Experiments in turbulent soap-film flows: Marangoni shocks, frictional drag, and energy spectra

We carry out unprecedented experimental measurements of the frictional drag in turbulent soap-film flows over smooth walls. These flows are effectively two-dimensional, and we are able to create soap-film flows with the two types of turbulent spectrum that are theoretically possible in two dimensions: the ``enstrophy cascade,'' for which the spectral exponent α=3, and the ``inverse energy cascade,'' for which the spectral exponent α=5/3. We find that the functional relation between the frictional drag $f$ and the Reynolds number Re depends on the spectral exponent: where α=3, f \propto {\Re^-1/2}; where α=5/3, f \propto {\Re^-1/4}. These findings cannot be reconciled with the classic theory of the frictional drag. The classic theory provides no means of distinguishing between one type of turbulent spectrum and another, and cannot account for the existence of a ``spectral link'' between the frictional drag and the turbulent spectrum. In view of our experimental results, we conclude that the classic theory must be considered incomplete. In contrast, our findings are consistent with a recently proposed spectral theory of the frictional drag. In this theory the frictional drag of turbulent flows on smooth walls is predicted to be f\propto {\rm Re}<sup>(1-α)/(1+α)</sup>, where α is the spectral exponent. This prediction is in exact accord with our experiments on soap-film flows. It is also in accord with the available experimental data on three-dimensional pipe flows, where a single type of spectrum is possible: the ``energy cascade,'' for which α=5/3 (the same as for the inverse energy cascade). In fact, for α=5/3 the prediction of the spectral theory coincides with the emprirical law of Blasius (f \propto {\Re^-1/4}), which gives the best representation of the available experimental results for three-dimensional pipe flows of moderate turbulent strength (starting from ${\rm Re} \approx 2,500$ and up to ${\rm Re}\approx 100,000$). In carrying out our experiments on the frictional drag, we discover the spontaneous occurrence in unobstructed soap-film flows of a type of shock related to the elasticity of the film. By means of extensive experimental measurements, we verify that these shocks are dissipative and diffusive; that they give rise to fluctuations independently from the boundaries, with a strong but circumscribed effect on the spatial distribution of turbulent intensity; and that they alter the structure of the turbulent spectrum downstream from the shock. We show that a simple one--dimensional model is capable of capturing the most salient features of our experimental measurements and observations on the shocks. In this model the steady-state equation of momentum balance contains four terms: the inertial force, the elastic force, the gravitational force, and the drag force of the ambient air. The elastic force consists of the gradient of the surface tension, and it can be computed under the assumption (which is satisfied in our experiments) that the film is in the Marangoni regime, i.e., that as the flow moves through the shock there is no time for diffusional exchange of soap molecules between the bulk and the faces of the film, so that the concentration of soap molecules in the bulk of the film remains invariant.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Theoretical & Applied Mechans

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Tran, Tuan A.

Contributors dc:contributor

• Gioia, Gustavo
• Christensen, Kenneth T.
• Freund, Jonathan B.
• Goldenfeld, Nigel D.

• Turbulence
• Two-dimensional turbulence
• Friction factor
• Frictional drag
• Energy spectrum
• Enstrophy cascade
• Inverse energy cascade
• Marangoni
• Shocks
• Soap-film flows
• Soap-film channel

Statement dc:rights

• Copyright 2010 Tuan Anh Tran.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/26017

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/26017