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Wake oscillator model

DeepVIV-Dyn is an optional module that calculates the Vortex Induced Vibrations of a line due to current loading using a time dependent wake oscillator. The module is fully integrated in DEEPLINES.

Wake Oscillator

DeepVIV-Dyn contains a time domain VIV model. The effect of the near wake on the structure is simulated with wake oscillators distributed along the line on each structural discretisation point. The dynamic of the near wake is described by a couple of real variables (q,p) in cross-flow and in-line directions (see Fig 1).

Fig 1 Wake Oscillators along a line

The model is based on the van der Pol equation ([1], [2]) which can be written as (for cross-flow direction for example):

\(\frac {\partial^2 q}{\partial t^2} + \varepsilon \omega_{\nu}(z)(q^2-1)\frac {\partial q}{\partial t}+ \omega_{\nu}^2(z)q - \nu \frac {\partial^3 q}{\partial t \partial z^2} - \chi \frac {\partial^2 q}{\partial z^2}=f(y)\) (Eq. 1)

The dimensionless variable q is associated to the fluctuating lift (drag for the in line vibrations) on the structure. The dimensional lift force can be written as:

\(F_{lift} = \frac 1 2 \rho U^2 D C_L\) (Eq. 2)

where rho is the fluid density, U the free stream velocity, D the pipe diameter and is the instantaneous sectional lift coefficient on the structure. The fluid variable is therefore interpreted as a reduced lift coefficient:

\(q=2C_L/C_{L_0}\) (Eq. 3)

where the lift coefficient \(C_{L_0}\) is that on a fixed structure subjected to vortex shedding. From a physical point of view, the ratio \(C_L/C_{L_0}=q/2\) therefore describes the lift amplification with respect to a fixed structure experiencing vortex shedding. \(\omega_{\nu}\) is the basic angular frequency of vortex shedding:

\(\omega_{\nu}(z)=2\pi S_t \frac{U(z)}{D(z)}\) (Eq. 4)

St being the Strouhal number. The parameter \(\varepsilon\) tunes the magnitude of non-linearity. The diffusion coefficient \(\nu\) and the stiffness coefficient \(\chi\) model the phenomenological interactions which characterize the tri-dimensional vortex shedding.

Similar oscillators are used to model the in-line vibration, the angular frequency in the in-line direction is taken equal to the double of those considered for the cross-flow vibrations. The parameters \(A_0, \varepsilon, \nu\) and \(\chi\) used for the in-line model are the same as those tuned for the cross-flow.

Coupling of Wake Oscillator and Structure Calculation

Several choices may be considered for the action of the structure on the fluid wake oscillator. The method chosen is based on a linear inertial effort of the structure on the fluid, namely an acceleration coupling which has been proposed by Facchinetti ([1], [2]):

\(f = A_0 \frac {\partial ^2 y}{\partial t^2}\) (Eq. 5)

In the wake oscillator dynamics, Eq 1, the van der Pol parameter \(\varepsilon\) and the scaling of the coupling force \(f\), namely \(A_0\) , Eq 5, need to be set. This has been done by analyzing the near wake dynamics in the experimental cases of Bishop & al [3] and more recent investigations on vortex shedding timing [4].

The ability of the acceleration coupling has been analytically and numerically analyzed. This coupling succeeds in modeling the oscillation amplitude at lock-in and the extension of lock-in area [2]. This coupling term is integrated in the module DeepVIV-Dyn of DeepLines.

Model parameters

  • Strouhal Number: the angular frequency of the oscillators is those of Strouhal of the vortex shedding. The Strouhal number of a cylinder is around 0.2 in a wide range of Reynolds Number (see Fig 2).

Fig 2 Strouhal number as a function of Reynolds number

  • Number of time step per wake period: The time step used by DeepVIV-Dyn is the same as those used by DeepLines to compute the dynamics of the structure. This time-step is used to solve the van der Pol equation which represents the dynamic of the near wake. It should be short compared to the expected period of vortex shedding, the Strouhal period defined by \(T_{S_t} = D/U S_t\) . A time step \(\Delta t = T_{s_t}/100\) is a good rule, but on occasion a shorter time-step is necessary for accuracy. A sensitivity analysis is recommended.

  • VIV direction: DeepVIV-Dyn allows to compute VIV in cross-flow and/or in-line direction(s). The value of this switch is 0 for cross-flow VIV solely and 1 for cross-flow and in-line VIV.

  • Lift coefficient: The lift coefficient \(C_{L_0}\) , corresponds to that on a fixed structure subjected to vortex shedding. The value depends on the Reynolds number. This lift coefficient is used for cross-flow VIV computation.

  • Drag coefficient: The drag coefficient \(C_{D_0}\) , corresponds to that on a fixed structure subjected to vortex shedding. The value depends on the Reynolds number (Fig 3). This drag coefficient is used for in-line VIV computation.

Fig 3 Drag coefficient as a function of the Reynolds number [6]

  • Scaling of coupling force and van der Pol \(\varepsilon\) parameter: These parameters allows to set the scaling of the coupling force and non linearity of the model. A detailed analysis in comparison with experimental data [2] showed that the optimum couple is \((A_0;\varepsilon) = (12;0.3)\).

  • Diffusion and Stiffness coefficient: These coefficients allows to set the interaction between wake oscillators distributed along the line in order to represent the 3D effects of the wake. The diffusion term reads :\(\nu = 1/\varepsilon\) (Eq. 6)

The stiffness coefficient is given by the expression: \(\lvert \chi \rvert = (2 \pi S_t)^2\) (Eq. 7)

Note

The default values of the wake oscillator are the "best fit" values as defined in the published references. To facilitate further research and possible re-calibration of the model, these values can be changed by the user. However, for a normal use according to our experience, we recommend keeping the default values.

References

[1] M.L. Facchinnetti, Un module phnomnologique des vibrations induites par détachement tourbillonnaire, PhD Thesis, Ecole Polytechnique, 2003.

[2] M.L. Facchinetti, E. De Langre and F. Biolley, Coupling of structure and wake oscillators in vortex induced vibrations, Journal of Fluids and Structures, Vol. 9, pp.123-140, 2004.

[3] R.E.D Bishop and A.Y Hassan, The lift and drag forces on a circular cylinder oscillating in a flowing fluid, Proceedings of the Royal Society of London series A, 277:51-75, 1964.

[4] X.Y Lu and C. Dalton, Calculation of the timing of vortex formation from an oscillating cylinder, Journal of Fluids and Structures, Vol. 10, pp.527-541, 1996.

[5] R. Blevins, Applied fluid dynamics handbook, Van Nostrand Reinhold, 1990.

[6] S. Etienne, Contribution la modlisation de l'coulement de fluide visqueux autour de faisceaux de cylindres circulaires, PhD Thesis, Université d'Aix- Marseille II, 1999.