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Vortex Induced vibrations

The flow behind a cylinder becomes unstable as the velocity of the fluid increases. Vortices are shed from the structure with a characteristic frequency, the so-called Strouhal frequency. If the cylinder is flexible, its eigenmode can even influence the shedding frequency, leading to the lock-in phenomenon. In the more complex case of a sheared flow along the structure, various modes are potentially excited, and they have to be taken into account to obtain the response of the structure.

Formulation

We are interested in the crossflow response of a cylinder to the vortex shedding as indicated in the following figure.

The non-dimensional quantities used to describe the phenomenon are:

  • Reynolds number: \(R_e = \frac {UD}{\nu}\)

  • Strouhal number: \(S_t = \frac {f_sD}{U}\)

  • Reduced velocity: \(V_r = \frac {U}{f_iD}\)

Where U is the flow velocity perpendicular to the structure, D the diameter, \(\nu\) the kinematic viscosity, fs the shedding frequency and \(f_i\) the \(i^{th}\) mode of the structure.

A modal approach is used to compute the response of the cylinder. The steps for the calculations are:

  1. Calculations of the mode of the structure taking into account the added mass coefficient.

  2. Selection of the potentially excited modes with two criteria:

    The mode is mainly perpendicular to the flow and the structure (shape)

    The frequency \(f\) satisfy the reduced velocity criteria: given \(S_t\) , the reduced velocity corresponding to \(S_t\) is \(V_r = 1/S_t\) , and the mode is excited if its frequency is located between \(V_r - \Delta V_{min}\) and \(V_r - \Delta V_{max}\).

  3. At each point along the structure, one mode if any, is considered excited. The lift force is then applied on the structure. The hydrodynamic damping balances that force, and by iteration, the proper amplitude is obtained. The response of the structure to a given excitation frequency is computed. The hydrodynamic damping is based on a Morison formulation. The lift force is function of both the amplitude and the Reynolds number. The lift coefficient is described as:

    \[ C_l(R_e,A/D ) = g(R_e)C_{l_r}(A/D) \]

  4. The total response of the structure is the sum over all the excited frequencies.