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SUPERCOMPUTATIONS of LOW-PRANDTL-NUMBER CONVECTION FLOWS Jorg¨ Schumacher1 and Janet D

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SUPERCOMPUTATIONS of LOW-PRANDTL-NUMBER CONVECTION FLOWS Jorg¨ Schumacher1 and Janet D XXIV ICTAM, 21-26 August 2016, Montreal, Canada SUPERCOMPUTATIONS OF LOW-PRANDTL-NUMBER CONVECTION FLOWS Jorg¨ Schumacher1 and Janet D. Scheel2 1Department of Mechanical Engineering, Technische Universitat¨ Ilmenau, Ilmenau, Germany 2Department of Physics, Occidental College, Los Angeles, USA Summary Massively parallel supercomputations are an important analysis tool to study the fundamental local and global mechanisms of heat and momentum transfer in turbulent convection. We discuss the perspectives and challenges in this vital field of fundamental turbulence research for the case of convection at very low Prandtl numbers. MOTIVATION AND SIMULATION MODEL Turbulent convection is an important area of present research in fluid dynamics with applications to diverse phenomena in nature and technology. The turbulent Rayleigh-Benard´ convection (RBC) model is at the core of all these turbulent flows. It can be studied in a controlled manner, but has enough complexity to contain the key features of turbulence in heated fluids. This flow in cylindrical cells has been investigated intensively over the last few years in several laboratory experiments all over the world [1]. In RBC, a fluid cell or layer is kept at a constant temperature difference ∆T = Tbottom − Ttop between top and bottom plates which are separated by a vertical distance H. The Rayleigh and Prandtl numbers are given by gαH3∆T ν Ra = and P r = : (1) νκ κ The parameter Ra characterizes the thermal driving in convective turbulence with the acceleration due to gravity, g, the thermal expansion coefficient, α, the kinematic viscosity, ν, and the thermal diffusivity, κ. The hard turbulence regime in RBC is established for Ra ≥ 106. The Prandtl number P r characterizes the molecular properties the working fluid. The aspect ratio Γ, which is the ratio of cell diameter or cell length and cell height H, is the third input parameter. In this contribution, we will focus on the specific turbulence properties of convection at very low Prandtl numbers and the resulting challenges for supercomputer simulations. The three-dimensional Boussinesq equations (2)–(4) are used to model Rayleigh-Benard´ convection. They are given in dimensionless form by r · u = 0 ; (2) r @u P r + (u · r)u = −∇p + r2u + T e ; (3) @t Ra z @T 1 + (u · r)T = p r2T: (4) @t RaP r The variable u(r; t) is the velocity field, p(r; t) is the pressure, and T (r; t) is the temperature. We use no-slip velocity boundary conditions along all walls of the closed cylindrical cell. The temperature field obeys insulating sidewalls and constant values along the top and bottom plates. A spectral element method (SEM) is applied which is based on the Nek5000 software package [2]. The order of the Lagrangian interpolation polynomials for the expansion on each element (see the element mesh in the left panel of figure 1) which is used in each space direction is as high as N − 1 = 13. This code is a pure MPI code which scales very well up to ∼ 106 MPI tasks. Numerical details and comprehensive resolution tests are found in Scheel et al. [3]. The SEM is preferred if the fine-scale structure, i.e. gradients of the turbulent fields have to be analyzed. TURBULENT CONVECTION AT VERY LOW PRANDTL NUMBERS Compared to the vast number of investigations of turbulent convection in air or water, the very-low-P r regime appears almost as a “Terra incognita” despite many applications. Turbulent convection in the Sun is present at Prandtl numbers P r < 10−3, in the liquid metal core of the Earth one finds P r ∼ 10−2. Convection flows as present in material processing, nuclear engineering, or liquid metal batteries have Prandtl numbers between 5×10−3 and 2×10−2. One reason for significantly fewer studies of convection at low Prandtl numbers is that laboratory measurements have to be conducted in opaque and aggressive liquid metals when P r should become much smaller than 0.1. Direct numerical simulations (DNS) are currently the only way to gain access to the full three-dimensional convective turbulent fields in low-P r convection. However, these simulations become very demanding when the small-scale structure of turbulence has to be studied, even for moderate Rayleigh numbers Ra. To give an example: a simulation for convection Figure 1: Left: Spectral element grid for the RBC cell. Right: Snapshot of the dynamics in the vicinity of the bottom plate of the convection cell. We show field lines of the wall shear stress field and isocontours of the magnitude of the wall-normal vorticity component. The snapshot is for convection in liquid mercury at P r = 0:021 at Ra = 108. in liquid sodium at Ra = 2:38 × 106 and P r = 0:005 required the same SEM mesh of 4:1 × 109 cells as a simulation for convection in air at Ra = 1010 and P r = 0:7. The latter Rayleigh number is almost four orders of magnitude larger than in sodium. Both runs were conducted with 65,536 MPI tasks on BG/Q. What makes these simulations so expensive? While the heat transport is reduced in low–P r convection, the production of vorticity and shear are enhanced significantly. We found that this in turn amplifies the small-scale intermittency in these flows and makes them more comparable to classical Kolmogorov turbulence [4]. We have also shown in [4] that the cascade of the fluid turbulence is extended at the large-scale and small-scale end with decreasing Prandtl number thus enhancing the flow Reynolds number and consequently the turbulent momentum transport in the convection flow. The more vigorous fluid turbulence enhances in the enstrophy production in the fluid. Since the Rayleigh-Benard´ fluid is confined, tiny boundary layers of the temperature and velocity fields form in the vicinity of the walls. Figure 1 (right) displays a snapshot of the velocity boundary layer dynamics at the bottom plate for our biggest simulation in liquid mercury with 131,072 MPI tasks on BG/Q. This figure and our comprehensive analysis suggest that turbulence is also strongly amplified in the boundary layers although the Rayleigh number seems to be moderate. An increase from the present value of Ra = 108 to Ra = 5 × 108 requires 1:3 × 1010 mesh cells and 524,288 MPI tasks for a well-resolved run. We will also discuss the increasing challenges in the data analysis which one faces for these big simulations. Our research topic demonstrates clearly that supercomputing has become an essential tool for revealing the secrets of turbulence, and especially those of turbulent convection. ACKNOWLEDGEMENTS The work of JS is supported by the Deutsche Forschungsgemeinschaft. We want to thank the John von Neumann-Institute for Computing for the support with supercomputing resources by projects HIL07 and HIL09 as well as the Scientific Big Data Analytics Project SBDA003. This work is also supported by the DOE INCITE projects RayBen (2014) and LiquidGaNa (2016). References [1] Chilla` F. and Schumacher J.: New perspectives in turbulent Rayleigh-Benard´ convection. Eur. J. Phys. E 35: no. 58, 2012. [2] Nek5000 Website: https://www.nek5000.mcs.anl.gov/ [3] Scheel J. D., Emran M. S. and Schumacher J.: Resolving the fine-scale structure in turbulent Rayleigh-Benard´ convection. New J. Phys. 15: no. 113063, 2013. [4] Schumacher J., Gotzfried¨ P. and Scheel J. D.: Enhanced enstrophy generation for turbulent convection in low–Prandtl number fluids. Proc. Natl. Acad. Sci. USA 112: 9530–9535, 2015..
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