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Physical meaning of k in the logarithmic layer for Reynolds-averaged Navier-Stokes models

Xiang I. A. Yang1,*, Ruifeng Hu2, Rahul Deshpande3, Robert Kunz1, and George Huang4

  • 1Mechanical Engineering, Pennsylvania State University, State College, Pennsylvania 16802, USA
  • 2Center for Particle-Laden Turbulence, Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, Gansu, People's Republic of China
  • 3Department of Mechanical Engineering, The University of Melbourne, Parkville Victoria 3010, Australia
  • 4Wright State University, Dayton, Ohio 45435, USA

  • *Contact author: xzy48@psu.edu
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Phys. Rev. Fluids 11, 064606 – Published 8 June, 2026

DOI: https://doi.org/10.1103/wk3c-xrdn

Abstract

Two-equation models introduce auxiliary transport equations to estimate a velocity scale and a length scale, whose combination yields the eddy viscosity. The velocity scale is typically associated with the variable k, often interpreted as the turbulent kinetic energy (TKE). Using high-Reynolds-number data that were not available at the time of model development, we affirm that the modeled k does not, in fact, represent the total TKE. Rather, it exhibits scaling behavior similar to that of the kinetic energy contained in the active motions—those responsible for producing Reynolds shear stress. To separate active and inactive motions, we propose and validate decomposition methods that segregate turbulent motions into active and inactive components across different levels of data availability. The decompositions are shown to obey the expected scalings of active motions. We further derive the energy budgets of the active and inactive components under the attached eddy hypothesis, revealing that the former are governed by the usual balance between production and dissipation, while the latter contains mean convection, viscous diffusion, dissipation, and the energy transferred from active motions. Analysis of the k equation in two-equation models reveals that it admits both the total TKE and the energy in active motions as asymptotic solutions; the fact that two-equation models select the latter arises from the structure of the dissipation equation. Finally, we introduce a transport equation that governs the kinetic energy in the inactive motions in the logarithmic region, providing a path toward reconciling model outputs with physical turbulence quantities.

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Outline

Information

Phys. Rev. Fluids 11, 064606– Published 8 June, 2026
  • Received 6 January 2026
  • Accepted 11 May 2026
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