Effects of turbulence on the estimation of horizontal velocity gradients from Lagrangian measurements

Abstract

A sequence of 16 computer simulations of Lagrangian drifter dispersion in a shear flow are used to provide input data to a linear regression estimator for horizontal velocity gradients . The only source of error considered is turbulence, which is incorporated into the simulations through a specified Eulerian spatial correlation matrix for turbulent velocity in a plane. No attempt is made to explicitly incorporate temporal correlation into the simulations. Time series of estimated velocity gradients are then used for subsequent analyses. Each simulated drifter experiment has six realizations of 80 time steps duration (with [delta]t = 1800 s ) . From these, estimates of sample mean, sample variance, probability density function, and cumulative distribution function of the four horizontal velocity gradients are computed. Of particular interest is the temporal variability of the variance of estimates given by the unbiased linear regression estimator. By subdividing the time sequences of estimates into five nonoverlapping intervals, gradient-estimate variances are computed as functions of time for all simulations. Plots of these variances indicate power law -like dependence on time at large dispersion times. The functional forms for cross-stream gradient variances are found to differ significantly from those for downstream gradients. A model accounting for these differences is developed from an approximation to the covariance matrix of the regression estimator. Expressions valid at large times are derived in which both cross-stream and downstream gradient variances are shown to depend in a complex manner on number of drifters, elapsed time of dispersion, value of the mean shear, and the time scales of the Lagrangian correlation matrix. Comparisons of these expressions with values obtained from the simulations show very good agreement, indicating the essential validity of the model, subject to the constraints implicit in development of the asymptotic relations.