A LOOSE COUPLING NUMERICAL SIMULATION METHOD BETWEEN FLOW FIELD AND ELECTROMAGNETIC FIELD OF HYPERSONIC VEHICLE
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摘要: 在高超声速再入式飞行器通信黑障问题众多解决方案中, 电磁调控方法是最具可行性的方法之一. 一个细致可靠的电磁调控方案不仅需要准确考虑外加电磁场对导电流体作用, 还需要考虑因流体高速运动产生的感生电磁场对导电流体的作用. 对于高超声速飞行器典型飞行状态, 全磁流体力学方法因数值刚性问题很难直接用于流场和电磁场耦合模拟, 而低磁雷诺数近似方法无法考虑感生磁场对导电流体作用. 针对磁流体力学两种经典方法应用于高超声速飞行器电磁调控方案设计时存在的困难和不足, 文章在数值求解流动控制方程N-S方程和电磁控制方程双曲Maxwell方程的基础上, 将两者结合起来松耦合迭代求解, 建立适用于高超声速飞行器流场和电磁耦合的数值模拟方法. 通过磁流体激波反射和二维超声速磁流体喷管两个典型算例, 对文章的数值方法进行验证. 基于数值方法对外加偶极子磁场作用下某高超声速飞行器典型飞行状态下流场电磁耦合问题进行了研究, 数值模拟结果表明, 磁雷诺数为0.7557时, 不考虑感生磁场对流体运动影响时激波脱体距离相对于考虑感生磁场作用时大11.11%, 此误差已不容忽视, 对于该飞行状态, 应完整考虑流场和磁场相互作用而不是仅考虑外加磁场对流场运动影响.Abstract: Among the many solutions to the communication blackout problem of hypersonic re-entry vehicles, the electromagnetic control method is one of the most feasible methods. A precise electromagnetic control scheme not only needs to accurately consider the effect of the applied electromagnetic field on the conductive fluid, but also needs to consider the effect of the induced electromagnetic field caused by the high-speed motion of the fluid on the conductive fluid. For the typical flight state of a hypersonic vehicle, the full MHD method is difficult to be directly used in the simulation of the coupling between flow field and electromagnetic field due to the problem of numerical rigidity, and the low magnetic Reynolds number approximation method cannot consider the effect of the induced magnetic field on the conductive fluid. Aiming at the difficulties and deficiencies in the application of the two classic methods of MHD to the design of electromagnetic control methods for hypersonic vehicles, this paper numerically solves the flow governing equation NS equation and the electromagnetic governing equation hyperbolic Maxwell equation, and combines them together by a loosely coupled iterative solution, and finally, a numerical simulation method suitable for coupling between flow field and electromagnetic field of hypersonic vehicle is established. The numerical method is verified by two test cases, and that is MHD shock reflection and two-dimensional supersonic MHD nozzle. Based on the numerical method in this paper, the coupling problem between flow field and electromagnetic field in a typical flight state of a hypersonic vehicle under the action of an applied dipole magnetic field is studied. The numerical simulation results show that, at the state of the magnetic Reynolds number 0.7557, when the effect of induced magnetic field on the fluid is ignored, the shock stand-off distance is 11.11% larger than that when the induced magnetic field is considered. The error cannot be ignored. For the flight state, the interaction between the flow field and the magnetic field should be fully considered instead of only considering the influence of the applied magnetic field on the flow field motion.
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引 言
磁流体动力学(MHD)是一门将流体动力学理论和电磁理论结合起来研究导电流体在电磁场中运动规律的交叉学科[1], 在地球物理、天体物理、可控核聚变以及航空宇航中都有着广泛用途[2]. 在不同学科研究中, 因研究介质不同, 其名称有所差异, 比如MHD(magnetohydrodynamics), MFD(magneto fluid dynamics)或MGD(magneto gas dynamics). 磁流体力学在航空宇航中一个典型应用为高超声速飞行器电磁流动调控[3-8]. 高超声速飞行器电磁调控除了可以进行流动减阻、磁控热防护、进气道激波系控制、边界层分离转捩控制以及飞行姿态控制外[9-11], 还可能用于解决高超声速再入式飞行器通讯黑障问题[12-14]. 在高超声速飞行器附近施加外加电磁场, 通过洛伦兹力作用于高温电离的高速导电气体, 改变流体运动状态, 降低天线附近局部电子密度, 进而缓解或者解决通信黑障问题. 相关研究表明, 电磁调控方法是通信黑障众多解决方案中最具可行性的方法之一[15]. 而对该问题的研究就涉及到高超声速飞行器高速导电流场与外加/感生电磁场之间的耦合.
