Lifting-line Theory - Derivation

Derivation

Nomenclature:

• is the circulation over the entire wing (m²/s)
• is the 3D lift coefficient (for the entire wing)
• is the aspect ratio
• is the freestream angle of attack
• is the freestream velocity
• is the drag coefficient for induced drag
• is the planform efficiency factor

The following are all functions of the wings span-wise station (i.e. they can all vary along the wing)

• is the 2D lift coefficient (units/m)
• is the 2D circulation at a section (m/s)
• is the chord length of the local section
• is the local change in angle of attack due to geometric twist of the wing
• is zero-lift angle of attack of that section (depends on the airfoil geometry)
• is the 2D lift coefficient slope (units/m⋅rad, and depends on airfoil geometry, see Thin airfoil theory)
• is change in angle of attack due to downwash
• is the local downwash velocity

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 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 (acts like half an infinite vortex line):

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 (assuming small angles):

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, as this will lead to a singular matrix, which can't be solved.