## Periodic Reporting for period 2 - UltimateRB (Direct numerical simulations towards ultimate turbulence)

Reporting period: 2020-07-01 to 2021-12-31

Rayleigh-Bénard (RB) convection, i.e. the flow in a box heated from below and cooled from above, is one of the paradigmatic systems in fluid dynamics. The system is used to test new concepts, such as instabilities, non-linear dynamics, pattern formation, or turbulence. RB convection is a relevant model for countless phenomena ranging from thermal convection in the atmosphere, oceans, and the outer layer of the Sun, to heating and ventilation of buildings and convection in various industrial applications. Thus the problem is of interest in a wide range of sciences. The RB system is popular because

1. The system is mathematically well-defined by the extended Navier-Stokes equations and the appropriate boundary conditions;

2. there are exact relations between the driving and the dissipation;

3. RB convection is experimentally accessible with high precision; and

4. The boundaries and resulting boundary layers play a prominent role.

Therefore the RB system is ideally suited to study the interaction between the boundary layer and bulk dynamics.

This project develops and uses groundbreaking direct numerical simulations (DNS) to improve our understanding of very high Rayleigh (Ra) number convection and the transition to the ultimate regime. Specifically, we plan to develop a code to perform DNS of unprecedentedly high Ra number convection. So far, high Ra number simulations and experiments have focused on small diameter to height aspect ratio cells. However, many natural instances of convection have enormous, almost infinite aspect ratios. Consequently, the influence of the diameter to height aspect ratio of the domain is less well understood. Therefore, we plan to study the effect of large-scale convective structures, known as superstructures, on the fundamental properties of thermal convection. Many natural and technical problems are also influenced by rotation or additional shear, and therefore it is essential to better understand the influence of these effects on heat transfer in turbulent thermal convection. For example, an intriguing phenomenon observed in rotating convection is that the heat transport is enhanced by Ekman pumping. For relatively low Ra, this process is reasonably well understood. In this project, we aim to explore the effect of shear and rotation at high Ra.

1. The system is mathematically well-defined by the extended Navier-Stokes equations and the appropriate boundary conditions;

2. there are exact relations between the driving and the dissipation;

3. RB convection is experimentally accessible with high precision; and

4. The boundaries and resulting boundary layers play a prominent role.

Therefore the RB system is ideally suited to study the interaction between the boundary layer and bulk dynamics.

This project develops and uses groundbreaking direct numerical simulations (DNS) to improve our understanding of very high Rayleigh (Ra) number convection and the transition to the ultimate regime. Specifically, we plan to develop a code to perform DNS of unprecedentedly high Ra number convection. So far, high Ra number simulations and experiments have focused on small diameter to height aspect ratio cells. However, many natural instances of convection have enormous, almost infinite aspect ratios. Consequently, the influence of the diameter to height aspect ratio of the domain is less well understood. Therefore, we plan to study the effect of large-scale convective structures, known as superstructures, on the fundamental properties of thermal convection. Many natural and technical problems are also influenced by rotation or additional shear, and therefore it is essential to better understand the influence of these effects on heat transfer in turbulent thermal convection. For example, an intriguing phenomenon observed in rotating convection is that the heat transport is enhanced by Ekman pumping. For relatively low Ra, this process is reasonably well understood. In this project, we aim to explore the effect of shear and rotation at high Ra.

We study the flow organization and the effect of shear and rotation on high Ra number thermal convection.

Flow organization

We studied the flow organization in thermal convection by investigating the interplay between large-scale flow patterns, known as superstructures, in the temperature and vertical velocity fluctuation fields. Previous studies suggested that the velocity superstructures are smaller than their thermal counterparts. However, using a scale-by-scale analysis of the correlation between the temperature and vertical velocity fields, we find that superstructures of the same size exist in both fields. The issue is clarified by observing that superstructures do not result in a peak in the vertical velocity power spectrum. We also developed a conditional averaging technique, which allows us to extract statistics of the large-scale circulation (LSC) even when their orientation and structure varies throughout the domain. Remarkably, we find that the LSC characteristics do not depend much on the horizontal extent of the LSC.

