﻿ 考虑几何设计参数不确定性影响的涡轮叶栅稳健性气动设计优化
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Robust aerodynamic design optimization of turbine cascades considering uncertainty of geometric design parameters
LUO Jiaqi, CHEN Zeshuai, ZENG Xian
School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

1 跨声速叶栅数值模拟

 图 1 HS1A几何示意图及计算网格 Fig. 1 Configuration and computation grid for HS1A

 参数 数值 弦长/mm 40.0 轴向弦长/mm 36.98 叶型前缘(后缘)角/(°) 50.5(59) 叶型安装角/(°) 25.1 几何进口角/(°) 46 栅距/mm 29.14

 $\xi = \frac{{{p_{0,1}} - {p_{0,2}}}}{{{p_{0,2}} - {p_2}}} = \frac{{{p_{0,1}} - {p_2}}}{{{p_{0,2}} - {p_2}}} - 1$ （1）

 网格 流向 周向 y+ ζ NS1 128 64 2.1 0.092 9 NS2/NS5 160 64 2.1 0.094 8 NS3 192 64 2.2 0.095 6 NS4 160 48 5.2 0.093 3 NS6 160 80 0.9 0.095 4 Eu4 160 32 0.073 8 Eu5 160 48 0.076 2 Eu6 160 64 0.077 1

 图 2 叶片表面等熵马赫数分布 Fig. 2 Isentropic Mach number distributions on blade surface

 $\begin{array}{*{20}{l}} {{\rm{Min }}}&{I(\mathit{\boldsymbol{w}},\mathit{\boldsymbol{b}})}\\ {{\rm{Sub}}}&{{c_i}(\mathit{\boldsymbol{w}},\mathit{\boldsymbol{b}}) \le 0,i = 1,2, \cdots ,{n_{\rm{c}}}} \end{array}$ （2）

 $\begin{array}{*{20}{l}} {{\rm{Min }}}&{{\mu _I},{\sigma _I}}\\ {{\rm{Sub}}}&{{c_i}(\mathit{\boldsymbol{w}},\mathit{\boldsymbol{b}}) \le 0,i = 1,2, \cdots ,{n_{\rm{c}}}} \end{array}$ （3）

2.2 叶栅气动优化设计

 $I = {\mu _\xi } + \lambda {\sigma _\xi }$ （4）

 ${\rm{ \mathsf{ δ} }} y(x) = \sum\limits_{j = 1}^N {{V_j}} {b_j}(x)$ （5）

 $\left\{ {\begin{array}{*{20}{l}} {{b_j}(x) = {\rm{si}}{{\rm{n}}^4}(\pi {x^{{p_j}}})}\\ {{p_j} = {\rm{ln}}(0.5)/{\rm{ln}}({x_{{\rm{c}},j}})\quad j = 1,2, \cdots ,N} \end{array}} \right.$ （6）

 $\begin{array}{l} f(\Delta V) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\frac{1}{{\sqrt {2\pi } {\sigma _V}}}{\rm{exp}}\left( { - \frac{{{{(\Delta V)}^2}}}{{2\sigma _V^2}}} \right)}&{\frac{{\Delta V}}{{{\sigma _V}}} \in [ - E,E]}\\ 0&{{\rm{ otherwise }}} \end{array}} \right. \end{array}$ （7）

 $\left\{ {\begin{array}{*{20}{l}} {{\mu _\xi } = \frac{1}{m}\sum\limits_{i = 1}^m {({\xi _0} + \Delta {\xi _i})} = {\xi _0} + \frac{1}{m}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^N {{g_j}} } \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Delta {V_{ij}} = {\xi _0} + \sum\limits_{j = 1}^N {{g_j}} \left( {\frac{1}{m}\sum\limits_{i = 1}^m \Delta {V_{ij}}} \right) = {\xi _0}}\\ {\sigma _\xi ^2 = \frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {{\xi _0} + \sum\limits_{j = 1}^N {{g_j}} \Delta {V_{ij}} - {\mu _\xi }} \right)}^2}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 1}^N {g_j^2} \left( {\frac{1}{m}\sum\limits_{i = 1}^m \Delta V_{ij}^2} \right) = \sigma _V^2\sum\limits_{j = 1}^N {g_j^2} } \end{array}} \right.$ （8）

 ${g_{I,i}} = \frac{{{\rm{ \mathsf{ δ} }} I}}{{{\rm{ \mathsf{ δ} }} {V_i}}} = \frac{{{\rm{ \mathsf{ δ} }} {\mu _\xi }}}{{{\rm{ \mathsf{ δ} }} {V_i}}} + \lambda \frac{{{\rm{ \mathsf{ δ} }} {\sigma _\xi }}}{{{\rm{ \mathsf{ δ} }} {V_i}}} = {g_i} + \lambda \frac{{\sigma _V^2}}{{{\sigma _\xi }}}\sum\limits_{j = 1}^N {{g_j}} {h_{ij}}$ （9）

