alberto verga- singularity formation in vortex sheets and interfaces
TRANSCRIPT
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 −→−→ ,