Vortex lattice method

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The Vortex lattice method, (VLM), is a numerical, Computational fluid dynamics, method used 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 a infinitely thin sheet of discrete vortices to compute lift and induced drag. The influence of the thickness, viscosity and other things, is neglected.

VLM:s can compute the flow around a wing with rudimentary geometrical definition. For a rectangular wing it is enough to know the span and chord. 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 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 aircrafts 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. Additionally, while the VLM cannot compute the viscous drag, the induced drag stemming from the production of lift can be estimated. 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.

1 Historical background
2 Theory
3 References
4 Sources

[edit] Historical background

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 lifting line theory ^[2], where the wing of an aircraft is modeled as a Horseshoe vortex. The name was coined by V.M. Falkner in his aeronautical research council paper of 1946^[3]. The method has since then been developed and refined further by W.P. Jones, H. Schlichting, G.N. Ward and others.

Although the computations needed can be carried out by hand, the VML benefited from the advent of computers for the large amounts of computations that are required.

Instead of only one horseshoe vortex per wing, as in the lifting line theory, the VLM utilizes a lattice of horseshoe vortices. The number of vortices used vary with the required pressure distribution resolution, and with required accuracy in the computed aerodynamic coefficients. A typical number of vortices would be around 100 for an entire aircraft wing.

The method is comprehensibly described in all major aerodynamic textbooks, such as Katz & Plotkin^[4], Anderson^[5], Bertin & Smith^[6] or Houghton & Carpenter^[7]

[edit] Theory

The vortex lattice methods is build 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. The method neglect all viscous effects. Turbulence, dissipation and boundary layers are not resolved at all. However, lift induced drag can be assessed and, taking special care, some stall phenomena can be modelled.

[edit] Assumptions

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

The flow field is incompressible, inviscous and irrotational.
The lifting surfaces are thin.
The angle of attack is small, small angle approximation.

[edit] Method

By the above assumptions the flowfield is Conservative vector field, which means that there exists a velocity potential:
$\bar{v} =\nabla \phi$

and that Laplace's equation hold.

Laplace’s equation is a second order linear equation, and being so it is subject to the principle of superposition. Which means that if $y 1$ and $y 2$ are two solutions of the differential equation $L (Y)$ , then the linear combination $c 1 y 1 + c 2 y 2$ is also a solution for any values of the constants $c 1$ and $c 2$ . As Anderson^[8] 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, the point 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.

[edit] Aircraft Model

The lifting surfaces of an aircraft is divided into several panels. A horseshoe vortex is applied on each of these panels and the velocity vector generated by the vortices at the collocation points of each panel is computed. The vortex is places at the 1/4 chord point of each panel, and the collocation point at 3/4 chord. For a problem with $n$ panels, the induced velocity of each unit strength vortex on each panel is collected in the influence matrix $W$

$\mathbf{W} = \begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n}\\ w_{11} & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ w_{n1} & \cdots & \cdots & w_{nn} \end{bmatrix}$

A Neumann boundary condition is applied, which prescribes that the normal velocity at the boundary is zero. It is also known as the flow tangency condition, or no cross flow condition. It means that at the boundary (e.g the surface of a wing) the flow must be parallel to the surface. The following system of equations may be set up. The right hand side is formed by the freestream and the angle of attack so that $b=v_{\infty} sin(\alpha)$ .

$\begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n}\\ w_{21} & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ w_{n1} & \cdots & \cdots & w_{nn} \end{bmatrix} \begin{bmatrix} \Gamma_{1} \\ \Gamma_{2} \\ \vdots \\ \Gamma_{n} \end{bmatrix}= \begin{bmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{n} \end{bmatrix}$

From this system of equations the strength $Γ$ of the vortices can be solved for, and the forces acting on the panels be computed with

$F=\rho_{air} \Gamma (V_{\infty}+V_{induced}) l$

[edit] References

^ NASA, Vortex-lattice utilization. Nasa SP-405, NASA-Langley, Washington, 1976.
^ Prandtl. L, Applications of modern hydrodynamics to aeronautics, NACA-TR-116, NASA, 1923.
^ Falkner. V.M., The Accuracy of Calculations Based on Vortex Lattice Theory, Rep. No. 9621, British A.R.C., 1946.
^ J. Katz, A. Plotkin, Low-Speed Aerodynamics, 2nd ed., Cambridge University Press, Cambridge, 2001.
^ J.D. Anderson Jr, Fundamentals of aerodynamics, 2nd ed., McGraw-Hill Inc, 1991.
^ J.J. Bertin, M.L. Smith, Aerodynamics for Engineers, 3rd ed., Prentice Hall, New Jersey, 1998.
^ E.L. Houghton, P.W. Carpenter, Aerodynamics for Engineering Students, 4th ed., Edward Arnold, London, 1993.
^ J.D. Anderson Jr, Fundamentals of aerodynamics, 2nd ed., McGraw-Hill Inc, 1991.

[edit] Sources

NASA, Vortex-lattice utilization. Nasa SP-405, NASA-Langley, Washington, 1976.
Prandtl. L, Applications of modern hydrodynamics to aeronautics, NACA-TR-116, NASA, 1923.
Falkner. V.M., The Accuracy of Calculations Based on Vortex Lattice Theory, Rep. No. 9621, British A.R.C., 1946.
J. Katz, A. Plotkin, Low-Speed Aerodynamics, 2nd ed., Cambridge University Press, Cambridge, 2001.
J.D. Anderson Jr, Fundamentals of aerodynamics, 2nd ed., McGraw-Hill Inc, 1991.
J.J. Bertin, M.L. Smith, Aerodynamics for Enginners, 3rd ed., Prentice Hall, New Jersey, 1998.
E.L. Houghton, P.W. Carpenter, Aerodynamics for Engineering Students, 4th ed., Edward Arnold, London, 1993.