7. Results of numerical investigation of hypersonic flow past space vehicle in Martian atmosphere

The investigations of a hypersonic flow past a frontal part of MSRO (Mars Sample Return Orbiter) and MARS EXPRESS vehicles descending in an atmosphere of Mars are shown below. The hypersonic Mach number means that appreciable quantity of molecules in hightemperature region began to dissociate. For an Earth atmosphere such numbers is equal to M <sup>∞</sup> ≥ 6. For an atmosphere of Mars in which main component is carbon dioxide as hypersonic numbers, the values M<sup>∞</sup> ≥ 10 are considered. The typical regimes of the entry in a Martian atmosphere are considered (Figure 2). 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/m3 , 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 [29].

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 the Earth.

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.

#### 7.1. Some features of a reacting gas mixtures flow

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

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

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

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

7. Results of numerical investigation of hypersonic flow past space

of flight of space vehicles in an atmosphere of Mars (V<sup>∞</sup> ≤ 6 km/s, r<sup>∞</sup> ≈ 10<sup>5</sup> kg/m3

The investigations of a hypersonic flow past a frontal part of MSRO (Mars Sample Return Orbiter) and MARS EXPRESS vehicles descending in an atmosphere of Mars are shown below. The hypersonic Mach number means that appreciable quantity of molecules in hightemperature region began to dissociate. For an Earth atmosphere such numbers is equal to M <sup>∞</sup> ≥ 6. For an atmosphere of Mars in which main component is carbon dioxide as hypersonic numbers, the values M<sup>∞</sup> ≥ 10 are considered. The typical regimes of the entry in a Martian atmosphere are considered (Figure 2). The conditions of a flow corresponding to the last stage

, H < 60 km)

various statements and in a wide range of numbers of Reynolds are used.

methods is described in [26–28] for the shock layer equations.

vehicle in Martian atmosphere

and benchmark problems, and so on.

60 Advances in Some Hypersonic Vehicles Technologies

is applied [16].

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 degree of accuracy is estimated in limits between values <sup>p</sup> <sup>¼</sup> <sup>r</sup>∞V<sup>2</sup> <sup>∞</sup>ð Þ 1 � k behind a direct shock wave and <sup>p</sup> <sup>¼</sup> <sup>r</sup>∞V<sup>2</sup> <sup>∞</sup>ð Þ 1 � 0:5 � k in a stagnation point [29]. Here value k ¼ r∞=r<sup>s</sup> is the characteristic 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 highspeed pressure r∞V<sup>∞</sup> 2 . We shall notice that for the perfect gas with a parameter of an adiabatic ratio γ = 1.4 at the given velocities, the pressure in a stagnation point takes ~0.92 from a highspeed pressure.

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

The change of specific heat capacity ratio γ ¼ сp=cv (с<sup>p</sup> is the specific heat capacity at constant pressure and cv is the specific heat capacity at constant volume) is shown in Figure 13.


Table 1. Trajectory parameters of MARS-EXPRESS.

Figure 13. Changing ratio of specific heat capacity for flows past MSRO.

The ratio is always greater than 1 and its value is an important indication of the atomicity of the gas. The laminar-to-turbulent transition of flow on the frontal surface proceeds at the altitude below 20 km. Thus the taking into account of the boundary layer transition does not affect on the results of heating the thermal protection. In Figure 14, distributions of shock layer temperature along a stagnation line near spherically blunted body (radius R = 1 m) under various conditions of a flow are shown. It includes the regimes from completely viscous shock layer until flow with a thin boundary layer. Parameters of flow in a shock layer is obtained in approach of a viscous shock layer by numerical computation and with help of the physical and chemical models submitted in work [10]. Thus Reynolds's number—Re<sup>∞</sup> varied (due to change of density of an external flow) from 5 103 to 1.5 <sup>10</sup><sup>5</sup> . In the shock layer, the pressure determined by the velocity and density of the external flow equal to <sup>р</sup> <sup>=</sup> 0.22 atm (Re<sup>∞</sup> = 1.5 105 ) and in the most part of a shock layer close to equilibrium value (a curve 4).

physical and chemical processes in a shock layer is essential that affects distribution of tem-

Figure 14. Distribution of temperature along stagnation line for two points MARS EXPRESS vehicle. 1: model of [30], 2:

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One of the most important problems of a hypersonic flow is the account of the real physical and chemical transformations in a shock layer. In theoretical works, the different authors used models of the chemical reactions essentially differing by reaction rate constants. Let us carry out comparison of the basic models used for calculation of chemical reaction rate constants in a high-temperature flow of carbon dioxide gas. We can estimate the influence of these models on character of a flow and heat transfer to the wall. The basic models of chemical reactions have essentially different reaction rate constants in a high-temperature flow of carbon dioxide gas. The corresponding dissociation reaction rate constants in a considered range of temperatures

In works [10, 13, 17–19], numerical research of a non-equilibrium flow of the bodies modeling the form of Martian vehicles MARS EXPRESS and MSRO with use of these models is carried out. The surface of the vehicles was considered or as ideal catalytic (the maximal velocities heterogeneous recombination a component of dissociated carbon dioxide gas), or non-catalytic (velocities of heterogeneous recombination of component is equal to zero). We shall consider

7.2. Influence of various models of chemical kinetics on a hypersonic flow past bodies

perature and does inapplicable many results of gas dynamics of the perfect gas.

can differ up to two orders in dependence on used models [30–35].

some results of numerical researches.

model of [31], 3: model of [32–34].

With reduction of pressure and also Reynolds's numbers the length of non-equilibrium region increases and at p = 0.007 atm character of flow in a shock layer becomes closer to frozen (curve 1). For comparison in Figure 14, the structure of temperature on a stagnation line of a flow without taking into account physical and chemical transformations is shown also (curve 5). This calculation is carried out under condition of laws of the perfect gas. It is visible that behind a boundary layer, the temperature leaves on "slop" and its value in some times higher than in case of a flow with chemical reactions. The estimations show that in case of chemical reactions under the given conditions up to 75% of full energy of an external flow it can be spent on dissociation molecules. As a result, the temperature in a shock layer essentially goes down and the density of gas increases. Thus the share of kinetic energy spent on realization of

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Figure 14. Distribution of temperature along stagnation line for two points MARS EXPRESS vehicle. 1: model of [30], 2: model of [31], 3: model of [32–34].

The ratio is always greater than 1 and its value is an important indication of the atomicity of the gas. The laminar-to-turbulent transition of flow on the frontal surface proceeds at the altitude below 20 km. Thus the taking into account of the boundary layer transition does not affect on the results of heating the thermal protection. In Figure 14, distributions of shock layer temperature along a stagnation line near spherically blunted body (radius R = 1 m) under various conditions of a flow are shown. It includes the regimes from completely viscous shock layer until flow with a thin boundary layer. Parameters of flow in a shock layer is obtained in approach of a viscous shock layer by numerical computation and with help of the physical and chemical models submitted in work [10]. Thus Reynolds's number—Re<sup>∞</sup> varied (due to change

determined by the velocity and density of the external flow equal to <sup>р</sup> <sup>=</sup> 0.22 atm (Re<sup>∞</sup> = 1.5 105

With reduction of pressure and also Reynolds's numbers the length of non-equilibrium region increases and at p = 0.007 atm character of flow in a shock layer becomes closer to frozen (curve 1). For comparison in Figure 14, the structure of temperature on a stagnation line of a flow without taking into account physical and chemical transformations is shown also (curve 5). This calculation is carried out under condition of laws of the perfect gas. It is visible that behind a boundary layer, the temperature leaves on "slop" and its value in some times higher than in case of a flow with chemical reactions. The estimations show that in case of chemical reactions under the given conditions up to 75% of full energy of an external flow it can be spent on dissociation molecules. As a result, the temperature in a shock layer essentially goes down and the density of gas increases. Thus the share of kinetic energy spent on realization of

. In the shock layer, the pressure

)

of density of an external flow) from 5 103 to 1.5 <sup>10</sup><sup>5</sup>

Figure 13. Changing ratio of specific heat capacity for flows past MSRO.

62 Advances in Some Hypersonic Vehicles Technologies

and in the most part of a shock layer close to equilibrium value (a curve 4).

physical and chemical processes in a shock layer is essential that affects distribution of temperature and does inapplicable many results of gas dynamics of the perfect gas.

#### 7.2. Influence of various models of chemical kinetics on a hypersonic flow past bodies

One of the most important problems of a hypersonic flow is the account of the real physical and chemical transformations in a shock layer. In theoretical works, the different authors used models of the chemical reactions essentially differing by reaction rate constants. Let us carry out comparison of the basic models used for calculation of chemical reaction rate constants in a high-temperature flow of carbon dioxide gas. We can estimate the influence of these models on character of a flow and heat transfer to the wall. The basic models of chemical reactions have essentially different reaction rate constants in a high-temperature flow of carbon dioxide gas. The corresponding dissociation reaction rate constants in a considered range of temperatures can differ up to two orders in dependence on used models [30–35].

In works [10, 13, 17–19], numerical research of a non-equilibrium flow of the bodies modeling the form of Martian vehicles MARS EXPRESS and MSRO with use of these models is carried out. The surface of the vehicles was considered or as ideal catalytic (the maximal velocities heterogeneous recombination a component of dissociated carbon dioxide gas), or non-catalytic (velocities of heterogeneous recombination of component is equal to zero). We shall consider some results of numerical researches.

As an example of the distribution of СО<sup>2</sup> concentration in Figure 15 in shock layer for two type of cone; (а) <sup>θ</sup> = 60, (b) <sup>θ</sup> = 10; <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> is shown.

In Figure 16, structures of mass concentration component СО<sup>2</sup> and СО obtained in case of use one-temperature (Tv = T) and two-temperature [36] reaction rate constants of chemical reactions. It is evident that for considered flow conditions an influence of non-equilibrium vibration on dissociation process is insignificant: structures of mass concentration СО<sup>2</sup> and СО coincide almost in all region of a shock layer. The small divergence is observed near a shock wave. From data, it is followed that process vibration non-equilibrium does not affect on parameters of flow near a body.

With use of these models, chemically non-equilibrium flow is considered and their influence on parameters of flow and heat exchange is established. The significant differences in distributions of temperature, concentration of a component of a gas mixture in a shock layer is observed at a variation of model chemical kinetics.

