﻿ 混合式CRP面元法计算对比<sup>1)</sup>
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Wang Rui, Xiong Ying, Wang Zhanzhi. COMPARATIVE STUDY ON SURFACE PANEL METHOD FOR THE HYDRODYNAMIC ANALYSIS OF HYBRID CONTRA-ROTATING SHAFT POD PROPULSOR1)[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1425-1436. DOI: 10.6052/0459-1879-16-199.
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### 基金项目

1) 国家自然科学基金资助项目（51479207）

### 文章历史

2016-07-15 收稿
2016-08-31 录用
2016-09-03网络版发表

1 数值方法 1.1 迭代面元法

 $\phi _{i{\rm F}} = \sum\limits_{k_1 = 1}^{Z_1 } \sum\limits_{j = 1}^M \phi _j C_{ij}^{k_1 } - \sum\limits_{k_1 = 1}^{Z_1 } \sum\limits_{j = 1}^M [({\boldsymbol U} +{\boldsymbol V}_{\rm FP} ) \cdot {\boldsymbol n}]_j {\boldsymbol B}_{ij}^{k_1 } + \\ \qquad \sum\limits_{k_1 = 1}^{Z_1 } \sum\limits_{j = 1}^{M_W } \Delta \phi_j W_{ij}^{k_1 }, \ \ i = 1, 2, \cdots, M$ (1)
 $\phi _{i{\rm P}} = \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{j = 1}^N \bar\phi _j C_{ij}^{k_2} + \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{j= 1}^N [({\boldsymbol U} +{\boldsymbol V}_{\rm PF} ) \cdot {\boldsymbol n}]_j {\boldsymbol B}_{ij}^{k_2 } + \\ \;\;\;\;\;\;\sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{j = 1}^{N_W } \overline{\Delta \phi}_j W_{ij}^{k_2}+\dfrac 1{Z_2} \sum\limits^{Z_2}_{\delta=1} \sum\limits^Y_{l=1} \bar \phi_{l\delta} C_{il\delta}+\\ \qquad \dfrac 1{Z_2} \sum\limits^{Z_2}_{\delta=1} \sum\limits^Y_{l=1} [({\boldsymbol U} +{\boldsymbol V}_{\rm PF} ) \cdot {\boldsymbol n}]_{l\delta} {\boldsymbol B}_{il\delta}, \\ \qquad i = 1, 2, \cdots, N, \ N+1, \cdots, N+Y$ (2)

 $C_{ij}^k = \dfrac{1}{2\pi }\mathop{{\iint}\mkern-22.5mu \bigcirc}\limits_{S_{B_j } } \dfrac{\partial }{\partial n_{q_j } } \Big(\dfrac{1}{r(p_i, q_{jk} )} \Big ) {\rm{d}} S_{q_j } \\ B_{ij}^k = - \dfrac{1}{2\pi }\mathop{{\iint}\mkern-22.5mu \bigcirc}\limits_{S_{B_j } } \Big(\dfrac{1}{r(p_i, q_{jk} )} \Big) {\rm{d}} S_{q_j } \\ W_{ij}^k = \dfrac{1}{2\pi }\mathop{{\iint}\mkern-22.5mu \bigcirc}\limits_{S_{W_j } } \dfrac{\partial }{\partial n_{q_j } } \Big(\dfrac{1}{r(p_i, q_{jk} )} \Big){\rm{d}} S_{q_j }$

 $\dfrac{1}{S_0 }\sum\limits_{S = 1}^{S_0 } {\sum\limits_{k_2 = 1}^{Z_2 } {\sum\limits_{l = 1}^N {\phi _{lS}^{k_2 } } } C_{ilS}^{k_2 } } = \dfrac{1}{S_1 }S_1 \sum\limits_{k_2 = 1}^{Z_2 } {\sum\limits_{l = 1}^N {\phi _{l1}^{k_2 } E_{il}^{k_2 } } } =\\ \qquad \sum\limits_{k_2 = 1}^{Z_2 } {\sum\limits_{l = 1}^N {\phi _{l1}^{k_2 } E_{il}^{k_2 } } }$ (11)

 $\left. \begin{array}{l} E_{il}^1 = (C_{il1}^1 + C_{il6}^1 + C_{il11}^1 + C_{il16}^1)/{S_2}\\ E_{il}^2 = (C_{il2}^1 + C_{il7}^1 + C_{il12}^1 + C_{il17}^1)/{S_2}\\ E_{il}^3 = (C_{il3}^1 + C_{il8}^1 + C_{il13}^1 + C_{il18}^1)/{S_2}\\ E_{il}^4 = (C_{il4}^1 + C_{il9}^1 + C_{il14}^1 + C_{il19}^1)/{S_2}\\ E_{il}^5 = (C_{il5}^1 + C_{il10}^1 + C_{il15}^1 + C_{il20}^1)/{S_2} \end{array} \right\}$ (12)

