Introduction to Lifting Line Theory

The first practical theory for predicting the aerodynamic properties of a finite length blade was developed by Ludwig Prandtl (see (Anderson, 2001)). The key assumption is that a finite blade can be represented as a lifting line where the lift force is a direct function of the circulation strength of the bound vortex (see (Anderson, 2001)). The lift per unit length on a blade is traditionally computed based on the lift coefficient $C_L$ using the equation

\[ \begin{equation} L = \frac{1}{2}\rho V_0^2 c C_L \label{eqn:classiclift} \end{equation} \]

where $\rho$ is the air density, $V_{\infty}$ the local inflow wind speed that the blade experiences and $c$ the chord length of the aerofoil. However, there also exists a relationship between the bound vortex of strength $\Gamma_i$ of the $i$th lifting line (or vortex tube) that induces a lift force $L$ given by the Kutta-Joukowski theorem

\[ \begin{equation} L = \rho V_0\Gamma_i. \label{eqn:kuttajoukowski} \end{equation} \]

Based on Helmholtz’ theorem a vortex tube cannot start or end in a flow (Saffman, 1992). Therefore, the bound vortex has trailing vortices that extend infinitely far downwind of the blade, which leads to the well-known horseshoe vortex representation for a finite blade of length $b$.

In reality, the bound vortex strength on a blade is not constant as the incoming wind speed and blade geometry are not uniform across the span of the blade. Therefore, the single horseshoe vortex is replaced with a superposition of multiple horseshoe vortices which then capture the continuously varying bound vortex strength $\Gamma(y)$ as shown in Figure 1.

Finite wind and horseshoe vortices — Figure 1: Representation of lift and drag forces on a blade using a superposition of 5 horseshoe vortices of varying strength $\Gamma_i$ resulting in spanwise bound vortex strength $\Gamma(y)$.

Vortex Elements and the Biot-Savart Law

Each horseshoe vortex emanating from the lifting line will result in an induced velocity $\bvector{v}_{\text{induced}}(t) = \bvector{v}_{\text{induced}}(x,y,z,t)$ throughout the flow field or computational domain. To compute the free wake lifting line solution numerically it is convenient to express the horseshoe vortex structures as a series of interconnected vortex elements. Then the contribution to the induced velocity can be computed per element and summed across all elements. For a time varying solution vortex elements will also be used to represent shed vortices as well. Each vortex element in the wake (trailing and shed) will influence the flow field. This computation needs to be repeated at each time step to compute the time-varying induced velocity.

The start and end point of each vortex element is denoted by the points A and B respectively. The induced velocity due to this single vortex element will be computed at an arbitrary point C. The position of A and B relative to C in 3 dimensional space is denoted by the vectors $\bvector{r}_1$ and $\bvector{r}_2$ respectively as shown in Figure 2. The vector along the vortex element from A to B is represented by $\bvector{r}_2 - \bvector{r}_1$. The length of each vector is written: $r_1 = |\bvector{r}_1|$, $r_2 = |\bvector{r}_2|$ and $L = |\bvector{r}_2 - \bvector{r}_1|$.

The flow induced on point C due to a vortex element A-B is computed using the Biot-Savart law (Anderson, 2001) and can be expressed mathematically as follows

\[ \begin{equation} \bvector{v}_{\text{induced}} = \frac{\Gamma}{4\pi} \frac{ ( r_1 + r_2 ) ( \bvector{r}_1 \times \bvector{r}_2 ) }{ r_1 r_2 ( r_1 r_2 + ( \bvector{r}_1 \cdot \bvector{r}_2 ) )}. \end{equation} \]

The above equation contains a singularity in case $\bvector{r}_1$ or $\bvector{r}_2$ tends to zero. This singularity is removed by introducing a viscous core model as defined by (Leishman, 2006). The Biot-Savart equation is now modified to

\[ \begin{equation} \bvector{v}_{\text{induced}} = K_v \frac{ \Gamma}{4\pi} \frac{ ( r_1 + r_2 ) ( \bvector{r}_1 \times \bvector{r}_2 ) }{ r_1 r_2 ( r_1 r_2 + ( \bvector{r}_1 \cdot \bvector{r}_2 ) + (\delta \cdot L)^{2} )} \end{equation} \]

which include the viscous core model according to Lamb-Oseen (Leishman, 2006) to define $K_v$ and a cut-off radius from (van Garrel, 2003) to define $\delta$. The equation $K_v$ is written as follows

\[ \begin{equation} K_{v} = \frac{h^{2}}{\sqrt{r_{c}^{4} + h^{4}}}. \end{equation} \]

The parameters governing $K_v$ are defined as follows:

$h$, is the normal distance from the vortex element to the point where induction is computed (point C).
$r_c$ is the viscous core radius.
$\delta \cdot L$ is a small fraction of the vortex element length introduced for numerical stability where $\delta = 0.001$.

In reality this calculation must be completed for all vortex elements in the vortex wake.

Last updated 30-08-2024

Table of Contents

Introduction to Lifting Line Theory

Vortex Elements and the Biot-Savart Law