From the data shown in Figure 17, it is evident that for conditions of flight H = 43 km, the significant differences of values of mass concentration of carbon dioxide and an withdrawal of the shock wave from a surface is observed. For Н = 32 km, corresponding values practically coincide. This fact can be explained at an altitude H = 43 km, the mode of flow in a shock layer is far from chemical equilibrium. In this case, the parameters of flow depend on reaction rate constants of direct and reverse chemical reactions. For different models under these conditions of a flow, it differs essentially. Therefore in considered case, if you used models of Park then the dissociation reaction СО<sup>2</sup> in the disturbed region goes with much more rate than it is proposed by another two models. The chemical components in a shock layer affects on distributions of temperature and also on size of a withdrawal of a shock wave from a surface of a body.

Figure 16. Distribution of mass concentration СО<sup>2</sup> and СО along the stagnation line, 1—with, 2—without taking into

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Figure 17. Profile of mass concentration of СО<sup>2</sup> along the stagnation line for two point of trajectories MARS EXPRESS

vehicle. 1: model of [30], 2: model of [31], 3: model of [32–34].

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account the influence of vibration relaxation on process dissociation, <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>

Figure 15. Distribution of СО<sup>2</sup> concentration in shock layer for two type of cone; (а) θ = 60, (b) θ = 10; V<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> .

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As an example of the distribution of СО<sup>2</sup> concentration in Figure 15 in shock layer for two type

In Figure 16, structures of mass concentration component СО<sup>2</sup> and СО obtained in case of use one-temperature (Tv = T) and two-temperature [36] reaction rate constants of chemical reactions. It is evident that for considered flow conditions an influence of non-equilibrium vibration on dissociation process is insignificant: structures of mass concentration СО<sup>2</sup> and СО coincide almost in all region of a shock layer. The small divergence is observed near a shock wave. From data, it is followed that process vibration non-equilibrium does not affect on

With use of these models, chemically non-equilibrium flow is considered and their influence on parameters of flow and heat exchange is established. The significant differences in distributions of temperature, concentration of a component of a gas mixture in a shock layer is

From the data shown in Figure 17, it is evident that for conditions of flight H = 43 km, the significant differences of values of mass concentration of carbon dioxide and an withdrawal of the shock wave from a surface is observed. For Н = 32 km, corresponding values practically coincide. This fact can be explained at an altitude H = 43 km, the mode of flow in a shock layer is far from chemical equilibrium. In this case, the parameters of flow depend on reaction rate constants of direct and reverse chemical reactions. For different models under these conditions of a flow, it differs essentially. Therefore in considered case, if you used models of Park then the dissociation reaction СО<sup>2</sup> in the disturbed region goes with much more rate than it is proposed by another two models. The chemical components in a shock layer affects on distributions of temperature and also on size of a withdrawal of a shock wave from a surface of a body.

Figure 15. Distribution of СО<sup>2</sup> concentration in shock layer for two type of cone; (а) θ = 60, (b) θ = 10; V<sup>∞</sup> = 5223 m/s,

of cone; (а) <sup>θ</sup> = 60, (b) <sup>θ</sup> = 10; <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> is shown.

parameters of flow near a body.

64 Advances in Some Hypersonic Vehicles Technologies

<sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

.

observed at a variation of model chemical kinetics.

Figure 16. Distribution of mass concentration СО<sup>2</sup> and СО along the stagnation line, 1—with, 2—without taking into account the influence of vibration relaxation on process dissociation, <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> .

Figure 17. Profile of mass concentration of СО<sup>2</sup> along the stagnation line for two point of trajectories MARS EXPRESS vehicle. 1: model of [30], 2: model of [31], 3: model of [32–34].

With reduction of velocity flight, it corresponds with an increase of density in a shock layer and flows more close to equilibrium case. The parameters of flow and structure of a gas mixture are defined by conditions of chemical equilibrium. The reaction rate constants in equilibrium will be the same for all used models. In this connection, corresponding structures of temperature and concentration, and also value of a withdrawal of a shock wave from a surface received for different models under conditions of a flow at height Н = 32 km well enough coincide.

case non-catalytic surfaces the difference in values of the heat fluxes received for different

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In Figure 20, for two points of a trajectory of MARS EXPRESS vehicle (altitude of flight equal to Н = 43 km and Н = 32 km) the distributions of temperature obtained with the help of model

Figure 19. Profiles of mass concentration СО<sup>2</sup> (––––) and СО (- - - - ) along stagnation line MSRO vehicle, non-catalytic

Figure 20. The temperature along the stagnation line for sphere R = 1 m in Mars atmosphere. 1: pressure р = 0.007 atm,

, perfect СО<sup>2</sup> gas.

; 3: <sup>р</sup> = 0.22 atm, Re∞= 1.5 105

; 4: <sup>р</sup> = 0.22 atm, Re∞= 1.5 105

,

models can be essential up to 30%.

wall, 1: model of [30], 2: model of [31], 3: model of [32–34].

; 2: 'frozen' flow: <sup>р</sup> = 0.07 atm, Re∞= 5 <sup>10</sup><sup>4</sup>

equilibrium case; 5: <sup>р</sup> = 0.065 <sup>а</sup>tm, Re<sup>∞</sup> = 1.3 <sup>10</sup><sup>5</sup>

Re<sup>∞</sup> = 5 103

Influence of chemical models on flow parameters and heat exchange is determined in a wide range of parameters of a flow of MSRO vehicle in cases ideally catalytic and non-catalytic wall. In Figures 18 and 19, mass concentrations of component СО<sup>2</sup> and СО along stagnation line for different models and types of condition on wall are submitted. The data are resulted for conditions of flow <sup>V</sup><sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m3 that corresponds approximately to altitude of flight Н ≈ 60 km; the temperature of a surface equal to Т<sup>w</sup> = 1500 K.

Comparing the data of Figures 18 and 19, it is possible to notice that influence of catalytic surface properties affects profiles of concentration basically near to a wall. In these figures, the significant divergence in the distributions of concentration the components obtained for different models is observed. It is evident that it is greater for degree СО<sup>2</sup> dissociation in a shock layer when it used of model of Park [31], and is smaller—when the model of Kenzie-Arnold [30] used. The model of Research Institute of Mechanics (NIIMekh) of Moscow State University [32–34] gives the intermediate results. It is established that change of reaction rates practically does not influence on value of a heat flux to ideal catalytic wall of the vehicle. In a

Figure 18. Profiles of mass concentration СО<sup>2</sup> (––––) and СО (- - - - ) along stagnation line MSRO vehicle, ideal-catalytic wall, 1: model of [30], 2: model of [31], 3: model of [32–34].

case non-catalytic surfaces the difference in values of the heat fluxes received for different models can be essential up to 30%.

With reduction of velocity flight, it corresponds with an increase of density in a shock layer and flows more close to equilibrium case. The parameters of flow and structure of a gas mixture are defined by conditions of chemical equilibrium. The reaction rate constants in equilibrium will be the same for all used models. In this connection, corresponding structures of temperature and concentration, and also value of a withdrawal of a shock wave from a surface received for different models under conditions of a flow at height Н = 32 km well

Influence of chemical models on flow parameters and heat exchange is determined in a wide range of parameters of a flow of MSRO vehicle in cases ideally catalytic and non-catalytic wall. In Figures 18 and 19, mass concentrations of component СО<sup>2</sup> and СО along stagnation line for different models and types of condition on wall are submitted. The data are resulted for conditions of flow <sup>V</sup><sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m3 that corresponds approximately to

Comparing the data of Figures 18 and 19, it is possible to notice that influence of catalytic surface properties affects profiles of concentration basically near to a wall. In these figures, the significant divergence in the distributions of concentration the components obtained for different models is observed. It is evident that it is greater for degree СО<sup>2</sup> dissociation in a shock layer when it used of model of Park [31], and is smaller—when the model of Kenzie-Arnold [30] used. The model of Research Institute of Mechanics (NIIMekh) of Moscow State University [32–34] gives the intermediate results. It is established that change of reaction rates practically does not influence on value of a heat flux to ideal catalytic wall of the vehicle. In a

Figure 18. Profiles of mass concentration СО<sup>2</sup> (––––) and СО (- - - - ) along stagnation line MSRO vehicle, ideal-catalytic

wall, 1: model of [30], 2: model of [31], 3: model of [32–34].

altitude of flight Н ≈ 60 km; the temperature of a surface equal to Т<sup>w</sup> = 1500 K.

enough coincide.

66 Advances in Some Hypersonic Vehicles Technologies

In Figure 20, for two points of a trajectory of MARS EXPRESS vehicle (altitude of flight equal to Н = 43 km and Н = 32 km) the distributions of temperature obtained with the help of model

Figure 19. Profiles of mass concentration СО<sup>2</sup> (––––) and СО (- - - - ) along stagnation line MSRO vehicle, non-catalytic wall, 1: model of [30], 2: model of [31], 3: model of [32–34].

Figure 20. The temperature along the stagnation line for sphere R = 1 m in Mars atmosphere. 1: pressure р = 0.007 atm, Re<sup>∞</sup> = 5 103 ; 2: 'frozen' flow: <sup>р</sup> = 0.07 atm, Re∞= 5 <sup>10</sup><sup>4</sup> ; 3: <sup>р</sup> = 0.22 atm, Re∞= 1.5 105 ; 4: <sup>р</sup> = 0.22 atm, Re∞= 1.5 105 , equilibrium case; 5: <sup>р</sup> = 0.065 <sup>а</sup>tm, Re<sup>∞</sup> = 1.3 <sup>10</sup><sup>5</sup> , perfect СО<sup>2</sup> gas.

of Park [31], models of Kenzie-Arnold [30] and the model developed in Research Institute of Mechanics (NIIMekh) of Moscow State University [32–34] are shown. The data are resulted along a stagnation line. The surface of the vehicle it is an ideal—catalytic wall with the constant temperature.

