EAGER: A Non-Boussinesq, Non-Incompressible Framework for Studying Atmospheric Turbulence

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A new theoretical framework for studying atmospheric turbulence is introduced, which challenges two of the most widely-used assumptions: the incompressibility of air and the Boussinesq approximation. The incompressible assumption is represented mathematically by the conservation of density following an air parcel and implies that the flow field is non-divergent. With the Boussinesq approximation, air density can be replaced by a constant, uniform value in all terms of the momentum equation, except for the gravity term. Both assumptions are generally valid in atmospheric flows and have the great benefit that they simplify noticeably the system of equations and their closures. However, the real atmosphere is neither incompressible nor homogeneous. Compressibility effects may be especially important in the proximity of wind turbines, whose rotor tips move at speeds that are close to Mach number of ~0.3, the limit for the incompressible assumption. Also, the majority of numerical mesoscale models used for routine weather forecasting are compressible. Given today's powerful and inexpensive computer systems, it seems time to step away from un-necessary assumptions and attempt to resolve "real" turbulent air flows, which are compressible and not homogeneous.In this new framework, the air incompressibility and the Boussinesq approximation are tossed and replaced by a simple, less-restrictive, new assumption: that air density is not turbulent. Whereas wind velocity, temperature, and pressure are turbulent and therefore they can be decomposed into a mean (or resolved) and a fluctuating (or eddy) component, air density is assumed to be varying slowly enough in time that its eddy component is zero. This simply means that air density responds to flow changes slower than the other meteorological variables, in such a way that it is fully resolved at the grid resolution and time step of the simulations and without chaotic, turbulent fluctuations below those temporal and spatial scales. Because under such an assumption density would remain 4-D (as opposed to 0-D as with the Boussinesq approximation), such a flow is not incompressible by design, because air density changes with time following a parcel, and the Boussinesq approximation becomes moot. The relevant equations need to be rewritten entirely, but, surprisingly, they remain very similar to their incompressible and Boussinesq counterparts, except for one additional term each in the momentum and in the continuity equations. These terms are easy to calculate numerically because they depend only on mean-flow properties and do not require any new closure or parameterization. The Large-Eddy Simulation (LES) model WiTTS (Wind Turbine and Turbulence Simulator), recently developed in-house at the University of Delaware, will be modified and tested with and without the new framework to evaluate the magnitude of the compressible effects.Impacts can occur in the wind industry, which currently relies on incompressible and Boussinesq LES codes to understand the deleterious effects of turbine wakes on the performance of wind turbines located further downstream. Even if incompressibility errors are just ~5-10%, understanding compressibility in turbulent wakes will allow them to reduce wake losses and optimize wind turbine layout, which could introduce savings of the order of millions of dollars over the lifetime of a wind project and therefore lower energy prices. Lastly, as wind farms are becoming increasingly widespread and more people live near a wind farm, understanding mixing of temperature, humidity, pollutants and dust induced by turbine wakes, as well as their sound waves, can positively impact the health and well-being of people who live in the proximity of wind turbines. Because this framework is transformative and challenges the "status-quo" of turbulence theory, it fits well with EAGER.