Three-dimensional receptivity of high-speed blunt cone to different types of freestream disturbances

Wan Bing-Bing; Hu Wei-Bo; Li Xiao-Hu; Huang Wen-Feng; Chen Jian-Qiang; Tu Guo-Hua

doi:10.7498/aps.73.20241383

Abstract

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1. Introduction
2. 数值方法与参数设置
2.1 控制方程
2.2 来流扰动波
2.3 钝锥感受性计算模型
3. 不同来流扰动入射的三维感受性对比分析
3.1 频率ω = 1.1来流扰动的三维感受性
3.2 频率ω = 5来流扰动的三维感受性
4. 结　论

References

Abstract

Receptivity to freestream disturbances is the initial stage of the boundary-layer transition process, which can determine the final path of boundary-layer disturbance triggering transition. At present, there is relatively sufficient research on the receptivity of two-dimensional boundary layers to zero incident angle disturbances. In fact, the freestream disturbances often propagate into the boundary layer in the form of non-zero incident angle, resulting in a component of spatial disturbance in the circumferential direction of rotating body (such as a cone). It is a receptivity problem with distinct three-dimensional features. However, there is relatively little research on this three-dimensional receptivity issue. The preliminary work only studied the three-dimensional receptivity to low-frequency incident slow acoustic waves. There has not been a systematic study on the three-dimensional receptivity to different types of freestream disturbances. The three-dimensional receptivity of a blunt cone to different freestream disturbances is studied in this work. Firstly, a high-resolution numerical simulation method is used to investigate the three-dimensional receptivity process by introducing freestream disturbances with an incident angle of 15°. The freestream disturbances include fast acoustic wave, slow acoustic wave, entropy wave, and vortex wave. Their frequencies are chosen as dimensionless 1.1 and 5, corresponding to the first mode frequency and the second mode frequency, respectively. Then, the phase velocity and shape function of the boundary-layer disturbances at each position of circumference for the numerical results are obtained by Fourier transform. To explain the receptivity mechanisms, the corresponding results by linear stability analysis are obtained for comparisons. The results are shown below. The first mode and the second mode of the boundary layer can be effectively excited by the incident slow acoustic waves; it is difficult for the incident fast acoustic waves to excite unstable modes in the boundary layer; the incident entropy wave and vortex wave are difficult to excite the first mode at low frequency, but can excite the second mode at high frequency. Furthermore, the incident angle of the freestream disturbances can lead to the differences in the receptivity at different circumferential positions of the cone, which can be reflected in two ways. One is the difference in the dominant disturbance form at different circumferential positions, and the other is the difference in the amplitude of boundary-layer disturbances. Under different disturbance types and frequencies, these differences between different circumferential positions exhibit different results. The strongest receptivity may occur on the incident front, the incident back, and the incident side. These phenomena may result from the combined action of the upstream head disturbance and the disturbance on the incident front.

Keywords:

PACS:

47.15.Fe	(Stability of laminar flows)
47.20.Ib	(Instability of boundary layers; separation)
47.20.Pc	(Flow receptivity)
47.40.Ki	(Supersonic and hypersonic flows)

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State Key Laboratory of Aerodynamics, Mianyang 621000, China

Corresponding author: Tu Guo-Hua, ghtu@skla.cardc.cn

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1. Introduction

Boundary layer transition is a process in which the flow changes from laminar to turbulent. How to predict the transition is one of the problems that have not been completely solved in classical mechanics. This process can lead to dramatic changes in friction, heat exchange, noise, etc. It has been pointed out that the heat flux and friction in turbulent state after transition are 3-5 times that in laminar state. ^{[ 1 ]} Therefore, the transition problem is an important factor that has to be considered in the fine design of aircraft. Usually, the boundary layer transition process goes through three stages, namely, receptivity, disturbance instability, and transition to turbulence. Among them, the receptivity stage mainly solves the problem of how external disturbances excite unstable disturbance waves in the boundary layer, which can provide information such as initial amplitude and phase for the disturbance evolution in the latter two stages. Saric et al. ^{[ 2 ]} Five transition paths have been summarized according to the different receptivity processes caused by different external disturbance amplitudes. When the external disturbance amplitude is low, the boundary layer instability disturbance wave excited by receptivity generally evolves according to linear instability, nonlinear instability and secondary instability. When the external disturbance amplitude is high, the boundary layer disturbance evolution after receptivity may skip the linear instability or even the second stage of the whole transition and directly reach the third stage. Therefore, ^{[ 3 ]} Zeng pointed out that "in order to solve the problem of transition prediction, the most needed basic research is the problem of receptivity under supersonic/hypersonic conditions".

The research on receptivity has a long history. In the early stage, it mainly focused on the field of low-speed flow, and the research is more mature, such as Goldstein. ^{[ 4 , 5 ]} 、Ruban ^{[ 6 ]} Choudhari and Streett ^{[ 7 , 8 ]} 、Duck ^{[ 9 ]} Hammerton and Kerschen ^{[ 10 ]} Wanderley and Corke ^{[ 11 ]} 、Wu ^{[ 12 , 13 ]} , Dong et al. ^{[ 14 ]} Lu Changgen et al. ^{[ 15 ]} By means of the front of the body and surface roughness elements, the temporal or spatial scale of the inflow disturbance is converted into the scale of the instability wave in the boundary layer, thus exciting the corresponding disturbance wave, which is not described in detail in this paper. For the field of high-speed flow, the research has been carried out relatively late, and some progress has been made after nearly 30 years of efforts. Fedorov ^{[ 16 ]} , Zhong and Wang ^{[ 17 ]} Jiang Xianyang and Li Cunbiao ^{[ 18 ]} Su Caihong ^{[ 19 ]} Some summaries were made. In the 1990s, Fedorov and Khokhlov ^{[ 20 – 22 ]} A breakthrough has been made in the study of high-speed boundary layer receptivity. They proposed the receptivity "synchronization" theory, that is, when the frequency and wavenumber (or phase velocity, the ratio of frequency to wavenumber) of the excited disturbance and the excited disturbance are very close, the action of the excitation will occur. This theory includes the leading edge "synchronization" theory and the downstream intermediate mode conversion "synchronizing" theory. They found that there are two modes in the boundary layer near the leading edge, according to their phase velocity. ^{[ 23 – 25 ]} The receptivity theory comprehensively describes the receptivity process of incoming flow disturbance, and has been applied and enriched in the receptivity of sharp flat plate.

