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 哈尔滨工程大学学报  2019, Vol. 40 Issue (8): 1414-1419  DOI: 10.11990/jheu.201709124 0

### 引用本文

LI Yulong, LIAO Honglie, HU Zhan, et al. High accuracy deep sea riser VIV numerical analysis[J]. Journal of Harbin Engineering University, 2019, 40(8), 1414-1419. DOI: 10.11990/jheu.201709124.

### 文章历史

1. 中山大学 海洋工程与技术学院, 广东 珠海 519082;
2. 广州船舶及海洋工程设计研究院, 广东 广州 119077;
3. 中山大学 海洋科学学院, 广东 广州 119077

High accuracy deep sea riser VIV numerical analysis
LI Yulong 1, LIAO Honglie 2, HU Zhan 3, LUO Xiangxin 1
1. College of Marine Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China;
2. Guangzhou Marine Engineering Corporation, Guangzhou 119077, China;
3. College of Marine Science, Sun Yat-sen University, Guangzhou 119077, China
Abstract: This work proposes a high-precision numerical prediction method to solve the vortex-induced vibration (VIV) problem. The three-dimensional (3-D) Common Refinement method (CRM) is applied to investigate the spatial interpolation for the coupling contact surface between the non-overlapping subdomains of a discretized incompressible fluid and a nonlinear super-elastic structure. Incompressible fluid flow is discretized through the Petrov-Galerkin finite element method, and the large deformation elastic structure is discretized by using the continuous Galerkin finite element method. An arbitrary Lagrangian-Eulerian formulation is used to treat the large deformation of the fluid-solid grid, and the fluid and solid domains are solved through fully decoupled implicit partitioning procedures. The accuracy and reliability of the spatial interpolation of the CRM to meet the balanced condition is adopted. The result reveals that the common refinement grid across nonmatching fluid-structure grids enables the accurate transfer of the physical quantities across the fluid-structure system. Finally, CRM is applied to the deep-sea riser VIV problem, and the results of this study are then compared with those of relevant works. Results illustrate that the presented method can accurately and reliably solve the fluid-solid coupling problem encountered in marine engineering.
Keywords: fluid-structure interaction    Common Refinement    non-matching mesh    vortex induced vibration    finite element method

Common-Refinement方法是一种精度很高，并能够保证数值计算稳定性和守恒性的非匹配网格技术。由于其数据结构的困难性和复杂程度，只有部分学者在一维和二维假设下对此进行了精度和稳定性的分析与研究[3-5]。研究表明，常规的节点投影，以及单元投影方法产生的非匹配网格两侧的局部误差会非常大，且会随着时间不断放大，并且对网格两侧的空间划分比例的依赖性很高，进而影响数值计算的稳定性。

1 控制方程

 $\begin{array}{*{20}{c}} {{\rho ^{\rm{f}}}\frac{{\partial {{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}}}}{{\partial t}}\left| {_{\hat x}} \right. + {\rho ^{\rm{f}}}\left( {{{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}} - {{\mathit{\boldsymbol{\bar w}}}^{\rm{f}}}} \right) \cdot \nabla {{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}} = \nabla \cdot {{\mathit{\boldsymbol{\bar \sigma }}}^{\rm{f}}} + }\\ {\nabla \cdot {\mathit{\boldsymbol{\sigma }}^{{\rm{sgs}}}} + {\mathit{\boldsymbol{b}}^{\rm{f}}},{\rm{on}}\;{\mathit{\Omega }^{\rm{f}}}\left( t \right)} \end{array}$ (1)
 $\nabla \cdot {\mathit{\boldsymbol{\bar u}}^{\rm{f}}} = 0,{\rm{on}}\;{\mathit{\Omega }^{\rm{f}}}\left( t \right)$ (2)

 ${\mathit{\boldsymbol{\bar \sigma }}^{\rm{f}}} = - {\bar p^{\rm{f}}}I + {\mu ^{\rm{f}}}\left( {\nabla {{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}} + {{\left( {\nabla {{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}}} \right)}^{\rm{T}}}} \right)$ (3)

