哈尔滨工业大学学报  2021, Vol. 53 Issue (6): 54-61  DOI: 10.11918/201909018
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引用本文 

张宇, 王晓亮. 参数化艇身阻力特性的全局敏感度及设计空间[J]. 哈尔滨工业大学学报, 2021, 53(6): 54-61. DOI: 10.11918/201909018.
ZHANG Yu, WANG Xiaoliang. Global sensitivity and design space of drag characteristics of parametric airship hull[J]. Journal of Harbin Institute of Technology, 2021, 53(6): 54-61. DOI: 10.11918/201909018.

基金项目

国家自然科学基金(61733017);上海市自然科学基金(18ZR1419000)

作者简介

张宇(1996—),男,硕士研究生

通信作者

王晓亮,wangxiaoliang@sjtu.edu.cn

文章历史

收稿日期: 2019-09-02
参数化艇身阻力特性的全局敏感度及设计空间
张宇, 王晓亮    
上海交通大学 航空航天学院,上海 200240
摘要: 为有效降低飞艇艇身外形设计参数的维度、提高设计效率,并给予飞艇艇身初期设计一定的参考和指导.结合PARSEC(parametric section)参数化方法、计算流体力学(Computational fluid dynamics, CFD)方法和基于方差的Sobol全局敏感度分析方法,形成了一套艇身外形参数关于阻力系数敏感度的评价体系.首先,采用物理意义明确的PARSEC方法描述飞艇艇身外形轮廓线.然后,由拉丁超立方抽样(Latin hypercube sampling, LHS)所产生样本艇身的阻力系数通过二维轴对称模型的CFD数值方法得到,CFD数值方法求解精度通过6个典型流线型旋成体的实验数据得到了验证,在保证雷诺数一致的情况下,计算和实验所得艇身阻力系数的平均相对误差为1.5%.最后,通过Sobol全局敏感度分析方法对艇身外形参数进行了敏感度排序.研究结果表明,与艇身阻力系数最敏感的3个参数分别为头部半径rh、最大半径rd和最大半径位置xd.在此工作基础上形成了飞艇艇身外形的设计空间,所得设计空间对提高飞艇外形设计效率、降低飞艇气动阻力系数具有积极意义.
关键词: Sobol全局敏感度    PARSEC    拉丁超立方抽样    平流层飞艇    数值模拟    
Global sensitivity and design space of drag characteristics of parametric airship hull
ZHANG Yu, WANG Xiaoliang    
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Abstract: To effectively reduce the dimension of the design parameters of airship hulls, improve the design efficiency, and provide reference and guidance for the design of airship hulls in the initial stage, an evaluation system was constructed for the airship hull shape parameters associated with drag coefficient sensitivity, combined with parametric section (PARSEC) method, computational fluid dynamics (CFD) method, and Sobol global sensitivity analysis method based on variance. First, the outline of airship hull was described by PARSEC method which has clear physical significance. Then, the drag coefficient of airship hull samples produced by Latin hypercube sampling (LHS) was obtained by 2D axisymmetric CFD method. The accuracy of CFD method was verified by the experimental data of six typical streamlined bodies of revolution. Under the condition that the Reynolds number was consistent, the average relative error between the calculated and experimental drag coefficient was 1.5%. Finally, the sensitivity of the hull shape parameters was ranked by the Sobol global sensitivity analysis method. Research results show that the three parameters which were the most sensitive to the hull drag coefficient were the head radius rh, the maximum radius rd, and the maximum radius position xd. On the basis of this study, the design space of the airship hull shape was formed, and the design space has positive significance for improving the efficiency of designing process and reducing the aerodynamic drag of airships.
Keywords: Sobol global sensitivity analysis    PARSEC    Latin hypercube sampling (LHS)    stratospheric airship    numerical simulation    

平流层一般指海拔高度处于10~55 km的空间,由于平流层的垂直对流效应小,气流稳定,适合布置平流层飞行器完成诸如中继通信、监察和运输等任务,具有极大的军事及民用价值[1].平流层飞艇是一种能定点飞行、效费比高的平流层飞行器,其结构一般包括艇身、吊舱、尾翼和支撑骨架(如图 1所示).对平流层飞艇而言,是否具备长航时是一项重要性能指标[2-3].平流层飞艇飞行过程中受到的阻力由动力系统平衡,阻力系数与动力消耗近似为正比关系,加之飞艇表面积巨大,驻留风阻很高,因此即便阻力系数略微减小,也能极大的减小动力系统的能量消耗,从而提高执行任务时间.通常,艇身阻力占到了全艇阻力的约2/3[4-5],因此找到使艇身具有最小阻力系数的外形具有十分重要的意义[6-7].

