Using theoretical arguments, we present two novel spectrum models of the streamwise velocity component with robust correlation structures, which account for and decouple the fractal dimension and Hurst effect. The formulations that use isotropic concepts are adapted from the modern probability theory using the so-called generalized Cauchy and Dagum models, which belong to wide-sense-stationary random fields. A complementary inspection of these two models with field data from a met-tower-mounted sonic anemometer located within the atmospheric surface layer reveals good agreement and better performance than other conventionally used isotropic-based models of streamwise velocity spectra. The fractal dimension, D, of both models is consistent with the well-known Kolmogorov −5/3 power law in the inertial sub-range. For completeness, the study includes a derivation of the explicit forms of the energy spectral densities of the Cauchy and Dagum covariances.
References
1.
T.
von Kármán
, “Progress in the statistical theory of turbulence
,” Proc. Natl. Acad. Sci. U. S. A.
34
(11
), 530
–539
(1948
).2.
G. K.
Batchelor
, The Theory of Homogeneous Turbulence
(Cambridge University Press
, 1953
).3.
A. S.
Monin
and A. M.
Yaglom
, Statistical Fluid Mechanics: Mechanics of Turbulence
(MIT Press
, Cambridge, MA
, 1975
).4.
E.
Cheynet
, J. B.
Jakobsen
, and J.
Reuder
, “Velocity spectra and coherence estimates in the marine atmospheric boundary layer
,” Boundary-Layer Meteorol.
169
, 429
–460
(2018
).5.
J. C.
Kaimal
, “Turbulence spectra, length scales and structure parameters in the stable surface layer
,” Boundary-Layer Meteorol.
4
, 289
–309
(1973
).6.
H. R.
Olesen
, S. E.
Larsen
, and J.
Hojstrup
, “Modelling velocity spectra in the lower part of the planetary boundary layer
,” Boundary-Layer Meteorol.
29
, 285
–312
(1984
).7.
S.
Leibovich
and Z.
Warhaft
, “John Leask Lumley: Whither turbulence?
,” Annu. Rev. Fluid. Mech.
50
, 1
–23
(2018
).8.
J.
Mann
, “The spatial structure of neutral atmospheric surface-layer turbulence
,” J. Fluid Mech.
273
, 141
–168
(1994
).9.
A.
Segalini
, J.
Arnqvist
, I.
Carlén
, H.
Bergström
, and P. H.
Alfredsson
, “A spectral model of stably stratified surface-layer turbulence
,” J. Phys.: Conf. Ser.
625
, 012003
(2015
).10.
L.
Mortarini
and D.
Anfossi
, “Proposal of an empirical velocity spectrum formula in low-wind speed conditions
,” Q. J. R. Meteorol. Soc.
141
, 85
–97
(2015
).11.
J. C.
Kaimal
and J. J.
Finnigan
, Atmospheric Boundary Flows
(Oxford University Press
, Oxford, UK
, 1994
).12.
N.
Tobin
, H.
Zhu
, and L. P.
Chamorro
, “Spectral behaviour of the turbulence-driven power fluctuations of wind turbines
,” J. Turbul.
16
(9
), 832
–846
(2016
).13.
T.
Gneiting
and M.
Schlather
, “Stochastic models that separate fractal dimension and the Hurst effect
,” SIAM Rev.
46
, 269
–282
(2004
).14.
J.
Mateu
, E.
Porcu
, and O.
Nicolis
, “A note on decoupling of local and global behaviours for the Dagum random field
,” Probab. Eng. Mech.
22
, 320
–329
(2007
).15.
L.
Shen
, M.
Ostoja-Starzewski
, and E.
Porcu
, “Harmonic oscillator driven by random processes having fractal and Hurst effects
,” Acta Mech.
226
, 3653
–3672
(2015
).16.
V. V.
Nishawala
and M.
Ostoja-Starzewski
, “Acceleration waves on random fields with fractal and Hurst effects
,” Wave Motion
74
, 134
–150
(2017
).17.
I.
Arad
, B.
Dhruva
, S.
Kurien
, V. S.
L'vov
, I.
Procaccia
, and K. R.
Sreenivasan
, “Extraction of anisotropic contributions in turbulent flows
,” Phys. Rev. Lett.
81
, 5330
(1998
).18.
S.
Kurien
, V. S.
L'vov
, I.
Procaccia
, and K. R.
Sreenivasan
, “Scaling structure of the velocity statistics in atmospheric boundary layers
,” Phys. Rev. E
61
, 407
(2000
).19.
L.
Kristensen
, D. H.
Lenschow
, P.
Kirkegaard
, and M.
Courtney
, “The spectral velocity tensor for homogeneous boundary-layer turbulence
,” Boundary-Layer Meteorol.
47
, 149
–193
(1989
).20.
D. J.
Daley
and E.
Porcu
, “Dimension walks and Schoenberg spectral measures
,” Proc. Am. Math. Soc.
142
, 1813
–1824
(2014
).21.
S. C.
Lim
and L. P.
Teo
, “Generalized Whittle–Matérn random field as a model of correlated fluctuations
,” J. Phys. A: Math. Theor.
42
, 105202
(2009
).22.
M.
Bevilacqua
and T.
Faouzi
, “Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics
,” Electron. J. Stat.
13
, 3025
–3048
(2019
).23.
C.
Berg
, J.
Mateu
, and E.
Porcu
, “The Dagum family of isotropic correlation functions
,” Bernoulli
14
, 1134
–1149
(2008
).24.
M.
Schlather
, Random Fields: Simulation and Analysis of Random Fields, R Package Version 3.1.50, 2015
.25.
L. P.
Chamorro
, S.-J.
Lee
, D.
Olsen
, C.
Milliren
, J.
Marr
, R. E. A.
Arndt
, and F.
Sotiropoulos
, “Turbulence effects on a full-scale 2.5 MW horizontal-axis wind turbine under neutrally stratified conditions
,” Wind Energy
18
, 339
–349
(2015
).26.
T.
Gneiting
, “Power-law correlations, related models for long-range dependence and their simulation
,” J. Appl. Probab.
37
, 1104
–1109
(2000
).27.
28.
H. W.
Borchers
, Practical Numerical Math Routines, R Package Version 2.1.8, 2018
.29.
A.
Malyarenko
and M.
Ostoja-Starzewski
, Tensor-Valued Random Fields for Continuum Physics
(Cambridge University Press
, 2019
).30.
L.
Giraitis
, H. L.
Koul
, and D.
Surgailis
, Large Sample Inference for Long Memory Processes
(Chapman and Hall/CRC
, 2012
).31.
M.
Stein
, Interpolation of Spatial Data: Some Theory of Kriging
(Springer
, New York
, 1999
).32.
M.
Ostoja-Starzewski
, Microstructural Randomness and Scaling in Mechanics of Materials
(CRC Press
, 2007
).33.
A.
Papoulis
and S. U.
Pillai
, Probability, Random Variables and Stochastic Processes
, 4th ed. (McGraw-Hill
, Boston
, 2001
).© 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
