|
Vortex tracking models |
Two vortex tracking models are available, which we refer to by number: vortex tracking (1) and vortex tracking (2). Both are based on the underlying physical equations of boundary layer theory and the Navier-Stokes equation. As a result they introduce physical realism that is absent from the wake oscillator models.
Vorticity is a measure of a fluid's rotation and it is often advantageous to analyse fluid dynamics in terms of vorticity. The reason for this is that the important fluid forces on a body in the flow are intimately related to the vorticity, and the vorticity is often confined to narrow sheet-like regions. Focussing solely on these regions is far more efficient from a computational point of view.
The vortex tracking models are much more computationally demanding than the wake oscillator models. In fact they are a type of computational fluid dynamics (CFD) model, but are much less computationally demanding than 'full' CFD. Work to date shows considerable promise, and we hope they will offer a practical analysis technique which gives much of the realism of full CFD without the associated extremely long run times.
Vortex tracking models represent the entire fluid flow field. In OrcaFlex, the vortex tracking models are used to give the force acting on the line, but other results can also be obtained from them, such as fluid velocity and pressure at any point – for example, we have used them experimentally to calculate the pressure variations on the line surface due to VIV. If other results such as these interest you then please contact Orcina for further details.
The vortex tracking models are based on the relative velocity of the flow past the line. They can therefore be used both for cases where the excitation is due to current or waves, and also where the excitation is due to the line moving, for example towed cases.
The models involve calculating and tracking many vortices for each node of the line. This can make the simulation file very large, but this can be controlled by limiting the maximum number of vortices logged.
| Note: | We have carried out validation work comparing the VIV models with real measured results – see the VIV validation section on the OrcaFlex validation page of the Orcina website. This work showed that the vortex tracking models tend to substantially over-predict VIV amplitudes. However we believe they are useful for qualitative investigations, especially of inline VIV in low to moderate shear conditions, since the other VIV models cannot model this situation. |
Here, we describes the basic vortex tracking idea on which both models are based.
Essentially, vortex tracking is a two-dimensional fluid model associated with a particular node on the line. It operates in the 2D 'slice' through the fluid normal to the line axis, which we call the vortex tracking plane. In this 2D slice the node is a disc.
To apply this 2D model to OrcaFlex's three dimensions, we attach a separate vortex tracking plane to each node on the line where VIV is enabled. Each node therefore handles its own fluid interaction, generating vortices, shedding them, and tracking them in its associated vortex tracking plane.
The drawing below shows a typical vortex plane, i.e. it is a cross section through the line, normal to the line axis. The line itself is represented by the grey disc and the fluid flow is coming from the left.
| Figure: | Vortex tracking plane |
The model has two main elements:
When the flow meets the disc it has to flow around the disc circumference and a boundary layer is formed. Boundary layer theory is used to model this region, where viscosity plays a crucial role. Sarpkaya and Shoaff originally used the Polhausen boundary layer method, but since then this method has been superseded by simpler and more accurate methods. OrcaFlex uses Thwaites' method (see Young 1989).
Some of the fluid flows around one side of the disc and some around the other, and the point where the flow splits is called the stagnation point. As the fluid flows around the disc it initially remains in contact with the disc, but it typically then reaches a point on each side where the flow separates. These are called the separation points, and at these points vorticity is shed from the disc.
The boundary layer theory gives the position of each separation point and the strength of vorticity shed there in one time step. A new vortex of this strength is then created at the separation point. The new vortex is placed at the separation point, but at a distance $\lambda r$ from the disc surface, where $r$ is the line radius. In our model 2, $\lambda$ is the user-defined creation clearance. In model 1, and in model 2 if the creation clearance is '~', $\lambda$ is calculated to be the value that results in the tangential velocity contribution of the new vortex just cancelling out the existing tangential velocity at the separation point.
In the drawing above (and in the OrcaFlex 3D view) the stagnation point is shown as a small triangle, and the separation points as small blobs part way around the disc circumference. The vorticity has different directions of rotation on each side of the disc: clockwise for vorticity shed from the upper side and anti-clockwise from the lower side, as seen in the drawing. The two are distinguished by being drawn (here and on the 3D view) in different colours.
After being shed from the boundary layer, the vorticity flows downstream. In reality, the vorticity is shed continuously and it is shed along the neighbouring parts of the line at the same time, so as it flows away it forms sheets of vorticity, one on each side. In the above drawing these vortex sheets are shown as red and green lines – lines because what the drawing shows is the intersection between the sheets and the vortex plane. The colour denotes the sign of the vorticity.
For computation purposes, the continuous vorticity must be represented by discrete vortex points. The vortex sheet is therefore represented by a sequence of vortex points, each one of which represents the vorticity of a short length of vortex sheet.
In the 3D view these individual vortex points are drawn as circles with their centres joined to represent the vortex sheet line, with the colour indicating the sign of vorticity. A vortex sheet therefore appears as a linked sequence of circles flowing away from the separation point.
Sarpkaya and Shoaff originally used singular vortex points, but in both of the OrcaFlex vortex tracking models we use smeared vortex points.
The vortex tracking models do not have any Strouhal number built in. Rather, the Strouhal period emerges directly from the physics of the vortex dynamics and boundary layer.