In Figure 23, comparison of total rotation energy Е<sup>v</sup> = Е<sup>12</sup> + Е<sup>3</sup> and the vibration energy Е<sup>v</sup> obtained with the help of three-temperature [10] and two-temperature models [36] for two conditions of a flow of the vehicle is carried out: V∞ = 5223 m/s (altitude of flight Н ~ 40 km)

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The vibration energy difference near a shock wave in relaxation zone is visible. The structures received with the help of two-temperature model have more "smearing" type than the structures

Figure 22. Distribution of vibration energy component along stagnation line MSRO vehicle, three-temperature model [10],

Figure 23. Distribution of vibration energy along stagnation line MSRO vehicle; 1: Е<sup>v</sup> = Е<sup>12</sup> + Е3, three-temperature model

; (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>

.

.

and V∞ = 5687 km/s (Н ~ 60 km).

1: <sup>E</sup>12, 2: <sup>E</sup>3. <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

of [10]; 2: <sup>Е</sup>v, model of [37]; (а) <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>

Figure 21 shows the density distribution along the stagnation line for three conditions of a flow of MSRO vehicle in an atmosphere of Mars. In case of non-equilibrium flows, the density along stagnation line considerably changes.

#### 7.3. Influence of non-equilibrium vibration kinetics on parameters of a flow

Let us consider results of numerical calculation of a non-equilibrium flow from point of view of different vibration relaxation models. The influence of non-equilibrium excitation of vibration degrees of freedom of carbon dioxide was investigated on a basis of three-temperature kinetic model and two simplified case: in two-temperature approach when it introduced uniform vibration temperature for all types of vibration of СО<sup>2</sup> molecule and in one temperature approach when translational and vibration temperature was the same [10, 36].

Structures of two specific vibration energy E<sup>12</sup> and E<sup>3</sup> for conditions of a flow past 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 (altitude of flight ~40 km) along the stagnation line in shock layer obtained with use of three-temperature model of a vibration relaxation and presented in Figure 22.

These profiles are characterized by significant flow gradients in relaxation region near a shock wave and their behavior reflects of features of considered vibration relaxation model of molecules СО<sup>2</sup> which is taking into account internal structure of molecules СО<sup>2</sup> and extra mode exchanges by vibration energy.

Figure 21. Density distribution along the stagnation line for different condition of flow past MSRO vehicle. 1: V<sup>∞</sup> = 5223 m/ s, <sup>r</sup><sup>∞</sup> = 2.933 <sup>10</sup><sup>4</sup> kg/m3 ; 2: <sup>V</sup><sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.141 <sup>10</sup><sup>4</sup> kg/m3 ; 3: <sup>V</sup><sup>∞</sup> = 3536 m/s, <sup>r</sup><sup>∞</sup> = 2.819 <sup>10</sup><sup>5</sup> kg/m<sup>3</sup> .

In Figure 23, comparison of total rotation energy Е<sup>v</sup> = Е<sup>12</sup> + Е<sup>3</sup> and the vibration energy Е<sup>v</sup> obtained with the help of three-temperature [10] and two-temperature models [36] for two conditions of a flow of the vehicle is carried out: V∞ = 5223 m/s (altitude of flight Н ~ 40 km) and V∞ = 5687 km/s (Н ~ 60 km).

of Park [31], models of Kenzie-Arnold [30] and the model developed in Research Institute of Mechanics (NIIMekh) of Moscow State University [32–34] are shown. The data are resulted along a stagnation line. The surface of the vehicle it is an ideal—catalytic wall with the constant

Figure 21 shows the density distribution along the stagnation line for three conditions of a flow of MSRO vehicle in an atmosphere of Mars. In case of non-equilibrium flows, the density

Let us consider results of numerical calculation of a non-equilibrium flow from point of view of different vibration relaxation models. The influence of non-equilibrium excitation of vibration degrees of freedom of carbon dioxide was investigated on a basis of three-temperature kinetic model and two simplified case: in two-temperature approach when it introduced uniform vibration temperature for all types of vibration of СО<sup>2</sup> molecule and in one tempera-

Structures of two specific vibration energy E<sup>12</sup> and E<sup>3</sup> for conditions of a flow past 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 (altitude of flight ~40 km) along the stagnation line in shock layer obtained with use of three-temperature model of a vibration relaxation and

These profiles are characterized by significant flow gradients in relaxation region near a shock wave and their behavior reflects of features of considered vibration relaxation model of molecules СО<sup>2</sup> which is taking into account internal structure of molecules СО<sup>2</sup> and extra mode

Figure 21. Density distribution along the stagnation line for different condition of flow past MSRO vehicle. 1: V<sup>∞</sup> = 5223 m/

; 3: <sup>V</sup><sup>∞</sup> = 3536 m/s, <sup>r</sup><sup>∞</sup> = 2.819 <sup>10</sup><sup>5</sup> kg/m<sup>3</sup>

.

; 2: <sup>V</sup><sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.141 <sup>10</sup><sup>4</sup> kg/m3

7.3. Influence of non-equilibrium vibration kinetics on parameters of a flow

ture approach when translational and vibration temperature was the same [10, 36].

temperature.

presented in Figure 22.

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

exchanges by vibration energy.

along stagnation line considerably changes.

68 Advances in Some Hypersonic Vehicles Technologies

The vibration energy difference near a shock wave in relaxation zone is visible. The structures received with the help of two-temperature model have more "smearing" type than the structures

Figure 22. Distribution of vibration energy component along stagnation line MSRO vehicle, three-temperature model [10], 1: <sup>E</sup>12, 2: <sup>E</sup>3. <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 .

Figure 23. Distribution of vibration energy along stagnation line MSRO vehicle; 1: Е<sup>v</sup> = Е<sup>12</sup> + Е3, three-temperature model of [10]; 2: <sup>Е</sup>v, model of [37]; (а) <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> ; (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> .

received with use of more rigorous three-temperature approach. You can see also that the maximum values of vibration energy obtained with the help of three-temperature model on 15–20% surpass corresponding values of two-temperature model. In the most part of a shock layer, conditions of thermodynamic equilibrium are realized and both models give identical result.

It is possible to estimate the non-equilibrium vibration zone value considering the structures translational and vibration temperatures Т12, Т3. The various modes received with help of three-temperature model and for two variants of a flow are displayed in Figure 24. It is visible that the size of this zone is more for the second variant of a flow (V∞ = 5687 m/s) corresponding to altitude ~ 60 km.

Comparison of values of translational temperature for two conditions of a flow in the assumption of a weak deviation from thermal equilibrium at Tv = T (a curve 1) and in non-equilibrium gas (curves 2, 3) along a stagnation line is presented in Figure 25(a) and (b). Corresponding to relaxation models profiles of temperature (curves 2 and 3) for these conditions practically coincide, a divergence no more than 5%.

It is visible that the account of non-equilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> leads to insignificant increase in a withdrawal of a shock wave from a surface of a body and to essential increase (~ on 25–30%) translational temperature in the field of a shock wave in comparison with thermally equilibrium case. It does not influence on a gas mixture temperature near to a body surface. The fact of translational temperature increase in relaxation zone is connected with transition of internal degrees energy of freedom of molecules СО<sup>2</sup> in translational energy of others components.

surfaces (Figure 27) is carried out. It is visible that the thermal flux from point of view of nonequilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> can surpass on 10% a

Figure 25. The translational temperature along the stagnation line for MSRO vehicle, 1: equilibrium flow, 2: non-equilibrium flow, three-temperature model of [10], 3: non-equilibrium flow, two-temperature model [38], (а) V<sup>∞</sup> = 5223 m/s,

.

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, (b) <sup>V</sup><sup>∞</sup> <sup>=</sup> 5687 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 3.14 <sup>10</sup><sup>5</sup> kg/m3

Comparing the data in Figures 26 and 27 for the same conditions of a flow, we shall notice that the heat transfer to ideal-catalytic surface in 3–4 times surpasses a corresponding flux to non-

This fact can be explained by that in a case when on a surface the maximal rates of heterogeneous recombination of a component of carbon dioxide gas is observed (ideal catalytic), the chemical energy spent on dissociation is allocated and transferred to a surface. If recombination does not occurs (low catalytic activity), gas components pass down flow, carrying away

As it was already observed, the heat transfer to ideal catalytic wall does not depend on used model of chemical reaction rates that the curves show in Figure 26. For non-catalytic walls, the heat transfer to a surface with using of Kenzie-Arnold model [30] approximately on 30% exceeds a heat transfer obtained with help of model of Park [31]. It shows that at non-catalytic surfaces of the vehicle does not occur of recombination of a component. In this case, reactions go only in one direction, and with different velocities for different models. As result for considered models, the different chemical composition of a gas mixture near surfaces (Figure 29) obtained. Also divergences in values of a heat transfer are observed. Besides the recombination reactions on a surface of additional body heating does not occur. Therefore value of a heat transfer for non-catalytic walls in 3–3.5 times less than corresponding values for ideal catalytic surfaces (Figure 27). A heat transfer distribution of along a frontal surface of MARS EXPRESS vehicle (it is ideal catalytic surface) with use of three models chemical kinetics is resulted in Figure 29. In spite of various rates of reactions, the good correlation of results observed for a heat transfer to a surface for considered models. Really in a case, ideally catalytic wall last plays a role of the catalyst. It is promotes the reactions of recombination and as a result the

with itself the dissociation energy and the additional heat transfer does not occur.

flux for thermally equilibrium gas.

catalytic wall.

<sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 <sup>10</sup><sup>4</sup> kg/m3

#### 7.4. Processes of heat transfer in the multi-component mixture

Comparison of a heat transfer in the assumption thermally equilibrium (curves 1) and nonequilibrium gas (curves 2, 3) in cases ideal catalytic (Figures 26 and 27) and non-catalytic

Figure 24. Profiles translational (Т) and vibration (Т12, Т3) temperature for MSRO vehicle, three-temperature model of [10]; (а) <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> ; (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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received with use of more rigorous three-temperature approach. You can see also that the maximum values of vibration energy obtained with the help of three-temperature model on 15–20% surpass corresponding values of two-temperature model. In the most part of a shock layer, conditions of thermodynamic equilibrium are realized and both models give identical

It is possible to estimate the non-equilibrium vibration zone value considering the structures translational and vibration temperatures Т12, Т3. The various modes received with help of three-temperature model and for two variants of a flow are displayed in Figure 24. It is visible that the size of this zone is more for the second variant of a flow (V∞ = 5687 m/s) corresponding

Comparison of values of translational temperature for two conditions of a flow in the assumption of a weak deviation from thermal equilibrium at Tv = T (a curve 1) and in non-equilibrium gas (curves 2, 3) along a stagnation line is presented in Figure 25(a) and (b). Corresponding to relaxation models profiles of temperature (curves 2 and 3) for these conditions practically

It is visible that the account of non-equilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> leads to insignificant increase in a withdrawal of a shock wave from a surface of a body and to essential increase (~ on 25–30%) translational temperature in the field of a shock wave in comparison with thermally equilibrium case. It does not influence on a gas mixture temperature near to a body surface. The fact of translational temperature increase in relaxation zone is connected with transition of internal degrees energy of freedom of molecules СО<sup>2</sup> in

Comparison of a heat transfer in the assumption thermally equilibrium (curves 1) and nonequilibrium gas (curves 2, 3) in cases ideal catalytic (Figures 26 and 27) and non-catalytic

Figure 24. Profiles translational (Т) and vibration (Т12, Т3) temperature for MSRO vehicle, three-temperature model of

; (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>

.

result.

to altitude ~ 60 km.

coincide, a divergence no more than 5%.

