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

Dynamics of non-neutrally buoyant particles

Particles with r vanishing intertia r r d X r V u ( X , t ) = = (fluid elements) dt Impurities: r r r 2 particles with d X d V F = = 2 finite inertia dt dt m and/or finite size

r r d X Impurities: V dt = particles with finite inertia r r d V F and/or finite size dt = m

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 r D r r V − r 6 a 2 ∇ 2 r r   d V Dt − 9 µ u u − 1 ( ) u g ρ p dt = ρ f  + ρ p − ρ f  2 a 2   r r r r 2 R ρ p − ρ f ( ) + ( ) − 2 V − ρ f u Ω × ρ p Ω r V − r 10 a 2 ∇ 2 r − ρ f   d u − 1 u   2 dt   r t V − r 6 a 2 ∇ 2 r   − 9 µ 1 d u − 1 ∫ u  d τ  1/ 2 [ ] 2 a πν ( t − τ ) d τ   0 D r r r r Ω × r 2 R + µ ∇ 2 r u Dt = −∇ p + ρ f g − 2 ρ f u + ρ f u ρ f Ω µ = ρ f ν

Difference between D/DT and d/dt D r ∂ u = + ⋅ ∇ Dt t ∂ r d ∂ V = + ⋅ ∇ dt t ∂

Discard Faxen corrections r dt = δ D r r V − r ) r d V Dt − 1 u ( ) + 1 − δ ( u g τ a r r r V − δ r 2 ( ) + ( ) − 2 u R 1 − δ Ω × Ω r r t ) − Re 1/ 2 V − r V − r d a 1 d − δ ( ( ) d τ ∫ u u 1/ 2 [ ] 2 dt L π ( t − τ ) d τ τ a 0 δ = ρ f ρ p 2    τ a = 2 a Re = St Re = UL ,  9 δ  L  δ ν

2D horizontal motion, no rotation, no gravity discard addedd mass and Basset term r r r d V D u 1 r ( ) V u = δ − − dt Dt τ a ρ f δ = ρ p 2 2 a UL   Re , Re τ = =   a 9 L δ ν  

2D horizontal motion, no rotation: formally, a dissipative system r dt = δ D r r V − r d V Dt − 1 u ( ) u τ a Θ = ( X , Y , U , V ) d Θ dt = Φ ( X , Y , U , V ) ∇ 4 ⋅ Φ = − 2 τ a

Example: r dt = δ D r r V − r d V Dt − 1 u ( ) u τ a   r u = ( u , v ) = − ∂ ψ ∂ y , ∂ ψ   ∂ x   ( ) ψ = 2 cos x + cos y

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 r dt = D r r V − r d V Dt − 1 u ( ) u τ a δ = ρ f = 1 ρ p 2    τ a = 2 a Re = UL Re = St ,  9  L  ν

Neutrally-bouyant impurities with finite size: What if D/Dt=d/dt r r V − r V − r d ) = − 1 ( ( ) u u dt τ a r r   V = r V − r ) 0 exp − 1 ( u + u   τ a   δ = ρ f = 1 ρ p 2    τ a = 2 a Re = UL Re ,  9  L  ν

d ≠ D But dt Dt r r r V − r V − r ) ⋅ ∇ r V − r d u − 1 ( ) = − ( ( ) u u u dt τ a δ = ρ f = 1 ρ p 2    τ a = 2 a Re = UL Re ,  9 L   ν Babiano, Cartwright, Piro, Provenzale, PRL (2000)

r r r V − r V − r ) ⋅ ∇ r V − r d u − 1 ( ) = − ( ( ) u u u dt τ a r r V − r A = u r r   d dt = − J + 1 A A  ⋅  τ a     ∂ x u ∂ y u J =   ∂ x v ∂ y v  

r r V − r A = u r r   d dt = − J + 1 A A  ⋅  τ a     ∂ x u ∂ y u J =   ∂ x v ∂ y v   4 s 2 − ζ 2 ( ) = − det J = λ 2 Q = 1 Trace ( J ) = 0   λ − 1 r 0   r d A τ a D   A dt = ⋅ D − λ − 1   0   τ a  

( ) cos y ψ = A cos x + B sin ω t

( ) cos y ψ = A cos x + B sin ω t

2D turbulence

Falling impurities (no rotation) r dt = δ D r r V − r ) r d V Dt − 1 u ( ) + 1 − δ ( u g τ a δ = ρ f ρ p 2    τ a = 2 a Re = UL Re ,  9 δ L   ν

Very heavy impurities δ =0 r r V − r ) + r d dt = − 1 V ( u g τ a

Stommel (1949): permanent suspension r r V − r ) + r 0 = d dt = − 1 V ( u g τ a r r V = r u + r τ a = r g u + W   r u = ( u , v ) = − ∂ψ ∂ y , ∂ψ   ∂ x   r    = − ∂ ˜  ∂ y , ∂ ˜  V = ( u , v − g τ a ) = − ∂ψ ∂ y , ∂ψ ψ ψ ∂ x − g τ a    ∂ x     ˜ ψ = ψ − g τ a x Time dependence: Smith and Spiegel (1985)

Maxey and Corrsin (1986): permanent suspension is not possible r r V − r ) + r 0 ≠ d dt = − 1 V ( u g τ a

permanent suspension is possible in more complex flow fields r r V − r ) + r 0 ≠ d dt = − 1 V ( u g τ a 2D stationary random field with Kolmogorov energy spectrum k -5/3 (Pasquero, Provenzale, Spiegel, PRL, 2003)

Distribution of fall velocities r r V − r ) + r d dt = − 1 V ( u g τ a terminal velocity in still air : W = g τ 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 = ν 1/ 2 ε − 1/ 2 E ( k ) = ε 2/ 3 k − 5/ 3 for

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 on particle collisions and reactions

The role of rotation r dt = δ D r r V − r ) r d V Dt − 1 u ( ) + 1 − δ ( u g τ a r r r V − δ r 2 ( ) + ( ) − 2 u R 1 − δ Ω × Ω δ = ρ f ρ p 2    τ a = 2 a Re = UL Re ,  9 δ L   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) r r r r r V − r ) + r d dt = − 1 V 2 ( u g − 2 V + R Ω × Ω τ a

Dust grains in the solar nebula r r r r V − r d dt = − 1 V ( ) − 2 u V Ω × τ E − GM r + Ω 2 r ˆ r 2 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 of their motion is not that easy: Even the form of the equations of motion is not really known.