基于CFD的数值模拟是研究流场与电磁耦合的可行方法之一. 当前对该问题的数值模拟方法主要有两种: 全磁流体力学方法(full MHD equations)和低磁雷诺数近似方法(low magnetic Reynolds number approximation). 理论上, 任何流场和电磁场耦合问题可以使用全磁流体力学方法解决, 然而, 对于高超声速飞行器真实流动, 整个流场中, 部分区域电导率几乎为0, 部分区域存在一定电导率但仍比较有限, 此时全磁流体力学方法数值刚性问题突出[16-18], 且目前尚缺乏切实可行的解决方法, 致使全磁流体力学方法很难直接应用于高超声速飞行器流场和电磁耦合问题. 低磁雷诺数近似方法主要基于低磁雷诺数假设, 即感生磁场相对于外加磁场可忽略不计, 只考虑外加磁场对流体微团的电磁力作用而忽略感生磁场对流体微团作用. 该方法只考虑外加磁场而忽略感生磁场, 因此无需求解电磁相关方程, 只需求解考虑电磁源项的N-S方程. 此外, 该方法在已有的N-S方程求解器上实现起来较为简单方便, 相关研究多基于该方法进行[4, 8, 17, 19-25]. 然而, 对于典型高超声速飞行器飞行状态, 特征飞行速度$6000\;{\text{m/s}}$, 流场特征电导率$500\;{\text{S/m}}$, 特征长度$L = 1\;{\text{m}}$, 可得磁雷诺数$ {{Re} _m} = 3.77 $, 此时感生磁场相对于外加磁场是否可以忽略存在争议[16], 仍然使用该方法进行高超声速飞行器电磁调控方案分析和设计可能会引起较大误差, 影响飞行安全.
综上所述, 对于高超声速飞行器流场和电磁耦合问题, 全磁流体力学方法几乎无法直接求解, 而低磁雷诺数方法虽可用于分析, 但是其适用性不易保证, 误差难以衡量, 这给高超声速飞行器电磁调控精细化分析和设计带来了诸多挑战. 针对这样的不足, 本文将考虑电磁作用的流动控制方程N-S方程和电磁场控制方程双曲Maxwell方程结合起来并使用松耦合的方式进行迭代求解, 建立高超声速飞行器流场和电磁耦合数值模拟方法. 该方法可以充分考虑流场与外加/感生磁场之间的相互作用, 满足高超声速飞行器电磁调控精细化分析需求.
1. 数值模拟方法
1.1 数值方法框架及求解策略
本文对高超声速飞行器流场和电磁场耦合的数值模拟基于多年来持续开发的F2M框架[26-29]. F2M软件架构如图1所示, 主要包含公共类库、基础类库、数值方法类库以及各个具体应用模块. F2M早期主要是在使用计算流体力学方法数值求解流动控制方程N-S方程的基础上对柔性飞行器飞行动力学和气动弹性力学问题进行数值模拟和分析. 近年来, 经过不断改进, 已经可以用于求解电磁场相关问题. 本文将流场求解部分和电磁场求解部分结合起来建立高超声速飞行器流场和电磁场耦合数值模拟方法.
