The natural transitions of bow boundary layers over underwater axisymmetric bodies are investigated using numerical methods. The laminar flow fields over the underwater axisymmetric bodies are first calculated, and then the linear stability of the boundary layers is analyzed considering both the streamwise and circumferential curvatures of the wall. Based on the stability results, the eN method is employed to predict the transition locations. Numerical calculations are performed for seven forebody shapes under six oncoming flow velocities, allowing the influences of the forebody shapes and the oncoming flow velocities on the transition to be investigated. For the different forebody shapes, the boundary layer stability is generally the same behind the streamwise location of twice the forebody length, but varies within in the range of twice the forebody length. The transition locations are significantly different for the different forebody shapes. As the oncoming flow velocity increases, the dimensional unstable zone expands significantly, and the transition location moves upstream. The SUBOFF forebody shape proposed by Groves et al. [“Geometric characteristics of DARPA SUBOFF models (DTRC model numbers 5470 and 5471),” Report No. DTRC/SHD-1298-01 (David Taylor Research Center, West Bethesda, MD, 1989)] has a particularly late transition location and a large diameter close to the leading edge. This delayed transition location is caused by two separated unstable zones. Considering multiple factors, our analyses indicate that the SUBOFF forebody shape is quite valuable for practical engineering problems.

1.
Arakeri
,
V. H.
, “
A note on the transition observations on an axisymmetric body and some related fluctuating wall pressure measurements
,”
J. Fluids Eng.
97
,
82
86
(
1975
).
2.
Borodulin
,
V. I.
,
Ivanov
,
A. V.
,
Kachanov
,
Y. S.
,
Mischenko
,
D. A.
,
Örlü
,
R.
,
Hanifi
,
A.
, and
Hein
,
S.
, “
Experimental and theoretical study of swept-wing boundary-layer instabilities. Three-dimensional Tollmien–Schlichting instability
,”
Phys. Fluids
31
,
114104
(
2019
).
3.
Bountin
,
D.
,
Maslov
,
A.
, and
Gromyko
,
Y.
, “
Analysis of disturbances in a hypersonic boundary layer on a cone with heating/cooling of the nose tip
,”
Phys. Fluids
30
,
054103
(
2018
).
4.
Cebeci
,
T.
, and
Bradshaw
,
P.
,
Momentum Transfer in Boundary Layer
(
McGraw-Hill Book Co
.,
New York
,
1977
).
5.
Chen
,
J.-Q.
,
Dong
,
S.-W.
,
Chen
,
X.
,
Yuan
,
X.-X.
, and
Xu
,
G.-L.
, “
Stationary cross-flow breakdown in a high-speed swept-wing boundary layer
,”
Phys. Fluids
33
,
024108
(
2021
).
6.
Craik
,
A. D. D.
, “
Nonlinear resonant instability in boundary layers
,”
J. Fluid Mech.
50
,
393
413
(
1971
).
7.
Drazin
,
P. G.
, and
Reid
,
W. H.
,
Hydrodynamic Stability
(
Cambridge University Press
,
Cambridge
,
1981
).
8.
Granville
,
P. S.
, “
The calculation of viscous drag of bodies of revolution
,”
Report No. 849
(
Navy Department, The David Taylor Model Basin
,
1953
).
9.
Groves
,
N. C.
,
Huang
,
T. T.
, and
Chang
,
M. S.
, “
Geometric characteristics of DARPA SUBOFF models (DTRC model numbers 5470 and 5471)
,” Report No.
DTRC/SHD-1298-01
(
David Taylor Research Center
,
West Bethesda, ML
,
1989
).
10.
He
,
H.-S.
, “
Calculation of pressure distribution of blunt axisymmetric body
,”
Report No. 85092
(
China Ship Scientific Research Center
,
Wuxi
,
1984
) (in Chinese).
11.
Heisenberg
,
W.
, “
Über Stabilität und Turbulenz von Flüssigkeitsströmen
,”
Ann. Phys.
379
,
577
627
(
1924
).
12.
Herbert
,
T.
, “
Secondary instability of plane channel flow to subharmonic three-dimensional disturbances
,”
Phys. Fluids
26
,
871
874
(
1983
).
13.
Huang
,
C.-L.
,
Yang
,
K.-D.
,
Li
,
H.
, and
Zhang
,
Y.-K.
, “
The flow noise calculation for an axisymmetric body in a complex underwater environment
,”
J. Mar. Sci. Eng.
7
,
323
(
2019
).
14.
Ivanov
,
R. I.
, and
Martin
,
C.