图 1 飞艇结构示意 Fig. 1 Airship structure diagram

目前,飞艇减阻一般采用优化方法进行. Lutz等[8]通过在艇身布置周向分布的点源/汇从而得到了压力场和速度场,并利用边界层模型得到了不同雷诺数下的最小阻力外形.Wang等[9]采用势流-边界层耦合方法和混合遗传算法对平流层飞艇艇身外形进行了优化. Geruti等[10]基于粒子群优化算法(PSO)提出了适用于考虑附加质量的非常规布局飞艇的优化框架. Kanikdale等[11]采用GNVR作为飞艇基础外形,提出了飞艇外形的多变量约束方法,并用模拟退火算法对外形进行了优化. Wang等[12]将气动、结构、能源和质量作为优化对象,通过综合目标方程对飞艇外形进行了多学科优化.一般地,飞艇艇身外形廓线可由八参数、四参数、GNVR、椭球形和“海豚”形描述[13-15].但这些方法能包含的外形范围较小,表达过于数学化,效率较低. 翼型参数化方法较上述飞艇外形描述方法丰富很多,能表征更为多样的外形空间.翼型参数化方法主要有Hick-Henne Bump Functions、B-splines、PARSEC和Class/Shape function Transformation (CST)等4种.其中,Hick-Henne Bump Functions方法通过叠加形状函数并改变形状函数因子αi对已有的基础外形进行扰动,例如: Zhong等[16]以RAE2822为基础翼型,通过引入10个形状函数来表示新的翼型;Yang等[17]利用开源软件SU2,引入5个形状函数分别对NACA0012和RAE2822翼型进行了优化;总的来说,Hick-Henne Bump Functions方法适用于对已有外形进行优化,难以形成设计空间.B-Splines方法通过在翼型周围改变控制点位置来更新翼型形状,例如Pérez等[18]使用B-spline方法控制螺旋桨不同展向处叶素的外形,通过设定不同的控制点数和容错系数得到不同的三维螺旋桨CAD模型;Martin等[19]应用扩展的B-spline方法对三维机翼形状进行了优化;该方法具有很强的凸包性和稳定性,但上述控制点没有具体的物理含义,不便于形成明确的设计参考. CST方法通过类函数C(x)和形状函数S(x)的乘积定义外形曲线,例如Masdari等[20]应用CST方法描述翼型形状,结合离散涡方法,以功率系数为目标,对Savonius涡轮进行了优化. 虽然该方法的适应性较强,但表达过于数学化,同样不利于形成明确的设计参考. PARSEC方法通过11个参数控制翼型形状,Chen等[21]通过PARSEC方法描述翼型形状,以功率系数为目标对垂直轴流风力机进行了优化,使功率系数比NACA0015翼型的结果提高了13.26%;PARSEC方法每个参数均有清晰的物理含义,利于形成明确的设计空间. Sripawadkul等[22]系统地研究了上述4种翼型参数化方法的特性,并给出了每种方法在不同指标下的评价结果,其中的正交性指标体现了各参数化方法中参数与外形是否一一对应,该指标在形成设计空间的过程中十分重要,结果表明只有PARSEC方法与CST方法能完全满足正交性.考虑到CST方法表达过于繁复,可选PARSEC方法来描述飞艇外形.

一般而言,优化过程是面向所有参数的,这往往导致在次要参数上会消耗许多不必要的时间.因此提取对艇身阻力系数最敏感的参数,降低优化设计维度,并由这些参数形成飞艇艇身外形设计空间,从而给予艇身初期设计一定的参考是十分必要的.基于此,本文以艇身阻力系数为目标,通过PARSEC参数化方法、CFD方法和Sobol全局敏感度分析方法得到对阻力系数最敏感的参数,进而形成了由这些参数构成的飞艇艇身外形设计空间.

1 数值方法及其验证 1.1 数值方法

艇身阻力系数可由势流理论、风洞实验和计算流体力学(CFD)方法得到[23].风洞实验的高耗时和高成本不适用于本文工作,针对飞艇发展的势流理论通常只对细长体预测的较准确,且黏性作用往往使势流理论的结果不准确.随着计算机技术的发展,CFD成为解决气动优化设计和流固耦合问题的重要手段.低速不可压缩流动的NS方程为

$ \frac{\partial \boldsymbol{u}}{\partial t}+(\boldsymbol{u} \cdot \nabla) \boldsymbol{u}=-\frac{1}{\rho} \nabla p+\mu \nabla^{2} \boldsymbol{u}. $ (1)

式中:ρ为流体密度; p为压力; μ为动力黏度; u为速度矢量.对式(1)进行无量纲化可得

$ \frac{\partial \boldsymbol{u}^{*}}{\partial t^{*}}+\left(\boldsymbol{u}^{*} \cdot \nabla^{*}\right) \boldsymbol{u}^{*}=-\nabla^{*} p^{*}+\frac{1}{R e_{v}} \nabla^{* 2} \boldsymbol{u}^{*}. $ (2)

式中:u*= u/Ut*=tU/Lp*=p/ρU2,▽*=L▽,其中L为模型特征长度, U为来流速度. 且从式(2)可知,低速流动状态能由雷诺数Rev表征.

