Vortex lattice method
The Vortex lattice method, (VLM), is a numerical method used in computational fluid dynamics, mainly in the early stages of aircraft design and in aerodynamic education at university level.
The VLM models the lifting surfaces, such as a wing, of an aircraft as an infinitely thin sheet of discrete vortices to compute lift and induced drag.
VLMs can compute the flow around a wing with rudimentary geometrical definition.
On the other side of the spectrum, they can describe the flow around a fairly complex aircraft geometry (with multiple lifting surfaces with taper, kinks, twist, camber, trailing edge control surfaces and many other geometric features).
By simulating the flow field, one can extract the pressure distribution or as in the case of the VLM, the force distribution, around the simulated body.
This knowledge is then used to compute the aerodynamic coefficients and their derivatives that are important for assessing the aircraft's handling qualities in the conceptual design phase.
With an initial estimate of the pressure distribution on the wing, the structural designers can start designing the load-bearing parts of the wings, fin and tailplane and other lifting surfaces.
Hence as the drag must be balanced with the thrust in the cruise configuration, the propulsion group can also get important data from the VLM simulation.
John DeYoung provides a background history of the VLM in the NASA Langley workshop documentation SP-405.
[1] The VLM is the extension of Prandtl's lifting-line theory,[2] where the wing of an aircraft is modeled as an infinite number of Horseshoe vortices.
Falkner in his Aeronautical Research Council paper of 1946.
Instead of only one horseshoe vortex per wing, as in the Lifting-line theory, the VLM utilizes a lattice of horseshoe vortices, as described by Falkner in his first paper on this subject in 1943.
A typical number of vortices would be around 100 for an entire aircraft wing; an Aeronautical Research Council report by Falkner published in 1949 mentions the use of an "84-vortex lattice before the standardisation of the 126-lattice" (p. 4).
[5] The method is comprehensibly described in all major aerodynamic textbooks, such as Katz & Plotkin,[6] Anderson,[7] Bertin & Smith[8] Houghton & Carpenter[9] or Drela,[10] The vortex lattice method is built on the theory of ideal flow, also known as Potential flow.
Ideal flow is a simplification of the real flow experienced in nature, however for many engineering applications this simplified representation has all of the properties that are important from the engineering point of view.
However, lift induced drag can be assessed and, taking special care, some stall phenomena can be modelled.
The following assumptions are made regarding the problem in the vortex lattice method: By the above assumptions the flowfield is Conservative vector field, which means that there exists a perturbation velocity potential
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.
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.
is given by summing the contributions of all the horseshoe vortices in terms of an Aerodynamic Influence Coefficient (AIC) matrix
A Neumann boundary condition is applied at each collocation point, which prescribes that the normal velocity across the camber surface is zero.
Alternate implementations may also use the Dirichlet boundary condition directly on the velocity potential.
By evaluating the dot products above the following system of equations results.
, and the right hand side is formed by the freestream speed and the two aerodynamic angles
This system of equations is solved for all the vortex strengths
about the origin are then computed by summing the contributions of all the forces
The preliminary design of airplanes requires unsteady aerodynamic models, usually written in the frequency domain for aeroelastic analyses.
Commonly used is the Doublet Lattice Method, where the wing system is subdivided into panels.
Each panel has a load point where the lifting force is assumed applied and a control point where the aeroelastic boundary condition is enforced.