Most of the components of aircraft are generally designed as blunt body models, so the study of blunt body problems can better reflect the actual receptivity process. Under the condition that wind tunnel experiments can not be widely carried out, direct numerical simulation technology is the most commonly used means to study receptivity. The research models include blunt plate, blunt wedge and blunt cone. Generally, two-dimensional problems are studied by Zhong et al. ^{[ 26 – 28 ]} And Balakumar et al. ^{[ 29 – 31 ]} Zhang Yudong and others in China ^{[ 32 ]} Qin and Wu ^{[ 33 ]} Ba, etc ^{[ 34 , 35 ]} And Wan et al. ^{[ 36 – 38 ]} Due to the bluntness and the bow shock, there is an inviscid thin layer attached to the boundary layer in the flow field behind the shock, which is characterized by high temperature and high entropy gradient, called the entropy layer. ^{[ 39 – 43 ]} Therefore, the incoming flow disturbance does not directly act on the boundary layer, but excites the boundary layer mode after passing through the bow shock wave and the entropy layer. At this time, the boundary layer is no longer faced with the incoming flow disturbance. This phenomenon can be described by direct numerical simulation to realize the representation of the receptivity, but it can not answer the questions of the receptivity mechanism, such as the disturbance components between the shock wave and the boundary-layer, the propagation path of the disturbance, and the response mechanism of the boundary-layer to the disturbance.

In response to this problem, Wan et al. ^{[ 36 – 38 ]} In this paper, a receptivity path analysis method is proposed, in which the receptivity process of a bluff body is divided into two stages. In the first stage, the bow shock is impacted by an incoming flow disturbance, which can generally produce all kinds of disturbance waves, namely, acoustic waves (including fast and slow acoustic waves), entropy waves and vortex waves, which can be proved by the solution of the linear disturbance Rankine-Hugoniot (R-H) relation. ^{[ 44 – 46 ]} In the second stage, the disturbance waves behind these shock waves propagate downstream to excite the boundary layer modes, and then the receptivity problem can return to the "synchronization" problem of Fedorov and Khokhlov. However, the disturbance waves need to pass through the entropy layer before entering the boundary layer, so the receptivity process will also be affected by the disturbance of the entropy layer. Take the slow sound wave in the inflow as an example, Wan et al. ^{[ 36 – 38 ]} In this paper, the sensitivity path mechanism of blunt cone and blunt wedge to low and high frequency waves is studied, and it is found that the disturbance wave behind the shock wave can promote the excitation of boundary layer modes, in which the slow sound wave behind the shock wave dominates the excitation of the first mode, while the entropy layer disturbance dominates the excitation of the second mode. There is an obvious phase velocity difference between the entropy layer disturbance and the excited second mode. This excitation process is completed by the forced excitation of the entropy layer perturbation to the boundary layer, so its mechanism is different from the classical ^{[ 34 ]} The receptivity path mechanism of different bluntness is further investigated, and the same conclusion is obtained. It is also found that when the receptivity process of incoming flow disturbance is considered, e. ^N The dominant frequency of transition analyzed by the method will be reduced, and even the low-frequency first mode may replace the high-frequency second mode to dominate the transition, which also shows that whether the receptivity is considered or not will be related to the accuracy of transition prediction. ^{[ 35 ]} .

In wind tunnel experiments, most of the incoming flow disturbance waves are radiated from the nozzle surface, and there is often an angle between the radiation direction and the incoming flow, which is called the incident angle of the incoming flow disturbance waves. It is generally stipulated that the downward direction is positive and the upward direction is negative. Fedorov ^{[ 22 ]} Based on the theory of scattering and diffraction of sound wave at the leading edge of an incoming flow, the influence of the incident angle of the sound wave on the receptivity is studied, and the receptivity coefficients of the fast and slow modes of the boundary layer excited by the fast and slow sound waves at different incident angles are obtained. It is found that the receptivity coefficients are larger when the absolute value of the incident angle is smaller, and the receptance coefficients decrease with the increase of the absolute value. However, Fedorov ^{[ 22 ]} The theoretical study is aimed at the ideal basic flow, that is, the shock wave of the bluff body model is not considered. For the bluff body model with zero angle of attack (the angle between the basic flow direction of the incoming flow and the symmetry plane/axis of the model is zero), it can be divided into an infinite span body (such as a blunt wedge and a blunt plate) and a rotating body (such as a blunt cone). For the blunt wedge and a blunt plate, according to the incident direction of the disturbance wave of the incoming flow, the upper ^{[ 25 ]} 、Balakumar ^{[ 30 ]} And Malik and Balakumar ^{[ 47 ]} The influence of the incident angle on the sensitivity is studied, and it is found that the influence of the incident wave on the sensitivity of the upper and lower surfaces of the blunt plate is similar to Fedorov's. ^{[ 22 ]} The conclusion is basically the same: when the absolute value of the incident angle increases, the receptivity intensity on the windward side decreases, and the receptivity intensity on the leeward side also decreases, but the change is relatively small.

对于钝锥模型, 由于其展向旋转, 在考虑来流扰动的入射角时, 每个子午面的扰动感受性及演化情况都不一样. 子午面在这里指的是钝锥对称轴与经线或母线构成的所在平面, 如图1所示. 这变成一个较为复杂的三维问题, 目前研究比较少. Balakumar^[48]曾开展过钝锥对三维声波的感受性研究, 一方面尽管是三维问题, 但不是入射角导致的; 另一方面研究不够细致, 没有针对来流扰动与激波作用、激波后新扰动下游传播、进入边界层激发边界层模态的感受性全过程. Wan等^[49]考虑了激波以及激波后扰动的作用, 考察了来流慢声波入射角对激发钝锥边界层第一模态的三维感受性的影响问题, 发现入射角的影响规律与钝板和钝楔的不一样. 入射角越大, 在钝锥的入射正面激发第一模态越弱, 但在钝锥入射背面的感受性不再是变弱, 而是变强, 原因是主导激发第一模态的激波后新慢声波在入射背面时产生的位置相比入射正面更早.

$图 1 钝锥子午面示意图\r\nFig. 1. Schematic diagram of blunt-cone meridian plane.$

图 1 钝锥子午面示意图

Fig. 1. Schematic diagram of blunt-cone meridian plane.

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来流扰动一般可以分解为声波(包括快声波和慢声波)、熵波和涡波, 作者前期工作^[49]仅考察了入射来流慢声波激发第一模态的三维感受性, 尚未考察不同来流扰动波类型、更高频扰动以非零入射角入射时的三维感受性问题. 本文将在前期工作认识基础上, 进一步开展不同频率、不同来流扰动波入射的高速钝锥三维感受性问题研究, 对比分析不同来流扰动波对激发钝锥边界层模态的影响规律, 包括激发第一模态和第二模态. 本文第2节将介绍研究方法和计算模型; 第3节通过在来流分别引入入射角15°条件下不同频率的不同类型扰动波, 对比分析对三维感受性的影响规律; 最后是结论.

