Whady F. Flórez

A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations

The two-dimensional Navier-Stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions (MAPS) in its global formulation. The fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a Stokes non-homogeneous system of equations with multiquadric (MQ) radial basis function as the source term. The nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field, and the resulting linear system of equations is solved by a simple direct iterative scheme (Picard iteration), with the velocity guess given by the solution at the previous iteration. Although the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation. The proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the Kovasznay flow problem at different Reynolds numbers, Re. From this analysis, it is observed that the MAPS results are stable and accurate for a wide range of shape parameter values. In addition, lid-driven cavity flow problems in rectangular and triangular domains up to Re=3200 and Re=1000, respectively, and the backward-facing step at Re=800 are solved, and the results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement.

Modelling and simulation of time-dependent coal combustion processes in stacks

Abstract A time-dependent mathematical model and a numerical algorithm have been developed to simulate the combustion of piled coal particles. The model can predict the evolution profiles of unburned solid fraction along the bed height, the gas composition, heat of reaction, gas temperature and the coal (solid phase) temperature. Also, it predicts the radial temperature profile inside the particles. The model includes a system of six differential equations derived from the mass and energy balances for all phases at any point along the bed height and within each particle. For the numerical solution, implicit collocation and relaxation techniques were used with finite differences for the time advance. Additionally, the model can be used to optimise the combustion process varying the excess of air, particle size distribution, coal type, geometry of the reactor, different types of ignition mechanisms and the velocity of the grate.

The global approximate particular solution meshless method for two-dimensional linear elasticity problems

The two-dimensional linear elasticity equations are solved by the global method of approximate particular solution as a new meshless option to the conventional finite element discretization. The displacement components are approximated by a linear combination of the elasticity particular solutions and the stress tensor is obtained by differentiating the displacement expressions in terms of the particular solutions. The multiquadric radial basis function (RBF) is employed as the non-homogeneous term in the governing equation to compute the particular solutions. The cantilever beam and the infinite plate with a hole problem are solved to verify the implemented meshless method. For each situation, the trend of the root mean square error is assessed in terms of the shape parameter and the number of nodes. Unlike most of the RBF collocation strategies, it is found that numerical results are in good agreement with the analytical solutions for a wide range of shape parameter values.

A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

ABSTRACT The unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

Multi‐domain DRM boundary element method for non‐isothermal non‐Newtonian Stokes flow with viscous dissipation

This paper presents a boundary element method (BEM) based on a subdomain approach for the solution of non‐Newtonian fluid flow problems which include thermal effects and viscous dissipation. The volume integral arising from non‐linear terms is converted into equivalent boundary integrals by the multi‐domain dual reciprocity method (MD‐DRM) in each subdomain. Augmented thin plate splines interpolation functions are used for the approximation of field variables. The iterative numerical formulation is achieved by viewing the material as divided into small elements and on each of them the integral representation formulae for the velocity and temperature are applied and discretised using linear boundary elements. The final system of non‐linear algebraic equations is solved by a modified Newtons method. The numerical examples include non‐Newtonian problems with viscous dissipation, temperature‐dependent viscosity and natural convection due to bouyancy forces.

The control volume radial basis function method CV-RBF with Richardson extrapolation in geochemical problems

The aim of this paper is to present how to implement a control volume approach improved by Hermite radial basis functions (CV-RBF) for geochemical problems. A multi-step strategy based on Richardson extrapolation is proposed as an alternative to the conventional dual step sequential non-iterative approach (SNIA) for coupling the transport equations with the chemical model. Additionally, this paper illustrates how to use PHREEQC to add geochemical reaction capabilities to CV-RBF transport methods. Several problems with different degrees of complexity were solved including cases of cation exchange, dissolution, dissociation, equilibrium and kinetics at different rates for mineral species. The results show that the solution and strategies presented here are effective and in good agreement with other methods presented in the literature for the same cases. HighlightsA control volume Hermite RBF scheme has been implemented for groundwater reactive problems.The standard SNIA approach was improved by a Richardson extrapolation.The PHREEQ modules are coupled with a transport solution by a local RBF method.The solution by CV-RBF is effective and in agreement with other numerical methods.

Modeling and Simulation of a Co-Current Rotary Dryer Under Steady Conditions

A model which joins the overall design algorithm of a rotary dryer with the drying kinetics equations derived from experimental data and with a finite segment algorithm is implemented in order to verify the dryer dimensions obtained from a basic sizing procedure. Total energy and mass balances and well-known correlations for the overall heat transfer coefficient are employed to develop it. Moreover, a one-dimensional finite segment model is solved to obtain the length profiles of temperature and water content for the air and solid phases. An experimental correlation for the mass transfer coefficient between solid and air phases is included in the finite segment model. The chosen heat transfer unit number for the basic sizing is verified with the outlet temperature and water content calculated by the finite segment scheme.

Convective Heat Transfer in Nanofluids: A Computational Approach

Nanofluids are a novel strategy to increase heat transfer characteristics of fluids by the addition of solid particles with diameters below 100 nm. Experimental measurements have shown that this approach can greatly increase heat conductivity, even above that predicted by Maxwell’s theory. This paper shows a direct numerical simulation of the flow and thermal behaviour of a nanofluid loaded with alumina nanoparticles. The Boundary Element Method is used given its capabilities to deal with moving boundary problems. Results showed strong convective currents caused by the presence of the nanoparticles, which in time increase total heat flow in the cavity.