Numerical scheme for riser motion calculation during 3-D VIV simulation

doi:10.1016/j.jfluidstructs.2011.06.010

Journal of Fluids and Structures

Volume 27, Issue 7, October 2011, Pages 947-961

https://doi.org/10.1016/j.jfluidstructs.2011.06.010 Get rights and content

Abstract

This paper presents a numerical scheme for riser motion calculation and its application to riser VIV simulations. The discretisation of the governing differential equation is studied first. The top tensioned risers are simplified as tensioned beams. A centered space and forward time finite difference scheme is derived from the governing equations of motion. Then an implicit method is adopted for better numerical stability. The method meets von Neumann criteria and is shown to be unconditionally stable. The discretized linear algebraic equations are solved using a LU decomposition method. This approach is then applied to a series of benchmark cases with known solutions. The comparisons show good agreement. Finally the method is applied to practical riser VIV simulations. The studied cases cover a wide range of riser VIV problems, i.e. different riser outer diameter, length, tensioning conditions, and current profiles. Reasonable agreement is obtained between the numerical simulations and experimental data on riser motions and cross-flow VIV a/D. These validations and comparisons confirm that the present numerical scheme for riser motion calculation is valid and effective for long riser VIV simulation.

Highlights

► We develop a riser motion solver suitable for long riser VIV simulation. ► We apply the riser motion solver to various benchmarking and application cases. ► The motion solver accommodates the parallel computation of fluid domain. ► The long riser VIV tends to be multi-modal, with more than one dominant mode. ► Higher harmonics response has been observed during the VIV simulations.

Introduction

Riser VIV numerical simulation has been an interesting area for many years. Many of these investigations are limited to 2-D or rigid risers with low L/D. Some of them extend the 2-D methods to simulate 3-D VIV through strip theory (Meneghini et al., 2004, Newman and Karniadakis, 1996, Willden and Graham, 2001, Willden and Graham, 2004, Willden and Graham, 2005, Willden and Graham, 2006; Yamamoto et al., 2004). A list of the popularly used CFD tools for riser VIV simulation was discussed by Chaplin et al. (2005). However, only a few are capable of full 3-D time domain simulation of the riser VIV with large L/D. One of them is the CFD approach used by Holmes et al. (2006) to simulate riser VIV in fully 3-D using 10 million unstructured elements. In their approach the riser motions were calculated through a general-purpose structural motion solver. Constantinides and Oakley, 2008a, Constantinides and Oakley, 2008b used the same approach to VIV simulations of long cylinders. Chen and Kim (2010) used ANSYS MFX package, a newly released feature by ANSYS Inc., to study the VIV of a vertical riser in 3-D. In this paper we present a simple yet effective riser motion solver and its numerical scheme for riser 3-D VIV simulation. There are many general-purpose structural programs available in the market, and several others are specialized in riser dynamics analysis. Most of them are based on finite element analysis (FEA) methods. One of the advantages of the FEA is that a complex riser system could be decomposed into simple components. The dynamic equations of each component are then assembled into a global matrix, which typically has a dimension of 6N×6N (N is the total element number) if all six-degrees-of-freedom are included for each element; while in VIV simulation, usually only the cross-flow and in-line motion of the riser are of interest. This would enable the possibility of calculating the riser motions in a more efficient way. In what follows, a finite difference approach is developed to solve the riser motions during VIV simulation.

In this paper, we first present a finite difference scheme for solving the riser motion equation. This is a fourth-order PDE with derivatives in space and time. Then we study the numerical stability of the scheme. A riser motion solver is established based on this numerical scheme. To validate the solver, we apply it to a series of benchmark cases, and compare the results to theoretical values or other popularly used FEA tools. After that, we integrate the riser motion solver into a flow field solving code, and apply it to riser VIV simulations with L/D up to 3350. The VIV simulation results, including rms a/D, motion trajectories, VIV induced stresses, VIV induced fatigue, are compared to experimental data or other software showing good agreement. It is concluded that the present riser motion numerical scheme is valid and effective for riser VIV simulations.

