Streamwise dispersion and mixing in quasi-two-dimensional steady turbulent jets

We investigate experimentally and theoretically the streamwise transport and dispersion properties of steady quasi-two-dimensional plane turbulent jets discharged vertically from a slot of width d into a fluid confined between two relatively close rigid boundaries with gap W similar to O(d). We model the evolution in time and space of the concentration of passive tracers released in these jets using a one-dimensional time-dependent effective advection-diffusion equation. We make a mixing length hypothesis to model the streamwise turbulent eddy diffusivity such that it scales like b(z)(W) over bar (m)(z), where z is the streamwise coordinate, b is the jet width, (W) over bar (m) is the maximum time-averaged vertical velocity. Under these assumptions, the effective advection-diffusion equation for phi(z, t), the horizontal integral of the ensemble-averaged concentration, is of the form partial derivative(t)phi + KaM01/2 partial derivative(z) (phi/z(1/2)) = KdM01/2 partial derivative(z) (z(1/2)partial derivative(z)phi), where t is time, K-a (the advection parameter) and K-d (the dispersion parameter) are empirical dimensionless parameters which quantify the importance of advection and dispersion, respectively, and M-0 is the source momentum flux. We find analytical solutions to this equation for phi in the cases of a constant-flux release and an instantaneous finite-volume release. We also give an integral formulation for the more general case of a time-dependent release, which we solve analytically when tracers are released at a constant flux over a finite period of time. From our experimental results, whose concentration distributions agree with the model, we find that K-a = 1.65 +/- 0.10 and K-d = 0.09 +/- 0.02, for both finite-volume releases and constant-flux releases using either dye or virtual passive tracers. The experiments also show that streamwise dispersion increases in time as t(2/3). As a result, in the case of finite-volume releases more than 50% of the total volume of tracers is transported ahead of the purely advective front (i.e. the front location of the tracer distribution if all dispersion mechanisms are ignored and considering a 'top-hat' mean velocity profile in the jet); and in the case of constant-flux releases, at each instant in time, approximately 10% of the total volume of tracers is transported ahead of the advective front.