Simulations Of Two-dimensional Turbulence Research Articles

Simulation of two-dimensional turbulence in a planar channel

A study is made of the two-dimensional flow of an incompressible viscous fluid in a planar channel at supercritical Reynolds number Re = 104. Calculation of the flow over an appreciable time interval leads to the establishment of a statistically steady flow regime and stabilization of its averaged characteristics (the profile of the mean velocity, the mean pressure gradient, the pulsation energy, etc.). The calculations show that numerical simulation of turbulent flow of an incompressible fluid on the basis of the Navier-Stokes equations leads to qualitatively correct characteristics of this flow regime.

Numerical simulation of two-dimensional turbulence in a plane channel

Two-dimensional flow of an incompressible viscous fluid in a plane channel is studied under fixed flux and supercritical Reynolds number, Re = 10 4. For numerical simulation, and original algorithm, possessing good stability and accuracy properties, is used. Calculation of the flow over a large interval of time leads to a statistically steady-state condition of the flow and to stabilization of the averaged characteristics, e.g., profiles of averaged velocity, averaged gradient of pressure, energy, etc. The computations show that qualitatively proper characteristics of “two-dimensional” turbulence can be obtained by numerical simulation of the Navier-Stokes equations.

Ergodic Boundary in Numerical Simulations of Two-Dimensional Turbulence

In numerical calculations, we observe a dichotomy in the time evolution of solutions of the high-Reynolds-number, two-dimensional, incompressible Navier-Stokes equations using power-law initial modal energy spectra $E(\ensuremath{\kappa},0)\ensuremath{\sim}{\ensuremath{\kappa}}^{\ensuremath{-}{\ensuremath{\mu}}_{0}}$. The boundary is expressed in terms of critical values of viscosity ${\ensuremath{\nu}}_{\mathrm{cr}}$ and microscale ${\ensuremath{\lambda}}_{\mathrm{cr}}$. For both parameters initially above critical, $\ensuremath{\mu}$ approaches 4. For both parameters initially below critical, $\ensuremath{\mu}$ approaches 1 at large $\ensuremath{\kappa}$, consistent with equipartition of the vorticity spectrum. In the former case, large-scale vortex states form after a long time; and, in both cases, energy flows to the lowest wave numbers.

Numerical Simulation of Two-Dimensional Turbulence

The two-dimensional incompressible Navier-Stokes equations are numerically integrated in order to test the validity of Kraichnan's predictions on the structure of two-dimensional turbulence. The numerically simulated turbulence is produced by an artificial forcing function, constructed from a randomly varying set of Fourier modes lying within a narrow spectral band near scalar wavenumber ke. For the case of turbulence maintained by a constant rate of energy injection, Kraichnan predicts the development of two power-law energy spectra, a k−5/3 range for frequencies less than ke, and a k−3 range for those greater than ke. The numerical results are consistent with these predictions, although truncation and aliasing errors restrict the useful spectral range of the calculations to less than that desirable for a fully satisfactory test. The predictions and results are apparently relevant to several areas of geophysical fluid dynamics and to the correct interpretation of other numerical simulation experiments.