Two-particle diffusion in homogeneous turbulence with stable stratification by linear RTD spectral space analysis and application of a simplified model

This article paper investigates the numerical study of relative two-particle diffusion in homogeneous stably stratified turbulence. The focus is on obtaining components of tensors \(\varDelta ({\text{t}}^{{\prime }},\text{t})\) and \({\varDelta }^{\text{r}}\left(\text{t}\right)\)which are associated with one-particle and two-particle diffusion, respectively. This is achieved by utilizing Lagrangian two-time velocity correlations for both individual particles and two particles initially separated by \(\overrightarrow{{\delta }{\text{x}}_{0}}\). To facilitate the analysis, a simple two-particle diffusion model is developed based on assumptions similar to previous works by Kaneda and Ishida, as well as Ishihara and Kaneda, to elaborate one-particle and two-particle diffusion models. Using this model, Lagrangian two-time velocity correlations are calculated from numerically obtained two-time correlations spectra using a linear analysis RDT in the spectral space. The components of the two-particle diffusion tensor are then deduced directly. The application of this two-particle diffusion model considers an initial turbulence with a Reynolds number \({\text{R}}_{{\lambda }}=\text{2022,80}\) associated with the Taylor micro-scale of 2022.80. Two values of the Froude number (Fr) are selected: Fr = 0.0026 and Fr = 0.0045, in accordance with previously adopted criteria. Additionally, different values of the vertical displacement \({\delta }{\text{x}}_{0}/{\eta }\)are chosen, where (η) represents the Kolmogorov length-scale. Ultimately, two numerical simulation series are conducted: \({\text{R}}_{{\lambda }}\)=2022.80, Fr = 0.0045, (\({\delta }{\text{x}}_{0}/{\eta }\)=15, 17, 20, 24, and \({\text{R}}_{{\lambda }}\)=2022.80, Fr = 0.0026, \({\delta }{\text{x}}_{0}/{\eta }\)=15, 20, 25. The primary objective of this study is to delve into the fundamental problem of highlighting the \({\text{t}}^{3}\)Richardson law and the 4/3 diffusivity Richardson law in the horizontal plane during the intermediate range of times, focusing on two-particle diffusion.