2. Basic equations and numerical method

#### 2.1. Navier-Stokes equation

complex problem based on hypersonic aerodynamics and involving multidisciplinary and multi-domain. Problems of hypersonic aerodynamics and aerodynamic heating, structures and materials technology are the main technological difficulties. There remains much variance between hypersonic flow and supersonic flow about the problem of aerodynamics. For the hypersonic boundary layer, entropy layer and shock wave layer overlapping each other, while it is high temperature and low density flow in boundary layer, which undoubtedly makes the hypersonic boundary layer flow more complex. Therefore, hypersonic aerodynamics is a key technology in the research and development of hypersonic vehicle. Various flow disturbance problems, usually exists in the flying environment of vehicle. For instance, the explosive blast wave [1], reverse jet [2, 3], the non-uniformity flow, the instability of flight, the rough wall and so on, which would happen in flying. It can be seen disturbances are common in flow fields. It's significantly different about the ideal hypothesis state of the steady flowfield and the flow condition existing in disturbance waves. The disturbance in the flowfield will have a significant influence on aerothermodynamics characteristics. The disturbance, whether it is strong or weak, will interfere with the flowfield, especially the shock and boundary layers. After the disturbance in the flowfield interference with shock wave and boundary layer, the disturbance wave will be induced. Then the induced wave will cause further interaction with boundary layer, and create new unstable waves. The stability characteristic of boundary layer and the laminar-turbulent transition mechanism will be significantly changed due to the induced unstable waves. Laminar-turbulent transition not only affects the aerodynamic heating of the wall of hypersonic vehicle, but directly changes the aerodynamic force. Especially, laminarturbulent transition will greatly increases aerodynamic drag, which reduces the lift drag ratio of hypersonic vehicle significantly and increases the requirement of thermal protection.

22 Advances in Some Hypersonic Vehicles Technologies

Therefore, it is necessary to accurately predict the hypersonic unstable flowfield and the flowfield response characteristic induced disturbance waves during the design and development process of hypersonic aircraft. Considering the complexity and expensiveness of the hypersonic vehicle wind tunnel test, it is of practical significance to conduct the numerical simulation of hypersonic unsteady flowfield. In recent years, the hypersonic flowfield response induced by different disturbance waves and the influences of the disturbance wave on the stability of the boundary layer are studied by many scholars using numerical or experimental methods. Ma and Zhong [4] investigated the response of hypersonic boundary layer over a blunt cone to freestream acoustic waves at Mach 7.99. The receptivity of a flat plate boundary layer to a freestream axial vortex is discussed by Boiko [5]. Zhong [6] investigated the leading-edge receptivity to freestream disturbance waves for hypersonic flow over a parabola. The effect of wall disturbances on hypersonic flowfield, and the response of hypersonic boundary layer to wall disturbances are also widely studied [7–10]. Literature [11] points out that, after the interaction between any form of freestream disturbance and the shock wave as well as the boundary layer in hypersonic flow field, three independent forms of disturbance waves, including acoustic disturbance (fast and slow acoustic disturbance), entropy wave disturbance, and vortex wave disturbance will be generated. Among these investigations, most scholars are committed to study the effects of freestream continuous disturbance or the effects of wall disturbance on the stability of boundary layer and laminar-turbulent transition [12–14]. There is still less research on the effects of freestream pulse disturbance on the stability of boundary layer, and the mechanism in this field is still not fully understood. So, this paper

According to the forms of conservation equation, momentum equation and energy equation, the three basic equations of fluid governing equations can be written as a general form, that is, the two-dimensional unsteady compressive N-S equation can be expressed as:

$$\frac{\partial \mathbf{Q}}{\partial t} + \frac{\partial (\mathbf{f} \cdot \mathbf{f}\_v)}{\partial \mathbf{x}} + \frac{\partial (\mathbf{g} \cdot \mathbf{g}\_v)}{\partial y} = \mathbf{0} \tag{1}$$

Where the state vector Q can be expressed as:

$$\mathbf{Q} = \begin{bmatrix} \rho & \rho u\_x & \rho u\_y & \rho E \end{bmatrix}^T \tag{2}$$

Similarly, f, fv , g, gv can be expressed respectively as:

$$\mathbf{f} = \begin{bmatrix} \rho u\_x & \rho u\_x \end{bmatrix}^2 + P \quad \rho u\_x u\_y \quad \left(\rho E + P\right) u\_x \Big|^T \tag{3}$$

$$\mathbf{f}\_v = \begin{bmatrix} \mathbf{0} & \tau\_{\text{xx}} & \tau\_{\text{xy}} & u\_{\text{x}}\tau\_{\text{xx}} + u\_{\text{y}}\tau\_{\text{xy}} + k \frac{\partial T}{\partial \mathbf{x}} \end{bmatrix}^\text{T} \tag{4}$$

$$\mathbf{g} = \begin{bmatrix} \rho u\_y & \rho u\_x u\_y & \rho u\_y^2 + p & (\rho E + p)u\_y \end{bmatrix}^T \tag{5}$$

The variables a<sup>i</sup> and b<sup>i</sup> is the specific coefficients. b1, b<sup>2</sup> and b<sup>3</sup> are equal to 1, 3/4 and 1/3,

