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

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∫−

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∆≈

Experimental setup

1

max

+

=

αα

ϕϕωϕr

&

Time evolution

222 )()())(),((δ+−+−

−−−Γ= ∑

jiji

jiji

jj

i

yyxxxxyy

dtdx

)2sin()1(),0( Γ−+Γ=Γ πiaz

Kelvin-Helmholtz instability and topological transition

Secondary vortex formation

Contour plot of vorticity

Instability growth rate

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

Vortex breakdownside view

Top view

Drop in a oil-water interface

Driven interface deformation

• The interface between two fluids is driven by a fixed dipole

• Gravity and inertia:Froude number

5/ gdFr α=

Small Fr: wedge formation

Zoom showing the wedge region

Moderate Fr: cavity formation

Strong Fr: Cusp formation

The splash: convergence of a capillary "shock"

Continuity equation

0)( =∂∂

+∂∂ uh

xth

Momentum equation

hx

Sxuu

tu

3

3

∂∂

=∂∂

+∂∂

Modulationnal instability: NLS-like behavior

NLS-like

Maximum of the height amplitude showing "almost" recurrence

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

Convergence and collapse of a capillary wave front

Similarity solution

Constraints in planar and axisymmetric geometries

Equations in the similarity variable:

5/2/ tx=ξ

Symmetries:

UU −→−→ ,ξξuutt −→−→ ,