2. 数值方法与参数设置

2.1 控制方程

控制方程采用完全气体模型下的三维Navier-Stokes方程, 用于计算层流场和扰动场, 形式如下:

$\frac{{\partial {\boldsymbol{U}}}}{{\partial t}} + \frac{{\partial {\boldsymbol{E}}}}{{\partial x}} + \frac{{\partial {\boldsymbol{F}}}}{{\partial y}} + \frac{{\partial {\boldsymbol{G}}}}{{\partial z}} = \frac{{\partial {{\boldsymbol{E}}_v}}}{{\partial x}} + \frac{{\partial {{\boldsymbol{F}}_v}}}{{\partial y}} + \frac{{\partial {{\boldsymbol{G}}_v}}}{{\partial z}}.$

(1)

式中, t为时间变量, x, y, z分别为轴向、法向和展向的空间坐标. U为守恒型变量, $\dfrac{{\partial {\boldsymbol{E}}}}{{\partial x}}$ , $\dfrac{{\partial {\boldsymbol{F}}}}{{\partial y}}$ , $\dfrac{{\partial {\boldsymbol{G}}}}{{\partial z}}$ 为对流项, 包含压力项, $\dfrac{{\partial {{\boldsymbol{E}}_v}}}{{\partial x}}$ , $\dfrac{{\partial {{\boldsymbol{F}}_v}}}{{\partial y}}$ , $\dfrac{{\partial {{\boldsymbol{G}}_v}}}{{\partial z}}$ 为黏性项, 包含热传导项.

数值计算过程分为两个部分, 一是给定来流参数计算至定常获得层流场; 二是在层流场外边界上引入来流扰动波, 再计算达到周期性定常, 通过减去流动参数平均量获得感受性过程的扰动场. 计算方法均采用有限差分法进行求解, 方程时间项采用三阶Runge-Kutta离散格式, 空间项中黏性项采用六阶中心差分格式, 无黏项在计算层流场时采用五阶WENO格式^[50,51], 在计算扰动场时采用五阶迎风格式. 为提高计算的稳定性, 无黏项需根据流动的传播方向分裂正负通量, 本文选用Lax-Friedrich方法分裂.

边界条件: 计算层流场时, 壁面采用无滑移、无渗透的绝热边界条件, 外边界给定来流条件, 极轴上选择下游第4排点求和取平均得到的值, 出口采用三点外推方法; 计算扰动场时, 壁面上速度和温度脉动均取零, 极轴同上, 出口处采用嵌边区吸收扰动, 外边界引入平面来流扰动波.

2.2 来流扰动波

来流扰动波有3类, 分别为声波、熵波和涡波, 其中声波进一步分为快声波和慢声波. 平面来流扰动波的形式:

${{\boldsymbol{q}}'_\infty }(x,y,t) = |{\boldsymbol{q}}{|_\infty }{{\text{e}}^{{\text{i}}(\alpha x + ky - \omega t)}} + {\text{c}}{\text{.c}}. ,$

(2)

其中, q = (ρ, u, v, w, T, p)^T.

对于声波, +/–表示快/慢声波:

$\begin{split} |{\boldsymbol{q}}{|_\infty } = \;&{A_\infty }( {Ma_\infty ^2,}\;\;{ \pm M{a_\infty }\cos\theta ,}\;\;{ \pm M{a_\infty }\sin \theta ,}\\ &{0,}\;\;{(\gamma - 1)Ma_\infty ^2,}\;\;1 )^{\text{T}}. \end{split}$

(3)

对于熵波:

$|{\boldsymbol{q}}{|_\infty } = {A_\infty }Ma_\infty ^2{\left( {\begin{array}{*{20}{l}} { - 1,}&{0,}&{0,}&{0,}&{1,}&0 \end{array}} \right)^{\text{T}}}.$

(4)

对于涡波, 采用平面波:

$|{\boldsymbol{q}}{|_\infty } = {A_\infty }M{a_\infty }{\left( {\begin{array}{*{20}{l}} {0,}&{0,}&{1,}&{0,}&{0,}&0 \end{array}} \right)^{\text{T}}}.$

(5)

其中A_∞表示来流扰动波的初始幅值, 设定为10^–5, θ为来流扰动波入射角, 设定为15°, ω和α分别为无量纲圆频率和流向复波数, 涉及的特征长度均为头部半径, i为虚数单位, c.c.表示共轭项. 对于声波, +/–分别表示快/慢声波, ω与α满足如下关系:

$\frac{\omega }{\alpha } = \left(1 \pm \frac{1}{{M{a_\infty }\cos \theta }}\right).$

(6)

对于熵波和涡波, ω与α满足如下关系:

$\frac{\omega}{\alpha}=\frac{1}{\mathrm{cos}\ \theta}.$

(7)

2.3 钝锥感受性计算模型

本文以钝锥半模作为研究对象, 如图2所示. 来流基本流参数选择风洞工况, 马赫数为6, 单位雷诺数为2.6×10⁷ m^–1, 温度为63 K, 具体参数半径r = 5 mm, 半锥角ϕ = 5°, 马赫数Ma_∞ = 6, 单位雷诺数Re_∞ = 2.6×10⁷ m^–1, 温度T_∞ = 63 K. 来流扰动方向与基本流存在入射夹角θ, 定义入射正面的周向位置φ = 0°, 入射背面的周向位置φ = 180°.

$图 2 钝锥计算域示意图\r\nFig. 2. Schematic diagram of blunt-cone computation domain.$

图 2 钝锥计算域示意图

Fig. 2. Schematic diagram of blunt-cone computation domain.

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通过线性稳定性理论(LST)分析, 钝锥边界层存在低频第一模态和高频第二模态. 图3给出这两个模态的中性曲线, 横坐标ξ表示贴体坐标系下沿钝锥表面方向的坐标, 以头部半径无量纲化. 图中曲线以内表示模态不稳定区域, 曲线以外表示模态稳定区域. 此外, 在边界层外的熵层区域中还分析出熵层不稳定模态. 与边界层模态集中在下游的现象不一样, 熵层不稳定模态主要集中在上游, 不稳定的频率很小, 甚至比边界层第一模态还小. 为了研究来流扰动激发边界层第一模态和第二模态的感受性问题, 本文选择两个频率考察4种来流入射扰动, 即低频33 kHz和高频150 kHz, 分别对应无量纲频率为ω = 1.1和ω = 5, 入射角为θ = 15°.

