High-Schmidt number turbulent mixing as an aggregation process
Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
Final Report Abstract
In this project, we studied the mixing of a passive scalar field in a three-dimensional turbulent flow at Schmidt numbers Sc ≥ 1. The project addressed general and fundamental aspects of scalar mixing which is the reason of why this study was restricted to simple flow configurations and geometries. We showed that different spatio-temporal correlations of the flow affect the statistics of passive scalar turbulence at the small scales in the flows with Sc ≥ 1. These correlations were for example varied by an increase of the simulation box size and thus of the flow Reynolds number. In contrast to the original plan, we did not study many different driving mechanisms. It was however shown that for a strongly anisotropic driving by a constant mean scalar gradient in a very high Reynolds number flow, the passive scalar builds up a pronounced ramp-cliff structure that dominates the statistical properties, in particular the highest-order statistical moments. The major focus of the project was on the development, test and subsequent analysis of our Lagrangian mixing model and its comparison with the standard Eulerian model of passive scalar turbulence. The direct comparison of freely decaying Lagrangian and Eulerian passive scalars gives a good agreement of the scalar variance for times t ≲ 10(ts) and for the probability density functions P (Θ, t) taken with respect to the whole simulation domain. We also showed how the multi-layer aggregations of scalar filaments and sheets in the Lagrangian frame are increasingly influenced by the noise due to discreteness with progressing dilution of the initially high tracer particle concentration. This limits the Lagrangian approach in its present form and for the obtainable Schmidt numbers to studies of shorter time periods. A simple one-dimensional advection-diffusion model of a solitary strip is finally applied to the problem at hand to derive the probability density function of the scalar concentration, P (Θ, t), from the one of the compressive local finite-time Lyapunov exponent, p(λ3, t). Model prediction with and without self-convolution and numerical data of the scalar distributions agree qualitatively, however with quantitative differences particularly for small scalar concentrations. The present Lagrangian approach to passive scalar mixing in turbulence opens the application of new and more flexible passive scalar injection and boundary conditions and allows to relax the resolution constraints for high-Schmidt number mixing studies. Although we were facing several numerical difficulties in our Lagrangian approach, we can finally state that the project could be finished successfully. Also, we would like to underline that the DFG-ANR support has contributed to the progress of the field of Mixing in a broader sense beyond, and in parallel to this precise project. We have been able to contribute to several other closely related topics such as mixing in disordered porous media, or to the physics of dense sprays evaporation, for example. Our collaboration will be hopefully successfully continued by the participation in an ITN Network on Scalar Mixing.
Publications
- Experimental measurement of the Melnikov function, Physics of Fluids, 27, 077103 (2015)
P. Meunier, P. Huck, C. Nobili, and E. Villermaux
(See online at https://doi.org/10.1063/1.4927036) - Dense spray evaporations as a mixing process, Physical Review Fluids 1, 014201 (2016)
A. De Rivas and E. Villermaux
(See online at https://doi.org/10.1103/PhysRevFluids.1.014201) - Droplet dynamics and fine-scale structure in a shearless turbulent mixing layer with phase changes, Journal of Fluid Mechanics 814, 452–483 (2017)
P. Götzfried, B. Kumar, R. A. Shaw, and J. Schumacher
(See online at https://doi.org/10.1017/jfm.2017.23) - Fine structure of the vapor field in evaporating dense sprays, Physical Review Fluids 2, 074501 (2017)
E. Villermaux, A. Moutte, M. Amielh, and P. Meunier
(See online at https://doi.org/10.1103/PhysRevFluids.2.074501) - Scalar Gradients in stirred mixtures and the deconstruction of random fields, Journal of Fluid Mechanics 812, 578–610 (2017)
T. Le Borgne, P. Huck, M. Dentz, and E. Villermaux
(See online at https://doi.org/10.1017/jfm.2016.799) - Scalar mixtures in porous media, Physical Review Fluids 2, 104502 (2017)
M. Kree and E. Villermaux
(See online at https://doi.org/10.1103/PhysRevFluids.2.104502) - Stretching and Mixing in sheared particulate suspensions, Journal of Fluid Mechanics 812, 611–635 (2017)
M. Souzy, H. Lhuissier, E. Villermaux, and B. Metzger
(See online at https://doi.org/10.1017/jfm.2016.828) - On Shapes and Forms: Population balance dynamics of corrugated stirred fronts, Comptes Rendus Physique 19, 306–315 (2018)
E. Villermaux
(See online at https://doi.org/10.1016/j.crhy.2018.10.009) - Scale dependence of cloud microphysical response to turbulent entrainment and mixing, Journal of Advances in Modeling Earth Systems 10, 2777–2785 (2018)
B. Kumar, P. Gotzfried, N. Suresh, J. Schumacher, and R. A. Shaw
(See online at https://doi.org/10.1029/2018MS001487) - Steep cliffs and saturated exponents in three-dimensional scalar turbulence, Physical Review Letters 121, 264501 (2018)
K. Iyer, J. Schumacher, K. R. Sreenivasan, and P. K. Yeung
(See online at https://doi.org/10.1103/PhysRevLett.121.264501) - The diffusive sheet method for scalar mixing, Journal of Fluid Mechanics 837, 230–257 (2018)
D. Martínez-Ruiz, P. Meunier, B. Favier, L. Duchemin, and E. Villermaux
(See online at https://doi.org/10.1017/jfm.2017.862) - Mixing versus stirring, Annual Reviews of Fluid Mechanics 51, 245–273 (2019)
E. Villermaux
(See online at https://doi.org/10.1146/annurev-fluid-010518-040306)
