**2. Mathematical and physical modeling**

The two-dimensional, steady, compressible, Navier Stokes equation set was applied as the governing equations. The fundamental governing equations are represented in Eqs. (1)–(6). The density was calculated using the ideal gas law. For viscous and compressible flow, the viscosity is generally [25] calculated as a function of temperature as defined by Armally and Sutton law as in Eq. (7), and hence it is used in the present work.

#### **2.1 Species conservation**

For species in a mixture, the mass conservation equation is dictated by:

$$\underbrace{\frac{\partial \rho\_a}{\partial t}}\_{1} + \underbrace{\frac{\partial}{\partial x} \left(\rho\_a u - \left(-q\_{ax}^D\right)\right)}\_{2} + \underbrace{\frac{\partial}{\partial y} \left(\rho\_a u - \left(-q\_{ay}^D\right)\right)}\_{3} = \underbrace{\dot{\mathbf{w}}\_a}\_{4} \tag{1}$$

The global continuity equation is given by

$$
\underbrace{\frac{\partial\rho}{\partial t}}\_{1} + \underbrace{\frac{\partial}{\partial x}(\rho u)}\_{2} + \underbrace{\frac{\partial}{\partial y}(\rho u)}\_{3} = 0
\tag{2}
$$

#### **2.2 Global momentum conservation**

The mixture's momentum balance is given by

$$\underbrace{\frac{\partial \rho u}{\partial t}}\_{1} + \underbrace{\frac{\partial}{\partial \mathbf{x}} \left(\rho u^2 + p - \tau\_{\mathbf{x}\mathbf{x}}\right)}\_{2} + \underbrace{\frac{\partial}{\partial \mathbf{y}} \left(\rho uv - \tau\_{\mathbf{x}\mathbf{y}}\right)}\_{3} = \mathbf{0} \tag{3}$$

*Hypersonic Vehicles - Applications, Recent Advances, and Perspectives*

$$\underbrace{\frac{\partial \rho v}{\partial t}}\_{1} + \underbrace{\frac{\partial}{\partial \mathbf{x}} \left(\rho uv - \tau\_{\mathbf{xy}}\right)}\_{2} + \underbrace{\frac{\partial}{\partial \mathbf{y}} \left(\rho v^{2} + p - \tau\_{\mathbf{yy}}\right)}\_{3} = \mathbf{0} \tag{4}$$

#### **2.3 Vibrational energy conservation**

Vibrational energy conservation is a phenomenological characterization of the average energy in each molecular species' vibrational mode. The conservation of vibrational energy is regulated by:

$$\underbrace{\frac{\partial}{\partial t}\left(\rho\_m e\_m^{vib}\right)}\_{1} + \underbrace{\frac{\partial}{\partial \mathbf{x}}\left(\rho\_m e\_m^{vib} u - q\_{m\mathbf{x}}^{vib}\right)}\_{2} + \underbrace{\frac{\partial}{\partial \mathbf{y}}\left(\rho\_m e\_m^{vib} v - q\_{m\mathbf{y}}^{vib}\right)}\_{3} = \underbrace{\Omega\_m}\_{4} \tag{5}$$

#### **2.4 Total energy conservation**

Total energy conservation (internal + kinetic) is regulated by

$$\underbrace{\frac{\partial(\rho E)}{\partial t} + \underbrace{\frac{\partial}{\partial x}(\rho E + p)u - (u\tau\_{\rm{xx}} + v\tau\_{\rm{yx}} + qt\_{\rm{x}})}\_{2}}\_{2} + \underbrace{\frac{\partial}{\partial y}\left((\rho E + p)v - \left(u\tau\_{\rm{xx}} + v\tau\_{\rm{xy}} + q\_{\rm{y}}t\right)\right)}\_{3} \tag{6}$$