$图 3 第一模态、第二模态和熵层模态的中性曲线[37]\r\nFig. 3. Neutral curves of the first mode, the second mode, and the entropy-layer mode[37].$

图 3 第一模态、第二模态和熵层模态的中性曲线^[37]

Fig. 3. Neutral curves of the first mode, the second mode, and the entropy-layer mode^[37].

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针对钝锥半模, 计算网格在流向和法向上均为非均匀网格, 周向网格为均匀网格(见图2). 在法向上壁面附近网格要求足够密以保证边界层有足够的网格点, 一般要求至少100个点, 同时为减少扰动过激波时的数值耗散, 激波附近也适当加密, 且激波与流向网格尽可能平行; 在流向上头部区域网格较密, 并沿下游方向变稀, 同时保证一个扰动波长内至少20个点; 在周向上半模网格均匀分布, 且也要保证一个扰动波长内至少20个点.

本文分别针对来流扰动的两个频率ω = 1.1和ω = 5, 设置两套网格. 对于频率ω = 1.1, 计算网格采用文献[49]中研究带入射角的低频慢声波三维感受性问题的网格设置方案, 即流向×法向×展向为5001×351×61, 这里不做说明. 对于频率ω = 5, 为捕捉边界层第二模态的小尺度扰动, 需要更大的网格量, 设置为10081×251×121. 图4是说明选择该套网格的网格无关性验证, 对比的对象是网格量更大的计算网格(13001×301×151). 图4显示二者在模拟慢声波感受性得到的结果差别较小, 因此按照节省计算成本原则, 选择网格量较低且保证感受性模拟不失真, 即频率ω = 5条件下计算网格为10081×251×121.

$图 4 频率ω = 5条件下感受性数值模拟的网格无关性验证\r\nFig. 4. Grid independent verification of receptivity numerical simulation for ω = 5.$

图 4 频率ω = 5条件下感受性数值模拟的网格无关性验证

Fig. 4. Grid independent verification of receptivity numerical simulation for ω = 5.

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3. 不同来流扰动入射的三维感受性对比分析

3.1 频率ω = 1.1来流扰动的三维感受性

首先对比分析不同低频来流扰动波激发钝锥边界层第一模态的三维感受性. 图5(a)—(d)给出入射角θ = 15°的不同来流扰动波感受性数值计算结果, 包括快声波、慢声波、熵波和涡波. 图中显示扰动场的密度脉动云图, 可以看出激波后扰动演化相比激波前来流扰动发生变化. 对比入射正面(周向角φ = 0°)和入射背面(周向角φ = 180°)的扰动演变, 入射正面激波后扰动幅值较小而入射背面较大, 因此边界层内扰动幅值也将会有相同的差异. 此外, 对于来流涡波, 激波外的密度脉动值基本为零, 但激波后(特别是熵层内)存在明显的扰动演变. 图6(a)—(d)给出4种来流扰动感受性数值计算在x = 30的横向截面结果, 可以看出扰动演化存在展向分量, 说明带入射角的来流扰动感受性具有明显的三维特征.

$图 5 不同来流低频扰动波的三维感受性数值模拟\r\nFig. 5. Three dimensional receptivity numerical simulations for different freestream disturbances with low frequency.$

图 5 不同来流低频扰动波的三维感受性数值模拟

Fig. 5. Three dimensional receptivity numerical simulations for different freestream disturbances with low frequency.

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$图 6 不同来流低频扰动波条件下x = 30处的横截面显示\r\nFig. 6. Cross section displays at x = 30 for different freestream disturbances with low frequency.$

图 6 不同来流低频扰动波条件下x = 30处的横截面显示

Fig. 6. Cross section displays at x = 30 for different freestream disturbances with low frequency.

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图7给出来流快声波、慢声波、熵波和涡波入射条件下壁面压力脉动沿贴体流向ξ的变化, 图中不同颜色曲线代表不同周向位置上的扰动. 可以看出, 不同周向位置上的扰动演化规律存在差异, 且不同来流扰动入射时的影响规律也不尽相同. 对于入射慢声波, 根据前期工作研究^[49], 入射角使钝锥不同周向位置的感受性强度不一样, 从入射正面(φ = 0°)到入射背面(φ = 180°), 感受性强度逐渐增大, 激发得到边界层第一模态的幅值逐渐增大. 对于快声波、熵波和涡波, 与慢声波感受性不一样, 边界层扰动演化规律基本上是扰动幅值随下游微弱增大或者减小. 观察不同周向位置的感受性, 快声波入射时, 周向位置φ = 90°的下游边界层扰动幅值最大(见图7(a)); 慢声波和熵波入射时, 周向位置φ = 180°的下游边界层扰动幅值最大(见图7(b), (c)); 涡波入射时, 周向位置φ = 0°的下游边界层扰动幅值最大(见图7(d)).

$图 7 不同来流低频扰动波条件下壁面压力脉动扰动演化　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波\r\nFig. 7. Evolution of the wall pressure fluctuations for different freestream disturbances with low frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.$

图 7 不同来流低频扰动波条件下壁面压力脉动扰动演化　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波

Fig. 7. Evolution of the wall pressure fluctuations for different freestream disturbances with low frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

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总体上当低频来流扰动波入射时, 激发边界层扰动幅值从大到小依次为慢声波>快声波>涡波>熵波. 需要说明的是, 慢声波入射时激发出来的边界层第一模态沿下游继续增长, 将大幅领先快声波入射时边界层扰动幅值.

图8给出不同来流扰动条件下壁面扰动相速度沿贴体流向的变化, 不同颜色的曲线代表不同周向位置上的扰动, 同时给出稳定性理论LST分析得到的快模态、慢模态(或者第一模态)和熵层模态的相速度(黑色曲线)与之对比. 为了更明确感受过程中扰动主导成分的演变, 如图9所示, 还对比了扰动的形状函数(红色曲线)与LST得到的不同模态特征函数(蓝色曲线).

