Hello all,
I am studying the laminar-turbulent phase transition. It is well known that in pipe flows, turbulence first manifests itself in localized turbulent patches surrounded by laminar background. On the other that, it is noteworthy that at transitional flow regime, the Reynolds number is relatively low and the flow is known to be dominated by coherent structures of some kind. Since coherent structures can not survive in high Reynolds numbers, I suppose that coherent structures are essentially a transient phenomenon governing the transitional flow regime.
However, these two pictures for transitional flow regime:
1) turbulence start with a well-defined spot embedded in the laminar environment and grows bigger and bigger until occupying the entire flow domain; and
2) Initially, transitional flow regime is overwhelmingly dominated by coherent structures; coherence is lost as Re increases and vanishes in fully developed turbulence.
to me, are contradicting. I would like to know if there is a connection between coherent structures and turbulent patches. And how about other confusing concepts such as traveling waves and laminar streaks?
Wish all a Merry Christmas and a prosperous new year ahead!
Thank you.
Very kind wishes,
Wang Zhe
It depends on the molecular composition of the fluid, the boundary conditions and the scales you are looking at. See the attached file for an introduction.
Article: Transitional boundary layer spot in a fully turbulent environment M. Zilberman · I. Wygnanski · R. E. Kaplan
[Show abstract] [Hide abstract]ABSTRACT: A spark was used to initiate and mark in time a turbulent spot in an initially laminar boundary layer. This marked spot of turbulence merged and interacted with the natural turbulent boundary layer generated by a row of spherical trips. By using a digital technique to align individual spot signatures, thus correcting for variations in the transit time to a given measurement station, a structure was tracked over a streamwise extent of 70 average turbulent boundary-layer thicknesses. The scale of the structure is of the order of 10 boundary layer thicknesses in the streamwise direction becoming 2-3 boundary layer thicknesses in the interface region of the boundary layer and is less than 4 boundary layer thicknesses in the spanwise direction, in spite of the fact that no spanwise alignment was performed. The structure is characterized by a convection speed of 0.9 streamline flow. It exhibits features in detailed agreement with those at the outer region of the turbulent boundary layer (interface region) and is consistent with existing two- and three-point space-time correlations.
Article · Oct 1977
The above mentioned article was published in the Physics of Fluids S1977.
It exhibits the identification of a turbulent spot that penetrates an existing turbulent boundary layer .
It also supports the idea that the turbulent boundary layer is formed by the superposition of interacting sport that are formed in the transitional region, following the 2-D Tollmien-Schlichting waves.
The coherent structures that are identified within the turbulent region are resembling the features of a turbulent spot, though , following the process of superposition the scale of the structures is limited to the boundary layer thickness where the turbulent spot growth in a laminar region is only limited in the vertical sense to the same scale but free to grow laterally.
See M. Zilberman Ph. D. Thesis 1981. "On the interaction of transitional spots and generation of a synthetic turbulent boundary layer".
Dear Adrian and Moshe,
Thank you for sharing your works, I will read them carefully.
In the study of laminar-turbulent phase transition, I mainly consider the pipe flow and the Couette flow because they are linearly stable so that the instability must be inherent in the nonlinear inertia term. Instead of perturbing laminar base flows, I perturb the nonlinear inertia term and study their bifurcations in the phase space. In this sense, I am still working under the continuum hypothesis, I am afraid.
I found that the original laminar flow can become chaotic by undergoing the following bifurcation sequence: Counterrotating vortical dipoles (by reflectional symmetry) - periodic solutions (reflectional symmetry breaking, generation of helicity from original helicity-free flow) - 2-torus (a slow frequency is born, possibly associated with coherent structures) - homoclinic orbit breaks up - Smale horseshoe (chaos is born, turbulent spots occur).
Now I am trying to find physical evidence and interpretation for the mathematical model and associated bifurcation sequence. I was thinking about the possibility of existing a coherent state between the laminar and fully developed turbulent states. I work at the school of electrical engineering, so I actually took an analog from the phase transition between ferromagnetic and paramagnetic materials.
I am aware that the dynamical system theory studies the temporal chaos, while coherent structures and turbulent spots are spatially organized structures, but the resemblance is so seductive that I cannot resist conjecturing there exists certain unknown mechanism synchronize fluids with coherent state and chaotic state respectively into coherent structures and turbulent packets. During the transition, the coherent and turbulent states compete for their appearance, until reaching the critical Reynolds number, coherent states vanishes, and turbulence becomes self-sustaining. Such that, the "coherent structure" here may not be certain well-defined structures restricted in wall-bounded shear flows, instead, they are long-lasting periodic solutions of the Navier-Stokes equation, coexisting with turbulent spots....
Merry Christmas and happy holidays!
Best wishes from sunny Singapore,
Wang Zhe
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