LIGHTHILL TRANSPIRATION VELOCITY REVISITED - AN EXACT FORMULATION

The Lighthill transpiration-velocity correction is commonly used in th e analysis of viscous flows of aeronautical interest, in order to take into account the perturbation to the potential flow caused by the pre sence of the vorticity in the boundary layer. This correction consists of considering the boundary as a permeable surface from which the flu id flows through the boundary surface, with a velocity (named the tran spiration velocity) determined on the basis of the local boundary-laye r characteristics. Here, we use a new potential-vorticity decompositio n in order to derive an exact representation of the effects of the vor ticity on the external flow. The relationship between approximate tran spiration-velocity representation and the exact one presented here is analyzed: it is shown that, under typical boundary-layer assumptions, the new representation reduces to that by Lighthill, except far a corr ective field term. Finally, in order to quantify, in a simple case, th e contribution of the corrective terms which arise in the new formulat ion, we examine, as a numerical test case, the problem of an attached boundary-layer flow over a flat plate: the numerical results indicate that the corrective term is negligible for Reynolds numbers above 10(4 ).