, “
On the time-evolution of resonant triads in rotational capillary-gravity water waves
,”
Phys. Fluids
31
,
117103
(
2019
).
15.
Johnson
,
H. B.
,
Seipp
,
T. G.
, and
Candler
,
G. V.
, “
Numerical study of hypersonic reacting boundary layer transition on cones
,”
Phys. Fluids
10
,
2676
2685
(
1998
).
16.
Kachanov
,
Y. S.
, “
On the resonant nature of the breakdown of a laminar boundary layer
,”
J. Fluid Mech.
184
,
43
74
(
1987
).
17.
Kegelman
,
J. T.
,
Nelson
,
R. C.
, and
Mueller
,
T. J.
, “
The boundary layer on an axisymmetric body with and without spin
,”
AIAA J.
21
,
1485
1491
(
1983
).
18.
Knisely
,
C. P.
, and
Zhong
,
X.
, “
Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. I. Linear stability theory
,”
Phys. Fluids
31
,
024103
(
2019
).
19.
Kumar
,
C.
, and
Prakash
,
A.
, “
Secondary subharmonic instability of hypersonic boundary layer in thermochemical equilibrium over a flat plate
,”
Phys. Fluids
33
,
024107
(
2021
).
20.
Kumar
,
P.
, and
Mahesh
,
K.
, “
Large-eddy simulation of flow over an axisymmetric body of revolution
,”
J. Fluid Mech.
853
,
537
563
(
2018
).
21.
Ladd
,
D. M.
, “
Control of natural laminar instability waves on an axisymmetric body
,”
AIAA J.
28
,
367
369
(
1990
).
22.
Lee
,
S. K.
, and
Jones
,
M. B.
, “
Surface-pressure pattern of separating flows over inclined slender bodies
,”
Phys. Fluids
32
,
095123
(
2020
).
23.
Li
,
F.-X.
,
Duan
,
P.-X.
, and
Zhang
,
Y.-W.
, “
Fore-body shaping design of an axisymmetric body for minimum noise
,”
Acta Acust.
27
,
258
262
(
2002
) (in Chinese).
24.
Li
,
F.-X.
,
Shi
,
X.-H.
, and
Zhang
,
Y.-W.
, “
Effect of length of the transition zone on the sound radiation from the boundary layer of an axisymmetric body
,”
Appl. Acoust.
19
,
28
32
(
2000
) (in Chinese).
25.
Li
,
F.-X.
,
Zhang
,
Y.-W.
, and
Shi
,
X.-H.
, “
Flow noise of an axisymmetric body–the radiated noise due to boundary layer transition
,”
Acta Acust.
24
,
536
543
(
1999
) (in Chinese).
26.
Li
,
X. G.
, and
Yang
,
K.-D.
, “
An analysis of the diffracted sound field from the transition region of an axisymmetric body in water
,”
Tech. Acoust.
30
,
73
74
(
2011
) (in Chinese).
27.
Li
,
X.-G.
,
Yang
,
K.-D.
, and
Wang
,
Y.
, “
The power spectrum and correlation of flow noise for an axisymmetric body in water
,”
Chin. Phys. B
20
,
064302
(
2011a
).
28.
Li
,
X.-G.
,
Yang
,
K.-D.
, and
Wang
,
Y.
, “
The diffracted sound field from the transition region of an axisymmetric body in water
,”
Chin. Phys. B
20
,
074301
(
2011b
).
29.
Liu
,
B.
, and
Zhang
,
Y.-M.
, “
A numerical study on the natural transition locations in the flat-plate boundary layers on superhydrophobic surfaces
,”
Phys. Fluids
32
,
124103
(
2020
).
30.
Liu
,
L.-W.
,
Chen
,
M.-X.
,
Yu
,
J.-W.
,
Zhang
,
Z.-H.
, and
Wang
,
X.-Z.
, “
Full-scale simulation of self-propulsion for a free-running submarine
,”
Phys. Fluids
33
,
047103
(
2021
).
31.
Liu
,
Y.-W.
,
Li
,
Y.-L.
, and
Shang
,
D.-J.
, “
The generation mechanism of the flow-induced noise from a sail hull on the scaled submarine model
,”
Appl. Sci.
9
,
106
(
2018
).
32.
Luo
,
J.-S.
, “
Transition and prediction for hypersonic boundary layers
,”
Acta Aeronaut. Astronaut. Sin.
36
,
357
372
(
2015
) (in Chinese).
33.
Lv
,
S.-J.
,
Miao
,
J.-L.
, and
Zhang
,
X.-W.