70 Advances in Some Hypersonic Vehicles Technologies

translational energy of others components.

[10]; (а) <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>

7.4. Processes of heat transfer in the multi-component mixture

Figure 25. The translational temperature along the stagnation line for MSRO vehicle, 1: equilibrium flow, 2: non-equilibrium flow, three-temperature model of [10], 3: non-equilibrium flow, two-temperature model [38], (а) V<sup>∞</sup> = 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.93 <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> 3.14 <sup>10</sup><sup>5</sup> kg/m3 .

surfaces (Figure 27) is carried out. It is visible that the thermal flux from point of view of nonequilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> can surpass on 10% a flux for thermally equilibrium gas.

Comparing the data in Figures 26 and 27 for the same conditions of a flow, we shall notice that the heat transfer to ideal-catalytic surface in 3–4 times surpasses a corresponding flux to noncatalytic wall.

This fact can be explained by that in a case when on a surface the maximal rates of heterogeneous recombination of a component of carbon dioxide gas is observed (ideal catalytic), the chemical energy spent on dissociation is allocated and transferred to a surface. If recombination does not occurs (low catalytic activity), gas components pass down flow, carrying away with itself the dissociation energy and the additional heat transfer does not occur.

As it was already observed, the heat transfer to ideal catalytic wall does not depend on used model of chemical reaction rates that the curves show in Figure 26. For non-catalytic walls, the heat transfer to a surface with using of Kenzie-Arnold model [30] approximately on 30% exceeds a heat transfer obtained with help of model of Park [31]. It shows that at non-catalytic surfaces of the vehicle does not occur of recombination of a component. In this case, reactions go only in one direction, and with different velocities for different models. As result for considered models, the different chemical composition of a gas mixture near surfaces (Figure 29) obtained. Also divergences in values of a heat transfer are observed. Besides the recombination reactions on a surface of additional body heating does not occur. Therefore value of a heat transfer for non-catalytic walls in 3–3.5 times less than corresponding values for ideal catalytic surfaces (Figure 27). A heat transfer distribution of along a frontal surface of MARS EXPRESS vehicle (it is ideal catalytic surface) with use of three models chemical kinetics is resulted in Figure 29. In spite of various rates of reactions, the good correlation of results observed for a heat transfer to a surface for considered models. Really in a case, ideally catalytic wall last plays a role of the catalyst. It is promotes the reactions of recombination and as a result the

Figure 27. Heat transfer to MSRO vehicle surface in case of non-catalytic wall; 1: equilibrium flow, 2: non-equilibrium flow for three-temperature model [10], 3: non-equilibrium flow for two-temperature model [38]. V<sup>∞</sup> = 5223 m/s,

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Figure 28. Heat transfer to MSRO vehicle surface in case of ideal catalytic wall; 1: equilibrium flow, 2: non-equilibrium flow for three-temperature model [10], 3: non-equilibrium flow for two-temperature model [38]. V<sup>∞</sup> = 5687 m/s,

<sup>r</sup><sup>∞</sup> = 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

<sup>r</sup><sup>∞</sup> = 3.141 <sup>10</sup><sup>5</sup> kg/m3

.

.

Figure 26. Heat transfer to MSRO vehicle surface in case of ideal catalytic wall; 1: equilibrium flow, 2: non-equilibrium flow for three-temperature model [10], 3: non-equilibrium flow for two-temperature model [38]. V<sup>∞</sup> = 5223 m/s, <sup>r</sup><sup>∞</sup> = 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

chemical energy spent on dissociation is transferred to wall. This process is determined mainly by conditions of equilibrium of reactions.

Influence of non-equilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> on distribution of a heat flux along a surface of a body is displayed in Figures 26–29.

In Figure 29 for comparison (a triangular marker) the results of works [6] in which Navier-Stokes equations solved by method of finite volume are submitted also. In this case, the model of chemical reactions [32–34] was used. For some conditions of a flow the discrepancy reached (up to 20%) for the obtained values of heat fluxes. The corresponding data of [6] obtained for different boundary conditions: in Ref. [6] the condition conservation of heat balance used on a surface, and in our case—a condition of a constant of surface temperature is considered.

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Figure 27. Heat transfer to MSRO vehicle surface in case of non-catalytic wall; 1: equilibrium flow, 2: non-equilibrium flow for three-temperature model [10], 3: non-equilibrium flow for two-temperature model [38]. V<sup>∞</sup> = 5223 m/s, <sup>r</sup><sup>∞</sup> = 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

chemical energy spent on dissociation is transferred to wall. This process is determined mainly

Figure 26. Heat transfer to MSRO vehicle surface in case of ideal catalytic wall; 1: equilibrium flow, 2: non-equilibrium flow for three-temperature model [10], 3: non-equilibrium flow for two-temperature model [38]. V<sup>∞</sup> = 5223 m/s,

Influence of non-equilibrium excitation of vibration degrees of freedom of molecules СО<sup>2</sup> on

In Figure 29 for comparison (a triangular marker) the results of works [6] in which Navier-Stokes equations solved by method of finite volume are submitted also. In this case, the model of chemical reactions [32–34] was used. For some conditions of a flow the discrepancy reached (up to 20%) for the obtained values of heat fluxes. The corresponding data of [6] obtained for different boundary conditions: in Ref. [6] the condition conservation of heat balance used on a surface, and in our case—a condition of a constant of surface temperature is considered.

distribution of a heat flux along a surface of a body is displayed in Figures 26–29.

by conditions of equilibrium of reactions.

.

72 Advances in Some Hypersonic Vehicles Technologies

<sup>r</sup><sup>∞</sup> = 2.93 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

Figure 28. Heat transfer to MSRO vehicle surface in case of ideal catalytic wall; 1: equilibrium flow, 2: non-equilibrium flow for three-temperature model [10], 3: non-equilibrium flow for two-temperature model [38]. V<sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.141 <sup>10</sup><sup>5</sup> kg/m3 .

In Figure 33, the heat transfer to non-catalytic wall is shown (conditions of flow V<sup>∞</sup> = 5223 m/s,

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

Figure 30. Heat transfer to MSRO vehicle surface in case (–––) idealcatalytic wall; ( - - - - ) non-catalytic wall; V<sup>∞</sup> = 5687 m/s,

; 1: model of [30], 2: model of [31], 3: model of [32–34].

Figure 31. Heat transfer to MSRO vehicle surface; non-catalytic surface <sup>V</sup><sup>∞</sup> = 3998 m/s, <sup>r</sup><sup>∞</sup> = 3.0 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

[30], 2: model of [31], 3: model of [32–34].

. 1: model of

) for different values of Prandtl's number: Pr = 0.66 and Pr = 0.75. In a case

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<sup>r</sup><sup>∞</sup> <sup>=</sup> 2.9 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

<sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m3

Figure 29. Heat transfer to MARS EXPRESS vehicle surface in case of non-catalytic (a) and ideal catalytic wall (b); 1: model [30], 2: model of [31], 3: model of [32–34], ▲: data of [6].

Catalytic surface have selectively but significant effect on species and reactions on the surfaces. During entry in Martian atmosphere molecule of carbon dioxide (CO2) dissociated on CO and O. The most part of energy of flow (¾) spends on dissociation. For high catalytic surface kw (surface assist to recombination the atoms to molecules), chemical reactions that spend on dissociation transfer (partially or fully) their energy back to the surface and produce on an additional heating. For low catalytic surface, recombination does not occurs and atoms move downstream and take away the energy of dissociation and additional heating does not happen. It is possible to diminish temperature of surface on 300–500 K when temperature of flow is about 1800–2000 K.

In Figure 30, the heat transfer to ideal catalytic and non-catalytic surface of the vehicle for conditions of flow MSRO <sup>V</sup><sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m3 and <sup>Т</sup><sup>w</sup> = 1500 K are displayed.

In Figure 31, the heat transfer to non-catalytic wall of the vehicle obtained for conditions of flow <sup>V</sup><sup>∞</sup> = 3998 m/s, <sup>r</sup><sup>∞</sup> = 3.0 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> . In Figure 31, the distinction between data of corresponding curves pick up to 15%. In this case, the flow in a shock layer is characterized by smaller temperatures (velocity in an external flow was less) than the regime which correspond to curves of Figure 30.

Therefore, it is marked a smaller divergence in values of reaction rate constants and as consequence at values of a heat fluxes for different models.

In Figure 32, the heat flux to non-catalytic wall for numbers Sc = 0.45; 0.65 and two conditions of a flow are shown. It is evident that for non-catalytic surfaces the value of Schmidt's number practically does not take an influence on the heat transfer. The heat flux is determined basically just by heat conductivity. In this case near the surface of a body there are not recombination reactions. The products of dissociation are carrying out with themselves the energy of dissociation. As a result of additional heating, the body caused by diffusion processes it does not occur.