选取Spalart-Allmaras模型求解湍流流动,Spalart-Allmaras模型适合求解航空外流场问题,其计算开销低,稳定性好[24].求解器采取不可压缩形式的压力基求解器.压力-速度耦合格式为“coupled”,空间梯度离散方法为“least squares cell based”,压力项采用二阶格式离散,动量和修正湍流黏度采用二阶迎风格式离散.Spalart-Allmaras模型通过求解关于修正湍流黏度的输运方程获得流场信息:

$ \begin{aligned} \rho \frac{\partial}{\partial t}(\tilde{\nu})+\rho \frac{\partial}{\partial x_{i}}\left(\tilde{\nu} u_{i}\right)=& G_{\nu}+\frac{1}{\sigma_{\tilde{\nu}}}\left\{\frac{\partial}{\partial x_{j}}\left[(\mu+\tilde{\rho \nu}) \frac{\partial \tilde{\nu}}{\partial x_{j}}\right]+\right.\\ &\left.C_{b 2} \rho\left(\frac{\partial \tilde{\nu}}{\partial x_{j}}\right)^{2}\right\}-Y_{\nu}+S_{\tilde{\nu}} . \end{aligned} $ (3)

式中:Gν为湍流黏度生成项; Yν为近壁面处产生的湍流黏度损耗项; ${\sigma _{\tilde v}}$Cb2为常数,其值分别为0.667和0.622;$\tilde v$为分子运动黏度;${S_{\tilde v}}$为用户自定义源项,此处取0.

1.2 网格无关性验证

因艇身外形一般为旋成体,故简化为二维轴对称模型.艇身长度为L,艇身与入口和出口的距离均为60L,远场边界与旋转轴的距离也为60L,如图 2所示.模型经过归一化处理,模型长度L取为1.边界条件设置见表 1.流场的数值求解精度主要由y+和艇身沿轴向的网格种子数量决定.设置艇身表面的第1层网格高度为10-6L,计算得到的流线型旋成体模型轮廓线上的y+分布如图 3所示,横坐标为归一化的艇身轴向位置,可见本文设置能使y+满足要求.如图 4所示,对比了在不同Rev下艇身轴向网格种子数量与体积阻力系数Cdv的关系,可见当艇身轴向网格种子数量取为600时,Cdv几乎不再变化.后续数值分析均基于以上述网格划分模式进行.

图 2 计算域几何示意图 Fig. 2 Computational domain geometry
表 1 边界条件类型 Tab. 1 Boundary condition types
图 3 不同Rev下Model 4154表面的y+分布 Fig. 3 The y+ distribution of Model 4154 along the hull with different Rev
图 4 网格无关性检验 Fig. 4 Grid independence verification

文中以飞艇体积的立方根值作为特征长度,因此Rev定义如下:

$ R e_{v}=\frac{\rho U_{\infty} V^{\frac{1}{3}}}{\mu}, $ (4)

式中,V为飞艇体积.

总阻力系数Cdv由压差阻力系数Cdpv和黏性阻力系数Cdfv构成:

$ C d_{v} =\frac{F_{d}}{\frac{1}{2} \rho U_{\infty}^{2} V^{\frac{2}{3}}} , $ (5)
$ F_{d} =F_{p}+F_{f}, $ (6)
$ C d_{v} =C d p_{v}+C d f_{v} . $ (7)

式中:Fd为总阻力; Fp为压差阻力; Ff为黏性阻力.

1.3 算法验证

文献[25]中对于流线型旋成体做了详细的流动实验分析记录,在此选取6组模型作为验证对象,模型材质为红木,长度统一为2.74 m,通过在水洞中拖曳模型获取阻力,模型中轴线距离水面深度为2.74 m,模型标签名分别为4 154、4 156、4 158、4 162、4 164和4 175,对应长细比分别为4、6、8、7、7和5. 流体介质的密度和动力黏度分别为998.2 kg/m3和0.001 003 kg/m · s,确保在数值计算中使CFD与实验的雷诺数相等. 对比结果如图 5所示,计算和实验结果的平均相对误差为1.5%,最大相对误差为3.8%,满足工程误差许可,表明本文数值方法是可靠的.

图 5 数值方法验证结果 Fig. 5 Verification of numerical simulation method

为支撑数值方法中雷诺数能表征流动情况这一结论,现从具体算例上进行结果分析.以4 154模型为对象,取流动雷诺数Rev为5×106,固定来流速度U为10 m/s,通过调整流体密度ρ与流体动力黏度μ使Rev保持不变.各种参数配合下4 154模型受到的阻力情况见表 2,可见当雷诺数Rev一定时,模型的阻力系数可视为常数.因此不论从控制方程还是计算方法来看,雷诺数均能表征流动情况.

