Dynamics of non-neutrally buoyant particles Particles with r - - PowerPoint PPT Presentation

Dynamics of non-neutrally buoyant particles

m F dt V d dt X d t X u V dt X d r r r r r r r = = = =

2 2

) , (

Particles with vanishing intertia (fluid elements) Impurities: particles with finite inertia and/or finite size

d r X dt = r V d r V dt = r F m

Impurities: particles with finite inertia and/or finite size

Boussinesq (1885) Basset (1888) Faxen (1922) Taylor (1928) Stommel (1949) Maxey and Riley (1983) Auton, Hunt and Prud’homme (1988) Michaelides (1997)

A simplified equation

for the dynamics of impurities

ρp d r V dt = ρ f Dr u Dt − 9µ 2a2 r V − r u − 1 6 a2∇2r u       + ρp − ρ f

( )

r g −2 r Ω × ρp r V − ρ f r u

( ) +

r Ω

2R ρp − ρ f

( )

− ρ f 2 d dt r V − r u − 1 10 a2∇2r u       − 9µ 2a 1 πν(t − τ)

[ ]

1/ 2

d dτ r V − r u − 1 6 a2∇2r u       dτ

t

∫

ρ f Dr u Dt = −∇p + ρ f r g − 2ρ f r Ω × r u + ρ f r Ω

2R + µ∇2r

u µ = ρ f ν

Difference between D/DT and d/dt

∇ ⋅ + ∂ ∂ = ∇ ⋅ + ∂ ∂ = V t dt d u t Dt D r r

Discard Faxen corrections

d r V dt = δ Dr u Dt − 1 τ a r V − r u

( ) + 1−δ

( ) r

g −2 r Ω × r V −δ r u

( ) +

r Ω

2

R 1−δ

( )

− δ 2 d dt r V − r u

( ) − Re1/ 2

τ a a L 1 π(t − τ)

[ ]

1/ 2

d dτ r V − r u

( )dτ

t

∫

δ = ρ f ρp τ a = 2 9δ a L      

2

Re = St δ , Re = UL ν

2D horizontal motion, no rotation, no gravity discard addedd mass and Basset term

( )

ν δ τ ρ ρ δ τ δ UL L a u V Dt u D dt V d

a p f a

=       = = − − = Re , Re 9 2 1

2

r r r r

2D horizontal motion, no rotation: formally, a dissipative system

d r V dt = δ Dr u Dt − 1 τ a r V − r u

( )

Θ = (X,Y,U,V) dΘ dt = Φ(X,Y,U,V) ∇4 ⋅ Φ = − 2 τ a

Example:

d r V dt = δ Dr u Dt − 1 τ a r V − r u

( )

r u = (u,v) = − ∂ψ ∂y ,∂ψ ∂x       ψ = 2 cos x + cosy

( )

Example:

Crisanti, Falcioni, Provenzale, Tanga, Vulpiani, Phys. Fluids (1992)

2D horizontal motion, no rotation: formally, a dissipative system Consequences:

• 1. Possibility of chaotic motion

also for stationary 2D flow

• 2. An initially

homogeneous particle distribution can become non-homogeneous

Neutrally-bouyant impurities with finite size: δ=1

d r V dt = Dr u Dt − 1 τ a r V − r u

( )

δ = ρ f ρp =1 τ a = 2 9 a L      

2

Re = St , Re = UL ν

Neutrally-bouyant impurities with finite size: What if D/Dt=d/dt

d dt r V − r u

( ) = − 1

τ a r V − r u

( )

r V = r u + r V − r u

( ) 0 exp − 1

τ a       δ = ρ f ρp =1 τ a = 2 9 a L      

2

Re , Re = UL ν

But

d dt r V − r u

( ) = −

r V − r u

( ) ⋅ ∇ r

u − 1 τ a r V − r u

( )