In Figure 33, the heat transfer to non-catalytic wall is shown (conditions of flow V<sup>∞</sup> = 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.9 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> ) for different values of Prandtl's number: Pr = 0.66 and Pr = 0.75. In a case

Catalytic surface have selectively but significant effect on species and reactions on the surfaces. During entry in Martian atmosphere molecule of carbon dioxide (CO2) dissociated on CO and O. The most part of energy of flow (¾) spends on dissociation. For high catalytic surface kw (surface assist to recombination the atoms to molecules), chemical reactions that spend on dissociation transfer (partially or fully) their energy back to the surface and produce on an additional heating. For low catalytic surface, recombination does not occurs and atoms move downstream and take away the energy of dissociation and additional heating does not happen. It is possible to diminish temperature of surface on 300–500 K when temperature of flow

Figure 29. Heat transfer to MARS EXPRESS vehicle surface in case of non-catalytic (a) and ideal catalytic wall (b); 1:

In Figure 30, the heat transfer to ideal catalytic and non-catalytic surface of the vehicle for conditions of flow MSRO <sup>V</sup><sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m3 and <sup>Т</sup><sup>w</sup> = 1500 K are displayed. In Figure 31, the heat transfer to non-catalytic wall of the vehicle obtained for conditions of

corresponding curves pick up to 15%. In this case, the flow in a shock layer is characterized by smaller temperatures (velocity in an external flow was less) than the regime which corre-

Therefore, it is marked a smaller divergence in values of reaction rate constants and as

In Figure 32, the heat flux to non-catalytic wall for numbers Sc = 0.45; 0.65 and two conditions of a flow are shown. It is evident that for non-catalytic surfaces the value of Schmidt's number practically does not take an influence on the heat transfer. The heat flux is determined basically just by heat conductivity. In this case near the surface of a body there are not recombination reactions. The products of dissociation are carrying out with themselves the energy of dissociation. As a result of additional heating, the body caused by diffusion processes it does not

. In Figure 31, the distinction between data of

is about 1800–2000 K.

spond to curves of Figure 30.

occur.

flow <sup>V</sup><sup>∞</sup> = 3998 m/s, <sup>r</sup><sup>∞</sup> = 3.0 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

model [30], 2: model of [31], 3: model of [32–34], ▲: data of [6].

74 Advances in Some Hypersonic Vehicles Technologies

consequence at values of a heat fluxes for different models.

Figure 30. Heat transfer to MSRO vehicle surface in case (–––) idealcatalytic wall; ( - - - - ) non-catalytic wall; V<sup>∞</sup> = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m3 ; 1: model of [30], 2: model of [31], 3: model of [32–34].

Figure 31. Heat transfer to MSRO vehicle surface; non-catalytic surface <sup>V</sup><sup>∞</sup> = 3998 m/s, <sup>r</sup><sup>∞</sup> = 3.0 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> . 1: model of [30], 2: model of [31], 3: model of [32–34].

of non-catalytic wall, a leading role has another dimensionless factor—Prandtl's number. You can see that the increase in value of Prandtl's number from 0.66 till 0.75 leads to reduction of a heat flux value approximately on 10%.

7.5. Influence of catalytic wall on a heat transfer

account of diffusion processes has great importance.

surface.

constants.

10�<sup>4</sup> kg/m3

temperature approach is determined as:

The heat flux to a surface of the vehicle without taking into account radiation effect in one-

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

N

i¼1

where hi is the enthalpy of ith components of a mixture, λ is the coefficient of heat conductivity of all degrees of freedom which are taking place in a condition of local thermal balance. The second component with in the right part (5) defines the diffusion component of a heat flux and

As show results of calculations the contribution of the third component on value of a total heat transfer makes less than 1%. It means that influence of thermo-diffusion on heat exchange is small. The mass transfer processes play the important role in definition of a heat transfer to a surface. Really diffusion component for a heat transfer for the considered conditions of a flow makes a significant part (50–75%) from value of a full heat flux. In this connection, the correct

As already it has been shown that the main factors influencing heat transfer to a surface of the vehicle, it is heterogeneous recombination of component in dependence on catalytic properties of a surface. We try to explain the influence of chemical reactions on parameters of flow in a shock layer and value of a heat transfer for MSRO vehicle in a case of non-catalytic surfaces. And then it is compared with received results with the corresponding data for ideal catalytic

For calculation, diffusion fluxes were used: Fick's law with the adjusting amendment; Fick's law in standard form in which effective diffusion coefficients are calculated through binary diffusion coefficients and concentrations from point of view of the formula; and Fick's law in the usual form in which Schmidt's number was considered equal to all a component and to be

Depending on the manner of representation of diffusion, the heat transfer can differ essentially up to 30%. Let us notice that more correct way of the diffusion definition that is used for the Fick's law with the amendment gives the values of a heat transfer exceeding on 10% corresponding values received from point of view of Fick's law in the standard form. It is evident that the variant with Schmidt's number Sc = 0.45 results most close to corresponding

For non-catalytic surface, main effect will be play another parameter—Prandtl's number. In Figure 34, you can see the heat transfer to non-catalytic surface (V<sup>∞</sup> = 5223 m/s, r<sup>∞</sup> = 2.9 �

) for different Prandtl's number: Pr = 0.66 and Pr = 0.75. Increase of Prandtl's

data that it is obtained at a correct way of the diffusion account [38, 39].

number from 0.66 until 0.75 leads to decreasing heat transfer on 10%.

hiJ<sup>i</sup> � p

X N

d<sup>i</sup> (35)

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i¼1 DTi

<sup>q</sup> ¼ �λ∇<sup>T</sup> <sup>þ</sup><sup>X</sup>

the third term characterizes influence of thermo-diffusion on heat exchange.

Figure 32. Heat transfer to vehicle surface; <sup>Т</sup>1: <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> ; Т2: V∞ = 5687 m/s, <sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m<sup>3</sup> .

Figure 33. Heat transfer to non-catalytic wall for different Prandtl's number; <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.9 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup> .

## 7.5. Influence of catalytic wall on a heat transfer

of non-catalytic wall, a leading role has another dimensionless factor—Prandtl's number. You can see that the increase in value of Prandtl's number from 0.66 till 0.75 leads to reduction of a

Figure 33. Heat transfer to non-catalytic wall for different Prandtl's number; <sup>V</sup><sup>∞</sup> <sup>=</sup> 5223 m/s, <sup>r</sup><sup>∞</sup> <sup>=</sup> 2.9 <sup>10</sup><sup>4</sup> kg/m<sup>3</sup>

Figure 32. Heat transfer to vehicle surface; <sup>Т</sup>1: <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>

.

; Т2: V∞ = 5687 m/s,

heat flux value approximately on 10%.

76 Advances in Some Hypersonic Vehicles Technologies

<sup>r</sup><sup>∞</sup> = 3.14 <sup>10</sup><sup>5</sup> kg/m<sup>3</sup>

.

The heat flux to a surface of the vehicle without taking into account radiation effect in onetemperature approach is determined as:

$$\mathbf{q} = -\lambda \nabla T + \sum\_{i=1}^{N} h\_i \mathbf{J}\_i - p \sum\_{i=1}^{N} D\_{T\_i} \mathbf{d}\_i \tag{35}$$

where hi is the enthalpy of ith components of a mixture, λ is the coefficient of heat conductivity of all degrees of freedom which are taking place in a condition of local thermal balance. The second component with in the right part (5) defines the diffusion component of a heat flux and the third term characterizes influence of thermo-diffusion on heat exchange.

As show results of calculations the contribution of the third component on value of a total heat transfer makes less than 1%. It means that influence of thermo-diffusion on heat exchange is small. The mass transfer processes play the important role in definition of a heat transfer to a surface. Really diffusion component for a heat transfer for the considered conditions of a flow makes a significant part (50–75%) from value of a full heat flux. In this connection, the correct account of diffusion processes has great importance.

As already it has been shown that the main factors influencing heat transfer to a surface of the vehicle, it is heterogeneous recombination of component in dependence on catalytic properties of a surface. We try to explain the influence of chemical reactions on parameters of flow in a shock layer and value of a heat transfer for MSRO vehicle in a case of non-catalytic surfaces. And then it is compared with received results with the corresponding data for ideal catalytic surface.

For calculation, diffusion fluxes were used: Fick's law with the adjusting amendment; Fick's law in standard form in which effective diffusion coefficients are calculated through binary diffusion coefficients and concentrations from point of view of the formula; and Fick's law in the usual form in which Schmidt's number was considered equal to all a component and to be constants.

Depending on the manner of representation of diffusion, the heat transfer can differ essentially up to 30%. Let us notice that more correct way of the diffusion definition that is used for the Fick's law with the amendment gives the values of a heat transfer exceeding on 10% corresponding values received from point of view of Fick's law in the standard form. It is evident that the variant with Schmidt's number Sc = 0.45 results most close to corresponding data that it is obtained at a correct way of the diffusion account [38, 39].

For non-catalytic surface, main effect will be play another parameter—Prandtl's number. In Figure 34, you can see the heat transfer to non-catalytic surface (V<sup>∞</sup> = 5223 m/s, r<sup>∞</sup> = 2.9 � 10�<sup>4</sup> kg/m3 ) for different Prandtl's number: Pr = 0.66 and Pr = 0.75. Increase of Prandtl's number from 0.66 until 0.75 leads to decreasing heat transfer on 10%.

In Figure 35, the heat flux for two types of a surface and two flow conditions is shown. You can see that value of a heat transfer to non-catalytic surfaces is approximately 3–4 times less than corresponding value of a heat transfer in a case ideal catalytic surface.

