Ph.D. project description
Linear model for vortex-induced vibrations of slender structures

This text gives some details on the research that I conducted in my Ph.D. The general purpose of my work was to show that the major aspects of VIV for slender structures in both uniform and non uniform flows can be understood and reproduced qualitatively by using the concept of wake oscillator and linear stability. This work was done at both Ecole Polytechnique in France in the hydrodynamic laboratory (LadHyX) under the supervision of prof. Emmanuel de Langre and at the French Petroleum Institute (IFP). This work was financed by the French Petroleum Institute (IFP). The results of this research is an interesting model for early design stage. Its should be use to get insights into the main dynamics of complex configurations. Modification on the structure configuration can thus be made efficiently, the cost of computation being remarkably low. This text summarises quickly some important points concerning my thesis. For more detail, the reader can refer to this pdf containing a paper that we wrote about this subject and that is currently under consideration for publication at the Journal of Fluid Mechanics.


The flow profile behind a fixed cylinder in a uniform cross-flow is unstable past a certain critical Reynolds number. This instability takes the form of periodic shedding of vortices that are advected away from the bluff body (sketched on Figure 1).

Figure 1 - Vortex shedding
This alternating vortex shedding produces periodic pressure fluctuations with predominant component in the cross-flow direction (Y direction on Fig. 1). The structure, if flexible, undergoes what is commonly called vortex-induced vibrations (VIV). For velocities in the lock-in range (lock-in range being the range of velocities for which the shedding frequency deviates from the Strouhal law and "locks-on" to the cylinder vibration frequency), the amplitude of vibrations of the structure can attain important values, typically of the order of the diameter of its cross section.

The high VIV amplitudes are a major concern in regards to the service life of offshore structures that are connecting the seabed to the sea surface, such as risers of mooring cables (Figure 2). Sustained VIV cause material fatigue and lock-in phenomena implies a higher drag force on the structure, causing structure displacements and higher attachment load at the oil rig.

Figure 1 - Risers exposed to complex sea currents.
In  my thesis, I have studied a simplified version of a wake-oscillator model in order to bring some insights to complex VIV features for slender structures. The basic idea being wake oscillator models is that part of the wake is modeled using a mechanical oscillator (usally one that includes non-linearities, such as the van der Pol equation). This is shown on Figure 3 for an elastically supported rigid cylinder under upcoming uniform flow. Following Facchinetti et al. 2004, the cylinder forces the wake by its acceleration.
Figure 3 - Wake oscillator principle.
A classical modal analysis of the simplest form of the model presented above is done in order to identify the elementary mechanism that allows it to reproduce VIV physics. In that goal, the nonlinearities are left out. A typical results of such analysis is showed on Figure 4. It can be seen that for low and high velocities, the system possess two stable frequencies, one for which the movement of the cylinder is dominant (noted C) and the other for which the wake part is dominant (noted W). However, the two frequencies merge for a range of flow velocities (noted CMF) leading to a positive growth rate: the resulting mode amplitude grows exponentially. The main characteristics of lock-in (beside the amplitudes) can be recover by this simple analytical development. Lock-in of the wake with the cylinder can be interpreted as a linear coupled-mode instability between the wake oscillator and the cylinder itself. For more detail, see de Langre 2006.
Figure 4 - Results of modal analysis as a function of the reduced velocity, frequency (left) and growth rate (right). 
We have perform a stability analysis of the same fashion on the case of flexible structures, such as tension cables or beams subjected to uniform flow fields. Figure 5 shows, as an example, the evolution of the frequency and growth rate of the two first modes of a tension cable for heavy and light structural density (the notion of light and heavy being here in comparison to the fluid density). The results shown here (intability ranges, frequency evolution and growth rate evolution) are obtained by simple analytical calculations. For the light structure, the theory predicts an overlap of the range of instability of both mode. This result shows that at a given uniform flow velocity, more than one frequency is unstable, and  thus grows in amplitude with time. As general results, we have found that a given mode lock-in range corresponds to the range of velocities where the latter is the most unstable. Similar linear modes growth rate leads to strong modulation of the saturated response (phenomenom called mode switching in Chaplin et al. 2005 and time sharing in Swithenbanks 2007).
Figure 5 - Frequency (top images) and growth rate (lower images) evolution with flow velocity for the two first mode of a tension cable in the case of high structural mass (left figures) and low structural mass (right figures). Mode 1 in solide line and Mode 2 in dashed line.
In non-uniform flows, the generic forms of response were identified using a simple non-uniform flow configuration that is composed of an infinite tension cable subjected to two uniform flow profiles (as shown on Figure 6). Those forms are called here Primary Wave System (P) and Secondary Wave System (P). The two forms display similarities concerning their shape: they are both charaterised by a wave whose amplitude is modulated in space in one part of the spatial domain and is exponentially decreasing in the other part. This wave is travelling downward for (P) and upward for (S).     
Figure 6 - Generic forms of response for non-uniform flows: Primary Wave System (P) and Secondary Wave System (S). For (P) and (S), the system response evolution with time and space (left) and the maximum amplitude envelop (right) are shown. The growth of amplitude in time has been removed.
Two important characteristics concerning the (P) and (S) systems need to be identified here. First the frequency of (P) is always greater than (S). This also results always in smaller wavelength for (P). Second, the growth rate of (P) is always greater than (S). This results in a region of high frequency expanding spatially over a low frequency region when the whole system responses to an initial condition. This is illustrated in Figure 7. Here, both wave system are given an equal initial amplitude at t = 0. The intersection of maximum amplitude is marked here by a green dot. This dot seperates both high and low frequency regions. As time goes on, this dot moves downward as the high frequency zone expands over the low frequency.   

