A theory of three-dimensional incompressible flow separation is presented in terms of the on-wall signatures of the flow. Some long-standing controversial issues are revisited and answers given, such as the inconsistency of the separation criteria based on the topological theory and “open separation,” and whether a separation line is an asymptote or envelope of neighboring skin-friction lines. General criteria for identifying an “open” or “closed” flow separation zone and separation line (including the initial point of the latter), steady and unsteady, are obtained, which apply to a generic smooth curved wall at any Reynolds numbers. The criteria are found to be most clearly given in terms of on-wall signatures of vorticity dynamics. These are then specified to steady boundary layer separation at large Reynolds numbers. A scale analysis under mild assumptions leads to a three-dimensional triple-deck structure near a generic boundary layer separation line. Criteria are presented for “separation watch,” which tells that a boundary-layer may soon break away, and for “separation warning,” which identifies the vorticity characteristics in an already formed boundary-layer separation zone and along a boundary-layer separation line. Flow behavior and its dependence on outer flow conditions are examined qualitatively. A numerical example is given which confirms the predictions of the theory.

Topics
Differential geometry, Incompressible flow, Fluid drag, Vortex dynamics
1.
L.
Li
,
J. D. A.
Walker
,
R. I.
Bowles
, and
F. T.
Smith
, “
Short-scale break-up in unsteady interactive layers: Local development of normal pressure gradient and vortex wind-up
,”
J. Fluid Mech.
374
,
335
(
1998
).
Google Scholar
Crossref
Search ADS
 
2.
M. J. Lighthill, “Introduction. Boundary layer theory,” in Laminar Boundary Layers, edited by L. Rosenhead (Clarendon, Oxford, 1963), pp. 46–113.
3.
M.
Tobak
and
D. J.
Peake
, “
Topology of three-dimensional separated flows
,”
Annu. Rev. Fluid Mech.
14
,
61
(
1982
).
Google Scholar
Crossref
Search ADS
 
4.
H. X.
Zhang
, “
The separation criteria and flow behavior for three-dimensional steady separated flow
,”
Acta Aerodyn. Sinica
1
(
1
),
1
(
1985
) (in Chinese).
Google Scholar
 
5.
J. Z.
Wu
,
J. W.
Gu
, and
J. M.
Wu
, “
Steady three-dimensional fluid-particle separation from arbitrary smooth surface and formation of free vortex layers
,”
Z. Flugwiss. Weltraumforsch.
12
,
89
(
1988
).
Google Scholar
 
6.
P. G. Bakker, Bifurcations in Flow Patterns (Kluwer, Dordrecht, 1991).
7.
X. Q. Xia and X. Y. Deng, Dynamics of Engineering Separated Flows (Beijing University Aero. Astro. Press, Beijing, 1991) (in Chinese).
8.
K. C.
Wang
, “
Three-dimensional boundary layer near the plane of symmetry of a spheroid at incidence
,”
J. Fluid Mech.
43
,
187
(
1970
).
Google Scholar
Crossref
Search ADS
 
9.
K. C.
Wang
,
H. C.
Zhou
,
C. H.
Hu
, and
S.
Harrington
, “
Three-dimensional separated flow structure over prolate spheroids
,”
Proc. R. Soc. London, Ser. A
421
,
73
(
1990
).
Google Scholar
 
10.
K. C. Wang, “Features of three-dimensional separation and separated flow structure,” Flow at Large Reynolds Numbers, edited by H. Schmitt (Comput. Mech. Publications, 1997), pp. 1– 33.
11.
V. V. Sychev, A. I. Rubin, Vic. V. Sychev, and G. L. Korolev, Asymptotic Theory of Separated Flows (Cambridge University Press, Cambridge, 1998).
12.
F. T.
Smith
, “
Steady and unsteady boundary-layer separation
,”
Annu. Rev. Fluid Mech.
18
,
197
(
1986
).
Google Scholar
Crossref
Search ADS
 
13.
F. T. Smith, “Three-dimensional viscous and inviscid separation of a vortex sheet from a smooth non-slender body,” RAE Tech. Report No. 78095, 1978.
14.
F. T. Smith, “On physical mechanisms in two- and three-dimensional separations,” Philos. Trans. R. Soc. London, Ser. A (2000) (in press).
15.
K. C.
Wang
, “
On the dispute about open separation
,”
AIAA Pap.
83
–
0296
(
1983
).
Google Scholar
 
16.
J. Z.
Wu
and
J. M.
Wu
, “
Vorticity dynamics on boundaries
,”
Adv. Appl. Mech.
32
,
119
(
1996
).
Google Scholar
Crossref
Search ADS
 
17.
J. Z.
Wu
and
J. M.
Wu
, “
Boundary vorticity dynamics since Lighthill’s 1963 article: Review and development
,”
Theor. Comput. Fluid Dyn.
10
,
459
(
1998
).
Google Scholar
Crossref
Search ADS
 
18.
L.
Prandtl
, “
Über Flüssigkeitsbewegung bei sehr kleiner Reibung
,”
Verh. III Int. Math. Kongr.
,
484
(
1904
).
Google Scholar
 
19.
S.
Goldstein
, “
On laminar boundary-layer flow near a position of separation
,”
Q. J. Mech. Appl. Math.
1
,
43
(
1948
).
Google Scholar
Crossref
Search ADS
 
20.
R. L. Panton, Incompressible Flow, 2nd ed. (Wiley, New York, 1996).
21.
J. M.
Hamilton
,
J.
Kim
, and
F.
Waleffe
, “
Regeneration mechanisms of near-wall turbulent structures
,”
J. Fluid Mech.
287
,
317
(
1995
).
Google Scholar
Crossref
Search ADS
 
22.
G. T.
Chapman
and
L. A.
Yates
, “
Topology of flow separation on three-dimensional bodies
,”
Appl. Mech. Rev.
44
,
329
(
1991
).
Google Scholar
Crossref
Search ADS
 
23.
U. Dallmann and B. Schulte-Werning, “Topological changes of axisymmetric and non-axisymmetric vortex flows,” in Proceedings of the IUTAM Symposium on Topological Fluid Mechanics, edited by H. K. Moffat and A. Tsinober (Cambridge University Press, Cambridge, 1990), pp. 372–383.
24.
In addition to a couple of algebraic errors which have been corrected here, Ref. 5 is limited by the concept that any separation line needs to initiate from a fixed point. The authors thus proposed that a separation line may initiate from a higher-order saddle-nodal point so that it looks like an open separation line, although they were aware of the inherent instability and transient nature of such a fixed point. This attempt was unsuccessful and not necessary.
25.
M.
Tobak
, “
Topologically derived separation onset conditions for two- and three-dimensional laminar flows
,”
AIAA Pap.
97
-
0866
(
1997
).
Google Scholar
 
26.
U.
Dallmann
,
Th.
Herberg
,
H.
Gebing
,
W. H.
Su
, and
H. Q.
Zhang
, “
Flow field diagnostics: Topological flow changes and spatio-temporal flow structure
,”
AIAA Pap.
95
–
0791
(
1995
).
Google Scholar
 
27.
W. H. Su (private communication).
28.
F. T. Smith (private communication).
29.
H. X.
Zhang
, “
The behavior of separation line for three-dimensional steady viscous flow—based on boundary layer equations
,”
Acta Aerodyn. Sinica
1
(
4
),
1
(
1985
) (in Chinese).
Google Scholar
 
30.
R. W.
Tramel
and
R. H.
Nochols
, “
A high efficient numerical method for overset-mesh moving-body problems
,”
AIAA Pap.
97
–
2040
(
1997
).
Google Scholar
 
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.