Spectra of Decaying Turbulence in a Soap Film

Abstract

Grid turbulence in a flowing soap film is investigated with ambient pressure P as one of the control parameters. Reducing P significantly decreases air drag thus minimizing three-dimensional effects on the flowing film. It was found that while the total kinetic energy 1 2 ͗y 2 ͘ R E͑k͒ dk of velocity fluctuations increased noticeably as P was reduced, the energy spectrum E͑k͒ for wave numbers greater than the injection scale was remarkably insensitive to the pressure change, and obeyed a power law E͑k͒ ϳ k 23.360.3 . [S0031-9007(98)05979-1] PACS numbers: 47.27.Gs, 67.40.Vs, 68.15. + e, 92.60.EkWhen a laminarly flowing fluid rushes through a grid, a certain amount of kinetic energy in the mean flow is transferred to rotational motion. If the mean flow speedV is sufficiently large, the rotational motion loses its spatiotemporal coherence, and the flow can then be characterized as turbulent. In three-dimensional grid turbulence nonlinear interactions and viscous damping cause the rms velocity fluctuations to decay downstream [1]. In spite of this decay, the structure of turbulence remains self-similar and is in quantitative agreement with Kolmogorov scaling predictions if there exists an initial subrange [2]. In two dimensions, on the other hand, one expects the self-similar form of turbulence to be markedly different for the forced and the freely evolving cases.The experiment reported herein explores decaying turbulence behind a grid in a two-dimensional (2D) flow channel [3,4]. The system used is a freely suspended flowing soap film driven by gravity and rendered turbulent by insertion of a comb perpendicular to the mean flow [3]. Previous experiments on flowing soap films have revealed certain features that can be associated with 2D hydrodynamic behavior [3,5,6]. However, since all these measurements were performed in air, its effect on a turbulently flowing film was a concern. Air drag has been shown to be a significant dissipative force in laminarly flowing films [4], making it clear that in order to fully understand turbulence in these films, experiments in the presence and "absence" of air are needed. To accomplish this we reduced the air pressure surrounding the film.It is helpful to recall some of the main predictions of 2D turbulence. A distinct feature of 2D flow is that in the limit of small viscosity n, local vorticity v ϵ = 3 y is a conserved quantity. Therefore at a large Reynolds number Re~n 21 one has two quadratic constants of motion: the energy 1 2 ͗y 2 ͘ and the enstrophy 1 2 ͗v 2 ͘. With these invariants of motion, one expects that most kinetic energy injected into the system at a scale ᐉ i is transferred to large scales ᐉ . ᐉ i , while a small fraction of it is transferred to small scales ᐉ , ᐉ i . The former process is known as the inverse energy cascade, whereas the latter is referred to as the direct enstrophy cascade [7]. Using a Kolmogorovlike argument [8], it follows that each cascade regime is self-similar in that the energy spectrum E͑k͒ (or energy per wave number k ϵ 2p͞ᐉ) obeys a powe...

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