Starting with irrotational flow, which is defined as a flow where the vorticity is zero at every point.
(3.1)
If ϕ is a scalar function, following the vector identity, we get
. (3.2)
Which means that the curl of the gradient of a scalar function is equal to zero. combining (3.1) and (3.2), gives
. (3.3)
Equation (3.3) states that for irrotational flow, there exists a scalar function ϕ such that the velocity is given by the gradient of ϕ. From now on, ϕ is denoted as the velocity potential.
From the principle of mass conservation for an incompressible flow, Equation (3.4) is obtained.
(3.4)
With the definition of velocity potential ϕ, for a flow that is both incompressible and irrotational, (3.3) and (3.4) can be combined to
, (3.5)
or
.
Equation (3.5) is the well known Prandtl-Glauert Equation (similar to Laplace’s Equation), which governs irrotational, incompressible flow. Because the Laplace’s Equation is linear, (Anderson 2011) states that:
A complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows which are also irrotational and incompressible.
These different elementary flows include point/line source, point/line sink, point/line doublet and point/line vortex. The VLM is based on these line vortices (Liu 2007).
3.1.2 Boundary Conditions
VLM linearizes and transfers the boundary condition (shown in Figure 3.2) and makes a linear approximaton between velocity and pressure by using the so-called “thin airfoil boundary condition”. After the linear approximation on the lower and upper surfaces of the airfoil, (Mason 2015) states:
For cases where the linearized pressure coefficients relation is valid, thickness does not contribute to lift to first order in the velocity disturbance!
This means that the cambered surface boundary conditions can be applied on a flat coordinate surface and result in a much more easy way to apply the boundary conditions (Liu 2007).
Considering a symmetrical airfoil/wing, the camber effect can also be neglected, after applying this boundary condition to Laplace’s Equation, the problem can easily be solved by including the effect of angle of attack on a flat surface. This is what VLM uses (Liu 2007).
Consider a wing, placed on the x-y plane. The boundary condition states that normal flow across the thin wing’s solid surface is zero (Liu 2007).
(3.6)
Which means that the sum of the normal velocity component induced by the wing’s bound vortices wb, by the wake wi and by the free-stream velocity V∞ will be zero (Liu 2007).
(3.7)
Figure 3.2 Decomposition of a general airfoil at a certain incidence (Mason 2015).
3.1.3 Biot-Savart Law
As stated, the point vortex singularity is one of the possible solutions for Laplace’s Equation.
The vortex flow is shown in Figure 3.3. This vortex induces a tangential velocity defined by
(3.8)
Where Γ is the vortex circulation strength. Note that this is constant around the circle of radius to the flow center r. The circulation has the same sign as it’s vorticity, so it’s positive in the clockwise direction (Liu 2007).
The idea of a point vortex can be extended to a general three-dimensional vortex filament.
The flowfield induced by this vortex filament can be seen in Figure 3.4.
The mathematical description of the flow induced by this filament is given by the Biot-Savart law. It states that the increment of the velocity dV at a point P due to a segment of a vortex filament dl at a point q is (Liu 2007).
(3.9)
This can then be integrated over the entire length of the vortex filament to obtain the induced velocity in point P (Liu 2007).
(3.10)
Figure 3.3 Representation of the point vortex (Mason 2015).
Figure 3.4 Three-dimensional vortex filament (Liu 2007).
3.1.4 Horseshoe Vortex
In VLM, a special form of vortex is used. This is the horeshoe vortex, shown in Figure 3.5, where the vortex line is assumed to be placed in the x-y plane, making the horseshoe vortex a simplified case of the vortex ring. It consist of four vortex filaments. Two trailing segments ab and cd of the vortex are placed parallel to the direction of the free-stream velocity and start at infinity. The other two segments bc and ad are finite. Normally, the effect of ad can be neglected because of the infinite distance. So, in practice, the horseshoe vortex only contains three parts. The straight bound vortex segment bc models the lifting properties and the two semi-infinite trailing vortex lines model the wake. In general, the expression of the induced velocity at a point by one horseshoe vortex is (Liu 2007)
. (3.8)
Figure 3.5 Schematic of one single horseshoe vortex, which is part of a vortex system on a wing (Anderson 2011).
For the finite length vortex segment bc in the horseshoe vortex, the induced velocity at a certain point can be calculated using (3.13), where r1 and r2 are the distances from this certain point to the two end points of the segment and r0 is the length of the segment (Liu 2007).
(3.12)
As mentioned, the horseshoe vortex is going to represent a lifting surface. Important here, is the location of the vortex and where the location of a control point has to be, to satisfy the surface boundary condition. The answer to this problem is called the “¼ – ¾ rule”. The vortex is located at the ¼ chord point, and the control point is located at the ¾ chord point. The rule was discovered by Pistolesi, has proven to be sufficiently accurate in practice, and has
become a rule of thumb (Mason 2015). Mathemetical derivations of more precise vortex/control point locations are available in (Lan 1974).
Note that the lift is on the bound vortices. This is because of the Kutta – Joukowsky theorem (eq. 3.13) (Mason 2015), which states that a vortex of certain circulation Γ moving with free-stream velocity Q∞ creates lift L (Budziak 2015).
(3.9)
As said, the surface of the model is divided into a finite number of panels (chordwise and spanwise). On each of these panels there is a horseshoe vortex, as shown in Figure 3.6. Each vortex has his own circulation and thus, to get the total aerodynamic force, the contribution of all panels have to be summarized (Anderson 2011).
Figure 3.6 Vortex Lattice System on a finite wing (Anderson 2011).