Surface Singularity Methods

Surface singularity or panel methods are based on the assumption of small dis­turbance, incompressible irrotational flow, for which the governing equation is Laplace’s equation (Eq. 14.36) for the velocity potential. The basic approach is described by Hess & Smith (1962), Rubbert & Saaris (1969), Morino (1985), Hess (1990), and Katz & Plotkin

(1991) . These methods can be used to model the flow over helicopter fuselages, and with
extensions to account for the vorticity distribution in the wake behind the body, the flow over wings and rotor blades too. The idea is that surface singularities of unknown strength can be distributed over the body surface. After determining the strengths of the singularities by imposing a condition of flow tangency on the surface, the surface velocities, and hence the aerodynamic loading on the body, can be calculated. For an engaging modem treatment of the various types of panel methods, see Katz & Plotkin (1991). Large computer codes that use panel methods are commercially available and are in widespread use in the helicopter industry. See also Maskew (1982) for a good summary of the general approach.

Classical panel methods have found considerable use in helicopter applications for airfoil and fuselage design. The principles of both 2-D and 3-D panel methods are similar. To illustrate the principles, consider the flow about a 2-D airfoil. The airfoil surface is replaced by distribution of N discrete panels, and each panel і contains a singularity distribution that can be described by a velocity potential function фі. This means that the total potential induced by all the singularities in the flow is

N

Ф =

i = l

where the constants y, must be found subject to the boundary condition that the flow must be tangential to the surface at N discrete collocation points xj on the surface. That is,

(V0(*7-) + Voo) ■ nj = 0, (14.38)

where n j is the outward pointing normal unit vector at the j th collocation point. Usually the collocation points are selected to be the midpoints of the panels. If the panels contain vortex singularities that vary linearly in strength over the area of the panel, then with N panels there are N equations containing N + 1 unknowns. The Kutta condition at the trailing edge of the airfoil provides the auxiliary equation required to solve the problem uniquely and to define the airloads on the airfoil. The equations governing the problem can be expressed in the form

where A = A-j = Wфi(xj) • nj is a (N + 1) x (iV + 1) matrix of influence coefficients. The influence coefficient А,-j is effectively the normal component of the velocity induced at the collocation point on panel j by the singularity distribution 0, on panel і and depends on the relative distance between, and orientation of, the two panels. Notice that solving Eq. 14.39 involves finding the inverse of the matrix of influence coefficients, and, besides the calculation of the influence coefficients themselves, this inversion is the primary com­putational expense involved in the panel method. If the angle of attack, a, is included in the vector on the right-hand side of Eqs. 14.39, then the influence coefficients need only be calculated once, however. Once the singularity strengths у,- are known, the loading on the airfoil follows immediately.

Panel methods have also been used to model the aerodynamics of helicopter airframes. In such an application, the fuselage surface might be represented by N small, flat, quadrilateral panels onto which singularities in the form of sources and sinks and/or doublets are placed

Discretization of body Free-stream Va

surface (source/sink or doublet panels)

(see Fig. 14.4 for an example). The principles are very similar to the 2-D case, but the size of the numerical problem becomes much larger. Assume for illustration that only source singularities are used. If the singularity strength on the ith panel is cr, , then the velocity induced by this panel at the control point on the yth panel can be written as A, j a,-. At the jth control point, the component normal to the surface of the velocity induced by all N sources is then

N

Vj = Y A‘j ai for/ Ф J – (14.40)

/=i

The boundary condition of flow tangency is imposed at the control point of each panel to allow the unknown singularity strengths о,- to be evaluated. So, for instance, if is the velocity at panel j as a result of the motion of the fuselage, the rotor-induced velocity and so on, the component of VCXj normal to panel j will be • hj, where hj is the unit normal vector to panel j. Therefore, at the jth panel

N

Y, Aij a і + Voo. • Hj = 0. (14.41)

/=1

One equation such as this applies on each of the N panels. This leads to a set of linear simultaneous equations that can be used to solve for the singularity strengths, a, , using standard numerical methods of linear algebra. The governing equations can be written in matrix form as

[Aij]{ai} = -{Voo – nj},

with solution

{ct,) = [Aij]_1 {-Voo-ny},

which involves finding the inverse of the influence coefficient matrix. Once the values of ot are obtained, the velocity components tangential to the panels can be obtained, and hence the pressure distribution over the fuselage follows using Bernoulli’s equation.

Normally thousands of panels are required to resolve adequately the complex shape of a helicopter fuselage and care must be taken, for instance, to concentrate a sufficient number of panels in regions of high surface curvature such as near the rotor hub. Because the numerical cost of panel methods is of order N3, their use is by no means inexpensive even on a modem computer. But they are considerably less computer intensive than most other types of CFD methods, however. As an enhancement to the basic panel method, the solution of the momentum integral equation from boundary layer theory allows displacement corrections to be made to solutions obtained from the panel method, thereby allowing viscous effects (but not flow separation) to be modeled. The effects of flow separation can be included, however, by extending the surface singularities off the airframe as free shear layers – see Polz (1982), Le et al. (1987), de Bruin (1987), Ahmed et al. (1988), Ahmed (1990), and Katz & Plotkin (1991). Overall, if suitable care is taken in the distribution of panels on the surface of the fuselage, the 3-D surface singularity method can be used to obtain quite credible predictions of the loading on an airframe – certainly to the fidelity required for preliminary design (see also Section 14.10.4). See Chaffin & Berry (1994) for a further discussion of the predictive capabilities of panel methods compared to measurements and other forms of analysis.