Axisymmetric description of the scale-by-scale scalar transport

Torino, October 27, 2009

CNRS – UNIVERSITE et INSA de Rouen

Axisymmetric description of the scale-by-scale

scalar transport

Luminita Danaila

Context:

ANR ‘ANISO’: F. Godeferd, C. Cambon, J.B. Flor

ANR ‘Micromixing’: B. Renou, J.F. Krawczynski,G. Boutin, F. Thiesset CORIA, Saint-Etienne-du-Rouvray, FRANCE



OUTLINE

II. Analytical development

III. Validation with experimental data

I. Context, previous work and motivation

V. Conclusions



Kolmogorov : Local isotropy

Universality

I. Context and motivation

FLOW

Re

)()( xurxuu LLL

Real space:

x

rx

22 )()( xurxuu LLL

A simple analytical plateform: relation between the second-and the third-order moments at a scale r

energy flux at a scale r



Scale-by-scale energy budget: Kolmogorov (1941)

3 4δu ε r

5

λR

Non-universality for moderate Reynolds numbers


3 24< δu > = ε r 6ν δu

5 r

Antonia & Burattini, 2006



• For different REAL flows (moderate Reynolds, locally isotropic or anisotropic …)

• Necessity to account for explicitly the non-negligible correlation between large-and small scales




2

3 2 2

2

1 4 4 4+ δu ε 2ν + δu δu

3 r r 3 r r r t

Isotropy (local) and integration with respect to r

r

244

0

3+ s δu ds

r t

3 24< δu > = ε r 6ν δu

5 r

Method: Navier-Stokes in 2 space points:

Increments:


)()( xurxuu LLL



fuI

< ε > r 24 6ν

δu5 ε r r

3δu

ε r

*r

Finite Reynolds numbers- flows:

Grid turbulence, round jet,

channel flow (axis, near wall) ...

Conclusion:

Energy transferred at a scale r= turbulent diffusion + molecular effects+

large-scale effects: shear, decay, mean temperature gradient …


Kolmogorov, 1941Saffman 1968, Danaila et al. 1999, Lindborg 1999



fuI

< ε > r 24 6ν

δu5 ε r r

3δu

ε r

Real- Finite Reynolds numbers- flows: Slightly heated grid turbulence, grid turbulence with a Mean scalar gradient ..

Same conclusion : Energy transferred at a scale r … large-scale effects


Similar questions hold for scalars and turbulent kinetic energy

r

I

r

u

dr

dk fsL

2

22

3

4

r

I

r

quq

dr

d fqL

2

22

3

4 R.A. Antonia et al. 1997Danaila et al., 2004Burattini et al., 2005

Yaglom, 1949Danaila et al. 1999

Kolmogorov, 1941




r

I

r

quq

dr

d fqL

2

22

3

4

Shortcome: local isotropy was supposed …

Question: which is the anisotropic/axisymmetric form of

Scalar equation simpler development

Note: The axisymmetric form of Kolmogorov equation

Chandrasekar 1950, Lindborg 1996, Antonia et al. 2000, Ould-Rouis 2001 ….

Problem: a large number of scalars which are difficult/impossible to be determined from experiments

?




),(2 rq

)()(2)(2 2 rqur

rux

uuru

x

uU i

ii

i

From:

rxxrq

r

2)(2 2

2

2

x

rx

n

)cos(

r

r

…. All the terms in Eq. (1) depend on 2 variables

Several ways:

isotropy dependence on r

integration over a sphere of radius r dependence on r

(1)




nrNrrMrqu aa ),(),()(2

Chandrasekar, 1950

Development similar to Shivamoggi and Antonia, (Fluid Dyn. Res., 2000)

22 1**),(),( rrMrqu a

),(**),(),(2 rNrrMrqu aaII

Measurable:

u andIIu

aa Nrr

Mr

r

21

3

),(21

22 2

2

2

2

2

2

2

rIqrr

Injection: decay, production..




rr

r

dsPDs

sr

dsqs

sr

dsqs

sr

rG

0 2

22

30

2

2

22

2

22

3

0

2

2

22

3

1)1(2

12

12

3

2),(

C. Cambon, L. Danaila, F. Godeferd, Y. Gagne and J. Scott, in preparation

With: Vrrr

Na21

and

Vrr

Na

2

* 1

*aa NMG

Eq. (2)



III. Validation with experimental data: EXPERIMENTAL SET-UP*

Volume PaSR: V = 11116 cm3

Injection velocity: UJ = 4.5 - 47 m/s Return flow= porous top/bottom plates Residence time: tR = 8 -46 ms

Reynolds number 60 Rl 1000 (center)

*Prof. P.E. Dimotakis of Caltech was responsible for the conceptual and detailed design of the PaSR and contributed to the initial experiments.



