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“Ⅳ型”激波干扰中流-热-固耦合问题一体化计算分析

Integrated numerical analysis of fluid-thermal-structural problems in "Type Ⅳ" shock wave interference
LI Jiawei, WANG Jiangfeng, YANG Tianpeng, LI Longfei, WANG Ding
Key Laboratory of Unsteady Aerodynamics and Flow Control of Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: An integrated fluid-thermal-structural numerical method based on the finite volume method is proposed to study the thermal behavior of the multi-physical coupling of aerodynamic heating and thermal structural heat transfer for hypersonic cylindrical leading edge with "Type Ⅳ" shock wave interference. In this method, the external high-speed flow field and the internal structural temperature field are unified into the integrated governing equations, and the finite volume method is used to perform unified discretization and solution, avoiding the cumbersome data exchange strategy of the traditional partitioned coupling approach. In addition, a new dual temperature resistance model is developed to calculate the physical parameters on the flow-solid interface to ensure the calculation accuracy. The LU-SGS implicit time-stepping scheme and the adaptive unsteady time-step size are used to improve the calculation efficiency. The aerodynamically heated cylinder leading edge is in good agreement with the references and the experimental data, demonstrating the capability and reliability of the integrated method. Issues of the thermal structural response are studied for "Type Ⅳ" shock wave interference on hypersonic cylindrical leading edge. The property distributions of temperature and heat flux are obtained and the time-variant characteristics are presented and analyzed. The numerical simulation results show that the jet produced in "Type Ⅳ" shock interference impinges perpendicularly to the surface, and shock interference increases the maximum pressure coefficient by about 9 times and the maximum heat flux by about 4.7 times, posting severe challenges to the design and selection of thermal protection for high-speed vehicle. These results also show that the integrated method can provide theoretical and technical support for the comprehensive performance evaluation and optimization of hypersonic thermal protection systems under long-endurance flight conditions and complex flight environments.
Keywords: "Type Ⅳ" shock wave interference     fluid-thermal-structural coupling     integrated numerical method     hypersonic     finite volume method

 图 1 前体-进气道激波干扰示意图 Fig. 1 Schematic diagram of forebody-inlet shock wave interference

Edney[3]根据激波干扰点的位置，将激波干扰分为6种类型(Ⅰ~Ⅵ型)，其中Ⅳ型干扰形成超声速射流冲击壁面，产生极大的压强和热流[4-5]，对飞行器安全影响最大，因此备受关注，也是本文主要研究内容。另外，高速飞行器外部流场的气动加热与热防护结构内的热传导相互作用不可分割，因此要准确预测“Ⅳ”型激波干扰下飞行器的气动热环境，必须考虑气动加热与结构传热的相互耦合作用[6]

1 一体化数值计算方法 1.1 一体化控制方程及求解方法

 $\frac{\partial }{{\partial t}}\int_\mathit{\Omega } {\rho {C_{\rm{s}}}T{\rm{d}}\mathit{\Omega }} - \oint_{\partial \mathit{\Omega }} {k\nabla T \cdot \mathit{\boldsymbol{\hat n}}{\rm{d}}S} = 0$ （1）

 $\frac{\partial }{{\partial t}}\int_\mathit{\Omega } \mathit{\boldsymbol{W}} {\rm{d}}\mathit{\Omega } + \oint_{\partial \mathit{\Omega }} {\left( {{\mathit{\boldsymbol{F}}_{\rm{c}}} - {\mathit{\boldsymbol{F}}_{\rm{v}}}} \right){\rm{d}}S} = {\bf{0}}$ （2）

 $\mathit{\boldsymbol{W}} = \left[ {\begin{array}{*{20}{c}} \rho \\ {\rho u}\\ {\rho v}\\ {\rho w}\\ {\rho E} \end{array}} \right],{\mathit{\boldsymbol{F}}_{\rm{c}}} = \left[ {\begin{array}{*{20}{c}} {\rho V}\\ {\rho uV + {n_x}p}\\ {\rho uV + {n_y}p}\\ {\rho wV + {n_z}p}\\ {\rho HV} \end{array}} \right]$
 ${\mathit{\boldsymbol{F}}_{\rm{v}}} = \left[ {\begin{array}{*{20}{c}} 0\\ {{n_x}{\tau _{xx}} + {n_y}{\tau _{xy}} + {n_z}{\tau _{xz}}}\\ {{n_x}{\tau _{yx}} + {n_y}{\tau _{yy}} + {n_z}{\tau _{yz}}}\\ {{n_x}{\tau _{zx}} + {n_y}{\tau _{zy}} + {n_z}{\tau _{zz}}}\\ {{n_x}{\mathit{\Theta }_x} + {n_y}{\mathit{\Theta }_y} + {n_z}{\mathit{\Theta }_z}} \end{array}} \right]$
 $\left\{ \begin{array}{l} {\mathit{\Theta }_x} = u{\tau _{xx}} + v{\tau _{xy}} + w{\tau _{xz}} + k\frac{{\partial T}}{{\partial x}}\\ {\mathit{\Theta }_y} = u{\tau _{yx}} + v{\tau _{yy}} + w{\tau _{yz}} + k\frac{{\partial T}}{{\partial y}}\\ {\mathit{\Theta }_z} = u{\tau _{zx}} + v{\tau _{zy}} + w{\tau _{zz}} + k\frac{{\partial T}}{{\partial z}} \end{array} \right.$