对于高超声速飞行器流场和电磁耦合问题, 本文采用松耦合的方式进行迭代求解, 求解步骤如图2所示, 主要步骤如下:
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图 2 高超声速飞行器流场电磁松耦合求解步骤Figure 2. Solution steps of loose coupling between flow field and electromagnetic field of hypersonic vehicle(1) 准备好数值模拟需要的计算网格, 给定流场电磁场初始和边界条件以及外加电磁场;
(2) 根据外加和感生电磁场(感生电磁场第一步为0, 后续为上一步值)计算总电磁场, 结合当前流场速度使用欧姆定律计算电流密度;
(3) 根据电流密度和总电磁场计算流场电磁源项, 求解考虑电磁力源项的流动控制方程N-S方程, 获得总电磁场作用下的高超声速飞行器流场;
(4) 结合当前总电磁场, 使用新的流场速度依据欧姆定律再次更新传导电流密度, 求解双曲Maxwell方程获得感生电磁场;
(5) 检查流场和电磁场方程残差是否收敛, 若不收敛, 则跳转到步骤(2)继续执行, 若收敛, 则退出迭代, 结束. 此时获得的流场和电磁场即充分考虑了流场与外加/感生电磁场相互作用. 同时可以发现, 流场方程和电磁场方程通过交替求解、多次迭代的方式进行耦合求解, 是比较典型的松耦合求解方式.
此外, 需要说明的是, 第一次执行到步骤(5)时, 如果不跳转到步骤(2)执行而是直接退出, 如图3所示, 此时只是计算了外加磁场作用下的流场以及因流体运动产生的感生磁场而没有考虑感生磁场对流体运动的影响, 为单向耦合. 若步骤(2) ~ 步骤(5)循环多次直至残差收敛, 如图4所示, 此时既考虑到因流体运动而产生的感生磁场, 又考虑到感生磁场对流体自身的作用, 为双向耦合. 从本质上来讲, 单向耦合是在使用低磁雷诺数近似方法求解外加磁场作用下流场的基础上, 进一步求解流场诱导产生的感生磁场. 而双向耦合则进一步考虑感生磁场对流场作用. 双向耦合相对于单向耦合更充分地考虑了流场和电磁之间的相互作用. 本文的数值方法可以同时进行单向耦合和双向耦合模拟分析. 数值方法原理主要涉及流场计算方法和电磁场计算方法, 分别详细介绍如下.
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图 3 流场电磁耦合问题单向耦合Figure 3. One-way coupling between flow field and electromagnetic field![]()
图 4 流场电磁耦合问题双向耦合Figure 4. Two-way coupling between flow field and electromagnetic field1.2 考虑电磁作用的流动控制方程及数值模拟方法
本文对高超声速飞行器流场的求解基于无量纲形式的N-S方程, 电磁力对导电流体的作用以体积力源项形式添加, 即
$$ \frac{{\partial \rho }}{{\partial t}} + \nabla \cdot (\rho {\boldsymbol{V}}){\text{ = 0}} $$ (1) $$ \frac{{\partial \rho {\boldsymbol{V}}}}{{\partial t}} + \nabla \cdot (\rho {\boldsymbol{VV}} + p\bar{\bar {\boldsymbol{I}}} ){\text{ = }}\frac{{Ma}}{{Re}}\nabla \cdot \bar{\bar {\boldsymbol{\pi }}} + \frac{{R{e_m}}}{{Ma}}{{\boldsymbol{J}}_{{\mathrm{total}}}} \times {{\boldsymbol{B}}_{{\mathrm{total}}}} $$ (2) $$ \begin{split} & \frac{\partial\rho\varepsilon}{\partial t}+\nabla\cdot(\rho\varepsilon\boldsymbol{V}+p\boldsymbol{V})\text{ = } \\ & \qquad\frac{Ma}{Re}\nabla\cdot(\boldsymbol{V}\cdot \bar{\bar {\boldsymbol{\pi }}})-\frac{Ma}{Re}\nabla\cdot\boldsymbol{q}+\frac{Re_m}{Ma}\boldsymbol{J}_{\mathrm{total}}\cdot\boldsymbol{E}_{\mathrm{total}}\end{split} $$ (3) 式中, $ \rho $为气体密度, $t$为时间, ${\boldsymbol{V}}$为流体运动速度, $ p $和$ \varepsilon $分别表示气体压强以及单位质量总能. $ \bar{\bar {\boldsymbol{\pi }}} $为黏性应力张量, $ {\boldsymbol{q}} $为热流密度矢量, $ \bar{\bar {\boldsymbol{I}}} $为二阶单位张量. $ Ma $为来流马赫数, $ Re $为雷诺数. ${{\boldsymbol{B}}_{{\mathrm{total}}}}$和${{\boldsymbol{E}}_{{\mathrm{total}}}}$分别表示总的磁感应强度和电场强度, 即
