Integral solution methods

Instead of computing the detailed u, v, т(s, n) fields, integral methods determine only the integral thicknesses and key shear quantities, namely 5*, 6, Cf, cD(s), etc. This relatively small number of unknowns makes the integral methods very economical, but a drawback is that their solutions must always be approximate, in
that they cannot produce theoretically exact solutions of the laminar boundary layer equations (4.21). For turbulent flows this is not really an issue, since even nominally “exact” solutions of equations (4.21) still require turbulence models for jt which are inherently approximate. In practice, the simpler and much more economical integral methods are sufficiently accurate for a large majority of aerodynamic flow prediction applications, for both laminar and turbulent flows.

To compute the integral thicknesses 5*, 6, 6*(s), integral methods solve either the von Karman equation (4.28) or the kinetic energy equation (4.35), or both as in some advanced methods. In effect, they seek to evaluate the integrals in (4.36) and/or (4.38) in some manner. Here we will focus on solving only the von Karman equation (4.28). This equation is not integrable as written, because it contains the terms Cf and H which are additional unknowns, and therefore require two additional closure relations or functions to relate them to the primary variables 6,ue, v. How these additional unknowns are determined is primarily what distinguishes the many different integral calculation methods which have been developed to date.