 $\sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^N \phi _{l1}^{k_2 } E_{il}^{k_2 } = \sum\limits_{l = 1}^N \Big (\sum\limits_{k_2 = 1}^{Z_2 } \phi _{l1}^{k_2 } \Big)E_{il}^1 =\\ \qquad \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^N \Big ( \sum\limits_{k_2 = 1}^{Z_2 } \phi _{l1}^{k_2 } \Big / {Z_2 } \Big )E_{il}^1 = \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^N \Big( \sum\limits_{k_2 = 1}^{Z_2 } \phi _{l1}^{k_2 } \Big / {Z_2 } \Big )E_{il}^{k_2 }$ (13)

 $E_{ix} = \dfrac{1}{S_0 }\sum\limits_{S = 1}^{S_0 } {C_{ixS} }$ (16)

 $\overline \phi _i = \sum\limits_{k_1 = 1}^{Z_1 } \sum\limits_{j = 1}^M \overline \phi _j C_{ij}^{k_1 } + \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^N \overline \phi _l E_{il}^{k_2 } + \\ \qquad \sum\limits_{x = 1}^Y \overline \phi _x E_{ix} + \cdots, \qquad i = 1, 2, \cdots, M$ (17)

 $\sum\limits_{S = 1}^{S_0 } {\sum\limits_{k_1 = 1}^{Z_1 } {\sum\limits_{j = 1}^M {\varphi _{jS}^{k_1 } } } C_{ijS}^{k_1 } } = \sum\limits_{k_1 = 1}^{Z_1 } {\sum\limits_{j = 1}^M {\overline \varphi _j E_{ij}^{k_1 } } }$ (18)

 $\left. \begin{array}{l} E_{ij}^1 = (C_{ij1}^1 + C_{ij5}^1 + C_{ij9}^1 + C_{ij13}^1 + C_{ij17}^1)/{S_1}\\ E_{ij}^2 = (C_{ij2}^1 + C_{ij6}^1 + C_{ij10}^1 + C_{ij14}^1 + C_{ij18}^1)/{S_1}\\ E_{ij}^3 = (C_{ij3}^1 + C_{ij7}^1 + C_{ij11}^1 + C_{ij15}^1 + C_{ij19}^1)/{S_1}\\ E_{ij}^4 = (C_{ij4}^1 + C_{ij8}^1 + C_{ij12}^1 + C_{ij16}^1 + C_{ij20}^1)/{S_1} \end{array} \right\}$ (19)

 $\sum\limits_{S = 1}^{S_0 } {\sum\limits_{k_2 = 1}^{Z_2 } {\sum\limits_{l = 1}^N {\phi _{lS}^{k_2 } } } C_{ilS}^{k_2 } } + \dfrac{1}{Z_2 }\sum\limits_{S = 1}^{S_0 } {\sum\limits_{k_3 = 1}^{Z_2 } {\sum\limits_{x = 1}^Y {\phi _{xS}^{k_3 } } } C_{ixS}^{k_3 } } =\\ \qquad \sum\limits_{k_2 = 1}^{Z_2 } {\sum\limits_{l = 1}^N {\overline \phi _l } } C_{il}^{k_2 } + \sum\limits_{x = 1}^Y {\overline \phi _x } E_{ix}$ (20)

 $E_{ix} = \dfrac{1}{Z_2 }\sum\limits_{k_3 = 1}^{Z_2 } {C_{ix}^{k_3 } }$ (21)

 $\overline \phi _i = \sum\limits_{k_1 = 1}^{Z_1 } {\sum\limits_{j = 1}^M {\overline \phi _j E_{ij}^{k_1 } } } + \sum\limits_{k_2 = 1}^{Z_2 } {\sum\limits_{l = 1}^N {\overline \phi _l } } C_{il}^{k_2 } + \sum\limits_{x = 1}^Y {\overline \phi _x E_{ix} } + \cdots, \\ i = M + 1, M + 2, \cdots, M + N + 1, \cdots, M + N + Y$ (22)