Section snippets

Riser motion equations

A top tensioned riser can be simplified as a tensioned beam with varying sectional properties along the riser. Its lateral motion is described in terms of Eq. (1). This is a parabolic system of PDEs, with a fourth-order derivative in space and a second-order derivative in time $T \frac{d^{2} x}{d z^{2}} + \frac{d x}{d z} \frac{d T}{d z} - \frac{d^{2}}{d z^{2}} (E I \frac{d^{2} x}{d z^{2}}) + f_{x} = m \ddot{x} + D {}_{s}{\dot{x}}, T \frac{d^{2} y}{d z^{2}} + \frac{d y}{d z} \frac{d T}{d z} - \frac{d^{2}}{d z^{2}} (E I \frac{d^{2} y}{d z^{2}}) + f_{y} = m y + D {}_{s}y,$ where z is along the axial direction of the undistorted riser in calm water, while x and y denote the in-line and cross-flow

Numerical scheme stability check

The stability of the numerical scheme is checked through the von Neumann method. The stability check considers only the finite difference solution of the structural response, and does not include the fluid–structure interaction. With an initial error vector $ξ^{0}$ , the error distribution at time step n and node j is expressed as $ξ_{j}^{n} = {(G)}^{n} e^{i θ \cdot j}$ , where $θ = k π Δ x$ . Substituting in Eq. (7), we have $\frac{E I}{h^{4}} {(G)}^{n} e^{i θ (j - 2)} - (\frac{T_{j}}{h^{2}} - \frac{w_{j}}{2 h} + \frac{4 E I}{h^{4}}) {(G)}^{n} e^{i θ (j - 1)} + (\frac{2 T_{j}}{h^{2}} + \frac{6 E I}{h^{4}} + \frac{m}{τ^{2}} + \frac{D_{s}}{τ}) {(G)}^{n} e^{i θ j} - (\frac{T_{j}}{h^{2}} + \frac{w_{j}}{2 h} + \frac{4 E I}{h^{4}}) {(G)}^{n} e^{i θ (j + 1)} + \frac{E I}{h^{4}} ($

Numerical simulation approach

The flow field around a riser is calculated by numerically solving the unsteady Navier-Stokes equations with a Smagorinsky subgrid-scale turbulence model. The flow field solver is a Finite-Analytic Navier-Stokes (FANS) code that has been validated for Reynolds numbers up to 1×10⁷ (Pontaza et al., 2005c). A detailed description of the governing equations can be found in our previous papers (Chen et al., 2006, Huang and Chen, 2006, Huang et al., 2009a). The governing equations were transformed

Discussion and conclusions

In this paper a riser motion solver has been developed and applied to long riser VIV 3-D simulations. The governing tensioned beam equations are discretized by a finite difference numerical scheme, which is consistent and unconditionally stable for these equations alone. Its stability was confirmed using the von Neumann criteria. This riser motion solver was integrated with a finite analytical Navier-Stokes code to simulate long riser VIV. Three different risers with L/D ranging from 482 to

Acknowledgements

The project was sponsored in part by the U. S. Department of Interior, Minerals Management Service (MMS) and in part by the industry funding through the Offshore Technology Research Center (OTRC). The support and input from Dr. Richard Mercier, Director of OTRC, is also greatly appreciated.

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Numerical scheme for riser motion calculation during 3-D VIV simulation

Abstract

Highlights

Introduction

Section snippets

Riser motion equations

Numerical scheme stability check

Numerical simulation approach

Discussion and conclusions

Acknowledgements

Journal of Computational Physics

Journal of Fluids and Structures

Journal of Fluids and Structures

Journal of Fluids and Structures

Journal of Fluids and Structures

Journal of Fluids and Structures

European Journal of Mechanics B—Fluids

Journal of Fluids and Structures

Journal of Fluids and Structures

Journal of Fluids and Structures

Journal of Fluids and Structures

European Journal of Mechanics B/Fluids

Journal of Fluids and Structures

European Journal of Mechanics B/Fluids

European Journal of Mechanics B—Fluids

Journal of Fluids and Structures

Journal of Computers and Fluids

Flow-Induced Vibrations

Near-wall turbulence models for complex flows including separation

AIAA Journal