Decaying turbulence in a stratified fluid of high Prandtl number


In nature, the transportation of scalars, such as: salt, heat and pollutants, by fluid flow is a universal phenomenon that is of crucial significance in the fields of geophysics and engineering. Ideally, the key parameter that characterizes the diffusivity of these scalars is the Prandtl number (Pr); a non-dimensional number defined by the ratio of the kinematic viscosity v of fluid to the diffusion coefficient κ of the scalar. Generally, the diffusion coefficient of a substance in a liquid is normally much smaller than that of heat, hence the corresponding Prandtl number much higher. Numerous studies have been undertaken to investigate this phenomenon. Nonetheless, there is need to know the effect of molecular diffusivity on the structure of stratified turbulence, especially at high Prandtl numbers. The behavior of small-scale fluctuations of a high-Prandtl-number scalar was investigated theoretically by Batchelor (J. Fluid Mech., vol. 5, 1959, pp.113-133). In recent times, studies based on Batchelor’s work have been undertaken but all are based on a passive scalar in isotropic turbulence.

In stratified fluids, the distribution of the active scalars such as salt generates the buoyancy force, and the scalars behave differently from passive scalars. In the studies of decaying stratified turbulence, the suppression of vertical motion due to the exchange between the vertical kinetic energy and the potential energy has been observed. A thorough review of existing literature reveals that there have been only a few numerical studies intended for a Prandtl number larger than unity, due to the difficulty in resolving the very small Batchelor scales at high Prandtl numbers, while at the same time, it is crystal clear that the simulation at Pr ∼ 700 is necessary for direct comparison with the salt-water experiments. As such, there is need for further investigation on the decaying turbulence in a density-stratified fluid with a Prandtl number up to Pr = 70.

In a recent publication, Dr. Shinya Okino and Professor Hideshi Hanazaki from the Department of Mechanical Engineering and Science at Kyoto University in Japan, developed a direct numerical simulation to investigate the decaying stratified turbulence at Prandtl numbers up to 70, with a higher resolution in comparison with previous numerical simulations. They anticipated that their work would help realize conditions more similar to the salt-stratified experiments (Pr ∼ 700) compared to the previous numerical studies, which were intended mostly for Pr = 7. Their work is currently published in Journal of Fluid Mechanics.

In their approach, the two scholars considered a turbulent flow of a density-stratified fluid whose stratification is generated by a buoyant scalar of Prandtl number unity or larger. To begin with, they first focused on demonstrating the Prandtl-number dependence of the statistical quantities such as energies and vertical density flux; following which they investigated the spatial distribution of density fluctuations, which exhibits vertically small-scale structures at the highest Prandtl number of 70. All in all, they examined the relevant spectra following which the generation mechanism of the vertically thin structures of the density fluctuations, modifying the theory for a passive scalar (Batchelor 1959) into a theory for a buoyant scalar was discussed.

The researchers reported that the kinetic and potential energies decayed more slowly for higher Prandtl numbers, since the scalar fluctuations of a higher Prandtl number dissipated more slowly and more potential energy could be converted into vertical kinetic energy through the vertical density flux, which was more negative and more counter-gradient. In addition, it was realized that the difference due to the increase of the Prandtl number became less significant for larger Pr, although there was no upper threshold value of Pr at which further Pr dependence completely disappeared.

In summary, the study presented an in-depth investigation on the decaying turbulence in a density-stratified fluid of a high Prandtl number up to Pr = 70 through a direct numerical simulation. The effect of the Prandtl number is significant in the density fluctuation at small scales, which shows vertically thin structures at the late stage of decaying stratified turbulence for the highest Pr (= 70). The theoretical analysis presented showed that the vertically sheared horizontal flow at the Kolmogorov scale reduced the vertical scale of density fluctuations, but did not change the horizontal scale.

Decaying turbulence in a stratified fluid of high Prandtl number - Advances in Engineering

About the author

Shinya Okino is an assistant professor at the Department of Mechanical Engineering and Science, Kyoto University. He received his PhD degree at the Department of Aeronautics and Astronautics, Kyoto University in 2011.

His research interests include stratified turbulence, flow past an obstacle in stratified fluids and transition to turbulence in wall-bounded shear flows.

About the author

Hideshi Hanazaki is a professor in the Department of Mechanical Engineering and Science at Kyoto University. He received PhD degree in Applied Physics from the University of Tokyo (1989). He also worked at the National Institute for Environmental Studies and Tohoku University.

His research interests are the flows in stratified/rotating fluids, especially the flow around an obstacle, nonlinear internal/inertial waves, and turbulence. He also works on surface water waves with capillary effects.


S. Okino, H. Hanazaki. Decaying turbulence in a stratified fluid of high Prandtl number. Journal of Fluid Mechanics (2019), volume 874, page 821–855.

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