#### **2.5 Armaly Sutton law**

$$\mu\_a = 0.1 \exp\left[ \left( A\_a^{\mu} \ln \left( T \right) + B\_a^{\mu} \right) \ln \left( T \right) + C\_a^{\mu} \right] \tag{7}$$

### **2.6 Chemical kinetic model**

A thermo-chemically non-equilibrium flow of a five-component air model consisting of species N2, O2, NO, N and O was considered. We have made the hypothesis of a chemical flow at vibrational equilibrium; the ionization phenomena have been neglected. The most important chemical reactions between these species are: [26, 27].

$$\mathbf{N}\_2 + \mathbf{M} \rightleftharpoons \mathbf{2N} + \mathbf{M} \tag{8}$$

$$\text{O}\_2 + \text{M} \rightleftharpoons 2\text{O} + \text{M} \tag{9}$$

$$\rm{NO} + \rm{M} \rightleftharpoons \rm{N} + \rm{O} + \rm{M} \tag{10}$$

$$\text{NO} + \text{O} \rightleftharpoons \text{N} + \text{O}\_2 \tag{11}$$

$$\text{N}\_2 + \text{O} \rightleftharpoons \text{NO} + \text{N} \tag{12}$$

The chemical source names are formed from reactions that take place between the gas's constituents. A mass transfer mechanism occurs between species as reactions occur so the formulas for these mass transfer rates are determined. Several separate elementary chemical reactions between species in the gas can take place at the same time. Consider the rth chemical reaction of Nr elementary reactions between Ns chemically reacting species:

$$\sum v\_{a,r}' X\_a = \sum v\_{a,r}'' X\_a \tag{13}$$

*Aero Heating Optimization of a Hypersonic Thermochemical Non-Equilibrium Flow… DOI: http://dx.doi.org/10.5772/intechopen.101659*

There is a forward and backward portion to the chemical reaction equation,

Eq. (13). The forward and backward reaction rates are calculated as follows: **Forward:**

$$\frac{d[X\_a]\_r^f}{dt} = \left(\upsilon\_{a,r}^{\prime\prime} - \upsilon\_{a,r}^{\prime}\right) \left[k\_{\,f,r} \prod\_{a=1}^{N\_\iota} \left[X\_a\right]^{\upsilon\_{a,r}^{\prime}}\right] \tag{14}$$

**Backward:**

$$\frac{d[X\_a]\_r^b}{dt} = \left(\upsilon\_{a,r}^{\prime\prime} - \upsilon\_{a,r}^{\prime}\right) \left[k\_{b,r} \prod\_{a=1}^{N\_t} \left[X\_a\right]^{\upsilon\_{a,r}^{\prime\prime}}\right] \tag{15}$$

Where

• *kf, r* and *kb, r* are the forward and backward reaction rate coefficients of reaction r, which are both affected by the temperature of the reaction.

The net rate for the above general reaction r can be written as

$$\frac{d[\mathbf{X}\_a]}{dt} = \frac{d[\mathbf{X}\_a]\_r^f}{dt} - \frac{d[\mathbf{X}\_a]\_r^b}{dt} = \left(v\_{a,r}^{\prime\prime} - v\_{a,r}^{\prime}\right) \left[k\_{f,r} \prod\_{a=1}^{N\_\prime} [\mathbf{X}\_a]^{v\_{a,r}^{\prime}} - k\_{b,r} \prod\_{a=1}^{N\_\prime} [\mathbf{X}\_a]^{v\_{a,r}^{\prime\prime}}\right] \tag{16}$$

The equation above is a general form of the law of mass action, which assures that total mass is preserved during a chemical reaction.

The FLUENT uses the expression given by the law of ARRHENIUS to calculate the direct speed constant. The expression of ARRHENIUS is given by:

$$K\_{f,r} = \mathbb{C}\_{f,r} T\_{f,r}^{\mathfrak{n}\_{f,r}} e^{\overline{\mathfrak{a}^{\mathfrak{F}}\_{f,r}}} \tag{17}$$