Figure 36 shows values of a heat flux to a surface having various catalytic property. The curve 2 (a variant of a surface with final catalytic) is obtained for the following parameters:

Figure 34. Heat transfer to vehicle surface. Black line—total heat flux; red line—diffusion part; (а) V<sup>∞</sup> = 5687 m/s, (b) V<sup>∞</sup> = 5223 m/s.

probability of recombination reactions—γ<sup>w</sup> = 2.7 <sup>10</sup><sup>3</sup>

significant part of a trajectory in some times.

non-catalytic wall, V<sup>∞</sup> = 5223м/c, H = 40 km.

shock layer.

angles.

surface (Figure 37).

kwco = 0.77 m/s (see the formula 30). From the presented data, you can see that due to use low catalytic coverings it is possible to lower a heat transfer to a surface of the vehicle on a

Figure 36. Heat transfer to MSRO vehicle surface. 1: Ideal catalytic wall, 2: finite catalytic, kwco = 0.77 m/s, kwo = 1 m/s, 3:

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

The submitted data show an insignificant influence of processes of vibration relaxation on a heat transfer to a surface of MSRO vehicle. It is also possible to note that taking into account of complex internal structure of molecules СО<sup>2</sup> and exchanges of vibration energy between modes does not take an influence on a heat transfer to a surface. It testifies to legitimacy of application of the simplified models of vibration kinetics at the solution of the given class of problems. The fact of weak influence of a vibration relaxation on a heat transfer to a surface of the vehicle is possible to explain for considered conditions of a flow and the considered form of a surface intensive process dissociation molecules of carbon dioxide gas in a shock layer is observed, and also by fast process of an vibration relaxation of molecules СО<sup>2</sup> as a result of which thermodynamic equilibrium is present almost in all

We would like to consider the influence of the form of the blunted body on parameters of a non-equilibrium flow in modeling an atmosphere of Mars a flow of carbon dioxide gas. Let us consider the heat transfer along surface of spherically blunted cones with various half opening

In a vicinity of the stagnation point on a spherical part of cone surface with various angle of opening, the solution practically coincides. On a conic surface, the size of parameters of a flow essentially depends on an angle of opening a cone that affects on value of a heat flux to a

; catalytic constants—kwo = 1 m/s,

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Figure 35. Heat transfer to MSRO vehicle surface. 1: V<sup>∞</sup> = 5223 m/s, H = 40 km; 2: V<sup>∞</sup> = 5687 m/s, Н = 60 km.

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In Figure 35, the heat flux for two types of a surface and two flow conditions is shown. You can see that value of a heat transfer to non-catalytic surfaces is approximately 3–4 times less than

Figure 36 shows values of a heat flux to a surface having various catalytic property. The curve 2 (a variant of a surface with final catalytic) is obtained for the following parameters:

Figure 34. Heat transfer to vehicle surface. Black line—total heat flux; red line—diffusion part; (а) V<sup>∞</sup> = 5687 m/s, (b)

Figure 35. Heat transfer to MSRO vehicle surface. 1: V<sup>∞</sup> = 5223 m/s, H = 40 km; 2: V<sup>∞</sup> = 5687 m/s, Н = 60 km.

corresponding value of a heat transfer in a case ideal catalytic surface.

78 Advances in Some Hypersonic Vehicles Technologies

V<sup>∞</sup> = 5223 m/s.

Figure 36. Heat transfer to MSRO vehicle surface. 1: Ideal catalytic wall, 2: finite catalytic, kwco = 0.77 m/s, kwo = 1 m/s, 3: non-catalytic wall, V<sup>∞</sup> = 5223м/c, H = 40 km.

probability of recombination reactions—γ<sup>w</sup> = 2.7 <sup>10</sup><sup>3</sup> ; catalytic constants—kwo = 1 m/s, kwco = 0.77 m/s (see the formula 30). From the presented data, you can see that due to use low catalytic coverings it is possible to lower a heat transfer to a surface of the vehicle on a significant part of a trajectory in some times.

The submitted data show an insignificant influence of processes of vibration relaxation on a heat transfer to a surface of MSRO vehicle. It is also possible to note that taking into account of complex internal structure of molecules СО<sup>2</sup> and exchanges of vibration energy between modes does not take an influence on a heat transfer to a surface. It testifies to legitimacy of application of the simplified models of vibration kinetics at the solution of the given class of problems. The fact of weak influence of a vibration relaxation on a heat transfer to a surface of the vehicle is possible to explain for considered conditions of a flow and the considered form of a surface intensive process dissociation molecules of carbon dioxide gas in a shock layer is observed, and also by fast process of an vibration relaxation of molecules СО<sup>2</sup> as a result of which thermodynamic equilibrium is present almost in all shock layer.

We would like to consider the influence of the form of the blunted body on parameters of a non-equilibrium flow in modeling an atmosphere of Mars a flow of carbon dioxide gas. Let us consider the heat transfer along surface of spherically blunted cones with various half opening angles.

In a vicinity of the stagnation point on a spherical part of cone surface with various angle of opening, the solution practically coincides. On a conic surface, the size of parameters of a flow essentially depends on an angle of opening a cone that affects on value of a heat flux to a surface (Figure 37).

radiation flux can be strong that play an essential role at a stage of a choice of the of descent

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

Thermo-chemical model of the CO2-N2 mixtures for the calculation of the non-equilibrium ultra-violet (UV) molecular band radiation in the high-temperature shock layer around the

In the frontal part of a thin shock layer radiation, absorption is small enough. Thus, the gas is assumed to be transparent to radiation in the relaxation zone of the shock layer. Radiation intensity is calculated in the approximation of volume luminescence. Radiation processes involving excited particles are considered as spontaneous radiating transitions, excitation and deactivation of the electronic states of the molecules, impacts of electron and heavy particles, etc. [40, 41]. Corresponding equation of excitation and deactivation of the electronic states determine a concentration of each component. Flow parameters across the shock layer are calculated and the spectral structure of radiation is obtained. For the analysis of the nonequilibrium radiation, the results of some theoretical and experimental studies behind shock waves in CO2-N2 mixtures used. In the relaxation zone of the vehicle shock layer, the Boltzmann distribution of the electronically excited states of atoms and molecules does not exist under the considered Martian entry conditions. It leads to a significant deviation of the

The suggested thermo-chemical model of the СО<sup>2</sup> + N2 mixture contains 10 neutral chemical species: СО2, СО, CN, NO, N2, О2, C2, С, N, О, 4 molecular ions and free electrons: СО<sup>+</sup>

Σ+

, e, and 12 electronically excited states of diatomic molecules: СО(A'П), СО(b<sup>3</sup>

П), NO(D<sup>2</sup>

Σ+

), N2(A<sup>3</sup>

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

Σ<sup>u</sup> +

), NO(С<sup>2</sup>

The thermo-physical properties of chemical species are taken from Ref. [42]. Thus, 19 chemical reactions and 33 reactions of the excitation of the electronic states of molecules are taken into account. The reactions with the participation of the neutral and charged particles in a hightemperature Martian atmosphere are considered. The rate constants of the basic chemical

Practically, it is convenient to use simplified radiation models that are capable to estimate radiation emission with sufficient accuracy. The estimations have shown that the gas is transparent to UV molecular radiation in the shock layer under the considered conditions. And it is possible to calculate radiation intensity with the approximation of volume luminescence. The "just overlapping line model" model is used to calculate spectral distribution of nonequilibrium molecular band radiation. The model considers a spectrum consisting of only one branch of rotational lines. The shock layer is optically thin for spectral range considered so the process of light absorption is not taken into account. It is shown that the depletion of electronically excited states of molecules due to spontaneous radiation transitions has a great effect on excited state populations and must be necessarily accounted for under the MSRO trajectory conditions. Spontaneous radiation emission leads to violation of Boltzmann approximation for excited state populations. The molecular band radiation results obtained in the local equilibrium approximation (i.e. the supposition of the Boltzmann distribution of molecules on

, NO+ ,

), N2(B<sup>3</sup>

Σ+ ), 81

Пg),

radiation intensity from local equilibrium especially for low gas density.

П), NO(A<sup>2</sup>

reactions have been collected from the literature [29, 31–34, 43].

trajectory and heat protection system of space vehicle.

capsule during an entry is considered [19].

N2 + , О<sup>2</sup> +

СN(В<sup>2</sup> Σ+

O2(B2 Σ<sup>u</sup> ), CN(A<sup>2</sup>

Пg).

), C2(d3

П),NO(B<sup>2</sup>

Figure 37. Heat transfer to the surface of blunted cones with different opening angle θ for Mars atmosphere; 1: θ = 60, 2: 50, 3: 40, 4: 30, 5: 20, 6: 10. <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> .

As a result of the carried out researches, it is received (calculations were carried out with help of two-temperature model of a vibration relaxation) that for conditions of flows in an atmosphere of Mars corresponding to velocities 4–6 km/s and to altitude Н <60 km flight, the vibration relaxation of molecules on a spherical part of a body occurs quickly and thermodynamic equilibrium is present almost in all cross section of a shock layer.

As show numerical calculations for altitude of flight Н ≤ 60 km the blunted bodies in an atmosphere of Mars the increase of a heat transfer to a surface taking in account the mechanism of an vibration relaxation makes no more than 20%.

#### 7.6. The role of non-equilibrium radiation

The role of radiation is rather insignificant for descent space vehicle in an atmosphere of Mars with the characteristic sizes 1–2 m and at entrance velocity of U<sup>∞</sup> ≈ 5 6 km=s. On vehicles which sizes will be at 5–10 times more, the radiant flux can be comparable with convective. The pilot and automatic expeditions with use of diving is perspective due to aerodynamic braking devices in the top layers of Mars atmosphere with U 6–8 km/s up to 3.3 km/s with the subsequent exit into basic orbits around of Mars. In this case the radiant flux is determined by non-equilibrium radiation as the vehicle penetrates an atmosphere of Mars at heights H > 30 km in which physical and chemical processes in a shock layer is essentially nonequilibrium. At H < 30 km, the radiation flux to space vehicle is determined by equilibrium radiation and its level is insignificant. At hyperbolic velocities of an entrance of flight, the radiation flux can be strong that play an essential role at a stage of a choice of the of descent trajectory and heat protection system of space vehicle.

Thermo-chemical model of the CO2-N2 mixtures for the calculation of the non-equilibrium ultra-violet (UV) molecular band radiation in the high-temperature shock layer around the capsule during an entry is considered [19].

In the frontal part of a thin shock layer radiation, absorption is small enough. Thus, the gas is assumed to be transparent to radiation in the relaxation zone of the shock layer. Radiation intensity is calculated in the approximation of volume luminescence. Radiation processes involving excited particles are considered as spontaneous radiating transitions, excitation and deactivation of the electronic states of the molecules, impacts of electron and heavy particles, etc. [40, 41]. Corresponding equation of excitation and deactivation of the electronic states determine a concentration of each component. Flow parameters across the shock layer are calculated and the spectral structure of radiation is obtained. For the analysis of the nonequilibrium radiation, the results of some theoretical and experimental studies behind shock waves in CO2-N2 mixtures used. In the relaxation zone of the vehicle shock layer, the Boltzmann distribution of the electronically excited states of atoms and molecules does not exist under the considered Martian entry conditions. It leads to a significant deviation of the radiation intensity from local equilibrium especially for low gas density.

