A brief history of simple invariant solutions in turbulence

@article{Veen2018ABH,
  title={A brief history of simple invariant solutions in turbulence},
  author={Lennaert van Veen},
  journal={arXiv: Fluid Dynamics},
  year={2018},
  pages={217-231}
}
  • L. V. Veen
  • Published 6 April 2018
  • Mathematics
  • arXiv: Fluid Dynamics
When studying fluid mechanics in terms of instability, bifurcation and invariant solutions one quickly finds out how little can be done by pen and paper. For flows on sufficiently simple domains and under sufficiently simple boundary conditions, one may be able to predict the parameter values at which the base flow becomes unstable and the basic properties of the secondary flow. On more complicated domains and under more realistic boundary conditions, such questions can usually only be… 

Can preferential concentration of finite-size particles in plane Couette turbulence be reproduced with the aid of equilibrium solutions?

This work employs for the first time invariant solutions of the Navier-Stokes equations to study the interaction between finite-size particles and near-wall coherent structures. We consider

Spontaneous Periodic Orbits in the Navier–Stokes Flow

A general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier-Stokes equations on the three-torus using a Newton-Kantorovich theorem is proposed.

On the classification of the laminar-turbulent transition process using the methods of nonlinear dynamics: general analysis and the future

The definition of a turbulent flow is still not mathematically defined. Different methods exists that provide different definitions resulting in different approaches and classifications of the

About mathematics and reality, Albert Einstein (1921 Nobel Prize in Physics), in his Geometry and Experience talk at the Prussian Academy of Sciences in Berlin

— Turbulence is a long-standing mystery. We survey some of the existing (and sometimes contradictory) results and suggest eight natural questions whose answers would increase the mathematical

References

SHOWING 1-10 OF 56 REFERENCES

The Significance of Simple Invariant Solutions in Turbulent Flows

Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. One of the significant advances in this respect has

Bifurcation phenomena in steady flows of a viscous fluid. I. Theory

  • T. Benjamin
  • Engineering
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1978
The investigation deals with questions relating to the existence of multiple solutions to hydrodynamic problems, especially questions about bifurcations of solutions and about stability. Although the

Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

This work shows that this method, called Stokes preconditioning, is 10–50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow and explains the convergence rate as a function of the integration period and Reynolds number.

Unstable periodic orbits in plane Couette flow with the Smagorinsky model

We aim at a description of the logarithmic velocity profile of wall turbulence in terms of unstable periodic orbits (UPOs) for plane Couette flow with a Smagorinsky-type eddy viscosity model. We

Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst

Two time-periodic solutions with genuine three-dimensional structure are numerically discovered for the incompressible Navier–Stokes equation of a constrained plane Couette flow. One solution with

Forecasting Fluid Flows Using the Geometry of Turbulence.

The existence and dynamical role of particular unstable solutions (exact coherent structures) of the Navier-Stokes equation is revealed in laboratory studies of weak turbulence in a thin,

Bifurcation phenomena in steady flows of a viscous fluid II. Experiments

  • T. B. Benjamin
  • Physics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1978
The experimental work is concerned with several phenomena studied theoretically in part I (Benjamin 1977). Observations on Taylor-vortex flows between cylinders of comparatively small but variable

Finite Amplitude Free Convection as an Initial Value Problem—I

Abstract The Oberbeck-Boussinesq equations are reduced to a two-dimensional form governing “roll” convection between two free surfaces maintained at a constant temperature difference. These equations

Bifurcation and stability analysis of laminar isothermal counterflowing jets

We present a numerical study of the structure and stability of laminar isothermal flows formed by two counterflowing jets of an incompressible Newtonian fluid. We demonstrate that symmetric

Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow

Unstable equilibrium solutions in a homogeneous shear flow with sinuous (streamwise-shift-reflection and spanwise-shift-rotation) symmetry are numerically found in large-eddy simulations (LES) with
...