boundary_layer_equations

# boundary_layer_equations - B.1 Appendix B Closure for...

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Unformatted text preview: B.1 Appendix B Closure for Three-Dimensional Boundary Layer Equations 1-2 Coordinate Deﬁnitions 1 = Streamwise Direction 2 => Crossﬂow Direction P 1‘1 6“=/(1——) —d 1 P: q: n 103 912 = f (1 — '2) il‘ldn ‘1: Pa 4c 11. u = / <——2> n ‘1: P: ‘1: 6; = f—ﬁ-ﬂdn P: qe 9; = E12 + E22 B.2 93-2 Coordinate Deﬁnitions Using a. rotation matrix, the 2-2 thicknesses may be determined in terms of the 1-2 thick- nesses and the angle between the two coordinate systems (cosa = “4, sina = ﬁ‘) [38, 49]. 9: chezoza: = P613911 + 99193622 Peqezezz = peu3912 _ Pew3021 - peucwe (912 + 921) + Peuewz(611 - 922) P613924: = P21113021 — PHI-73912 Pengzz = P2153022 + P2193011 + Peuewe (911 - 922) + Peuewe (912 + 921) Peqeﬁ; = Peue‘s; _ Pewe‘s; Pageg; = Panza; + Peweﬁi peq36: : Peue‘sl” _ Peweag‘ P4136? = pines? + Pewe‘sl. 104139; = 43 (Peue9i‘ - pewe03) peq30: = q? (Pew-Z + M59?) “2 we 11., w, T3w=—T1__T2 Tm: ‘7'2+—7'1 qe Q¢ q: q: D = /7'1du1 + /T2d’u2 B.3 Crossﬂow Model The crossﬂow model is Johnston’s triangular proﬁle [28] U2 1‘1 I]: = A, (1 - q—e) (13.1) where A; is the crossﬂow parameter. Streamwise-crossﬂow thicknesses may now be deﬁned. 104 B.4 Derived Thicknesses 5% II Ac(9p — 6f) 6" 1 Ac = —2 911 Hap - H 3“ ll \ I LE I‘D IE ll \ I h». n A H l | "1) £2 d1, q: Pa 11; II I it. n Q: ... N 105 (3.2) (3.3) (3.4) (3.5) (B.6) 6;“ = [(1 — ﬂ>’-‘ldn P: g: = “I _ 1) Ac (1 _ ﬂ) dn (13.7) pe qe = Aclgp ‘ 61"] E12 = 912 + Ac (2911 — E11) (3-8) E21 = -022 - AcEIZ (3-9) E22 = -Ac(E21 - 922) (3'10) B.5 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 6‘ H E -—1- 911 0 Hop E E:- (B.11) _ 51" H51? = 0—11- E11 H‘ E — 911 106 ...
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boundary_layer_equations - B.1 Appendix B Closure for...

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