Longitudinal dispersion within a two-dimensional turbulent shear flow

Abstract

This paper describes some laboratory and numerical experiments made on the longitudinal dispersion in an open channel flow. Particular attention has been paid to the initial stages of the process.Physical arguments suggest that the streamwise dispersion of a line of marked fluid elements across a two-dimensional turbulent shear flow occurs in three distinct stages. These stages are identified by a change in the form of the distribution of marked fluid elements in the flow direction. The skewed distribution of the first stage is readily identified by a constant value (approximately 1·1) for the ratio of the peak velocity (V1) of the distribution to the mean-flow velocity ; experiments using dyed fluid, made at this stage of the process, have revealed six identifiable features of the suggested distribution. The distributions suggested for the second and the third stage are consistent with the experimental findings of Elder (1959) for the second stage and Taylor (1954) for the third stage.An attempt has been made to simulate the process numerically using a Markovian model. The results of the simulation confirm features suggested by physical arguments and are in agreement with the open channel experiments.The Lagrangian autocorrelation function is found to be related to the Lagrangian velocity-history of marked fluid released from extreme positions on the flow cross-section. The correlation function, as expressed in terms of the velocity-history function provided by the numerical simulation, is \begin{eqnarray*} && R(t^{\prime}) = \exp (-bt^{\prime})\int_0^{1}U^{+2}dy^{\prime};\\ && t^{\prime} = tu_{*}/d,\quad U^{+} = \frac{U(y^{\prime})-\overline{U}}{u_{*}}, \end{eqnarray*} where u* is the friction velocity and U(y′) is the temporal mean velocity at a (non-dimensionalized) distance y′ from the flow boundary. In an open channel flow at a Reynolds number (based on friction velocity and channel depth) of 500, the numerical simulation provides the value of b = 0·536.The results of an experiment, in which the three-dimensional motion of small neutrally buoyant spheres was recorded in many small discrete time intervals, corroborate the theoretical suggestions and simulation results.