International Journal of Aerospace Engineering

Volume 2016, Article ID 3196057, 14 pages

http://dx.doi.org/10.1155/2016/3196057

## Receptivity of Boundary Layer over a Blunt Wedge due to Freestream Pulse Disturbances at Mach 6

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

Received 23 July 2015; Revised 28 February 2016; Accepted 14 March 2016

Academic Editor: Saad A. Ahmed

Copyright © 2016 Jianqiang Shi 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

Direct numerical simulation (DNS) of a hypersonic compressible flow over a blunt wedge with fast acoustic disturbances in freestream is performed. The receptivity characteristics of boundary layer to freestream pulse acoustic disturbances are numerically investigated at Mach 6, and the frequency effects of freestream pulse wave on boundary layer receptivity are discussed. Results show that there are several main disturbance mode clusters in boundary layer under acoustic pulse wave, and the number of main disturbance clusters decreases along the streamwise. As disturbance wave propagates from upstream to downstream direction, the component of the modes below fundamental frequency decreases, and the component of the modes above second harmonic components increases quickly in general. There are competition and disturbance energy transfer between different boundary layer modes. The nose boundary layer is dominated by the nearby mode of fundamental frequency. The number of the main disturbance mode clusters decreases as the freestream disturbance frequency increases. The frequency range with larger growth narrows along the streamwise. In general, the amplitudes of both fundamental mode and harmonics become larger with the decreasing of freestream disturbance frequency. High frequency freestream disturbance accelerates the decay of disturbance wave in downstream boundary layer.

#### 1. Introduction

The flow separation, lift, drag, and heat-load on surface are highly dependent on the stare of boundary layer flow when the aircraft flies through the atmosphere. However, laminar and turbulent are usually considered as types of boundary layer flow state [1]. And the aerodynamic forces and aerothermal characteristics for laminar state are significantly different from those for turbulent [2]. Also, it is known from [3] that the aerodynamic heating and forces of the aircraft are significantly affected by laminar-turbulent transition of flow state, which is the process of flow instability. Thus, it is of great importance to simulate the process from laminar condition to turbulent one. It is meaningful for the optimization design of aerodynamic shape, thermal protection, and safety improvement. Given the importance of investigations on boundary layer’s stability, a series of experimental and numerical investigations on the subject are conducted [4–8]. At the same time, the research of the stability and transition mechanisms for compressible flow becomes popular in the field of aerodynamics. Compared to incompressible flow, the investigations on the flow characteristics of compressible flow started later, and compressible flow is more complex than incompressible flow [9–11]. Fedorov and Khokhlov [12] confirmed that several unstable modes exist in hypersonic boundary layer. Some modes (mainly the first mode) which correspond to squire mode in incompressible boundary layer are subordinate to viscous unstable modes. Additional inviscid disturbances are called high mode disturbance, and it is believed that the first additional disturbances are the least stable modes for two-dimensional disturbances. Besides, a lot of works on the stability of hypersonic boundary layer were performed by many researchers, including the impact parameter and the theoretical explanation of stability characteristic which are widely studied [13–17]. The receptivity of the boundary layer over a flat plate is studied by Ma and Zhong [13], and the effects of different types of freestream disturbances are considered. Layek et al. [14] numerically studied the response of laminar flow in a symmetric sudden expanded channel to wall blowing-suction and found the critical Reynolds number for symmetry breaking of the flow decreased with the increasing values of suction speeds. The effects of nose bluntness on hypersonic boundary layer receptivity and stability over cones are discussed in [15]. Lee et al. [16], based on two different wall temperatures, investigated the effects of wall heating on turbulent boundary layers with temperature-dependent viscosity. Krogstad and Antonia [17] investigated the effects of surface roughness on a turbulent boundary layer by experiment; the turbulent energy production and the turbulent diffusion are significantly affected by rough surfaces. As is shown in previous studies, most of these researches focused on the receptivity of freestream disturbance and wall blowing-suction, effects of nose bluntness on boundary layer stability and wall temperature on disturbance modes evolution in boundary layer, and response of hypersonic boundary layer to roughness wall and turbulence degree in inlet flow. Investigations on this area are usually based on the time period state of unsteady flow under disturbance wave [18]. However, few works were conducted on the interactions between freestream pulse wave and flowfield as well as boundary layer and the stability characteristics of boundary layer for pulse disturbance, whereas flowfield characteristic especially boundary layer state under freestream pulse wave is rather different from that under freestream continuous wave, and the initial generation and development of boundary layer disturbance wave for the former are rather different from the latter. However, the disturbance production and development produced significant effect on boundary layer receptivity [11], which is a significant role in the transition from laminar flow to turbulent. Systematic investigations on the receptivity of boundary layer under freestream pulse are still rare, which not only are helpful to understand the stability characteristic of boundary layer under pulse wave, but also can give a new perspective for boundary layer stability mechanism studies. Therefore, it is necessary to investigate the boundary layer’s receptivity under freestream pulse wave. The significant impact of disturbance wave frequency in freestream is confirmed in the investigation on receptivity of hypersonic boundary layer to freestream continuous and weak disturbance wave [19]. It is believed that freestream pulse disturbance frequency had influenced the receptivity of hypersonic boundary layer. Thus, it is necessary to complete much more investigations on the effects of pulse wave frequency in freestream receptivity.