表 2 Rev=5×106时的阻力系数 Tab. 2 Drag coefficients at Rev=5×106
2 飞艇外形参数化方法

PARSEC方法通过11个参数控制翼型形状,每个参数均有清晰的物理含义.鉴于飞艇艇身的旋成体特性,只需选取上半部分的参数,于是可通过8个参数描述飞艇外形,如图 6所示.

图 6 基于“PARSEC”的8参数飞艇艇身外形 Fig. 6 Airship hull geometry defined by eight parameters based on PARSEC

图中:rh为头部半径;xd为最大半径位置;rd为最大半径;y″d为最大半径处的曲率;αt为尾部偏移角;βt为尾部张开角;Δyt为尾部厚度;yt为尾部高度.

飞艇外形可由6阶多项式表达:

$ y(x)=\sum\limits_{n=1}^{6} a_{n} x^{n-\frac{1}{2}}, $ (8)

式中的待定系数可通过求解下式获得[21]xte为弦长:

$ \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ x_{\mathrm{te}}^{1 / 2} & x_{\mathrm{te}}^{3 / 2} & x_{\mathrm{te}}^{5 / 2} & x_{\mathrm{te}}^{7 / 2} & x_{\mathrm{te}}^{9 / 2} & x_{\mathrm{te}}^{11 / 2} \\ x_{\mathrm{d}}^{1 / 2} & x_{\mathrm{d}}^{3 / 2} & x_{\mathrm{d}}^{5 / 2} & x_{\mathrm{d}}^{7 / 2} & x_{\mathrm{d}}^{9 / 2} & x_{\mathrm{d}}^{11 / 2} \\ 1 / 2 x_{\mathrm{te}}^{-1 / 2} & 3 / 2 x_{\mathrm{d}}^{1 / 2} & 5 / 2 x_{\mathrm{te}}^{3 / 2} & 7 / 2 x_{\mathrm{d}}^{5 / 2} & 9 / 2 x_{\mathrm{te}}^{7 / 2} & 11 / 2 x_{\mathrm{te}}^{9 / 2} \\ 1 / 2 x_{\mathrm{d}}^{-1 / 2} & 3 / 2 x_{\mathrm{d}}^{1 / 2} & 5 / 2 x_{\mathrm{d}}^{3 / 2} & 7 / 2 x_{\mathrm{d}}^{5 / 2} & 9 / 2 x_{\mathrm{d}}^{7 / 2} & 11 / 2 x_{\mathrm{d}}^{9 / 2} \\ -1 / 4 x_{\mathrm{d}}^{-3 / 2} & 3 / 4 x_{\mathrm{d}}^{-1 / 2} & 15 / 4 x_{\mathrm{d}}^{1 / 2} & 35 / 4 x_{\mathrm{d}}^{3 / 2} & 63 / 4 x_{\mathrm{d}}^{5 / 2} & 99 / 4 x_{\mathrm{d}}^{7 / 2} \end{array}\right]\left[\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \\ a_{5} \\ a_{6} \end{array}\right]=\left[\begin{array}{c} \sqrt{2 r_{h}} \\ y_{\mathrm{t}}+\Delta y_{\mathrm{t}} / 2 \\ r_{\mathrm{d}} \\ \tan \left(\alpha_{\mathrm{t}}-\beta_{\mathrm{t}}\right) \\ 0 \\ y_{\mathrm{d}}^{\prime \prime} \end{array}\right]. $ (9)
3 全局敏感度分析方法

敏感度体现了变量自身的改变对系统的影响程度.一般地,敏感度分析方法分为局部敏感度分析和全局敏感度分析.例如,龙卷风图, 基于分化的方法和筛查法属于前者; 基于回归的方法, 基于方差的方法, 转换不变方法和基于密度的方法属于后者[26].局部敏感度能体现输入空间一点附近的特性,全局敏感度还考虑了参数之间的相互影响,结果更合理可靠.

3.1 Sobol全局敏感度分析方法

假设物理模型能被方程f(x)描述,输入x=(x1, x2, …, xn) 是n维空间的一点且xi(i=1, 2, …,n)遵循[0, 1]上的均匀分布.全体x构成一个n维立方体:

$ C^{n}=\left\{x \mid 0 \leqslant x_{i} \leqslant 1 ; i=1,2, \cdots, n\right\}. $ (10)

输出项f(x)能被分解为

$ \begin{aligned} f(x)=& f_{0}+\sum\limits_{i=1}^{n} f_{i}\left(x_{i}\right)+\sum\limits_{1 \leqslant i<j \leqslant n} f_{i, j}\left(x_{i}, x_{j}\right)+\\ & \sum\limits_{1 \leqslant i<j<k \leqslant n} f_{i, j, k}\left(x_{i}, x_{j}, x_{k}\right)+\\ & \sum\limits_{1 \leqslant i<j<k<l \leqslant n} f_{i, j, k, l}\left(x_{i}, x_{j}, x_{k}, x_{l}\right)+\cdots+\\ & f_{1,2, \cdots, n}\left(x_{1}, x_{2}, \cdots, x_{n}\right), \end{aligned} $ (11)

式中f0为常数,且f1, 2, …, s(x1, x2, …, xs)关于它自身变量的积分为0.

