A time domain model for prediction of cross-flow vortex-induced vibrations (VIV) of slender structures with circular cross section has been under development since 2012. As an extension of this work, a time domain model for pure in-line VIV is here proposed, with the same underlying theory. The in-line force model, consisting of added mass, damping and excitation, is based on empirical data from forced oscillation tests of rigid cylinders. Damping and excitation is tuned to give the best fit of the excitation force coefficient calculated from experiments, whereas a strip theory approach is utilized to determine the force is phase with cylinder acceleration, i.e. added mass. The excitation force model represents the time varying drag force induced by alternating vortex shedding, and consists of two frequency-regions with positive excitation. Within these regions the excitation force is able to synchronize with the response vibrations, so that energy is transferred to the cylinder. Numerical simulations are performed to compare the present model with experimental results of free oscillations of rigid and flexible pipes with circular cross section, in uniform current. For the flexible cylinder case, a simple linear finite element structural model is combined with the in-line force model. The numerical simulations and the experiments are seen to match fairly well, both concerning frequency content, amplitude ratio and dominating vibration mode. Some discrepancies are observed, mostly concerning amplitude ratio. However, due to the complexity of VIV as a phenomenon, and the simplicity of the present model, it is concluded that the results are satisfactory. Consequently, this paper shows that the original idea of synchronization between excitation force and cylinder response is seen to work, not only for cross-flow VIV, but for pure in-line VIV as well.

Semi-empirical models are commonly used to predict vortex-induced vibrations (VIV). Empirical parameters are often based on forced harmonic motion tests of short rigid cylinders in one degree of freedom, which is a significant simplification of the three dimensional VIV experienced by flexible cylinders. Still, the models may provide satisfactory estimates of the response of slender marine structures exposed to current. At least in terms of root mean square (r.m.s.) of displacements and strains, dominating frequency and dominating mode. However, VIV of long slender beams, especially in sheared current, show large response variability in time and space, which appears to be of a random nature. Semi-empirical models that predict steady-state VIV are hence unable to reflect the non-stationary response observed in experiments, even though the time and space averaged result might be well represented.In this work, a previously published semi-empirical time domain model for cross-flow vortex-induced vibrations is modified to describe the stochastic nature of the response of long slender beams subjected to stationary current. The mean non-dimensional frequency of the synchronization model (previously constant) is taken to be a simplified Gaussian process, where standard deviation and spectral frequencies are input. The stochastic synchronization model allows the response to jump between different eigenfrequencies and corresponding modes, and still predict mean and r.m.s. values close to the deterministic model. Its performance is verified through simulation of an experiment with a long riser in sheared flow. The response sensitivity of standard deviation and spectral frequencies is investigated. It indicates that for a proper choice of empirical coefficients, the chaotic response of the riser can be quite realistically simulated in terms of frequency variation and, to some extent, amplitude modulation.

A semi-empirical prediction tool for pure in-line vortex-induced vibrations is under development. The long-term goal is to be able to realistically model the dynamic behavior of free spanning pipelines exposed to arbitrary time dependent external flows at low velocities. Most VIV programs operate in frequency domain, where only steady currents and linear structural models can be simulated. In contrast, the proposed model predicts hydrodynamic forces as function of time, enabling a time integration scheme to solve the equation of motion. Non-linear time domain simulations allow for modelling of excitation from non-steady currents. In addition, non-linear effects such as soil-pipe interaction, varying tension, and response dependent material, stiffness and damping properties may be included in the analysis, when combining the hydrodynamic force model with a structural non-linear finite element model. Hydrodynamically, the proposed prediction tool consists of the general Morison equation plus two vortex shedding forcing terms. The latter two are able to synchronize with the structural motion for a given frequency band, to induce vibrations in lock-in regimes. In this paper, the proposed pure in-line VIV model is compared to the frequency domain model VIVANA and DNV Recommended Practice, simulating experiments with a model-scale flexible pipe exposed to current velocities at which cross-flow vibrations have not yet developed. A few experimental data points are included in verifying the performance of the newly developed time domain model. The effect of changing empirical coefficients in the vortex shedding forcing terms, and allowing only one of the terms to excite structural vibrations during a simulation, is numerically investigated. A goal is to obtain increased understanding of how the proposed time domain model performs when simulating VIV of a flexible pipe, which is more complex than that of an elastically mounted rigid cylinder since several natural frequencies and corresponding modes might be excited.

A semi-empirical time domain force model for combined cross-flow and in-line vortex-induced vibrations (VIV) is proposed, based on a series of earlier publications. The new feature is a term which represents the effect of vortex shedding in the flow direction, referred to as the in-line vortex shedding load. The latter is added to Morison's equation and a cross-flow vortex shedding force. The fundamental idea of how to model the effect of vortex shedding is the same as in previous works. An algorithm for synchronization is applied between the vortex shedding loading terms themselves and the structural response, which excites vibrations for a pre-determined frequency interval. The originality of the present study is rather how the VIV-terms are integrated as part of Morison's equation, which is there to provide a description of inertia and drag forces. All the loading terms combined enables simultaneous simulation of VIV and other response phenomena, such as wave induced motion and static drag displacement.The performance of the time domain model is verified against measurements of a vertical riser subjected to two different external flow cases. The first one is simply steady uniform current whereas the second flow case combines the latter with irregular waves. To simulate the experiments, a linear finite element model of the riser is made, and the proposed hydrodynamic force model is applied to the translation degrees of freedom along the structure. It is to be noted that the same empirical coefficients are used to simulate both experiments. In uniform flow, the results are good. Dominating frequency and vibration amplitude agree well in both cross-flow and in-line directions. The simulated time series show more regular/less tendency of amplitude modulations than the experiments, but the overall agreement is still acceptable for engineering purposes. For the second flow case, where the riser was towed in irregular waves, the model provides a highly realistic representation of the riser motion. Both when the total response (containing wave and VIV frequencies) and the filtered signal (including only VIV frequencies) are analysed, the predictions follow the measurements closely. From before, the cross-flow part of the VIV model has been tested in oscillating flow and regular waves, and the performance of the former was experimentally proven. It is hence concluded that the proposed prediction tool is applicable to a variety of non-stationary flow conditions, implying that the synchronization model captures parts of the underlying physics, despite its simple form.

A promising time domain model for calculation of cross-flow vortex induced vibrations (VIV) is under development at the Norwegian University of Science and Technology. Time domain, as oppose to frequency domain, makes it possible to include non-linearities in the structural model. Pipelines that rest on an irregular seabed will experience free spans. In these areas VIV is a concern with respect to the fatigue life. In this paper, a time domain model for calculation of VIV on free spanning pipelines is proposed. The model has non-linear interaction properties consisting of discrete soil dampers and soil springs turning on or off depending on the pipeline response. The non-linear model is compared to two linear models with linear stiffness and damping properties. One linear model is based on the promising time domain VIV model, while the other one is based on RIFLEX and VIVANA, which calculates VIV in frequency domain. Through four case studies the effect of seabed geometry, current velocity and varying soil damping and soil stiffness is investigated for a specific pipeline. The results show that there is good agreement between the results produced by VIVANA and the linear model. The non-linear model predicts smaller stresses at the pipe shoulders, which is positive for the life time estimations. Soil damping does not influence the response significantly.