$$\left.\begin{split} & {{\boldsymbol{B}}_{{\mathrm{total}}}} = {{\boldsymbol{B}}_{{\mathrm{applied}}}} + {\boldsymbol{B}} \\ & {{\boldsymbol{E}}_{{\mathrm{total}}}} = {{\boldsymbol{E}}_{{\mathrm{applied}}}} + {\boldsymbol{E}} \end{split}\right\} $$ (4) 其中, ${\boldsymbol{B}}$和${\boldsymbol{E}}$分别表示感生磁感应强度和电场强度, ${{\boldsymbol{B}}_{{\mathrm{applied}}}}$和${{\boldsymbol{E}}_{{\mathrm{applied}}}}$分别表示外加磁感应强度和电场强度. $R{e_m}$为磁雷诺数, 满足
$$ {{Re} _m} = {\mu _{{m_\infty }}}{\sigma _\infty }{V_\infty }L $$ (5) 式中, $ L $为参考长度, $ {V_\infty } $为来流速度, ${\sigma _\infty }$为来流电导率, $ {\mu _{{m_\infty }}} $为来流磁导率. 另外, $ {{\boldsymbol{J}}_{{\mathrm{total}}}} $表示总电流密度, 可通过欧姆定律进行计算, 即
$$ {{\boldsymbol{J}}_{{\mathrm{total}}}} = \sigma ({{\boldsymbol{E}}_{{\mathrm{total}}}} + {\boldsymbol{V}} \times {{\boldsymbol{B}}_{{\mathrm{total}}}}) $$ (6) 式中, $ \sigma $表示导电流体的电导率. 此外, 还需补充无量纲形式的完全气体状态方程, 即
$$ p = \frac{{\rho T}}{\gamma } $$ (7) 其中, $ \gamma $表示气体比热比, $ T $表示静温. 式(1)为连续方程, 式(2)和式(3)分别为考虑了总的电磁场对导电流体作用的动量方程和能量方程. 式(1) ~ 式(3)可以写成守恒形式, 随后在非结构混合网格上基于格心有限体积法进行空间离散, 进一步可使用时间推进方法[30]求解. 其中对流通量使用AUSM + 迎风分裂格式, 黏性通量使用中心格式, 时间推进采用隐式LU-SGS方法.
1.3 电磁控制方程及数值模拟方法
外加电磁场满足Maxwell方程, 无需进行求解, 只需求解感生电磁场即可. 具体求解时不考虑介质的电极化和磁极化效应, 控制方程采用无量纲形式的纯双曲Maxwell方程[31-32], 即
$$ \left.\begin{split} & \frac{{\partial {\boldsymbol{B}}}}{{\partial t}} + \nabla \times {\boldsymbol{E}} + {\gamma ^2}\nabla \psi = 0 \\ & \frac{{\partial {\boldsymbol{E}}}}{{\partial t}} - {c^2}\nabla \times {\boldsymbol{B}} + {\chi ^2}{c^2}\nabla \phi = - \frac{{R{e_m}}}{{Ma}}\frac{{{{\boldsymbol{J}}_{{\mathrm{total}}}}}}{{{\varepsilon _0}}} \\ & \frac{{\partial \psi }}{{\partial t}} + {c^2}\nabla \cdot {\boldsymbol{B}} = 0 \\ & \frac{{\partial \phi }}{{\partial t}} + \nabla \cdot {\boldsymbol{E}} = \frac{{R{e_m}}}{{Ma}}\frac{{{\rho _c}}}{{{\varepsilon _0}}} \end{split} \right\}$$ (8) 式中, $c$, ${\mu _{m0}}$和${\varepsilon _0}$分别为自由空间中的光速、磁导率和介电常数, 且满足
$$ c{\text{ = }}{1 \mathord{\left/ {\vphantom {1 {\sqrt {{\mu _{m0}}{\varepsilon _0}} }}} \right. } {\sqrt {{\mu _{m0}}{\varepsilon _0}} }} $$ (9) 其中, ${\rho _c}$为体电荷密度, 对于本文算例, 恒为0. $\gamma $和$\chi $为波速调节因子, 为无量纲数, 一般都可以取1. $\psi $和$\phi $为额外引入的变量, 可以使得方程在时间推进过程中对感生磁场散度和电场散度进行约束, 避免数值震荡[31]. 类似于式(1) ~ 式(3), 式(8)也可以写成守恒形式, 可使用与N-S方程类似的方式进行求解. 这里, 对流通量使用Steger-Warming格式, 时间推进采用隐式LU-SGS方法.
