). <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

Vi ¼ �<sup>X</sup> j D∗

.

Figure 4. Coefficient of heat conductivity along a stagnation line for MSRO vehicle. 1: model of [8], 2: model of [21, 22], 3:

Vi ¼ �D<sup>∗</sup>

into account. The temperature in a shock layer changes essentially.

Vi ¼ �<sup>X</sup> j D∗

From the data of Figure 7, it is obtained that for considered flow conditions, it is necessary to take into account and non-diagonal elements of diffusion matrix. Influence of thermo-diffusion and pressure diffusion on parameters of a flow was considered also. As pressure across a shock layer is equal practically constant then process of pressure diffusion can be not taken

The temperature gradients are observed near a body surface and near a shock wave. In Figure 8, distribution along a stagnation line of sizes of thermo-diffusion coefficient DT for separate component of a gas mixture is shown. In the first case, diffusion velocity speed was

ij∇xj � DTi

and relation

calculated under the formula

model of [23, 24]. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 � <sup>10</sup>�<sup>5</sup> kg/m<sup>3</sup>

52 Advances in Some Hypersonic Vehicles Technologies

ii∇xj (23)

ii∇xi: (24)

∇ln T (25)

In the second case, without taking into account the second term in the right part of above expression. The data resulted in Figure 8 confirm that influence of thermo-diffusion effect is small.

In Figure 9, the obtained values of diffusion velocities for various component of a mixture with taking in account and without of thermo-diffusion are presented. It is evident that these values basically are much lower than corresponding parameters of mass diffusion. It allows suppose that thermo-diffusion influences are negligible. However for full clearness, it is necessary to take into account change of temperature.

In Figure 10, comparison of effective diffusion coefficients Di for a component of mixture СО<sup>2</sup> and СО is determined in two ways—with the help of binary diffusion coefficients Dij and multi-component coefficients Dij is shown. The data in Figure 10 are presented along a stagnation line across a shock layer for conditions of a flow of the vehicle: V<sup>∞</sup> = 5292 m/s, <sup>r</sup><sup>∞</sup> = 2.5 <sup>10</sup><sup>4</sup> kg/m3 in a case of ideal catalytic surfaces. It is shown that the effective diffusion coefficient determined with the help of two methods is very close.

Figure 6. Coefficient self-diffusion D\* ij (<sup>i</sup> 6¼ <sup>j</sup>) along stagnation line, <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 � <sup>10</sup>�<sup>4</sup> kg/m<sup>3</sup> .

The Schmidt number is characterized the ratio of processes of momentum and mass transfer. For multi-component gas mixtures Schmidt's, Lewis's numbers depend on temperature and species fraction. For multi-component gas mixtures, Schmidt's numbers are defined for every pair of gas mixture. In practice during numerical calculations of chemically non-equilibrium flow, Schmidt's number are chosen be equal to constant. Sometimes to all components of a mixture Schmidt's number is used as identical. In this connection, it is important to estimate the influence of this supposition on the received results. Let us remind the definition of

Figure 8. Distribution of thermo-diffusion coefficient DT along the stagnation line. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 � <sup>10</sup>�<sup>4</sup> kg/m<sup>3</sup>

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

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The Schmidt's number distribution along the stagnation line are shown in Figure 11.

Distribution of Lewis numbers along the stagnation line (Le = Pr/Sc, Pr is the Prandtl's number) for one of variants resulted in Figure 12. We shall notice, that near the surface of the vehicle (n = 0), Lewis's number considerably differs from unit that testifies discrepancy of mass

ij , C3 <sup>¼</sup> <sup>8</sup>:256<sup>∗</sup>10�<sup>7</sup>

T<sup>3</sup>=<sup>2</sup>

, Dij <sup>¼</sup> <sup>m</sup><sup>2</sup>=c, p½ �¼ atm,

.

55

Schmidt's number Sij <sup>¼</sup> <sup>μ</sup>=rDij, Dij <sup>¼</sup> C3C4=pQ<sup>1</sup>, <sup>1</sup>

velocities due to heat conductivity and diffusion in this area.