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {{\mu _\xi } = {\xi _0} + \frac{1}{m}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^N {{g_j}} } \Delta {V_{ij}} + }\\ {{\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} \frac{1}{m}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^N {\sum\limits_{k = 1}^N {\frac{1}{2}} } } {h_{jk}}\Delta {V_{ij}}\Delta {V_{ik}} = }\\ {{\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} {\xi _0} + \frac{1}{2}\sum\limits_{j = 1}^N {{h_{jj}}} \left( {\frac{1}{m}\sum\limits_{i = 1}^m \Delta V_{ij}^2} \right) = {\xi _0} + \frac{1}{2}\sigma _V^2\sum\limits_{j = 1}^N {{h_{jj}}} } \end{array}\\ \sigma _\xi ^2 = \frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {\sum\limits_{j = 1}^N {{g_j}} \Delta {V_{ij}} - {\mu _\xi }} \right)}^2}} + \\ {\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} \frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {\sum\limits_{j = 1}^N {\sum\limits_{k = 1}^N {\frac{1}{2}} } {h_{jk}}\Delta {V_{ij}}\Delta {V_{ik}} - {\mu _\xi }} \right)}^2}} = \\ {\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} \sigma _V^2\sum\limits_{j = 1}^N {g_j^2} + \frac{1}{m}\sum\limits_{i = 1}^m {\left( {\sum\limits_{j = 1}^N {\sum\limits_{k = 1}^N {\frac{1}{2}} } {h_{jk}}\Delta {V_{ij}} \cdot } \right.} \\ {\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} {\left. {\Delta {V_{ik}}} \right)^2} = \sigma _V^2\sum\limits_{j = 1}^N {g_j^2} + \frac{1}{2}\sigma _V^4\sum\limits_{j = 1}^N {\sum\limits_{k = 1}^N {h_{jk}^2} } \end{array} \right.$ （10）

3 伴随灵敏度计算 3.1 伴随方法基本原理

 ${I = I(\mathit{\boldsymbol{w}},\mathit{\boldsymbol{b}}),\frac{{{\rm{ \mathsf{ δ} }} I}}{{{\rm{ \mathsf{ δ} }} {b_i}}} = \frac{{\partial I}}{{\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} + \frac{{\partial I}}{{\partial {b_i}}}}$ （11）
 ${\mathit{\boldsymbol{R}}(\mathit{\boldsymbol{w}},\mathit{\boldsymbol{b}}) = {\bf{0}},\frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{R}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} = \frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} + \frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial {b_i}}} = {\bf{0}}}$ （12）

 $\frac{{{\rm{ \mathsf{ δ} }} I}}{{{\rm{ \mathsf{ δ} }} {b_i}}} = \left( {\frac{{\partial I}}{{\partial \mathit{\boldsymbol{w}}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial \mathit{\boldsymbol{w}}}}} \right)\frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} + \left( {\frac{{\partial I}}{{\partial {b_i}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial {b_i}}}} \right)$ （13）

 $\frac{{\partial I}}{{\partial \mathit{\boldsymbol{w}}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial \mathit{\boldsymbol{w}}}} = 0$ （14）

 $\frac{{{\rm{ \mathsf{ δ} }} I}}{{{\rm{ \mathsf{ δ} }} {b_i}}} = \frac{{\partial I}}{{\partial {b_i}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial {b_i}}}$ （15）

3.2 二阶灵敏度计算

 $\begin{array}{l} \frac{{{{\rm{ \mathsf{ δ} }} ^2}I}}{{{\rm{ \mathsf{ δ} }} {b_i}{\rm{ \mathsf{ δ} }} {b_j}}} = \frac{{{\partial ^2}I}}{{\partial {b_i}\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_j}}} + \frac{{{\partial ^2}I}}{{\partial {b_j}\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} + \frac{{{\partial ^2}I}}{{\partial {b_i}\partial {b_j}}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\partial ^2}I}}{{\partial {\mathit{\boldsymbol{w}}^2}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_j}}} + \frac{{\partial I}}{{\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{{\rm{ \mathsf{ δ} }} ^2}\mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}{\rm{ \mathsf{ δ} }} {b_j}}} \end{array}$ （16）
 $\begin{array}{*{20}{c}} {\frac{{{{\rm{ \mathsf{ δ} }} ^2}\mathit{\boldsymbol{R}}}}{{{\rm{ \mathsf{ δ} }} {b_i}{\rm{ \mathsf{ δ} }} {b_j}}} = \frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {b_i}\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_j}}} + \frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {b_j}\partial w}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} + \frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {b_i}\partial {b_j}}} + }\\ {\frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {\mathit{\boldsymbol{w}}^2}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_j}}} + \frac{{\partial \mathit{\boldsymbol{R}}}}{{\partial \mathit{\boldsymbol{w}}}} \cdot \frac{{{{\rm{ \mathsf{ δ} }} ^2}\mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}{\rm{ \mathsf{ δ} }} {b_j}}} = \mathit{\boldsymbol{0}}} \end{array}$ （17）