2. 算例验证与分析
这里使用两个经典的磁流体算例对本文的流场电磁松耦合数值模拟方法进行验证. 第1个算例为磁流体激波反射问题, 其几何外形较为简单, 而第2个算例为二维超声速磁流体喷管问题, 几何外形稍复杂一些.
2.1 磁流体激波反射
磁流体激波反射(MHD shock reflection)是一个非常经典的磁流体算例, 诸多学者曾基于全MHD方法对该问题进行研究[33-35]. 流场计算域为$2\;{\text{m}} \times 1\;{\text{m}}$, 网格单元总数为$100 \times 100$, 均匀分布. 来流为量热完全气体, 比热比$\gamma = 1.4$, 不考虑黏性效应, 且为理想导体. 左侧和上侧为入口, 来流条件分别为
$$ {{\boldsymbol{U}}}_{l} = [1.0,2.9,0.0,0.0,5/7\text{, }0.5,0.0,0.0,0.0,0.0,0.0] $$ (10) $$ \begin{split} &{{\boldsymbol{U}}}_{u} = [1.459\;8,2.717\;9\text{, }-0.404\;9,0.0,1.222\;9,\\ &\qquad 0.683\;8\text{, }-0.101\;9,0.0,0.0,0.0,-8.32 \times 10^{-6}]\end{split} $$ (11) 下侧为无穿透理想导体壁面, 右侧为出口, 具体边界条件可参考文献[30-31]. 其中
$$ {\boldsymbol{U}} = [\rho ,\;u,\;v,\;w,\;p,\;Bx,\;By,\;Bz,\;Ex,\;Ey,\;Ez] $$ (12) 图5给出了本文方法计算出的流场压力云图, 可以发现, 斜激波角度为${29^ \circ }$, 与文献一致, 表明本文的数值方法具有良好的准确性.
2.2 二维超声速磁流体喷管
磁流体喷管同样是一个非常经典的磁流体算例, 类似地, 之前的研究多基于全MHD方法[18, 36]. 喷管长3 m, 上表面为一曲线, 满足$y = 1 - 0.3{\sin ^2}({{\text{π} x} \mathord{\left/ {\vphantom {{\text{π} x} 3}} \right. } 3})$, 下表面为一直线, 网格单元总数为$400 \times 160$, 均匀分布, 无特殊加密处理. 来流为量热完全气体, 比热比$\gamma = {5 \mathord{\left/ {\vphantom {5 3}} \right. } 3}$, 不考虑黏性效应, 且为理想导体. 左侧为入口, 来流压强$p = 1$, 密度$\rho = 1$, 速度$u = 3.5$, 磁感应强度${B_x} = 2{{\sqrt 3 }/ {\sqrt 5 }}$, 可得来流马赫数$Ma = {{3.5\sqrt 3 } / {\sqrt 5 }}$, 来流磁马赫数$M{a_m} = 1.75$. 另外, 需要说明的是, 原始文献[36]在无量纲化时参考速度为来流速度, 本文在无量纲化时参考速度为来流声速, 而磁场无量纲化时的参考量与速度参考量有关, 因此这里给出的磁感应强度与原始文献不同. 上壁面和下壁面都为无穿透理想导体壁面, 右侧为出口边界, 具体边界条件可参考文献[30-31]. 图6给出了使用本文方法计算所得的流场密度云图, 可以发现, 激波在上下壁面各反射一次和两次, 最后从出口上半部分流出. 参考文献[36]中激波在下壁面两个反射位置分别为1.00和2.31, 本文计算结果为0.97和2.36, 与参考文献基本一致, 表明本文的数值计算方法具有良好的准确性.
