Vorticity and Dynamics

Vorticity and Dynamics

In Navier-Stokes equation Nonlinear term ω u the Lamb vector is related to the nonlinear term u 2 (u ) u = + ω u 2 Sort of Coriolis force in a rotation frame Viscous term ν u = ν (.u) ( u) = ν ω

u t u 2 + ω u + = [ p 2 ρ ] ν ω taking the rotational ω t + (ω u) = ν ω Dynamical equation for Vorticity

Pressure is eliminated in the equation p =2ρ Q Q = 1 2 [1 2 ω2 e 2 ] Q>0 Q-Criterium to define a vortex region

e yx = e xy = 1 2 e xx = e yy =0 u = u x (y) e x ; u y Mixing layer u = u θ (r) e θ ; Vortex ω = u y ω 2 =2e ij e ij pressure is uniform ω z = 1 r e rr =0, e θθ =0, e rθ = r 2 (ru θ ) r r (u θ r ) Q =0

p =2ρ Q Q = 1 2r u 2 θ r Pressure can be computed using the first component of the Navier-Stokes equation in polar coordinates u 2 θ r = 1 p ρ r u 2 θ p(r) =p( ) ρ r r dr pressure decreases as we get near the vortex center : it is a pressure low.

Texte Q-vortex from Delcayre (2000)

Cavitation (Cadot et al)

Dissipation and Vorticity Internal dissipation Φ per unit volume particle deformation internal energy Φ =2µe ij e ij > 0 incompressibility assumed Φ = µ ω 2 +2µ. ω u + 12 (u 2 )

Φ dτ = µ v Dissipation and Vorticity v ω 2 dτ + µ surface [2 ω u + (u 2 )] nds the dissipation per unit mass used in turbulence = 1 ρ V v Φ dτ = ν V v ω 2 dτ There exists of a zone of dissipation around a vortex

Chaoticity and Vorticity u = u BS + φ Potential only depends on the boundary conditions since it satisfies at each time φ =0 n φ =(u u BS ) n Past history does not play a role for φ Vorticity precisely brings this historical aspect. Vorticity ω modifies velocityu Velocity u modifies with vorticity ω

This implies feedback loop Chaoticity and vortex Turbulence and vortex

Transport Equation for Vorticity : Incompressible Newtonian Fluid [ω u] = (u )ω (ω )u Dω Dt =[ t + u. ]ω = ω. u + ν ω Lagrangian Transport Stretching Viscous Diffusion Circulation theorem dγ dt = ν ω.dl C

Vorticity Diffusion #$%+ (-* #$+ (.* #$% #$#+ '! #!#$#+!#$%!#$%+!#$"!!!" # "! & " #!#$+!!!" # "! & U θ (r, t) = Lamb-Oseen Vortex Γ r2 [1 exp( 2πr a 2 )] Ω(r, t) = Γ r2 exp( πa2 a 2 ) a 2 (t) =a 2 (0) + 4νt

Inviscid incompressible flows Dω Dt =[ t + u. ]ω = ω. u Consider a vector element of a material line A B = δ l During time dt, points A and B move respectively to points A' and B' B B A B = δ l = δ l +(u B u A )δt A A Dδ l Dt = δ l δ l δt and Dδ l Dt =(δ l )u u B u A =( A B )u

Inviscid incompressible flows By comparing equations Dω Dt = ω. u Dδ l Dt =(δ l )u In the inviscid fluid, for a given fluid particle, vorticity evolves in time the same way as does a small material line with the same direction. (a) u (b) it may be tilted ω j (t) ω j (t + δt) ω i (t) ω i (t + δt) x j x i it may be stretched/compressed Concept used in numerical vortex methods.

Vorticity Enhancement by Strain Field Dω Dt =[ t + u. ]ω = ω. u + ν ω Stretching

Law of conservation : Helmholtz Laws for inviscid incompressible flows

First Law of conservation Circulation around a material loop Γ = C u.dl = d Du u dx = dt C(t) C(t) Dt dx S ω.ds d dt Γ = ν ω dx C(t) ν =0 d dt Γ =0 Circulation along a material line remains constant when convected by the fluid

Second Law of conservation Consider a fluid particle without vorticity All small loop around the particle would have zero circulation d dt Γ =0 all small loop remain of zero circulation ω =0 U y C b x C a A potential region remains potential when convected by the fluid in the inviscid context

Third Law of conservation. Vortex lines are material lines i.e. convected by the fluid Tilting and Stretching of Vortex lines First and Third laws imply that: Circulation of a vortex tube does not change with time

Helmholtz conservation laws in 2D For the two-dimensional case u(x,y), v(x,y), w=0 ω = ωe z = v y u e z y Vorticity lines are straight and there is no stretching (u ) ω =0 In 2D Euler, vorticity is such that Dω Dt =0 Vorticity of each particle is hence conserved

Helmholtz conservation laws in axisymmetric case axisymmetric vortex ring u = u r (r, z, t) e r + u z (r, z, t) e z ω = ω θ (r, z) e θ ω θ = u r z u z r Vorticity lines are rings and there is stretching In Euler, vorticity is such that D ωθ =0 Dt r ω θ r is hence conserved