﻿ 基于模式函数和变分法的螺旋桨最佳环量计算方法
 舰船科学技术  2019, Vol. 41 Issue (3): 47-50 PDF

1. 中国船舶科学研究中心，江苏 无锡 214082;
2. 中船重工（上海）节能技术发展有限公司，上海 200011

Propeller optimal circulation calculation based on mode function and variational method
HAN Yong-bo1,2, ZHOU Wei-xin1, DONG Zheng-qing1,2, FENG Jun1, DONG Shi-tang1
1. China Ship Scientific Research Center, Wuxi 214082, China;
2. CSIC Shanghai Marine Energy Saving Technology Development Co., Ltd., Shanghai 200011, China
Abstract: In order to improve the numerical stability of propeller optimal circulation method, an improved calculation method based on the sine series mode function and variational method is present in this paper. Be different from the traditional method that solves the circulation at discrete points directly, the radial circulation distribution was expressed by the sine series at first. Then, variational method is adopted to establish a system of linear equations about the coefficients of this sine series to get the optimal Circulation. The program for solving the optimal circulation is compiled by using this method. The calculated results show that this improved method is feasible and correct, and the numerical results are stable.
Key words: mode function     optimal circulation     variational method
0 引　言

1 基本理论 1.1 正弦级数模式函数表达螺旋桨径向环量分布

 \begin{aligned} & \left\{ \begin{array}{*{20}{c}} {G(\varphi ) = \left( {1 - \displaystyle\frac{\varphi }{{\text{π}} }} \right){G_0} + \displaystyle\sum\limits_{m = 1}^{M - 1} {{G_m}\sin (m\varphi )} + \frac{\varphi }{{\text{π}}}{G_M}}{\text{，}}\\ {\bar r = \displaystyle\frac{1}{2}(1 + {{\bar r}_H}) - \displaystyle\frac{1}{2}(1 - {{\bar r}_H})\cos \varphi{\text{，}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}\right.\\ & \quad\quad({{\bar r}_H} \text{≤} \bar r \text{≤} 1.0,0 \text{≤} \varphi \text{≤} \text{π} ){\text{。}} \end{aligned} (1)

1.2 螺旋桨诱导速度的计算

 $\frac{{{u_{a,t}}(\varphi )}}{{{v_s}}} = \frac{1}{{1 - {{\bar r}_H}}}\int_0^{\text{π}} {\frac{{{\rm d}G}}{{{\rm d}{\varphi _0}}}} \frac{{{i_{a,t}}(\varphi ,{\varphi _0})}}{{\cos {\varphi _0} - \cos \varphi }}{\rm d}{\varphi _0}{\text{。}}$ (2)

 $\begin{split} \frac{{{\rm d}G}}{{{\rm d}{\varphi _0}}} =& \frac{{\rm d}}{{{\rm d}{\varphi _0}}}\left[ {\left( {1 - \frac{{{\varphi _0}}}{{\text{π}} }} \right){G_0} + \sum\limits_{m = 1}^{M - 1} {{G_m}\sin (m{\varphi _0}) + \frac{{{\varphi _0}}}{\text{π} }{G_M}} } \right] =\!\!\! \\ & - \frac{{{G_0}}}{{\text{π}} } + \sum\limits_{m = 1}^{M - 1} {m{G_m}\cos (m{\varphi _0})} + \frac{{{G_M}}}{\text{π}}{\text{，}} \end{split}$ (3)
 $\begin{split} \frac{{{u_{a,t}}(\varphi )}}{{{v_s}}} =& \frac{1}{{1 - {{\bar r}_H}}}\int_0^\text{π} {\left( - \frac{{{G_0}}}{\text{π}} + \sum\limits_{m = 1}^{M - 1} {m{G_m}\cos (m{\varphi _0})} + \frac{{{G_M}}}{\text{π} }\right)}\times\\ & \frac{{{i_{a,t}}(\varphi ,{\varphi _0})}}{{\cos {\varphi _0} - \cos \varphi }}{\rm d}{\varphi _0}{\text{。}} \end{split}$ (4)

 $\frac{{{u_{a,t}}(\varphi )}}{{{v_s}}} = \sum\limits_{m = 0}^M {{G_m}} H_m^{a,t}(\varphi ){\text{。}}$ (5)