$图 8 不同来流低频扰动波条件下壁面扰动相速度与稳定性理论比较　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波\r\nFig. 8. Phase speeds of the wall disturbances compared with the stability theory for different freestream disturbances with low frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.$

图 8 不同来流低频扰动波条件下壁面扰动相速度与稳定性理论比较　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波

Fig. 8. Phase speeds of the wall disturbances compared with the stability theory for different freestream disturbances with low frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

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$图 9 不同扰动形状函数与稳定性理论(快模态、熵层模态和第一模态)比较　(a) 快声波, ξ = 200, φ = 90; (b)慢声波, ξ = 60, φ = 90; (c)慢声波, ξ = 350, φ = 90\r\nFig. 9. Disturbance shape functions (fast mode, entropy-layer mode, and first mode) compared with the stability theory: (a) Fast acoustic wave, ξ = 200, φ = 90; (b) slow acoustic wave, ξ = 60, φ = 90; (c) slow acoustic wave, ξ = 350, φ = 90.$

图 9 不同扰动形状函数与稳定性理论(快模态、熵层模态和第一模态)比较　(a) 快声波, ξ = 200, φ = 90; (b)慢声波, ξ = 60, φ = 90; (c)慢声波, ξ = 350, φ = 90

Fig. 9. Disturbance shape functions (fast mode, entropy-layer mode, and first mode) compared with the stability theory: (a) Fast acoustic wave, ξ = 200, φ = 90; (b) slow acoustic wave, ξ = 60, φ = 90; (c) slow acoustic wave, ξ = 350, φ = 90.

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对于入射慢声波, 边界层扰动的相速度在上游与熵层模态接近(见图8(b)), 其形状函数也与熵层模态定性上符合(见图9(b)), 说明上游扰动以熵层模态形式主导. 随后, 下游扰动的相速度与第一模态匹配, 其形状函数与下游第一模态基本符合(见图9(c)), 表明在下游边界层第一模态被激发出来. 从φ = 180°到φ = 0°, 匹配位置向下游移动, 说明第一模态的激发逐渐延迟, 这是导致图7(b)下游边界层扰动幅值逐渐减小的原因.

对于入射快声波, 从上游到下游, 边界层扰动相速度基本上与快模态匹配(见图8(a)), 其形状函数也与快模态基本吻合(见图9(a)), 说明在该计算域中快声波入射只能激发出快模态, 没有激发出第一模态, 导致边界层扰动幅值没有显著增长; 从入射正面到入射背面, 边界层扰动始终以快模态主导. 对于入射熵波(见图8(c)), 靠近入射正面(φ = 0°和φ = 60°), 边界层扰动也是以快模态主导; 靠近入射背面(φ = 90°, φ = 120°和φ = 180°), 边界层扰动在下游开始以熵波为主(相速度接近1). 对于入射涡波(见图8(d)), 结果与熵波入射时相似, 靠近入射正面(φ = 0°, φ = 60°和φ = 90°), 边界层扰动以快模态主导; 靠近入射背面(φ = 120°和φ = 180°), 边界层扰动在下游开始以涡波为主(相速度接近1).

综上, 当低频慢声波入射时, 在下游激发出边界层第一模态, 主导边界层扰动幅值增长; 当低频快声波、熵波和涡波入射时, 没有激发出边界层第一模态, 导致边界层扰动在下游没有增长起来.

3.2 频率ω = 5来流扰动的三维感受性

图10(a)—(d)给出频率ω = 5、入射角θ = 15°的不同来流扰动波感受性数值计算结果, 包括快声波、慢声波、熵波和涡波, 可以看出, 激波后扰动相比激波前扰动也发生了变化. 对比入射正面(周向角φ = 0°)和入射背面(周向角φ = 180°)的扰动演变, 入射正面激波后扰动幅值较小、背面较大, 因而边界层内扰动幅值也将会有相同的差异. 与低频扰动感受情况相似, 来流涡波入射时, 激波外的密度脉动值基本为零, 但激波后(特别是熵层内)存在明显的扰动演变. 图11(a)—(d)给出4种来流扰动感受性数值计算在x = 30的横向截面结果, 可以看出扰动也会在展向方向上演变, 说明带入射角的高频来流扰动感受性也具有明显的三维特征. 其中, 来流慢声波和熵波入射时边界层外空间扰动在展向演化时波长逐渐增大, 快声波入射时扰动波长变化相对不大, 涡波时空间扰动幅值偏小, 主要是熵层中(边界层之外)的扰动.

$图 10 不同来流高频扰动波的三维感受性数值模拟\r\nFig. 10. Three dimensional receptivity numerical simulations for different freestream disturbances with high frequency.$

图 10 不同来流高频扰动波的三维感受性数值模拟

Fig. 10. Three dimensional receptivity numerical simulations for different freestream disturbances with high frequency.

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$图 11 不同来流高频扰动波条件下x = 30的横截面显示\r\nFig. 11. Cross section displays at x = 30 for different freestream disturbances with high frequency.$

图 11 不同来流高频扰动波条件下x = 30的横截面显示

Fig. 11. Cross section displays at x = 30 for different freestream disturbances with high frequency.

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图12给出高频快声波、慢声波、熵波和涡波入射时壁面压力脉动沿贴体流向的变化, 图中不同颜色曲线代表不同周向位置上的扰动. 可以看出, 不同周向位置上的扰动演化规律也不尽相同, 主要表现如下.

$图 12 不同来流高频扰动波条件下壁面压力脉动扰动演化　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波\r\nFig. 12. Evolution of the wall pressure fluctuations for different freestream disturbances with high frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.$

图 12 不同来流高频扰动波条件下壁面压力脉动扰动演化　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波

Fig. 12. Evolution of the wall pressure fluctuations for different freestream disturbances with high frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

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1)前期研究表明^[37], 来流慢声波能有效激发边界层第二模态, 基于前期认识, 首先介绍来流慢声波的感受性结果. 在来流慢声波条件下(见图12(b)), 尽管不同周向位置上的扰动演化规律基本一致, 但其扰动幅值存在差异. 其中在上游(如ξ = 50—65), 扰动幅值从入射正面到入射背面呈逐渐增大的趋势; 在中游(如ξ = 160—230), 所有周向位置上的扰动幅值基本相同; 在下游(如ξ = 300—350), 扰动幅值沿周向角不呈单调变化, 而是先增大后减小, 其中周向角φ = 90°, 即入射侧面处的扰动最大. 根据前期工作认识, 上游扰动主要是快模态, 中游扰动主要是外界扰动涌入边界层内并激励边界层扰动, 下游扰动主要是第二模态.

2)在来流快声波条件下(见图12(a)), 周向角φ = 0°—120°的扰动演化规律基本一致, 且直到下游都未出现像图12(b)那样的第二模态幅值增长规律, 但其扰动幅值随着周向角增大而减小. 然而, 入射背面φ = 180°的扰动演化规律发生了变化, 并且在下游出现了第二模态一样的幅值增长规律, 推测是由于入射正面扰动向背面汇聚从而激励出第二模态. 中游未出现外界扰动激励边界层扰动的现象, 上游不同周向位置的扰动幅值差别不大.

