# Induced Drag from Span Load Distribution

The induced drag may be computed from the span load distribution on a planar wing. Most books on aerodynamics show how to do this if you know the analytical form of the loading function. This algorithm enables you to solve the same problem if you only know a few values of this loading function.

In most textbooks on aerodynamics that treat the subject of finite-span wing theory, one usually sees a development of the concept of computing the induced drag of the wing from a knowledge of the spanwise load distribution. The load distribution is a function that defines ccl, i.e., the local chord multiplied by the local lift coefficient at each chord of the wing. This is done by developing the loading as a Fourier sine series and showing that the drag may be calculated from the coefficients of the series.

When a practising engineer is faced with making a numerical calculation of induced drag and plans to use this technique, the textbooks don't give very much support because one rarely has a mathematical expression for the span loading. Instead one usually has a few sparsely spaced values from a wind tunnel or flight test or a theoretical result from a grid that is usually more coarse than desired. The few examples where a numerical example is worked out usually suggest that the number of terms in the series expansion must be the same as the number of data points, and this is rarely enough to insure convergence.

In a note in the Journal of Aircraft, 1977, p309, Jerry Lundry described a simple algorithm for computing a curve that in one sense is the smoothest that exactly matches the data points and produces the Fourier sine coefficients as part of the solution. This technique and other similar algorithms are used widely by specialists but do not seem to be well documented and available to students. The routines given here are a coding of Lundry's equations 3 and 5 for asymmetric and symmetric loadings.

The essential routines are encapsulated in a Fortran module. The coefficients are computed with a call to subroutine ComputeFourierCoefficients. The drag may then be computed from the coefficients by use of the function DragFromCoefficients. Lift coefficient may also be computed. It is simply PI*span*span*coeff(1)/sref. If you just want the induced drag and don't want to be bothered with the coefficients, you may use the functions AsymmetricLoadingInducedDrag and SymmetricLoadingInducedDrag. They take the loadings and return D/q. Be sure to divide by the reference area if you want drag coefficient. Once you have compiled the module, you have access to any of the routines by putting the statement USE InducedDrag in your application programs.

For those folks who want a stand-alone program and don't want to compile their own, I have supplied a program called induced.exe that reads the essential loading information from a file and produces a file with the essential results that is designed to be printed.

You may note that the code is written in Fortran 90. This was done to make use of dynamic storage allocation . If the loads are known at N points and you use M terms in the Fourier series, then you need to build a square matrix with M+N rows and columns to perform the solution. If we were to code this with Fortran 77, we would need some technique of passing this work array into the various routines that require it. With modern Fortran, you just allocate the memory when you need it and deallocate it when you are finished. First class!