AbstractWe 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 $W\ensuremath{\sim} O(d)$. 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 $b(z){ \overline{w} }_{m} (z)$, where $z$ is the streamwise coordinate, $b$ is the jet width, ${ \overline{w} }_{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 ${\partial }_{t} \phi + {K}_{a} {\text{} {M}_{0} \text{} }^{1/ 2} {\partial }_{z} \left(\phi / {z}^{1/ 2} \right)= {K}_{d} {\text{} {M}_{0} \text{} }^{1/ 2} {\partial }_{z} \left({z}^{1/ 2} {\partial }_{z} \phi \right)$, 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 ${K}_{a} = 1. 65\pm 0. 10$ and ${K}_{d} = 0. 09\pm 0. 02$, 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.
Optimal error estimate of the Legendre spectral approximation for space-fractional reaction–advection–diffusion equation
