Part A | 7
174 Part A Fundamentals
Reynolds number Re ı D Uı== as
T
ı
2:3Re ı
1=2
:
(7.97)
so that the larger the Reynolds number, the greater the
range of eddies and length scales involved.
Turbulent Jets and Wakes. The maximum speed in
a turbulent circular jet of momentum J D
’
S u
2 ds,
where S is the cross-sectional area of the nozzle exit,
decays linearly with downstream distance x, where x is
measured from a virtual origin as
U max
7:4
x
Â
J
à 1=2
:
(7.98)
The cross-sectional extent of the jet varies as b 0:4x
and self-similar mean flow profile in the jet may be approximated [7.16, 39] as
U U max sech
2
 10:4r
x
Ã
:
(7.99)
For an axisymmetric wake, the maximum defect in the
mean velocity in the wake is given by
U e U min C 0 x
2=3
;
(7.100)
and the width of the wake varies as
b.x/ C 1 x
2=3
;
(7.101)
where C 0 and C 1 are constants. These results may be
compared with the corresponding cases of laminar flow.
The various parameters characterizing the behavior of
the turbulent jets or wakes are fairly independent of flow
Reynolds number.
References
7.1
B.J. Cantwell: Introduction to Symmetry Analysis
(Cambridge Univ. Press, Cambridge 2002)
7.2
W. Valentine, K.D. von Ellenrieder: Model scaling of
ocean hydrokinetic renewable energy systems, IEEEJOE 40(1), 27–36 (2015)
7.3
O.M. Faltinsen: Hydrodynamics of High-Speed Marine Vehicles (Cambridge Univ. Press, New York 2005)
7.4
M. Ferrando, M. Viviani, S. Crotti, P. Cassella, S. Caldarella: Influence of Weber number on surface
piercing propellers model tests scaling, Proc 7th Int.
Conf. Hydrodyn., Ischia (2006)
7.5
J.N. Newman: Marine Hydrodynamics (MIT Press,
Cambridge 1977)
7.6
R.D. Blevins: Flow-induced Vibration (Van Nostrand
Reinhold, New York 1990)
7.7
G.S. Triantafyllou, M.S. Triantafyllou, R. Gopalkrishnan: Wake mechanics for thrust generation in oscillating foils, Phys. Fluids A 3(12), 2835–2837 (1991)
7.8
J.H. VanZwieten, N. Vanrietvelde, B.L. Hacker: Numerical simulation of an experimental ocean current
turbine, IEEE-JOE 38(1), 131–143 (2013)
7.9
A. Roshko: Experiments on the flow past a circular
cylinder at very high Reynolds number, J. Fluid Mech.
10(3), 345–356 (1961)
7.10 O.M. Griffin, S.E. Ramberg: Vortex shedding from
a cylinder vibrating in line with an incident uniform
flow, J. Fluid Mech. 75(2), 257–271 (1976)
7.11 R.A. Dalrymple, R.G. Dean: Water Wave Mechanics for
Engineers and Scientists (Prentice-Hall, Hackensack
1991)
7.12 M. Miche: Le pouvoir reflechissant des ouvrages maritimes exposes a l’action de la houle, Ann. Ponts
Chaussées 121, 285–319 (1951)
7.13 I.A. Hunt: Design of seawalls and breakers, J. Waterw. Harb. Coast. Eng. Div. 85(3), 123–152 (1959)
7.14 R.E. Randall: Elements of Ocean Engineering (Society
of Naval Architects and Marine Engineers, Jersey City
2010)
7.15 M.S. Longuet-Higgins: Longshore currents generated
by obliquely incident sea waves: 1, J. Geophys. Res.
75(33), 6778–6789 (1970)
7.16 F.M. White: Viscous Fluids Flow (McGraw-Hill, New
York 1991)
7.17 B. Cantwell, D. Coles, P. Dimotakis: Structure and entrainment in the plane of symmetry of a turbulent
spot, J. Fluid Mech. 87(4), 641–672 (1978)
7.18 C.M. De Silva, E.P. Gnanamanickam, C. Atkinson,
N.A. Buchmann, N. Hutchins, J. Soria, I. Marusic:
High spatial range velcity measurements in a high
Reyonolds number turbulent boundary layer, Phys.
Fluids 26(2), 025117 (2004)
7.19 J.D. van Manen, P. van Oossanen: Resistance and
propulsion. In: Principles of Naval Architecture, Vol.
2, ed. by E.V. Lewis (Society of Naval Architects
and Marine Engineers, New York 1988) pp. 228–
462
7.20 H. Schlichting: Boundary Layer Theory (McGraw-Hill,
New York 1968)
7.21 J.D. Anderson: Fundamentals of Aerodynamics
(McGraw-Hill, New York 2001)
7.22 I.H. Abbott, A.E. von Doenhoff, L.S. Stivers: Summary
of Airfoil Data, Rep. 824 (NACA, Langley 1945)
7.23 S.F. Hoerner: Fluid Dynamic Drag (Hoerner Fluid Dynamics, Alburqueque 1965)
7.24 P.F. Rynne, K.D. von Ellenrieder: Unmanned autonomous sailing: Current status and future role in
sustained ocean observations, Mar. Technol. Soc. J.
