### Abstract

Probabilistic methods and computer simulation are used to analyze streamline properties of flows defined by a general class of random, incompressible velocity fields. Such fields are stochastically modeled by a superposition of simple shear-flow layers. The resulting flow is governed by a nonstationary, random Hamiltonian with Hurst exponent H = 0.5 and having the form H_{0} = Σ^{N}_{k=1} b_{k}-W_{k}(n_{k} · r) for constants b_{k}, where the W_{k} are two-sided continuous-time random walks. The statistical topography of the flow, as characterized by the level sets or streamlines of H_{0}, is analyzed via an associated Brownian walk space from which asymptotic results are determined. The flow consists of a hierarchy of nested, closed streamlines. For a given box of side length 2L centered at the particle's starting position, an upper bound on the particle's probability of exit, or "noncycling" probability, p_{nc}, is shown to have a power law dependence on the box size, p_{nc}(L) ∼ L^{-α}, for L ≫ 1 and positive constant α. We also introduce a constant, nonzero mean flow and denote its relative strength with respect to the r.m.s. fluctuations of the random field by ρ. In the case 0 < ρ < 1, the fraction p_{nc}(ρ) of "percolating" or noncycling particles (in an infinitely large box) satisfies the relation ρ/ρ + 1 ≤ p_{nc} (ρ) < 1. All particles percolate in the case ρ ≥ 1. Computer simulations for various values of N agree well with earlier work on the N = 2 case by Avellaneda, Elliott, and Apelian, thereby confirming and validating both studies. Numerical results also show the power law exponent α to be remarkably robust with respect to changes in topology, including the existence of traps, irregularly spaced modes, and the value of N. All runs yield a common value of α ≈ 0.22. Likewise, the mean length of streamlines exiting boxes of size 2L, 〈λ(L)〉, scales like L^{γ} with γ ≈ 1.28 for all N. These exponent values contrast with those predicted by Isichenko and Kalda yet consistently satisfy a "sum rule," α + γ = 2 - H, relating α, γ, and H, the Hurst exponent of the flow.

Original language | English (US) |
---|---|

Pages (from-to) | 1053-1088 |

Number of pages | 36 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 50 |

Issue number | 11 |

State | Published - Nov 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*50*(11), 1053-1088.

**A turbulent transport model : Streamline results for a class of random velocity fields in the plane.** / Apelian, Christopher; Holmes, Richard L.; Avellaneda, Marco.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 50, no. 11, pp. 1053-1088.

}

TY - JOUR

T1 - A turbulent transport model

T2 - Streamline results for a class of random velocity fields in the plane

AU - Apelian, Christopher

AU - Holmes, Richard L.

AU - Avellaneda, Marco

PY - 1997/11

Y1 - 1997/11

N2 - Probabilistic methods and computer simulation are used to analyze streamline properties of flows defined by a general class of random, incompressible velocity fields. Such fields are stochastically modeled by a superposition of simple shear-flow layers. The resulting flow is governed by a nonstationary, random Hamiltonian with Hurst exponent H = 0.5 and having the form H0 = ΣNk=1 bk-Wk(nk · r) for constants bk, where the Wk are two-sided continuous-time random walks. The statistical topography of the flow, as characterized by the level sets or streamlines of H0, is analyzed via an associated Brownian walk space from which asymptotic results are determined. The flow consists of a hierarchy of nested, closed streamlines. For a given box of side length 2L centered at the particle's starting position, an upper bound on the particle's probability of exit, or "noncycling" probability, pnc, is shown to have a power law dependence on the box size, pnc(L) ∼ L-α, for L ≫ 1 and positive constant α. We also introduce a constant, nonzero mean flow and denote its relative strength with respect to the r.m.s. fluctuations of the random field by ρ. In the case 0 < ρ < 1, the fraction pnc(ρ) of "percolating" or noncycling particles (in an infinitely large box) satisfies the relation ρ/ρ + 1 ≤ pnc (ρ) < 1. All particles percolate in the case ρ ≥ 1. Computer simulations for various values of N agree well with earlier work on the N = 2 case by Avellaneda, Elliott, and Apelian, thereby confirming and validating both studies. Numerical results also show the power law exponent α to be remarkably robust with respect to changes in topology, including the existence of traps, irregularly spaced modes, and the value of N. All runs yield a common value of α ≈ 0.22. Likewise, the mean length of streamlines exiting boxes of size 2L, 〈λ(L)〉, scales like Lγ with γ ≈ 1.28 for all N. These exponent values contrast with those predicted by Isichenko and Kalda yet consistently satisfy a "sum rule," α + γ = 2 - H, relating α, γ, and H, the Hurst exponent of the flow.

AB - Probabilistic methods and computer simulation are used to analyze streamline properties of flows defined by a general class of random, incompressible velocity fields. Such fields are stochastically modeled by a superposition of simple shear-flow layers. The resulting flow is governed by a nonstationary, random Hamiltonian with Hurst exponent H = 0.5 and having the form H0 = ΣNk=1 bk-Wk(nk · r) for constants bk, where the Wk are two-sided continuous-time random walks. The statistical topography of the flow, as characterized by the level sets or streamlines of H0, is analyzed via an associated Brownian walk space from which asymptotic results are determined. The flow consists of a hierarchy of nested, closed streamlines. For a given box of side length 2L centered at the particle's starting position, an upper bound on the particle's probability of exit, or "noncycling" probability, pnc, is shown to have a power law dependence on the box size, pnc(L) ∼ L-α, for L ≫ 1 and positive constant α. We also introduce a constant, nonzero mean flow and denote its relative strength with respect to the r.m.s. fluctuations of the random field by ρ. In the case 0 < ρ < 1, the fraction pnc(ρ) of "percolating" or noncycling particles (in an infinitely large box) satisfies the relation ρ/ρ + 1 ≤ pnc (ρ) < 1. All particles percolate in the case ρ ≥ 1. Computer simulations for various values of N agree well with earlier work on the N = 2 case by Avellaneda, Elliott, and Apelian, thereby confirming and validating both studies. Numerical results also show the power law exponent α to be remarkably robust with respect to changes in topology, including the existence of traps, irregularly spaced modes, and the value of N. All runs yield a common value of α ≈ 0.22. Likewise, the mean length of streamlines exiting boxes of size 2L, 〈λ(L)〉, scales like Lγ with γ ≈ 1.28 for all N. These exponent values contrast with those predicted by Isichenko and Kalda yet consistently satisfy a "sum rule," α + γ = 2 - H, relating α, γ, and H, the Hurst exponent of the flow.

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UR - http://www.scopus.com/inward/citedby.url?scp=0031534279&partnerID=8YFLogxK

M3 - Article

VL - 50

SP - 1053

EP - 1088

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 11

ER -