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Interaction of a vortex couple with a free surface

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Abstract

The characteristics of three-dimensional flow structures (scars and striations) resulting from the interaction between a heterostrophic vortex pair in vertical ascent and a clean free surface are described. The flow features at the scar-striation interface (a constellation of whirls or coherent vortical structures) are investigated through the use of flow visualization, a motion analysis system, and the vortex-element method. The results suggest that the striations are a consequence of the short wavelength instability of the vortex pair and the helical instability of the tightly spiralled regions of vorticity. The whirls result from the interaction of striations with the surface vorticity. The whirl-merging is responsible for the reverse energy cascade leading to the formation and longevity of larger vortical structures amidst a rapidly decaying turbulent field.

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Abbreviations

A c :

Area of a vortex core (Fig. 6b)

AR :

Aspect ratio of the delta wing model

B :

base width of delta wing

b 0 :

initial separation of the vortex couple

d 0 :

depth at which the vortex pair is generated

c :

average whirl spacing in the x-direction

E :

energy density

Fr :

Froude number (\(\left( {{{ = V_0 } \mathord{\left/ {\vphantom {{ = V_0 } {\sqrt {g b_0 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {g b_0 } }}} \right)\))

g :

gravitational acceleration

L :

length of the scar band

L ko :

length of the Kelvin oval

N w :

number of whirls in each scar band

P c :

Perimeter of a vortex core

q :

surface velocity vector

r c :

core size of the whirl ( = 2A c/P c)

Re :

Reynolds number ( = \(\left( {{{ = V_0 } \mathord{\left/ {\vphantom {{ = V_0 } {{{v = \Gamma _0 } \mathord{\left/ {\vphantom {{v = \Gamma _0 } {2\pi v}}} \right. \kern-\nulldelimiterspace} {2\pi v}}}}} \right. \kern-\nulldelimiterspace} {{{v = \Gamma _0 } \mathord{\left/ {\vphantom {{v = \Gamma _0 } {2\pi v}}} \right. \kern-\nulldelimiterspace} {2\pi v}}}}} \right)\))

Rnd :

a random number

s :

inboard edge of the scar front (Fig. 6 a)

t :

time

u :

velocity in the x-direction

υ:

velocity in the y-direction

V b :

velocity imposed on a scar by the vortex couple (Fig. 6 a)

V 0 :

initial mutual-induction velocity of the vortex couple (=Γ0/2πb 0)

V t :

tangential velocity at the edge of the whirl core

w :

width of the scar front (Fig. 6 a)

z :

complex variable

z k :

position of the whirl center

α :

half included angle of V-shaped scar band

β :

wave number

Г m :

initial mean circulation of the whirls

Г 0 :

initial circulation of the vortex pair

Г w :

circulation of a whirl

γ min :

minimum survival strength of a whirl

Δt :

time step

gDz :

increment of z

gDζ :

change in vorticity

δ :

cut-off distance in velocity calculations

ɛ :

critical merging distance

κ :

curvature of the surface

λ :

wavelength

ν :

kinematic viscosity

ω :

angular velocity of a whirl core

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Authors and Affiliations

  1. Department of Mechanical Engineering, Naval Postgraduate School, 93943, Monterey, CA, USA

    T. Sarpkaya

  2. Submarine Officer School, 06877, Groton, CT, USA

    P. Suthon

Authors
  1. T. Sarpkaya
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  2. P. Suthon
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Sarpkaya, T., Suthon, P. Interaction of a vortex couple with a free surface. Experiments in Fluids 11, 205–217 (1991). https://doi.org/10.1007/BF00192746

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