Numerical analysis of an averaged model of turbulent transport near a roughness layer

Abstract

We formulate and numerically analyse the averaged model of dispersion in turbulent canopy flows. The averaging is carried out across the flow, for example over the river depth. To perform the averaging, we use the general approach suggested by Roberts and co-authors in the late 1980s, which is based on centre manifold theory. We derive an evolution partial differential equation for the depth averaged concentration, involving first, second and higher order derivatives with respect to the downstream coordinate. The coefficients of the equation are expressed in terms of parameters characterising the turbulent flow. Preliminary numerical results are demonstrated. In particular, it is shown that the advection and diffusion coefficients coincide with their values obtained earlier for the flow over a smooth bottom in the limit of large depths. References Mercer, G. N. and Roberts, A. J., A centre manifold description of contaminant dispersion in channels with varying flow properties, SIAM J. Appl. Math., 50, 1990, 1547--1565. doi:10.1137/0150091 Roberts, A. J. and Strunin, D. V., Two-zone model of shear dispersion in a channel using centre manifolds, Quarterly J. Mech. Appl. Math., 57, 2004, 363--378. doi:10.1.1.102.854 Macdonald, R. W., Modelling the mean velocity profile in the urban canopy layer, Boundary-Layer Meteorol., 97, 2000, 25--45. doi:10.1023/A:1002785830512 Harman, I. N. and Finnigan, J. J., A simple unified theory for flow in the canopy and roughness sublayer, Boundary-Layer Meteorol., 123, 2007, 339--363. doi:10.1007/s10546-006-9145-6 H. Tennekes and J. L. Lumley. A First Course in Turbulence. MIT Press, Cambridge, MA, 1972. Cionco, R. M, Mathematical model for air flow in a vegetative canopy, J. Appl. Meteorol., 4, 1965, 517--522. doi:10.1175/1520-0450(1965)004<0517:AMMFAF>2.0.CO:2 Cionco, R. M., A wind-profile index for canopy flow, Boundary-Layer Meteorol., 3, 1972, 255--263. doi:10.1007/BF0203923 Barenblatt, G. I., Transfer of a passive additive in a turbulent boundary layer at very large Reynolds numbers, Proc. Nat. Acad. Sci., 100, 2003, 1481--1483. doi:10.1073/pnas.0337426100 Strunin, D. V., Universality of turbulent dispersion in a steady flow in an open channel, Quarterly J. Mech. Appl. Math., 64, 2011, 197--214. doi:10.1093/qjmam/hbr002