$ \int\limits_{0}^{1} f_{1,2, \cdots, s}\left(x_{1}, x_{2}, \cdots, x_{s}\right) \mathrm{d} x_{k}=0,1 \leqslant k \leqslant s. $ (12)

积分式(11)有

$ f_{0}=\int\limits_{C^{n}} f(x) \mathrm{d} x. $ (13)

由于式(11)右侧至少有一个变量是不同的,故具有正交性,再由式(12)可知:

$ \begin{array}{l} \int\limits_{C^{n}} f_{i_{1}, i_{2}, \cdots, i_{s}}\left(x_{i_{1}}, x_{i_{2}}, \cdots, x_{i_{s}}\right) f_{j_{1}, j_{2}, \cdots, j_{l}}\left(x_{j_{1}}, x_{j_{2}}, \cdots, x_{j_{l}}\right) \mathrm{d} x=\\ 0,\left(i_{1}, i_{2}, \cdots, i_{s}\right) \neq\left(j_{1}, j_{2}, \cdots, j_{l}\right). \end{array} $ (14)

对式(11)积分和平方, 即

$ \begin{array}{c} \int\limits_{C^{n}} f^{2}(x) \mathrm{d} x=f_{0}^{2}+ \\ \sum\limits_{s=1}^{n} \sum\limits_{<\cdots<s}^{n} \int f_{1,2, \cdots, s}^{2} \mathrm{~d} x_{1} \mathrm{~d} x_{2} \cdots \mathrm{d} x_{s} . \end{array} $ (15)

式(15)右侧第2项被称为总方差,写为

$ D=\int\limits_{C^{n}} f^{2}(x) \mathrm{d} x-f_{0}^{2}, $ (16)

偏方差定义如下:

$ D_{1,2, \cdots, s}=\int f_{1,2, \cdots, s}^{2} \mathrm{~d} x_{1} \mathrm{~d} x_{2} \cdots \mathrm{d} x_{s}, $ (17)

偏方差和总方差的关系为

$ D=\sum\limits_{s=1}^{n} \sum\limits_{<\cdots<s}^{n} D_{1,2, \cdots, s}, $ (18)

引入比率S1, 2, …, s, 即

$ S_{1,2, \cdots, s}=\frac{D_{1,2, \cdots, s}}{D}. $ (19)

对于具有n个输入变量的离散样本,本文需要计算每个S1, 2…, s的值,但对于需要借助CFD软件才能获得的输出,这样势必造成极大的时间开销.基于此,本文将所有输入变量分为两个子集,子集Xi仅仅包含变量xi, 另一子集Xei包含除了xi的所有变量. 因此总方差可写为

$ D=D_{i}+D_{\mathrm{e} i}+D_{i, \mathrm{e} i}, $ (20)

引入新的参量STi,即

$ S_{\mathrm{T}i}=S_{i}+S_{i, \mathrm{e} i}=1-S_{\mathrm{e} i}, $ (21)

Si表征了变量xi单独对总方差的影响,STi体现了变量xi对系统的“总体”影响,即不仅仅包含变量xi自身也包含了变量xi和余下变量的任意组合对输出的影响.因此,本文称SiSTi为变量xi的一阶敏感度和全局敏感度.

3.2 离散数据的敏感度分析

事实上,本文经常面对的是离散的输入与输出数据,其中没有明确的函数关系.现在假设有一具有n个输入变量的系统,抽样产生N个样本.计算之前,本文需要准备两组输入数据x1, x2, …, xnx′1, x′2, …, x′n,写成矩阵形式如下:

$ \begin{array}{*{20}{l}} {\left[ {\begin{array}{*{20}{c}} {{x_{11}}}&{{x_{12}}}& \cdots &{{x_{1n}}}\\ {{x_{21}}}&{{x_{22}}}& \cdots &{{x_{2n}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{x_{N1}}}&{{x_{N2}}}& \cdots &{{x_{Nn}}} \end{array}} \right] \to {{\left[ {\begin{array}{*{20}{c}} {{f_1}}\\ {{f_2}}\\ \vdots \\ {{f_N}} \end{array}} \right]}^\prime }\begin{array}{*{20}{c}} |\\ |\\ |\\ | \end{array}\left[ {\begin{array}{*{20}{c}} {x_{11}^\prime }&{x_{12}^\prime }& \cdots &{x_{1n}^\prime }\\ {x_{21}^\prime }&{x_{22}^\prime }& \cdots &{x_{2n}^\prime }\\ \vdots & \vdots & \ddots & \vdots \\ {x_{N1}^\prime }&{x_{N2}^\prime }& \cdots &{x_{Nn}^\prime } \end{array}} \right] \to \left[ {\begin{array}{*{20}{c}} {f_1^\prime }\\ {f_2^\prime }\\ \vdots \\ {f_N^\prime } \end{array}} \right].}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {{x_1}} \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {{x_2}} \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {{x_n}} \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ f \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {x_1^\prime } \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {x_2^\prime } \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {x_n^\prime } \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} \uparrow \\ {{f^\prime }} \end{array}} \end{array} $ (22)

