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Universal Equilibrium and Beyond - Challenging the Richardson-Kolmogorov Paradigm

Periodic Reporting for period 2 - UniEqTURB (Universal Equilibrium and Beyond - Challenging the Richardson-Kolmogorov Paradigm)

Reporting period: 2020-10-01 to 2022-03-31

The objective of the project is to test the range of validity of the classical theory of turbulence. When e.g. mixing cream into coffee, one can observe whorls of various sizes (scales). The classical theory assumes that turbulence at the small and intermediate scales behaves similarly to molecules in a thermal equilibrium, i.e. statistically they are at a state of equilibrium and even universal (i.e. same for all turbulent flows) - no matter what the dynamics at the large scales.

However, the analogy halts at several instances. Not least since even molecules (which are much smaller than turbulent scales) can be pushed out of equilibrium in common high-speed flow applications such as e.g. jet engines. The breakdown of the classical theory of turbulence is believed to happen in many technologically important turbulent flows, not least where there exist more or less rapid changes in space or time. Prominent examples of potentially significant non-equilibrium turbulence of technological importance is gas turbines, wind turbine wakes, propeller wakes and unsteady fuel injection. However, non-equilibrium turbulence is believed to also exist in less fiercely accelerated flows. The current project sets out to investigate the limits for which turbulence can be considered to be in equilibrium and how sensitive turbulence is to accelerations in time and/or space.

The study is being carried out using
- A theoretical framework that describes turbulence mathematically based on the governing equations.
- Direct measurements. Two types of measurements are being carried out: one covering the large scales and one covering the smallest relevant scales.
- Numerical simulations that resolve all relevant scales in time and space.

The theoretical framework allows us to describe turbulence based on the governing equations, while the large scale measurements will provide the actual dynamics of the turbulence that can fulfill the equations. This will allow us to test whether the energy exchange between scales is in fact dominatingly between scales of similar size, as predicted by the classical theory, which is believed to lead to equilibrium in the small and intermediate scales, or if other types of energy exchange can be possible. This is particularly interesting to study in turbulence being pushed out of equilibrium. We will compare the experimental findings to further develop and elaborate upon the theoretical description. The small scale measurements resolve the dissipative scales (where kinetic energy predominating is converted to heat through friction-based processes). This will quantify directly how much the flow is pushed out of equilibrium and provide modeling parameters that can be directly implemented. The simulations have inherently limited capacity since they can only provide limited amounts of the vast data necessary to carry out the analysis. Although they can be used for analysis only to some extent, they are an invaluable tool for designing the experiments and building the laboratory necessary to properly carry out the necessary experiments. The establishment of the laboratory is a major task in the current project.

If we can better understand and model non-equilibrium turbulence, which is of immense technological importance, then it will have a broad impact in engineering and natural science. This is very much basic research indeed, but the results will be directly applicable to analytical and computational modeling. Understanding turbulence is vital e.g. in almost all forms of transportation, designs of machinery operating in gasses (e.g. air) and liquids (e.g. water) and in the predictions of weather and climate. Turbulence is omnipresent in liquids and gasses and the impact of a successfully carried out project is difficult to overestimate.
Despite the challenges that have been presented by the global pandemic and consequent lock-downs of the university, laboratory and supporting functions such as workshops, support with instrumentation development etc., we are now in the finalizing stages of establishing the laboratory, flow generating facilities and developing the necessary laser diagnostics to carry out the experimental work associated with the project. The flow generating facilities will be able to produce turbulent flows of controllable degrees of non-equilibrium. Measurements will be carried out in two separate experimental campaigns: one resolving the smallest relevant scales to quantify the degree of non-equilibrium and one capturing the global turbulence dynamics in time and all three dimensions in space. This has required significant efforts in developing suitable laser diagnostics tools for properly carrying out these challenging measurements.

The establishment of the laboratory, flow facilities and laser diagnostics is a major result in itself, constituting a world-leading facility for basic turbulence research for both the global and local measurements. This is one of few, if not the only, facility where the full dissipation tensor and the full global dynamics (in time and three dimensions in space) of high intensity turbulence can both be measured. The measured signals are observed to be of high quality and the significant challenge of big data handling has been essentially resolved.

Theory intensive Direct Numerical Simulations have been finalized at an early stage in the project and have been used for developing theory, methods of analysis and for designing the experimental laboratory facilities. These simulations are computationally intensive and require significant storage capabilities, since they resolve all relevant scales of the turbulent flow in question both in time and space. In these simulations, we thus do not have to model any parts of the turbulence, but rather simulate the flow in accordance with the equations that govern fluid flow.

A theoretical framework has been developed to analyze the computational and experimental data generated. By revisiting the mathematical foundations for how to analyze turbulence dynamically, the classical description has been reformulated and expanded to being able to analyze and describe more challenging flow cases than the classical ones (e.g. stationary, homogeneous and isotropic turbulence). With this framework, we will not only be able to quantify the degree of non-equilibrium in turbulence, but also describe the interactions between different scales and how they are altered by the degree of non-equilibrium.

Even in global equilibrium turbulence (and even directly from the theory framework without the need to consider experimental data), a closer investigation has revealed the presence of dynamics that are clearly of non-equilibrium character. So far, theoretical and experimental results thus support the hypothesis that turbulence can indeed behave very differently from the classical theory.
Progress beyond state of the art:
- Development of unprecedented, sophisticated measurement system to measure dissipation (the smallest relevant turbulent scales)
- Development of laser diagnostics to cover larger volumes than previously reported in particle tracking measurements
- Development of a theoretical framework to decompose non-equilibrium turbulence in realistic settings in a mathematically sound manner. This will allow correct analysis of energy exchange between scales.

Expected results until end of the project:
- Quantitative measurements of the effect of various degrees of non-equilibrium and an understanding of the physical processes leading to the obtained non-equilibrium state.
Turbulence measurement setup