# The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Modelling

_{p}) and flexural rigidity (EI), conveying multiphase flow, flowing parallel to the pipe’s centerline. The centerline axis of the pipe in its undeformed state is assumed to overlap with the Y-axis, and the cylinder is assumed to vibrate in the (Y, X) plane (see Figure 1). To derive the system’s governing equations of motion, the following basic assumptions were made for the cylinder and the fluid: (i) the mean flow velocity is constant; (ii) the cylinder is slender, so that the Euler–Bernoulli beam theory is applicable; (iii) although the deflections of the cylinder may be large, the strains are small; and (iv) the cylinder centerline is extensible.

#### 2.1. Derivation of the Equation of Motion

- ${M}_{j}$ is the mass of the phases in the fluid;
- ${U}_{j}$ is the flow velocity of the phases in the fluid; and
- $\mathcal{L}$ is the Lagrangian operator expressed in Equation (2):$$\mathcal{L}={\mathcal{T}}_{f}+{\mathcal{T}}_{p}-{\mathcal{V}}_{f}-{\mathcal{V}}_{p}$$

#### 2.1.1. Kinetic Energy

#### 2.1.2. Potential Energy

#### 2.1.3. Non-Conservative Work Done

#### 2.2. Equation of Motion for Multiphase Flow

#### Dimensionless Equation of Motion for Two-Phase Flow

#### 2.3. Empirical Gas–Liquid Two-Phase Flow Model

## 3. Method of Solution

#### 3.1. Linear Analysis

#### 3.1.1. Natural Frequencies and Modal Functions

#### 3.1.2. Solution to Axial Vibration Problem

**In matrix form:**

#### 3.1.3. Solution to Transverse Vibration Problem

#### 3.2. Nonlinear Analysis

#### 3.2.1. Nonlinear Axial and Transverse Vibration Problem

_{1}, $\left(CC\right)$ is the complex conjugate, $\varphi \left(x\right)\text{}\mathrm{and}\text{}\eta \left(x\right)$ are the modal functions for the axial and transverse vibrations, and $\omega =Re\left(\omega \right)\text{}\mathrm{and}\text{}\lambda =Re\left(\lambda \right)$. (the real parts of the complex frequencies) are the natural frequencies for the axial and transverse vibrations.

#### 3.2.2. When $\omega $ Is Far from $2\lambda $

#### 3.2.3. When $\omega $ Is Close to $2\lambda $

## 4. Numerical Results

#### 4.1. Effects of Two-Phase Flow on the Dynamic Behavior of the Pipe

#### 4.1.1. Effects of Temperature Difference on the Dynamic Behavior

#### 4.1.2. Effects of Flow Pressure on the Dynamic Behavior

#### 4.1.3. Effects of Top Tension on the Dynamic Behavior

#### 4.2. Time History and Phase Plots

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**First four modes axial dimensionless complex frequency as a function of dimensionless single-phase flow velocity.

**Figure 3.**First four modes transverse dimensionless complex frequency as a function of dimensionless single-phase flow velocity.

