Environmental Fluid Mechanics

, Volume 16, Issue 2, pp 373–399 | Cite as

Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: a direct numerical simulation approach

  • Ping HeEmail author
  • Sukanta Basu
Original Article


In this study, a newly developed direct numerical simulation (DNS) solver is utilized for the simulations of numerous stably stratified open-channel flows with bulk Reynolds number (Re b ) spanning 3400–16,900. Overall, the simulated bulk Richardson number (\(Ri_b\)) ranges from 0.08 (weakly stable) to 0.49 (very stable). Thus, both continuously turbulent and (globally) intermittently turbulent cases are represented in the DNS database. Using this comprehensive database, various flux-based and gradient-based similarity relationships for energy dissipation rate (ε) and temperature structure parameter (\(C_T^2\)) are developed. Interestingly, these relationships exhibit only minor dependency on Re b . In order to further probe into this Re b -effect, similarity relationships are also estimated from a large-eddy simulation (LES) run of an idealized atmospheric boundary layer (very high Re b ) case study. Despite the fundamental differences in the estimation of ε and \(C_T^2\) from the DNS- and the LES-generated data, the resulting similarity relationships, especially the gradient-based ones, from these numerical approaches are found to be remarkably similar. More importantly, these simulated relationships are also comparable, at least qualitatively, to the traditional observational data-based ones. Since these simulated similarity relationships do not require Taylor’s hypothesis and do not suffer from mesoscale disturbances and/or measurement noise, they have the potential to complement the existing similarity relationships.


Direct numerical simulation Energy dissipation rate Large-eddy simulation Similarity theory Stable boundary layer Temperature structure parameter 



The authors acknowledge financial support received from the Department of Defense (AFOSR grant under award number FA9550-12-1-0449) and the National Science Foundation (grant AGS-1122315). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Department of Defense or the National Science Foundation. The authors also acknowledge computational resources obtained from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation (grant number ACI-1053575). Finally, the authors thank Adam DeMarco and Patrick Hawbecker for their valuable comments and suggestions to improve the quality of the paper.


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Marine, Earth, and Atmospheric SciencesNorth Carolina State UniversityRaleighUSA

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