International Journal of Aerospace Engineering

Volume 2018, Article ID 7234706, 13 pages

https://doi.org/10.1155/2018/7234706

## Receptivity of the Boundary Layer over a Blunt Wedge with Distributed Roughness at Mach 6

^{1}College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China^{2}China Academy of Space Technology, Beijing 100094, China

Correspondence should be addressed to Xiaojun Tang; moc.anis@78gnatnujoaix

Received 8 December 2017; Accepted 28 March 2018; Published 29 April 2018

Academic Editor: William W. Liou

Copyright © 2018 Zhenqing Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A hypersonic flow field over a blunt wedge with or without roughness is simulated by a direct numerical simulation method. The effect of isolated and distributed roughnesses on the steady and unsteady hypersonic flow field and boundary layer is analyzed. The shape of roughness is controlled by cubic polynomial. The evolution of disturbance waves caused by slow acoustic wave in the boundary layer is investigated by fast Fourier spectrum analysis. The results show that there is a great influence of roughness on the evolution of disturbance waves in the hypersonic boundary layer. The disturbance waves are promoted in the upstream-half region of roughness while suppressed in the downstream-half region of roughness. There is always a mode competition among different modes both in the temporal domain and in the frequency domain in the boundary layer, and mode competition is affected by roughness. The location of the dominant mode which is changed to a second-order harmonic mode from the fundamental mode moves upstream. The vortices caused by roughness also impact the evolution of disturbance waves in the boundary layer. The fundamental mode is suppressed in the vortex region while other harmonic modes are promoted.

#### 1. Introduction

Boundary-layer laminar-turbulent transition effectively affects the aerodynamic lift and drag of hypersonic vehicles and has a great effect on the design of hypersonic vehicles [1–3]. However, the process of transition is very complex and affected by many factors. In general, there are four phases of transition under freestream with small disturbances: (1) receptivity, (2) linear instability, (3) nonlinear stability and saturation, and (4) secondary instability and breakdown to turbulence. The receptivity is crucial in the process of transition because it defines the initial state of the disturbances in the boundary layer. Receptivity of the boundary layer is the response process of the boundary layer to external disturbances that include freestream disturbances, wall temperature, and roughness.

Investigating the effect of freestream disturbances is significant to understand the transition mechanism. The type and parameters of freestream disturbances both affect the process of transition, and it has been investigated by many scholars [4–11]. As the amplitude of freestream disturbances increases to finite amplitude, the second and third phases of transition may be short or even disappear. The transition mechanism is essentially changed by the change of amplitude of freestream disturbances. Ma and Zhong [4, 5] investigate the receptivity over a plate flat under the freestream with small disturbances by DNS and linear stability theory (LST). Balakumar and Kegerise [6] analyze the response of the boundary layer over straight and flared cones to small disturbances. Small disturbances and finite amplitude disturbances (10^{−4}–10^{−1}) in the boundary layer over the blunt wedge are both investigated to reveal the leading edge receptivity by Cerminara and Sandham [7], and the Fourier analysis method is used to analyze the evolution of disturbances in the boundary layer. D. Park and S. O. Park [8] and Olaf et al. [9] also investigate the evolution process under the wall roughness in nonlinear instability by Fourier spectrum analysis. As for the type of freestream disturbances, they can be decomposed into entropy, vortical, and acoustic waves which include slow acoustic waves and fast acoustic waves. Qin and Wu [10], Zhang et al. [11], and Ma and Zhong [5] research the receptivity characteristic of a supersonic/hypersonic boundary layer under small disturbances with different types of freestream disturbances and found that there is a great difference between the four kinds of disturbances on the transition of the boundary layer.

In view of the fact that the change of flow state in the hypersonic boundary layer directly affects the propulsion efficiency and maneuverability of the hypersonic vehicle, and results in aerodynamic drag and significantly increased wall heating, laminar flow control (LFC) technologies are addressed. Laminar flow control technologies can be divided into three categories [12]: (1) active techniques: such as reverse jet [13], local heating, or cooling; (2) passive techniques: such as smoothing and shaping the wall; and (3) reactive techniques: such as actuators and microelectromechanical systems. However, due to complicated working conditions and serious environment, active techniques and reactive techniques are difficult to be applied in engineering practice, and passive technology is widely used in hypersonic vehicle design [14].

Passive techniques include a variety of technologies such as shaping and passive coatings [15–17]. There is a very effective method to arrange roughness on the wall. Roughness can be classified by many methods, such as two-dimensional roughness and three-dimensional roughness and isolated roughness and distributed roughness [18].

The effects of different types of roughness on the receptivity and stability of the boundary layer vary greatly. A large number of investigations have been carried out to investigate the effects of roughness on the receptivity and stability of the hypersonic boundary layer, and some achievements have been made [19–28]. Wang and Zhong and Fong et al. [20, 21] analyzed the effect of the height of isolated roughness on the receptivity of the hypersonic boundary layer by direct numerical simulation. They found that the height of isolated roughness made the disturbance grow instantaneously, and the growth rate increased with the increase of the height of isolated roughness. Zhao et al. [22] found that the transition position moved forward with the increase of the height of isolated roughness. Holloway and Sterett and Fujii [23, 24] found that the position of the isolated roughness would also affect the transition. Duan et al. [25, 26] found that the change of the position of isolated roughness directly affects the propagation of mode S in the boundary layer by a direct numerical simulation method and then affected the receptivity mechanism of the hypersonic boundary layer.