3. 数值模拟结果与分析
这一节以某高超声速飞行器为例, 使用本文的数值方法对流场和电磁场耦合问题进行数值模拟和分析.
3.1 飞行状态与网格无关性验证
选用的高超声速钝头体头部为一半球, 半球后为一圆柱, 如图7所示. 该外形可通过旋转生成, 截面几何形状如图8所示. 头部半球半径$a = 10{\text{ mm}}$, 圆柱长$50{\text{ mm}}$, 整个钝头体不导电、不导磁. 来流为量热完全气体, 具有均匀电导率, 比热比$\gamma = 1.4$, 马赫数$Ma = 5$, 无迎角和侧滑角, 基于头部半球半径的雷诺数$Re = 80\;000$. 壁面为等温壁, 温度为来流温度的3倍. 不考虑湍流黏性效应, 使用层流模型进行计算, 层流黏性系数使用Sutherland经验公式[37]获得.
使用本文方法对该高超声速钝头体在外加偶极子磁场作用下的流动进行计算. 其中, 外加偶极子磁场在头部半球球心位置施加, 方向沿x轴, 磁感应强度为
$$ {{\boldsymbol{B}}_{{\mathrm{applied}}}} = - {B_0}\frac{{{a^3}}}{{{{{r}}^3}}}\left( {\cos \theta {{\boldsymbol{e}}_r} + \frac{1}{2}\sin \theta {{\boldsymbol{e}}_\theta }} \right) $$ (13) 式中, $ {B_0} $为外加磁场在壁面位置的幅值, $r$为任一位置到球心距离, 角度$\theta $为连接该位置与球心的连线与$x$轴正方向夹角, $ {{\boldsymbol{e}}_r} $和$ {{\boldsymbol{e}}_\theta } $分别为球坐标系中的单位径向矢量和$\theta $角度矢量. 由于几何外形的对称性, 计算所用网格可通过二维平面网格旋转若干层获得, 因此下文只给出截面位置的网格和流场云图. 数值模拟结果需要保证与网格无关. 这里选择3套网格, 即稀网格、中等网格与密网格对该问题进行分析, 截面网格单元数和结点数如表1所示. 3套网格计算域尺寸相同, 且保证足够大, 网格单元和结点合理分布, 尽量平滑过渡, 保证网格质量. 壁面第一层高度均按照${{{y}}^{\text{ + }}}{\text{ = }}1$估算获得, 壁面附近网格分布基本相同, 区别在于距离壁面稍远位置网格疏密. 流场与电磁场双向耦合时, 不同网格所得驻点线上感生磁感应强度Bx如图9所示. 可以发现, 3套网格计算所得磁场基本相同, 仅在峰值附近存在较小差异. 稀网格计算的磁场峰值略小, 而中等网格和密网格计算结果基本相同. 考虑到计算准确性, 最终选择密网格进行数值模拟, 网格如图10所示.
表 1 截面网格数量Table 1. Number of grid cells and nodesGrid/domain Number of cells Number of nodes coarse fluid 7300 7474 solid 4096 4210 medium fluid 19050 19328 solid 12225 12426 fine fluid 42000 42431 solid 42300 42671 ![]()
图 9 双向耦合时不同疏密网格计算的驻点线感生磁感应强度BxFigure 9. Induced magnetic induction intensity Bx along axial line calculated on different grid by two-way coupling3.2 外加偶极子磁场作用下高超声速钝头体流场和电磁场分析
首先看一下外加偶极子磁场对导电流体运动的影响. 图11给出了来流是否导电时计算所得的流场压力云图. 其中上半部分流体不导电, 下半部分来流存在均匀电导率, 对应的磁雷诺数为$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$. 当流体不导电时, 外加磁场对流体没有洛伦兹力作用, 此时激波脱体距离(相对于半球半径进行无量纲化)为0.16. 而当流体导电时, 流体微团受到洛伦兹力作用, 使用双向耦合方法得到的激波脱体距离为0.27. 对比两个状态, 可得磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$时, 外加偶极子磁场使得激波脱体距离相对于非导电流体增加68.75%, 这体现出了高超声速飞行器外加磁场对流体运动的效果.
