department of applied physics, university of …lagr05/pdf/calzavarini.pdfslide 17 1-4 sept 05...

Enrico Calzavarini1-4 Sept 05 Castel GandolfoSlide 1

Department of Applied Physics, University of Twente, The Netherlands.

Physics of Fluids Group

Bubbly turbulence in KFset-up of a 3D DNS, problems and perspectives

Enrico Calzavarini

with Federico Toschi, Detlef Lohse, Luca Biferale




The problem Drag reduction can be induced by bubbles/micro-bubbles.

Experiments:-McCormick & Bhattacharyya, Nav.Eng.J. (1973) Turbulent Boundary layer-Mandavan, Deutsch & Merke JFM (1985) TBL on a flat plane 80% drag reduction.- Kodama et al. (1999) TBL on a flat plane ship 20% drag reduction- R. van den Berg et al. Vertical channel and Taylor-Couette PRL (2005) up to 20% drag reduction

Numerical studies:Different approaches, different kind of flows.- Maxey, Xu & Karnidachis Force Coupling Method Channel Flow few bubbles- A.Ferrante & S.Elgobashi FCM Turbulent Boundary layer on a flat plate- Trygvasson et al. Front tracking method Channel flow few bubbles- K.Sugyyama et al. FCM and FT Turbulent Boundary layer or transient micro-bubble flow- I.Mazzitelli, F.Toschi & D.Lohse FCM Homogeneous Isotropic turbulence-Theory:- L’vov et al. PRL (2005) Microbubbles in a Channel Flow

Up to now not clear if Drag Reduction is due tolocal compressibility of the flow, bubble deformations or wall effect,or if it is a transient or statistically steady effect.




Our approachStudy the effect of bubbles on a basic turbulent Flowwith mean velocity profile:

Kolmogorov FlowIn particular:- Detect statistically steady effect (if any) on the mean velocity,shear stress, turbulent velocity fluctuations profiles.- Main features of the mean bubble concentration at changing therelevant dynamical parameters. An Eulerian-Lagrangian approch with Force Coupling Method is used

Outline of the talk:- description of the model- discussion of some preliminary results




The flowKolmogorov Flow

Studied by V.Borue & S.Orszag JFM (1996)and in A.Celani, G.Boffetta & A.Mazzino PRE (2005) coupled to Oldroyd-Bviscoelastic polymer model .

z

y

x

g




Laminar and Turbulent KF

Laminar:

Fully turbulent:

Drag coefficient:Cf

log Re

Re-1

50

β

We are mainly interested to the mean profiles: Where:

Sin profile both in laminar and fully turbulent regime, here kf=1

Similarity/differences with channel flow

γ




Mean velocity profile

Given F and ν we can estimate U, ε and:

From a turbulentKF flowat Re = 117.

Newtoniancase




Bubbles equation of motions

Fluid inertia/added mass term in clean water

Drag Buoyancy Lift force

Maxey & Riley Phys Fluid (1983)Thomas et al. (1984)Auton et al JFM (1988)

Relaxation time Terminal velocity

Bubble radius

Range of validity:

Therefore it applies for airmicro-bubble in water.

The history forceis neglected.




Bubbles equationdimensionless form

The phase space of the parameters for the 1 wayproblem is 3-dimensional:

Re, St, β

new units:




Bubble feedback on the flowA small bubble is viewed as a point-like source of momentum in the flow.A multi-pole expansion of forces can be adopted Saffman (1973)

Force Coupling Method

Monopole : transfer of momentum

Dipole: torque and strain rate from the particle on the fluid

When:

Maxey et al. Flu Dyn Res (1997)Maxey & Patel Int J Mult Flow (2001)Lomholt, Stenu & Maxey Int J Mult Flow(2003)monopole dipole

Gaussian function modeling the bubble shape

Dirac delta function

Numerical implementation: every δ contribution is spread on the 8 nearest grid points.




2-Way Couplingthe full set of equations

4 parameters:

Re, St, β, α




Some explorative runs

Corresponding to real bubbles in waterof size rb = 0.14 mmvT ≈ 6 cm/sand Reb ≈ 8.6

40 20010%00.16

96 20010%00.09

524 28810%00.03

40 20010%10.16

96 20010%10.09

524 28810%10.03

NbαβSt ReL = 117 ( Reλ= 23 )

Spectral codeN x N x N = 64 x 64 x 64

“Large” number of eddy turnovertimes collected:

τL ≈ 102

NbαβSt

65 5361.25%10.03

65 5361.25%00.03

1 wa

y2

way

At St =0.03 -> D/ηK = 0.835




Instantaneous bubble distribution (1)β=0

St =0.03 St =0.09 St =0.16

z

y

Projections on the y-z plane of 4 x 104 bubbles center points,( same forcing amplitude, different times and bubble initial conditions)

Bubbles concentrate in filaments




Instantaneous bubble distribution (2)

1.68 ± 0.020.16

1.44 ± 0.010.09

1.15 ± 0.010.03

<Ωb>/<Ω>St

time (a.u. ~ 18 large eddy turnover times)

Mean Enstrophy at bubble positions

Long time averages:

As for the Homogeneous turbulence caseMazzitelli, Lohse & Toschi JFM (2003)




Mean bubble concentration (1)

β=0

Why this shapes?

Fluid Inertia anddrag

-> right-left simmetry -> half of the cell

Normalized local mean void fraction

Weakly non homogeneousprofile




Mean bubble concentration (2)

Mean |v-u| ∝ StFluid Inertia & drag




Bubbles concentration (problems…)Pdf of the local void fraction,for the three runs.

mean α = 0.1 (or 10%)

( if α

> 0

)

Max geometricalpacking limit of spheres

Nb ≈ 5·105

Main technical problem: strong bubble clustering produces numerical instability !!!




Effect of gravity

0.533 ± 0.0410.16

0.499 ± 0.0200.03

0.497 ± 0.0200.09

0

1

1

β

0.501 ± 0.040.16

0.521 ± 0.020.09

0.508± 0.010.03

N-/NSt

Mazzitelli, Lohse & ToschiPhys Fluid (2003)

Here: vT/U≈ 1/6

β=1

If β ≤ 1 (small bubbles) trapping in all turbulent structures.Note that vT/U << 1 .

If β >> 1 (large bubbles) weak interaction with the flow.Bubbles move rapidly.

N-/Nrelative

number ofbubbles

in down-flowregions




Effect of gravity (2)The left-right symmetry in the mean concentration isbroken by the effect of the lift force

Mean lift force?

β=1

0




2Way coupling preliminary results, α = 1.25 %

Bubble Energy input

Weak increase but still within statistical error bars!( Same uncertain result if the mean stress< uy uz > are compared. )

Weak non-homogeneityin the mean void fraction

Very weak injection of energy

mean uzprofiles

β=0




2Way coupling preliminary results , α = 1.25 %

Bubble Energy input

Wea

k m

ean

conc

entr

atio

n

<u2>

dominated by the gravity.

mean uzprofiles

β=1




Further developments

-Explore the parameter space: Re, St, β, α

-Study more active bubbles, avoid numerical instabilities and locallarge (not physical) bubbles concentrations by collisions

Additional energy dissipation/injection effects can be furtherconsidered:-Inelastic collisions may be adopted to model shape oscillation in bubble-bubble interaction (4way coupling).-Size oscillations may be implemented through including the effectof the local pressure on the bubble radius.

department of applied physics, university of …lagr05/pdf/calzavarini.pdfslide 17 1-4 sept 05...

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