Formulations
Please note that this ‘page’ originally did not exist in the table of contents, but I combined the various formulations as presented in there on one page here.
Table of contents
Inviscid
The inviscid formulation of XFOIL is a simple linear-vorticity stream function panel method. A finite trailing edge base thickness is modeled with a source panel. The equations are closed with an explicit Kutta condition. A high-resolution inviscid calculation with the default 160 panels requires seconds to execute on a RISC workstation. Subsequent operating points for the same airfoil but different angles of attack are obtained nearly instantly.
A Karman-Tsien compressibility correction is incorporated, allowing good compressible predictions all the way to sonic conditions. The theoretical foundation of the Karman-Tsien correction breaks down in supersonic flow, and as a result accuracy rapidly degrades as the transonic regime is entered. Of course, shocked flows cannot be predicted with any certainty.
Inverse
There are two types of inverse methods incorporated in XFOIL: Full-Inverse and Mixed-Inverse. The Full-Inverse formulation is essentially Lighthill’s and van Ingen’s complex mapping method, which is also used in the Eppler code and Selig’s PROFOIL code. It calculates the entire airfoil geometry from the entire surface speed distribution. The Mixed-Inverse formulation is simply the inviscid panel formulation (the discrete governing equations are identical) except that instead of the panel vortex strengths being the unknowns, the panel node coordinates are treated as unknowns wherever the surface speed is prescribed. Only a part of the airfoil is altered at any one time, as will be described later. Allowing the panel geometry to be a variable results in a non-linear problem, but this is solved in a straightforward manner with a full-Newton method.
Viscous
The boundary layers and wake are described with a two-equation lagged dissipation integral BL formulation and an envelope e^n transition criterion, both taken from the transonic analysis/design ISES code. The entire viscous solution (boundary layers and wake) is strongly interacted with the incompressible potential flow via the surface transpiration model (the alternative displacement body model is used in ISES). This permits proper calculation of limited separation regions. The drag is determined from the wake momentum thickness far downstream. A special treatment is used for a blunt trailing edge which fairly accurately accounts for base drag.
The total velocity at each point on the airfoil surface and wake, with contributions from the freestream, the airfoil surface vorticity, and the equivalent viscous source distribution, is obtained from the panel solution with the Karman-Tsien correction added. This is incorporated into the viscous equations, yielding a nonlinear elliptic system which is readily solved by a full-Newton method as in the ISES code. Execution times are quite rapid, requiring about 10 seconds on a RISC workstation for a high-resolution calculation with 160 panels. For a sequence of closely spaced angles of attack (as in a polar), the calculation time per point can be substantially smaller.
If lift is specified, then the wake trajectory for a viscous calculation is taken from an inviscid solution at the specified lift. If alpha is specified, then the wake trajectory is taken from an inviscid solution at that alpha. This is not strictly correct, since viscous effects will in general decrease lift and change the trajectory. This secondary correction is not performed, since a new source influence matrix would have to be calculated each time the wake trajectory is changed. This would result in unreasonably long calculation times. The effect of this approximation on the overall accuracy is small, and will be felt mainly near or past stall, where accuracy tends to degrade anyway. In attached cases, the effect of the incorrect wake trajectory is imperceptible.