Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method

Abstract

Centre manifold method is an accurate approach for analytically constructing an advection-diffusion equation (and even more accurate equations involving higher-order derivatives) for the depth-averaged concentration of substances in channels. This paper presents a direct numerical verification of converge to each other, with their velocities becoming practically equal. The obtained numerical results also demonstrate that the longitudinal diffusion can be neglected compared to the advection.

A symmetric integrated radial basis function method for solving differential equations

“…The stencil, which is associated with node 5, consists of regular nodes: (1,4,5,6,7,8,9) and irregular nodes: (10,11,12). The PDE is imposed at side nodes: (11,4,6,8) [Color figure can be viewed at wileyonlinelibrary.com] nodes are simply the intersection points of the stencil grid lines, while irregular nodes are generated from the intersection of the boundary and the stencil grid lines. As a result, for boundary stencils, the number of nodes are typically greater than 9.…”

“…In Kansa's method, the PDE is discretised by means of point collocation and the field variable is approximated by a set of multiquadric functions with a differentiation process being employed to obtain basis functions for derivative terms in the PDE (DRBFs). Since then, the RBF solution to differential problems has received a great deal of attention (see, e.g., [4][5][6][7][8][9][10][11][12][13][14]). For a given node distribution, the solution accuracy can still be improved by changing the value of the RBF width.…”

Dynamics of curved reaction fronts under a single-equation model

“…The 1d-irbf and irbf-based methods were successfully verified through several engineering problems such as turbulent flows [4], laminar viscous C401 flows [5,6,7], structural analysis [8], and fluid-structure interaction [9].…”

“…This step-like structure is a consequence of the balance between the energy release, represented by the term −A(∂ x u) 2 ∂ 2 x u , and the dissipation, represented by the term C∂ 6 x u . The term B(∂ x u) 4 plays the role of a bridge between the two, as explained by Strunin [1,2,3]. By re-scaling t, x and u, equation (1) can always be transformed into canonical form where all the coefficients, A, B and C become unity.…”

A reduced model for solute transport in compliant blood vessels with arbitrary axial velocity profile

“…The authors presented the method for the case of a straight channel and generalized it for a channel with varying radius based on a procedure given in [18]. A numerical verification of the resulting equation for the case of laminar and turbulent flow was carried out in [13]. Marbach and Alim recently used the generalized center manifold method to the case of a general velocity profile in a straight pipe, where the radial component of the velocity is neglected.…”