有关量可由蒙特卡洛方法近似得到[27]

$ f_{0} \approx \frac{1}{N} \sum\limits_{i=1}^{N} f_{i}, $ (23)
$ D+f_{0}{}^{2} \approx \frac{1}{N} \sum\limits_{i=1}^{N} f_{i}^{2}, $ (24)
$ D_{j} \approx D-\sum\limits_{i=1}^{N} \frac{\left(f_{i}-f_{i, j}\right)^{2}}{2 N} , $ (25)
$ D_{T_{j}} \approx \sum\limits_{i=1}^{N} \frac{\left(f_{i}-f_{i, j}^{\prime}\right)}{2 N}. $ (26)

式中:fi, j为将变量xj替换为x′ j所得到的输出;f′i, j为将变量x′ j替换为xj所得到的输出.于是本文能通过上述算法和定义获得输入变量的一阶或全局敏感度.

4 敏感度分析和艇身阻力特性研究 4.1 敏感度分析

表 3给出了基于PARSEC方法的飞艇外形参数范围.头部半径rh不超过艇身长度1;最大半径rd处于0.05~0.25之间,对应艇身长细比处于2~10之间,能涵盖大多数常规艇身;最大半径位置xd处于0.3~0.5之间,以避免艇身最大截面位置过于靠前或靠后,以此减少艇身外形出现畸变的概率;尾部张开角βt处于20°~40°之间,以保证艇身尾部光滑收缩;尾部偏移角αt,尾部厚度Δyt和尾部高度yt被设定为0,以确保表征的外形为旋成体.如图 7所示,不恰当的参数组合会导致病态外形出现,实线外形包含了负值,虚线外形有多个极值,因此这些病态外形要在抽样时被剔除.基于表 3中的参数范围,得到所有非病态外形样本构成的范围如图 8所示,可见所取的参数能很好地保证艇身廓线的多样性.

表 3 飞艇外形参数范围 Tab. 3 Ranges of airship shape parameters
图 7 不合理的参数组合造成的病态外形 Fig. 7 Ill-conditioned shapes caused by inappropriate combinations of parameters
图 8 艇身廓线构成的样本空间 Fig. 8 Sample space formed by airship outlines

飞艇飞行高度设定为20 km,当地空气密度为0.088 9 kg/m3,空气动力黏度为1.421 61×10-5 kg/ m · s,大气压强为5 529.31 Pa.一般地,平流层飞艇的绕流雷诺数在6.0×106和1.2×107之间,对一典型体积为3.0×105 m3的飞艇而言,当来流速度为20 m/s时,其雷诺数为8.37×106.故本文取雷诺数为8.0×106进行后续研究,进行数值分析时调整流体密度使所有工况的绕流雷诺数一致.

图 9显示了每个外形参数对各阻力系数的总敏感度.可以发现,参数rhrd对总阻力系数的敏感度分别达到68.8%和66.0%,除了xd以外的其余参数对总阻力系数几乎没有影响,且xd的总敏感度仅为2.1%,故剩余参数可设为确定值.观察由参数rhrd组成的设计空间,该空间可近似看作被3条线段和1条三次曲线包围形成,如图 10所示.该空间的数学表述为

$ \text { s. t. }\left\{\begin{array}{l} r_{\mathrm{h}} \leqslant 3.720\ 7 r_{\mathrm{d}}^{3}+6.702\ 7 r_{\mathrm{d}}^{2}+0.420\ 9 r_{\mathrm{d}}-0.038, \\ 0 \leqslant r_{\mathrm{h}} \leqslant 0.50, \\ 0.05 \leqslant r_{\mathrm{d}} \leqslant 0.25. \end{array}\right. $ (27)
图 9 不同参数的总敏感度 Fig. 9 Global sensitivity of different parameters
图 10 病态和设计空间 Fig. 10 Ill-conditioned space and design space
4.2 阻力特性分析

分别设置参数xdy″dβt的值为0.35,-0.5和30°.然后将设计空间离散,由同样的计算条件得到参数rhrd与各阻力系数的关系,如图 11所示,随着设计空间向右上方推进压差阻力系数逐渐升高,而黏性阻力系数与此趋势恰好相反,二者的综合效果使总阻力系数在设计空间上存在极小区域.由此可得到参数rhrd的最佳组合,另外由图 12可知,在其他参数确定的情况下随着xd的增加,总阻力系数先减小再增大,当xd=0.435时,总阻力系数取最小值.