The suggested thermo-chemical model of the СО<sup>2</sup> + N2 mixture contains 10 neutral chemical species: СО2, СО, CN, NO, N2, О2, C2, С, N, О, 4 molecular ions and free electrons: СО<sup>+</sup> , NO+ , N2 + , О<sup>2</sup> + , e, and 12 electronically excited states of diatomic molecules: СО(A'П), СО(b<sup>3</sup> Σ+ ), СN(В<sup>2</sup> Σ+ ), CN(A<sup>2</sup> П),NO(B<sup>2</sup> П), NO(A<sup>2</sup> Σ+ ), NO(С<sup>2</sup> П), NO(D<sup>2</sup> Σ+ ), N2(A<sup>3</sup> Σ<sup>u</sup> + ), N2(B<sup>3</sup> Пg), O2(B2 Σ<sup>u</sup> ), C2(d3 Пg).

As a result of the carried out researches, it is received (calculations were carried out with help of two-temperature model of a vibration relaxation) that for conditions of flows in an atmosphere of Mars corresponding to velocities 4–6 km/s and to altitude Н <60 km flight, the vibration relaxation of molecules on a spherical part of a body occurs quickly and thermody-

Figure 37. Heat transfer to the surface of blunted cones with different opening angle θ for Mars atmosphere; 1: θ = 60,

.

As show numerical calculations for altitude of flight Н ≤ 60 km the blunted bodies in an atmosphere of Mars the increase of a heat transfer to a surface taking in account the mecha-

The role of radiation is rather insignificant for descent space vehicle in an atmosphere of Mars with the characteristic sizes 1–2 m and at entrance velocity of U<sup>∞</sup> ≈ 5 6 km=s. On vehicles which sizes will be at 5–10 times more, the radiant flux can be comparable with convective. The pilot and automatic expeditions with use of diving is perspective due to aerodynamic braking devices in the top layers of Mars atmosphere with U 6–8 km/s up to 3.3 km/s with the subsequent exit into basic orbits around of Mars. In this case the radiant flux is determined by non-equilibrium radiation as the vehicle penetrates an atmosphere of Mars at heights H > 30 km in which physical and chemical processes in a shock layer is essentially nonequilibrium. At H < 30 km, the radiation flux to space vehicle is determined by equilibrium radiation and its level is insignificant. At hyperbolic velocities of an entrance of flight, the

namic equilibrium is present almost in all cross section of a shock layer.

nism of an vibration relaxation makes no more than 20%.

2: 50, 3: 40, 4: 30, 5: 20, 6: 10. <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>

80 Advances in Some Hypersonic Vehicles Technologies

7.6. The role of non-equilibrium radiation

The thermo-physical properties of chemical species are taken from Ref. [42]. Thus, 19 chemical reactions and 33 reactions of the excitation of the electronic states of molecules are taken into account. The reactions with the participation of the neutral and charged particles in a hightemperature Martian atmosphere are considered. The rate constants of the basic chemical reactions have been collected from the literature [29, 31–34, 43].

Practically, it is convenient to use simplified radiation models that are capable to estimate radiation emission with sufficient accuracy. The estimations have shown that the gas is transparent to UV molecular radiation in the shock layer under the considered conditions. And it is possible to calculate radiation intensity with the approximation of volume luminescence. The "just overlapping line model" model is used to calculate spectral distribution of nonequilibrium molecular band radiation. The model considers a spectrum consisting of only one branch of rotational lines. The shock layer is optically thin for spectral range considered so the process of light absorption is not taken into account. It is shown that the depletion of electronically excited states of molecules due to spontaneous radiation transitions has a great effect on excited state populations and must be necessarily accounted for under the MSRO trajectory conditions. Spontaneous radiation emission leads to violation of Boltzmann approximation for excited state populations. The molecular band radiation results obtained in the local equilibrium approximation (i.e. the supposition of the Boltzmann distribution of molecules on electronic states) strongly overestimates non-equilibrium radiation values and cannot be used even for preliminary predictions.

The radiation intensities are obtained for the 11 strongest systems of molecular bands: 3rd and 4th positive band of СО molecule, the red and violet band of CN molecule, the Schuman-Runge system of О<sup>2</sup> molecule, the β, γ, δ, ε systems NO molecule, the Swan band of the С<sup>2</sup>

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In Figure 38b, distributions of mass concentrations of neutral chemical species along the stagnation line are shown for 5687 m/s. Note a maximum of concentration CN in the relaxation zone behind the shock wave. With increasing velocity, the degree of dissociation of the mole-

In Figure 39, distributions of volume concentration of the electronically excited states along the stagnation line are shown. Continuous lines show the values obtained with non-equilibrium approach, while dotted lines correspond to the local equilibrium approximation. Practically, all the excited levels, except О2(В), reach a maximum near the shock wave. Such a fact is due to the larger values of the temperature in this region. Populations of the molecules containing carbon, except CN (A) and CO (d3) are essentially smaller (approximately in 100 times), the corresponding Boltzmann distributions. Thus, in the relaxation zone of the shock layer, the Boltzmann distribution of atoms and molecules on the electronically excited states is violated. It leads to a significant deviation of the radiation intensity from that corresponding to local

For the molecules forming behind the shock waves, the populations of NO (D), NO (C) states are close to equilibrium. The populations of the rest electronic states differ from their equilib-

Intensity of radiation strongly decreases near the body surface. Thus for rather low value of temperature across the boundary layer, it does not bring an appreciable contribution to the radiation heat transfer. Boundary conditions on the surface of the body, in particular, the

cules СО<sup>2</sup> increases and concentration of CN molecules become larger.

rium values but in a less degree than for the molecules containing carbon.

catalytic condition does not affect value of radiation heat transfer.

Figure 39. Populations of electronically excited states along stagnation line, V = 3998 m/s.

molecule, 1st positive band of N2 molecule.

equilibrium case especially for low gas density.

Calculations of the convective heat flux and the non-equilibrium radiation were carried out for the MSRO vehicle entering into the Martian atmosphere. The wall is assumed non-catalytic. Trajectory parameters are presented in Table 2.

In Figure 38a, distributions of temperature along the stagnation line are shown. In all cases, it follows from a result the model viscous shock layer is realized. Boundary layer thickness takes approximately 1/4 from the shock layer thickness.

The main contribution to a radiation in the shock layer is produced by the bands of the molecular systems as found from theory and experiments. Main source of the shock layer radiation are the molecules that form as a result of chemical reactions.

Molecules СО, О2, СN, С<sup>2</sup> are formed only as a result of the chemical reactions and the information about vibration states of these molecules are absent. For these molecules, their vibration modes are in thermal equilibrium. Besides, there is a significant amount of oxygen atoms due to the fast dissociation of the molecules СО<sup>2</sup> behind shock wave. Oxygen atoms have large enough cross sections for V-T processes energy exchange. In the free stream, СО<sup>2</sup> and N2 molecules have almost zero vibration energy, therefore, for them in a shock layer there is an area with non-equilibrium vibration.


Table 2. Trajectory parameters.

Figure 38. Translational temperature (a) and mass concentrations, V = 5687 m/s, (b) along stagnation line.

The radiation intensities are obtained for the 11 strongest systems of molecular bands: 3rd and 4th positive band of СО molecule, the red and violet band of CN molecule, the Schuman-Runge system of О<sup>2</sup> molecule, the β, γ, δ, ε systems NO molecule, the Swan band of the С<sup>2</sup> molecule, 1st positive band of N2 molecule.

electronic states) strongly overestimates non-equilibrium radiation values and cannot be used

Calculations of the convective heat flux and the non-equilibrium radiation were carried out for the MSRO vehicle entering into the Martian atmosphere. The wall is assumed non-catalytic.

In Figure 38a, distributions of temperature along the stagnation line are shown. In all cases, it follows from a result the model viscous shock layer is realized. Boundary layer thickness takes

The main contribution to a radiation in the shock layer is produced by the bands of the molecular systems as found from theory and experiments. Main source of the shock layer

Molecules СО, О2, СN, С<sup>2</sup> are formed only as a result of the chemical reactions and the information about vibration states of these molecules are absent. For these molecules, their vibration modes are in thermal equilibrium. Besides, there is a significant amount of oxygen atoms due to the fast dissociation of the molecules СО<sup>2</sup> behind shock wave. Oxygen atoms have large enough cross sections for V-T processes energy exchange. In the free stream, СО<sup>2</sup> and N2 molecules have almost zero vibration energy, therefore, for them in a shock layer there

V, m/s T∞, K r∞, kg/m<sup>3</sup> <sup>140</sup> 2.82 <sup>10</sup><sup>5</sup> <sup>140</sup> 3.07 <sup>10</sup><sup>4</sup> <sup>140</sup> 2.93 <sup>10</sup><sup>4</sup> <sup>140</sup> 3.125 <sup>10</sup><sup>5</sup>

Figure 38. Translational temperature (a) and mass concentrations, V = 5687 m/s, (b) along stagnation line.

even for preliminary predictions.

82 Advances in Some Hypersonic Vehicles Technologies

Trajectory parameters are presented in Table 2.

approximately 1/4 from the shock layer thickness.

is an area with non-equilibrium vibration.

Table 2. Trajectory parameters.

radiation are the molecules that form as a result of chemical reactions.

In Figure 38b, distributions of mass concentrations of neutral chemical species along the stagnation line are shown for 5687 m/s. Note a maximum of concentration CN in the relaxation zone behind the shock wave. With increasing velocity, the degree of dissociation of the molecules СО<sup>2</sup> increases and concentration of CN molecules become larger.

In Figure 39, distributions of volume concentration of the electronically excited states along the stagnation line are shown. Continuous lines show the values obtained with non-equilibrium approach, while dotted lines correspond to the local equilibrium approximation. Practically, all the excited levels, except О2(В), reach a maximum near the shock wave. Such a fact is due to the larger values of the temperature in this region. Populations of the molecules containing carbon, except CN (A) and CO (d3) are essentially smaller (approximately in 100 times), the corresponding Boltzmann distributions. Thus, in the relaxation zone of the shock layer, the Boltzmann distribution of atoms and molecules on the electronically excited states is violated. It leads to a significant deviation of the radiation intensity from that corresponding to local equilibrium case especially for low gas density.