 $\begin{array}{*{20}{c}} {\int_{{\mathit{\Omega }^{\rm{f}}}\left( t \right)} {{\rho ^{\rm{f}}}\left( {{\partial _t}{{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}} + \left( {{{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}} - {\mathit{\boldsymbol{w}}^{\rm{f}}}} \right) \cdot \nabla {{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}}} \right) \cdot {\phi ^{\rm{f}}}\left( x \right){\rm{d}}\mathit{\Omega }} + }\\ {\int_{{\mathit{\Omega }^{\rm{f}}}\left( t \right)} {\left( {{{\mathit{\boldsymbol{\bar \sigma }}}^{\rm{f}}} + {\mathit{\boldsymbol{\sigma }}^{{\rm{sgs}}}}} \right):\nabla {\phi ^{\rm{f}}}\left( x \right){\rm{d}}\mathit{\Omega }} = }\\ {\int_{{\mathit{\Omega }^{\rm{f}}}\left( t \right)} {{\mathit{\boldsymbol{b}}^{\rm{f}}} \cdot {\phi ^{\rm{f}}}\left( x \right){\rm{d}}\mathit{\Omega }} + \int_{\mathit{\Gamma }_{\rm{h}}^{\rm{f}}\left( t \right)} {{h^{\rm{f}}} \cdot {\phi ^{\rm{f}}}\left( x \right){\rm{d}}\mathit{\Gamma }} } \end{array}$ (4)
 $\int_{{\mathit{\Omega }^{\rm{f}}}\left( t \right)} {\nabla \cdot {{\mathit{\boldsymbol{\bar u}}}^{\rm{f}}}q\left( x \right){\rm{d}}\mathit{\Omega }} = 0$ (5)

 $\begin{array}{*{20}{c}} {\int_{\mathit{\Omega }_0^{\rm{s}}} {\tau _{ij}^{\rm{s}}\delta L_{ij}^{\rm{s}}{\rm{d}}\mathit{\Omega }} - \int_{\mathit{\Omega }_0^{\rm{s}}} {{\rho ^{\rm{s}}}b_i^{\rm{s}}\delta v_i^{\rm{s}}{\rm{d}}\mathit{\Omega }} + \int_{\mathit{\Omega }_0^{\rm{s}}} {\frac{{\partial u_i^{\rm{s}}}}{{\partial t}}\delta v_i^{\rm{s}}{\rm{d}}\mathit{\Omega }} - }\\ {\int_{\mathit{\Gamma }_2^{\rm{s}}} {t_i^{\rm{s}}\delta v_i^{\rm{s}}{\eta ^{\rm{s}}}{\rm{d}}\mathit{\Gamma }} = 0} \end{array}$ (6)

 $\sigma _{ij}^{\rm{s}} = \frac{{{\mu ^{\rm{s}}}}}{{{{\left( {{J^{\rm{s}}}} \right)}^{5/3}}}}\left( {B_{ij}^{\rm{s}} - \frac{1}{3}B_{kk}^{\rm{s}}{\delta _{ij}}} \right) + {K^{\rm{s}}}\left( {{J^{\rm{s}}} - 1} \right){\delta _{ij}}$ (7)

 $F_{i j}^{\mathrm{s}}=\delta_{i j}+\partial d_{i}^{\mathrm{s}} / \partial x_{j}$ (8)

 ${\mathit{\boldsymbol{\bar u}}^{\rm{f}}}\left( {{\mathit{\boldsymbol{\varphi }}^{\rm{s}}}\left( {{\mathit{\boldsymbol{x}}^{\rm{s}}},t} \right),t} \right) = {\mathit{\boldsymbol{u}}^{\rm{s}}}\left( {{\mathit{\boldsymbol{x}}^{\rm{s}}},t} \right)$ (9)
 $\int_{{{\rm{ \mathsf{ φ} }}^s}\left( {\gamma ,t} \right)} {{\mathit{\boldsymbol{\sigma }}^{\rm{f}}}\left( {{\mathit{\boldsymbol{x}}^{\rm{f}}},t} \right)} \cdot \mathit{\boldsymbol{n}}{\rm{d}}\mathit{\Gamma }\left( {{x^{\rm{f}}}} \right) + \int_\gamma {{t^{\rm{s}}}{\rm{d}}\mathit{\Gamma }} = 0$ (10)