Q<sup>1</sup>, <sup>1</sup> ij <sup>¼</sup> <sup>A</sup>02.

Figure 7. Diffusion velocity along the stagnation line: 1: results with taking into account of all diffusion coefficients (formula (24)); 2: results with taking into account coefficients self-diffusion (formula (25)); (а) red line—СО<sup>2</sup> component; (b) black line—СО component. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 � <sup>10</sup>�<sup>4</sup> kg/m3 .

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Figure 8. Distribution of thermo-diffusion coefficient DT along the stagnation line. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 � <sup>10</sup>�<sup>4</sup> kg/m<sup>3</sup> .

Figure 6. Coefficient self-diffusion D\*

54 Advances in Some Hypersonic Vehicles Technologies

ij (<sup>i</sup> 6¼ <sup>j</sup>) along stagnation line, <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 � <sup>10</sup>�<sup>4</sup> kg/m<sup>3</sup>

Figure 7. Diffusion velocity along the stagnation line: 1: results with taking into account of all diffusion coefficients (formula (24)); 2: results with taking into account coefficients self-diffusion (formula (25)); (а) red line—СО<sup>2</sup> component;

.

(b) black line—СО component. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 � <sup>10</sup>�<sup>4</sup> kg/m3

.

The Schmidt number is characterized the ratio of processes of momentum and mass transfer. For multi-component gas mixtures Schmidt's, Lewis's numbers depend on temperature and species fraction. For multi-component gas mixtures, Schmidt's numbers are defined for every pair of gas mixture. In practice during numerical calculations of chemically non-equilibrium flow, Schmidt's number are chosen be equal to constant. Sometimes to all components of a mixture Schmidt's number is used as identical. In this connection, it is important to estimate the influence of this supposition on the received results. Let us remind the definition of Schmidt's number Sij <sup>¼</sup> <sup>μ</sup>=rDij, Dij <sup>¼</sup> C3C4=pQ<sup>1</sup>, <sup>1</sup> ij , C3 <sup>¼</sup> <sup>8</sup>:256<sup>∗</sup> 10�<sup>7</sup> T<sup>3</sup>=<sup>2</sup> , Dij <sup>¼</sup> <sup>m</sup><sup>2</sup>=c, p½ �¼ atm, Q<sup>1</sup>, <sup>1</sup> ij <sup>¼</sup> <sup>A</sup>02.

The Schmidt's number distribution along the stagnation line are shown in Figure 11.

Distribution of Lewis numbers along the stagnation line (Le = Pr/Sc, Pr is the Prandtl's number) for one of variants resulted in Figure 12. We shall notice, that near the surface of the vehicle (n = 0), Lewis's number considerably differs from unit that testifies discrepancy of mass velocities due to heat conductivity and diffusion in this area.

Figure 9. Distribution of diffusion velocity of different mixture component along the stagnation line with influence and without thermo-diffusion. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

Figure 10. Effective diffusion coefficient along the stagnation line for СО<sup>2</sup> and СО, is obtained by two methods: <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

5. Boundary conditions

<sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

.

<sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m3

The solution must be found out in the region restricted by: (a) body surface; (b) inflow: surface of external flow, where the conditions are known—V∞, p∞, r∞, ci∞; (c) axis of symmetry:

Figure 12. Values of Lewis's number along the stagnation line for different component of mixture, ideal catalytic surface.

Figure 11. Values of Schmidt number along the stagnation line, ideal catalytic surface, (а) V<sup>∞</sup> = 5223 m/s,

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

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57

.

, (b) <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 <sup>10</sup><sup>5</sup> kg/m<sup>3</sup>

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Figure 11. Values of Schmidt number along the stagnation line, ideal catalytic surface, (а) V<sup>∞</sup> = 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m3 , (b) <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.141 <sup>10</sup><sup>5</sup> kg/m<sup>3</sup> .

Figure 12. Values of Lewis's number along the stagnation line for different component of mixture, ideal catalytic surface. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

#### 5. Boundary conditions

Figure 10. Effective diffusion coefficient along the stagnation line for СО<sup>2</sup> and СО, is obtained by two methods:

Figure 9. Distribution of diffusion velocity of different mixture component along the stagnation line with influence and

.

<sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

.

without thermo-diffusion. <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.933 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

56 Advances in Some Hypersonic Vehicles Technologies

The solution must be found out in the region restricted by: (a) body surface; (b) inflow: surface of external flow, where the conditions are known—V∞, p∞, r∞, ci∞; (c) axis of symmetry: symmetrical or anti-symmetrical reflection depending on functions; (d) outflow: some surface in down part of flow, where usually ones use so called "soft" boundary extrapolation conditions.

The boundary conditions at the thermo-chemically stable surface include no slip conditions for component of velocities. Scott's wall slip conditions applied to velocity, species mass fractions for modeling flow fields in high altitude [25].

Appropriate boundary conditions at thermally stable surface include conditions for the diffusive fluxes of element at the wall, mass balance equations for the reaction product. When the temperature of the wall is done (T ¼ T∞), then boundary conditions at the surface include L conditions for the elemental diffusive fluxes at the wall

$$J\_{\slash w}^\* = 0, \langle j = 1, \ldots, L \rangle \tag{26}$$

JO<sup>2</sup> ¼ �JO, JCO<sup>2</sup> ¼ �JCO � JO (30)

Kwi (31)

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59

hkJ<sup>k</sup> (32)

Jqw ¼ �λ∂T=∂y (33)

<sup>w</sup>, (34)

Above boundary condition for the mass concentration on the body surface can be expended <sup>r</sup>Dij∂ci=∂<sup>y</sup> <sup>¼</sup> ð Þ <sup>r</sup>wciw <sup>ν</sup>

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

If surface posses by catalytic properties then surface provoke to recombine the atoms in molecules. On absolutely catalytic surface concentration of atoms, it is equal to zero: cA ¼ 0.

An expression for the heat flux to the surface may be deduced (for simplicity, relaxation is

The heat flux depends essentially on the boundary conditions for the species concentrations at

For reusable vehicles, the catalyze quality of heat-protective coverings become very important. The heat flux increases as the diffusion contributes to the maximum total heat flux. Since homogenous recombination and neutralization occur slowly at high altitudes, exothermic heterogeneous processes at the body surface become crucial to the magnitude of the convection heat flux.

Next equation may be used to find the temperature with the boundary condition of heat

<sup>J</sup>qw <sup>¼</sup> εσT<sup>4</sup>

Rotational temperatures of molecules are equal to the translational temperature of heavy atomic particles due to a fast translational-rotational energy exchange requiring only several collisions to establish the Boltzmann distribution. In the free stream, СО<sup>2</sup> molecules have almost zero vibration energy, therefore, for them in a shock layer, there is an area with nonequilibrium vibration. Vibration temperatures of all the electronically exited molecules are

Hypersonic flows over real space configurations represent a substantial problem from the point of view of the development of new and more effective mathematical models, numerical

balance at the wall between flux to surface and reflection. The energy equation yields

where ε is a measure of the surface blackness and σ is the Stefan-Boltzmann constant.

considered to be equal to the translation temperature of heavy atomic particles.

6. Mathematical models and numerical methods

algorithms and the use of modern computer systems.

N

k¼Lþ1

<sup>J</sup>qw ¼ �λ∂T=∂<sup>y</sup> � <sup>X</sup>

For absolutely non-catalytic wall—∂cA=∂y ¼ 0.

the wall. Thus at Kwi = 0, we obtain

considered to be already completed at the wall), that is.

and the mass balance equations for the reaction products

$$J\_{\rm inv} = (\rho\_w c\_{\rm inv})^\text{\prime} K\_{\rm wi} \left( i = L + 1, \ldots, N \right), \tag{27}$$

where Kwi ¼ γð Þ kTw=2πmi <sup>1</sup>=<sup>2</sup> is the effective catalytic constant. Here recombination is qualitatively characterized by an effective probability 0 < γ < 1 or by rate constant Kwi(Kwi = 0 for an non-catalytic wall, Kwi= ∞ for a perfect catalytic wall). Value ν is the order of the reaction, mi is the atomic mass. The cases of γ = 1 and 0 correspond to absolutely catalytic and absolutely non-catalytic materials. The catalytic property of the wall has an important effect on the heat transfer of reusable vehicles over the considerable interval of the trajectory. The structure of the surface (contamination, roughness, porosity, etc.) affects the rates of the atomic adsorption and de-sorption processes.