For the molecules forming behind the shock waves, the populations of NO (D), NO (C) states are close to equilibrium. The populations of the rest electronic states differ from their equilibrium values but in a less degree than for the molecules containing carbon.

Intensity of radiation strongly decreases near the body surface. Thus for rather low value of temperature across the boundary layer, it does not bring an appreciable contribution to the radiation heat transfer. Boundary conditions on the surface of the body, in particular, the catalytic condition does not affect value of radiation heat transfer.

Figure 39. Populations of electronically excited states along stagnation line, V = 3998 m/s.

Spectral distributions of radiation intensity from ultra-violet up to near infra-red are estimated. The data of non-equilibrium radiating processes and the local equilibrium approach are compared. Use of the local equilibrium modifies drastically the spectral distribution of radiation intensity.

In Figure 40, the convective heat fluxes for different trajectory points are shown. It is possible to divide the distribution of the convective heat flux along the surface in three distinct regions. The maximum heat transfer occurs at the stagnation point. Then, along the spherical part, the heat transfer decreases as the pressure drops. The heat transfer along the conic part is almost constant. At last, there is a local increase of the heat transfer in the shoulder region connected with an increase of the velocity gradient.

In Figure 41, convective and radiation heat transfer values are compared for different trajectory points. Convective heat flux is predominant compared to the non-equilibrium radiation flux. With account of non-equilibrium character of collision-radiation processes in the shock layer values of radiation heat transfer are several orders (from 102 to 107 times) below convective ones for the considered trajectory points. For a correct prediction of heat transfer and surface temperatures near space vehicle at entry conditions in the Martian atmosphere, the careful examination of theoretical and experimental catalytic properties results of a of thermal protection covering are required.

However, for the local equilibrium approach, the radiation flux is close to the convective value only for the trajectory point (<sup>V</sup> = 5687 km/s, <sup>r</sup><sup>∞</sup> = 3.125 <sup>10</sup><sup>5</sup> kg/m3 ). The non-equilibrium radiation flux is one order of magnitude smaller less than the flux obtained under the local equilibrium assumption.

To assess the influence of the catalytic wall on heat transfer and radiation equilibrium temperature of the surface thermal protection calculations for 120 degree cone blunted on sphere of radius R = 0.7 m at speed V = 6150 km/s considering a pure CO2 atmosphere at an altitude of 40 km have been carried out. Calculations were made for four values of the recombination

probability that are typical for heat-shielding materials of different types. Results of calcula-

γ Kw, m/s Qw, kW/m<sup>2</sup> Тw, K <sup>4</sup> 0.035 281 1530 <sup>3</sup> 0.37 380 1650 <sup>2</sup> 4 692 1915 <sup>1</sup> 40 897 2045

Figure 41. Heat flux at the stagnation point along trajectory, 1: convective, 2: local equilibrium radiation, 3: non-

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

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85

According to the results, the ratio between the lowest heat flux and largest one is close to three. The equilibrium surface temperatures when blackness of a surface ε = 0.9 can differ more than 500 K. For a correct prediction of heat transfer and surface temperatures, careful experimental researches about the catalytic properties of the thermal protection covering are required.

Non-equilibrium flows of the reacting mixture CO2/CO/O2/C/O in a viscous shock layer near a spacecraft entering the Mars atmosphere are studied using the accurate three-temperature model developed on the basis of the kinetic theory methods. Gas dynamic parameters, transport coefficients in a shock layer, and heat fluxes to the body surface are calculated for non-catalytic

tions are presented in Table 3.

8. Conclusions

Table 3

equilibrium radiation.

Figure 40. Convective heat flux distributions.

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules http://dx.doi.org/10.5772/intechopen.71666 85

Figure 41. Heat flux at the stagnation point along trajectory, 1: convective, 2: local equilibrium radiation, 3: nonequilibrium radiation.


#### Table 3

Spectral distributions of radiation intensity from ultra-violet up to near infra-red are estimated. The data of non-equilibrium radiating processes and the local equilibrium approach are compared. Use of the local equilibrium modifies drastically the spectral distribution of radiation

In Figure 40, the convective heat fluxes for different trajectory points are shown. It is possible to divide the distribution of the convective heat flux along the surface in three distinct regions. The maximum heat transfer occurs at the stagnation point. Then, along the spherical part, the heat transfer decreases as the pressure drops. The heat transfer along the conic part is almost constant. At last, there is a local increase of the heat transfer in the shoulder region connected

In Figure 41, convective and radiation heat transfer values are compared for different trajectory points. Convective heat flux is predominant compared to the non-equilibrium radiation flux. With account of non-equilibrium character of collision-radiation processes in the shock layer values of radiation heat transfer are several orders (from 102 to 107 times) below convective ones for the considered trajectory points. For a correct prediction of heat transfer and surface temperatures near space vehicle at entry conditions in the Martian atmosphere, the careful examination of theoretical and experimental catalytic properties results of a of thermal

However, for the local equilibrium approach, the radiation flux is close to the convective value

radiation flux is one order of magnitude smaller less than the flux obtained under the local

To assess the influence of the catalytic wall on heat transfer and radiation equilibrium temperature of the surface thermal protection calculations for 120 degree cone blunted on sphere of radius R = 0.7 m at speed V = 6150 km/s considering a pure CO2 atmosphere at an altitude of 40 km have been carried out. Calculations were made for four values of the recombination

). The non-equilibrium

only for the trajectory point (<sup>V</sup> = 5687 km/s, <sup>r</sup><sup>∞</sup> = 3.125 <sup>10</sup><sup>5</sup> kg/m3

intensity.

with an increase of the velocity gradient.

84 Advances in Some Hypersonic Vehicles Technologies

protection covering are required.

Figure 40. Convective heat flux distributions.

equilibrium assumption.

probability that are typical for heat-shielding materials of different types. Results of calculations are presented in Table 3.

According to the results, the ratio between the lowest heat flux and largest one is close to three. The equilibrium surface temperatures when blackness of a surface ε = 0.9 can differ more than 500 K. For a correct prediction of heat transfer and surface temperatures, careful experimental researches about the catalytic properties of the thermal protection covering are required.

#### 8. Conclusions

Non-equilibrium flows of the reacting mixture CO2/CO/O2/C/O in a viscous shock layer near a spacecraft entering the Mars atmosphere are studied using the accurate three-temperature model developed on the basis of the kinetic theory methods. Gas dynamic parameters, transport coefficients in a shock layer, and heat fluxes to the body surface are calculated for non-catalytic and fully catalytic surfaces. The results are compared with the ones obtained in the simplified two-temperature approximation and in the one-temperature approach for weak deviations from thermal equilibrium. A considerable influence of CO2 vibration excitation on the flow parameters and transport properties in a shock layer is found. The difference between the results obtained using the accurate and simplified vibration non-equilibrium models are weak under conditions considered in the paper. This justifies the validity of the approximate twotemperature model under the re-entry conditions. It is shown that difference in reaction rate constants practically has small influence on value of a heat transfer to ideal catalytic surfaces of the vehicle. In a case of non-catalytic surfaces difference in value of the heat transfer obtained by different models can be essential up to 30%. The effect of bulk viscosity in a shock layer is studied. Including this coefficient to the fluid dynamics equations improves the accuracy of the heat flux calculation up to 10%.

[4] Problems of Aerothermoballistics, Radiation Gasdynamics, Heat and Mass Transfer for Planet Sample Return Missions. Project Technical Report of ISTC No 1549-00; 2003. http://

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

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[5] Rouzaud O, Zhlukhtov S, Egorov I, Fletcher D, Gromov V, Nagnibeda E, Shevelev Yu, Omaly P. Numerical, analytical and experimental investigation of convective and radiative heating of a martian descent module. European Space Agency, ESA SP (583); 2005.

[6] Afonina NE, Gromov VG. Numerical Modeling of Martian Descent Module Flow. Institute of Mechanics in Moscow State University. Aeromechanics and Gas Dynamics: Mos-

[7] Nagnibeda E, Kustova E. Kinetic Theory of Transport and Relaxation Processes in Nonequilibrium Reacting Gas Flows. Saint Petersburg: Saint Petersburg University Press; 2003

[8] Kustova E, Nagnibeda E. On a correct description of a multi-temperature dissociating

[9] Kustova EV, Nagnibeda EA, Shevelev YD, Syzranova NG. The influence of CO2 kinetics on the hypersonic flow near blunt bodies. American Institute of Physics, AIP Conference

[10] Shevelev YD, Syzranova NG, Kustova EV, Nagnibeda Numerical EA. Investigation of hypersonic fluid discending flow in Mars atmosphere. Mathematical Models and Com-

[11] Kustova EV, Nagnibeda EA, Shevelev YD, Syzranova NG. Non-equilibrium kinetics and transport processes in a hypersonic flow of CO2/CO/O2/C/O mixture. American Institute

[12] Kustova EV, Nagnibeda EA, Shevelev YD, Syzranova NG. Comparison of different models for non-equilibrium CO2 flows in a shock layer near a blunt body. Shock Waves. 2011;21(3):

[13] Shevelev YD, Syzranova NG. The effect of multi-component diffusion on supersonic flow

[14] Kustova EV, Nagnibeda EA, Shevelev YD, Syzranova NG. Comparison of non-equilibrium supersonic CO2 flows with real gas effects near a blunt body. European Space Agency, ESA

[15] Shevelev YD. Three-Dimensional Problems of Computational Fluid Dynamics. Moscow:

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The heat flux to ideal catalytic surface for the considered conditions of a flow can up to four times surpasses a heat transfer to non-catalytic wall.

The influence of the different chemical reactions models: (1) Mc. Kenzie and Arnold chemistry model, (2) Park's model, amd (3) model of S. Losev and others, on component concentrations and heat flux are presented. Numerical calculations of the coefficients of viscosity and heat conductivity give the close result for the heat flux for different models comparing with results obtained by exact kinetic theory. The diffusion parameters affects on the magnitude of the heat flux especially in the case of a catalytic wall. The pressure- and thermo-diffusion influence on heat flux are small. The different models of the vibration relaxation of CO2 considered. They give approximately the same values of main properties of the flow. The catalytic properties of the surface are most important for a valid determination of the heat flux to the wall. The insertion of bulk viscosity into the equations leads to the small increase of the heat fluxes.
