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<p>In this article we consider a damped version of the incompressible Navier–Stokes equations in the whole three-dimensional space with a divergence-free and time-independent external force. Within the framework of a well-prepared force and with a particular choice of the damping parameter, when the Grashof numbers are large enough, we are able to prove some estimates from below and from above between the fluid characteristic velocity and the energy dissipation rate according to the Kolmogorov dissipation law. Precisely, our main contribution concerns the estimate from below which is not often studied in the existing literature. Moreover, we address also some remarks which open the door to a deep discussion on the validity of this theory of turbulence. Finally, we study the Kolmogorov dissipation law in the setting of the stationary solutions.</p>

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A study on turbulence and open problems for a damped Navier–Stokes equation

  • 期刊论文

  • 发表日期：

    2025年4月17日

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  • Oscar Jarrín,
  • Diego Chamorro

摘要

In this article we consider a damped version of the incompressible Navier–Stokes equations in the whole three-dimensional space with a divergence-free and time-independent external force. Within the framework of a well-prepared force and with a particular choice of the damping parameter, when the Grashof numbers are large enough, we are able to prove some estimates from below and from above between the fluid characteristic velocity and the energy dissipation rate according to the Kolmogorov dissipation law. Precisely, our main contribution concerns the estimate from below which is not often studied in the existing literature. Moreover, we address also some remarks which open the door to a deep discussion on the validity of this theory of turbulence. Finally, we study the Kolmogorov dissipation law in the setting of the stationary solutions.