**Figure 12.**(

**a**) Time history and (

**b**) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3.

**Figure 13.**(

**a**) Time history and (

**b**) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4.

**Figure 14.**(

**a**) Time history and (

**b**) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5.

**Figure 15.**(

**a**) Time history and (

**b**) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3.

**Figure 16.**(

**a**) Time history and (

**b**) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4.

**Figure 17.**(

**a**) Time history and (

**b**) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5.

**Figure 18.**(

**a**) Time history and (

**b**) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3, σ of 2.0.

**Figure 19.**(

**a**) Time history and (

**b**) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3, σ of 2.0.

**Figure 20.**(

**a**) Time history and (

**b**) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4, σ of 2.0.

**Figure 21.**(

**a**) Time history and (

**b**) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4, σ of 2.0.

**Figure 22.**(

**a**) Time history and (

**b**) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5, σ of 2.0.

**Figure 23.**(

**a**) Time history and (

**b**) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5, σ of 2.0.

Parameter Name | Parameter Unit | Parameter Values |
---|---|---|

External Diameter | D_{o} (m) | 0.0113772 |

Internal Diameter | D_{i} (m) | 0.00925 |

Length | L (m) | 0.1467 |

Pipe density | ρ_{pipe} (kg/m^{3}) | 7800 |

Gas density | ρ_{Gas} (kg/m^{3}) | 1.225 |

Water density | ρ_{Water} (kg/m^{3}) | 1000 |

Tensile and compressive stiffness | EA (N) | 7.24 × 10^{6} |

Bending stiffness | EI (N) | 1.56 × 10^{3} |

Fluid | Void Fraction | $\mathit{\beta}$ Liquid | $\mathit{\beta}$ Gas | $\mathit{\Psi}$ Liquid | $\mathit{\Psi}$ Gas | Critical Velocity | |
---|---|---|---|---|---|---|---|

Transverse | Axial | ||||||

Single Phase | NA | 0.2 | 0.0 | 1.0 | 0.0 | 5.653 | 14.149 |

Fluid | Void Fraction | $\mathit{\beta}$ Liquid | $\mathit{\beta}$ Gas | $\mathit{\Psi}$ Liquid | $\mathit{\Psi}$ Gas | Critical Mixture Velocity | |
---|---|---|---|---|---|---|---|

Transverse * | Axial | ||||||

Two-Phase | 0.3 | 0.19998 | 0.00010 | 0.99948 | 0.00052 | 12.505 | 31.634 |

Two-Phase | 0.4 | 0.19997 | 0.00016 | 0.99918 | 0.00082 | 13.349 | 33.750 |

Two-Phase | 0.5 | 0.19995 | 0.00024 | 0.99878 | 0.00122 | 14.613 | 36.966 |

**Table 4.**Summary of the linear two-phase solution of critical flow velocities for varying temperature difference.

Parameter | Void Fraction | Thermal Expansivity $\mathit{\alpha}$ | Critical Mixture Velocity | |
---|---|---|---|---|

Transverse * | Axial | |||

DT = 0 | 0.3 | 0.002 | 12.505 | 31.634 |

DT = 40 | 0.3 | 0.002 | 9.253 | 31.634 |

DT = 50 | 0.3 | 0.002 | 8.237 | 31.634 |

**Table 5.**Summary of the linear two-phase solution of critical flow velocities for varying pressurization.

Parameter | Void Fraction | Critical Mixture Velocity | |
---|---|---|---|

Transverse * | Axial | ||

$\mathit{\Pi}2=0$ | 0.3 | 12.505 | 31.634 |

$\mathit{\Pi}2=5$ | 0.3 | 10.596 | 31.634 |

$\mathit{\Pi}2=10$ | 0.3 | 8.237 | 31.634 |

**Table 6.**Summary of the linear two-phase solution of critical flow velocities for varying top tensions.

Parameter | Void Fraction | Critical Velocity | |
---|---|---|---|

Transverse * | Axial | ||

$\mathit{\Pi}0=0$ | 0.3 | 12.505 | 31.634 |

$\mathit{\Pi}0=5$ | 0.3 | 14.155 | 31.634 |

$\mathit{\Pi}0=-5$ | 0.3 | 10.596 | 31.634 |

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**MDPI and ACS Style**

Adegoke, A.S.; Oyediran, A.A. The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading. *Math. Comput. Appl.* **2017**, *22*, 44.
https://doi.org/10.3390/mca22040044

**AMA Style**

Adegoke AS, Oyediran AA. The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading. *Mathematical and Computational Applications*. 2017; 22(4):44.
https://doi.org/10.3390/mca22040044

**Chicago/Turabian Style**

Adegoke, Adeshina S., and Ayo A. Oyediran. 2017. "The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading" *Mathematical and Computational Applications* 22, no. 4: 44.
https://doi.org/10.3390/mca22040044