2 流固耦合系统的离散化

 $\begin{array}{*{20}{c}} {\int_{\mathit{\Omega }_{\rm{h}}^{\rm{f}}\left( {{t^{n + 1}}} \right)} {{\rho ^{\rm{f}}}} \left( {{\partial _t}\mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + \alpha _m^{\rm{f}}} + \left( {\mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}} - \mathit{\boldsymbol{w}}_h^{{\rm{f}},n + {\alpha ^{\rm{f}}}}} \right) \cdot \nabla \mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}}} \right) \cdot }\\ {{\phi ^{\rm{f}}}{\rm{d}}\mathit{\Omega } + \int_{\mathit{\Omega }_h^{\rm{f}}\left( {{t^{n + 1}}} \right)} {\left( {\mathit{\boldsymbol{\bar \sigma }}_h^{{\rm{f}},n + {\alpha ^{\rm{f}}}} + \mathit{\boldsymbol{\sigma }}_h^{{\rm{sgs}},n + {\alpha ^{\rm{f}}}}} \right):\nabla {\phi ^{\rm{f}}}{\rm{d}}\mathit{\Omega }} - }\\ {\int_{\mathit{\Omega }_h^{\rm{f}}\left( {{t^{n + 1}}} \right)} {\nabla \cdot \mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}}q{\rm{d}}\mathit{\Omega }} + \sum\limits_{e = 1}^{{n_{{\rm{el}}}}} {\int_{{\mathit{\Omega }^{\rm{e}}}} {\tau _m^{\rm{f}}\left( {{\rho ^{\rm{f}}}\left( {\mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}} - \mathit{\boldsymbol{w}}_h^{{\rm{f}},n + {\alpha ^{\rm{f}}}}} \right) \cdot } \right.} } }\\ {\nabla {\phi ^{\rm{f}}} + \nabla q) \cdot \left( {{\rho ^{\rm{f}}}{\partial _t}\mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}_{\rm{m}}}} + {\rho ^{\rm{f}}}\left( {\mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}} - \mathit{\boldsymbol{w}}_h^{{\rm{f}},n + {\alpha ^{\rm{f}}}}} \right)} \right. \cdot }\\ {\nabla \mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}} - \nabla \cdot \mathit{\boldsymbol{\overline \sigma }} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}} - \nabla \cdot \mathit{\boldsymbol{\sigma }}_h^{{\mathop{\rm sgs}\nolimits} ,n + {\alpha ^{\rm{f}}}} - {\mathit{\boldsymbol{b}}^{\rm{f}}}\left( {{t^{n + {\alpha ^{\rm{f}}}}}} \right)){\rm{d}}{\mathit{\Omega }^e} + }\\ {\sum\limits_{e = 1}^{{n_{{\rm{el}}}}} {\int_{{\mathit{\Omega }_{\rm{e}}}} {\nabla \cdot {\phi ^{\rm{f}}}\tau _c^{\rm{f}}{\rho ^{\rm{f}}}\nabla \cdot \mathit{\boldsymbol{\overline u}} _h^{{\rm{f}},n + {\alpha ^{\rm{f}}}}{\rm{d}}{\mathit{\Omega }^{\rm{e}}}} } = \int_{\mathit{\Omega }_h^{\rm{f}}\left( {{t^{n + 1}}} \right)} {{b^{\rm{f}}}\left( {{t^{n + {\alpha ^{\rm{f}}}}}} \right)} \cdot }\\ {{\phi ^{\rm{f}}}{\rm{d}}\mathit{\Omega } + \int_{\mathit{\Gamma }_h^{\rm{f}}\left( {{t^{{\rm{n}} + 1}}} \right)} {{\mathit{\boldsymbol{h}}^{\rm{f}}} \cdot {\phi ^{\rm{f}}}{\rm{d}}\mathit{\Gamma }} } \end{array}$ (11)

