Study of steady two-dimensional advection–diffusion equation with stratification using second-kind shifted Chebyshev polynomials

This study investigates the steady two-dimensional (2D) distribution of suspended sediment concentration in an open channel turbulent flow, utilizing five eddy viscosity profiles incorporating the stratification effect. In addition to three well-known eddy viscosity profiles, two recently proposed profiles, based on the concepts of velocity and length scale concepts, are also considered. The most realistic bottom boundary condition is taken, where the net flux is a function of deposition velocity and equilibrium bottom concentration. The steady 2D transport equation is solved using the collocation method with second-kind shifted Chebyshev polynomials. The spectral radius of the scheme’s iteration matrix is found to be less than unity, ensuring convergence regardless of inlet concentration, downstream position or eddy viscosity profiles. Furthermore, it is observed that the sum of squared residual errors (SSRE) decreases to zero as the number of second-kind shifted Chebyshev polynomials increases, confirming the accuracy of the method. Although the focus is given on second-kind shifted Chebyshev polynomials, comparisons are made with other kinds (first, third and fourth) in solving the governing equation. The effect of stratification on concentration profiles is analyzed through the stratification correction parameter and it is found that the inclusion of the effect reduces sediment concentration when the inlet concentration is zero. Conversely, for uniform inlet concentration, the concentration increases near the inlet but decreases as the downstream distance increases. The sensitivity of the inverse turbulent Schmidt number on vertical sediment concentration is also examined at various downstream positions. Finally, the solution is validated against existing solutions, laboratory data and field measurements from the Rio Grande conveyance channel, Missouri River and Mississippi River.