Evolution of the Vortex Wake Geometry

This article describes how the vortex wake geometry is formed from a combination of shed and trailing vortices and how the wake node positions evolve in time.

Wake Geometry

For turbine rotor applications, the aerodynamic inflow properties typically vary along the blade span and in time. Therefore, the model of a horseshoe vortex with constant trailing vortex strength is extended. Figure 1 shows the use of a time-marching model where after each wake (time) step new trailing and shed vortex elements are introduced in the wake. Each element is connected to two vortex wake nodes. The vortex wake nodes are convected downwind by the local wind speed by the time-marching model. The local wind speed is a function of the free flow wind speed, the wake rotation and induction due to the vortex wake. Consequently, the length of each element will change in time.

The first set of shed vortex elements are bound to the blade quarter chord line and are therefore referred to as the bound vortex elements. The first set of trailing elements are always connecting the quarter chord point with the trailing edge. Finally, the second set of shed elements are connected to the trailing edge of the aerofoil. The bound vortex strength \(\Gamma(r)\) of the bound vortices at the quarter chord point is depicted in Figure 1 to illustrate the non-uniform nature of the flow field and blade geometry in the spanwise direction.

Figure 1: Schematic of blade and formation of wake geometry by addition of vortex elements representing the trailing and shed vortices in a free wake time marching lifting line model.

Initial Shed Node Position

The first set of wake nodes that are not connected to the blade are treated differently. The position of these nodes is rigidly attached at the 1/4 chord length behind the trailing edge. The wake node lies on a direction vector pointing from the quarter chord point to the trailing edge along the chord line Figure 2. The camber of the aerofoil is not included as typically the geometry of the aerofoil is not known.

Figure 2: Placement of first wake nodes relative to trailing edge.

Incrementing Wake Node Position in Time

After each vortex wake (time) step the positions of the free vortex wake nodes \(\bvector{x}\) are updated. The free vortex wake nodes do not include nodes attached to the blade or the initial shed node. The update of free wake nodes is done using a simple Euler forward method. The future, current and previous wake steps are denoted by the superscripts \((t+1)\), \((t)\) and \((t-1)\) respectively. The wake node is convected downwind using the following equation

\[ \begin{equation} \bvector{x}^{(t + 1)} = \bvector{x}^{(t)} + \frac{1}{2} \left\lbrack \bvector{v}^{(t)} + \bvector{v}^{(t - 1)} \right\rbrack \delta t \end{equation} \]

where

\[ \begin{equation} \bvector{v}^{(t)} = \bvector{v}_{\text{wind}} + \bvector{V}_{s}(t) + \bvector{v}_{\text{induced}}(t). \end{equation} \]

Only upflow and wind shear are included in the wind velocity vector \(\bvector{v}_{\text{wind}}\). In case a turbulent wind field is used, then the turbulent part of the wind is ignored. The induction \(\bvector{v}_{\text{induced}}(t)\) is updated until a user defined number of free wake steps \(N_w\) when the free wake is frozen. After this time the last computed induced velocity \(v^{(t-N_w)}\) is stored and no longer updated. A diagram demonstrating the convection of the trailing vortices from the blade tip is shown in Figure 3. The trailing vortex nodes (dissipate) are no longer convected after \(N\) vortex wake steps.

The velocity \(\bvector{V}_{s}(t)\) is the velocity tangential to the circular path swept out by the blade section from which the wake node was originally released. This gives the trailing vortex wake a helical shape.

Figure 3: Illustration of different stages of a helical trailing vortex shed from the blade tip and the velocity vectors used to convect the wake nodes. The wind velocity used to convect the wake node is computed once and then remains fixed per wake node. The induction calculation is not updated at a wake node after a defined number of free wake steps.

Last updated 12-09-2024