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Direct numerical simulations towards ultimate turbulence

Periodic Reporting for period 4 - UltimateRB (Direct numerical simulations towards ultimate turbulence)

Reporting period: 2023-07-01 to 2023-12-31

What problem is addressed?
Rayleigh-Bénard (RB) convection, which involves fluid flow in a box heated from below and cooled from above, serves as a fundamental model in fluid dynamics. It is used to test concepts such as non-linear dynamics, pattern formation, and turbulence. RB convection is particularly ideal for examining the interactions between boundary layers and bulk fluid dynamics. Additionally, turbulent thermal convection is crucial in a wide range of natural and industrial settings, from astrophysical and geophysical flows to process engineering. The paradigmatic representation of thermal convection is RB flow. A major challenge is to determine the scaling relation of the Nusselt number (Nu), i.e. the dimensionless heat transport, with the Rayleigh number (Ra), which is the dimensionless temperature difference between the two plates, expressed as Nu~Ra^gamma. Theory predicts that the scaling exponent gamma increases for extremely strong driving when the boundary layers transition from laminar to turbulent. Understanding the transition to this so-called ‘ultimate’ regime is crucial since an extrapolation of results from lab-scale experiments and simulations to astro- and geophysical phenomena becomes meaningless when the transition to this ‘ultimate’ state is not understood. So far, there is no consensus among experimental efforts for obtaining the ‘ultimate’ regime. We propose using direct numerical simulations (DNS) to gain a better understanding of the transition towards the ‘ultimate’ regime. While obtaining ‘ultimate’ thermal convection in simulations has been elusive, new developments make this feasible now. The benefit of simulations is that they allow full access to the flow and temperature fields, while all boundary conditions are set exactly and independently. This allows us to test various physical effects at full dynamic similarity. To trigger the excitation of the ‘ultimate’ regime at lower Ra than in standard small aspect ratio cells, we want to study the effect of roughness, additional shear, and large domains in which a stronger flow can develop than in confined small aspect ratio cells that are traditionally considered. The addition of rotation will be studied to disentangle the complicated effect of rotation on high Ra number thermal convection.

Why is it important for society?
Fluid dynamics and turbulence are key areas in physics, applied mathematics, and engineering with applications in industries like process technology, automotive, and transportation. They are essential in diverse environments, including the atmosphere and oceans, as well as in geophysical and astrophysical studies. The simulation tool AFiD, which is publicly available, to explore turbulent flows with great precision. AFiD supports simulations of canonical flows. Given its capacity to handle the exact balance of driving forces and dissipation rates, the RB system is highly effective for developing new criteria for simulating highly turbulent flows. The methodologies and insights from this project will significantly impact the broader field of fluid dynamics.

What are the overall objectives?
Developing groundbreaking DNS to enhance our understanding of very high Ra number convection; project objectives include
* Developing a code to perform DNS of 'ultimate' RB convection
* Study impact of superstructures on the fundamental properties of thermal convection
* Model heat transfer in rotating high Ra number convection

Conclusions
The methodologies, simulation methods, and scientific discoveries related to the physical phenomena have been thoroughly documented and are accessible to the scientific communities. The developed software is publicly available at:
* AFiD software: github.com/PhysicsofFluids/AFiD
* GPU-version: github.com/PhysicsofFluids/AFiD_GPU_opensource
* AFiD-MuRPhFi: github.com/chowland/AFiD-MuRPhFi
* We conduct direct numerical simulations to study heat transfer and flow dynamics in turbulent RB convection across varying domain shapes and aspect ratios (0.25≤Γ≤32). Focusing on a Prandtl number of Pr=1 and Rayleigh numbers Ra=2×10^7 and Ra=10^8, we observe distinct effects of confinement on heat transfer in both laterally periodic and cylindrical domains. For smaller aspect ratios (Γ≲0.75) heat transfer decreases in periodic domains but increases in cylindrical ones. This distinction fades as Γ approaches 0.75 equalizing in both domain types around Γ=4. At this point, the boundary layer thickness and volume-integrated Reynolds number reach their large-Γ limit.

* For Γ=0.23 and Γ=0.50 with Ra up to 1×10^{14}, we highlight the importance of resolving the sidewall boundary layer, revealing that the structure of the velocity boundary layer within the thermal boundary layer shifts notably with increasing Ra. Simulated results align closely with experimental data, demonstrating changes in flow strength and heat transfer dynamics across aspect ratios.

* Extending the Grossmann & Lohse theory, we provide scaling relations for the Nusselt number and friction coefficient in sheared RB convection, involving Couette or Poiseuille type shear. Our simulations confirm Prandtl's logarithmic friction law across different shear conditions, suggesting a universality in shear-driven and thermal convection flows.

* Exploring rotating RB convection, we find the heat transfer is influenced significantly by Pr, Ra, and Ro numbers. Notably, heat transfer peaks at an optimal rotation rate, which varies across different Ra regimes. The optimal rotation rate calculation, derived from boundary layer thicknesses, indicates a complex relationship with Pr and Ra.

* In spherical RB convection under rotation, we identify three distinct flow regions characterized by varying convection styles and their corresponding heat transport. From high to equatorial latitudes, flow structures and heat transport mechanisms differ, with mid-latitude regions showing bulk-dominated convection, indicating an ultimate regime.

* Lastly, considering icy moons' subglacial oceans, our simulations of rotating spherical RB convection reveal a potential for enhanced polar heat transport, particularly in polar tangent cylinders. This polar enhancement, which could exceed equatorial heat transport by up to 50%, appears less affected by variations in gravity profiles and is more pronounced in thicker shells, aligning with principles observed in planar RB convection.
* The UltimateRB project set a new benchmark in RB convection simulations with nearly 100 billion grid points and requiring over 50 million CPU hours, highlighted on the cover of Physics Today last November.

* Our spherical convection simulations showed that flow dynamics and heat transport are primarily influenced by mid-latitude regions.

* In spherical rotating RB convection, we found enhanced heat transfer at the poles due to specific flow patterns.

* We extended the famous Grossmann and Lohse's unifying theory of thermal convection to include shear convection, enhancing our understanding of thermal convection.

* Our studies revealed that the impact of rotation on thermal convection varies with Ra number and the system's aspect ratio. We studied the transition to geostrophic turbulence at high Ra number thermal convection.
Flow structure for optimal rotation rate (highest heat transport) for low (left) and high (right) Ra
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Temperature (left) and vertical velocity (right) in thermal convection
Visualization of the large scale circulation in thermally driven turbulence.
Temperature (left) and vertical velocity (right) in thermal convection after high pass filter
Temperature field at thermal boundary layer height for a very high Rayleigh number simulation.
Vertical cross section of the temperature field of a very high Rayleigh number simulation