In the case of flow-induced vibration, the equation of motion of the cable for its two directions of motion (i.e. in the x and y-directions) is given by a slightly modified forced wave equation:
where ξ(z,t) = (ζ(z,t), η(z,t)) gives the cable displacement in the streamwise and crossflow directions and gives the phase speed of waves in the cable. The cable has mass per unit length m and tension T. To maintain a mean displacement, the cable is lightly elastically supported by linear springs with spring constant k, giving a natural frequency of The spring constant is selected sufficiently small to have negligible effect on the cable response. The fluid force on the cable is denoted by F(z, t). The components of F(z, t) in the streamwise and crossflow directions are the drag and lift force on the cable. Internal damping is neglected here as it does not significantly influence the response.
To simplify the solution of the fluid equations we use a coordinate system attached to the cable. This maps the time-dependent and deforming problem domain to a stationary and non-deforming one as shown in Figure 1. This mapping is described by the following transformation:
Accordingly, the velocity components and pressure are transformed as follows:
The Navier-Stokes equation and continuity equation are transformed to
where the forcing term A(u,p,ξ) is the extra acceleration introduced by the transformation, consisting of both inviscid and viscous contributions.
To solve the three-dimensional Navier-Stokes equations, we use a parallel spectral element/Fourier method . Spectral elements are used to discretize the x-y planes, while a Fourier expansion is used in the z-direction (i.e. along the cable). Consequently, all flow and cable variables are assumed to be periodic in the spanwise direction. The computational domain extends 35d (cable diameters) downstream, and 15d above and below the cable. The “cable span”, i.e. wavelength of vibrations in the cable was L/d=12.6 for the 3-d simulations. Each x-y plane is discretized by a 110 element mesh, with each element