﻿ 波浪前缘静子叶片对高速轴流风扇单音噪声的影响
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Tonal noise reduction of a high-speed single axial fan with wavy leading-edge stator
TONG Hang, LI Lin, MAO Luqin, XIANG Kangshen, QIAO Weiyang
School of Power and Energy, Northwestern Polytechnical University, Xi'an 710129, China
Keywords: wavy leading-edge blade    tonal noise    acoustic analogy theory    fan    FW-H equations    hybrid method    noise reduction

1 计算方法

 图 1 单音噪声混合模型计算流程 Fig. 1 Computational steps of tonal noise hybrid method
1.1 Goldstein管道声学基本方程

 $p(\mathit{\boldsymbol{x}}, t) = \int_{ - T}^T {\int_S {\frac{{\partial G}}{{\partial {\mathit{\boldsymbol{y}}_i}}}} } {f_i}{\rm{d}}S(\mathit{\boldsymbol{y}}){\rm{d}}\tau$ （1）

 $\begin{array}{l} G(\mathit{\boldsymbol{y}}, \tau /\mathit{\boldsymbol{x}}, t) = \frac{{\rm{i}}}{{4\pi }}\sum\limits_{m, n} {\frac{{{\varPsi _m}({\kappa _{mn}}r)\varPsi _m^*({\kappa _{mn}}{r^\prime })}}{{{\varGamma _{mn}}}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}[ {\rm{i}}m (\varphi - \bar \varphi )] \cdot \int_{ - \infty }^\infty {\left\{ {\frac{{{\rm{exp}}[{\rm{i}}\omega (\tau - t)]}}{{{k_{mn}}}}} \right.} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\rm{exp}}\left[ {{\rm{i}}\frac{{Ma\omega }}{{{\beta ^2}{c_0}}}({y_1} - {x_1})} \right]}}{{{k_{mn}}}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\frac{{{\rm{exp}}\left[ {{\rm{i}}\frac{{{k_{mn}}}}{{{\beta ^2}}}|{y_1} - {x_1}|} \right]}}{{{k_{mn}}}}} \right\}{\rm{d}}w \end{array}$ （2）

 ${\varPsi _m}({\kappa _{nn}}r) = a{J_m}({\kappa _{mn}}r) + b{Y_m}({\kappa _{mn}}r)$ （3）

 $\begin{array}{l} p(\mathit{\boldsymbol{x}}, t) = \int_{ - T}^T {\int_{{S_F}} \mathit{\boldsymbol{n}} } (\mathit{\boldsymbol{y}}) \cdot \mathit{\boldsymbol{\nabla}} G(\mathit{\boldsymbol{x}}, \mathit{\boldsymbol{y}}, t - \tau ) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} P(\mathit{\boldsymbol{y}}, \tau ){\rm{d}}S(\mathit{\boldsymbol{y}}){\rm{d}}\tau \end{array}$ （4）

 $\begin{array}{l} p(\mathit{\boldsymbol{x}}, \omega ) = \sum\limits_m {\sum\limits_n {{A_{mn}}} } (\omega ){\varPsi _{mn}}({\kappa _{mn}}r) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}( {\rm{i}}m \varphi - {\rm{i}}{\gamma _{mn}}{x_1}) \end{array}$ （5）

 $\begin{array}{l} {A_{mn}}(\omega ) = \frac{1}{{2{\rm{i}}{\varGamma _{mn}}{\kappa _{mn}}}}\int_{{S_F}} {\{ {\varPsi _m}(} {\kappa _{mn}}{r^\prime }) \cdot \mathit{\boldsymbol{n}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\nabla}} [{\rm{exp}}( - {\rm{i}}m{\varphi ^\prime } + {\rm{i}}{\gamma _{mn}}{y_1})] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} P(\mathit{\boldsymbol{y}}, \omega - m\varOmega )\} {\rm{d}}S(\mathit{\boldsymbol{y}}) \end{array}$ （6）

 $\begin{array}{l} {A_{mn}}(\omega ) = \frac{1}{{2{\rm{i}}{\varGamma _{mn}}{\kappa _{mn}}}}\int_{{S_F}} {{\varPsi _m}(} {\kappa _{mn}}{r^\prime }) \cdot \mathit{\boldsymbol{n}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\nabla}} [{\rm{exp}}( - {\rm{i}}m{\varphi ^\prime } + {\rm{i}}{\gamma _{mn}}{y_1})] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\sum\limits_{s = 0}^{V - 1} {{P_s}} (\mathit{\boldsymbol{y}}, \omega - m\varOmega ){\rm{exp}}({\rm{i}}2\pi ms/V)} \right]{\rm{d}}S(\mathit{\boldsymbol{y}}) \end{array}$ （7）