A phenomenological model for catalytic reactions used that accounts for physical and chemical absorption, the interaction between the impinging atoms and ad-atoms (adsorbed atoms), and between the ad-atoms themselves. A model of the Rideal-Eley and Langmuir-Hinshelwood layer with ideal adsorption applied. Let us consider the heterogeneous catalytic reactions on surfaces [6]:

$$\text{O} + \text{O} \rightarrow \text{O}\_2 \bullet \text{O} + \text{CO} \rightarrow \text{CO}\_2 \tag{28}$$

For a surface with final catalytic properties, it is applicable the simplified boundary conditions with use of effective probabilities of heterogeneous recombination that are equal among themselves γ<sup>О</sup> = γСО = γw. Diffusion fluxes on a surface for a molecule СО and atoms O can be written as follows:

$$-I\_{\rm CO} = \rho k\_{\rm wCO} \varepsilon\_{\rm CO}, \quad -I\_O = \rho k\_{\rm wO} \varepsilon\_{\rm O}, \\ k\_{\rm wi} = \frac{2\gamma\_w}{2 - \gamma\_w} \sqrt{\frac{RT\_w}{2\pi m\_i}}, \\ i = \text{CO,O} \tag{29}$$

And for molecules O2 and СО<sup>2</sup>

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules http://dx.doi.org/10.5772/intechopen.71666 59

$$J\_{\rm O\_2} = -J\_{\rm O^\*} J\_{\rm CO\_2} = -J\_{\rm CO} - J\_{\rm O} \tag{30}$$

Above boundary condition for the mass concentration on the body surface can be expended

symmetrical or anti-symmetrical reflection depending on functions; (d) outflow: some surface in down part of flow, where usually ones use so called "soft" boundary extrapolation conditions. The boundary conditions at the thermo-chemically stable surface include no slip conditions for component of velocities. Scott's wall slip conditions applied to velocity, species mass fractions

Appropriate boundary conditions at thermally stable surface include conditions for the diffusive fluxes of element at the wall, mass balance equations for the reaction product. When the temperature of the wall is done (T ¼ T∞), then boundary conditions at the surface include L

tively characterized by an effective probability 0 < γ < 1 or by rate constant Kwi(Kwi = 0 for an non-catalytic wall, Kwi= ∞ for a perfect catalytic wall). Value ν is the order of the reaction, mi is the atomic mass. The cases of γ = 1 and 0 correspond to absolutely catalytic and absolutely non-catalytic materials. The catalytic property of the wall has an important effect on the heat transfer of reusable vehicles over the considerable interval of the trajectory. The structure of the surface (contamination, roughness, porosity, etc.) affects the rates of the atomic adsorption and

A phenomenological model for catalytic reactions used that accounts for physical and chemical absorption, the interaction between the impinging atoms and ad-atoms (adsorbed atoms), and between the ad-atoms themselves. A model of the Rideal-Eley and Langmuir-Hinshelwood layer with ideal adsorption applied. Let us consider the heterogeneous catalytic

For a surface with final catalytic properties, it is applicable the simplified boundary conditions with use of effective probabilities of heterogeneous recombination that are equal among themselves γ<sup>О</sup> = γСО = γw. Diffusion fluxes on a surface for a molecule СО and atoms O can

�JCO <sup>¼</sup> <sup>r</sup>kwCOcCO, � JO <sup>¼</sup> <sup>r</sup>kwOcO, kwi <sup>¼</sup> <sup>2</sup>γ<sup>w</sup>

jw ¼ 0, jð Þ ¼ 1;…; L (26)

Kwi, ið Þ ¼ L þ 1;…; N , (27)

<sup>1</sup>=<sup>2</sup> is the effective catalytic constant. Here recombination is qualita-

О þ O ! O2, O þ CO ! CO2 (28)

ffiffiffiffiffiffiffiffiffiffiffi RTw 2πmi

, i ¼ CO, O (29)

s

2 � γ<sup>w</sup>

for modeling flow fields in high altitude [25].

