A 1D "Navier-Stokes Machine" and its application to turbulence studies

Preben Buchhave, Clara M. Velte

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In the present work, we investigate a numerical one‐dimensional solution to the Navier‐Stokes equation that retains all terms in the Navier‐Stokes equation, including both pressure and dissipation. The calculations take the full 4D flow as its
starting point and continuously projects the forces acting on the fluid at a fixed Eulerian point onto the direction of the instantaneous velocity. The pressure is included through modeling. Adhering to the requirement that time must in general
be considered an independent variable, the time development of the time records and power spectra of the velocity fluctuations are studied. It is found that the actions of the nonlinear term in the Navier‐Stokes equation manifests itself as sharp pulses in the time traces, where the sharpness is bounded by the finite viscosity. The sharp variations are interpreted as high shear regions, where large scales can interact to directly generate and energize smaller scales through nonlocal interactions. In the spectral domain, the sharp gradients in the pulses generate energy contributions at high frequencies that yields a ‐2 slope across the inertial range. The ‐2 (or ‐6/3) slope is explained through a simple example and the classically expected ‐5/3 slope in the inertial range can be recovered from the pressure fluctuations of the full field, which can be considered as a noise contribution in the point of consideration. We also observe that the spectrum can in principle keep spreading to higher frequencies or wavenumbers without upper bound, as the viscosity is approaching the zero limit. Furthermore, the large‐scale features of the unforced time trace are observed to be preserved throughout the flow development, ranging even into the soliton‐like state of decay – both in experiments and simulations.
Original languageEnglish
Publication statusSubmitted - 2021


  • Turbulence
  • Triad interactions
  • Navier‐Stokes
  • Velocity power spectrum
  • Numerical simulations


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