dorsal/arxiv
View SchemaAnalysis of a Symmetry leading to an Inertial Range Similarity Theory for Isotropic Turbulence
| Authors | Mogens V Melander |
|---|---|
| Categories | |
| ArXiv ID | physics/0702073 |
| URL | https://arxiv.org/abs/physics/0702073 |
Abstract
We present a theoretical attack on the classical problem of intermittency and anomalous scaling in turbulence. Our focus is on an ideal situation: high Reynolds number isotropic turbulence driven by steady large scale forcing. Moreover, the fluid is incompressible and no confining boundaries are present. We start from a good set of basis functions for the velocity field. These are real and divergence-free. To each wave-vector k in Fourier space there is one pair of basis functions with respectively left and right-handed polarity. Isotropy makes all k on the shell of constant |k| statistically equivalent. Consequently, the coefficients, X+ and X-, to the basis functions in that shell become two random variables whose joint pdf describes the statistics at scale L =2*pi/|k|. Moreover, (X+)**2+(X-)**2 becomes a random variable for the energy. Switching to polar coordinates, the joint pdf expands in azimuthal modes. We focus on the axisymmetric mode which is itself a pdf and characterized by it radial profile P(r;L). Observations from both shell model and DNS data indicate that (1) the moments of P(r;L) scale as power laws in L, and (2) the profile obeys an affine symmetry P(r;L)=C(L)*f((lnr-mu(L))/sigma(L)). We raise the question: What statistics agree with both observation? The answer is pleasing. We find the functions f, mu, sigma C analytically in terms of a few constants. Moreover, we obtain closed form expressions for both scaling exponents and coefficients in the power laws. A virtual origin also emerges as an intrinsic length scale L0 for the inertial range.
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"abstract": "We present a theoretical attack on the classical problem of intermittency and\nanomalous scaling in turbulence. Our focus is on an ideal situation: high\nReynolds number isotropic turbulence driven by steady large scale forcing.\nMoreover, the fluid is incompressible and no confining boundaries are present.\nWe start from a good set of basis functions for the velocity field. These are\nreal and divergence-free. To each wave-vector k in Fourier space there is one\npair of basis functions with respectively left and right-handed polarity.\nIsotropy makes all k on the shell of constant |k| statistically equivalent.\nConsequently, the coefficients, X+ and X-, to the basis functions in that shell\nbecome two random variables whose joint pdf describes the statistics at scale L\n=2*pi/|k|. Moreover, (X+)**2+(X-)**2 becomes a random variable for the energy.\nSwitching to polar coordinates, the joint pdf expands in azimuthal modes. We\nfocus on the axisymmetric mode which is itself a pdf and characterized by it\nradial profile P(r;L). Observations from both shell model and DNS data indicate\nthat (1) the moments of P(r;L) scale as power laws in L, and (2) the profile\nobeys an affine symmetry P(r;L)=C(L)*f((lnr-mu(L))/sigma(L)). We raise the\nquestion: What statistics agree with both observation? The answer is pleasing.\nWe find the functions f, mu, sigma C analytically in terms of a few constants.\nMoreover, we obtain closed form expressions for both scaling exponents and\ncoefficients in the power laws. A virtual origin also emerges as an intrinsic\nlength scale L0 for the inertial range.",
"arxiv_id": "physics/0702073",
"authors": [
"Mogens V Melander"
],
"categories": [
"physics.flu-dyn"
],
"title": "Analysis of a Symmetry leading to an Inertial Range Similarity Theory for Isotropic Turbulence",
"url": "https://arxiv.org/abs/physics/0702073"
},
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