Global energy budgets in turbulent plane Couette and Poiseuille flows

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Andrea Andreolli

A general framework for comparing different turbulent wall flows in terms of their integral budget of kinetic energy is introduced and applied to plane Couette and Poiseuille flows. The works of Hasegawa et al. (2014) and Gatti et al. (2018) discussing pressure-driven Poiseuille flows are extended to account for the externally-imposed shear typical of a Couette flow. The mean velocity is decomposed into a Stokes solution and a deviation field, which leads to a similar decomposition of the integral budget of kinetic energy as well. Thanks to a power-based non-dimensionalisation, several quantities (including some velocity scales and the decomposed budget terms) can be expressed as functions of a power-based Reynolds number and integrals of the shear stress.

The framework is then applied to a database of Poiseuille and Couette flows, built with new Direct Numerical Simulations and integrated with literature data. Terms of the integral budget for the kinetic energy are found to be comparable when the friction-based Reynolds number Reτ for Poiseuille is roughly three times the one for Couette. Among these budget terms, the laminar dissipation ΦL is of particular importance, as it can be interpreted as the ratio between the power required by the Stokes solution and the actual mean flow. Since the Stokes component can be demonstrated to have the least power requirements, ΦL can be regarded as an efficiency. Lastly, the wall-normal profiles of the turbulent kinetic energy budget terms for Couette and Poiseuille are compared under different conditions, such as keeping Reτ or ΦL constant. It is found that a consistently improved correspondence in the buffer layer (y+ ≈ 10) of several statistics is obtained in the latter case, both in terms of peak position and amplitude. All these findings suggest that laminar dissipation ΦL should be preferred to Reτ as a means of comparing different flows.