43(1), 21–30 (2009)
7.25 P.F. Rynne, K.D. von Ellenrieder: Development and
preliminary experimental validation of a wind and
174 Part A Fundamentals
Reynolds number Re ı D Uı== as
T
ı
2:3Re ı
1=2
:
(7.97)
so that the larger the Reynolds number, the greater the
range of eddies and length scales involved.
Turbulent Jets and Wakes. The maximum speed in
a turbulent circular jet of momentum J D
’
S u
2 ds,
where S is the cross-sectional area of the nozzle exit,
decays linearly with downstream distance x, where x is
measured from a virtual origin as
U max
7:4
x
Â
J
à 1=2
:
(7.98)
The cross-sectional extent of the jet varies as b 0:4x
and self-similar mean flow profile in the jet may be approximated [7.16, 39] as
U U max sech
2
 10:4r
x
Ã
:
(7.99)
For an axisymmetric wake, the maximum defect in the
mean velocity in the wake is given by
U e U min C 0 x
2=3
;
(7.100)
and the width of the wake varies as
b.x/ C 1 x
2=3
;
(7.101)
where C 0 and C 1 are constants. These results may be
compared with the corresponding cases of laminar flow.
The various parameters characterizing the behavior of
the turbulent jets or wakes are fairly independent of flow
Reynolds number.
References
7.1
B.J. Cantwell: Introduction to Symmetry Analysis
(Cambridge Univ. Press, Cambridge 2002)
7.2
W. Valentine, K.D. von Ellenrieder: Model scaling of
ocean hydrokinetic renewable energy systems, IEEEJOE 40(1), 27–36 (2015)
7.3
O.M. Faltinsen: Hydrodynamics of High-Speed Marine Vehicles (Cambridge Univ. Press, New York 2005)
7.4
M. Ferrando, M. Viviani, S. Crotti, P. Cassella, S. Caldarella: Influence of Weber number on surface
piercing propellers model tests scaling, Proc 7th Int.
Conf. Hydrodyn., Ischia (2006)
7.5
J.N. Newman: Marine Hydrodynamics (MIT Press,
Cambridge 1977)
7.6
R.D. Blevins: Flow-induced Vibration (Van Nostrand
Reinhold, New York 1990)
7.7
G.S. Triantafyllou, M.S. Triantafyllou, R. Gopalkrishnan: Wake mechanics for thrust generation in oscillating foils, Phys. Fluids A 3(12), 2835–2837 (1991)
7.8
J.H. VanZwieten, N. Vanrietvelde, B.L. Hacker: Numerical simulation of an experimental ocean current
turbine, IEEE-JOE 38(1), 131–143 (2013)
7.9
A. Roshko: Experiments on the flow past a circular
cylinder at very high Reynolds number, J. Fluid Mech.
10(3), 345–356 (1961)
7.10 O.M. Griffin, S.E. Ramberg: Vortex shedding from
a cylinder vibrating in line with an incident uniform
flow, J. Fluid Mech. 75(2), 257–271 (1976)
7.11 R.A. Dalrymple, R.G. Dean: Water Wave Mechanics for
Engineers and Scientists (Prentice-Hall, Hackensack
1991)
7.12 M. Miche: Le pouvoir reflechissant des ouvrages maritimes exposes a l’action de la houle, Ann. Ponts
Chaussées 121, 285–319 (1951)
7.13 I.A. Hunt: Design of seawalls and breakers, J. Waterw. Harb. Coast. Eng. Div. 85(3), 123–152 (1959)
7.14 R.E. Randall: Elements of Ocean Engineering (Society
of Naval Architects and Marine Engineers, Jersey City
2010)
7.15 M.S. Longuet-Higgins: Longshore currents generated
by obliquely incident sea waves: 1, J. Geophys. Res.
75(33), 6778–6789 (1970)
7.16 F.M. White: Viscous Fluids Flow (McGraw-Hill, New
York 1991)
7.17 B. Cantwell, D. Coles, P. Dimotakis: Structure and entrainment in the plane of symmetry of a turbulent
spot, J. Fluid Mech. 87(4), 641–672 (1978)
7.18 C.M. De Silva, E.P. Gnanamanickam, C. Atkinson,
N.A. Buchmann, N. Hutchins, J. Soria, I. Marusic:
High spatial range velcity measurements in a high
Reyonolds number turbulent boundary layer, Phys.
Fluids 26(2), 025117 (2004)
7.19 J.D. van Manen, P. van Oossanen: Resistance and
propulsion. In: Principles of Naval Architecture, Vol.
2, ed. by E.V. Lewis (Society of Naval Architects
and Marine Engineers, New York 1988) pp. 228–
462
7.20 H. Schlichting: Boundary Layer Theory (McGraw-Hill,
New York 1968)
7.21 J.D. Anderson: Fundamentals of Aerodynamics
(McGraw-Hill, New York 2001)
7.22 I.H. Abbott, A.E. von Doenhoff, L.S. Stivers: Summary
of Airfoil Data, Rep. 824 (NACA, Langley 1945)
7.23 S.F. Hoerner: Fluid Dynamic Drag (Hoerner Fluid Dynamics, Alburqueque 1965)
7.24 P.F. Rynne, K.D. von Ellenrieder: Unmanned autonomous sailing: Current status and future role in
sustained ocean observations, Mar. Technol. Soc. J.
43(1), 21–30 (2009)
7.25 P.F. Rynne, K.D. von Ellenrieder: Development and
preliminary experimental validation of a wind and