The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability in the calculation of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the appropriate inf–sup conditions, whereas the dominant convection requires a stabilized formulation in any case. This paper proposes two stabilized schemes of Sub-Grid Scale (SGS) type, differing in the definition of the space of the sub-grid scales, and both allowing to use the same interpolation for the variables σ–u–p (deviatoric stress, velocity and pressure), even in problems where the convection component is dominant and the velocity–stress gradients are high. Another aspect considered in this work is the non-linearity of the viscosity, modeled with constitutive models of quasi-Newtonian type. This paper includes a description of the proposed methods, some of their implementation issues and a discussion about benefits and drawbacks of a three-field formulation. Several numerical examples serve to justify our claims.
Abstract
The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. The [...]
In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian–Eulerian formulation to account for the motion of the free surface because of demand and supply of water, and a coupling of the hydrodynamics with a convection–diffusion–reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection‐dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed.
Abstract
In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. [...]