Midchord laminar separation bubbles, which act as a transition mechanism on low Reynolds number airfoils, make a contribution to wing section profile drag that becomes increasingly important at low Reynolds number. A model for the analysis of the boundary layer through the bubble is needed. The model developed here, which is based on Horton's method, provides a simple computationally efficient analysis that matches the integral boundary-layer analysis methods used on most existing boundary-layer codes. The bubble calculation is initiated by the detection of laminar separation. Transition location and boundary-layer growth in the laminar region are determined using Van Ingen's shortcut e" method and Schmidt's correlations, respectively. Following Horton, the turbulent region is calculated using an iterative scheme that also functions as a bursting criterion, but the original linear velocity distribution has been replaced by Wortmann's concave velocity distribution. Both computation efficiency and prediction accuracy were improved after this change. Testing against experimental data showed that the bubble model greatly improved the drag prediction accuracy of the analysis in the Reynolds number range from 0.2 to 1.5 x 10 6 , especially in cases when the midchord bubble was dominant. The validity of the bubble model was further confirmed by accurate prediction of bubble size and reattachment velocity gradient.
B-Nomenclature laminar separation angle parameter drag coefficient dissipation integral, 2 /« r(du/drj) skin friction coefficient, [2r/(pw^)] TJ=0 lift coefficient chord shape factor, 81/82 shape factor, 8 3 /8 2 maximum amplification integral total bubble length, l v + / 2 length of bubble laminar region, S T -s s length of bubble turbulent region, S R --S T velocity distribution exponent defined in Eq. (9) exponential growth of Tollmien-Schlichting waves in the e" method chord Reynolds number, u^clv distance Reynolds number, u^slv momentum thickness Reynolds number, u e 8 2 /v distance along surface from stagnation point freestream turbulence level velocity component, tangential to surface external velocity position along the chord line angle of attack velocity distribution parameter used in Eq. (8) displacement thickness, /o (1 -u/u e ) drj momentum thickness, Jo (u/u e )[l -(u/u e )] dr; energy thickness, /" (ulu e ][l -(ulu e ) 2 ] dr/ distance normal to airfoil surface velocity gradient parameter, (8 2 /u e )(du e /ds) Pohlhausen velocity gradient parameter, P T = density = shear stress v = kinematic viscosity Subscripts m = mean value R = reattachment S = separation T = transition Superscripts = lengths and velocities normalized by 8 2S and u es , respectively A = lengths and velocities normalized by 8 2T and u eT , respectively
