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Large Eddy Simulations of Bubble Convection

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					                                                        LECTURES IN ENVIRONMENTAL TURBULENCE
                                                        Babiano A., Fraunie P., Redondo J.M. and Vassilicos J.C
                                                                                 Ed. CIMNE, Barcelona, (2006)




               Structure Functions and Intermittency
               in Non-homogeneous Turbulent Flows
                                O. B. Mahjoub and J.M. Redondo
                                 Departament de Fisica Aplicada
                                Campus Nord, B5, UPC, Barcelona


                                                  Abstract

This work is developed in two parts. First, we show the results of some coastal experiments, where three
dimensional velocity measurements of high quality were obtained in the Sea with ADV sonic
velocimetry. The scale to scale transfer and the structure functions are calculated and from these the
intermittency parametrs. The estimates of turbulent diffusivity could also be measured using neutrally
buoyant Pliolite particles without significant external forces. Some two point correlations and time lag
calculations are used to investigate the time and spatial integral length scales obtained from both
Lagrangian and Eulerian correlations and functions, and we compare these results with both theoretical
and experimentals ones in the Laboratory with a 100 m long Wave tank and in the field measurements.
The second part of the work is complemented with a theoretical description of how to simulate
intermittency following the beta-model and the role of locality in higher order exponents.




          Figure 1. Experimental configuration in the Wind tunnel of Warsav Politechnical University.
            Fig 2. Description of the intermittency as a function of the order of the structure functions



1. Introduction

 In this case, under non homogeneous conditions, we are able to obtain a better
quantification of the intermittency than in homogeneous situation. It can be obtained
using the sixth order structure function and the β-model:

                                                             3  D (1  )
                                                          p               p
                                        2 6                 6 
                                                          3               3
where p is the order of the structure function , in this case p=6, in a similar way, the
fourth order structure function may also be used as described by Frish(1992)




   Fig 3. Experimental parameters downstream of a grid, It-Turbulent Intensity, Reynolds number , Kolmogorov
                                     Lengthscale η and average Disipation, ε.
In figure 3 we show the conditions of the flow across a 5 cm Mesh grid in a 4 meter long water
channel, the experimental parameters are shown as functions of the nondimensional position
downstream of the grid, x/M. The parameter It = u’/U is theTurbulent Intensity, the Reynolds
number is calculated in terms of the Taylor microscale λ as Reλ = λ u’ / ν , the Kolmogorov
Lengthscale η and the average Disipation, ε.




            Figure 4. The relative structure functions for two downstream positions x/D =7 and 20




                      Figure 6 Corrected power spectra with -5/3 showing the influence
                           of the von Karman vortices behind a cilinder




                                        References

Ben-Mahjoub O., Babiano A. y Redondo J.M. (1998) Velocity structure and Extended Self
Similarity in non-homogeneous Turbulent Jets and Wakes. Applied Scientific Research. 59 ,
299-313.

Ben-Mahjoub, O. (2000) Intermittency and non-local dynamics PhD Thesis UPC. Barcelona

P.Fraunie, S Grilli, J.M. Redondo,V. Rey, S. Arnoux-Chiavassa, S. Berrabaa y S. Guignard
(1999)
Proceedings of Hydralab workshop in Hannover. experimental research and synergy effects
with mathematical models 209-215.

 Mahjoub O.B., Redondo J.M. y Babiano A. (2000) Self similarity and intermittency in a
turbulent non-homogeneous wake. Proceedings of the Eighth European Turbulence Conference.
(Eds. Dopazo et al.) CIMNE, Barcelona. 2000. p.783-786.
            Large Eddy Simulations of Bubble Convection
                  Claudia Benitez, J.M. Redondo, Peter Furmanek*
                   Univ. Politecnica de Catalunya, Barcelona Spain
                   *Technical Universitry, Prague, Czech Republic.


                                            Abstract

A Large Eddy Simulation model (LES) may used to simulate the evolution of the rise of
a bubble driven convective structure. This is an interesting problem to investigate the
mixing process in the ocean surface layers when an injection of air or CO2 is used as an
imput of mechanical energy. Also taking into account the possible use of the Oceans as
a CO2 sink. The model solves the Boussinesq equations in a two dimensional grid and
is based in Rees(1987). Together with the momentum equations, the continuity equation
for incompresible flow is used taking into account only x and z components. Initially
the subgrid scaling assumed a constant turbulent viscosity defined in terms of
dimensional scaling as ν = l2 / ∆ t and taking the integral scale as a constant in terms
of the mesh size as l = 0.23 (∆x ∆ z)1/2 Different aspect ratios of the convective flow,
generated by the rising of the bubbles are considered. At 1:1 ratio, a single cell forms,
with updrafts and downdrafts clearly separated. At aspect ratios of 8:1 a complex
convective pattern developes showing some asymetry as in the experiments of Kimura
and Iga(1995).


1. Introduction




                  Figure 1. The iteration process (left) of the Koch curve (right)




Using the variance of the signal (t) defined from:

               V (T )   (  (t  T )   (t )) 2 
where < > denotes the average over the entire periode T and the dependence for fractal
time series V(T)  T2H (Voss 1985 a,b 1988).Using T = 1/f and the description of the
spectral density function, S(f), we have equivalently

                                    S(T)  T
and
                                 T
                          S  T   2 (t ) e ift dt  T V
                                 0
so we can relate
                              S ( f )  T V  T 2 H 1  T 2 E 12 D

so that considering both molecular and turbulent diffusivities we have

                                 dc       d 2 c d  dc 
                                    D 2          k 
                                 dt       dx    dx  dx 
In the case when a Bounday, such as a wave breaker transkers momentum to the coastal
flow through horizontal shear. The same hypothesis used by Von Karman for the
Atmospheric Bounday Layer may be used, defining a relevant scale of velocity, that
associated to the turbulent sidewise friction produced by the boundary as

                                        r τ=*u2
then assuming that eddies proportional to the distance from the breaker are most
efficient we model the lateral shear as

                                      dU/dz = u*/kz
being κ the so called Von Karman constant with value for the atmosphere of 0,41.
Integrating the diferential equation we have a logarithmic current profile such as

                                  U(z) = u* ln(z/zo) / κ




                           Figure 2. Description of the basic flow




2. Description of the experiments

                               2.1. Measurement system




                                     2.2 Data Analysis
3. Results




                  Figure 3. Evolution of the density contours as convection developes




4. Discussion




5 Conclusions




                                         Acknowledgments

The authors would like to thank Julia Rees from Sheffield University for providing the
initial version of the LES program used in this study. Thanks also to professor Ruijy
Kimura for his help and discussions comparing experiments and simulations, Peter
Furmanek had a Socrates EU felowship. Support was provided by ERCOFTAC and
grant 2001SGR00221 (2005) of the Generalitat de Catalunya.



                                            References

Kimura R. and Iga, I. 1995 in Mixing in Geophysical Flows
Rees , J. 1987
Feder, 1988. Fractals in Physics. Cambridge Univ. Press. Cambridge.
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Hentschel H.G.E. and Procaccia I 1983a. The infinite number of generalized dimensions of fractals and
strange attractors. Physica 8D, 435-444
Hentschel H.G.E. and Procaccia I 1983b. Fractal nature of turbulence as manifested in turbulent diffusion,
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Redondo, J.M., 1990. The structure of density interfaces. Ph.D. Thesis. Univers. of Cambridge.
Cambridge.
Richardson, L.F.,1926. Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc.
London. A 110, p. 709.
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turbulent flows". Annales Geophysicae. Abstract (16 ), 1133.
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