1
|
Gioia G, Chakraborty P. Spectral derivation of the classic laws of wall-bounded turbulent flows. Proc Math Phys Eng Sci 2017; 473:20170354. [PMID: 28878568 PMCID: PMC5582189 DOI: 10.1098/rspa.2017.0354] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2017] [Accepted: 07/05/2017] [Indexed: 11/12/2022] Open
Abstract
We show that the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law’—can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also show that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.
Collapse
Affiliation(s)
- Gustavo Gioia
- Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0495, Japan
| | - Pinaki Chakraborty
- Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0495, Japan
| |
Collapse
|
2
|
Katul GG, Manes C. Cospectral budget of turbulence explains the bulk properties of smooth pipe flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:063008. [PMID: 25615188 DOI: 10.1103/physreve.90.063008] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2014] [Indexed: 06/04/2023]
Abstract
Connections between the wall-normal turbulent velocity spectrum E(ww)(k) at wave number k and the mean velocity profile (MVP) are explored in pressure-driven flows confined within smooth walls at moderate to high bulk Reynolds numbers (Re). These connections are derived via a cospectral budget for the longitudinal (u') and wall-normal (w') velocity fluctuations, which include a production term due to mean shear interacting with E(ww)(k), viscous effects, and a decorrelation between u' and w' by pressure-strain effects [=π(k)]. The π(k) is modeled using a conventional Rotta-like return-to-isotropy closure but adjusted to include the effects of isotropization of the production term. The resulting cospectral budget yields a generalization of a previously proposed "spectral link" between the MVP and the spectrum of turbulence. The proposed cospectral budget is also shown to reproduce the measured MVP across the pipe with changing Re including the MVP shapes in the buffer and wake regions. Because of the links between E(ww)(k) and the MVP, the effects of intermittency corrections to inertial subrange scales and the so-called spectral bottleneck reported as k approaches viscous dissipation eddy sizes (η) on the MVP shapes are investigated and shown to be of minor importance. Inclusion of a local Reynolds number correction to a parameter associated with the spectral exponential cutoff as kη→1 appears to be more significant to the MVP shape in the buffer region. While the bulk shape of the MVP is reasonably reproduced in all regions of the pipe, the solution to the cospectral budget systematically underestimates the negative curvature of the MVP within the buffer layer.
Collapse
Affiliation(s)
- Gabriel G Katul
- Nicholas School of the Environment, Duke University, Durham, North Carolina 27708-0328, USA and Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina 27708, USA
| | - Costantino Manes
- Faculty of Engineering and the Environment, University of Southampton, Southampton SO171BJ, United Kingdom
| |
Collapse
|
3
|
Samanta D, Ingremeau F, Cerbus R, Tran T, Goldburg WI, Chakraborty P, Kellay H. Scaling of near-wall flows in quasi-two-dimensional turbulent channels. PHYSICAL REVIEW LETTERS 2014; 113:024504. [PMID: 25062190 DOI: 10.1103/physrevlett.113.024504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2014] [Indexed: 06/03/2023]
Abstract
The law of the wall and the log law rule the near-wall mean velocity profile of three-dimensional turbulent flows. These well-known laws, which are validated by legions of experiments and simulations, may be universal. Here, using a soap-film channel, we report the first experimental test of these laws in quasi-two-dimensional turbulent channel flows under two disparate turbulent spectra. We find that despite the differences with three-dimensional flows, the laws prevail, albeit with notable distinctions: the two parameters of the log law are markedly distinct from their three-dimensional counterpart; further, one parameter (the von Kármán constant) is independent of the spectrum whereas the other (the offset of the log law) depends on the spectrum. Our results suggest that the classical theory of scaling in wall-bounded turbulence is incomplete wherein a key missing element is the link with the turbulent spectrum.
Collapse
Affiliation(s)
- D Samanta
- Okinawa Institute of Science and Technology, Okinawa, Japan 904-0495
| | - F Ingremeau
- University of Bordeaux, LOMA (UMR 5798 CNRS), 351 cours de la Libération, 33405 Talence, France
| | - R Cerbus
- Okinawa Institute of Science and Technology, Okinawa, Japan 904-0495 and Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - T Tran
- Division of Thermal and Fluids Engineering, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
| | - W I Goldburg
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - P Chakraborty
- Okinawa Institute of Science and Technology, Okinawa, Japan 904-0495
| | - H Kellay
- University of Bordeaux, LOMA (UMR 5798 CNRS), 351 cours de la Libération, 33405 Talence, France
| |
Collapse
|
4
|
Katul GG, Porporato A, Shah S, Bou-Zeid E. Two phenomenological constants explain similarity laws in stably stratified turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:023007. [PMID: 25353571 DOI: 10.1103/physreve.89.023007] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2013] [Indexed: 06/04/2023]
Abstract
In stably stratified turbulent flows, the mixing efficiency associated with eddy diffusivity for heat, or equivalently the turbulent Prandtl number (Pr(t)), is fraught with complex dynamics originating from the scalewise interplay between shear generation of turbulence and its dissipation by density gradients. A large corpus of data and numerical simulations agree on a near-universal relation between Pr(t) and the Richardson number (R(i)), which encodes the relative importance of buoyancy dissipation to mechanical production of turbulent kinetic energy. The Pr(t)-R(i) relation is shown to be derivable solely from the cospectral budgets for momentum and heat fluxes if a Rotta-like return to isotropy closure for the pressure-strain effects and Kolmogorov's theory for turbulent cascade are invoked. The ratio of the Kolmogorov to the Kolmogorov-Obukhov-Corrsin phenomenological constants, and a constant associated with isotropization of the production whose value (= 3/5) has been predicted from Rapid Distortion Theory, explain all the macroscopic nonlinearities.
