The ``Law of the Wall” for turbulent convection Guenter Ahlers, University of California-Santa Barbara, DMR Ludwig Prandtl is viewed as the “Father of Fluid Mechanics”; his boundary-layer theory enabled the design of airplanes, ships and many other things involving flow over a surface. Around 1930 he and his former student Theodor von Karman found that the average down-stream velocity of the flow over a solid wall depends logarithmically on the distance z from that wall when z is not too large (the Law of the Wall). This enabled the calculation of the drag on objects (including planes and ships) inserted in a flowing liquid or gas. Recently, in collaboration with the group of Eberhard Bodenschatz, Director of the Max Planck Institute for Dynamics and Self-organization in Goettingen, Germany, we found an analogous ``Law of the Wall”  ~ log(z) for the temperature above a heated horizontal plate with a fluid above it, and provided a derivation based on a simple model of the system 1. We also found that the intensity  2 of the temperature fluctuations varies as log(z). We found from experiment, and showed theoretically based on a model analogous to one for shear flow, that the power spectrum of the temperature fluctuations varies as the inverse of the frequency f whenever  2 ~ log(z). 2 1 G. Ahlers and E. Bodenschatz and X. He, J. Fluid Mech., submitted (2014) (arXiv: ); P. Wei and G. Ahlers, J. Fluid Mech., submitted (2014). 2 X. He, D.P.M. van Gils, E. Bodenschatz, and G. Ahlers, Phys. Rev. Lett. 112, (2014); P. Wei and G. Ahlers, to be pub. Ludwig Prandtl Theodor von Karman The temperature  depends logarithmically on the distance z/L from the plate ! At many points within the sample the power P times the frequency f is constant !

Broad impact on fundamental science: The study of convection over a heated plate started with the work of Henri Benard in 1905; the study of turbulent shear flow started with the work of Ludwig Pradtl in the same year. Then the two fields developed quite separately, with little contact. We find it of particularly broad interest and of aesthetic appeal that our work shows a century later a remarkable commonality in the ``Laws of the Wall” of shear flow and of thermal convection. International collaboration: Our work was done both in Santa Barbara with laboratory-sized apparatus, and in Goettingen with the ``Uboot of Goettingen”, a major facility not available in the US (a pressure vessel of 25 m 3 volume, containing a convection cell of 2.2 m height and filled with up to two tons of sulfur hexafluoride at a pressure of up to 19 bars). Only this international collaboration made this project possible. Impact on other fields: Developing a better understanding of turbulent convection can lead to greater efficiency of industrial processes and a deeper knowledge of the Earth, our atmosphere, our oceans, our environment, our climate, and our bodies. The ``Law of the Wall” for turbulent convection Guenter Ahlers, University of California- Santa Barbara, DMR Left: Inserting the 2-m convection cell, weighing 2 tons, into the Uboot of Goettingen. Right: Our postdocs Ping Wei and Stephan Weiss are constructing a new apparatus in our Santa Barbara lab for the study of convection; the cell (the pink part) weighs only about 50 kg. Impact on education: We provide excellent training of undergraduate and graduate students, as well as post-doctoral scholars, who participate in our research.