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Turbulence and the dynamics of coherent structures. I. Coherent structures

High-order splitting methods for the incompressible Navier-Stokes equations

The theory of two-dimensional turbulence is reviewed and unified, and some hydrodynamic and plasma applications are considered. The topics covered include some equations of incompressible hydrodynamics, absolute statistical equilibrium, spectral transport of energy and enstrophy, turbulence on the surface of a rotating sphere, turbulent diffusion, MHD turbulence, and two-dimensional superflow. Finally, an attempt is made to assess the status and future of the principal research topics which have been discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/43/5/001/pdf

Two-dimensional turbulence

The nonlinear evolution of magnetized Keplerian shear ows is simulated in a local, three-dimensional model, including the eeects of compressibility and stratiication. Supersonic ows are initially generated by the Balbus-Hawley magnetic shear instability. The resulting ows regenerate a turbulent magnetic eld which, in turn, reinforces the turbulence. Thus, the system acts like a dynamo that generates its own turbulence. However, unlike usual dynamos, the magnetic energy exceeds the kinetic energy of the turbulence by a factor of 3{10. By assuming the eld to be vertical on the outer (upper and lower) surfaces we do not constrain the horizontal magnetic ux. Indeed, a large scale toroidal magnetic eld is generated, mostly in the form of toroidal ux tubes with lengths comparable to the toroidal extent of the box. This large scale eld is mainly of 1 even (i.e. quadrupolar) parity with respect to the midplane and changes direction on a timescale of about 30 orbits, in a possibly cyclic manner. The eeective Shakura-Sunyaev alpha viscosity parameter is between 0.001 and 0.005, and the contribution from the Maxwell stress is about 3-7 times larger than the contribution from the Reynolds stress.

Dynamo-generated Turbulence and Large-Scale Magnetic Fields in a Keplerian Shear Flow

Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112090000829

Three-dimensional finite-amplitude solutions in plane couette flow: bifurcation from infinity

Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112080002066

Transition to turbulence in plane Poiseuille and plane Couette flow

A statistical analysis is performed of various low‐order truncations of the two‐dimensional inviscid Navier–Stokes equations. The behavior of the time‐averaged enstrophy and time‐correlation function of each Fourier mode is analyzed and compared to the behavior expected in a random dynamical system. Empirical evidence is presented for the randomness of all isotropically truncated models for which the wavenumber cutoff is large enough to insure that every mode is involved in nontrivial interactions. This evidence is contrasted with the apparent nonrandomness found in other special truncations. The results strengthen the evidence for the validity of statistical descriptions of two‐dimensional turbulence.

Lawrence C. Kells

Papers

Transition to turbulence in plane Poiseuille and plane Couette flow

Randomness of low‐order models of two‐dimensional inviscid dynamics