Shear dispersion looked at from a new angle

It is shown that for a sudden uniform discharge at x = 0, t = 0 in a bounded shear flow, the asymptotic concentration distribution at moderately large times can be well approximated by the tilted Gaussian \[ c=\frac{\overline{q}}{(2\pi\langle\sigma^2\rangle)^{\frac{1}{2}}}\exp\left(-\frac{(x-\overline{u}t-g_0(y,z)+2\alpha D)^2}{2\langle \sigma^2\rangle }\right), \] with \begin{eqnarray*} & \langle \sigma^2\rangle = 2Dt + 2\alpha D(x-\overline{u}t)-3\overline{g^2_0}-4\alpha^2 D^2,\\ & \overline{g_0} = 0,\quad D = \overline{ug_0},\quad \alpha = \frac{(\overline{u-\overline{u}})g^2_0}{2D^2}. \end{eqnarray*} Here u(y, z) is the velocity profile, g0(y, z) the centroid displacement function, and the overbars denote cross-sectional averaging. The tilt parameter α makes the concentration distribution suitably skew. The effectiveness of this simple formula is demonstrated for two-layer flows and for plane Poiseuille flow.