Effect of shear

We showed that with increasing shear, different flow regimes are formed as the flow transitions from the buoyancy dominated regime, to a transitional regime, up to a shear dominated regime. In the buoyancy-dominated regime, the flow dynamics are dominated by the LSC. The transitional regime is characterized by rolls that are increasingly elongated with increasing shear. The flow in the shear-dominated regime consists of meandering rolls. Due to these different flow regimes, for fixed Ra and with increasing shear, the heat transfer first decreases due to the breakup of the thermal rolls and increases at the beginning of the shear-dominated regime.

Effect of rotation

Heat transfer is strongly influenced by the Ra, Prandtl (Pr), and Rossby numbers. For Pr>1 and intermediate rotation rates, the heat transfer is increased compared to the nonrotating case. We find that the regime of increased heat transfer is subdivided into a low and high Ra number regime. For low Ra, the heat transfer at a given Ra and Pr is highest when the thicknesses of the viscous and thermal boundary layers are equal. However, in the newly discovered high Ra number regime, the optimal rotation rate and the corresponding flow structures are very different than in the low Ra number regime.

Flow organization

We studied the flow organization in thermal convection by investigating the interplay between large-scale flow patterns, known as superstructures, in the temperature and vertical velocity fluctuation fields. Previous studies suggested that the velocity superstructures are smaller than their thermal counterparts. However, using a scale-by-scale analysis of the correlation between the temperature and vertical velocity fields, we find that superstructures of the same size exist in both fields. The issue is clarified by observing that superstructures do not result in a peak in the vertical velocity power spectrum. We also developed a conditional averaging technique, which allows us to extract statistics of the large-scale circulation (LSC) even when their orientation and structure varies throughout the domain. Remarkably, we find that the LSC characteristics do not depend much on the horizontal extent of the LSC.

Effect of shear

We showed that with increasing shear, different flow regimes are formed as the flow transitions from the buoyancy dominated regime, to a transitional regime, up to a shear dominated regime. In the buoyancy-dominated regime, the flow dynamics are dominated by the LSC. The transitional regime is characterized by rolls that are increasingly elongated with increasing shear. The flow in the shear-dominated regime consists of meandering rolls. Due to these different flow regimes, for fixed Ra and with increasing shear, the heat transfer first decreases due to the breakup of the thermal rolls and increases at the beginning of the shear-dominated regime.

Effect of rotation

Heat transfer is strongly influenced by the Ra, Prandtl (Pr), and Rossby numbers. For Pr>1 and intermediate rotation rates, the heat transfer is increased compared to the nonrotating case. We find that the regime of increased heat transfer is subdivided into a low and high Ra number regime. For low Ra, the heat transfer at a given Ra and Pr is highest when the thicknesses of the viscous and thermal boundary layers are equal. However, in the newly discovered high Ra number regime, the optimal rotation rate and the corresponding flow structures are very different than in the low Ra number regime.

So far, as described above, we have obtained novel insights into the structure of turbulent thermal convection. We have, for example, explained the apparent discrepancy between the size of the flow structures as measured from the temperature and vertical velocity fields. Furthermore, we have discovered a high Ra number regime in rotating thermal convection in which the heat transfer and flow structures are very different than in the low Ra number regime. In addition, we explored the effects of shear on thermal convection. We hope to build on these findings and obtain other fascinating results in the remainder of the project.

In the project, we also develop and execute landmark DNS of RB convection at very high Ra. We are currently performing simulations at Ra=10^13 and Ra=10^14 in a closed cylindrical domain. The simulation at Ra=10^14 is performed on a grid with nearly 100 billion nodes, a landmark simulation for turbulent flow in a fully closed domain. Comparison with the Göttingen measurements will allow a one-to-one comparison between experiments and simulations that will allow us to investigate the flow physics at unprecedentedly high Ra.

In the project, we also develop and execute landmark DNS of RB convection at very high Ra. We are currently performing simulations at Ra=10^13 and Ra=10^14 in a closed cylindrical domain. The simulation at Ra=10^14 is performed on a grid with nearly 100 billion nodes, a landmark simulation for turbulent flow in a fully closed domain. Comparison with the Göttingen measurements will allow a one-to-one comparison between experiments and simulations that will allow us to investigate the flow physics at unprecedentedly high Ra.