Kinetic equations for turbulent fields

• Applicant: Professor Dr. Ulrich Hansen, since 10/2012
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2012 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 211849969

• Final Report Year: 2016

Final Report Abstract

We analyzed the turbulent flow in Rayleigh–Bénard convection on the basis of statistical quantities like the temperature PDF and conditionally averaged fields. We derived that the mean path a fluid particle takes through phase space (spanned by temperature and spatial coordinates) is defined by so-called characteristics, i. e. trajectories in phase space that follow the conditionally averaged vector field composed of heat diffusion and fluid velocities. Thereby, we could characterize the dynamics and flow features that occur in turbulent convection cells from a statistical point of view, i. e. from averaged quantities like the temperature distribution and its moments as well as statistics conditioned on temperature and spatial position. By estimating the aforementioned vector fields for three different Rayleigh–Bénard geometries while utilizing their symmetries and then integrating the characteristics, we described the mean dynamics that fluid particles undergo – i. e. we could describe how fluid of different temperatures behaves in different regions of the convection volume. We also distinguished regions of high and low transport through phase space. For all geometries there are high phase space speeds for intense temperatures in the bulk (which we attribute to localized events of intense temperatures and high speeds, i. e. plumes) as well as high speeds near the horizontal plates for all temperatures, while for the case of cylindrical convection the phase space speed also takes high values near the wall of the cylinder. This we interpret as plumes that are directed along the insulating sidewalls. In the conditionally averaged vector field of the cylinder, we could furthermore identify corner flows near the sidewalls for fluid of different temperatures. Cold fluid experiences a corner flow near the bottom plate while showing no corner flow near the upper plate and vice versa. Additionally, we described the higher moments of the temperature distributions, where we could link features of the moments to coherent structures that appear in turbulent flows. When we then obtained the characteristics by integrating trajectories through the conditionally averaged vector field, we found that for all different convection setups, the characteristics form closed cycles in phase space. These cycles reconstruct the typical Rayleigh–Bénard cycle a fluid particle undergoes on average, i. e. fluid is heated up at the bottom and rises upwards while slightly cooling down until it hits the upper plate, where it cools down fast and falls down to the lower plate while slightly heating up, thus starting the cycle all over again. The method thus allows to further pin-point and quantify the differences and similarities between Rayleigh–Bénard convection in two- and three-dimensional periodic boxes and in three-dimensional convection in a cylindrical cell.

Publications

• The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence. Comptes Rendus Physique, 13(9–10):929–953, 2012
  Rudolf Friedrich, Anton Daitche, Oliver Kamps, Johannes Lülff, Michel Voßkuhle, and Michael Wilczek
  
• (2015). Describing the heat transport of turbulent Rayleigh–Bénard convection by POD methods
  J. Lülff
  
• (2015). Turbulent Rayleigh–Bénard convection described by projected dynamics in phase space. Journal of Fluid Mechanics, 781:276–297
  J. Lülff, M. Wilczek, R. J. A. M. Stevens, R. Friedrich, D. Lohse