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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 12 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. Grau, J., Platonov, A., Redondo, J. M., 2003. Análisis multifractal de procesos complejos. Revista Internacional de Métodos Numéricos en la Ingeniería. (In publication). 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, Vol 27, no 2 Hentschel H.G.E. and Procaccia I 1984. Relative diffusion in turbulent media: The fractal dimension of clouds. Physical Review A, Vol 29, No. 3, pp 1461-1470. Kolmogorov, A.N., 1941. Local structure of turbulence in an incompressible fluid at very high Reynolds numbers. Dokl. Academia de Ciencias de la URSS, 30:299-303. Platonov A., 2002. Aplicación de imágenes de satélite SAR en los estudios de contaminación marina y de dinámica de las aguas en el mediterráneo noroccidental. http://www.tdcat.cbuc.es/TDCat-0905102- 135541/index.html 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. Redondo J.M. and Cantalapiedra I.R. (1993) "Mixing in horizontally heterogeneous flows", Applied Scientific Research, 51, 217-222. Redondo J.M., M.A- Sanchez, I.R. Cantalapiedra and R. Castilla (1998) "Vortical structures in stratified turbulent flows". Annales Geophysicae. Abstract (16 ), 1133. Redondo,J.M. (1995). Diffúsion in the atmosphere and ocean, Eds. M. Velarde and C. Christos, 584-597

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Large-Eddy Simulations, Eddy Simulations, large-eddy simulation, eddy viscosity, Reynolds number, Large eddy simulation, scale model, turbulent flows, Cambridge University Press, turbulent flow

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posted: | 7/20/2010 |

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