﻿ 超/高超声速飞行器动态稳定性导数极快速预测方法
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1. 中国空气动力研究与发展中心 高超声速冲压发动机技术重点实验室, 绵阳 621000;
2. 南京航空航天大学 航空学院, 南京 210016

Extremely efficient prediction technique of dynamic derivatives for super/hypersonic flight vehicles
LI Zhengzhou1,2, GAO Chang1, XIAO Tianhang2, MA Zicheng1, XIAO Jiliang2, ZHU Jianhui2
1. Science and Technology on Scramjet Laboratory, China Aerodynamics Research and Development Center, Mianyang 621000, China;
2. Aeronautics Engineering College, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: In the early stage of aircraft design, it is necessary to predict the dynamic stability derivatives under a large number of operating conditions. An extremely efficient prediction technique of dynamic stability derivatives for high-speed flight vehicles is developed in this paper. Firstly, based on the local piston theory, the unsteady aerodynamics of high-speed flight vehicles can be divided into the non-disturbance aerodynamic component and the disturbance aerodynamic component. Secondly, the local surface inclination method and the isentropic flow relation after a shock wave are employed to solve the surface flow variables, and then the unsteady aerodynamic forces are calculated by combining the unsteady motion. Finally, the dynamic derivatives are extracted and identified from unsteady aerodynamic forces using the undetermined coefficient method. Our method overcomes the time-consuming of the traditional CFD method that is dependent and coupled with numerical simulation flow field variables. Typical test cases indicate that our method can predict the trend of dynamic derivatives well under both supersonic and hypersonic conditions. At last, our method is successfully applied to dynamic derivatives prediction of flight vehicles with complex shape, and the error sources with CFD method are discussed. The efficient dynamic derivatives prediction technique in this paper can be used to screen layout of high-speed flight in conceptual design stage.
Keywords: dynamic derivatives    forced vibration motion    dynamic stability    local piston theory    hypersonic    supersonic

1 动导数高效预测策略及流程

 图 1 高速飞行器动导数高效预测流程 Fig. 1 Efficient prediction process of dynamic derivatives for high-speed flight vehicles

2 非定常气动力快速预测 2.1 当地流活塞理论的推导

 图 2 当地流活塞理论示意图 Fig. 2 Sketch of local piston theory

 ${\rm{d}}P \cdot S \cdot {\rm{d}}t = \rho aS{\rm{d}}t \cdot {\rm{d}}W$ （1）

 ${\rm{d}}P = {\rho _1}{a_1}{\rm{d}}W$ （2）

 $P = {\rho _1}{a_1}W + C$ （3）

 $\left\{ {\begin{array}{*{20}{l}} {P = {P_{{\rm{steady}}}} + {\rho _1}{a_1}W}\\ {W = {\mathit{\boldsymbol{V}}_1} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{n}} + {\mathit{\boldsymbol{V}}_{\rm{b}}} \cdot \mathit{\boldsymbol{n}}}\\ {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{n}} = {\mathit{\boldsymbol{n}}_0} - \mathit{\boldsymbol{n}}} \end{array}} \right.$ （4）

2.2 气动力无附加扰动项计算

 ${C_p} = {C_{p,\max}}\left( {\frac{{1 - B}}{{0.32Ma}} + B \sin \theta } \right)\sin\theta$ （5）

 ${C_p} = \frac{{ - \left( {\gamma - 1} \right){\theta ^2}}}{2}\left[ {\sqrt {1 + {{\left( {\frac{4}{{\left( {\gamma + 1} \right)Ma\theta }}} \right)}^2}} - 1} \right]$ （6）

2.3 气动力附加扰动项计算

2.3.1 正激波完全气体边界层外缘参数

 ${P_2} = {P_\infty }\left( {\frac{{2\gamma Ma_\infty ^2}}{{1 + \gamma }} - \frac{{\gamma - 1}}{{\gamma + 1}}} \right)$ （7）
 ${\rho _2} = {\rho _\infty }\left[ {\frac{{\left( {1 + \gamma } \right)Ma_\infty ^2}}{{2 + \left( {\gamma - 1} \right)Ma_\infty ^2}}} \right]$ （8）

 ${\rho _{\rm{e}}} = {\left( {\frac{{{P_{\rm{e}}}}}{{{P_{\rm{2}}}}}} \right)^{\frac{1}{\gamma }}}{\rho _2}$ （9）

 ${a_{\rm{e}}} = \sqrt {\gamma \frac{{{P_{\rm{e}}}}}{{{\rho _{\rm{e}}}}}}$ （10）

2.3.2 斜激波完全气体边界层外缘参数

 ${P_2} = {P_\infty } + {P_\infty }\frac{{2\gamma }}{{\gamma + 1}}\left( {Ma_\infty ^2{{\sin }^2}\beta - 1} \right)$ （11）
 ${\rho _2} = {\rho _\infty }\frac{{\left( {\gamma + 1} \right)Ma_\infty ^2{{\sin }^2}\beta }}{{\left( {\gamma - 1} \right)Ma_\infty ^2{{\sin }^2}\beta + 2}}$ （12）

3 气动导数提取及辨识

 $\alpha = {\alpha _0} + {\alpha _{\rm{m}}}\sin \left( {\omega t} \right)$ （13）

 ${C_m} = {C_{m\alpha }}\Delta \alpha + {C_{m\dot \alpha }}\Delta \bar {\dot \alpha} + {Q_{mq}}\Delta \bar q + {C_{m\ddot \alpha }}\Delta \bar {\ddot \alpha} + \cdots$ （14）

 ${C_m} = {C_{m0}} + {C_{m\alpha }}\Delta \alpha + \left( {{C_{m\dot \alpha }} + {C_{mq}}} \right)\Delta \bar {\dot \alpha}$ （15）