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图 11 是否导电时流场压力云图(上: 不导电, 下: 导电, 磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}})$Figure 11. Flow field pressure when conducting or not (top: non-conductive, bottom: conductive, magnetic Reynolds number $R{e_{\text{m}}} = {\text{0}}{\text{.755 7}})$图12给出了磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$时是否考虑感生磁场对导电流体微团作用时流场压力云图. 上半部分不考虑感生磁场对导电流体作用, 即单向耦合, 也即低磁雷诺数方法, 此时无量纲激波脱体距离为0.30. 而下半部分与图11下半部分相同, 考虑了感生磁场对导电流体作用, 即双向耦合, 也即完整流场−电磁耦合方法. 真实的流场应同时考虑外加磁场和感生磁场作用, 即后者应为真实流场, 而前者计算的流场为低磁雷诺数近似流场. 当严格满足低磁雷诺数假设时, 感生磁场可以忽略不计, 此时两者流场应该相同. 然而, 对于该算例, 磁雷诺数达到$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$, 低磁雷诺数假设是否满足已存在争议. 若以激波脱体距离作为标准, 低磁雷诺数近似方法过高估计了电磁调控效果, 相对于完整流场−电磁耦合方法计算误差达到11.11%, 这个误差在典型工程应用中已经偏大, 反映了低磁雷诺数方法的不足. 对于磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$状态, 为保证数值模拟结果准确可信, 应使用完整流场−电磁耦合方法而不是低磁雷诺数近似方法.
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图 12 磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$时是否考虑感生磁场对流场作用时压力云图(上: 不考虑, 下: 考虑)Figure 12. Flow field pressure with/without considering the effect of induced magnetic field on flow at the magnetic Reynolds number $R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$ (top: with, bottom: without)图13和图14分别给出了是否考虑感生磁场对流场作用时感生磁感应强度Bx和By云图. 可以发现, 两者等势线趋势基本相同, 峰值和谷值位置比较接近, 但仍存在差异. 对于感生磁场Bx, 在钝头体半球头部前侧区域存在峰值(红色区域), 且考虑感生磁场作用时峰值略大, 此外, 该峰值更加靠近壁面. 在钝头体圆柱区域上侧存在谷值(蓝色区域), 且考虑感生磁场作用时谷值在数值上略大(绝对值略小). 对于感生磁场By, 在钝头体半球上侧存在峰值(红色区域), 且考虑感生磁场时峰值略小. 在钝头体半球前侧近似45°方向存在谷值(蓝色区域), 且考虑感生磁场时谷值在数值上略小(绝对值略大). 图12 ~ 图14给出的流场和电磁场云图便于从整体上把握流场电磁耦合作用效果, 可是却不便于定量分析. 为此, 图15给出了是否考虑感生磁场对流场作用时驻点线上压力. 从图中可以看出, 在整个激波后侧区域, 考虑感生磁场作用时驻点线上压力较低, 这反映出在该状态下低磁雷诺数方法会过高估计驻点线上压力. 图16给出了是否考虑感生磁场对流场作用时驻点线上感生磁感应强度Bx. 在壁面附近, 考虑感生磁场耦合作用时, 磁感应强度略大; 而在离壁面较远区域, 考虑感生磁场耦合作用时, 磁感应强度略小.