 $H_m^{a,t}(\varphi ) \!=\! \left\{ {\begin{split} & {\frac{1}{{1 \!-\! {{\bar r}_H}}}\!\int_0^{\text{π}} {\frac{{ - {i_{a,t}}(\varphi ,{\varphi _0})}}{{(\cos {\varphi _0} - \cos \varphi ){\text{π}} }}{\rm d}{\varphi _0}} ,\; m \!=\! 0}{\text{；}}\\ & \frac{1}{{1 \!-\! {{\bar r}_H}}}\int_0^{\text{π}} {\frac{{m{i_{a,t}}(\varphi ,{\varphi _0})\cos (m{\varphi _0})}}{{\cos {\varphi _0} - \cos \varphi }}} {\rm d}{\varphi _0},\\ & \; m \!=\! 1,2, \cdots ,M \!-\! 1{\text{；}}\\ & {\frac{1}{{1 - {{\bar r}_H}}}\int_0^{\text{π}} {\frac{{{i_{a,t}}(\varphi ,{\varphi _0})}}{{(\cos {\varphi _0} - \cos \varphi ){\text{π}} }}{\rm d}{\varphi _0}} ,\; m \!=\! M \!+\! 1}{\text{。}}\!\!\!\!\! \end{split}} \right.$ (6)
1.3 功率系数和推力系数的计算

 $G(\varphi ) = \sum\limits_{m = 0}^M {{G_m}{P_m}(\varphi )}{\text{，}}$ (7)

 $\begin{split} {C_P} \!=\!& \frac{{4Z}}{{{\lambda _s}}}\int_{{{\bar r}_H}}^1 {G(\bar r)\bar r\left[ {1 - {w_x}(\bar r) \!+\! \displaystyle\frac{{{u_a}(\bar r)}}{{{v_s}}}} \right]} {\rm d}\bar r=\\ & \frac{{2Z\left( {1 \!-\! {{\bar r}_H}} \right)}}{{{\lambda _s}}}\int_0^{\text{π}} {G(\varphi )\bar r(\varphi )\left[ {1 - {w_x}(\varphi ) + \frac{{{u_a}(\varphi )}}{{{v_s}}}} \right]} \sin (\varphi ){\rm d}\varphi =\\ & \frac{{2Z\left( {1 - {{\bar r}_H}} \right)}}{{{\lambda _s}}}\int_0^{\text{π}} \left( {\sum\limits_{m = 0}^M {{G_m}{P_m}(\varphi )} } \right)\times\\ & \left[ {1 - {w_x}(\varphi ) + \sum\limits_{m = 0}^M {{G_m}} H_m^a(\varphi )} \right] \bar r(\varphi )\sin (\varphi ){\rm d}\varphi {\text{。}}\quad\quad\quad\;\;(8) \end{split}$

 ${A_m} = \frac{{2Z\left( {1 - {{\bar r}_H}} \right)}}{{{\lambda _s}}}\int_0^{\text{π}} {\bar r(\varphi ){P_m}(\varphi )\left[ {1 - {w_x}(\varphi )} \right]\sin (\varphi ){\rm d}\varphi } {\text{，}}$ (9)

 $\left( {\sum\limits_{m = 0}^M {{G_m}{P_m}(\varphi )} } \right)\left[ {\sum\limits_{m = 0}^M {{G_m}} H_m^a(\varphi )} \right] = \sum\limits_{m = 0}^M {\sum\limits_{n = 0}^M {{P_m}(\varphi )} } H_n^a(\varphi ){G_m}{G_n}{\text{，}}$ (10)

 ${B_{mn}} = \frac{{2Z\left( {1 - {{\bar r}_H}} \right)}}{{{\lambda _s}}}\int_0^{\text{π}} {\bar r(\varphi ){P_m}(\varphi )H_n^a(\varphi )\sin (\varphi ){\rm d}\varphi }{\text{，}}$ (11)

 ${C_P} = \sum\limits_{m = 0}^M {{A_m}{G_m} + } \sum\limits_{m = 0}^M {\sum\limits_{n = 0}^M {{B_{mn}}} } {G_m}{G_n}{\text{。}}$ (12)