Receptivity is the process that the freestream disturbances react with the shock wave, and then the disturbance entering into the boundary layer leads to the unstable disturbance wave in the boundary layer. To make a survey of the receptivity the theoretical method, the experimental research, and the numerical simulation are all the available tools. The flow stability theory is mainly including linear stability theory and nonlinear stability theory which is mainly used in the theoretical method. The main theoretical analysis tool for small perturbation wave evolution analysis is linear stability theory (LST). When the external disturbance is weak, there will be some linear growth areas with a long distance, and this distance also includes the process of receptivity. Thus, the linear stability theory and the receptivity theory are commonly used to analyze the growth and evolution of small disturbances. That can be used to identify the main components of stability characteristics of boundary layer disturbances [20]. LST is also the basic starting point of the method to predict the transition point. But when the flow disturbance is large, nonlinear effects become significant, and the linear stability theory will no longer play a part. The nonlinear stability theory is formed to explain the nonlinear flow transition mechanism. Although the theoretical analysis is reliable and suitable for some typical problems of flow stability, it is of great complexity and can provide only limited flowfield information and its function is limited for broader flow stability problems. The experimental research can provide physical flowfield information in the flow stability investigations. However, it is a huge project to make an experimental research in the aircraft engineering. It will cost a lot of money and energy even in a small experiment and is easily affected by the surroundings [21]. Considering the weak point of the experimental research and the theoretical method a new practical method exists. It is the numerical simulation that dominates in the hypersonic boundary layer stability research. Numerical simulations have high calculation accuracy and waste less energy and the calculation results suit well with the actual situation. DNS is one of the practical methods to simulate the receptivity of the boundary layer. It is of high accuracy and does not need any turbulence model [22].

As is stated above, the unsteady flowfield over a blunt wedge with 8° half-wedge-angle at Mach 6 under the action of freestream pulse acoustic wave is computed in the present paper. The receptivity characteristics of boundary layer to freestream pulse wave are analyzed, and the effects of pulse wave frequency in freestream on boundary layer receptivity are investigated.

#### 2. Solution Algorithms

The flowfield governing equations are the two-dimensional Navier-Stokes (N-S) equations, which can be written as follows:The variable in the equations is the vectors terms; the variable in the equations is inviscid terms, and the variable is viscous flux terms. The variables , , and are given byThe variables , , , , , , , and in (2) are density, velocity, total energy, Kronecker symbol, shear stress, pressure, and heat conductivity coefficients, respectively. The viscosity coefficient is determined by the Sutherland law; it is shown in Here, the Sutherland temperature K. Under the perfect gas assumption, the pressure is obtained byIn the present work, the compressible Navier-Stokes equations are solved by using a high order finite difference method for space discretization and time integration. The convection terms are split into positive convection terms and negative convection terms by Steger-Warming splitting method [23]. Assume the Jacobian is and the eigenvalue is , where . In this case, the inviscid flux terms can be split as follows:A 5th-order accurate weighted essentially nonoscillatory (WENO) scheme [24] is used for the space discretization of both positive convection terms and negative convection terms:A 6th-order center difference scheme [25] is employed for viscous terms space discretion:Here, the variables , , and are weighting coefficient.

A third-order, total variation diminishing (TVD) Runge-Kutta scheme [26] is introduced to time integrationIn (8), is time increment and and are the weighting coefficient.

These methods are suitable in the flows with discontinuities or high gradient regions. In particular, the high order weighted essentially nonoscillatory (WENO) and the improved WENO methods are widely implemented in the DNS of compressible turbulent flows, in order to keep higher order approximations in smooth regions and to eliminate or suppress oscillatory behavior near the discontinuities [27–31]. The solution of the unsteady flowfield can be described as follows: in a given direction, the spatial derivatives at the nodes are approximated by a higher order interpolation and the neighboring nodal values in that direction. The resulting equations are then integrated into time to get the instantaneous parameter values at the point as a function of time. To validate this numerical method, the similar situation of the numerical simulation is performed by the numerical scheme, which is a Mach 8 flow over a 5.3° half-angle wedge under disturbances wave. Figure 1 shows the comparison of the pressure disturbance amplitude of the fourth harmonic mode development along streamwise with Wang et al.’s result [32], indicating that the numerical scheme used in this paper is available.