3)在来流熵波条件下(见图12(c)), 周向位置φ = 0°—90°的扰动演化规律基本相同, 现象与快声波条件下(图12(a))入射背面φ = 180°的一致; 周向角φ = 120°—180°的扰动演化是另一种规律, 现象接近于慢声波条件下(图12(b))的结果. 周向角φ = 0°—90°和周向角φ = 120°—180°的结果区别在于中游处扰动规律不同, 后者出现外界扰动激励边界层扰动的现象, 而前者未出现. 从幅值方面看, 下游壁面扰动幅值随着周向角的增加而增大. 该现象是受上游和中游的扰动幅值的耦合影响所致, 其中中游扰动幅值在不同周向角的差别大于上游扰动幅值差别, 说明中游扰动对激发边界层第二模态的作用更强.

4)在来流涡波条件下(见图12(d)), 不同周向角的扰动演化规律与来流熵波条件下的结果(图12(c))基本一致, 说明来流涡波与熵波的感受性机理基本相同, 主要差别是感受性强度, 即激发边界层第二模态的幅值差别.

总体上, 当高频来流扰动波入射时, 激发边界层扰动幅值从大到小依次为慢声波>涡波>熵波>快声波. 与低频扰动入射情况不一样的地方在于, 高频熵/涡波入射激发的边界层扰动幅值比快声波大, 主要原因是高频熵/涡波能够显著激发边界层第二模态, 而高频快声波仍难以激发第二模态.

图13给出不同来流扰动入射时壁面扰动相速度沿贴体流向的变化, 不同颜色的曲线代表不同周向位置上的扰动, 同时给出稳定性理论LST分析得到的快模态(黑色曲线)和第二模态的相速度(紫色曲线)与数值计算结果进行对比. 这两个模态曲线相交叉(即所谓的“同步”)之前, 第二模态相速度那一支本质为慢模态. 同时, 还给出扰动的形状函数与快模态和第二模态特征函数的对比结果, 如图14所示. 可以看出, 不同来流扰动、不同周向位置上的扰动相速度变化也不尽相同, 主要表现如下.

$图 13 不同来流高频扰动波条件下壁面扰动相速度与稳定性理论比较　(a) 快声波; (b)慢声波; (c) 熵波; (d)涡波\r\nFig. 13. Phase speeds of the wall disturbances compared with the stability theory for different freestream disturbances with high frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.$

图 13 不同来流高频扰动波条件下壁面扰动相速度与稳定性理论比较　(a) 快声波; (b)慢声波; (c) 熵波; (d)涡波

Fig. 13. Phase speeds of the wall disturbances compared with the stability theory for different freestream disturbances with high frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

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$图 14 不同扰动形状函数与稳定性理论(快模态和第二模态)比较　(a) 慢声波, ξ = 60, φ = 90; (b) 慢声波, ξ = 350, φ = 90\r\nFig. 14. Disturbance shape functions (fast mode and second mode) compared with the stability theory: (a) Slow acoustic wave, ξ = 60, φ = 90; (b) slow acoustic wave, ξ = 350, φ = 90.$

图 14 不同扰动形状函数与稳定性理论(快模态和第二模态)比较　(a) 慢声波, ξ = 60, φ = 90; (b) 慢声波, ξ = 350, φ = 90

Fig. 14. Disturbance shape functions (fast mode and second mode) compared with the stability theory: (a) Slow acoustic wave, ξ = 60, φ = 90; (b) slow acoustic wave, ξ = 350, φ = 90.

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1)在来流慢声波条件下(见图13(b)), 上游的扰动相速度接近快模态, 其形状函数与快模态基本符合(见图14(a)), 中游(ξ = 160—230)的扰动相速度接近于1, 以熵/涡波为主, 下游的扰动相速度接近第二模态, 其形状函数也与第二模态基本符合(见图14(b)). 值得注意的是, φ = 180°的扰动相速度相比其他周向位置波动很大, 说明此处扰动除了上述扰动波, 还存在其他扰动形式, 可能源自入射正面的扰动.

2)在来流快声波条件下(见图13(a)), 周向角φ = 0°时, 扰动相速度约为1.18. 由于激波后流场马赫数在下游约为5.0—5.6, 根据快声波定义(即1+1/Ma_e, Ma_e为激波后边界层外的马赫数), 快声波相速度为1.18—1.2, 说明图12(a)显示的周向角φ = 0°处扰动演化是以快声波为主. 随着周向角增大, 上游扰动相速度逐渐接近快模态, 说明上游快模态开始被激发出来. 继续增大到φ = 180°, 下游的扰动相速度开始接近第二模态, 边界层第二模态也开始被激发出来. 由于φ = 180°处整体扰动相速度未按稳定性理论的快慢模态同步激发第二模态的规律演化, 因此此时第二模态可能是由入射正面扰动汇聚的能量激励作用的, 这是正是图12(a)中φ = 180°处的扰动演化规律异于其他周向位置的原因.

3)在来流熵波条件下(见图13(c)), 周向角φ = 0°—90°时, 扰动相速度变化规律与来流快声波条件下(图13(a))周向角φ = 180°时一致; 而周向角φ = 120°—180°时, 扰动相速度变化规律与来流慢声波条件下(图13(b))一致. 这两者差别的原因也正是中游的边界层外熵/涡波扰动是否起作用导致的.

4)在来流涡波条件下(见图13(d)), 不同周向角扰动相速度变化规律与来流熵波入射时的基本一致, 主要差别是周向角φ = 90°时扰动相速度变化规律与来流慢声波入射时接近, 而不再是来流快声波条件下周向角φ = 180°时的相速度变化规律. 对比来流熵波和涡波入射的结果, 周向角φ = 90°的结果是一种临界情形.

4. 结　论

本文以零攻角Ma6钝锥为研究对象, 通过数值计算与理论分析, 开展不同类型来流扰动波以非零入射角形式的三维感受性研究, 考察不同来流扰动分别对激发边界层低频第一模态和高频第二模态的不同作用机制, 来流扰动波包括快声波、慢声波、熵波和涡波, 得到结论如下.

1)当低频慢声波入射时, 在下游激发出边界层第一模态, 主导边界层扰动幅值增长; 当低频快声波、熵波和涡波分别入射时, 下游主要以快模态或熵/涡波为主, 没有激发出边界层第一模态, 导致边界层扰动在下游没有增长起来. 总体上, 激发边界层低频扰动幅值从大到小依次为慢声波、快声波、涡波和熵波.

2)当高频慢声波、熵波和涡波分别入射时, 在下游可以激发出边界层第二模态; 当高频快声波入射时, 下游扰动主要以快声波/快模态为主, 很难激发出边界层第二模态, 整体上扰动不增长, 仅在入射背面存在扰动增长现象. 总体上, 激发边界层高频扰动幅值从大到小依次为慢声波、涡波、熵波和快声波.