 $\begin{array}{*{20}{l}} {{W_{mn}}(\omega ) = \frac{{\pi (r_{\rm{D}}^2 - r_{\rm{H}}^2)}}{{{\rho _0}U}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{ \mp M{a^2}{{(1 - M{a^2})}^2}(\omega /U){k_{mn}}(\omega )}}{{{{[\omega /{c_0} \pm Ma{k_{mn}}(\omega )]}^2}}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [{A_{mn}}(\omega ) \cdot {{({A_{mn}}(\omega ))}^*}]} \end{array}$ （8）

 ${W_{\rm{t}}}(\omega ) = \sum\limits_{m = - \infty }^\infty {\sum\limits_{n = 0}^\infty {{W_{mn}}} } (\omega )$ （9）

 $\omega = jB{\varOmega _{\rm{R}}}\quad j = 1, 2, 3, \cdots$ （10）

1.2 声源流场数值计算

2 流场/声场混合模型运用说明

2.1 高速轴流风扇预测结果

Tsuchiya等[27]利用混合模型与三维线性理论对叶片通过频率处单音噪声进行了预测。同时Tsuchiya将预测结果与试验对比后发现混合模型的精度明显更高。

 Parameter Value Number of rotor blade 18 Number of stator vane 45 Hub to tip ratio 0.55 Axial flow Mach number 0.51 Rotor tip Mach number 1.13 Stator chord/tip radius 0.192 5 Operating condition Design point
 图 3 高速风扇噪声谱[27] Fig. 3 Noise spectrum of high speed fan[27]

 图 4 声功率级对比结果[27] Fig. 4 Comparison results of PWL[27]
2.2 低速轴流风扇预测结果

Tong等[23]通过本文所用URANS/DBAA混合模型准确地预测了西北工业大学单级轴流风扇气动噪声试验台(NPU-fan)的转/静干涉单音噪声，并与试验结果做了对比。西北工业大学单级轴流风扇气动噪声试验台设计参数如表 2所示。

 Parameter Value Rotor blade number 19 Stator blade number 18 Shroud diameter/m 0.5 Hub-shroud rate 0.57 Rate of flow/(kg·s-1) 6.1 Design speed/(r·min-1) 3 000 Total pressure ratio 1.02 Airfoil shape NACA-65

 图 5 混合模型计算结果[23] Fig. 5 Hybrid method calculation results[23]

3 计算对象设置

 Parameter Value Rotor blade number 24 Stator blade number 36 Inlet shroud diameter/m 0.764 Outlet shroud diameter/m 0.764 Inlet hub-shroud rate 0.306 Outlet hub-shroud rate 0.581 Inlet axial Mach number 0.32 Outlet axial Mach number 0.41 60.5% design speed/(r·min-1) 7 000 Total pressure ratio 1.24
3.1 波浪前缘叶片构型方法

 $c{|_r} = \bar c{|_r} + \frac{A}{2}{\rm{sin}}\left( {\frac{{2\pi }}{W}r} \right)$ （11）

 $A{|_r} = \varepsilon \cdot \bar c{|_r}$ （12）

3.2 波浪前缘静子叶片结构

3.3 计算设置

NPU-HiFan风扇的转子和静子叶片数分别为24和36，可以用包含2个转子叶片和3个静子叶片的计算域来数值模拟，并准确地捕捉风扇级内部的流场信息，计算域示意图如图 8所示。

 图 8 计算域示意图 Fig. 8 Sketch of computational domain

 Stator kind Grids number Base blade 9 096 960 A9W15 blade 10 452 224 A9W7.5 blade 12 080 640 A13.5W15 blade 10 982 656

4 计算结果 4.1 管道声模态

 图 9 最大截通模态数 Fig. 9 Max cut-on mode number

 Frequency m=hB±kV n 1 BPF -12 0, 1 2 BPF -24 0, 1, 2, 3 12 0-7 3 BPF -36 0-5 0 0-13 36 0-5
4.2 气动性能

 Stator kind Total pressure ratio Isentropic efficiency Base blade 1.222 57 0.868 34 A9W15 blade 1.221 57 0.865 715 A9W7.5 blade 1.222 43 0.866 763 A13.5W15 blade 1.222 38 0.867 384
4.3 流场结果

 图 10 压力面极限流线 Fig. 10 Pressure surface limit streamline
 图 11 吸力面极限流线 Fig. 11 Suction surface limit streamline