A forced box turbulence

x

y z

Injection zone = impinging jets

« Mixing » zone = stagnation zone

Return flow (top/bottom porous)





I

II

Energy Isotropy? iiuuq2



3

PIV

( u) 4

r ε 5

JETS

The sign changes at

Large scales (inhomogeneity)


Third order ‘classical’ Structure functions : Selection of one particular direction




),(2 zxq ),(2 rq

H

H

V

Vr

r

x

x

r

),( r




04/ 2/

3

2),(rG

Eq. (3)




]2/,0[

d

drGrR

0

0

32

),()(



IV. Conclusions

Theory:

-Scale-by-scale energy budget equation for kinetic energy (scalar) in axisymmetric

turbulence (axis of a round jet, axis of two opposed jets ..)

-Measured quantities: u and v (components perpendicular and parallel to )

-all the terms in Eq. (2) can be determined experimentally

The flow:

• Pairs of impinging jets; Return flow by top/bottom porous locally axisymmetric flow

Results :

- better agreement with asymptotic predictions (high Reynolds, anisotropic flows) along the direction normal to the axisymmetry axis (homogeneous plane) than along the direction parallel to

n

n



Large pannel of structures

Large-scale instabilities (jets flutter)

Mechanisms controlling the mixing?

H/D=3 H/D=5

Instantaneous fields of the mixing fraction

IV. Description of the scalar mixing: fluctuating field



Re 104D (mm), 2H (mm), H/D

10, 60, 3

10, 60, 3

6, 60, 510, 60,

310, 60, 3 6, 60, 5 6, 60, 5 6, 60, 5

Qv (m3/h) 60 86 60 129 155.2 100 129 155.2

Vinj (m/s) 6.63 9.50 18.42 14.26 17.15 30.70 39.60 47.65

TR (ms) 43.56 30.39 43.56 20.26 17.15 26.14 20.26 17.15

P (bar) 1.40

(m²/s) x10-5 1.089

6089 8728 10149 13092 15751 16915 21821 26252

26252

6089

8 injection

conditions

DVinj Re


2 Geometries:

H/D=3

H/D=5



The other tests• 2-rd order SF with the Kolmogorov constant

• Normalized dissipation which L? Attention to initial conditions versus universality .. However, a reliable test

for • The most reliable test is the 1—point energy budget

equation, when the pressure-related terms might be neglected (point II).

III. DESCRIPTION of the FLOW: fluctuating field; PIV for determining small scale properties

3/23/22 rCu K

3'u

LC

5.0C

NDISSIPATIO

PRODUCTIONDIFFUSIONVISCOUS

DIFFUSIONPRESSUREDIFFUSIONTURBULENTDECAY

R

Uw

y

Vv

R

Uu

y

U

R

Vuv

y

q

R

q

RR

q

pvy

puRRR

vqy

uqRRRy

qV

R

qU

²²²1

111²

2

1²

2

11²²

22

2

22

150R

/r

3/2r



Velocity field: 1) Particle velocimetry– Resolution and noise limitations– PIV resolution linked to size of interrogation/correlation window, e.g., 16, 32,

and 64 pix2, and processing algorithm choices• Does not resolve small scales: the smallest 100% =1.7 mm• Problem to estimate energy dissipation directly

– Towards adaptive/optimal vector processing/filtering

2) LDV in (1 point and) 2 points – Simultaneous measurements of One velocity component in two points of the

space: spatial resolution 200 * 50 microns; sampling frequency= 20 kHZ

Scalar field: PLIF on acetone Small-scale limitations set by spatial resolution (pixel/laser-sheet size)

The smallest resolved scale 100% =0.7 mm

Signal-to-noise ratio per pixel– Adaptive/optimal image processing/filtering




JETS


Third order ‘classical’ Structure functions

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Axisymmetric description of the scale-by-scale scalar transport

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