Simulations of vortex-induced vibrations of long cylinders with two degrees of freedom

摘要

Long cylindrical risers are required for deep water exploration and production of petroleum or natural gas. The flow of seawater around these long cylinders is subject to vortex shedding. This is an unsteady oscillatory phenomenon, which causes the pressure distribution around the cylinders to fluctuate. If the vortex shedding frequency is equal to a natural frequency of the riser, then the vortex shedding will induce the riser to vibrate. These are known are Vortex Induced Vibrations or VIV. These vibrations cause premature fatigue or clashing between neighbouring risers. In the current contribution we carry out an unsteady two-dimensional numerical simulation of VIV. The numerical algorithm incorporates several desirable features for such simulations, namely, it uses an adaptive non-isotropic Cartesian grid and the Immersed Boundary Method for boundary condition specification around the cylinder. The current simulations use LES with a Smagorinsky model to calculate the effective viscosity. Its main advantage, however, is the ability to easily handle flows with moving boundaries. The cylinder is assumed to be 1800 m long with a diameter of 0.25 m and subjected to traction force of 10~6N, with a flow Reynolds number of 8640. The cylinder vibration is assumed to lock in to the 2nd natural mode with a frequency of 0.0473 Hz. At each time step the flow velocity and pressure distributions are calculated. The pressure distribution around the cylinder is used to calculate the drag and lift coefficients. This information is then used to solve two 2nd order simple harmonic motion ODEs, which give the velocity and displacement of the cylinder in cross flow and stream-wise directions. This information is used to update the position of the cylinder and its velocity. Most results available in the literature for cylinders subjected to vortex induced forces are limited to either stationary or one degree of freedom (usually in the cross flow direction). In the current study we compare the results for a rigid cylinder, a cylinder with one degree of freedom and two degrees of freedom. For the moving cylinders three mass damping factors are considered. The final results show the effect of the stream-wise motion on the vortex shedding pattern, the cross-stream vibration amplitude, the drag coefficient and the lift coefficient.