Linear Doublet Method

The method of the previous section can be further simplified by equating the source strengths to zero in Eq. (11.60). The value of the constant for the internal potential is selected to be zero and consequently the boundary condition describing the internal potential (Eq. (11.60)) reduces to

!сЛ + Ф. = 0 (11.81)

;=i

where Л/, is the number of singularity strength parameters and

Ф„ = LLx + W„z (11.82)

Again, note that now ju will represent the potential jump from zero to Фи on the boundary (see Fig. 11.26) and therefore Ф„ is the local total potential (whereas in the previous example ц was the jump in the perturbation potential only).

Equation (11.81) can be specified at each collocation point inside the body, providing a linear algebraic equation for this point. The steps toward establishing such a numeric solution are very similar to those of the previous method.

Selection of singularity element. The potential at an arbitrary point P due to the y’th linearly varying strength doublet element with the two edge values of yt, and ju/+1 is given by Eq. (11.116):

Recall that the superscripts ( )“ and ( )b represent the panel influence contributions due to the leading and trailing doublet strengths, respectively.

Discretization of geometry. The N + 1 panel corner points and N collocation points are generated in a manner similar to the previous examples (see Fig.

11.18) . In this case of the internal Dirichlet boundary condition the collocation points must be placed inside the body. This small inward displacement of the collocation point can be skipped if the panel self induced influence is specified in a separate formula.

Influence coefficients. The influence of this doublet panel is calculated exactly as in the previous section. The velocity potential at each point is the sum of all the individual panel influences and, for example, for the first panel is given by

Eq. (11.117):

фді = (Ф? іМі + Фи^г) + (Ф?2^2 + ФЇгМз)

Н—– 1-(Фїл^л/+ Філг/*л/+і) + Фiwl*w (11.117)

where fiw is the constant-strength wake doublet element (as in Section 11.3.1). The strength of this wake doublet element is set by applying the Kutta condition at the trailing edge and is given by Eq. (11.118).

Defining the influence coefficients c, y as in the previous section the following N + 1 influence relations and the (N + 2)th Kutta condition can be summarized in a matrix as in Eq. (11.119):

(11.123)

Establish RHS vector. The second term in this equation is known and can be transferred to the right-hand side of the equation. The RHS vector then becomes

RHSi
RHS2

|RHS„+I|
0

and the Ф^ term is calculated by using Eq. (11.82).

Solve equations. Substituting the influence coefficients of the doublets and the

RHS vector into the boundary condition of Eq. (11.81) we get

(11.125)

In the case that both of Eqs. (11.118) and (11.118a) are applied at the trailing edge then the boundary condition is specified only at the N collocation points (see previous section). Consequently with the use of Eq. (11.119a) the equations to be solved become

(11.125a)

Either of these full-matrix equations with N + 2 unknown values ju* can be solved by standard matrix solvers. Note that in this case (compared to Eq. (11.121)) the doublet represents the jump in the total potential (and not the perturbation only).

Calculation of pressures and loads. Once the values of the doublet parameters (fiu , nN+i) are known, the tangential velocity component at each colloca­tion point can be calculated by differentiating the local potential. For example, such a two-point formula is

g,. = ^— ^ (11.126)

‘ xi+l-xj

and the pressure coefficient can be calculated by using Eq. (11.18). The lift and pitching moment of the panel can be obtained by using the method described by Eqs. (11.78-11.80).

This method seems to involve less numerical calculations than the equivalent linear doublet/source method and therefore will require somewhat less computational time. (A computer program based on this method is presented in Appendix D, Program No. 9.)