δ = ρ f ρp =1 τ a = 2 9 a L      

2

Re , Re = UL ν Dt D dt d ≠

Babiano, Cartwright, Piro, Provenzale, PRL (2000)

d dt r V − r u

( ) = −

r V − r u

( ) ⋅ ∇ r

u − 1 τ a r V − r u

( )

r A = r V − r u d r A dt = − J + 1 τ a       ⋅ r A J = ∂xu ∂yu ∂xv ∂yv      

r A = r V − r u d r A dt = − J + 1 τ a       ⋅ r A J = ∂xu ∂yu ∂xv ∂yv       Q = 1

4 s2 −ζ 2

( ) = −det J = λ2

Trace(J) = 0 d r A

D

dt = λ − 1 τ a −λ − 1 τ a             ⋅ r A

D

ψ = Acos x + Bsinω t

( )cosy

ψ = Acos x + Bsinω t

( )cosy

2D turbulence

Falling impurities (no rotation)

d r V dt = δ Dr u Dt − 1 τ a r V − r u

( ) + 1−δ

( ) r

g δ = ρ f ρp τ a = 2 9δ a L      

2

Re , Re = UL ν

Very heavy impurities δ=0

d r V dt = − 1 τ a r V − r u

( ) + r

g

Stommel (1949): permanent suspension

0 = d r V dt = − 1 τ a r V − r u

( ) + r

g r V = r u + r g τ a = r u + r W r u = (u,v) = − ∂ψ ∂y ,∂ψ ∂x       r V = (u,v − gτ a) = − ∂ψ ∂y ,∂ψ ∂x − gτ a       = − ∂ ˜ ψ ∂y ,∂ ˜ ψ ∂x       ˜ ψ =ψ − gτ a x

Time dependence: Smith and Spiegel (1985)

Maxey and Corrsin (1986): permanent suspension is not possible

0 ≠ d r V dt = − 1 τ a r V − r u

( ) + r

g

permanent suspension is possible in more complex flow fields

0 ≠ d r V dt = − 1 τ a r V − r u

( ) + r

g

2D stationary random field with Kolmogorov energy spectrum k -5/3

(Pasquero, Provenzale, Spiegel, PRL, 2003)

Distribution of fall velocities

d r V dt = − 1 τ a r V − r u

( ) + r

g terminal velocity in still air : W = gτ a

What happens in the presence of a fluid flow ?

Faster than free fall: Maxey (1987), Wang and Maxey (1993) Slower than free fall: Fung (1993), Davila and Hunt (2001)

τ k = ν1/ 2ε−1/ 2 for E(k) = ε2/ 3k−5/ 3

What happens for a 2D time evolving random field ?

A wide distribution of suspension times can have important effects on particle growth by condensation: Broadening of cloud droplet spectra The prolonged suspension times lead to a wide distribution of fall velocities This can have important effects

• n particle collisions and reactions

The role of rotation

d r V dt = δ Dr u Dt − 1 τ a r V − r u

( ) + 1−δ

( ) r

g −2 r Ω × r V −δ r u

( ) +

r Ω

2

R 1−δ

( )

δ = ρ f ρp τ a = 2 9δ a L      

2

Re , Re = UL ν

Tanga, Doctorate Thesis Montabone, Doctorate Thesis Tanga et al., ICARUS (1996) Provenzale, Annu. Rev. Fluid Mech. (1999)

The role of rotation on a horizontal plane (discard the centrifugal term) Heavy grains concentrate in anticyclones

For very heavy particles (δ = 0)

d r V dt = − 1 τ a r V − r u

( ) + r

g − 2 r Ω × r V + r Ω

2

R

Dust grains in the solar nebula

d r V dt = − 1 τ E r V − r u

( ) − 2

r Ω × r V − GM r2 ˆ r + Ω2r ˆ r δ =10−8 Epstein regime

Tanga, Babiano, Dubrulle, Provenzale, ICARUS (1996) Bracco, Chavanis, Provenzale, Spiegel, Phys. Fluids (2000)

Non-neutral impurities have a richer behavior than fluid parcels. However, the prediction and interpretation

• f their motion is not that easy:

Even the form of the equations of motion is not really known.