3)来流扰动入射角给感受性过程带来两个方面影响: 一是感受性过程中不同周向位置上主导扰动形式的变化, 特别是下游是否激发边界不稳定模态的变化, 从而影响扰动的增长过程, 例如高频的快声波和高/低频的熵/涡波; 二是不改变扰动演化形式, 仅改变不同周向位置上边界层扰动的幅值, 例如低频的快/慢声波和高频的慢声波.

4)对比不同周向位置上的感受性结果, 主要有3种规律: 一是入射背面的感受性幅值最大, 正面最小, 例如低频慢声波和熵波, 高频熵波和涡波; 二是入射正面的感受性幅值最大, 例如低频涡波和高频快声波; 三是侧面的感受性幅值最大, 例如低频快声波和高频慢声波.

总之, 入射角能造成感受性过程在不同周向位置上扰动呈现出不同演变的三维特征, 而且不同频率、不同来流扰动类型的感受性规律不尽相同. 入射背面的边界层扰动激发往往是多个源头扰动的耦合作用结果, 即不仅源于上游头部扰动, 还源于入射正面向背面传播的扰动. 从上述结果来看, 这种耦合作用并不一定是正向叠加, 也有可能是负向作用, 如低频涡波和高频快声波入射时背面感受性反而比正面弱, 低频快声波和高频慢声波入射时背面感受性也不是最强. 解释这些问题需要对比分析不同来流扰动经过激波作用后的传播路径机制以及主导路径的差异, 未来进一步研究.

此外, 为了更有效地支撑预测边界层转捩, 未来计划针对更真实的风洞和飞行环境扰动, 开展三维外形边界层的感受性机制研究; 并且结合机器学习方法开展感受性建模研究, 改进转捩预测技术, 提高转捩预测准确度.

感谢天津大学赵磊老师对本文计算程序的帮助.

References

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[3]	周恒, 张涵信 2017 空气动力学学报 35 151 Google Scholar Zhou H, Zhang H X 2017 Acta Aero. Sin. 35 151 Google Scholar
[4]	Goldstein M E 1983 J. Fluid Mech. 127 59 Google Scholar
[5]	Goldstein M E 1985 J. Fluid Mech. 154 509 Google Scholar
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[7]	Choudhari M, Streett C 1992 Phys. Fluids 4 2495 Google Scholar
[8]	Choudhari M 1993 Theor. Comp. Fluid Dyn. 4 101 Google Scholar
[9]	Duck P W, Ruban A I, Zhikharev C N 1996 J. Fluid Mech. 312 341 Google Scholar
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图 1 钝锥子午面示意图

Figure 1. Schematic diagram of blunt-cone meridian plane.

DownLoad: Full-Size Img PowerPoint

图 2 钝锥计算域示意图

Figure 2. Schematic diagram of blunt-cone computation domain.

DownLoad: Full-Size Img PowerPoint

图 3 第一模态、第二模态和熵层模态的中性曲线^[37]

Figure 3. Neutral curves of the first mode, the second mode, and the entropy-layer mode^[37].

DownLoad: Full-Size Img PowerPoint

图 4 频率ω = 5条件下感受性数值模拟的网格无关性验证

Figure 4. Grid independent verification of receptivity numerical simulation for ω = 5.

DownLoad: Full-Size Img PowerPoint

图 5 不同来流低频扰动波的三维感受性数值模拟

Figure 5. Three dimensional receptivity numerical simulations for different freestream disturbances with low frequency.

DownLoad: Full-Size Img PowerPoint

图 6 不同来流低频扰动波条件下x = 30处的横截面显示

Figure 6. Cross section displays at x = 30 for different freestream disturbances with low frequency.

DownLoad: Full-Size Img PowerPoint

图 7 不同来流低频扰动波条件下壁面压力脉动扰动演化　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波

Figure 7. Evolution of the wall pressure fluctuations for different freestream disturbances with low frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

DownLoad: Full-Size Img PowerPoint

图 8 不同来流低频扰动波条件下壁面扰动相速度与稳定性理论比较　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波

Figure 8. Phase speeds of the wall disturbances compared with the stability theory for different freestream disturbances with low frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

DownLoad: Full-Size Img PowerPoint

图 9 不同扰动形状函数与稳定性理论(快模态、熵层模态和第一模态)比较　(a) 快声波, ξ = 200, φ = 90; (b)慢声波, ξ = 60, φ = 90; (c)慢声波, ξ = 350, φ = 90

Figure 9. Disturbance shape functions (fast mode, entropy-layer mode, and first mode) compared with the stability theory: (a) Fast acoustic wave, ξ = 200, φ = 90; (b) slow acoustic wave, ξ = 60, φ = 90; (c) slow acoustic wave, ξ = 350, φ = 90.

DownLoad: Full-Size Img PowerPoint

图 10 不同来流高频扰动波的三维感受性数值模拟

Figure 10. Three dimensional receptivity numerical simulations for different freestream disturbances with high frequency.

DownLoad: Full-Size Img PowerPoint

图 11 不同来流高频扰动波条件下x = 30的横截面显示

Figure 11. Cross section displays at x = 30 for different freestream disturbances with high frequency.

DownLoad: Full-Size Img PowerPoint

图 12 不同来流高频扰动波条件下壁面压力脉动扰动演化　(a) 快声波; (b)慢声波; (c)熵波; (d)涡波

Figure 12. Evolution of the wall pressure fluctuations for different freestream disturbances with high frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

DownLoad: Full-Size Img PowerPoint

图 13 不同来流高频扰动波条件下壁面扰动相速度与稳定性理论比较　(a) 快声波; (b)慢声波; (c) 熵波; (d)涡波

Figure 13. Phase speeds of the wall disturbances compared with the stability theory for different freestream disturbances with high frequency: (a) Fast acoustic wave; (b) slow acoustic wave; (c) entropy wave; (d) vortex wave.

DownLoad: Full-Size Img PowerPoint

图 14 不同扰动形状函数与稳定性理论(快模态和第二模态)比较　(a) 慢声波, ξ = 60, φ = 90; (b) 慢声波, ξ = 350, φ = 90

Figure 14. Disturbance shape functions (fast mode and second mode) compared with the stability theory: (a) Slow acoustic wave, ξ = 60, φ = 90; (b) slow acoustic wave, ξ = 350, φ = 90.