58 Advances in Some Hypersonic Vehicles Technologies

where Kwi ¼ γð Þ kTw=2πmi

de-sorption processes.

reactions on surfaces [6]:

be written as follows:

And for molecules O2 and СО<sup>2</sup>

conditions for the elemental diffusive fluxes at the wall

and the mass balance equations for the reaction products

J∗

<sup>J</sup>iw <sup>¼</sup> ð Þ <sup>r</sup>wciw <sup>ν</sup>

$$
\rho D\_{i\rangle} \partial \mathbf{c}\_i / \partial \mathbf{y} = (\rho\_w \mathbf{c}\_{iw})^\prime \mathbf{K}\_{wi} \tag{31}
$$

If surface posses by catalytic properties then surface provoke to recombine the atoms in molecules. On absolutely catalytic surface concentration of atoms, it is equal to zero: cA ¼ 0. For absolutely non-catalytic wall—∂cA=∂y ¼ 0.

An expression for the heat flux to the surface may be deduced (for simplicity, relaxation is considered to be already completed at the wall), that is.

$$J\_{qw} = -\lambda \partial T / \partial y - \sum\_{k=L+1}^{N} h\_k \mathbf{J}\_k \tag{32}$$

The heat flux depends essentially on the boundary conditions for the species concentrations at the wall. Thus at Kwi = 0, we obtain

$$J\_{qw} = -\lambda \mathfrak{d}T / \mathfrak{d}y\tag{33}$$

For reusable vehicles, the catalyze quality of heat-protective coverings become very important. The heat flux increases as the diffusion contributes to the maximum total heat flux. Since homogenous recombination and neutralization occur slowly at high altitudes, exothermic heterogeneous processes at the body surface become crucial to the magnitude of the convection heat flux.

Next equation may be used to find the temperature with the boundary condition of heat balance at the wall between flux to surface and reflection. The energy equation yields

$$J\_{qw} = \varepsilon \sigma T\_{w'}^{l} \tag{34}$$

where ε is a measure of the surface blackness and σ is the Stefan-Boltzmann constant.

Rotational temperatures of molecules are equal to the translational temperature of heavy atomic particles due to a fast translational-rotational energy exchange requiring only several collisions to establish the Boltzmann distribution. In the free stream, СО<sup>2</sup> molecules have almost zero vibration energy, therefore, for them in a shock layer, there is an area with nonequilibrium vibration. Vibration temperatures of all the electronically exited molecules are considered to be equal to the translation temperature of heavy atomic particles.

#### 6. Mathematical models and numerical methods

Hypersonic flows over real space configurations represent a substantial problem from the point of view of the development of new and more effective mathematical models, numerical algorithms and the use of modern computer systems.

During the past decade, a large number of computational codes have been developed that differ in the grid generation methods and numerical algorithms used. For numerical simulation of external flow fields, past real form bodies are necessary to construct the geometry, to design a discrete set-grid, to provide the mathematical model of the initial value problem, to approximate the governing equation by numerical ones, to design a computational algorithm, to realize the flow field, to establish a feed-back of obtained results with experiment, analytical and benchmark problems, and so on.

were studied. Determining process at such velocities is a process of dissociation. Up to 75% of

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

The region where non-equilibrium physical and chemical processes realized is a significant part from all considered regions. Velocity of physical and chemical processes, as a rule, grows together with density of gas. As the density of an atmosphere of Mars is much less than in atmosphere of the Earth, the equilibrium flows for bodies of the moderate sizes are observed at smaller altitude: Н <10–20 km—for an atmosphere of Mars, Н ≤ 30 km—for an atmosphere of

