Phase aligned ensemble averaging for environmental flow studies
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Phase aligned ensemble averaging for environmental flow studies Qiang Zhong1,2 · Fazle Hussain2,3 · Harindra J. S. Fernando2,4 Received: 4 December 2019 / Accepted: 19 September 2020 / Published online: 30 September 2020 © Springer Nature B.V. 2020
Abstract The quantification of turbulent mixing in nature is predicated by inherent randomness of causal events, and obtaining relevant turbulence statistics requires ensemble averaging of identical realizations that are unachievable in field observations or onerous in laboratory situations. Laboratory modeling is often used to study nonstationary natural processes, but jitters due to intrinsic variability of events as well as experimental uncertainties introduce additional (spurious) fluctuations that affect ensemble averaging of individual realizations. In this paper, the phase-aligned ensemble averaging technique (PAET), which aligns the events based on information on flow structures, is introduced in the context of environmental fluid mechanics studies. The accuracy and computational efficiency of PAET are investigated systematically for two cases: (1) synthetic density field alignment and (2) laboratory flows involving collision of counter flowing gravity currents. The latter is a frequent phenomenon in the stable atmospheric boundary layer in mountainous areas. In the synthetic density field case, the PAET aligns the complex structures precisely within the allowable range of measurement accuracy. For the experimental gravity current case, the precisely aligned result is unknown, and the results of PAET are compared with those obtained with the Monte Carlo method, a simulated annealing algorithm, and the gradient descent method; the PAET was found to be the most efficient. This study broaches the PAET as a versatile method for obtaining accurate turbulence statistics in laboratory experiments designed to mimic environmental flows where spatial and temporal inhomogeneities abound. Keywords Phase-aligned · Ensemble average · Jitter · Optimization problem · Colliding gravity currents
* Qiang Zhong [email protected] 1
College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, China
2
Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN 46556, USA
3
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409‑1021, USA
4
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
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Vol.:(0123456789)
1358
Environmental Fluid Mechanics (2020) 20:1357–1377
1 Introduction ̃ t) of any turbulent In statistical analysis of turbulence, the instantaneous value F(𝐱, quantity is separated into averaged F(𝐱, t) and fluctuating F(𝐱, t) components based on the Reynolds decomposition, ̃ t) + F(𝐱, t). ̃ t) = F(𝐱, F(𝐱,
(1)
Most of the classic decomposition methods are based on the ensemble-averaging technique [1], with +∞
F(𝐱, t) =
∫−∞
[ ] ̃ ̃ t) ⋅ p F(𝐱, ̃ t) dF, F(𝐱,
(2)
where p[.] is the probability den
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