Higher-order approximation of contaminant transport equation for turbulent channel flows based on centre manifolds and its numerical solution

- We accurately reduce the 2D contaminant transport equation to its 1D approximation.
- The analytical derivation is based on the centre manifold theory.
- Two types of the velocity profile are considered: logarithmic and power profiles.
- We develop a high-order numerical approximation for solving high-order PDEs.
- The comparison of the 1D and 2D models shows good agreement.

SummaryThe contaminant transport process governed by the advection-diffusion equation plays an important role in modelling industrial and environmental flows. In this article, our aim is to accurately reduce the 2-D advection-diffusion equation governing the dispersion of a contaminant in a turbulent open channel flow to its 1-D approximation. The 1-D model helps to quickly estimate the horizontal size of contaminant clouds based on the values of the model coefficients. We derive these coefficients analytically and investigate numerically the model convergence. The derivation is based on the centre manifold theory to obtain successively more accurate approximations in a consistent manner. Two types of the average velocity profile are considered: the classical logarithmic profile and the power profile. We further develop the one-dimensional integrated radial basis function network method as a numerical approach to obtain the numerical solutions to both the original 2-D equation and the approximate 1-D equations. We compare the solutions of the original models with their centre-manifold approximations at very large Reynolds numbers. The numerical results obtained from the approximate 1-D models are in good agreement with those of the original 2-D model for both the logarithmic and power velocity profiles.