 $\Delta t_{\rm{f}}^I = {N_{{\rm{cfl}}}}\frac{{{\mathit{\Omega }_I}}}{{{{\left( {{\mathit{\Lambda }_{\rm{c}}} + C{\mathit{\Lambda }_{\rm{v}}}} \right)}_I}}}$ （3）

 $\Delta t_{\rm{s}}^I = \frac{{{{\left( {{\rm{ \mathsf{ δ} }}x} \right)}^2}}}{{2\alpha }},\alpha = \frac{k}{{\rho {C_{\rm{s}}}}}$ （4）

1.2 流-固耦合界面参数定义

 $T = \frac{{{k_1}{T_1} + {k_{\rm{r}}}{T_{\rm{r}}}}}{{{k_1} + {k_{\rm{r}}}}}$ （5）
 图 2 流-固交界面左右单元意图 Fig. 2 Schematic diagram of left and right cells at fluid-solid interface

 $\left\{ {\begin{array}{*{20}{l}} {\nabla {T^*} = \left( {\nabla {T_1} + \nabla {T_{\rm{r}}}} \right)/2}\\ {\nabla T = \nabla {T^*} - \left( {\nabla {T^*} \cdot {\mathit{\boldsymbol{r}}_{{\rm{lr}}}} - \frac{{{T_{\rm{r}}} - {T_1}}}{{{L_{{\rm{lr}}}}}}} \right){\mathit{\boldsymbol{r}}_{{\rm{lr}}}}} \end{array}} \right.$ （6）

 ${R_{\rm{t}}} = \frac{L}{{kA}}$ （7）

 ${R_{{\rm{t}},{\rm{bnd}}}} = {R_{{\rm{t}},1}} + {R_{{\rm{t}},{\rm{r}}}}$ （8）

 $\frac{L}{{kA}} = \frac{{{L_1}}}{{{k_1}A}} + \frac{{{L_{\rm{r}}}}}{{{k_{\rm{r}}}A}}$ （9）

 $k = \frac{{{k_1}{k_{\rm{r}}}}}{{{k_1}{L_{\rm{r}}} + {k_{\rm{r}}}{L_1}}} \cdot \left( {{L_{\rm{r}}} + {L_1}} \right)$ （10）
1.3 非定常时间步长自适应计算方法

 图 3 时间步长PID控制器示意图 Fig. 3 Schematic diagram of PID control system in time step-size

 $\Delta t_{n + 1}^{\rm{r}} = {\left( {\frac{{{e_{n - 1}}}}{{{e_n}}}} \right)^{{k_{\rm{P}}}}}{\left( {\frac{1}{{{e_n}}}} \right)^{{k_{\rm{I}}}}}{\left( {\frac{{e_{n - 1}^2}}{{{e_n}{e_{n - 2}}}}} \right)^{{k_{\rm{D}}}}}\Delta {t_n}$ （11）

 ${e_n} = \frac{{e_n^*}}{{{\rm{tol}}}},e_n^* = \frac{{\left\| {{T^n} - {T^{n - 1}}} \right\|}}{{\left\| {{T^n}} \right\|}}$ （12）

 $\Delta {t_{\min }} \le \Delta {t_n} \le \Delta {t_{\max }}$ （13）
 $\Delta t_{n + 1}^\alpha = \frac{{{\alpha _{{\rm{ref}}}}}}{\alpha }\Delta {t_n}$ （14）
 $\Delta {t_{n + 1}} = \min \left( {\Delta t_{n + 1}^{\rm{r}},\Delta t_{n + 1}^\alpha } \right)$ （15）

 $\alpha = \mathop {\max }\limits_n \frac{{\left\| {{T^n} - {T^{n - 1}}} \right\|}}{{\left\| {{T^{n - 1}} - {T^{n - 2}}} \right\|}}$ （16）
2 验证算例 2.1 计算模型及网格

