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Wing of finite span and downwash: Theory Prandtl lifting line

Understand and master the Wing of finite span and downwash: Theory Prandtl lifting line:

Vortex sheet and losses wingtip shown schematically in software Heliciel:

Rectangular wing length of 2 meters Aspect ratio = 4 ratio (Lift / Drag) =16

Rectangular wing length of 9 meters Aspect ratio = 18 ratio (Lift / Drag) =32

Triangular wing length of 2 meters. Aspect ratio = 5 ratio (Lift / Drag) =19. Triangular wing length of 9 meters. Aspect ratio = 33 ratio (Lift / Drag) =36


We have seen in the page for general information on the wings and hydrofoils that drag and lif forces calculated using the 2D CL and Cd are theoretical values, not taking into account the vortex phenomena. These vortex phenomena are caused by the pressure difference between the upper and lower surfaces of the wing:

the overpressure zone and the depression zone of a wing causes flashover and losses at the wing tip

This is the pressure difference between the lower and upper surfaces, which generates lift. We can therefore conclude that the intensity of the vortices Γ (gamma) will change according to the lift..

lift-induced wing vortex

These vortices alter the trajectory of the fluid around the wing, and induce velocity components directed downward. The speeds induced vortices vary depending on the distance and the intensity of the vortices.

speed induced downwash

induced Speed calculation vortices can redefine the effective angle of attack at different regions (elements) of the wing.The actual angle of attack finally allows the use of 2D performance, to calculate the lift forces (dL) and drag (dD) taking into account the vortex phenomena. We will calculate this phenomenon for each element of our wing or blade.
Change of the effective angle generated by the induced down speed in the profile of an element:

velocities and wing angles finite span

How to calculate the induced velocities and the effective angle determining the actual performance of our wing?
Lifting line Prandt's ltheory Ludwig Prandtl has developed the first method for the analysis of a wing of finite span in 1918 equating all vortex filaments attached to a wing has a single filament called "lifting line." Laws and theorems defining vortices allow calculation of induced velocities:


Lift(2D)( Newtons)= CL . element area . r . V² /2

wing swirls

In reality, when the wing is flying, the "vortex departure to downstream infinity" is far enough away from the wing to overlook the induced velocities, it causes on the wing. (because the velocity induced by a vortex varies as the inverse of the distance to the axis of the vortex). So we consider only the free vortices marginal, and induced download speeds, as seen by the wing. Free vortices marginal are defined as semi-infinite, because it starts at the end of the element, and extend to the downstream infinity.

vortex system

System of horseshoe vortices, of constant intensity (Γ) equivalent to the lift, generating induced speeds down.

upwash and downwash induced speeds

Let the point P in the system of two horseshoe vortices, where the induced velocity is produced by two free vortices. Point A is located at -b/2 and +b/2 and the point P on the Y axis.Induced speed based the y-position is composed of two terms:
  1. terms - Γ/[4p(b/2+y)]representing the contribution of vortices located at point A with y=-b/2
  2. terms - Γ/[4p(b/2-y)] representing the contribution of vortices located at point A with y=b/2

speed induced horseshoe

Giving the expression of the induced velocity, depending on the position y for a single vortex system:
  • w(y) = -Γ/[4p(b/2+y)] - Γ/[4p(b/2+y)]
  • equivalent to :w(y) = - Γ/(4p) .{b/[(b/2)²-y²]}
In this latest expression of the induced velocities for a single isolated vortex system, when the position y approach to tip (y=b/2 ou y=-b/2), induced velocity tends to negative infinity. Which can not correspond to a realistic representation of the wing of finite span. to remedy this, Prandtl, introduces the concept of lifting line:
The first theorem Helmotz defining the laws of vortex motion states that the intensity (Γ) of an isolated vortex is constant and equivalent to the lift. A variation of the lift (therefore the intensity Γ) can not be represented by a vortex system isolated.


Variation of lift, therefore the vortex intensity, on a wing:

lift distribution

Helmotz first theorem shows that the variation of lift on the wing (therefore of vortex intensity) can only be done if the vortex filaments of intensity equal to the intensity variation add to or leave the wing:

vortex filament separation

A system of vortices superimposed is used to represent the variation of 'vortex intensity associée associated with the lift of a wing. At each point of the wing, in which is a variation of lift, a vortex of intensity equal to the change in lift is formed:

lifting line

In reality this vortices sheet forming on the trailing edges is combined ,

effective vortices

to form the two vortices sometimes visible:wing vortex But the variation of lift and intensity is continuous and progressive on the wing, the number of horseshoe vortices, intensity dΓ superimposed on the lifting line describing the intensity variation is infinite.

lifting line

  • Elementary dG intensity of horseshoe vortex, corresponding to the lifting line segment [dy] positioned at y on the figure above, is given by : dΓ= (deltaΓ/delta y).dy
  • the free vortex intensity dΓ will induce dw velocity at the point y0:
    using the law of Biot savart defining the velocity induced by a semi-infinite vortex:

induced velocity semi-finished vortices


where is substituted :
  • Γ by (deltaΓ/delta y).dy, and h by y0-y
we obtain the elementary induced velocity at the point y0, produced by free vortex intensity dΓ, passing through y :
  • dw= - {(deltaΓ/delta y).dy}/{4p.(y0-y)}
The total speed y0, induced by integrality of free vortex layer from -b/2 to b/2 is:
  • w(y0) = -1/(4p)(- +b/2) {(deltaΓ/delta y)/(y0-y)}.dy

wing span speed and angles

The angle of incidence induced (ai) is :
  • ai(y0)= atan[-w(y0) / V∞]


( references:"Aérodynamique subsonique" Ion Parashivoiu école polytechnique de Montréal .

You will also find a description of the theory of Prandtl lifting line on the site: eand in particular this document pdf :Theory of Prandtl lifting line