图 11 rhrd形成的关于各个阻力系数的云图 Fig. 11 Relationship between drag coefficient, rh, and rd
图 12 Cdvxd的关系 Fig. 12 Relationship between Cdv and xd

综上所述,可获得使艇身具有最小阻力系数的参数组合,该艇身外形的头部半径rh为0.012 2,最大半径rd为0.95,最大半径位置xd为0.435.上述所得艇身长细比为5.263,与文献[28]中的结论吻合.在实际设计阶段,头部半径rh的范围可取为[0, 0.06],最大半径rd的范围可取为[0.08, 0.12],该范围内的阻力系数大小浮动程度较小,均可较好的降低艇身阻力系数.

5 结论

1) 对平流层飞艇而言,动力消耗近似正比于阻力系数,艇身阻力占到了全艇阻力的约2/3,在初期设计阶段,快速找到最佳艇身外形具有十分重要的意义.为降低设计维度,有必要对外形参数进行全局敏感度分析,进一步所得设计空间可为类似设计提供参考.

2) 对流线外形的旋成体,宜采取2维轴对称模型和一方程Spalart-Allmaras湍流模型进行流场求解,所得阻力系数与实验值吻合度较高,具有很高的精度.

3) 在8.0×106这一典型雷诺数下,平流层飞艇阻力对外形参数的敏感度由高到低依次为:rhrdxd,其他参数的敏感度相较于这3个量可忽略不计,因此评估飞艇气动特性时应同时考虑以上3个参数,而不只考虑长细比这一参量.

4) 从减阻的角度出发,在实际设计中头部半径rh的范围可取为0~0.06,最大半径rd的范围可取为0.08~0.12.