 $\overline {\phi _i } = \sum\limits_{k_1 = 1}^{Z_1 } {\sum\limits_{j = 1}^M {\overline \phi _j E_{ij}^{k_1 } } } + \sum\limits_{k_1 = 1}^{Z_1 } {\sum\limits_{j = 1}^M {\Big (\overline {\dfrac{\partial \phi }{\partial n}} \Big )_j F_{ij}^{k_1 } } } + \\ \qquad \sum\limits_{k_1 = 1}^{Z_1 } \sum\limits_{j = 1}^{M_W } \overline {\Delta \phi } _j G_{ij}^{k_1 } + \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^N \overline \phi _l E_{il}^{k_2 } + \\ \qquad \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^N \Big(\overline {\dfrac{\partial \phi }{\partial n}} \Big)_l F_{il}^{k_2 } + \sum\limits_{k_2 = 1}^{Z_2 } \sum\limits_{l = 1}^{N_W } \overline {\Delta \phi } _l G_{il}^{k_2 } +\\ \qquad \sum\limits_{x = 1}^Y \overline \phi _x E_{ix} + \Big(\overline {\dfrac{\partial \phi }{\partial n}} \Big)_x F_{ix}, \\ \qquad i = 1, 2, \cdots, M, M + 1, \cdots, M + N, \cdots, \\ \qquad \qquad M + N + Y$ (23)

 $\left. \begin{array}{l} \Delta {p_m} = {p_{m{\rm{U}}}}-{p_{m{\rm{L}}}}\\ m = 1, 2, \cdots, {m_{\rm{F}}}, \\ \qquad {m_F} + 1, \cdots, {m_{\rm{F}}} + {m_A} \end{array} \right\}$ (26)

 $\Delta {\boldsymbol\phi}^{(k + 1)} = \Delta {\boldsymbol\phi}^{(k)}-{\boldsymbol J}^{ - 1}\Delta {\boldsymbol p}^{(k)}$ (27)

 $\left. \begin{array}{l} {J_{ij}} = \frac{{\partial {{(\Delta p)}_i}}}{{\partial {{(\Delta \phi )}_j}}}\\ i({\rm{or}}\;j) = 1, 2, \cdots, {m_F}, {m_F} + 1, \cdots, {m_F} + {m_A} \end{array} \right\}$ (28)

 $\left. {\begin{array}{*{20}{l}} {{\beta _{{\rm{HF}}}} = \left\{ {\begin{array}{*{20}{l}} {{\omega _{{\rm{F1}}}}{\beta _{{\rm{SF}}}}, }&{{x_{{\rm{wake}}}} \le {x_{{\rm{A-plane}}}}}\\ {{\omega _{{\rm{F2}}}}{\beta _{{\rm{SF}}}}, }&{{x_{{\rm{wake}}}} > {x_{{\rm{A-plane}}}}} \end{array}} \right.}\\ {{\beta _{{\rm{HA}}}} = {\omega _{\rm{A}}}{\beta _{{\rm{SA}}}}} \end{array}{\rm{ }}} \right\}$ (29)

 图 5 $l$取100时影响系数对比 Figure 5 The comparison of influence coefficient when $l$ is 100
 图 6 $l$取500时影响系数对比 Figure 6 The comparison of influence coefficient when $l$ is 500
 图 7 $j$取200时影响系数对比 Figure 7 The comparison of influence coefficient when $j$ is 200
 图 8 $j$取600时影响系数对比 Figure 8 The comparison of influence coefficient when $j$ is 600

 $\sum\limits_{i = 1}^n \dfrac{\phi_i^2 }{n} \approx \sum\limits_{i = 1}^n \Big(\dfrac{\phi _i }{n} \Big )^2$ (31)

4 总结

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COMPARATIVE STUDY ON SURFACE PANEL METHOD FOR THE HYDRODYNAMIC ANALYSIS OF HYBRID CONTRA-ROTATING SHAFT POD PROPULSOR1)
Wang Rui2), Xiong Ying, Wang Zhanzhi
Department of Naval Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: The present numerical study for the performance analysis of HCRSP (hybrid contra-rotating shaft pod) propulsor is based on the viscous flow method due to the structure complexity and the lacking of effectively potential flow method. In order to developing an effcient numerical method for the performance analysis of HCRSP propulsor, the HCRSP propulsor was divided into two parts, a single forward propeller and an aft podded propulsor. Then an iterative surface panel method was presented based on the single propeller surface panel method and podded propulsor integral panel method. The geometry characteristic and panel singularity strength of HCRSP propulsor were then analyzed, and an integral panel method was presented to treat HCRSP propulsor as a unit. The control equations of the integral panel method were derived in detail, and numerical solution program was developed. Based on these studies, the open water performance of an HCRSP propulsor was analyzed by iterative panel method and integral panel method. Numerical results were compared with experimental data and show that the relative result error of the two surface panel method are all within 5%, but the calculation time of integral panel method is about two thirds of that of iterative method. The integral panel method is more suitable in the design of HCRSP propulsor due to its calculation without iterative process. In the last, the error sources of integral panel method were discussed.
Key words: HCRSP propulsor    surface panel method    integral calculation model    iterative calculation model