Vortex lattice method 1

Vortex lattice methodThe Vortex lattice method, (VLM), is a numerical, Computational fluid dynamics, method used mainly in the earlystages of aircraft design and in aerodynamic education at university level. The VLM models the lifting surfaces, suchas a wing, of an aircraft as an infinitely thin sheet of discrete vortices to compute lift and induced drag. The influenceof the thickness, viscosity is neglected.VLMs can compute the flow around a wing with rudimentary geometrical definition. For a rectangular wing it isenough to know the span and chord. On the other side of the spectrum, they can describe the flow around a fairlycomplex aircraft geometry (with multiple lifting surfaces with taper, kinks, twist, camber, trailing edge controlsurfaces 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 forcedistribution, around the simulated body. This knowledge is then used to compute the aerodynamic coefficients andtheir derivatives that are important for assessing the aircraft's handling qualities in the conceptual design phase. Withan initial estimate of the pressure distribution on the wing, the structural designers can start designing theload-bearing parts of the wings, fin and tailplane and other lifting surfaces. Additionally, while the VLM cannotcompute the viscous drag, the induced drag stemming from the production of lift can be estimated. Hence as the dragmust be balanced with the thrust in the cruise configuration, the propulsion group can also get important data fromthe VLM simulation.

Historical backgroundBart Rademaker provides a background history of the VLM in the NASA Langley workshop documentationSP-405.[1]

The VLM is the extension of Prandtl lifting line theory,[2] where the wing of an aircraft is modeled as an infinitenumber of Horseshoe vortices. The name was coined by V.M. Falkner in his Aeronautical Research Council paper of1946.[3] The method has since then been developed and refined further by W.P. Jones, H. Schlichting, G.N. Wardand others.Although the computations needed can be carried out by hand, the VLM benefited from the advent of computers forthe 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 horseshoevortices, as described by Falkner in his first paper on this subject in 1943.[4] The number of vortices used vary withthe 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; an Aeronautical Research Councilreport by Falkner published in 1949 mentions the use of an "84-vortex lattice before the standardisation of the126-lattice" (p.4).[5]

The method is comprehensibly described in all major aerodynamic textbooks, such as Katz & Plotkin,[6] Anderson,[7]

Bertin & Smith[8] or Houghton & Carpenter[9]

TheoryThe vortex lattice methods is built on the theory of ideal flow, also known as Potential flow. Ideal flow is asimplification of the real flow experienced in nature, however for many engineering applications this simplifiedrepresentation has all of the properties that are important from the engineering point of view. This method neglectsall viscous effects. Turbulence, dissipation and boundary layers are not resolved at all. However, lift induced dragcan be assessed and, taking special care, some stall phenomena can be modelled.

Vortex lattice method 2

AssumptionsThe following assumptions are made regarding the problem in the vortex lattice method: The flow field is incompressible, inviscid and irrotational. However, subsonic compressible flow can be modeled

if the 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.

MethodBy the above assumptions the flowfield is Conservative vector field, which means that there exists a velocitypotential:

and that Laplace's equation hold.Laplaces equation is a second order linear equation, and being so it is subject to the principle of superposition.Which means that if and are two solutions of the differential equation , then the linear combination

is also a solution for any values of the constants and . As Anderson[7] put it "A complicatedflow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementaryflows, which are also irrotational and incompressible.. Such elementary flows are the point source or sink, thedoublet and the vortex line, each being a solution of Laplaces equation. These may be superposed in many ways tocreate the formation of line sources, vortex sheets and so on.

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

A Neumann boundary condition is applied, which prescribes that the normal velocity at the boundary is zero. It isalso known as the flow tangency condition, or no cross flow condition. It means that at the boundary (e.g. the surfaceof a wing) the flow must be parallel to the surface. The following system of equations may be set up. The right handside is formed by the freestream and the angle of attack so that .

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

Vortex lattice method 3

References[1] NASA, Vortex-lattice utilization. NASA SP-405, NASA-Langley, Washington, 1976.[2] Prandtl. L, Applications of modern hydrodynamics to aeronautics, NACA-TR-116, NASA, 1923.[3] Falkner. V.M., The Accuracy of Calculations Based on Vortex Lattice Theory, Rep. No. 9621, British A.R.C., 1946.[4] Falkner. V.M., The Calculations of Aerodynamic Loading on Surfaces of any Shape, R&M 1910 (http:/ / aerade. cranfield. ac. uk/ ara/ arc/

rm/ 1910. pdf), British A.R.C., 1943.[5] Falkner. V.M., A Comparison of Two Methods of Calculating Wing Loading with Allowance for Compressibility, R&M 2685 (http:/ / aerade.

cranfield. ac. uk/ ara/ arc/ rm/ 2685. pdf), British A.R.C., 1949.[6] J. Katz, A. Plotkin, Low-Speed Aerodynamics, 2nd ed., Cambridge University Press, Cambridge, 2001.[7] J.D. Anderson Jr, Fundamentals of aerodynamics, 2nd ed., McGraw-Hill Inc, 1991.[8] J.J. Bertin, M.L. Smith, Aerodynamics for Engineers, 3rd ed., Prentice Hall, New Jersey, 1998.[9] E.L. Houghton, P.W. Carpenter, Aerodynamics for Engineering Students, 4th ed., Edward Arnold, London, 1993.

External links http:/ / web. mit. edu/ drela/ Public/ web/ avl/

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 Engineers, 3rd ed., Prentice Hall, New Jersey, 1998. E.L. Houghton, P.W. Carpenter, Aerodynamics for Engineering Students, 4th ed., Edward Arnold, London, 1993. Lamar, J. E., Herbert, H. E., Production version of the extended NASA-Langley vortex lattice FORTRAN

computer program. Volume 1: User's guide, NASA-TM-83303, NASA, 1982 Lamar, J. E., Herbert, H. E., Production version of the extended NASA-Langley vortex lattice FORTRAN

computer program. Volume 2: Source code, NASA-TM-83304, NASA, 1982 Melin, Thomas, A Vortex Lattice MATLAB Implementation for Linear Aerodynamic Wing Applications, Royal

Institute of Technology (KTH), Sweden, December, 2000

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Article Sources and ContributorsVortex lattice method Source: http://en.wikipedia.org/w/index.php?oldid=554057972 Contributors: Brewcrewer, Chris the speller, EagleFan, Jmencisom, Salih, Share'n'serve, Squids andChips, Tomas melin, TraceyR, WereSpielChequers, YSSYguy, 16 anonymous edits

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