Numerical Study of Hypersonic Boundary Layer Receptivity Characteristics Due to Freestream Pulse Waves

http://dx.doi.org/10.5772/intechopen.70660

25

The model parameters and calculation conditions of flowfield calculation includes the freestream condition, model parameter, boundary conditions and meshing. For freestream conditions, the freestream temperature is equal to 69 K, and the Mach number is equal to 6. The Reynolds number, based on the nose radius, is equal to 10,000. The angle of attack is equal to 0. Figure 1 shows the computing model and schematic diagram. Calculation mode is a blunt wedge with the wedge angle of 16e; the nose radius r = 1 mm. the adiabatic wall, no-penetration and non-slip is introduced for wall condition; the symmetric boundary conditions is introduced for y = 0; the freestream conditions and exportation boundary conditions are introduced for the upstream of computing domain and the downstream of calculation domain, respectively. Figure 2 shows the computational grid. The local intensive grid method is carried out near the nose area and wall area, and the grid number is 300 120. The parameters of blunt wedge nose radius r, freestream velocity V∞, freestream temperature T∞, freestream viscosity coefficient μ∞, freestream conductivity coefficient k∞, freestream density ρ<sup>∞</sup> are chosen as the normalized basic measure that is the characteristic

In order to explore the influence of freestream pulse wave on hypersonic flow field, the interaction process between freestream pulse slow acoustic wave and hypersonic flowfield is direct numerical simulated. In present investigation, the stable flow over a blunt wedge at Mach 6 is calculated firstly, and then the simulation of hypersonic unsteady flowfield under the freestream pulse wave is conducted by introducing freestream pulse slow

The introducing time of the freestream pulse wave is recorded as t = 0. The form of pulse slow

acoustic wave at the upstream boundary of computing domain.

acoustic wave is expressed as follows:

Figure 1. Computational model and schematic diagram.

respectively; a1, a<sup>2</sup> and a<sup>3</sup> are equal to 1, 1/4 and 2/3, respectively.

3. Computing model and conditions

variable.

$$\mathbf{g}\_v = \begin{bmatrix} 0 & \tau\_{xy} & \tau\_{yy} & u\_x \tau\_{xy} + u\_y \tau\_{yy} + k \frac{\partial T}{\partial y} \end{bmatrix}^T \tag{6}$$

where ux, uy, P, ρ, T and E indicate the velocity in the x direction, the velocity in the y direction, pressure, density, temperature and total energy, respectively. k, τ and E are the thermal conductivity, the stress component and the total energy, respectively.

#### 2.2. Numerical method

In this paper, a high order finite difference method is used to solve the flowfield governing equation directly. The inviscid vector flux of the Navier-Stokes equation is divided into the positive and negative convection terms using S-W method [17]. The positive and negative convection terms are discretized by the 5th order upwind WENO scheme [18]. The viscous term is discretized by the 6th order central difference scheme [19]. In order to obtain the transient information of the flowfield and reduce computation time, the 3th TVD Runge-Kutta scheme is used for time advance [20].

The spatial discretization for positive and negative convection terms can be expressed as Eqs. (7) and (8), respectively:

$$\mathcal{W}' = \left(m\_1\mathcal{W}\_{j+3} + m\_2\mathcal{W}\_{j+2} + m\_3\mathcal{W}\_{j+1} + m\_4\mathcal{W}\_j + m\_5\mathcal{W}\_{j-1} + m\_6\mathcal{W}\_{j-2} + m\_7\mathcal{W}\_{j-3} + m\_8\mathcal{W}\_{j-4}\right)/\Delta. \tag{7}$$

$$H' = \left(n\_1H\_{j+4} + n\_2H\_{j+3} + n\_3H\_{j+2} + n\_4H\_{j+1} + n\_5H\_j + n\_6H\_{j-1} + n\_7H\_{j-2} + n\_8H\_{j-3}\right)/\Delta \tag{8}$$

where W and H are positive convection terms and negative convection terms, respectively; Δ is the grid spacing; W<sup>0</sup> and H<sup>0</sup> are the difference approximation of the derivative of W and H, respectively; m<sup>i</sup> and n<sup>i</sup> are specific coefficients, no longer list.

The spatial discretizations for viscous terms can be expressed as follow:

$$\mathbf{L}^{'} = \left( \mathbf{K}\_{1} \left( \mathbf{L}\_{\dot{\gamma}+1} - \mathbf{L}\_{\dot{\gamma}-1} \right) + \mathbf{K}\_{2} \left( \mathbf{L}\_{\dot{\gamma}+2} - \mathbf{L}\_{\dot{\gamma}-2} \right) + \mathbf{K}\_{3} \left( \mathbf{L}\_{\dot{\gamma}+3} - \mathbf{L}\_{\dot{\gamma}-3} \right) \right) / \Delta \tag{9}$$

where L, L', Δ and K<sup>i</sup> are viscous terms, the difference approximation of the derivative of viscous terms, the grid spacing and specific coefficients, respectively, which are similar to the symbols in Eqs. (7) and (8).

The TVD Runge-Kutta discretization can be expressed as follows

$$\begin{cases} \mathcal{U}^{(1)} = a\_1 \mathcal{U}^n + b\_1 \mathcal{f}^n \Delta t & \mathcal{U}^{(2)} = a\_2 \mathcal{U}^n + b\_2 \left( \mathcal{U}^{(1)} + f^1 \Delta t \right) \\ \mathcal{U}^{(3)} = a\_3 \mathcal{U}^n + b\_3 \left( \mathcal{U}^{(2)} + f^2 \Delta t \right) & \mathcal{U}^{n+1} = \mathcal{U}^{(3)} \end{cases} \tag{10}$$

where Δt is time increment for time advance; L(U) is the partial derivative of U relative to time.

The variables a<sup>i</sup> and b<sup>i</sup> is the specific coefficients. b1, b<sup>2</sup> and b<sup>3</sup> are equal to 1, 3/4 and 1/3, respectively; a1, a<sup>2</sup> and a<sup>3</sup> are equal to 1, 1/4 and 2/3, respectively.
