Lifting-line theory


The Prandtl lifting-line theory is a mathematical model that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory.
The theory was expressed independently by Frederick W. Lanchester in 1907, and by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk.
In this model, the vortex loses strength along the whole wingspan because it is shed as a vortex-sheet from the trailing edge, rather than just at the wing-tips.

Introduction

It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate.
When analyzing a three-dimensional finite wing, the first approximation to understanding is to consider slicing the wing into cross-sections, and analyzing each cross section independently as a wing in a two-dimensional world. Each of these slices is called an airfoil, and it is easier to understand an airfoil than a complete three-dimensional wing.
One might expect that understanding the full wing simply involves adding up the independently-calculated forces from each airfoil segment. However, it turns out that this approximation is grossly incorrect: on a real wing, the lift over each wing segment does not correspond simply to what two-dimensional analysis predicts. In reality, the local amount of lift on each cross section is not independent, and is strongly affected by neighboring wing sections.
The lifting-line theory corrects some of the errors in the naive two-dimensional approach, by including some of the interactions between the wing slices. It produces the lift distribution along the span-wise direction, based on the wing geometry and flow conditions.

Principle

The lifting-line theory applies the concept of circulation and the Kutta–Joukowski theorem,
so that instead of the lift distribution function, the unknown effectively becomes the distribution of circulation over the span,.
Modeling the local lift with the local circulation allows us to account for the influence of one section over its neighbors. In this view, any span-wise change in lift is equivalent to a span-wise change of circulation. According to Helmholtz's theorems, a vortex filament cannot begin or terminate in the air. Any span-wise change in lift can be modeled as the shedding of a vortex filament down the flow, behind the wing.
This shed vortex, whose strength is the derivative of the local wing circulation distribution,, influences the flow left and right of the wing section.
This sideways influence is the key to the lifting-line theory. Now, if the change in lift distribution is known at given lift section, it is possible to predict how that section influences the lift over its neighbors: the vertical induced velocity can be quantified using the velocity distribution within a vortex, and related to a change in effective angle of attack over neighboring sections.
In mathematical terms, the local induced change of angle of attack on a given section can be quantified with the integral sum of the downwash induced by every other wing section. In turn, the integral sum of the lift on each downwashed wing section is equal to the total desired amount of lift.
This leads to an integro-differential equation in the form of where is expressed solely in terms of the wing geometry and its own span-wise variation,. The solution to this equation is a function,, that accurately describes the circulation distribution over a finite wing of known geometry.

Derivation

Nomenclature:
The following are all functions of the wings span-wise station
To derive the model we start with the assumption that the circulation of the wing varies as a function of the spanwise locations. The function assumed is a Fourier function. Firstly, the coordinate for the spanwise location is transformed by, where y is spanwise location, and s is the semi-span of the wing.
and so the circulation is assumed to be:
Since the circulation of a section is related to the by the equation:
but since the coefficient of lift is a function of angle of attack:
hence the vortex strength at any particular spanwise station can be given by the equations:
This one equation has two unknowns: the value for and the value for. However, the downwash is purely a function of the circulation only. So we can determine the value in terms of, bring this term across to the left hand side of the equation and solve. The downwash at any given station is a function of the entire shed vortex system. This is determined by integrating the influence of each differential shed vortex over the span of the wing.
Differential element of circulation:
Differential downwash due to the differential element of circulation :
The integral equation over the span of the wing to determine the downwash at a particular location is:
After appropriate substitutions and integrations we get:
And so the change in angle attack is determined by :
By substituting equations 8 and 9 into RHS of equation 4 and equation 1 into the LHS of equation 4, we then get:
After rearranging, we get the series of simultaneous equations:
By taking a finite number of terms, equation 11 can be expressed in matrix form and solved for coefficients A. Note the left-hand side of the equation represents each element in the matrix, and the terms on the RHS of equation 11 represent the RHS of the matrix form. Each row in the matrix form represents a different span-wise station, and each column represents a different value for n.
Appropriate choices for are as a linear variation between. Note that this range does not include the values for 0 and, as this leads to a singular matrix, which can't be solved.

Lift and drag from coefficients

The lift can be determined by integrating the circulation terms:
which can be reduced to:
where is the first term of the solution of the simultaneous equations shown above.
The induced drag can be determined from
which can also be reduced to:
where are all the terms of the solution of the simultaneous equations shown above.
Moreover, this expression may be arranged as a function of in the following way :
where
is the span efficiency factor

Symmetric wing

For a symmetric wing, the even terms of the series coefficients are identically equal to 0, and so can be dropped.

Rolling wings

When the aircraft is rolling, an additional term can be added that adds the wing station distance multiplied by the rate of roll to give additional angle of attack change. Equation 3 then becomes:
where
Note that y can be negative, which introduces non-zero even coefficients in the equation that must be accounted for.

Control deflection

The effect of ailerons can be accounted by simply changing term in Equation 3. For non-symmetric controls such as ailerons the term changes on each side of the wing.

Elliptical wings

For an elliptical wing with no twist, with:
The chord length is given as a function of span location as:
Also,
This yields the famous equation for the elliptic induced drag coefficient:
where
A useful approximation is that
where
The theoretical value for is 2. Note that this equation becomes the thin airfoil equation if AR goes to infinity.
As seen above, the lifting-line theory also states an equation for induced drag:.
where
The lifting line theory does not take into account the following: