Vortex lattice method

The following assumptions are made regarding the problem in the vortex lattice method:

• The flow field is incompressible, inviscid and irrotational. However, small-disturbance subsonic compressible flow can be modeled if the general 3D Prandtl-Glauert transformation is incorporated into the method.
• The lifting surfaces are thin. The influence of thickness on aerodynamic forces are neglected.
• The angle of attack and the angle of sideslip are both small, small angle approximation.

By the above assumptions the flowfield is Conservative vector field, which means that there exists a perturbation velocity potential ${\displaystyle \varphi }$ such that the total velocity vector ${\displaystyle \mathbf {V} }$ is given by

${\displaystyle \mathbf {V} =\mathbf {V} _{\infty }+\nabla \varphi }$

and that ${\displaystyle \varphi }$ satisfies Laplace's equation.

Laplace’s equation is a second order linear equation, and being so it is subject to the principle of superposition. Which means that if ${\displaystyle \varphi _{1}}$ and ${\displaystyle \varphi _{2}}$ are two solutions of the linear differential equation, then the linear combination ${\displaystyle c_{1}\varphi _{1}+c_{2}\varphi _{2}}$ is also a solution for any values of the constants ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$. As Anderson[7] put it "A complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows, which are also irrotational and incompressible.”. Such elementary flows are the point source or sink, the doublet and the vortex line, each being a solution of Laplace’s equation. These may be superposed in many ways to create the formation of line sources, vortex sheets and so on. In the Vortex Lattice method, each such elementary flow is the velocity field of a horseshoe vortex with some strength ${\displaystyle \Gamma }$.

All the lifting surfaces of an aircraft are divided into some number of quadrilateral panels, and a horseshoe vortex and a collocation point (or control point) are placed on each panel. The transverse segment of the vortex is at the 1/4 chord position of the panel, while the collocation point is at the 3/4 chord position. The vortex strength ${\displaystyle \Gamma }$ is to be determined. A normal vector ${\displaystyle \mathbf {n} }$ is also placed at each collocation point, set normal to the camber surface of the actual lifting surface.

For a problem with ${\displaystyle N}$ panels, the perturbation velocity at collocation point ${\displaystyle i}$ is given by summing the contributions of all the horseshoe vortices in terms of an Aerodynamic Influence Coefficient (AIC) matrix ${\displaystyle \mathbf {w} _{ij}}$.

${\displaystyle \nabla \varphi _{i}=\sum _{j=1}^{N}\mathbf {w} _{ij}\Gamma _{j}}$

The freestream velocity vector is given in terms of the freestream speed ${\displaystyle V_{\infty }}$ and the angles of attack and sideslip, ${\displaystyle \alpha ,\beta }$.

${\displaystyle \mathbf {V} _{\infty }=V_{\infty }{\begin{bmatrix}\cos \alpha \cos \beta \\-\sin \beta \\\sin \alpha \cos \beta \end{bmatrix}}}$

A Dirichlet boundary condition is applied at each collocation point, which prescribes that the normal velocity across the camber surface is zero.

${\displaystyle \mathbf {v} _{i}\cdot \mathbf {n} _{i}=\left(\mathbf {V} _{\infty }+\sum _{j=1}^{N}\mathbf {w} _{ij}\Gamma _{j}\right)\cdot \mathbf {n} _{i}=0}$

This is also known as the flow tangency condition. By evaluating the dot products above the following system of equations results. The new normalwash AIC matrix is ${\displaystyle a_{ij}=\mathbf {w} _{ij}\cdot \mathbf {n} _{i}}$, and the right hand side is formed by the freestream speed and the two aerodynamic angles ${\displaystyle b_{i}=V_{\infty }[-\cos \alpha \cos \beta ,\sin \beta ,-\sin \alpha \cos \beta ]\cdot \mathbf {n} _{i}}$

${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1N}\\a_{21}&\ddots &&\vdots \\\vdots &&\ddots &\vdots \\a_{N1}&\cdots &\cdots &a_{NN}\end{bmatrix}}{\begin{bmatrix}\Gamma _{1}\\\Gamma _{2}\\\vdots \\\Gamma _{N}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{N}\end{bmatrix}}}$

This system of equations is solved for all the vortex strengths ${\displaystyle \Gamma _{i}}$. The total force vector ${\displaystyle \mathbf {F} }$ and total moment vector ${\displaystyle \mathbf {M} }$ about the origin are then computed by summing the contributions of all the forces ${\displaystyle \mathbf {F} _{i}}$ on all the individual horseshoe vortices, with ${\displaystyle \rho }$ being the fluid density.

${\displaystyle \mathbf {F} _{i}=\rho \Gamma _{i}(\mathbf {V} _{\infty }+\mathbf {v} _{i})\times \mathbf {l} _{i}}$

${\displaystyle \mathbf {F} =\sum _{i=1}^{N}\mathbf {F} _{i}}$

${\displaystyle \mathbf {M} =\sum _{i=1}^{N}\mathbf {F} _{i}\times \mathbf {r} _{i}}$

Here, ${\displaystyle \mathbf {l} _{i}}$ is the vortex's transverse segment vector, and ${\displaystyle \mathbf {v} _{i}}$ is the perturbation velocity at this segment's center location ${\displaystyle \mathbf {r} _{i}}$ (not at the collocation point).

The lift and induced drag are obtained from the ${\displaystyle x,y,z}$ components of the total force vector ${\displaystyle \mathbf {F} }$. For the case of zero sideslip these are given by

${\displaystyle {\begin{array}{rcl}D_{i}&=&\;\;F_{x}\cos \alpha +F_{z}\sin \alpha \\L&=&\!-F_{x}\sin \alpha +F_{z}\cos \alpha \end{array}}}$