Document Type


Publication Date


Journal Title

Physical Review Fluids

Volume Number


Issue Number


First Page


Last Page




Publisher PDF: the final published version of the article, with professional formatting and typesetting


Engineering | Mechanical Engineering | Physics


The Lagrangian and Eulerian acceleration properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from Ri=0, corresponding to unstratified shear flow, to Ri=1, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both Lagrangian and Eulerian accelerations have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the Eulerian acceleration are stronger than those observed for the Lagrangian acceleration. Geometrical statistics explain that the magnitude of the Eulerian acceleration is larger than its Lagrangian counterpart due to the mutual cancellation of the Eulerian and convective acceleration, as both vectors statistically show an antiparallel preference. A wavelet-based scale-dependent decomposition of the Lagrangian and Eulerian accelerations is performed. The tails of the acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence the flatness increases with decreasing scale, indicating stronger intermittency at smaller scales. The joint pdfs of the Lagrangian and Eulerian accelerations indicate a trend to stronger correlations with increasing Richardson number and at larger scales of the turbulent motion. A consideration of the terms in the Navier-Stokes equation shows that the Lagrangian acceleration is mainly determined by the pressure-gradient term, while the Eulerian acceleration is dominated by the nonlinear convection term. A similar analysis is performed for the Lagrangian and Eulerian time rates of change of both fluctuating density and vorticity. The Eulerian time rates of change are observed to have extreme values substantially larger than those of their Lagrangian counterparts due to the advection terms in the advection-diffusion equation for fluctuating density and in the vorticity equation, respectively. The Lagrangian time rate of change of fluctuating vorticity is mainly determined by the vortex stretching and tilting term in the vorticity equation. Since the advection-diffusion equation for fluctuating density lacks a quadratic term, the Lagrangian time rate of change pdfs of fluctuating density show a more Gaussian shape, in particular, for large Richardson numbers. Hence, the Lagrangian acceleration and time rates of change of fluctuating density and vorticity reflect the dominant physics of the underlying governing equations, while the Eulerian acceleration and time rates of change are mainly determined by advection.


Original publication information: F. Jacobitz, K. Schneider, Lagrangian and Eulerian accelerations in turbulent stratified shear flows, Phys. Rev. Fluids 6, 07469 (2021).