alberto verga- singularity formation in vortex sheets and interfaces
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Singularity formation inSingularity formation invortex sheets and interfacesvortex sheets and interfaces
AlbertoAlberto VergaVergaIRPHE IRPHE –– Université d’AixUniversité d’Aix--MarseilleMarseilleT.T. LewekeLeweke, M., M. AbidAbid, F., F. GrimalGrimal, T., T. FrischFrisch
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Vortex sheet roll-up, and secondary vortex formation,
A vortex sheet is separated from a moving plate. In 2D is shape is described by the Birkhoff-Rottequation:
plate
Spiral vortex sheet
∫ Γ−ΓΓ−
=∂Γ∂
)',(),('..
2),(
tztzdVPi
ttz
π
bordσ=Γ Udt
d
)V(),(v''),'( ttxxx
dxtxa
a
+=−σ∫−
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KH instabilityA vortex sheet is a tangential velocity discontinuity in a perfect fluid
The sheet is unstable: a periodic shape disturbance will grow:
λσ /U∆≈
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Experimental setup
1
max
+
=
αα
ϕϕωϕr
&
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Time evolution
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222 )()())(),((δ+−+−
−−−Γ= ∑
jiji
jiji
jj
i
yyxxxxyy
dtdx
)2sin()1(),0( Γ−+Γ=Γ πiaz
Kelvin-Helmholtz instability and topological transition
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Secondary vortex formation
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Contour plot of vorticity
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Instability growth rate
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Three dimensional sheets and vortex breakdown
Using a triangular plate a 3D sheet is generated. The resulting vortex has an axial flow. The appearance of a stagnation pointdestroys the vortex core. It is also a topological transition.
Vortex core
Axial flow
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Vortex breakdownside view
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Top view
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Drop in a oil-water interface
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Driven interface deformation
• The interface between two fluids is driven by a fixed dipole
• Gravity and inertia:Froude number
5/ gdFr α=
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Small Fr: wedge formation
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Zoom showing the wedge region
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Moderate Fr: cavity formation
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Strong Fr: Cusp formation
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The splash: convergence of a capillary "shock"
Continuity equation
0)( =∂∂
+∂∂ uh
xth
Momentum equation
hx
Sxuu
tu
3
3
∂∂
=∂∂
+∂∂
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Modulationnal instability: NLS-like behavior
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NLS-like
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Maximum of the height amplitude showing "almost" recurrence
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Modulation of a high frequency wave:Derivation of a Non-Linear Schrödinger equation.
h(x,t)u(x,t)
H0
Perturbation expansion:
xXtTtTuuuu
hhhHh
ε=ε=ε=
ε+ε+ε=
ε+ε+ε+=
,, 221
)3(3)2(2)1(
)3(3)2(2)1(0
Focusing NLS:
02 =++ AAgAiA XXT
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Convergence and collapse of a capillary wave front
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Similarity solution
Constraints in planar and axisymmetric geometries
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Equations in the similarity variable:
5/2/ tx=ξ
Symmetries:
UU −→−→ ,ξξuutt −→−→ ,