Energy spectrum of grid-generated He II turbulence

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 64, Issue 6, 2001, Pages 4-

  Skrbek, L ^a,b   Niemela, J J ^a   Sreenivasan, K R ^c  

^a UNIVERSITY OF OREGON   (United States)

^b INSTITUTE OF PHYSICS   (Czech Republic)

^c YALE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85035283256     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.64.067301     Document Type: Article

References (16)

1. K.W. Schwarz, Phys. Rev. B 38, 2398 (1988).
2. J. Maurer and P. Tabeling, Europhys. Lett. 43, 29 (1998).
3. S.R. Stalp, L. Skrbek, and R.J. Donnelly, Phys. Rev. Lett. 82, 4831 (1999).
4. L. Skrbek, J.J. Niemela, and R.J. Donnelly, Phys. Rev. Lett. 85, 2973 (2000).
5. W.F. Vinen, Phys. Rev. B 61, 1410 (2000).
6. L. Skrbek and S.R. Stalp, Phys. Fluids 12, 1997 (2000).
7. S.R. Stalp, J.J. Niemela, and R.J. Donnelly, Physica B 284, 75 (2000).
8. S.R. Stalp, J.J. Niemela, W.F. Vinen, and R.J. Donnelly (unpublished).
9. The equation for (Formula presented) is of the form (Formula presented), where (Formula presented) and the virtual origin time (Formula presented), where (Formula presented) denotes the vortex line density at (Formula presented). We are interested in the real solution for L that decays with time: (Formula presented), where (Formula presented) and (Formula presented). For given (Formula presented), the time dependent term (Formula presented) is a measure of the up or down deviation from the (Formula presented), which is the result for classical scaling.
10. D. Kivotides, J.C. Vassilicos, D.C. Samuels, and C.F. Barenghi, Phys. Rev. Lett. 86, 3080 (2001).
11. W.F. Vinen, Proc. R. Soc. London, Ser. A 242, 498 (1957);
12. Proc. R. Soc. London, Ser. AW.F. Vinen243, 400 (1957).
13. In the late stages of decay, it may be thought that experiments have reached the limits of sensitivity of resolution, and the remnant vorticity may have affected the observations. D.D. Awschalom and K.W. Schwarz, Phys. Rev. Lett. 52, 49 (1984) quote that the remnant vortex line density (Formula presented), where (Formula presented) is the vortex core parameter. For the geometry of Ref. 4 this formula gives a remnant vorticity of about (Formula presented), which is of order of the lower limit of the experimental sensitivity in Ref. 4. This level of remnant vorticity, included in the background attenuation of the second sound, was approximately stable in the experiment, and so should not affect measured values of L. However, the possibility that the late stages of decay are in some way affected by the remnant vorticity cannot be fully excluded.
14. The analytical solution for decaying vortex line density for (Formula presented) is of the same form as the classical solution for decaying vorticity [see Eq. (4) in Ref. 4], obtained when the exponential tail in the classical energy spectrum is approximated by a sharp cutoff at effectively the Kolmogorov wave number. The resulting solution (Formula presented) 9 describes experimental data on the temporal decay of the vortex line density after saturation of the energy containing length scale of 4 equally well, using classical value for C and the measured values of (Formula presented) 8, with (Formula presented) as the only parameter. However, this form of decay cannot occur beyond (Formula presented), when the growing quantum scale approaches the size of the channel and becomes saturated by it for the rest of the decay. It is easy to show that this happens when (Formula presented), which for the geometry of Ref. 4 corresponds roughly to (Formula presented) about (Formula presented), or to the average intervortex distance of about 1 mm.
15. K.R. Sreenivasan, Phys. Fluids 7, 2778 (1995).
16. The agreement is even better if one accounts for the existence of the boundary layer that ought to build up even in the case of zero mean flow grid turbulence, due to shear between the channel wall and the energy containing eddies. Full agreement with the experiment, assuming (Formula presented) would require introducing the boundary layer of thickness about 0.08 cm, or about (Formula presented) of the channel width. Boundary layers of similar fractional thickness have been observed by us in classical towed grid experiment in water.