Turbulent superstructures

A tremendous variety of physical phenomena involve turbulence, such as bird and airplane flight, propulsion of fish and boats, sailing, and even galaxy formation. Turbulent flow is characterized by chaotic swirling motions that vary widely in size, from submillimeter scales and storm clouds to phenomena on a galactic scale. The interaction of motions across these scales makes the underlying geometric structures challenging to simulate and understand. Resolving these patterns is necessary to connect small-scale transport to large-scale flow organization.

Turbulent thermal convection is a class of turbulent flows driven by temperature differences. Prime examples include convection in the atmosphere, thermohaline circulation in the oceans, heating and ventilation in buildings, and industrial processes. The model system is Rayleigh-Bénard flow, in which a fluid layer is heated from below and cooled from above. The temperature difference between the plates produces large-scale flow patterns in which hot fluid rises and cold fluid descends. Because the Rayleigh-Bénard system is mathematically well defined, simulations can be compared directly with theory and state-of-the-art laboratory experiments. For more than a century, it has therefore served as a test bed for experimental and numerical techniques in fluid dynamics.

A classical assumption is that sufficiently strong turbulent fluctuations reduce the influence of system geometry because the accessible states are explored statistically. This motivates the use of small-aspect-ratio domains, whose horizontal extent is small relative to their height. Such domains greatly reduce the experimental or computational cost of reaching high Rayleigh numbers. Consequently, many experiments and simulations at high Rayleigh number have focused on small-aspect-ratio cells. This approach enables high-Rayleigh-number heat-transfer studies at lower cost, but it restricts the large-scale horizontal organization that the domain can contain.

However, while heat transfer in industrial applications often occurs in confined systems, many natural instances of convection take place in horizontally extended systems. Previous experiments and simulations at relatively low Rayleigh numbers showed that large-scale horizontal flow patterns can emerge. Motivated by advances in computational capabilities, we performed AFiD simulations of high-Rayleigh-number convection in very wide domains. The simulations reveal that turbulent thermal superstructures survive in fully turbulent flow; see figure 1 [1]. Later work showed that temperature and vertical-velocity superstructures have comparable characteristic sizes and that this size increases with Rayleigh number [2]. The required domain size depends on the observable: heat transfer and total flow strength reach their wide-domain limits near an aspect ratio of 4, whereas temperature and velocity variance profiles converge near 8 in periodic domains and near 16 in cylindrical cells [3]. Thus, large-scale organization remains important for the flow statistics even after global transport appears converged.

References

3. R.J.A.M. Stevens, R. Hartmann, R. Verzicco, D. Lohse,
   How wide must Rayleigh-Bénard cells be to prevent finite aspect ratio effects in turbulent flow?
   J. Fluid Mech. 1000, A58 (2024).
4. D. Krug, D. Lohse, R.J.A.M. Stevens,
   Coherence of temperature and velocity superstructures in turbulent Rayleigh-Bénard flow,
   J. Fluid Mech. 887, A2 (2020).
5. R.J.A.M. Stevens, A. Blass, X. Zhu, R. Verzicco, D. Lohse,
   Turbulent thermal superstructures in Rayleigh-Bénard convection,
   Phys. Rev. Fluids 3, 041501(R) (2018)