Figure 7 - Seperation in space of the configuration dominant frequency. As time goes on, the high frequency domain expand over the low frequency.
The linear theory concerning the general form of response in non-uniform flows has been compared against numerical computation by Lucor et al. 2006. Those authors performed tree-dimensionnal DNS computations of the flow behind a very slender tensioned beam. In their computations, the structure forcing on the fluid is taken into account, the movement of the structure and the wake evolution being coupled. They studied two flow profiles, Case L and Case E on Figure 8. On the latter figure, a comparison between the linear theory and the DNS compuration is shown. For the DNS, the space and time evolution of the beam displacement is shown. As for the linear theory prediction, the superposition of the (P) and (S) modes is shown. For Case L, there is a good qualitative agreement between the DNS and the linear theory prediction concerning the (P) mode. However, no (S) mode is observed in the DNS. In Case E, in addition to the waves with high frequency and short wavelength similar to linear mode (P) that are still found in the DNS, low frequency waves travelling from bottom to top are found. Those low frequency waves share some similarities with the linear (S) mode. When computing the time linear mode (P) takes to overun the entire domain, we find that this time is much longer for Case E than Case L, providing some explanation on why a low frequency is persisting for Case E and not for Case L.

The linear computation is orders of magnitude lower in computationnal cost than the DNS in this case. Those linear mode computations take about only a couple of minutes on a standard office computer. Although the results do not inform us on the amplitude, the knowledge of the principal dynamical characteristics, i.e. dominant frequencies, wavelengths, linear growth rates, modes configuration in space, can help when performing a trial and error design. Those principal characteristics are quickly obtained by the method proposed here. 


Figure 8 - Comparison between linear theory prediction and DNS computations by Lucor et al. 2006.
References:

Coupling of structure and wake oscillators in vortex-induced vibrations.
Facchinetti M.L., de Langre E., Biolley F., Journal of Fluids and Structures, v. 19, pp. 123-140, 2004

Frequency Lock-in is caused by a coupled-mode flutter.
de Langre E., Journal of Fluids and Structures, v 22, pp.783-791, 2006

Riser modal identification in CFD and full-scale experiments.
Lucor D., Mukundan H. and Tryantafyllou M.S. Journal of Fluids and Structures, 22, pp. 905-917, 2006

Laboratory measurements of vortex-induced vibrations of a vertical tension riser in a stepped current. 
Chaplin et al., Journal of Fluid and Structures, 21, pp.3-24, 2005

Dynamics of long flexible cylinders at high-mode number in uniform and sheared flows.
Swithenbanks S.D., Ph.D. thesis, Massachusetts Insta 

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