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

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen OUTLINE II. Analytical development III. Validation with experimental data I. Context, previous work and motivation V. Conclusions

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen Kolmogorov : Local isotropy Universality I. Context and motivation FLOW Re Real space: A simple analytical plateform: relation between the second- and the third-order moments at a scale r  energy flux at a scale r

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen Scale-by-scale energy budget: Kolmogorov (1941) Non-universality for moderate Reynolds numbers I. Context and motivation Antonia & Burattini, 2006

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen For different REAL flows (moderate Reynolds, locally isotropic or anisotropic …) Necessity to account for explicitly the non-negligible correlation between large-and small scales I. Context and motivation

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen Isotropy (local) and integration with respect to r Method: Navier-Stokes in 2 space points: Increments: I. Context and motivation

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen 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 … I. Context and motivation Kolmogorov, 1941  Saffman 1968, Danaila et al. 1999, Lindborg 1999

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen 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 I. Context and motivation Similar questions hold for scalars and turbulent kinetic energy R.A. Antonia et al Danaila et al., 2004 Burattini et al., 2005 Yaglom, 1949 Danaila et al Kolmogorov, 1941

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen II. Analytical development 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 ?

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen II. Analytical development From: …. 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)

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen II. Analytical development Chandrasekar, 1950 Development similar to Shivamoggi and Antonia, (Fluid Dyn. Res., 2000) Measurable: and Injection: decay, production..

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen II. Analytical development C. Cambon, L. Danaila, F. Godeferd, Y. Gagne and J. Scott, in preparation With: and Eq. (2)

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen III. Validation with experimental data: EXPERIMENTAL SET-UP* Volume PaSR: V = 11  11  6 cm 3 Injection velocity: U J = m/s Return flow= porous top/bottom plates Residence time: t R = ms Reynolds number 60  R l  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.

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen A forced box turbulence x y z Injection zone = impinging jets « Mixing » zone = stagnation zone Return flow (top/bottom porous) III. Validation with experimental data

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen III. Validation with experimental data I II Energy Isotropy?

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen JETS The sign changes at Large scales (inhomogeneity) III. Validation with experimental data Third order ‘classical’ Structure functions : Selection of one particular direction

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen III. Validation with experimental data H H V V r r

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen III. Validation with experimental data Eq. (3)

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen III. Validation with experimental data

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen 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

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen Large pannel of structures Large-scale instabilities (jets flutter) Mechanisms controlling the mixing? H/D=3H/D=5 Instantaneous fields of the mixing fraction  IV. Description of the scalar mixing: fluctuating field

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen Re  10 4 D (mm), 2H (mm), H/D 10, 60, 3 6, 60, 510, 60, 3 6, 60, 5 Q v (m 3 /h) V inj (m/s) T R (ms) P (bar)1.40 (m²/s) x injection conditions III. Validation with experimental data 2 Geometries: H/D=3 H/D=5

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen 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

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen Velocity field: 1) Particle velocimetry –Resolution and noise limitations –PIV resolution linked to size of interrogation/correlation window, e.g., 16, 32, and 64 pix 2, 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 III. Validation with experimental data

Torino, October 27, 2009 CNRS – UNIVERSITE et INSA de Rouen JETS III. Validation with experimental data Third order ‘classical’ Structure functions