{
  "id": "2208.09561",
  "version": "v1",
  "published": "2022-08-19T22:35:16.000Z",
  "updated": "2022-08-19T22:35:16.000Z",
  "title": "The saturation of exponents and the asymptotic fourth state of turbulence",
  "authors": [
    "Katepalli R. Sreenivasan",
    "Victor Yakhot"
  ],
  "categories": [
    "physics.flu-dyn",
    "cond-mat.stat-mech",
    "physics.app-ph"
  ],
  "abstract": "A recent discovery about the inertial range of homogeneous and isotropic turbulence is the saturation of the scaling exponents $\\zeta_n$ for large $n$, defined via structure functions of order $n$ as $S_{n}(r)=\\overline{(\\delta_r u)^{n}}=A(n)r^{\\zeta_{n}}$. We focus on longitudinal structure functions for $\\delta_r u$ between two positions that are $r$ apart in the same direction. In a previous paper (Phys.\\ Rev.\\ Fluids 6, 104604, 2021), we developed a theory for $\\zeta_n$, which agrees with measurements for all $n$ for which reliable data are available, and shows saturation for large $n$. Here, we derive expressions for the probability density functions of $\\delta_r u$ for four different states of turbulence, including the asymptotic fourth state corresponding to the saturation of exponents for large $n$. This saturation means that the scale separation is violated in favor of a strongly-coupled quasi-ordered flow structures, which take the form of long and thin (worm-like) structures of length $L$ and thickness $l=O(L/Re)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-19T22:35:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probability density functions",
      "longitudinal structure functions",
      "asymptotic fourth state corresponding",
      "inertial range",
      "isotropic turbulence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}