The turbulent energy cascade built by a vortex burst

Yannis Cuypers, Philippe Petitjeans

Laboratoire de Physique et Mecanique des Milieux Heterogenes, UMR CNRS 7636, EcoleSuperieure de Physique et de Chimie Industrielles, Paris, France

Agnes Maurel

Laboratoire Onde et Acoustique, UMR CNRS 7587,Ecole Superieure de Physique et de Chimie Industrielles, Paris, France

A stretched vortex is generated in a low velocity hydrodynamic channel. A small stepadded to a laminar boundary layer profile in the bottom wall produces the initial vorticitythat is strongly enhanced by the stretching produced by sucking the flow through slots oneach lateral wall. Varying the experimental parameters make two regimes occur: in the firstone, the vortex is stable, in the second one the vortex is experiencing periodical bursts. Theflow is characterized by the following experimental values: Re = 4000, R = 3 cm (lateralextension of the burst), a coarse estimation of the stretching is given by a=[1-10 s−1]. Wefocused on the second regime, where we characterize the vortex evolution via synchronizedhot film and PIV measurements. The main quantitative results have been obtained fromhot film measurements, preliminary results obtained with PIV measurements are also pre-sented. The construction of the experimental turbulent spectrum with time is investigatedand compared with the Lundgren mechanism for the energy transfer towards small scales.

Hot film data processing and results

The hot film probe is set parallel to the z axis and measure U =√

u2r + u2

θ. A typical hotfilm velocity signal shows the quasi-periodical character of the bursts. Each cycle is com-posed of two parts: a laminar part, for which the vortex is still coherent, showing a smoothincrease of the velocity, and a turbulent part associated to the vortex burst, showing rapidvelocity fluctuations. Using an appropriate rescaling of the hot film data and a local Taylorhypothesis to obtain spatial scales, we compute the energy spectrum E(k) of the turbulentparts and averaged over the N bursts, (N=200). A neat Kolmogorov k−5/3 fall off is obtainedbetween km = 0.2 cm−1 and kM = 2 cm −1. In the Lundgren model, the inertial range can

be estimated as [km; kM ], with km = 1/R and kM =√

a/ν. With our experimental values

one gets km ' 0.3 cm−1 and 10 cm−1 ≤ kM ≤ 34 cm−1 that reasonably compare with exper-imental inertial range.

One remarkable property of the Lundgren model is that the time averaging over the life-time of the vortex always results in the k−5/3 spectrum, independently of the spiral structureconsidered. Indeed instantaneously a spiral vortex give a k−p energy spectrum, where p is

1

dependant on the flow field characteristics. To better understand the energy cascade buildup and to go further in the comparison with the Lundgren model, we focus on the tem-poral evolution of the energy spectrum during the vortex burst. We define the cumulativespectrum Ec(k, t) as the spectrum averaged over the N bursts and computed between t = 0and t, where t = 0 denotes the onset of the turbulent parts for each burst. The cumulativespectrum slope pc as a function of t is found to decrease from a value close to -1 and reaches-5/3 for t = Tv = 1.5s. The cumulative spectrum is expected to give the averaged spectralcontribution of the vortex burst between t=0 and t. Therefore, we can conclude from thisrepresentation that there is a temporal evolution in the spectrum and that averaging over[0, Tv] leads to the k−5/3 behavior. Tv can be compared with the lifetime tc of a Lundgrenvortex which is equal to the diffusion time of the spiral arms. The estimation of this timewith our experimental values gives 0.5s < tc < 2.5s in good agreement with Tv. In orderto to characterize the evolution in the spectra slopes, we compute the quasi-instantaneousspectra e∆t(k, t) over small windows (t, t + ∆t), (∆t << Tv) within the turbulent parts.This spectrum is expected to give a quasi-instantaneous picture of the vortex bust energydistribution among scales at different times. We find that the slope of the spectra is timedependant and varies between a value close to -1 at the beginning of the burst to a value closeto -2 at the end of the burst. We are actually computing a Lundgren vortex with a vorticitypatch as the initial condition. We expect that this condition closer to the experiment willallow a transition from -1 to -2.

PIV data processing, preliminary resultsTo characterize the experimental vortex structure, we have performed PIV measurements in across section of the vortex at the middle of the channel. The measurement area, 5 cm × 4 cm,and the resolution obtained (80 × 64 vectors) is sufficient to capture the scales in the inertialrange. We have computed the energy spectrum E(k) from 400 PIV velocity fields acquiredduring the time interval [0, Tv]. A good agreement is found between the PIV spectrumand the hot-film spectrum over the scales of the inertial range. For wavenumbers k of thedissipative range, the PIV spectrum departs from the hot film spectrum. This divergence isdue to the noise present in PIV measurements for small scales. This measurement confirmsboth the k−5/3 behavior and the ability of PIV measurements to characterize the burststructure.

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References

[1] Y. Cuypers, A. Maurel A and P. Petitjeans 2003 Vortex burst as a source of turbulence,Phys. Rev. Lett. 91, 194502.

[2] T. S. Lundgren 1982. Strained spiral vortex model for tubulent fine structures, Phys.Fluids 25, 2193.

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