Vortex Strength Calculations

This article provides an overview of how the vortex strength is calculated for all the elements defined in the vortex wake. The vortex strength is a key input into the calculation of flow induction using the Biot-Savart law.

Solving for the Bound Vortex Strength at each Wake Step

At each wake step, the bound vortex strength distribution on the blade is unknown and needs to be solved. The lift force is a function of the bound vortex strength, but also of the angle of attack, which in turn depends on the bound vortex strength and its resulting induced velocity. As with a blade element momentum (BEM) method, iteration is required to solve for the unknown bound vortex strength distribution. This can not be done for each blade station individually as blade stations are not independent.

The calculation relies on discretising the blades into a series of blade elements similar to the traditional BEM method. The calculation of angle attack \(\alpha\) depends on the aerofoil chordline position and flow properties such as the blade section structural velocity \(\bvector{V}_s\), free stream wind speed \(\bvector{V}_0\), the induction \(\bvector{v}\). The resultant wind velocity accounting for induction \(\bvector{W}\) are computed in the same manner for both the vortex wake and BEM method and shown in Figure 1.

Vortex sheet — Figure 1: Flow components used for calculation of angle of attack.

The following iterative scheme is applied:

Assume an initial distribution of the vortex strength \(\Gamma_i^{(t)}\) for all bound vortex elements. This is typically the solution from the previous time step.
Compute the strength of the first trailing vortex elements and the shed elements at the trailing edge.
Calculate the total flow velocity \(\bvector{W}\) at the centre of each blade section. Note here that \(\bvector{v}_{\text{induced,blade}}\) is the induced velocity due to the mutual induction of the blades and the current estimate of the bound vortex strength. The total induction due to the vortex wake at a blade section is \(\bvector{v} = \bvector{v}_{\text{wake}} + \bvector{v}_{\text{induced,blade}}\). A relaxation factor \(\beta\) is applied to the induced velocity to ensure a smooth solution.

\[ \begin{equation} \bvector{W} = \bvector{V}_{0} + \bvector{V}_{s} + ( \bvector{v}^{(t)} + \beta ( \bvector{v}^{(t)} + \bvector{v}^{(t-1)})) \end{equation} \]

Compute the angle of attack \(\alpha\) for each blade section and lookup the corresponding lift coefficient using the appropriate interpolated aerofoil data
Use Kutta-Joukowski theorem to compute a new estimate for the vortex strength:

\[ \begin{flalign} \rho\left| \bvector{W} \right|\Gamma^{(t)} = \frac{1}{2}\rho\left| \bvector{W} \right|^2 C_l(\alpha)c(r) \\[1ex] \Gamma^{(t)} = \frac{1}{2}\left| \bvector{W} \right|C_l(\alpha)c(r) \end{flalign} \]

Use a relaxation scheme to compute a new estimate of the bound vortex strength; this is done for numerical stability. The relaxation factor \(D\) is typically equal to 0.5. However, in case the convergence fails, a new attempt will be made with a lower relaxation factor.

\[ \begin{equation} \Gamma^{(t)} = \Gamma^{(t)} + D(\Gamma^{(t)} - \Gamma^{(t-1)}) \end{equation} \]

Continue the iteration until convergence is achieved.

Vortex Wake Element Strengths

Mathematically, the strength of the newly released trailing and shed elements can be computed as a function of the current bound vortex strength for each radial location along the blade span as follows

\[ \begin{flalign} \Gamma_{\text{shed}}\left( r_i, t \right) = \Gamma_{\text{bound}}(r,t - \Delta t) - \Gamma_{\text{bound}}(r,t) \label{eqn:shedVortexStrength} \\[1ex] \Gamma_{\text{trail}}\left( r_i, t \right) = \Gamma_{\text{bound}}(r,t) - \Gamma_{\text{bound}}(r + \Delta r,t). \label{eqn:trailVortexStrength} \end{flalign} \]

Note

The shed vortex strength is a function of the current and previous bound vortex strength. This effectively stores the time history of the bound vortex strength in the wake. This is equivalent to the attached flow states used in the Beddoes-Leishman style dynamic stall model that are used to compute an effective angle of attack that includes the effect of shed vorticity (Theodorsen theory). Therefore, these attached flow states are automatically turned off in case the Theodorsen theory is used for the vortex wake model as this effect is inherently included in the model. The Theodorsen theory is included in some dynamic stall models such as the Incompressible Beddoes-Leishman, Compressible Beddoes-Leishman, Pre4.8 Beddoes-Leishman and IAG dynamic stall models.

At the root and tip of the blade, the trailing vortex strength is computed using the first/last blade element only, such that

\[ \begin{flalign} \Gamma_{\text{trail}}\left( - \frac{b}{2}, t \right) = - \Gamma_{\text{bound}}( - b/2,t) \label{eqn:rootVortexStrength} \\[1ex] \Gamma_{\text{trail}}\left( \frac{b}{2}, t \right) = \Gamma_{\text{bound}}(b/2,t). \label{eqn:tipVortexStrength} \end{flalign} \]

Time-Dependent Viscous Vortex Core Models

A model is needed that determines the viscous core size for each element as a function of time. These models are well documented in (Leishman, 2006), [(Dixon,, 2008)] and [(Sant 2007)]. In Bladed, the model by (Ramasamy, 2007) is selected. Here the vortex core size (\(r_c\)) is computed as a function of time

\[ \begin{equation} r_c = \sqrt{r_0^2 + 4\alpha\nu\left( 1 + a_1 Re_v \right)t } \end{equation} \]

where:

\(r_0\) initial core size at \(t\)=0
\(\alpha\), laminar vortex core constant \(\alpha = 1.25643\)
\(\nu\), kinematic fluid viscosity
\(a_{1} = 6.5e - 5\)
\(Re_{\nu} = \frac{\Gamma}{\nu}\)

High Angle of Attack Treatment

When the angle of attack (\(\alpha\)) at the blade sections become extremely high, the solutions of vortex wake theory may become unstable and can lead to non-physical results. To avoid this issue, and due to consideration that strip theory may be applied for a fully perpendicular flow (\(\alpha \approx 90\degree{}\)), the vortex wake solutions are damped by applying a damping factor (\(\phi\)) starting from \(45 \degree{} \leq |\alpha| \leq 60 \degree{}\) and between \(120 \degree{} \leq |\alpha| \leq 135 \degree{}\), see Figure 2. This approach is similar to damping applied for the IAG dynamic stall model to treat the extremely high \(\alpha\) characteristics.