Numerical study of Lagrangian velocity structure functions using acceleration statistics and a spatial-temporal perspective

Abstract

A fundamental relation in Lagrangian Kolmogorov theory is concerned with inertial range scaling of the second-order velocity structure function over intermediate time lags at sufficiently high Reynolds numbers. Significant theoretical support for asymptotic constancy of the scaling constant ($C_0$) is known, but limitations in the range of time scales accessible in direct numerical simulation make unambiguous testing of the scaling challenging. In this paper, direct numerical simulations of forced isotropic turbulence at Taylor-scale Reynolds numbers between 140 and 1300 are used to improve understanding in this subject. Uncertainties arising from modest simulation time spans in the high Reynolds number data are addressed by expressing the velocity structure function in terms of the acceleration autocorrelation, which suggests that $C_0$ may be sensitive to effects of Lagrangian intermittency but does not rule out asymptotic constancy at Reynolds numbers beyond those that may be feasible in simulations in the foreseeable future. The Lagrangian velocity increment is examined further from a spatial-temporal perspective, as a combination of convective (spatial) and local (temporal) contributions, which are subject to a strong but incomplete mutual cancellation dependent on Reynolds number and time lag. The convective contribution is strongly influenced by the particle displacement, which is driven by large-scale dynamics and can thus grow into inertial range dimensions in space within just a few Kolmogorov time scales, without fully satisfying classical Lagrangian inertial-range requirements. An overall conclusion in this work is that both the limited range of time scales (narrower than that for length scales) and the effects of particle displacements have significant roles in the observed behavior of the second-order Lagrangian velocity structure function.