Resuspension by vortex rings - PDF Free Download

Resuspension by vortex rings Stuart Dalziel A prototype for resuspension in turbulent flows? UNIVERSITY OF CAMBRIDGE

collaborators Nastja Bethke (BNP Paribas) Ian Eames (UCL) Rick Munro (Nottingham) Anna Mujal (UPC)

outline Introduction resuspension, wakes Vortex rings modelling, solid boundaries Particle beds crater formation, shapes & scalings Time dependence shear at bed, impact on ring development Other configurations cross-flow, uncompacted beds, underlying structure Conclusions

resuspension How does dust get suspended? Ballistic mechanism Collision of moving particles with a particle layer Hydrodynamics Lift and drag forces due to velocity differences Buoyancy forces u Lift Drag Buoyancy

hydrodynamics u Lift Drag Lift ~ f u 2 a 2 Drag ~ f u 2 a 2 or ua Buoyancy ~( p f )ga 3 Cohesion Rolling/sliding resistance Critical velocity Buoyancy dust sand pebbles No motion until Large particles : Drag ~ Rolling/sliding Medium particles: Lift ~ Buoyancy Small particles: Lift ~ Cohesion

Shields parameter lift stress area ~ or buoyancy buoyancy Resuspension: c Generically Steady turbulent flow t p gd 2 u* gd p u 2

wakes

Shields parameter lift stress area ~ or buoyancy buoyancy Resuspension: c Generically Steady turbulent flow t p gd 2 u* gd p u 2 Impact i p U 2 gd

modelling Critical impact Shields parameter 2 fuc c ga Viscous sublayer p f U d U ~ 2 U d ~ U

modelling For small particles V s 1 p f 18 f g 2 a Re s Va s ga p 3 2 18 Critical Shields parameter c U 3 Re 1 2 f c s ~ 1 5 2 p f ga Res Res 1

modelling c Re s

wakes Re ~ 850

ideas Simplify the wake problem Resuspension by vortex rings Can raindrops have wakes? Resuspension by droplets

collision with particles

vortex rings

parameters 100 W 700 mm/s 0.55 LD t 1 0 410 Re W D 3010 3 3 0 t 0.25 ar0.35 90 d 1000 m

vortex rings Plug of fluid forced through an orifice

vortex rings Vortex sheet wraps up

vortex rings Circulation half what you might expect L U UL t udx udx 1 UL 2

vortex rings

propagation Circular line R Kelvin (1867), a << R R 2a W 8R 1 ln 4 R a 4 Assuming thin core gives an error of around 5% in W 0

propagation Spherical vortex (Hill 1894) i 3 U r b r 4b 2 2 2 h W 2 4b 15 5b r h

finite core size Norbury (1973) W R W 2 2 ˆ R A R a e 2 2 2 0 2

Norbury

Norbury W R W 2 2 ˆ Wˆ

solid wall 2a 2R Re 4800 ar0.35

approaching a wall

approaching wall Ek dz K k W R dt 4 R Z 2 2 12 2 2 2 R 2Z E k 2Z Kk dr dt 4 RZ R Z 2 2 12 Complete elliptic integrals 2 2 1 sin k 2 2 1 2 K k k x dx 0 2 2 2 1 2 1 sin E k k x dx 0 4Rr z Z rr 2 2

solid wall

solid wall Re 4800 ar 0.35

solid wall

bed velocity solid Theoretical

bed velocity solid Boundary layer unsteady forcing

Shields parameter lift stress area ~ or buoyancy buoyancy Resuspension: c Generically Steady turbulent flow Impact Bed t b i p gd 2 u* gd p u W 2 2 p gd 2 U b gd p

the particles Nominally spherical Nominally monodisperse Negligible cohesion Depth: 10 mm Scraped to level/compact

the particles 90 m 250 m 1000 m acrylic

the splash

the splash: cartoon D t

particle motion Bed-load transport or resuspension? Flow separation

crater formation

critical conditions c p U 2 c gd U c maxu r b c 3 Re 1 s ~ 1 5 Re 2 s Res 1 Re p Va p Re s Va s ga p 3 2 18 Actual settling velocity Stokes settling velocity

craters

measurement technologies (2)

craters 250m 2.9U c 5.3U c 7.3U c

crater volume 250m V hh da A 0 40 L 70 mm

PE 250m 1 E 1 2 p v p g hh0 da 2 E E E p k c 1.31 0.02 A k 3 2 E f ad DU

shape 250m R e h e

shape 250m

shape 250m R d R e h d 1 10 hd h e

shape 250m h E E e k c 0.51 002. E E E p k c 1.31 0.02 E ~ h R R R 2 p e 0 e 0 ~ 2 1.27 0.06 R R E E e 0 k c 0.25 0.02

craters 90m 2.2U c 3.5U c 250 m 2.9U c 5.3U c 5.7U c 7.3U c

craters 90m h E E e k c 0.42 0.04 E E E p k c 1.15 0.03 R R E E e 0 k c 0.32 0.03 250 250 0.5 0.25 E ~ h R R R 2 p e 0 e 0 ~ 2 1.16 0.11 250 1.31

craters 1000m 1.6U c 3.6U c 250 m 2.9U c 5.3U c 5.1U c 7.3U c

craters 1000m

craters 1000m Deep craters Exceed angle of repose? collapse? Max measured slope ~21 Angle of repose ~ 24

craters 1000m h E E e k c 0.21 0.05 E E E p k c 0.86 0.05 R R E E e 0 k c 0.15 0.05 E ~ h R R R 2 p e 0 e 0 ~ 2 0.57 0.15 250 1.31 90 1.15

the effect on the ring vectors & colour: 250μm contours: solid boundary

bed velocity 90m

bed velocity - solid

bed velocity particles 90 m 1000 m Boundary layer thinner? particle mobility? bed permeability? solid

flow within bed a Ub r w p h 1 w rur r r z 0 2 p 2 U u ~ ~ ~ c r Ub r r a 2 2 2 1 k Uc 1 Uc d wp ~ h~ h 2 2 a a p 2 1 10 m/s 10 w O O U b

reticulated foam

cross flow 250m

uncompacted particles

uncompacted 90m

corrugated boundary

conclusions Resuspension linked with separation Self-similarity of crater shapes Avalanching if too steep Flow within porous bed may contribute to resuspension Porosity/permeability influence bed velocity