 $\begin{array}{*{20}{l}} {{h_{ij}} = \mathit{\boldsymbol{A}}\frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} + \mathit{\boldsymbol{B}}\frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_j}}} + \mathit{\boldsymbol{C}}\frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}}} \cdot \frac{{{\rm{ \mathsf{ δ} }} \mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_j}}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{D}}\frac{{{{\rm{ \mathsf{ δ} }} ^2}\mathit{\boldsymbol{w}}}}{{{\rm{ \mathsf{ δ} }} {b_i}{\rm{ \mathsf{ δ} }} {b_j}}} + \mathit{\boldsymbol{E}}} \end{array}$ （18）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{A}} = \frac{{{\partial ^2}I}}{{\partial {b_j}\partial \mathit{\boldsymbol{w}}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {b_j}\partial \mathit{\boldsymbol{w}}}}}\\ {\mathit{\boldsymbol{B}} = \frac{{{\partial ^2}I}}{{\partial {b_i}\partial \mathit{\boldsymbol{w}}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {b_i}\partial \mathit{\boldsymbol{w}}}}}\\ {\mathit{\boldsymbol{C}} = \frac{{{\partial ^2}I}}{{\partial {\mathit{\boldsymbol{w}}^2}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {\mathit{\boldsymbol{w}}^2}}}}\\ {\mathit{\boldsymbol{D}} = \frac{{\partial I}}{{\partial \mathit{\boldsymbol{w}}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{\partial R}}{{\partial \mathit{\boldsymbol{w}}}}}\\ {\mathit{\boldsymbol{E}} = \frac{{{\partial ^2}I}}{{\partial {b_i}\partial {b_j}}} - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\frac{{{\partial ^2}\mathit{\boldsymbol{R}}}}{{\partial {b_i}\partial {b_j}}}} \end{array}} \right.$ （19）

4 优化设计

 参数 C1 C2 C3 C4 σV/mm 0.025 0.050 0.025 0.050 λ 1.0 1.0 5.0 5.0

 图 5 第1步优化设计梯度对比 Fig. 5 Comparisons of gradients in first optimization cycle

 图 6 优化设计收敛曲线 Fig. 6 Convergence history of optimization design

 图 7 总压损失系数方差收敛曲线 Fig. 7 Convergence history of standard deviation of total pressure loss coefficients

 图 8 优化对比 Fig. 8 Optimization comparisons

 ${\rm{ \mathsf{ δ} }} \xi = \left( {1 - \frac{\xi }{{{\xi _0}}}} \right) \times 100\%$ （20）

 图 9 不同叶片的总压损失系数变化量统计值 Fig. 9 Statistics of total pressure loss coefficient change for different blades

 图 10 通道等熵马赫数变化均值云图 Fig. 10 Contours of mean change of isentropic Mach number in blade passage
 图 11 通道等熵马赫数变化方差云图 Fig. 11 Contours of standard deviation of isentropic Mach number in blade passage

 ${f_{{\rm{RADO}}}} = {f_{{\rm{DADO}}}} + a({f_{{\rm{0,RADO}}}} - {f_{{\rm{DADO}}}})$ （21）
 图 12 叶面等熵马赫数变化统计均值分布 Fig. 12 Distributions of means of isentropic Mach number change on blade surface
 图 13 叶面等熵马赫数变化统计方差分布 Fig. 13 Distributions of variances of isentropic Mach number change on blade surface

5 结论

2) 随着设计参数公差的增加，总压损失系数方差也增加；随着统计均值、方差多目标优化设计中方差权重的增加，总压损失系数下降更明显，优化叶片的气动稳健性更优。

3) 无黏流动中，影响叶片气动稳健性的主要因素是激波；气动外形优化设计通过减弱激波强度降低流场对设计参数变化的敏感度，从而降低跨声速叶栅总压损失系数变化量的统计值，改善优化叶片的稳健性。

http://dx.doi.org/10.7527/S1000-6893.2020.23826

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#### 文章信息

LUO Jiaqi, CHEN Zeshuai, ZENG Xian

Robust aerodynamic design optimization of turbine cascades considering uncertainty of geometric design parameters

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 123826.
http://dx.doi.org/10.7527/S1000-6893.2020.23826