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

Bubbly turbulence in KF

set-up of a 3D DNS, problems and perspectives

Enrico Calzavarini with Federico Toschi, Detlef Lohse, Luca Biferale Slide 1

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

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 to local compressibility of the flow, bubble deformations or wall effect, or if it is a transient or statistically steady effect. Slide 2

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Our approach

Study the effect of bubbles on a basic turbulent Flow with mean velocity profile:

Kolmogorov Flow

In 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 the relevant 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 Slide 3

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

The flow

Kolmogorov Flow

z

x

g

y

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

Slide 4

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Laminar and Turbulent KF We are mainly interested to the mean profiles:

Laminar:

Where:

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

Fully turbulent:

Cf

Drag coefficient:

Re-1

γ Similarity/differences with channel flow

β 50

Slide 5

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

log Re

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

Mean velocity profile From a turbulent KF flow at Re = 117.

Newtonian case

Given F and ν we can estimate U,

Slide 6

1-4 Sept 05 Castel Gandolfo

ε and:

Enrico Calzavarini

Physics of Fluids Group

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

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

Fluid inertia/added mass term in clean water

Relaxation time

Drag

Terminal velocity

Bubble radius

Slide 7

Buoyancy

1-4 Sept 05 Castel Gandolfo

Lift force The history force is neglected.

Range of validity:

Therefore it applies for air micro-bubble in water.

Enrico Calzavarini

Physics of Fluids Group

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

Bubbles equation dimensionless form

new units:

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

Re, St, β

Slide 8

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Bubble feedback on the flow A 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

Monopole : transfer of momentum

dipole

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

Gaussian function modeling the bubble shape

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

Dirac delta function

When: Numerical implementation: every δ contribution is spread on the 8 nearest grid points. Slide 9

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

2-Way Coupling the full set of equations

4 parameters:

Re, St, β, α

Slide 10

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

2 way

1 way

Some explorative runs St

β

α

0.03

0

10%

524 288

0.09

0

10%

96 200

0.16

0

10%

40 200

0.03

1

10%

524 288

0.09

1

10%

96 200 40 200

0.16

1

10%

St

β

α

Nb

( Reλ= 23 )

Spectral code N x N x N = 64 x 64 x 64 “Large” number of eddy turnover times collected:

τL ≈ 102

Nb

0.03

0

1.25%

65 536

0.03

1

1.25%

65 536

At St =0.03 -> D/ηK = 0.835

Slide 11

ReL = 117

1-4 Sept 05 Castel Gandolfo

Corresponding to real bubbles in water of size rb = 0.14 mm vT ≈ 6 cm/s and Reb ≈ 8.6 Enrico Calzavarini

Physics of Fluids Group

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

Instantaneous bubble distribution (1) β=0

St =0.09

St =0.03

St =0.16

Projections on the y-z plane of 4 x 104 bubbles center points,

z

( same forcing amplitude, different times and bubble initial conditions) y

Bubbles concentrate in filaments Slide 12

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Instantaneous bubble distribution (2) Mean Enstrophy at bubble positions

Long time averages: St

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

/

0.03

1.15 ± 0.01

0.09

1.44 ± 0.01

0.16

1.68 ± 0.02

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

Slide 13

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Mean bubble concentration (1) β=0

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

Normalized local mean void fraction

Weakly non homogeneous profile

Why this shapes? Fluid Inertia and drag Slide 14

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Mean bubble concentration (2)

Fluid Inertia & drag Mean |v-u| ∝ St

Slide 15

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Bubbles concentration (problems…) ( if α > 0 )

Pdf of the local void fraction, for the three runs. mean

α = 0.1

(or 10%)

Nb ≈ 5·105 Max geometrical packing limit of spheres

Main technical problem: strong bubble clustering produces numerical instability !!! Slide 16

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Effect of gravity

β=1

If β ≤ 1 (small bubbles) trapping in all turbulent structures. Note that vT/U > 1 (large bubbles) weak interaction with the flow. Bubbles move rapidly. N-/N relative number of bubbles in down-flow regions Mazzitelli, Lohse & Toschi Phys Fluid (2003)

Slide 17

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

St

β

N-/N

0.03

1

0.508± 0.01

0.09

1

0.521 ± 0.02

0.16

1

0.533 ± 0.04

0.03

0

0.499 ± 0.02

0.09

0

0.497 ± 0.02

0.16

0

0.501 ± 0.04

Physics of Fluids Group

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

β=1

Effect of gravity (2) The left-right symmetry in the mean concentration is broken by the effect of the lift force Mean lift force?

0

Slide 18

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

2Way coupling

β=0 preliminary results, α = 1.25 % Weak increase but still within statistical error bars!

mean uz profiles

( Same uncertain result if the mean stress < uy uz > are compared. )

Bubble Energy input

Weak non-homogeneity in the mean void fraction

Very weak injection of energy Slide 19

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

2Way coupling

β=1

preliminary results , α = 1.25 %

mean uz profiles Bubble Energy input

Weak mean concentration

dominated by the gravity.

Slide 20

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group

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

Further developments -Explore the parameter space: Re, St, β, α -Study more active bubbles, avoid numerical instabilities and local large (not physical) bubbles concentrations by

collisions

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

Slide 21

1-4 Sept 05 Castel Gandolfo

Enrico Calzavarini

Physics of Fluids Group