参考文献
[1]
赵达, 刘东旭, 孙康文, 等. 平流层飞艇研制现状、技术难点及发展趋势[J]. 航空学报, 2016, 37(1): 45.
ZHAO Da, LIU Dongxu, SUN Kangwen, et al. Research status, technical difficulties and development trend of stratospheric airship[J]. Acta Aerinautica et Astronautical Sinica, 2016, 37(1): 45. DOI:10.7527/S1000-6893.2015.0332
[2]
ANDROULAKAKIS S P, JUDY R A. Status and plans of high altitude airship (HAATM) program[C]// AIAA Lighter-Than-Air Systems Technology (LTA) Conference. Daytona Beach: AIAA, 2013. DOI: 10.2514/6.2013-1362
[3]
张永栋, 翟嘉琪, 孟小君, 等. 基于行为逻辑的平流层飞艇试验自动测试方法[J]. 航空学报, 2018, 39(9): 322185.
ZHANG Yongdong, ZHAI Jiaqi, MENG Xiaojun, et al. Approach for automatic testing of stratospheric airship test based on behavior logic[J]. Acta Aerinautica et Astronautical Sinica, 2018, 39(9): 322185. DOI:10.7527/S1000-6893.2018.22185
[4]
李天娥, 孙晓颖, 武岳, 等. 平流层飞艇气动阻力的参数分析[J]. 工程力学, 2019, 36(1): 248.
LI Tiane, SUN Xiaoying, WU Yue, et al. Parameter analysis of aerodynamic drag force in stratospheric airship[J]. Engineering Mechanics, 2019, 36(1): 248. DOI:10.6052/j.issn.1000-4750.2017.11.0814
[5]
LI Yuwen, NABON M, SHARF I. Airship dynamics modeling: A literature review[J]. Progress in Aerospace Sciences, 2011, 47(3): 217. DOI:10.1016/j.paerosci.2010.10.001
[6]
CARRION M, BIAVA M, STEIJL R, et al. CFD studies of Hybrid air vehicles[C]//Proceedings of the 54th AIAA Aerospace Sciences Meeting. San Diego, CA: AIAA, 2016. DOI: 10.2514/6.2016-0059
[7]
冯凯, 赵达, 孟小君, 等. 带辅助安定面平流层飞艇的航向静稳定性分析[J]. 宇航学报, 2018, 39(7): 715.
FENG Kai, ZHAO Da, MENG Xiaojun, et al. Directional static stability analysis of an airship with auxiliary stabilizer[J]. Journal of Astronautics, 2018, 39(7): 715. DOI:10.3873/j.issn.1000-1328.2018.07.002
[8]
LUTZ T, WAGNER S. Drag reduction and shape optimization of airship bodies[J]. Journal of Aircraft, 1998, 35(3): 345. DOI:10.2514/2.2313
[9]
WANG Xiaoliang, SHAN Xuexiong. Shape optimization of stratosphere airship[J]. Journal of Aircraft, 2006, 43(1): 283. DOI:10.2514/1.18295
[10]
CERUTI A, GAMBACORTA D, MARZOCCA P. Unconventional hybrid airships design optimization accounting for added masses[J]. Aerospace Science and Technology, 2018, 72: 164. DOI:10.1016/j.ast.2017.10.042
[11]
KANIKDALE T S, MARATHE A G, PANT R S. Multi-disciplinary optimization of airship envelope shape[C]// Proceedings of the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Albany: AIAA, 2004. DOI: 10.2514/6.2004-4411
[12]
WANG Quanbao, CHEN Jian, FU Gongyi, et al. An approach for shape optimization of stratosphere airships based on multidisciplinary design optimization[J]. Journal of Zhejiang University Science A, 2009, 10(11): 1609. DOI:10.1631/jzus.A0820814
[13]
宋保维, 李福新. 回转体最小阻力外形优化设计[J]. 水动力学研究与进展, 1994, 9(5): 523.
SONG Baowei, LI Fuxin. Optimum shaping design of axisymmetric bodies for minimum drag[J]. Journal of Hydrodynamics, Series A, 1994, 9(5): 523. DOI:10.16076/j.cnki.cjhd.1994.05.003
[14]
ALAM M I, PANT R S. Multidisciplinary approach for solar area optimization of high altitude airships[J]. Energy Conversion and Management, 2018, 164: 301. DOI:10.1016/j.enconman.2018.03.009
[15]
MUELLER J B, PALUSZEK M A, ZHAO Yiyuan. Development of an aerodynamic model and control law design for a high altitude airship[C]//Proceedings of the 3rd AIAA "Unmanned Unlimited" Technical Conference, Workshop and Exhibit. Chicago: AIAA, 2004. DOI: 10.2514/6.2004-6479
[16]
ZHONG Xiaoping, DING Jifeng, LI Weiji, et al. Robust airfoil optimization with multi-objective estimation of distribution algorithm[J]. Chinese Journal of Aeronautics, 2008, 21(4): 289. DOI:10.1016/s1000-9361(08)60038-2
[17]
YANG Guangda, RONCH A D, DROFELNIK J, et al. Sensitivity assessment of optimal solution in aerodynamic design optimization using SU2[J]. Aerospace Science and Technology, 2018, 81: 362. DOI:10.1016/j.ast.2018.08.012
[18]
PÉREZ-ARRIBAS F, PÉREZ-FERNÁNDEZ R. A B-spline design model for propeller blades[J]. Advances in Engineering Software, 2018, 118: 35. DOI:10.1016/j.advancesoft.2018.01.005
[19]
MARTIN M J, ANDRES E, LOZANO C, et al. Volumetric b-spline shape parametrization for aerodynamic shape design[J]. Aerospace Science and Technology, 2014, 37: 26. DOI:10.1016/j.ast.2014.05.003
[20]
MASDARI M, TAHANI M, NADERI M H, et al. Optimization of airfoil based Savonius wind turbine using coupled discrete vortex method and salp swarm algorithm[J]. Journal of Cleaner Production, 2019, 222: 47. DOI:10.1016/j.jclepro.2019.02.237
[21]
CHEN Jian, PAN Xiong, WANG Canxing, et al. Airfoil parameterization evaluation based on a modified PARASEC method for a H-Darrious rotor[J]. Energy, 2019, 187: 115910. DOI:10.1016/j.enegy.2019.115910
[22]
SRIPAWADKUL V, PADULO M, GUENOV M. A comparison of airfoil shape parameterization techniques for preliminary design optimization[C]//Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference. Fort Worth: AIAA, 2010. DOI: 10.2514/6.2010-9050
[23]
谭惠丰, 康敬天, 卫剑征, 等. 三角形微沟槽飞艇蒙皮表面的流场分析[J]. 哈尔滨工业大学学报, 2014, 46(7): 32.
TAN Huifeng, KANG Jingtian, WEI Jianzheng, et al. Flow field analysis of Micro-V shape riblets airship surface[J]. Journal of Harbin Institute of Technology, 2014, 46(7): 32. DOI:10.11918/j.issn.0367-6234.2014.07.006
[24]
史亚云, 白俊强, 华俊, 等. 基于放大因子与Spalart-Allmaras湍流模型的转捩预测[J]. 航空动力学报, 2015, 30(7): 1670.
SHI Yayun, BAI Junqiang, HUA Jun, et al. Transition prediction based on amplification factor and Spalart-Allmaras turbulence model[J]. Journal of Aerospace Power, 2015, 30(7): 1670. DOI:10.13224/j.cnki.jasp.2015.07.019
[25]
MORTON G. Drag experiments on a systematic series of streamlined bodies of revolution for application to the design of high-speed submarines: C-297[R]. Washington, DC: Navy Department, 1950
[26]
BORGONOVO E, PLISCHKE E. Sensitivity analysis: A review of recent advances[J]. European Journal of Operational Research, 2016, 248(3): 869. DOI:10.1016/j.ejor.2015.06.032
[27]
SOBOL I M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates mathematics and computers in simulations[J]. Mathematics and Computers in Simulation, 2001, 55(1/2/3): 271. DOI:10.1016/S0378-4754(00)00270-6
[28]
KHOURY G A, GILLETT J D. Airship technology[M]. Cambridge: Cambridge University Press, 1999.