 图 12 波浪前缘位置涡结构示意图[28] Fig. 12 Sketch of vortex structure at wavyleading-edge[28]

 图 13 叶片压力面流向涡量分布 Fig. 13 Streamwise vorticity distribution onpressure surface of blade
4.4 声学结果

 Stator kind PWL/dB 1 BPF 2BPF 3BPF Base blade 142.182 2 141.474 2 136.741 1 A9W15 blade 141.220 2 138.079 1 132.322 2 A9W7.5 blade 141.011 8 138.587 4 133.422 9 A13.5W15 blade 140.675 2 136.553 6 132.001 1
 图 14 单音噪声声功率对比 Fig. 14 Comparison of PWL of tonal noise

1) 对比基准叶型和A9W15叶型，其中1BPF单音噪声声功率降低0.962 dB，2BPF单音噪声声功率降低3.395 1 dB，3BPF单音噪声声功率降低4.418 9 dB。

2) 对比A9W15叶型和A9W7.5叶型，其中1BPF单音噪声声功率降低0.208 4 dB，2BPF和3BPF单音噪声声功率反而有所提升，可能的原因在4.5节中进行分析。

3) 对比A9W15叶型和A13.5W15叶型，其中1BPF单音噪声声功率降低0.545 dB，2BPF单音噪声声功率降低1.525 5 dB，3BPF单音噪声声功率降低0.321 1 dB。

4.5 降噪机理

 $\begin{array}{*{20}{c}} {{P_{{\rm{ Normal }}}} = \frac{1}{{{P_0}}} \cdot {P_s}(\mathit{\boldsymbol{y}}, \omega ) \cdot {\varPsi _{mn}}({\kappa _{mn}}{r^\prime }) \cdot \mathit{\boldsymbol{n}} \cdot }\\ {\mathit{\boldsymbol{\nabla}} ({\rm{exp}}( - {\rm{i}}m{\varphi ^\prime } + {\rm{i}}{\gamma _{mn}}{y_1}))} \end{array}$ （13）

 图 15 1BPF静子叶片表面声源分布(abs(PNormal)) Fig. 15 Noise source distribution on stator blade surface at 1BPF(abs(PNormal))
 图 16 1BPF静子叶片表面声源分布(Re(PNormal)) Fig. 16 Noise source distribution on stator blade surface at 1BPF(Re(PNormal))
 图 17 2BPF静子叶片表面声源分布(abs(PNormal)) Fig. 17 Noise source distribution on stator blade surface at 2BPF(abs(PNormal))
 图 18 2BPF静子叶片表面声源分布(Re(PNormal)) Fig. 18 Noise source distribution on stator blade surface at 2BPF(Re(PNormal))
 图 19 3BPF静子叶片表面声源分布(abs(PNormal)) Fig. 19 Noise source distribution on stator blade surface at 3BPF(abs(PNormal))
 图 20 3BPF静子叶片表面声源分布(Re(PNormal)) Fig. 20 Noise source distribution on stator blade surface at 3BPF(Re(PNormal))

5 结论

1) 本文分别采用3种不同的波浪前缘静子叶片与基准静子叶片对比，对高速轴流风扇进行了气动和声学性能评估。结果表明，3种不同的波浪前缘静子叶片能够将前3阶BPF风扇单音噪声声功率级有效降低，此外，波浪型前缘对风扇的气动性能影响不大。

2) 除了确定了波浪前缘静子叶片的降噪效果外，还研究和分析了这种降噪的机理。波浪前缘静子叶片可以将大尺度涡旋打碎成为小尺度涡旋，并诱导出许多流向涡结构，可以显著地改变叶片前缘压力脉动的分布。

3) 与文献[23]中的低速轴流风扇相比，波浪前缘静子叶片对于高马赫数、高雷诺数工况的高速轴流风扇降噪显得较为困难，但是仍有不错的降噪效果。

4) 波浪前缘静子叶片是通过改变叶片表面压力脉动振幅和相位关系来减小噪声强度的，研究发现增加波浪前缘静子叶片幅值可以有效降低声源面积和强度，而单纯改变波长不一定对降噪有利，所以在设计波浪前缘静子叶片时应将相应频率下截通的最大径向模态数考虑在内。

http://dx.doi.org/10.7527/S1000-6893.2019.23565

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#### 文章信息

TONG Hang, LI Lin, MAO Luqin, XIANG Kangshen, QIAO Weiyang

Tonal noise reduction of a high-speed single axial fan with wavy leading-edge stator

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 123565.
http://dx.doi.org/10.7527/S1000-6893.2019.23565