Appendix B Closure for Three-Dimensional Boundary Layer Equations B.1 1-2 Coordinate Definitions s Streamwise Direction 2→ Crossflow Direction /(1-2)2g dn dn le/pe ge/pe e 6 Pe qe 6=/1-2d E 11 E 12 Pe ge pe qe E21 E ≌2)2P≌ 22 dn ge/pe ge 6i=E11+E21 62=E12+E22
B 2 -z Coordinate definitions Using a rotation matrix, the z-z thicknesses may be determined in terms of the 1-2 thick- nesses and the angle between the two coordinate systems(cos a= g sin a=9)(38,49 pe Pete Pe9202z=peu2 128 Pelee = pe 61 u22+p2u261 611-62) (612+621) p96=pte61-p2t22 Pe9e8:=Pele&2+Pewd1 :92"= Pe"" 92 5"=Pete?"+Pew S5" 9 0:=9(peue8i-Pe w 02 0:=92(Peu:+P. 8i) TIdu+T2d B 3 Crossflow Model The crossflow model is Johnston's triangular profile [28] 1) where As is the crossflow parameter. Streamwise-crossflow thicknesses may now be defined
B 4 Derived Thicknesses P pe qe (B2) A P≌)d+/A A(6,-6:) u2 p u1 le pe ge A e p ul (B4) Pe qe a=/(-2)2 12 pe ge 22砌n+ (B5) ge pe ge ge 62 12 p 12 ge pe ge =/-4:(1-2)2 (B6) ge/ pe ge A61
Pp A6-61] E12=612+A(2611-E1) (B8) E21=-622-AE12 E22=-A(E21-62) (B10) B5 Empirical Closure Relations The following closure relations are taken from subroutines of Drela. The references for these relations are labeled along with the formula Shape Parameters H≡ He (B11) H E 11 61