Computation Of Steady Solutions Research Articles

Axisymmetric rotating flow with free surface in a cylindrical tank

The flow induced by a disk rotating at the bottom of a cylindrical tank is characterised using numerical techniques – computation of steady solutions or time-averaged two-dimensional and three-dimensional direct simulations – as well as laser-Doppler velocimetry measurements. Axisymmetric steady solutions reveal the structure of the toroidal flow located at the periphery of the central solid body rotation region. When viewed in a meridional plane, this flow cell is found to be bordered by four layers, two at the solid boundaries, one at the free surface and one located at the edge of the central region, which possesses a sinuous shape. The cell intensity and geometry are determined for several fluid-layer aspect ratios; the flow is shown to depend very weakly on Froude number (associated with surface deformation) or on Reynolds number if sufficiently large. The paper then focuses on the high Reynolds number regime for which the flow has become unsteady and three-dimensional while the surface is still almost flat. Direct numerical simulations show that the averaged flow shares many similarities with the above steady axisymmetric solutions. Experimental measurements corroborate most of the numerical results and also allow for the spatio-temporal characterisation of the fluctuations, in particular the azimuthal structure and frequency spectrum. Mean azimuthal velocity profiles obtained in this transitional regime are eventually compared to existing theoretical models.

On a modified non-singular log-conformation formulation for Johnson–Segalman viscoelastic fluids

A modified log-conformation formulation of viscoelastic fluid flows is presented in this paper. This new formulation is non-singular for vanishing Weissenberg numbers and allows a direct steady numerical resolution by a Newton method. Moreover, an exact computation of all the terms of the linearized problem is provided. The use of an exact divergence-free finite element method for velocity–pressure approximation and a discontinuous Galerkin upwinding treatment for stresses leads to a robust discretization. A demonstration is provided by the computation of steady solutions at high Weissenberg numbers for the difficult benchmark case of the lid driven cavity flow. Numerical results are in good agreement, qualitatively with experiment measurements on real viscoelastic flows, and quantitatively with computations performed by others authors. The numerical algorithm is both robust and very efficient, as it requires a low mesh-invariant number of linear systems resolution to obtain solutions at high Weissenberg number. An adaptive mesh procedure is also presented: it allows representing accurately both boundary layers and the main and secondary vortices.

Asymptotic Approximation and Perturbation Stream Functions in Viscous Flow Calculations

An asymptotic approximation technique is shown to substantially decrease the number of interations necessary for the accurate computation of steady solutions for viscous incompressible flows. A numerical study of flows exterior to a circular cylinder indicates that the technique reduces the required number of iterations by a factor of 10. The flux formulation of the vorticity stream function solution for the flow of a viscous incompressible fluid external to a circular cylinder requires the solution of the Poisson equation for the stream function. The use of the FACR (1), fast direct Poisson equation solver leads to serious errors in the computation of the stream function near the cylinder. However it is shown that these errors are eliminated through the introduction of a perturbation stream function.