External Dissipation in Driven Two-Dimensional Turbulence

Abstract

Turbulence in a freely suspended soap film is created by electromagnetic forcing and measured by particle tracking. The velocity fluctuations are shown to be adequately described by the forced Navier-Stokes equation for an incompressible two-dimensional fluid with a linear drag term to model the frictional coupling to the surrounding air. Using this equation, the energy dissipation rates due to air friction and the film's internal viscosity are measured, as is the rate of energy injection from the electromagnetic forcing. Comparison of these rates demonstrates that the air friction is a significant energy dissipation mechanism in the system.

Statistical properties of turbulence: An overview

“…[123]; such DNS studies have also investigated the scaling properties of structure functions and have provided some evidence for conformal invariance in the inverse cascade inertial range [124]. We end with an illustrative example of a recent DNS study [89] that sheds light on the effect of the Ekman friction on the statistics of the forward cascade in wall-bounded flows that are directly relevant to laboratory soap-film experiments [125][126][127][128]. The details of this DNS are given in Ref.…”

“…In the range of parameters used in typical experimental studies [1,35,36,87] both these systems can be described quite well [88,89] by the 2D Navier Stokes equation (see Sec. 3) with an additional Ekman-friction term, induced typically by air drag; however, in some cases we must also account for corrections arising from fluctuations of the film thickness, compressibility effects, and the Marangoni effect.…”

“…Velocity and vorticity structure functions can be measured as in 3D turbulence; however, inertial ranges associated with inverse and forward cascades must be distinguished; the former shows simple scaling with an energy spectrum E(k) ∼ k −5/3 whereas the latter has an energy spectrum E(k) ∼ k −(3+δ) , with δ = 0 if there is no Ekman friction and δ > 0 otherwise. In the forward cascade velocity structure functions show simple scaling [87]; we are not aware of experimental measurements of vorticity structure functions (we will discuss these in the context of numerical simulations in Sec. 6.3).…”

“…Two-dimensional analogues of the PDFs described above for 3D turbulence have been measured [see, e.g., Refs. [87]]; we will touch on these briefly when we discuss numerical simulations of 2D turbulence in Sec. 6.3.…”

Inverse velocity statistics in two-dimensional turbulence

“…Recently, new results have clarified the problem in the special case of the large scale energy sink given by a linear (Eckman) friction [8,10]. We recall that the presence of an energy sink at large scales is conceptually justified by the necessity of avoiding the pile up of energy on the gravest mode as a result of the inverse energy cascade [4] and it is physically motivated in terms of the friction to which a fluid is subjected in the Eckman layer [18,19]. The strong influence of large-scale phenomena in the whole enstrophy cascade range is believed to be a consequence of non local interactions (in Fourier space).…”