Theory (concluded)

Boundary-Layer Method

The laminar and turbulent boundary-layer development is computed using integral momentum and energy equations. The approximate solutions obtained from the laminar boundary-layer method agree very well with exact solutions. The turbulent boundary-layer method is based on the best available empirical skin-friction, dissipation, and shape-factor laws.

Of special interest are the predictions of separation and transition. The prediction of separation is determined by the shape factor based on energy and momentum thicknesses. (Note that this shape factor has the opposite tendency of the shape factor based on displacement and momentum thicknesses.) For laminar boundary layers, there exists a constant and reliable lower limit of this shape factor, which equals 1.515 and corresponds to laminar separation. For turbulent boundary layers, no such unique and reliable limit exists. It has been determined, however, that the turbulent boundary layer will separate if the shape factor falls below 1.46 and will not separate if the shape factor remains above 1.58. It has also been determined that thicker boundary layers tend to separate at lower shape factors. The uncertainty is not a significant disadvantage because the shape factor changes rapidly near separation. Nevertheless, results must be checked carefully with respect to turbulent separation.

The prediction of transition is based on an empirical criterion that contains the Reynolds number, based on local conditions and momentum thickness, and the shape factor. Previously, the transition criterion used was a local criterion. Recently, a new empirical transition criterion has been implemented that considers the instability history of the boundary layer. The results predicted using the new criterion are comparable to those using the e^n method but the computing time is negligible. The criterion contains a "roughness factor" that allows various degrees of surface roughness or free-stream turbulence to be simulated. The prediction of transition results in a switch from the laminar skin-friction, dissipation, and shape-factor laws to the turbulent ones, without changing the shape factor or the momentum thickness. Also, a procedure has recently been incorporated into the code that empirically estimates the increase in the boundary-layer thickness due to laminar separation bubbles; this procedure yields an additional "bubble drag."

The code contains an option that allows the analysis of the effect of single roughness elements on a turbulent as well as a laminar boundary layer. For the laminar case, transition is assumed to occur at the position of the roughness element. This simulates fixing transition by roughness in a wind tunnel or in flight.

The lift and pitching-moment coefficients are determined from the potential flow. Viscous corrections are then applied to these coefficients. The lift-curve slope where no separation is present is reduced to 2*pi from its theoretical value. In other words, the potential-flow thickness effects are assumed to be offset by the boundary-layer displacement effects. A lift-coefficient correction due to separation is also included. As an option, the displacement effect on the velocity distributions and the lift and pitching-moment coefficients can be computed. The boundary-layer characteristics at the trailing edge are used for the calculation of the profile-drag coefficient by a Squire-Young type formula. In general, the theoretical predictions agree well with experimental measurements. (See ref. 3, for example.)

The code contains an option that allows aircraft-oriented boundary-layer developments to be computed, where the Reynolds number and the Mach number vary with aircraft lift coefficient and the local wing chord. In addition, a local twist angle can be input. Aircraft polars that include the induced drag and an aircraft parasite drag can also be computed.

© 2000 Airfoils, Incorporated.