The dispersion of solute from time-dependent releases in parallel flow

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Abstract
This paper examines an often overlooked point in the theory of dispersion of passive contaminants in parallel flows – the behaviour of a cloud of solute which has been injected into the flow over a period of time. The problem is linear if the solute has neutral buoyancy and the diffusivity is independent of concentration, and so it can be treated by a Fourier transform in time. For such a Fourier-transform method to be successful, a non-standard eigenvalue problem has to be solved to determine the concentration pattern, the speed at which it is convected downstream and its decay distance downstream. The eigenvalue problem is not self-adjoint, the eigenvalue enters it nonlinearly if longitudinal diffusion is included, and it involves both regular and singular perturbation aspects. The eigenvalue problem is examined generally and the conclusions of Chatwin (1973a) are re-assessed. The eigenvalue problem is then solved numerically for three cases of interest. Two of these cases (dispersion from a harmonically varying source in Poiseuille and plane Couette flow) reveal most unusual eigenvalue structure, whilst the third (dispersion from a harmonically varying source in turbulent channel flow) is not exceptional. The effect of weak longitudinal diffusion is examined theoretically (for general applications) and numerically in one instance (the application to Poiseuille flow). Longitudinal diffusion, even if weak, has a marked effect on the eigenvalue structure. The paper concludes with a suggestion for an alternative attack on the original problem of dispersion from time-dependent sources.