Math 2.0

Fluid mechanics Definition

Vortex lattice method

Builds on Biot–Savart law for vortex filamentsKutta–Joukowski theoremPotential flowSystem of linear equationsLifting-line theory

Definition

Consider a thin lifting surface SS in a steady, incompressible, inviscid and irrotational flow with free stream velocity 𝐕∞\mathbf{V}_\infty. The vortex lattice method approximates the flow as follows.

  1. The surface SS, or its projection on a mean plane, is divided into NN quadrilateral panels, arranged in rows along the span and the chord.
  2. On panel jj a horseshoe vortex of unknown constant circulation Γj\Gamma_j is placed. It has a bound segment on the quarter-chord line of the panel and two semi-infinite trailing segments from its ends, parallel to the free stream.
  3. On panel ii a control point PiP_i is placed at the middle of its three-quarter-chord line, with unit normal 𝐧i\mathbf{n}_i to the surface.
  4. Let aija_{ij} be the normal component at PiP_i of the velocity induced by horseshoe vortex jj with unit circulation, computed with the Biot–Savart law. The circulations solve the linear system ∑j=1Naij Γj=−𝐕∞⋅𝐧i,i=1,…,N.\sum_{j=1}^{N} a_{ij}\, \Gamma_j = -\mathbf{V}_\infty \cdot \mathbf{n}_i, \qquad i = 1, \dots, N.

Let 𝐥j\mathbf{l}_j be the bound segment of panel jj as a vector, and ρ\rho the density. The Kutta–Joukowski theorem gives the force ρ 𝐕∞×Γj𝐥j\rho\, \mathbf{V}_\infty \times \Gamma_j \mathbf{l}_j on this segment, and the total force is the sum.

Remarks

Example: take a rectangular flat wing at small angle of attack α\alpha, with one chordwise row of NN spanwise panels. It gives an N×NN \times N linear system whose right side has the same value, of size V∞sin⁡αV_\infty \sin\alpha, in each row. Its solution is a discrete span load that falls off towards the tips. Non-example: a scheme that puts each control point on the bound vortex of its own panel fails clause 3. It breaks down because the Biot–Savart velocity is singular on the vortex. With one chordwise panel the method is a discrete version of lifting-line theory. The placement of the vortex at 1/41/4 and the control point at 3/43/4 of the panel chord makes a single panel reproduce the thin airfoil lift slope 2π2\pi exactly.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Circulation, concept, no evidence levelCirculationconceptComplex potential, concept, no evidence levelComplex potentialconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptIncompressible flow, concept, no evidence levelIncompressible flowconceptLifting-line theory, concept, no evidence levelLifting-line theoryconceptPotential flow, concept, no evidence levelPotential flowconceptSystem of linear equations, concept, no evidence levelSystem of linear equa…conceptVorticity, concept, no evidence levelVorticityconceptKutta–Joukowski theorem, statement, no evidence levelKutta–Joukowski theor…statementFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptCurl, concept, no evidence levelCurlconceptBiot–Savart law for vortex filaments, statement, no evidence levelBiot–Savart law for v…statementVortex lattice method, definition, no evidence levelVortex lattice methoddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circulation concept
  2. Complex potential concept
  3. Euclidean space object
  4. Function concept
  5. Incompressible flow concept
  6. Lifting-line theory concept
  7. Potential flow concept
  8. System of linear equations concept
  9. Vorticity concept
  10. Kutta–Joukowski theorem statement
  11. Function of several variables concept
  12. Vector-valued function concept
  13. Partial derivative concept
  14. Vector field concept
  15. Curl concept
  16. Biot–Savart law for vortex filaments statement
  17. Vortex lattice method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Vortex_lattice_method
About this page
TagHTYCMVWPSM
URIm2://HTYCMVWPSM
Kinddefinition
WritingW0 stub
MSC76M23, 76G25
DefinesVortex lattice method
UsesBiot–Savart law for vortex filaments, Kutta–Joukowski theorem, Potential flow, System of linear equations, Lifting-line theory
Changed3 November 2026

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