 Parameter ρ/(kg·m-3) Cs/(J·kg-1·K-1) k/(W·m-1·K-1) T/K Value 8 030 502.48 16.27 294.4

 图 4 计算网格与边界条件 Fig. 4 Computational grids and boundary conditions

 Parameter Ma∞ T∞/K p∞ /Pa Re∞ /m-1 Value 6.47 241.5 648.1 1.31×106

2.2 计算结果与分析

 图 5 密度等值线图与试验纹影图[26]对比 Fig. 5 Comparison of predicted density contours with experimental schlieren photograph[26]
 图 6 圆管表面压力分布对比 Fig. 6 Comparison of surface pressure distributions on cylinder

 图 7 t=0 s圆管表面热流分布对比 Fig. 7 Comparison of surface heat flux distributions on cylinder at t=0 s

 图 8 不同时刻流场与圆管沿中心线温度分布 Fig. 8 Temperature distribution along centerline within fluid and cylinder domains at different time

 图 9 不同时刻圆管内部温度云图 Fig. 9 Temperature contours within cylinder domain at different time

 图 10 圆管驻点温度随时间的变化 Fig. 10 Temporal variation of stagnation point temperature on cylinder

 图 11 圆管驻点热流随时间的变化 Fig. 11 Temporal variation of stagnation point heat flux on cylinder

 Parameter Value Present Ref.[23] Ref.[26] Ref.[12] Ref.[31] T0(t=2 s)/K 389.2 388.8 385 387 q0(t=0 s)/(kW·m-2) 492.4 670 488.6 485.5 505

 Method Iterative step Calculation time/h Partition-coupling calculation 20 000 About 20 Integrated calculation (fixed time step) 20 000 About 12 Integrated calculation (adaptive time step) 800 About 0.8

3 “Ⅳ型”激波干扰算例 3.1 计算模型及网格

 图 12 计算模型 Fig. 12 Computational model

 Parameter Ma∞ U∞ T∞/K p∞/Pa Value 9.5 2 871.28 226.5 1 197

 图 13 “Ⅳ型”激波干扰算例的计算网格与边界条件 Fig. 13 Computational grids and boundary conditions of "Type Ⅳ" shock wave interference case
3.2 计算结果与分析

3.2.1 定常计算结果

 图 14 “Ⅳ型”激波干扰稳态流场结构 Fig. 14 Steady flow field structure of "Type Ⅳ" shock wave interference

 图 15 流场与圆管稳态温度云图 Fig. 15 Steady temperature contours within fluid and cylinder domains

 图 16 圆管表面稳态热流与压力系数分布曲线 Fig. 16 Curves of steady surface heat flux and pressure coefficient distributions on cylinder

3.2.2 非定常计算结果

 图 17 圆管内部温度云图随时间变化历程 Fig. 17 Temporal variation of temperature contours within cylinder

 图 18 圆管表面压力系数随时间变化 Fig. 18 Pressure coefficient variation with time along cylinder surface
 图 19 圆管表面热流随时间变化 Fig. 19 Heat flux variation with time along cylinder surface

4 结论

1) 采用经典高超声速二维圆管绕流非定常算例对本文发展的一体化计算方法进行验证分析，计算结果与参考文献和风洞试验结果吻合较好，证明了本文高超声速流-热-固一体化计算方法的可行性与可靠性，同时也证明了自适应时间步长控制方法的正确性。

2) 本文所提出的高超声速流-热-固一体化求解方法可以高效解决流场与结构传热稳态求解问题，较快计算出稳态结构与流场的温度分布。计算方法进行全物理场统一迭代，可以很好地改善传统分区耦合算法多次迭代计算效率低的不足。

3) 对高超声速前缘“Ⅳ型”激波干扰流-热-固耦合进行定常/非定常一体化计算分析，计算发现，“Ⅳ型”激波干扰作用产生的超声速“喷流”不断冲击壁面，使得壁面最大压力系数增大约9倍，壁面最大热流增大约4.7倍，给高速飞行器的热防护设计与选材带来严峻挑战。

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http://dx.doi.org/10.7527/S1000-6893.2019.23190

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#### 文章信息

LI Jiawei, WANG Jiangfeng, YANG Tianpeng, LI Longfei, WANG Ding
“Ⅳ型”激波干扰中流-热-固耦合问题一体化计算分析
Integrated numerical analysis of fluid-thermal-structural problems in "Type Ⅳ" shock wave interference

Acta Aeronautica et Astronautica Sinica, 2019, 40(12): 123190.
http://dx.doi.org/10.7527/S1000-6893.2019.23190