 $k = \frac{{\omega {L_{{\rm{ref}}}}}}{{2{V_\infty }}}$ （16）

 ${C_m} = A + B \sin\left( {\omega t} \right) + C \cos\left( {\omega t} \right)$ （17）

 $\left( {{C_{m\dot \alpha }} + {C_{mq}}} \right) = C/\left( {k{\alpha _{\rm{m}}}} \right)$ （18）

 ${\mathit{\boldsymbol{V}}_\infty } = {V_\infty }\left[ {\cos\alpha \cos\beta ,\sin \beta ,\cos \beta \sin \alpha } \right]$ （19）

 $\begin{array}{l} \left( {{V_ \bot }/{V_\infty }} \right) = {n_x} \cos \alpha \cos\beta + {n_y}\sin\beta + \\ \;\;\;\;\;{n_z}\cos \beta \sin \alpha \end{array}$ （20）

 ${\left( {{V_ \bot }/{V_\infty }} \right)_q} = \left( {x{n_z} - z{n_x}} \right)/l$ （21）

 $\begin{array}{l} {C_{{p_q}}} = {C_{p,\max }}\left[ {1/M{a_\infty } + 1.2\left( {{V_ \bot }/{V_\infty }} \right)} \right] \cdot \\ \;\;\;\;\;\;\;{\left( {{V_ \bot }/{V_\infty }} \right)_q} \end{array}$ （22）

 ${\rm{d}}{M_{yq}} = {\rm{d}}{F_{aq}} \cdot z - {\rm{d}}{F_{nq}} \cdot x$ （23）

 ${\rm{d}}{F_{aq}} = {C_{{p_q}}} \cdot {n_x}{\rm{d}}s,{\rm{d}}{F_{nq}} = {C_{{p_q}}} \cdot {n_z}{\rm{d}}s$ （24）

 $\left( {{C_{m\dot \alpha }} + {C_{mq}}} \right) = \frac{{\sum {\left( {{\rm{d}}{M_{y\dot \alpha }}} \right)} + \sum {\left( {{\rm{d}}{M_{yq}}} \right)} }}{{l \cdot S}}$ （25）

4 典型算例验证

4.1 超声速工况算例

 图 3 有翼导弹几何外形示意图 Fig. 3 Sketch of basic finner missile geometry

 图 4 有翼导弹面元网格 Fig. 4 Surface mesh of BFM

 图 5 有翼导弹动导数预测结果对比 Fig. 5 Comparison of dynamic derivatives prediction results for BFM

 方法 耗时 非定常CFD周期计算[32] 61 h 非定常CFD差分法[32] 12 h 本文方法 ~5 s
4.2 高超声速工况算例

HBS标模是AGARD用来测定高超声速飞行器稳定性的典型外形。如图 6所示，HBS由三段长均为1.5d′(d′为球体直径)的球柱、5°半锥角的圆锥段和15°半锥角的圆锥段组成，其俯仰动导数具有较为精确的试验结果[33]，通常被用来衡量高超声速工况下动导数计算结果的准确程度。

 图 6 弹道外形示意图 Fig. 6 Sketch of hyperballistic shape

 图 7 弹道外形面元网格 Fig. 7 Surface mesh of hyperballistic shape

 图 8 弹道外形动导数预测结果对比 Fig. 8 Comparison of dynamic derivatives prediction results for HBS

 网格类型 网格量 动导数计算结果 粗糙 ~4 000 -31.9341 中等 ~20 000 -31.6621 加密 ~100 000 -31.1773

5 复杂外形飞行器气动导数预测

X-37B是美国为了验证可重复使用空间技术和在轨空间飞行任务而启动的项目[35]，并计划在X-37B飞行器技术基础上继续进行能够投送6名宇航员进入太空的X-37C计划[36]。因此，以类X-37B飞行器为对象研究高速飞行器气动导数具有典型的意义。图 9为类X-37B飞行器示意图。

 图 9 类X-37B飞行器示意图 Fig. 9 Sketch of X-37B analog

 图 10 类X-37B飞行器动导数预测结果对比 Fig. 10 Comparison of dynamic derivatives prediction results for X-37B analog

 图 11 边界层外缘密度对比 Fig. 11 Comparison of boundary layer edge densities
 图 12 边界层外缘声速对比 Fig. 12 Comparison of boundary layer edge sound speeds
6 结论

1) 本文动导数预测方法是基于“当地流活塞理论”及“牛顿撞击理论”等相关理论推导而出，因此“当地流活塞理论”及“牛顿撞击理论”的成立条件决定了本文方法适用的速域范围，即扰动可近似为沿物面法向传播的超声速/高超声速流动。

2) 当地流活塞理论推导过程中不需要等熵假设，拓宽了经典活塞理论对飞行器外形和迎角的适用范围，因此本文方法可用于复杂外形飞行器的动导数预测。

3) 本文方法避免了传统动导数预测方法对CFD流场的依赖、耦合，从而大幅提高了计算效率；同时在面对复杂外形时也能较好地得出动导数随迎角的变化规律，精度能够满足飞行器总体设计阶段的要求，可作为飞行器布局选型阶段的工具。

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http://dx.doi.org/10.7527/S1000-6893.2019.23545

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#### 文章信息

LI Zhengzhou, GAO Chang, XIAO Tianhang, MA Zicheng, XIAO Jiliang, ZHU Jianhui

Extremely efficient prediction technique of dynamic derivatives for super/hypersonic flight vehicles

Acta Aeronautica et Astronautica Sinica, 2020, 41(4): 123545.
http://dx.doi.org/10.7527/S1000-6893.2019.23545