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图 13 是否考虑感生磁场对流场作用时感生磁感应强度Bx云图(上: 不考虑, 下: 考虑)Figure 13. Induced magnetic induction intensity Bx with/without considering the effect of induced magnetic field on flow (top: with, bottom: without)![]()
图 14 是否考虑感生磁场对流场作用时感生磁感应强度By云图(上: 不考虑, 下: 考虑)Figure 14. Induced magnetic induction intensity By with/without considering the effect of induced magnetic field on flow (top: with, bottom: without)![]()
图 15 是否考虑感生磁场对流场作用时驻点线上压力Figure 15. Pressure along stationary line with/without considering the effect of induced magnetic field on flow![]()
图 16 是否考虑感生磁场对流场作用时驻点线上感生磁感应强度BxFigure 16. Induced magnetic induction intensity Bx along stationary line with/without considering the effect of induced magnetic field on flow4. 总结
针对全MHD方法和低磁雷诺数近似方法在高超声速飞行器流场电磁耦合数值模拟上的困难和不足, 本文以考虑电磁力作用的流动控制方程N-S方程和考虑高速导电流体运动作用的双曲Maxwell方程为基础, 将两者松耦合迭代求解, 建立了高超声速流场电磁耦合数值模拟方法. 该方法不仅考虑了外加磁场对导电流体作用, 还考虑导电流体高速运动产生的感生电磁场以及感生电磁场对导电流体自身影响.
为对数值方法进行验证, 本文对磁流体激波反射算例和二维超声速磁流体喷管两个典型磁流体力学算例进行了数值模拟, 计算结果与文献中使用全MHD方法计算结果基本一致, 表明本文数值方法可用于流场电磁耦合问题分析, 且具有良好的准确性.
针对高超声速飞行器流场电磁耦合问题, 本文以某高超声速钝头体外形为例, 定量分析比较了外加偶极子磁场作用下是否考虑感生磁场对导电流体作用时的流场和电磁场. 数值模拟结果表明, 对于本文典型飞行状态, 磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$时, 不考虑感生磁场作用时(即低磁雷诺数近似方法)计算的激波脱体距离相对于考虑感生磁场作用(完整流场电磁耦合)时高11.11%. 此误差已不容忽视, 在高超声速飞行器电磁调控方案设计时, 应引起足够重视.
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图 2 高超声速飞行器流场电磁松耦合求解步骤
Figure 2. Solution steps of loose coupling between flow field and electromagnetic field of hypersonic vehicle
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图 3 流场电磁耦合问题单向耦合
Figure 3. One-way coupling between flow field and electromagnetic field
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图 4 流场电磁耦合问题双向耦合
Figure 4. Two-way coupling between flow field and electromagnetic field
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图 9 双向耦合时不同疏密网格计算的驻点线感生磁感应强度Bx
Figure 9. Induced magnetic induction intensity Bx along axial line calculated on different grid by two-way coupling
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图 11 是否导电时流场压力云图(上: 不导电, 下: 导电, 磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}})$
Figure 11. Flow field pressure when conducting or not (top: non-conductive, bottom: conductive, magnetic Reynolds number $R{e_{\text{m}}} = {\text{0}}{\text{.755 7}})$
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图 12 磁雷诺数$R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$时是否考虑感生磁场对流场作用时压力云图(上: 不考虑, 下: 考虑)
Figure 12. Flow field pressure with/without considering the effect of induced magnetic field on flow at the magnetic Reynolds number $R{e_{\text{m}}} = {\text{0}}{\text{.755 7}}$ (top: with, bottom: without)
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图 13 是否考虑感生磁场对流场作用时感生磁感应强度Bx云图(上: 不考虑, 下: 考虑)
Figure 13. Induced magnetic induction intensity Bx with/without considering the effect of induced magnetic field on flow (top: with, bottom: without)
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图 14 是否考虑感生磁场对流场作用时感生磁感应强度By云图(上: 不考虑, 下: 考虑)
Figure 14. Induced magnetic induction intensity By with/without considering the effect of induced magnetic field on flow (top: with, bottom: without)
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图 15 是否考虑感生磁场对流场作用时驻点线上压力
Figure 15. Pressure along stationary line with/without considering the effect of induced magnetic field on flow
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图 16 是否考虑感生磁场对流场作用时驻点线上感生磁感应强度Bx
Figure 16. Induced magnetic induction intensity Bx along stationary line with/without considering the effect of induced magnetic field on flow
表 1 截面网格数量
Table 1 Number of grid cells and nodes
Grid/domain Number of cells Number of nodes coarse fluid 7300 7474 solid 4096 4210 medium fluid 19050 19328 solid 12225 12426 fine fluid 42000 42431 solid 42300 42671 
下载: 导出CSV
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