﻿ 基于流固耦合分析的船舶离心泵转子模态设计
 舰船科学技术  2023, Vol. 45 Issue (24): 89-92    DOI: 10.3404/j.issn.1672-7649.2023.24.016 PDF

1. 武汉理工大学，湖北 武汉 430074;
2. 闽西职业技术学院，福建 龙岩 364021

Modal design of ship centrifugal pump rotor based on fluid structure coupling analysis
ZHAN Xiao-hua1,2
1. Wuhan University of Technology, Wuhan 430074, China;
2. Minxi Vocational and Technical College, Longyan 364021, China
Abstract: Fluid structure coupling analysis is an analysis method that combines fluid dynamics and structural dynamics. In the modal design of ship centrifugal pump rotors, fluid structure coupling analysis can determine the key vibration modes and frequencies of the pump rotor, evaluate the impact of fluid on pump rotor vibration, identify potential vibration problems, and carry out targeted structural optimization. This article systematically introduces the process of fluid structure coupling analysis of ship centrifugal pump rotors, and introduces the fluid structure coupling analysis algorithm from the perspectives of fluid mechanics principles, dynamics principles, etc. By establishing a finite element model of ship centrifugal pump rotors, the fluid structure coupling modal analysis of the rotors is carried out in the analysis software workbench, which is of great significance for improving the design level of ship centrifugal pump rotors.
Key words: fluid structure coupling     fluid mechanics     centrifugal pump     Workbench
0 引　言

1 流体力学的湍流理论和壁面函数理论 1.1 湍流模型

1）涡粘模型

 $\overline {\rho {u_i}^\prime u_j^\prime } = - \mu \left( {\frac{{ \partial {{\bar u}_i}}}{{ \partial {x_j}}} + \frac{{ \partial {{\bar u}_j}}}{{ \partial {x_i}}}} \right) + \frac{2}{3}\left( {\rho k + {\mu _t}\frac{{ \partial {{\bar u}_ + }}}{{ \partial {x_i}}}} \right){\delta _{ij}} \text{。}$

2）$k - \varepsilon$模型

$k - \varepsilon$模型是一种经典的湍流模型，该模型引入了湍流动力耗散率$\varepsilon$，定义如下：

 $\varepsilon = \frac{\mu }{\rho }\overline {\left( {\frac{{ \partial u_i^{}}}{{ \partial {x_k}}}} \right)\left( {\frac{{ \partial {u_j}}}{{ \partial {x_k}}}} \right)} \text{。}$

$k - \varepsilon$参数表征湍流粘度${\ \mu _t}$为：

 ${\mu _t} = \rho {C_\mu }\frac{{{\kappa ^2}}}{\varepsilon } 。$

$k - \varepsilon$模型如下式：

 $\begin{gathered} \frac{{ \partial (\rho k)}}{{ \partial t}} + \frac{{ \partial \left( {\rho {u_i}k} \right)}}{{ \partial {x_i}}} = \frac{ \partial }{{ \partial {x_j}}}\left[ {\left( {\mu + \frac{{{\mu _t}}}{{{\sigma _k}}}} \right)\frac{{ \partial k}}{{ \partial {x_j}}}} \right] + {G_k} - \rho \varepsilon \text{，} \\ \frac{{ \partial (\rho \varepsilon )}}{{ \partial t}} + \frac{{ \partial \left( {\rho {u_j}\varepsilon } \right)}}{{ \partial {x_i}}} = \frac{ \partial }{{ \partial {x_j}}}\left[ {\left( {\mu + \frac{{{\mu _r}}}{{{\sigma _s}}}} \right)\frac{{ \partial \varepsilon }}{{ \partial {x_j}}}} \right] + \frac{\varepsilon }{k}\text{。} \\ \end{gathered}$

3）$rank - \varepsilon$模型

$rank - \varepsilon$湍流模型在分析流动和涡旋效应时有较大的优势，其基本表达式为：

 $\begin{gathered} \rho \frac{{{{\rm{D}}_k}}}{{{{\rm{D}}_t}}} = \frac{ \partial }{{ \partial {x_i}}}\left( {{a_k}{\mu _{eff}}\frac{{ \partial k}}{{ \partial {x_i}}}} \right) + {G_k} - \rho \varepsilon n\text{，} \\ \rho \frac{{{{\rm{D}}_l}}}{{{{\rm{D}}_t}}} = \frac{ \partial }{{ \partial {x_i}}}\left( {{a_t}{\mu _{eft}}\frac{{ \partial \varepsilon }}{{ \partial {x_i}}}} \right) + \frac{\varepsilon }{k}{G_l} - \rho \frac{{{\varepsilon ^2}}}{k} \text{。} \\ \end{gathered}$

1.2 壁面函数理论

 图 1 壁面函数理论的示意图 Fig. 1 Diagram of wall function theory

 $\begin{gathered} {U_t} = \frac{1}{k}\ln \left( {{y_t}} \right) + C \text{，} \\ {y_t} = \frac{{\rho \Delta {u_s}}}{\mu }\text{，} \\ {u_s} = {\left( {\frac{{{\tau _p}}}{\rho }} \right)^{\frac{1}{2}}}\text{。} \\ \end{gathered}$
2 基于有限元的船舶离心泵转子流固耦合分析 2.1 流固耦合求解方法研究

1）前沿耦合法（Front-Coupling Method）[3]。该方法将流体和固体的运动方程分别求解，然后通过界面条件将两者耦合起来。这种方法的优点是能够处理复杂的流体和固体结构，但计算量较大，适用于小规模问题。

2）后沿耦合法（Back-Coupling Method）。该方法将流体和固体的运动方程同时求解，通过界面条件将两者耦合起来。这种方法的优点是计算效率高，适用于大规模问题，但对于复杂的流体和固体结构可能不够准确。

3）有限元法。该方法分别建立流体和固体的网格，然后通过耦合方程将两者结合起来。这种方法的优点是适用于复杂的流体和固体结构，这也是本文使用的方法。

 图 2 流固耦合的有限元法原理 Fig. 2 Principle of fluid-structure coupling finite element method

 图 3 固体的空间弹性体应力模型 Fig. 3 Spatial elastomer stress model of solid

 $\left\{ {\begin{array}{*{20}{l}} {\{ {\boldsymbol{u}}\} = {{[u,v,w]}^{\rm{T}}}} \text{，}\\ {\{ {\bf{\varepsilon }}\} = {{\left[ {{\varepsilon _x},{\varepsilon _y},{\varepsilon _z},{\gamma _{xy}},{\gamma _{xx}},{\gamma _{xx}}} \right]}^{\rm{T}}}}\text{，} \\ {\{ {\bf{\sigma }}\} = {{\left[ {{\sigma _x},{\sigma _y},{\sigma _z},{\tau _{xy}},{\tau _x},{\tau _{zx}}} \right]}^{\rm{T}}}} \text{。} \end{array}} \right.$

 $\left[ A \right]\{ {\bf{\sigma }}\} + \{ {{R}}\} = 0 \text{，}$

 $[{\boldsymbol{A}}] = \left[ {\begin{array}{*{20}{c}} {\dfrac{ \partial }{{ \partial x}}}&0&0&{\dfrac{ \partial }{{ \partial y}}}&{}&{\dfrac{ \partial }{{ \partial z}}} \\ {}&{\dfrac{ \partial }{{ \partial y}}}&{}&{\dfrac{ \partial }{{ \partial x}}}&{\dfrac{ \partial }{{ \partial z}}}&{} \\ {}&{\dfrac{ \partial }{{ \partial z}}}&{}&{\dfrac{ \partial }{{ \partial y}}}&{\dfrac{ \partial }{{ \partial x}}}&{} \end{array}} \right] \text{。}$

 $\left\{ {\begin{array}{*{20}{l}} {{T_x} = {n_x}{\sigma _x} + {n_y}{\tau _{yx}} + {n_z}{\tau _{zx}}}\text{，} \\ {{T_y} = {n_x}{\tau _{xy}} + {n_y}{\sigma _y} + {n_z}{\tau _{zy}}}\text{，} \\ {{T_z} = {n_x}{\tau _{xz}} + {n_y}{\tau _{xy}} + {n_z}{\sigma _z}} \text{。} \end{array}} \right.$

2.2 模态理论分析

 $[{\boldsymbol{M}}]\{ {\boldsymbol{\ddot x}}\} + [{\boldsymbol{C}}]\{ {\boldsymbol{\dot x}}\} + [{\boldsymbol{K}}]\{ {\boldsymbol{x}}\} = \{ {\boldsymbol{F}}({\boldsymbol{t}})\} \text{。}$

 $\begin{gathered} \left( {{s^2}[{\bf{M}}] + s[{\bf{C}}] + [{\bf{K}}]} \right)\{ X(s)\} = \{ F(s)\}\text{，} \\ X(s) = \int_{ - \infty }^{ + \infty } X (t){e^{ - st}}{\rm{d}}t\text{，} \\ F(s) = \int_{ - \infty }^{ + \infty } F (t){e^{ - st}}{\rm{d}}t\text{。} \\ \end{gathered}$

 图 4 基于有限元软件的模态分析流程 Fig. 4 Modal analysis process based on finite element software
2.3 基于有限元分析的船舶离心泵转子流固耦合模态分析

1）三维建模

 图 5 离心泵转子的三维建模 Fig. 5 Three-dimensional modeling of centrifugal pump rotor

2）有限元建模

 图 6 离心泵转子结构有限元模型 Fig. 6 Centrifugal pump rotor structure

3）载荷与求解

 图 7 离心泵转子模态仿真云图 Fig. 7 Cloud image of rotor mode simulation of centrifugal pump

3 结　语

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