319 research outputs found
Remarks on a quasi-linear model of the Navier-Stokes Equations
Dinaburg and Sinai recently proposed a quasi-linear model of the
Navier-Stokes equations. Their model assumes that nonlocal interactions in
Fourier space are dominant, contrary to the Kolmogorov turbulence phenomenology
where local interactions prevail. Their equation corresponds to the linear
evolution of small scales on a background field with uniform gradient, but the
latter is defined as the linear superposition of all the small scale gradients
at the origin. This is not self-consistent.Comment: 4 page
Shear-flow transition: the basin boundary
The structure of the basin of attraction of a stable equilibrium point is
investigated for a dynamical system (W97) often used to model transition to
turbulence in shear flows. The basin boundary contains not only an equilibrium
point Xlb but also a periodic orbit P, and it is the latter that mediates the
transition. Orbits starting near Xlb relaminarize. We offer evidence that this
is due to the extreme narrowness of the region complementary to basin of
attraction in that part of phase space near Xlb. This leads to a proposal for
interpreting the 'edge of chaos' in terms of more familiar invariant sets.Comment: 11 pages; submitted for publication in Nonlinearit
Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
The instability of a streak and its nonlinear evolution are investigated by
direct numerical simulation (DNS) for plane Poiseuille flow at Re=3000. It is
suggested that there exists a traveling-wave solution (TWS). The TWS is
localized around one of the two walls and notably resemble to the coherent
structures observed in experiments and DNS so far. The phase space structure
around this TWS is similar to a saddle point. Since the stable manifold of this
TWS is extended close to the quasi two dimensional (Q2D) energy axis, the
approaching process toward the TWS along the stable manifold is approximately
described as the instability of the streak (Q2D flow) and the succeeding
nonlinear evolution. Bursting corresponds to the escape from the TWS along the
unstable manifold. These manifolds constitute part of basin boundary of the
turbulent state.Comment: 5 pages, 6 figure
The helical decomposition and the instability assumption
Direct numerical simulations show that the triadic transfer function T(k,p,q) peaks sharply when q (or p) is much smaller than k. The triadic transfer function T(k,p,q) gives the rate of energy input into wave number k from all interactions with modes of wave number p and q, where k, p, q form a triangle. This observation was thought to suggest that energy is cascaded downscale through non-local interactions with local transfer and that there was a strong connection between large and small scales. Both suggestions were in contradiction with the classical Kolmogorov picture of the energy cascade. The helical decomposition was found useful in distinguishing between kinematically independent interactions. That analysis has gone beyond the question of non-local interaction with local transfer. In particular, an assumption about the statistical direction of triadic energy transfer in any kinematically independent interaction was introduced (the instability assumption). That assumption is not necessary for the conclusions about non-local interactions with local transfer recalled above. In the case of turbulence under rapid rotation, the instability assumption leads to the prediction that energy is transferred in spectral space from the poles of the rotation axis toward the equator. The instability assumption is thought to be of general validity for any type of triad interactions (e.g. internal waves). The helical decomposition and the instability assumption offer detailed information about the homogeneous statistical dynamics of the Navier-Stokes equations. The objective was to explore the validity of the instability assumption and to study the contributions of the various types of helical interactions to the energy cascade and the subgrid-scale eddy-viscosity. This was done in the context of spectral closures of the Direct Interaction or Quasi-Normal type
Visualizing the geometry of state space in plane Couette flow
Motivated by recent experimental and numerical studies of coherent structures
in wall-bounded shear flows, we initiate a systematic exploration of the
hierarchy of unstable invariant solutions of the Navier-Stokes equations. We
construct a dynamical, 10^5-dimensional state-space representation of plane
Couette flow at Re = 400 in a small, periodic cell and offer a new method of
visualizing invariant manifolds embedded in such high dimensions. We compute a
new equilibrium solution of plane Couette flow and the leading eigenvalues and
eigenfunctions of known equilibria at this Reynolds number and cell size. What
emerges from global continuations of their unstable manifolds is a surprisingly
elegant dynamical-systems visualization of moderate-Reynolds turbulence. The
invariant manifolds tessellate the region of state space explored by
transiently turbulent dynamics with a rigid web of continuous and discrete
symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic
Experimental scaling law for the sub-critical transition to turbulence in plane Poiseuille flow
We present an experimental study of transition to turbulence in a plane
Poiseuille flow. Using a well-controlled perturbation, we analyse the flow
using extensive Particule Image Velocimetry and flow visualisation (using Laser
Induced Fluorescence) measurements and use the deformation of the mean velocity
profile as a criterion to characterize the state of the flow. From a large
parametric study, four different states are defined depending on the values of
the Reynolds number and the amplitude of the perturbation. We discuss the role
of coherent structures, like hairpin vortices, in the transition. We find that
the minimal amplitude of the perturbation triggering transition scales like
Re^-1
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