 $\begin{split} {C_T} =& 4Z\int_{{{\bar r}_H}}^{1.0} {G(\bar r)\left[ {\frac{{\bar r}}{{{\lambda _s}}} + {w_t}(\bar r) - \frac{{{u_t}(\bar r)}}{{{v_s}}}} \right]} {\rm d}\bar r=\\ & 2Z(1 - {{\bar r}_H})\int_0^{\text{π}} {G(\varphi )\left[ {\frac{{\bar r(\varphi )}}{{{\lambda _s}}} + {w_t}(\varphi ) - \frac{{{u_t}(\varphi )}}{{{v_s}}}} \right]} \sin \varphi {\rm d}\varphi =\\ & \sum\limits_{m = 0}^M {{C_m}} {G_m} + \sum\limits_{m = 0}^M {\sum\limits_{n = 0}^M {{D_{mn}}} } {G_m}{G_n}{\text{。}}\;\;\;\quad\quad\quad\quad\quad\quad(13) \end{split}$

 ${C_m} = 2Z(1 - {\bar r_H})\int_0^{\text{π}} {{P_m}(\varphi )\left[ {\frac{{\bar r(\varphi )}}{{{\lambda _s}}} + {w_t}(\varphi )} \right]} \sin \varphi {\rm d}\varphi {\text{，}}$ (14)
 ${D_{mn}} = 2Z(1 - {\bar r_H})\int_0^{\text{π}} {{P_m}(\varphi )} H_n^t(\varphi )\sin \varphi {\rm d}\varphi {\text{。}}$ (15)
1.4 变分法求解最佳环量

 $\left\{ {\begin{split} & {\frac{{\partial H}}{{\partial {G_q}}} = \lambda \frac{{\partial {C_T}}}{{\partial {G_q}}} + \frac{{\partial {C_Q}}}{{\partial {G_q}}} = 0}\\ & {\frac{{\partial H}}{{\partial \lambda }} = {C_T} - {C_{Trequired}} = 0} \end{split}} \right.\quad(q = 0,1,2, \cdots ,M){\text{，}}$ (16)

 $\begin{split} \frac{{\partial {C_T}}}{{\partial {G_q}}} =& \frac{\partial }{{\partial {G_q}}}\left(\sum\limits_{m = 0}^M {{C_m}} {G_m} + \sum\limits_{m = 0}^M {\sum\limits_{n = 0}^M {{D_{mn}}} } {G_m}{G_n}\right)=\\ & {C_q} + \sum\limits_{m = 0}^M {({D_{qm}} + } {D_{mq}}){G_m}{\text{，}} \end{split}$ (17)
 $\begin{split} \frac{{\partial {C_Q}}}{{\partial {G_q}}} =& \frac{\partial }{{\partial {G_q}}}\left(\sum\limits_{m = 0}^M {A{}_m} {G_m} + \sum\limits_{m = 0}^M {\sum\limits_{n = 0}^M {{B_{mn}}} } {G_m}{G_n}\right) =\\ & {A_q} + \sum\limits_{m = 0}^M {({B_{qm}} + } B{}_{mq}){G_m}{\text{。}} \end{split}$ (18)

 $\begin{split} \frac{{\partial H}}{{\partial {G_q}}} =& \lambda \left( {{C_q} + \sum\limits_{m = 0}^M {({D_{qm}} + } {D_{mq}}){G_m}} \right) +\\ & \left( {{A_q} + \sum\limits_{m = 0}^M {({B_{qm}} + } B{}_{mq}){G_m}} \right) = 0{\text{，}} \end{split}$ (19)

 $\begin{split} & \sum\limits_{m = 0}^M {\left[ {\lambda ({D_{qm}} + {D_{mq}}) + ({B_{qm}} + B{}_{mq})} \right]} {G_m}= \\ & - (\lambda {C_q} + {A_q}), \;\; (q = 0,1,2, \cdots ,M){\text{。}} \end{split}$ (20)

2 算例验证

5叶螺旋桨，毂径比0.2，不考虑桨毂和粘性影响，不考虑尾涡变形和收缩，敞水工况，Ct=0.512，计算J=0.1，0.2，0.4，0.6，0.8，1.0，1.2和1.4时的最佳环量分布。计算对比结果见图1（a）

 图 1 与Coney计算的最佳环量分布结果比较 Fig. 1 Comparisons of the optimum circulation distribution calculated in this paper and by coney

5叶螺旋桨，毂径比0.2，不考虑桨毂和粘性影响，不考虑尾涡变形和收缩，Ct=0.307，J=1.377，计算周向平均的轴向伴流下螺旋桨最佳环量分布。计算对比结果见图1（b）。关于本案例中具体的伴流分布可参见文献[9]。

 图 2 不同网格数量和网格划分方式下最佳环量计算结果比较 Fig. 2 Comparisons of optimal circulation calculation results under different number of mesh grids and mesh methods

3 结　语

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