Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

304 Appendix E
In Fig. E.2, Cp is the coe cient of pressure; s the coordinate along the surface;
the boundary layer thickness; the momentum thickness; Cf the skin friction
coe cient; H the velocity pro®le shape parameter; the density; VN the transpiration
velocity; K is a factor developed from stability analysis; the subscript v
marks the viscous boundary layer region; the displacement thickness; the
superscript i indicates inviscid region and the superscript m indicates the current
iteration.
Following are useful relations for some of the above quantities:
H ˆ ; ˆ
… 1
0
1 ÿ u
vu v
dn; K ˆ 2
; ˆ 1 ÿ M2 p
…E:1†
where n is the normal direction from the wing surface.
We have the following equations to be solved in the integral boundary layer lagentrainment
model.
Continuity
Momentum
Lag-entrainment
dC e
ds
dH
ds
ˆ F
ˆ dH
dH C e ÿ H 1
Cf 2 ÿ…H ‡ 1† duv uv ds
d
ds ˆ Cf 2 ÿ…H ‡ 2 ÿ M2 duv †
uv ds
ÿ uv
2:8
…C †
H ‡ H1 0:5
EQ ÿ C o 0:5
ÿ
f
‡ uv
du v
ds …1 ‡ 0:075M2 † …1 ‡ 0:2M2 †
…1 ‡ 0:1M 2 †
where F is a function of C e and C f and given as
F ˆ
0:02C e ‡ C 2 e ‡ 0:8C f
3
…0:01 ‡ C e†
du v
ds EQ
…E:2†
…E:3†
…E:4†
…E:5†
In the above equations, H and H1 are the velocity pro®le shape parameters de®ned as
H ˆ 1 …1 0
1 ÿ u
uv dn; H1 ˆ ÿ
…E:6†
Ce is the entrainment coe cient; uv the mean component of the streamwise velocity at
the edge of the boundary layer; M the Mach number; C the shear stress coe cient;
the scaling factor on the dissipation length; the subscripts EQ and EQo denote
respectively the equilibrium conditions and equilibrium conditions in the absence
of secondary in¯uences on the turbulence structure.