, “
Prediction method of hydrodynamic self-noise and design of low noise bow profile for underwater high speed vehicle
,”
J. Hydrodyn.
27
,
303
310
(
2012
) (in Chinese).
34.
Malik
,
M. R.
, “
Numerical methods for hypersonic boundary layer stability
,”
J. Comput. Phys.
86
,
376
413
(
1990
).
35.
Manovski
,
P.
,
Jones
,
M. B.
,
Henbest
,
S. M.
,
Xue
,
Y.
,
Giacobello
,
M.
, and
Silva
,
D. S.
, “
Boundary layer measurements over a body of revolution using long-distance particle image velocimetry
,”
Int. J. Heat Fluid Flow
83
,
108591
(
2020
).
36.
Masad
,
A.
, and
Iyer
,
V.
, “
Transition prediction and control in subsonic flow over a hump
,”
Phys. Fluids
6
,
313
327
(
1994
).
37.
Miao
,
X.-H.
,
Li
,
Y.-H.
,
Pang
,
F.-Z.
,
Xiao
,
J.-P.
, and
Jia
,
D.
, “
Experimental investigation on pulsating pressure of a cone-cylinder-hemisphere model under different flow velocities
,”
Phys. Fluids
32
,
095106
(
2020
).
38.
Nayfeh
,
A. H.
, and
Reed
,
H. L.
, “
Stability of flow over axisymmetric bodies with porous suction strips
,”
Phys. Fluids
28
,
2990
2998
(
1985
).
39.
Orr
,
W. M. F.
, “
The stability or instability of steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid.; Part II: A viscous liquid
,”
Proc. R. Ir. Acad., Sect. A
27
,
9
138
(
1907
).
40.
Pavia
,
G.
,
Varney
,
M.
,
Passmore
,
M.
, and
Almond
,
M.
, “
Three dimensional structure of the unsteady wake of an axisymmetric body
,”
Phys. Fluids
31
,
025113
(
2019
).
41.
Posa
,
A.
, and
Balaras
,
E.
, “
A numerical investigation about the effects of Reynolds number on the flow around an appended axisymmetric body of revolution
,”
J. Fluid Mech.
884
,
A41
(
2020
).
42.
Saric
,
W. S.
,
Reed
,
H. L.
, and
Kerschen
,
E. J.
, “
Boundary layer receptivity to freestream disturbances
,”
Annu. Rev. Fluid Mech.
34
,
291
319
(
2002
).
43.
Schlichting
,
H.
, “
Berechnung der Anfachung kleiner Störungen bei der Plattenströmung
,”
ZAMM
13
,
171
174
(
1933
).
44.
Schmid
,
P. J.
, and
Henningson
,
D. S.
,
Stability and Transition of Shear Flows
(
Springer
,
New York
,
2001
).
45.
Shi
,
B.-J.
,
Yang
,
X.-L.
,
Jin
,
G.-D.
,
He
,
G.-W.
, and
Wang
,
S.-Z.
, “
Wall-modeling for large-eddy simulation of flows around an axisymmetric body using the diffuse-interface immersed boundary method
,”
Appl. Math. Mech.-Engl. Ed.
40
,
305
320
(
2019
).
46.
Smith
,
A.
, and
Gamberoni
,
N.
, “
Transition, pressure gradient and stability theory
,”
Report No. ES 26388
(
Douglas Aircraft Company
,
California
,
1956
).
47.
Sommerfeld
,
A.
, “
Ein Bertrag zur hydrodynamischen Erklarung der turbulenten Flussigkeitsbewegungen
,” in
Roma: Atti del 4. Congresso Internazionale Dei Matematica
(
1908
), Vol.
III
, pp.
116
124
.
48.
Song
,
R.-J.
,
Zhao
,
L.
, and
Huang
,
Z.-F.
, “
Improvement of the parabolized stability equation to predict the linear evolution of disturbances in three-dimensional boundary layers based on ray tracing theory
,”
Phys. Rev. Fluids
5
,
033901
(
2020
).
49.
Stuart
,
J. T.
, “
On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow
,”
J. Fluid Mech.
9
,
353
370
(
1960
).
50.
Taleb
,
A.
,
Haikel
,
B. H.
,
Ouarzazi
,
M.
, and
Hassen
,
B.
, “
Analytical and numerical analysis of bifurcations in thermal convection of viscoelastic fluids saturating a porous square box
,”
Phys. Fluids
28
,
053106
(
2016
).
51.
Tollmien
,
W.