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

At a supersonic flow, the main features of reacting gas mixture can be evidently shown by change of flow parameters across shock layer. The distribution of pressure, velocities in a shock layer depends on physical and chemical processes weakly. The pressure with high

teristic value of gas compression in the shock layer equal the ratio of density in an external flow and density behind a direct shock wave. For flow parameters of MARS EXPRESS vehicles presented in Table 1, the pressure in a stagnation point equals to values 0.95–0.96 of a high-

ratio γ = 1.4 at the given velocities, the pressure in a stagnation point takes ~0.92 from a high-

Main results are shown: (1) in shock layer across of stagnation line; (2) along of surface body for heat transfer; and (3) in shock layer along body. We used the orthogonal system of coordinates (ξ, ζ). One coordinate ξ directs from a forward stagnation point along a streamline

The change of specific heat capacity ratio γ ¼ сp=cv (с<sup>p</sup> is the specific heat capacity at constant

Н, km V∞, m/s r∞, kg/m<sup>3</sup> T∞, K Re<sup>∞</sup> P0/(r∞V<sup>∞</sup>

52.59 <sup>5923</sup> 7.61 � <sup>10</sup>�<sup>5</sup> <sup>140</sup> 1.7 � <sup>10</sup><sup>4</sup> 0.96 43.01 <sup>5292</sup> 2.51 � <sup>10</sup>�<sup>4</sup> <sup>149</sup> 5.0 � <sup>10</sup><sup>4</sup> 0.96 36.16 <sup>4259</sup> 5.58 � <sup>10</sup>�<sup>4</sup> <sup>158</sup> 9.1 � <sup>10</sup><sup>4</sup> 0.95 32.42 <sup>3433</sup> 8.45 � <sup>10</sup>�<sup>4</sup> <sup>163</sup> 1.1 � <sup>10</sup><sup>5</sup> 0.96

pressure and cv is the specific heat capacity at constant volume) is shown in Figure 13.

<sup>∞</sup>ð Þ 1 � 0:5 � k in a stagnation point [29]. Here value k ¼ r∞=r<sup>s</sup> is the charac-

. We shall notice that for the perfect gas with a parameter of an adiabatic

<sup>∞</sup>ð Þ 1 � k behind a direct shock

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

61

2 )

full gas flow, energy can be spent on it [29].

7.1. Some features of a reacting gas mixtures flow

2

Table 1. Trajectory parameters of MARS-EXPRESS.

degree of accuracy is estimated in limits between values <sup>p</sup> <sup>¼</sup> <sup>r</sup>∞V<sup>2</sup>

contour along the surface. The coordinate ζ is a normal to wall.

the Earth.

molecules is necessary.

wave and <sup>p</sup> <sup>¼</sup> <sup>r</sup>∞V<sup>2</sup>

speed pressure r∞V<sup>∞</sup>

speed pressure.

As mathematical model, the Navier-Stokes equations and the various sub-models obtained in frameworks of the asymptotic analysis sub- and supersonic flow past blunted bodies in various statements and in a wide range of numbers of Reynolds are used.

Traditional asymptotic analysis of Navier-Stokes equations for different regimes of viscous compressible flow depending on small parameter 1/Re make it possible to decouple the different types of gas flows. The next methods were used: Navier-Stokes equations in socalled approximation of a viscous shock layer and full Navier-Stokes (N-S) equations. For solution of governing equations, the implicit finite-difference monotone schemes of the second order are used [15, 16]. Generalized Rankine-Hugoniot's conditions are imposed in the shock wave. Special method of high stiffness resolution of non-equilibrium phenomena is applied [16].

The Navier-Stokes equations are written in a conservative form for arbitrary coordinate system. The implicit iterative scheme is based on a variant of Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme. At high altitudes (low Reynolds numbers) where the bow shock has a finite thickness, a shock capturing approach is used. So inflow boundary conditions are specified in the free stream. At lower altitudes, a shock fitting scheme is employed with the modified Rankine-Hugoniot conditions specified at the bow shock. Besides the Navier-Stokes equations at lower altitudes, the viscous shock layer equations are also solved. This implicit scheme leads to the scalar diagonal manipulation for a case of non-reacting perfect gas flow and does not require any time-consuming matrix inversion. In more details, the numerical methods is described in [26–28] for the shock layer equations.
