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

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

Tran, Tuan A.

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

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/26017/tran_tuan.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

We carry out unprecedented experimental measurements of the frictional drag inturbulent 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 spectrumthat are theoretically possible in two dimensions: the ``enstrophy cascade,'' for which the spectral exponent $\alpha=3$, and the ``inverse energy cascade,'' for which the spectral exponent $\alpha=5/3$. We find that the functional relation between the frictional drag $f$ and the Reynolds number Re depends on the spectral exponent:where $\alpha=3$, $f \propto {\Re^{-1/2}}$;where $\alpha=5/3$, $f \propto {\Re^{-1/4}}$. These findings cannot bereconciled with the classic theory of thefrictional 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 frictionaldrag.In this theory the frictional drag of turbulent flows on smooth walls is predicted to be $f\propto {\rm Re}^{(1-\alpha)/(1+\alpha)}$,where $\alpha$ is the spectral exponent.This prediction is in exactaccord 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 $\alpha=5/3$ (the same as for the inverse energy cascade).In fact, for $\alpha=5/3$ theprediction of the spectral theorycoincides with the emprirical law of Blasius ($f \propto {\Re^{-1/4}}$), which gives the best representation of the available experimental results for three-dimensionalpipe flows of moderate turbulent strength(starting from ${\rmRe} \approx 2,500$ andup to ${\rm Re}\approx 100,000$). In carrying out our experiments on the frictional drag, we discover the spontaneous occurrencein 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 fromthe boundaries, witha strong but circumscribed effecton the spatial distribution of turbulent intensity; and that theyalter the structureof the turbulent spectrum downstream from the shock. We show that a simple one--dimensional model is capable of capturing the most salient featuresof our experimental measurements and observations on the shocks. In this model thesteady-state equation of momentum balancecontains 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 becomputed under the assumption (whichis satisfied in our experiments) thatthe film is in the Marangoni regime, i.e.,that as the flow moves through the shockthere is no time for diffusional exchange of soap molecules between the bulk and thefaces of the film, so thatthe concentration of soap molecules in the bulk of the film remains invariant.