DownLoad: Full-Size Img PowerPoint

[1]	陈坚强, 涂国华, 张毅锋, 徐国亮, 袁先旭, 陈诚 2017 空气动力学学报 35 311 Google Scholar Chen J Q, Tu G H, Zhang Y F, Xu G L, Yuan X X, Chen C 2017 Acta Aero. Sin. 35 311 Google Scholar
[2]	Saric W S, Reed H L, Kerschen E J 2002 Annu. Rev. Fluid Mech. 34 291 Google Scholar
[3]	周恒, 张涵信 2017 空气动力学学报 35 151 Google Scholar Zhou H, Zhang H X 2017 Acta Aero. Sin. 35 151 Google Scholar
[4]	Goldstein M E 1983 J. Fluid Mech. 127 59 Google Scholar
[5]	Goldstein M E 1985 J. Fluid Mech. 154 509 Google Scholar
[6]	Ruban A I 1985 Fluid Dyn. 19 709 Google Scholar
[7]	Choudhari M, Streett C 1992 Phys. Fluids 4 2495 Google Scholar
[8]	Choudhari M 1993 Theor. Comp. Fluid Dyn. 4 101 Google Scholar
[9]	Duck P W, Ruban A I, Zhikharev C N 1996 J. Fluid Mech. 312 341 Google Scholar
[10]	Hammerton P W, Kerschen E J 1996 J. Fluid Mech. 310 243 Google Scholar
[11]	Wanderley J B V, Corke T C 2001 J. Fluid Mech. 429 1 Google Scholar
[12]	Wu X S 2001 J. Fluid Mech. 449 373 Google Scholar
[13]	Wu X S 2001 J. Fluid Mech. 431 91 Google Scholar
[14]	Dong M, Liu Y H, Wu X S 2020 J. Fluid Mech. 896 A23 Google Scholar
[15]	陆昌根, 朱晓清, 沈露予 2017 物理学报 66 204702 Google Scholar Lu C G, Zhu X Q, Shen L Y 2017 Acta Phys. Sin. 66 204702 Google Scholar
[16]	Fedorov A 2011 Annu. Rev. Fluid Mech. 43 79 Google Scholar
[17]	Zhong X, Wang X 2012 Annu. Rev. Fluid Mech. 44 527 Google Scholar
[18]	江贤洋, 李存标 2017 实验流体力学 31 1 Google Scholar Jiang X Y, Lee C B 2017 J. Exp. Fluid Mech. 31 1 Google Scholar
[19]	苏彩虹 2020 空气动力学学报 38 355 Google Scholar Su C H 2020 Acta Aero. Sin. 38 355 Google Scholar
[20]	Fedorov A V, Khokhlov A P 1991 Fluid Dyn. 26 531 Google Scholar
[21]	Fedorov A V, Khokhlov A P 2001 Theor. Comp. Fluid Dyn. 14 359 Google Scholar
[22]	Fedorov A V 2003 J. Fluid Mech. 491 101 Google Scholar
[23]	Ma Y B, Zhong X L 2003 J. Fluid Mech. 488 31 Google Scholar
[24]	Ma Y B, Zhong X L 2003 J. Fluid Mech. 488 79 Google Scholar
[25]	Ma Y B, Zhong X L 2005 J. Fluid Mech. 532 63 Google Scholar
[26]	Zhong X L, Ma Y B 2006 J. Fluid Mech. 556 55 Google Scholar
[27]	He S M, Zhong X L 2021 AIAA J. 59 3546 Google Scholar
[28]	He S M, Zhong X L 2022 Phys. Fluids 34 054104 Google Scholar
[29]	Balakumar P 2006 36th AIAA Fluid Dynamics Conference and Exhibit. San Francisco, California, June 5–8, p3053
[30]	Balakumar P 2015 49th AIAA Fluid Dynamics Conference Dallas, Texas, p247
[31]	Balakumar P, King R A, Chou A, Owens L R, Kegerise M A 2018 AIAA J. 56 510 Google Scholar
[32]	张玉东, 傅德薰, 马延文, 李新亮 2008 中国科学G辑, 物理学力学天文学 38 1246 Zhang Y D, Fu D X, Ma Y W, Li X L 2008 Sci. Sin. Phys. Mech. Astron. 38 1246
[33]	Qin F F, Wu X S 2016 J. Fluid Mech. 797 874 Google Scholar
[34]	Ba W T, Niu M H, Su C H 2023 AIAA J. 61 518 Google Scholar
[35]	Niu M H, Su C H 2023 Phy. Fluids 35 034109 Google Scholar
[36]	Wan B B, Luo J S, Su C H 2018 Appl. Math. Mech. (English Edition) 39 1643 Google Scholar
[37]	Wan B B, Su C H, Chen J Q 2020 AIAA J. 58 4047 Google Scholar
[38]	Chen Y F, Tu G H, Wan B B, Su C H, Yuan X X, Chen J Q 2021 Phys. Fluids 33 084114 Google Scholar
[39]	Stetson K F, Thompson E R, Donaldson J C, Siler L G 1984 AIAA 22nd Aerospace Sciences Meeting Reno, Nevada, January 9–12, 1984 p0006
[40]	Fedorov A V 1990 J. Appl. Mech. Tech. Phy. 31 722 Google Scholar
[41]	万兵兵, 罗纪生 2018 空气动力学学报 36 247 Google Scholar Wan B B, Luo J S 2018 Acta Aero. Sin. 36 247 Google Scholar
[42]	欧吉辉, 万兵兵, 刘建新, 曹伟 2018 空气动力学报 36 238 Google Scholar Ou J H, Wan B B, Liu J X, Cao W 2018 Acta Aero. Sin. 36 238 Google Scholar
[43]	Han Y F, Zhou J T, Cao W 2022 Phys. Fluids 34 026101 Google Scholar
[44]	McKenzie J F, Westphal K O 1968 Phys. Fluids 11 2350 Google Scholar
[45]	Su C H, Geng J L 2017 Appl. Math. Mech. (English Edition) 38 1601 Google Scholar
[46]	Huang Z F, Wang H L 2019 J. Fluid Mech. 873 1179 Google Scholar
[47]	Malik M R, Balakumar P 2007 Theor. Comp. Fluid Dyn. 21 323 Google Scholar
[48]	Balakumar P 2007 37th AIAA Fluid Dynamics Conference and Exhibit Miami, FL, June 25–28, 2007 p4491
[49]	Wan B B, Chen J Q, Yuan X X, Hu W B, Tu G H 2022 AIAA J. 60 4523 Google Scholar
[50]	Zhang S H, Zhu J, Shu C 2019 Adv. Aerod. 1 16 Google Scholar
[51]	Ma Y K, Mao M L, Yan Z G, Bai J W, Zhu H J 2024 J. Comp. Phy. 510 113064 Google Scholar

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