By using the RANS boundary layer equations, it will be shown that the outer part of an adverse pressure gradient turbulent boundary layer tends to remain in equilibrium similarity, even near and past separation. Such boundary layers are characterized by a single and constant pressure gradient parameter, Λ, and its value appears to be the same for all adverse pressure gradient flows, including those with eventual separation. Also it appears from the experimental data that the pressure gradient parameter, $Λθ,$ is also approximately constant and given by $Λθ=0.21±0.01.$ Using this and the integral momentum boundary layer equation, it is possible to show that the shape factor at separation also has to within the experimental uncertainty a single value: $Hsep≅2.76±0.23.$ Furthermore, the conditions for equilibrium similarity and the value of $Hsep$ are shown to be in reasonable agreement with a variety of experimental estimates, as well as the predictions from some other investigators.

1.
Simpson
,
R. L.
, and
Strickland
,
J. H.
,
1977
, “
Features of a separating turbulent boundary layer in the vicinity of separation
,”
J. Fluid Mech.
,
79
, pp.
553
594
.
2.
Simpson
,
R. L.
, and
Chew
,
Y. T.
,
1981
, “
The structure of a separating turbulent boundary layer. Part 1. Mean flow and Reynolds stresses
,”
J. Fluid Mech.
,
113
, pp.
23
51
.
3.
Kline
,
S. J.
,
Bardina
,
J. G.
, and
Strawn
,
R. C.
,
1983
, “
Correlation of the detachment of two-dimensional turbulent boundary layers
,”
AIAA J.
,
21
, pp.
68
73
.
4.
Simpson
,
R. L.
, and
Chew
,
Y. T.
,
1981
, “
The structure of a separating turbulent boundary layer. Part 2. Higher-order turbulence results
,”
J. Fluid Mech.
,
113
, pp.
53
73
.
5.
Dengel
,
P.
, and
Fernholz
,
H. H.
,
1990
, “
An experimental investigation of an incompressible turbulent boundary layer in the vicinity of separation
,”
J. Fluid Mech.
,
212
, pp.
615
636
.
6.
Alving
,
A. E.
, and
Fernholz
,
H. H.
,
1995
, “
Turbulence measurements around a mild separation bubble and down-stream of reattachment
,”
J. Fluid Mech.
,
322
, pp.
279
328
.
7.
Driver, D. M., 1991, “Reynolds Shear stress measurements in a separated boundary layer flow,” AIAA-91-1787, 22nd Fluid Dynamics, Plasma Dynamics, and Lasers Conference.
8.
Schofield
,
W. H.
,
1986
, “
Two-dimensional Separating Turbulent Boundary layers
,”
AIAA J.
,
24
, No.
10
, pp.
1611
1620
.
9.
Stratford
,
B. S.
,
1959
, “
The prediction of separation of the turbulent boundary layer
,”
J. Fluid Mech.
,
5
, pp.
1
16
.
10.
Tennekes, H., and Lumley, J. L., 1972, A First Course in Turbulence, MIT Press, Cambridge, MA.
11.
George, W. K., and Castillo, L., 1993, “Boundary Layers with Pressure Gradient: Another look at the equilibrium boundary layer,” Near Wall Turbulence (So, R. et al. eds.), pp. 901–910, Elsevier, NY.
12.
Skote
,
M.
, and
Henningson
,
D. S.
,
2002
, “
Direct numerical simulation of a separated turbulent boundary layer
,”
J. Fluid Mech.
,
471
, pp.
107
136
.
13.
Sandborn
,
V. A.
, and
Kline
,
S. J.
,
1961
, “
Flow models in boundary-layer stall inception
,”
J. Basic Eng.
,
83
, pp.
317
327
.
14.
Sajben
,
M.
, and
Liao
,
Y.
,
1995
, “
Criterion for the detachment of laminar and turbulent boundary layers
,”
AIAA J.
,
33
, No.
11
, pp.
2114
2119
.
15.
Cebeci, T., and Bradshaw, P., 1977, Momentum Transfer in Boundary Layers, McGraw Hill Book Co., New York, pp. 194.
16.
Senoo
,
Y.
, and
Nishi
,
M.
,
1975
, “
Prediction of flow separation in a diffuser by a boundary layer calculation
,”
ASME J. Fluids Eng.
,
99
, pp.
379
389
.
17.
Gustavsson, J., 1998, “Experiments on turbulent flow separation,” Master’s thesis, Royal Institute of Technology, Sweden.
18.
Perry
,
A. E.
, and
Schofield
,
W. H.
,
1973
, “
Mean velocity and shear stress distributions in turbulent boundary layers
,”
Phys. Fluids
,
16, 12
, pp.
2068
2074
.
19.
Angele, K., and Muhammad-Klingman, B. 2003 “Self-similarity velocity scalings,” in Ph.D. dissertation of K. Angele, ‘Experimental studies of turbulent boundary layer separation and control,’ Dept. of Mechanics, KTH, Stockholm, Sweden, 81–105.
20.
Simpson
,
R. L.
,
1989
, “
Turbulent boundary-layer separation
,”
Annu. Rev. Fluid Mech.
,
21
, pp.
205
234
.
21.
Castillo
,
L.
, and
George
,
W. K.
,
2001
, “
Similarity analysis for turbulent boundary layer with pressure gradient: outer flow
,”
AIAA J.
,
39
, No.
1
, pp.
41
47
.
22.
Elsberry
,
K.
,
Loeffler
,
J.
,
Zhou
,
M. D.
, and
Wygnanski
,
I.
,
2000
, “
An experimental study of a boundary layer that is maintained on the verge of separation
,”
J. Fluid Mech.
,
423
, pp.
227
262
.
23.
George
,
W. K.
, and
Castillo
,
L.
,
1997
, “
Zero-pressure-gradient turbulent boundary layer
,”
Appl. Mech. Rev.
,
50
, pp.
689
729
.
24.
George, W. K., 1995, “Some new ideas for similarity of turbulent shear flows,” Turbulence, Heat and Mass Transfer, ed. by K. Hanjalic and J. C. F. Pereira, Begell House, NY.
25.
Castillo, L., 1997, “Similarity Analysis of turbulent boundary layers,” Ph.D. Dissertation, Department of Mechanical and Aerospace Engineering, SUNY/Buffalo, Buffalo, NY.
26.
Schubauer, G. B., and Klebanoff, P. S., 1951, “Investigation of separation of the turbulent boundary layer,” NACA Report 1030, NACA Technical Note 2133.
27.
Newman, B. G., 1951, “Some Contributions to the Study of the Turbulent Boundary Near Separation,” Austr. Dept. Supply Rep. ACA-53.
28.
Ludweig, H., and Tillmann, W., 1950, “Investigations of the wall shearing stress in turbulent boundary layers,” NACA TM 1285.
29.
Ha˚ll, U., 2003 private communication to WKG by U. Ha˚ll, Professor of Turbomachinery at Chalmers Univ. of Technology, Gothenburg, Sweden.
30.
Ska˚re
,
P. E.
, and
,
P.
,
1994
, “
A turbulent equilibrium boundary layer near separation
,”
J. Fluid Mech.
,
272
, pp.
319
348
.
31.
Coles, D. E., and Hirst, E. A., 1968, Computational of Turbulent Boundary Layers-1968 AFOSR-IFP-stanford conference, Vol. II.
32.
Marusic
,
I.
, and
Perry
,
A. E.
,
1995
, “
A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support
,”
J. Fluid Mech.
,
298
, pp.
389
407
.
You do not currently have access to this content.