 $\begin{array}{*{20}{c}} {\tau _m^{\rm{f}} = \left[ {{{\left( {\frac{{2{\rho ^{\rm{f}}}}}{{\Delta t}}} \right)}^2} + {{\left( {{\rho ^{\rm{f}}}} \right)}^2}\left( {\mathit{\boldsymbol{\bar u}}_h^{\rm{f}} - \mathit{\boldsymbol{w}}_h^{\rm{f}}} \right) \cdot \mathit{\boldsymbol{G}}\left( {\mathit{\boldsymbol{\bar u}}_h^{\rm{f}} - \mathit{\boldsymbol{w}}_h^{\rm{f}}} \right) + } \right.}\\ {{{\left. {{C_I}{{\left( {{\mu ^{\rm{f}}} + {\mu ^{\rm{t}}}} \right)}^2}\mathit{\boldsymbol{G}}:\mathit{\boldsymbol{G}}} \right]}^{ - 1/2}}} \end{array}$ (12)

 $\mathit{\boldsymbol{G}} = \frac{{\partial {\mathit{\boldsymbol{\xi }}^{\rm{T}}}}}{{\partial {\mathit{\boldsymbol{x}}^{\rm{f}}}}}\frac{{\partial \mathit{\boldsymbol{\xi }}}}{{\partial {\mathit{\boldsymbol{x}}^{\rm{f}}}}},\;\;\;\;\tau _c^{\rm{f}} = \frac{1}{{\left( {{\rm{tr}}\mathit{\boldsymbol{G}}} \right)\tau _m^{\rm{f}}}}$ (13)

 $\mathit{\boldsymbol{d}}_i^{\rm{s}}\left( {{\mathit{\boldsymbol{x}}^{\rm{s}}}} \right) = \sum\limits_{a = 1}^n {{N^a}\left( {{\mathit{\boldsymbol{x}}^{\rm{s}}}} \right)\mathit{\boldsymbol{d}}_i^{{\rm{s}},a}} ,\delta v_i^{\rm{s}}\left( {{\mathit{\boldsymbol{x}}^{\rm{s}}}} \right) = \sum\limits_{a = 1}^n {{N^a}\left( {{\mathit{\boldsymbol{x}}^{\rm{s}}}} \right)\delta v_i^{{\rm{s,}}a}}$ (14)

 $\begin{array}{*{20}{c}} {\int_{\mathit{\Omega }_0^{\rm{s}}} {{\rho ^{\rm{s}}}{N^b}{N^a}\frac{{{\partial ^2}d_i^{{\rm{s}},b}}}{{\partial {t^2}}}{\rm{d}}\mathit{\Omega }} + \int_{\mathit{\Omega }_0^{\rm{s}}} {\tau _{ij}^{\rm{s}}\left[ {F_{pq}^{\rm{s}}\left( {d_k^{{\rm{s}},b}} \right)} \right]\frac{{\partial {N^a}}}{{\partial {x_m}}}F_{mj}^{{\rm{s}}, - 1}{\rm{d}}\mathit{\Omega }} - }\\ {\int_{\mathit{\Omega }_0^{\rm{s}}} {{\rho ^{\rm{s}}}b_i^{\rm{s}}{N^a}{\rm{d}}\mathit{\Omega }} - \int_{{\mathit{\Gamma }^{{\rm{fs}}}}} {t_i^{\rm{s}}{N^a}{\eta ^{\rm{s}}}{\rm{d}}\mathit{\Gamma }} = 0} \end{array}$ (15)

3 Common-Refinement方法

 $t^{\mathrm{f}}\left(x^{\mathrm{f}}\right) \approx \sum\limits_{i=1}^{m_{f}} N_{i}^{\mathrm{f}} t_{i}^{\mathrm{f}}, t^{\mathrm{s}}\left(x^{\mathrm{s}}\right) \approx \sum\limits_{j=1}^{m_{s}} N_{j}^{\mathrm{s}} t_{j}^{\mathrm{s}}$ (16)

 $\int_{{\varGamma ^{{\rm{fs}}}}} {N_i^{\rm{s}}} {\mathit{\boldsymbol{t}}^{\rm{s}}}{\rm{d}}\mathit{\Gamma } = \int_{{\varGamma ^{{\rm{fs}}}}} {N_i^{\rm{s}}} {t^{\rm{f}}}{\rm{d}}\mathit{\Gamma }$ (17)