In terms of isolated roughness, the research of distributed roughness is less [18, 27, 28]. Obviously, the effect of distributed roughness on the flow field is more complicated. Balakumar [18] used a numerical simulation method to analyze the influence of distributed roughness on the receptivity and stability of the hypersonic tip-cone flow boundary layer under slow acoustic disturbances, fast acoustic disturbances, and vortex wave. It was found that the second mode disturbance in the nose region was significantly suppressed. Desjouy et al. [27] discussed the effects of isolated and distributed roughnesses on the transition of the hypersonic boundary layer. They found that the transition of the boundary layer appeared more than that in the presence of a single roughness and the spacing between the two roughnesses has a great influence on the flow state in the boundary layer. Saric et al. [28] experimentally proved that the transition of the boundary layer can be effectively restrained by arranging the roughness array at the leading edge.

In this paper, a high-order accurate finite difference method is used to simulate the hypersonic flow field over the blunt wedge with a smooth wall and rough wall, and the effect of roughness on the flow state in the boundary layer is discussed. The shape of roughness is controlled by cubic polynomial, and the wall conditions are divided into a smooth wall and rough wall with isolated roughness and distributed roughness. The influence of roughness on the steady-state flow field is investigated firstly. On this basis, the influence of the roughness on the evolution process of disturbances in the boundary layer is analyzed by adding the slow acoustic disturbances to disturb the hypersonic flow field. The effect of roughness on evolution of different mode disturbances in the boundary layer is present by using the fast Fourier spectrum analysis (FFSA) method, and the influence mechanism of roughness on the receptivity of the hypersonic boundary layer is revealed.

#### 2. Governing Equations and Numerical Methods

##### 2.1. Governing Equations

A high-order finite difference method is employed to simulate the hypersonic flow field to analyze the evolution of the disturbance wave in the hypersonic boundary layer and to investigate the effect of roughness on receptivity of the boundary layer. The governing equations are two-dimensional conservation N-S equations which can be expressed as
where is the time and is the vector of conservative variables, and are the convective fluxes corresponding to *x* and *y* directions, respectively, and and are the viscous fluxes corresponding to the *x* and *y* directions, respectively. Their specific expressions are written as
where , , , , , and denote density, velocity along the *x*-axis and *y* direction, pressure, temperature, and the heat transfer coefficient, respectively. and is the shear stress and total energy, respectively. The state equation for ideal gas is written as
where is 286.94 J/(kg·K).

In order to be convenient for computation, the governing equations are transformed from the physical coordinates (*x*, *y*) to computational coordinates (*ξ*, *η*), and the coordinate transformation coefficient is the Jacobian matrix that can be expressed as

The governing equations in the computational coordinates are expressed as where

##### 2.2. Numerical Methods

In order to accurately simulate the hypersonic flow field over a blunt wedge and correctly capture the changes of the hypersonic flow field and the evolution of the disturbance wave in the boundary layer, a high-order finite difference method is used to directly simulate the hypersonic flow field. The governing equation can be decomposed into three parts for discretization. The convective flux is split by the Steger-Warming (S-W) splitting method into positive and negative flux terms, and then the fifth weighted essential nonoscillatory (WENO) scheme [29–31] is used to discretize the positive and negative flux terms. The central [25] difference method not only introduces unnecessary information but also causes calculation error. It is worse that they are not robust enough in the shock region, and there are very serious, nonphysical numerical oscillations generated in the shock region. So the upwind scheme is proper to discretize the convective flux, and there is a great advantage in this respect. The WENO scheme can effectively suppress the numerical oscillation in a discontinuous or large gradient region, such as the shock region, and the information of the flow field can be acquired stably and accurately. The physical information of the flow field is spread around caused by viscosity, so the viscous fluxes are discretized by a six-order central difference scheme. This method only introduces phase errors but not dissipative errors, which can ensure the accuracy of the calculation. The three-step third-order Runge-Kutta method is employed to advance time. This method is now widely used in the direct numerical simulation of hypersonic compressible flow fields, and the results achieved are good.

#### 3. Computational Condition

A computational model is a hypersonic flow over a blunt wedge. Freestream conditions are adopted at the inlet, and extrapolation boundary conditions are adopted at the outlet; symmetric boundary conditions are employed at due to the symmetry of the model. The wall surface has no slip and no penetration and is isothermal. The influence of roughness on the receptivity of the hypersonic boundary layer is analyzed, and the wall surface conditions are divided into three types: wall surface without roughness, with isolated roughness, and with distributed roughness. The shape of roughness is controlled by a third-order polynomial, and the specific expression is as follows:
where , , , and represent the nose radius, the half width of roughness, the height, and the half wedge angle, respectively, and is the center coordinate of roughness. . represents the number of roughness: when , the wall surface condition is the wall with isolated roughness, and when , it represents the wall with distributed roughness. For this paper, is 6. The space between two roughnesses is . The calculational conditions and grid are given in Figure 1, where represents the angle of attack. The subscripts “∞” and “*w*” denote the freestream and wall condition, respectively. The geometric parameters of roughness are given in Table 1. The number of meshes is 600 × 150, and the exponential stretching method is used to close meshes near the wall surface and head area properly where a strong shear flow is in the area.