Collapse
Affiliation(s)
- Gabriel G Katul
- Nicholas School of the Environment, Duke University, Durham, North Carolina 27708-0328, USA and Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina 27708, USA
| | - Amilcare Porporato
- Nicholas School of the Environment, Duke University, Durham, North Carolina 27708-0328, USA and Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina 27708, USA
| | - Stimit Shah
- Department of Civil and Environmental Engineering, Princeton, New Jersey 08544, USA
| | - Elie Bou-Zeid
- Department of Civil and Environmental Engineering, Princeton, New Jersey 08544, USA
| |
Collapse
|
5
|
Katul GG, Li D, Chameki M, Bou-Zeid E. Mean scalar concentration profile in a sheared and thermally stratified atmospheric surface layer. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:023004. [PMID: 23496607 DOI: 10.1103/physreve.87.023004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2012] [Revised: 10/28/2012] [Indexed: 06/01/2023]
Abstract
Using only dimensional considerations, Monin and Obukhov proposed a "universal" stability correction function φ(c)(ζ) that accounts for distortions caused by thermal stratification to the mean scalar concentration profile in the atmospheric surface layer when the flow is stationary, planar homogeneous, fully turbulent, and lacking any subsidence. For nearly 6 decades, their analysis provided the basic framework for almost all operational models and data interpretation in the lower atmosphere. However, the canonical shape of φ(c)(ζ) and the departure from the Reynold's analogy continue to defy theoretical explanation. Here, the basic processes governing the scalar-velocity cospectrum, including buoyancy and the scaling laws describing the velocity and temperature spectra, are considered via a simplified cospectral budget. The solution to this cospectral budget is then used to derive φ(c)(ζ), thereby establishing a link between the energetics of turbulent velocity and scalar concentration fluctuations and the bulk flow describing the mean scalar concentration profile. The resulting theory explains all the canonical features of φ(c)(ζ), including the onset of power laws for various stability regimes and their concomitant exponents, as well as the causes of departure from Reynold's analogy.
Collapse
Affiliation(s)
- Gabriel G Katul
- Nicholas School of the Environment, Duke University, Durham, North Carolina 27708-0328, USA
| | | | | | | |
Collapse
|
6
|
Katul GG, Konings AG, Porporato A. Mean velocity profile in a sheared and thermally stratified atmospheric boundary layer. PHYSICAL REVIEW LETTERS 2011; 107:268502. [PMID: 22243189 DOI: 10.1103/physrevlett.107.268502] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2011] [Indexed: 05/31/2023]
Abstract
A stability correction function φ(m)(ζ) that accounts for distortions to the logarithmic mean velocity profile (MVP) in the lower atmosphere caused by thermal stratification was proposed by Monin and Obukhov in the 1950s using dimensional analysis. Its universal character was established from many field experiments. However, theories that describe the canonical shape of φ(m)(ζ) are still lacking. A previous link between the spectrum of turbulence and the MVP is expanded here to include the effects of thermal stratification on the turbulent kinetic energy dissipation rate and eddy-size anisotropy. The resulting theory provides a novel explanation for the power-law exponents and coefficients already reported for φ(m)(ζ) from numerous field experiments.
Collapse
Affiliation(s)
- Gabriel G Katul
- Nicholas School of the Environment, Box 80328, Duke University, Durham, North Carolina 27708, USA
| | | | | |
Collapse
|
7
|
Calzetta E. Drag reduction by polymer additives from turbulent spectra. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:066310. [PMID: 21230737 DOI: 10.1103/physreve.82.066310] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2010] [Revised: 08/26/2010] [Indexed: 05/30/2023]
Abstract
We extend the analysis of the friction factor for turbulent pipe flow reported by G. Gioia and P. Chakraborty [Phys. Rev. Lett. 96, 044502 (2006)] to the case where drag is reduced by polymer additives.
Collapse
Affiliation(s)
- Esteban Calzetta
- Departamento de Física, FCEN, Universidad de Buenos Aires, and IFIBA-CONICET, Buenos Aires, Argentina.
| |
Collapse
|