3. Computing model and conditions

g ¼ ρuy ρuxuy ρuy

ductivity, the stress component and the total energy, respectively.

respectively; m<sup>i</sup> and n<sup>i</sup> are specific coefficients, no longer list.

The TVD Runge-Kutta discretization can be expressed as follows

n

¼ K<sup>1</sup> Ljþ<sup>1</sup> � Lj�<sup>1</sup>

<sup>U</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>a</sup>1U<sup>n</sup> <sup>þ</sup> <sup>b</sup>1<sup>f</sup>

<sup>U</sup>ð Þ<sup>3</sup> <sup>¼</sup> <sup>a</sup>3U<sup>n</sup> <sup>þ</sup> <sup>b</sup><sup>3</sup> <sup>U</sup>ð Þ<sup>2</sup> <sup>þ</sup> <sup>f</sup>

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

� � <sup>þ</sup> <sup>K</sup><sup>2</sup> Ljþ<sup>2</sup> � Lj�<sup>2</sup>

2 Δt � �

2.2. Numerical method

24 Advances in Some Hypersonic Vehicles Technologies

used for time advance [20].

L 0

8 < :

and (8), respectively:

H0

Eqs. (7) and (8).

W<sup>0</sup>

g<sup>v</sup> ¼ 0 τxy τyy uxτxy þ uyτyy þ k

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 con-

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

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

¼ m1Wjþ<sup>3</sup> þ m2Wjþ<sup>2</sup> þ m3Wjþ<sup>1</sup> þ m4Wj þ m5Wj�<sup>1</sup> þ m6Wj�<sup>2</sup> þ m7Wj�<sup>3</sup> þ m8Wj�<sup>4</sup> � �

¼ n1Hjþ<sup>4</sup> þ n2Hjþ<sup>3</sup> þ n3Hjþ<sup>2</sup> þ n4Hjþ<sup>1</sup> þ n5Hj þ n6Hj�<sup>1</sup> þ n7Hj�<sup>2</sup> þ n8Hj�<sup>3</sup>

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,

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

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

� �=Δ (8)

� � <sup>þ</sup> <sup>K</sup><sup>3</sup> Ljþ<sup>3</sup> � Lj�<sup>3</sup> � � � � =Δ (9)

<sup>Δ</sup>t Uð Þ<sup>2</sup> <sup>¼</sup> <sup>a</sup>2U<sup>n</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>U</sup>ð Þ<sup>1</sup> <sup>þ</sup> <sup>f</sup>

<sup>U</sup><sup>n</sup>þ<sup>1</sup> <sup>¼</sup> <sup>U</sup>ð Þ<sup>3</sup>

1 Δt � �

� �<sup>T</sup>

<sup>2</sup> <sup>þ</sup> <sup>p</sup> <sup>ρ</sup><sup>E</sup> <sup>þ</sup> <sup>p</sup> � �uy

� �<sup>T</sup> (5)

∂T ∂y

(6)

=Δ (7)

(10)

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 variable.

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 acoustic wave at the upstream boundary of computing domain.

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

Figure 1. Computational model and schematic diagram.

Figure 2. Mesh grid.

$$
\begin{bmatrix} u' \\ v' \\ p' \\ \rho' \end{bmatrix} = \begin{cases} \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}^T & (t < 0) \\ \begin{bmatrix} A & 0 & -\frac{A}{M\_{\bullet}} & -AM\_{\bullet} \end{bmatrix}^T e^{i\left(kx - \frac{R\alpha\_{\theta}}{10^{\theta}} + \frac{\pi}{2}\right)} & (0 \le t < 2) \\\ \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 \end{bmatrix}^T & (t \ge 2) \end{cases} \tag{11}
$$

deformed and protruded outward, and the surrounding pressure is also significantly affected. Due to the deformation of the shock wave, the pressure of outward convex region near shock wave is significantly increased, and the flowfield pressure below the bow shock wave is greatly reduced. That is to say, from Figure 3(a)–(d), it shows that the freestream disturbance waves significantly interact with the bow-shaped shock waves, which greatly changes the

Figure 3. Contours of pressure under freestream pulse wave at different times. (a) t=2.0, (b) ) t=4.0, (c)t=6.0, (d) ) t=8.0.

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

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

27

Figure 4 shows the contours of density under freestream pulse wave at different times. Figure 4(a)–(f) correspond to t = 0.0, 2.0, 4.0, 6.0, 8.0 and 10.0, respectively. Figure 5 shows the contours of temperature under freestream pulse wave at different times. Figure 5(a)–(f) correspond to t = 0.0, 2.0, 4.0, 6.0, 8.0 and 10.0, respectively. As can be seen from Figure 4, the pulse wave disturbance has a great impact on the density. The density of the flowfield changes significantly under the action of freestream pulse wave, especially in the disturbance area between the pulse wave and the bow-shaped shock wave, where the density of the region is significantly smaller. Compared with the significant changes of flowfield density around the shock wave, the pulse wave has much smaller effect on the density in the boundary layer. Meanwhile, the closer to the nasal area, the greater the density changes and the effects are after the action of pulse wave. Obviously, this is because the closer the bowshaped shock wave is to the nose area, the stronger the action is. Therefore, it can be concluded that the stronger the intensity of the shock wave is, the more intense the effect of the pulse slow acoustic wave and the bow-shaped shock wave are. As can be seen from

shock standoff distance and the distribution of flowfield pressure in the active region.

Where u', v', P<sup>0</sup> and ρ' indicate the velocity disturbance along x direction, the velocity disturbance along y direction, the pressure disturbance and the density disturbance, respectively; A indicates the amplitude, F indicates the generalized frequency, t indicates time. Here, <sup>A</sup> = 8 � <sup>10</sup>�<sup>2</sup> ; <sup>k</sup> = 3.1446 � <sup>10</sup>�<sup>4</sup> ; F = 50π; Ma∞ = 6.