, “
Über die Entstehung der Turbulenz
,”
Nachr. Ges. Wiss. Göttingen
21
44
(
1929
).
52.
Van Ingen
,
J.
, “
A suggested semi-empirical method for the calculation of boundary layer transition region
,”
Report No. VTH-74
(
Delft University of Technology, Department of Aeronautical Engineering
,
1956
).
53.
Wang
,
H.-F.
, and
Chen
,
C.
, “
Stability analysis of a rotor system with fluid applying wave resonance theory
,”
Phys. Fluids
32
,
054106
(
2020
).
54.
Wang
,
S.-Z.
,
Shi
,
B.-J.
,
Li
,
Y.-H.
, and
He
,
G.-W.
, “
A large eddy simulation of flows around an underwater vehicle model using an immersed boundary method
,”
Theor. Appl. Mech. Lett.
6
,
302
(
2016
).
55.
Wazzan
,
A. R.
,
Okamura
,
T. T.
, and
Smith
,
A. M. O.
, “
Spatial and temporal stability charts for the Falkner-Skan boundary layer profiles
,”
Report No. DAC67086
(
McDonnell Douglas Corp
.,
California
,
1968
).
56.
Yu
,
M.-S.
,
Lv
,
S.-J.
, and
Wu
,
Y.-X.
, “
An acoustic designing method of low noise profile for fore-body of underwater vehicle
,”
J. Hydrodyn.
17
,
529
537
(
2002
) (in Chinese).
57.
Zhang
,
K.
, and
Song
,
W.-P.
, “
Application of the full eN transition prediction method to aerodynamic characteristics calculation of accurate airfoils
,”
J. Northwest. Polytech. Univ.
27
,
294
299
(
2009
) (in Chinese).
58.
Zhang
,
W.
, and
Samtaney
,
R.
, “
BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack
,”
Phys. Fluids
28
,
044105
(
2016
).
59.
Zhang
,
Y.-M.
,
Chen
,
X.-Z.
, and
Luo
,
J.-S.
, “
Study of secondary instability of a streaky boundary layer under spanwise-localized free-stream vertical disturbances
,”
Adv. Appl. Math. Mech.
11
,
686
699
(
2019
).
60.
Zhang
,
Y.-M.
, and
Zhou
,
H.
, “
PSE as applied to problems of secondary instability in supersonic boundary layers
,”
Appl. Math. Mech.
29
,
1
8
(
2008
).
61.
Zhang
,
Y.-M.
,
Zhou
,
W.-Q.
, and
Chen
,
X.-Z.
, “
An investigation on the disturbance evolution and the transition by resonant-triad interactions with a side-frequency disturbance in a boundary layer
,”
Phys. Fluids
32
,
074101
(
2020
).
62.
Zhang
,
Y.
,
Zaki
,
T. A.
,
Sherwin
,
S. J.
, and
Wu
,
X.
, “
Nonlinear response of a laminar boundary layer to isotropic and spanwise localized free-stream turbulence
,” AIAA Paper No. 2011-3292 (
2011
).
63.
Zhao
,
J.-P.
,
Shang
,
Q.
, and
Li
,
B.
, “
Genetic algorithm based low flow-noise optimization design for bodies of revolution
,”
Tech. Acoust.
30
,
496
500
(
2011b
) (in Chinese).
64.
Zhao
,
J.-P.
,
Shi
,
X.-H.
, and
Du
,
X.-D.
, “
The pressure distribution rule and Its effect on transitional point of a head part of revolution
,”
Comput. Simul.
25
,
42
45
(
2008
) (in Chinese).
65.
Zhao
,
J.-P.
,
Shi
,
X.-H.
, and
Du
,
X.-D.
, “
An integrated prediction method of torpedo flow noise
,”
Torpedo Technol.
17
,
10
14
(
2009
) (in Chinese).
66.
Zhao
,
J.-P.
,
Wu
,
J.
, and
Nie
,
X.-M.
, “
Research on prediction methods for noise radiation from boundary layer transitions on revolution bodies
,”
Appl. Acoust.
30
,
138
144
(
2011a
) (in Chinese).
67.
Zhou
,
G.-J.
,
Yan
,
Z.-Y.
,
Xu
,
S.-X.
, and
Zhang
,
K.-B.
,
Fluid Mechanics
(
Higher Education Press
,
Beijing
,
2000
) (in Chinese).
68.
Zhou
,
H.
, and
Zhao
,
G.-F.
,
Hydrodynamic Stability
(
National Defense Industry Press
,
Beijing
,
2004
) (in Chinese).
You do not currently have access to this content.