 $\int_{{\mathit{\Gamma }^{{\rm{fs}}}}} {N_i^{\rm{s}}N_j^{\rm{s}}\mathit{\boldsymbol{\tilde t}}_j^{\rm{s}}{\rm{d}}\mathit{\Gamma }} = \int_{{\mathit{\Gamma }^{{\rm{fs}}}}} {N_i^{\rm{s}}N_j^{\rm{f}}\mathit{\boldsymbol{\tilde t}}_j^{\rm{f}}{\rm{d}}\mathit{\Gamma }}$ (18)

 $\tilde{\boldsymbol{t}}_{j}^{\mathrm{s}}=\boldsymbol{M}_{i j}^{\mathrm{s}-1} \boldsymbol{f}_{i}^{\mathrm{s}}$ (19)

 $\mathit{\boldsymbol{M}}_{ij}^{\rm{s}} = \int_{{\mathit{\Gamma }^{{\rm{fs}}}}} {N_i^{\rm{s}}} N_j^{\rm{s}}{\rm{d}}\mathit{\Gamma }$ (20)

 $\mathit{\boldsymbol{f}}_i^{\rm{s}} = \sum\limits_{j = 1}^{{m_f}} {\mathit{\boldsymbol{\widetilde t}}_j^{\rm{f}}} \int_{{\mathit{\Gamma }^{{\rm{fs}}}}} {N_j^{\rm{f}}} N_i^{\rm{s}}{\rm{d}}\mathit{\Gamma }$ (21)

 ${x^{\rm{f}}} \approx \sum\limits_{i = 1}^{{m_f}} {N_i^{\rm{f}}\left( x \right)x_i^{\rm{f}}} \;\;\;{\rm{on}}\;\;\;\;\mathit{\Gamma }_h^{\rm{f}}$ (22)
 ${x^{\rm{s}}} \approx \sum\limits_{j = 1}^{{m_s}} {N_j^{\rm{s}}\left( x \right)x_j^{\rm{s}}} \;\;\;{\rm{on}}\;\;\;\;\mathit{\Gamma }_h^{\rm{s}}$ (23)

 $\mathit{\boldsymbol{f}}_j^{\rm{s}} = \sum\limits_{i = 1}^{{e_c}} {\int_{\sigma _i^c} {N_j^{\rm{s}}{{\mathit{\boldsymbol{\tilde t}}}^{\rm{f}}}{\rm{d}}\mathit{\Gamma }} }$ (24)

4 深海立管的涡激振动

 $\mathit{Re} = \frac{{{\rho ^{\rm{f}}}UD}}{{{\mu ^{\rm{f}}}}} = 4\;000$ (25)
 $EI/\left( {{\rho ^{\rm{f}}}{U^2}{D^{\rm{4}}}} \right) = {\rm{2}}{\rm{.115}}\;{\rm{8}} \times {\rm{1}}{{\rm{0}}^{\rm{7}}}$ (26)
 $T/\left( {{\rho ^{\rm{f}}}{U^2}{D^2}} \right) = 5.1062\;5 \times {10^4}$ (27)
 ${\mathit{\boldsymbol{m}}^ * } = {\mathit{\boldsymbol{m}}^{\rm{s}}}\left( {\frac{{\rm{ \mathsf{ π} }}}{4}{D^2}L{\rho ^{\rm{f}}}} \right) = 2.23$ (28)

 Download: 图 1 立管动态响应谱分析结果 Fig. 1 Spectral analysis of riser motion response

 $\left\{ \begin{array}{l} {A_{x,{\rm{rms}}}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{\left( {{A_{x,i}} - {{\bar A}_x}} \right)}^2}} } \\ {A_{y,{\rm{rms}}}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{\left( {{A_{y,i}} - {{\bar A}_y}} \right)}^2}} } \end{array} \right.$ (29)

5 结论

1) Common-Refinement方法可以准确地对流体和结构区域的计算数据进行传输，并同时保证流固耦合问题的计算精度与准确性。

2) 通过将提出的基于三维Common-Refinement方法的流固耦合有限元求解器应用于实际大尺度海洋立管问题，能够看出三维Common-Refinement方法提供了稳定且准确的模拟方式，立管的物理特性和动态响应捕捉良好，更进一步说明了这种方法的在海洋工程领域的实用价值。

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