On the dissipation rate of temperature fluctuations in stably stratified flows
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On the dissipation rate of temperature fluctuations in stably stratified flows Sukanta Basu1 · Adam W. DeMarco2 · Ping He3 Received: 13 January 2020 / Accepted: 10 August 2020 © The Author(s) 2020
Abstract In this study, we explore several integral and outer length scales of turbulence which can be formulated by using the dissipation of temperature fluctuations ( 𝜒 ) and other relevant variables. Our analyses directly lead to simple yet non-trivial parameterizations for both spatially-averaged 𝜒 and the structure parameter of temperature ( CT2 ). For our purposes, we make use of high-fidelity data from direct numerical simulations of stratified channel flows. Keywords Integral length scale · Outer length scale · Ozmidov scale · Stable boundary layer · Structure parameter
1 Introduction The molecular dissipation of temperature fluctuations ( 𝜒 ) is an important variable for characterizing turbulent mixing in various environmental flows. It is frequently used in micrometeorology (e.g., [66]) and atmospheric optics (e.g., [43]). Furthermore, any higher-order closure model requires solving a prognostic equation or a diagnostic parameterization for ensemble-averaged 𝜒 (refer to [15, 36, 40, 65]) . Over the years, a number of studies focused on the correlation between turbulent kinetic energy dissipation rate ( 𝜀 ) and 𝜒 (e.g., [2, 5–7, 29, 69]). In addition, some papers reported on the probability density function, spatio-temporal intermittency and anomalous scaling of 𝜒 (e.g., [6, 52, 54]). Often, 𝜒 has been found to be more intermittent (commonly quantified by the multifractal scaling exponents) and non-Gaussian than 𝜀 (e.g., [50, 54]). Most of these previous studies primarily focused on the instantaneous, localized traits of the dissipation fields. Instead, we are interested to better quantify their spatially averaged characteristics. Towards this goal, we first investigate the statistical properties of several length scales which can be formulated based on 𝜒 and other relevant variables. Based on * Sukanta Basu [email protected] 1
Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands
2
United States Air Force, Washington, D.C., USA
3
Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA
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Vol.:(0123456789)
Environmental Fluid Mechanics
these findings, we then derive simple parameterizations for 𝜒 and temperature structure parameter ( CT2 ). For all the analyses, we utilize a direct numerical simulation (DNS) database of stably stratified flows which is discussed in the following section.
2 Direct numerical simulation Recently, for the parameterization of optical turbulence, He and Basu [31] created a DNS database using a massively parallel DNS code, called HERCULES [30]. The DNS runs were conducted by solving the normalized Navier–Stokes and temperature equations, as shown in Eqs. 1–3 (using Einstein’s summation notation):
𝜕un,i 𝜕xn,i 𝜕un,i 𝜕tn
+
𝜕un,i un,j 𝜕xn,j
=−
= 0,
𝜕pn 1 𝜕 𝜕un,i + + 𝛥P𝛿i1 + Rib 𝜃n 𝛿i3 , 𝜕xn,i
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