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Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels

Moarref, Rashad and Sharma, Ati S. and Tropp, Joel A. and McKeon, Beverley J. (2013) Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. Journal of Fluid Mechanics, 734 . pp. 275-316. ISSN 0022-1120.

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We study the Reynolds-number scaling and the geometric self-similarity of a gainbased, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier–Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (Re_τ ≈ 10^3–10^(10)). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper DOIArticle
Sharma, Ati S.0000-0002-7170-1627
Tropp, Joel A.0000-0003-1024-1791
McKeon, Beverley J.0000-0003-4220-1583
Additional Information:© 2013 Cambridge University Press. Received 8 February 2013; revised 20 August 2013; accepted 24 August 2013; first published online 9 October 2013. The support of Air Force Office of Scientific Research under grants FA 9550-09-1-0701 (P. M. J. Schmisseur) and FA 9550-12-1-0469 (P. M. D. Smith) is gratefully acknowledged.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA 9550-09-1-0701
Air Force Office of Scientific Research (AFOSR)FA 9550-12-1-0469
Subject Keywords:mathematical foundations; Navier-Stokes equations; turbulent boundary layers
Record Number:CaltechAUTHORS:20131121-132351952
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Official Citation:Rashad Moarref, Ati S. Sharma, Joel A. Tropp and Beverley J. McKeon (2013). Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. Journal of Fluid Mechanics, 734, pp 275-316 doi:10.1017/jfm.2013.457
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42622
Deposited By: Tony Diaz
Deposited On:21 Nov 2013 22:45
Last Modified:03 Oct 2019 06:00

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