On the relationship between the Lagrangian and Eulerian time scale

Abstract

The knowledge about the connection between the Lagrangian and the Eulerian time scales is very important primarily with regard to Lagrangian dispersion models. A very extensive summary of the literature is compiled by Pasquill and Smith (1983), who refer to statement T L /T E =β=α/ u ′2 / u ̄ where ū is the mean local velocity and u ′2 is the root-mean-square velocity. The factor α is considered as a “constant”, which can assume values between 0.35 and 0.8. A theoretical relationship between the Lagrangian and Eulerian time scale is derived in this paper. The relation is inversely proportional to the turbulence intensity and proportional to the ratio of the advection velocity (the velocity by which eddies are convected within the flow, there is no connection with the convective velocity scale w ∗ in the mixed layer) and the mean local velocity (u c2/ u ̄ ) at this height. Furthermore, the latter velocity ratio provides a criterion for validity of the Taylor hypothesis and is equivalent to the above-mentioned factor α up to a value of 0.8, that is wrongly regarded as “constant”. T L T E =0.8 u c2 u ̄ u ̄ u ′2 . Our experiments in the wind tunnel provided data about the advection velocity for different rough boundary layers above a flat plate. The wind tunnel thereby models the neutrally stratified atmosphere. The velocity profiles cover almost the whole range to exist in nature with regard to roughness (profile exponent from 0.17 to 0.28, roughness height from 10 -2 to 10 0m). As the ratio u c2/ u ̄ , the results showed a dependence on the height in the boundary layer as well as on the roughness height. In particular the lower part of the boundary layer pointed out a strange dependence on the height; here the Taylor-hypothesis is invalid. An estimate is presented for factor α in dependence on the roughness for the atmospheric boundary layer, which is valid above the layer of constant shear stress. In this connection, values of α are obtained which correspond to the data of Pasquill and Smith very well.