Drag Coefficients of Viscous Spheres at Intermediate and High Reynolds Numbers

Abstract

A finite-difference scheme is used to solve the Navier-Stokes equations for the steady flow inside and outside viscous spheres in a fluid of different properties. Hence, the hydrodynamic force and the steady-state drag coefficient of the spheres are obtained. The Reynolds numbers of the computations range between 0.5 and 1000 and the viscosity ratio ranges between 0 (inviscid bubble) and infinity (solid particle). Unlike the numerical schemes previously implemented in similar studies (uniform grid in a stretched coordinate system) the present method introduces a two-layer concept for the computational domain outside the sphere. The first layer is a very thin one [ORe−1/2] and is positioned at the interface of the sphere. The second layer is based on an exponential function and covers the rest of the domain. The need for such a double-layered domain arises from the observation that at intermediate and large Reynolds numbers a very thin boundary layer appears at the fluid-fluid interface. The computations yield the friction and the form drag of the sphere. It is found that with the present scheme, one is able to obtain results for the drag coefficient up to 1000 with relatively low computational power. It is also observed that both the Reynolds number and the viscosity ratio play a major role on the value of the hydrodynamic force and the drag coefficient. The results show that, if all other conditions are the same, there is a negligible effect of the density ratio on the drag coefficient of viscous spheres.

Coupling a local adaptive grid refinement technique with an interface sharpening scheme for the simulation of two-phase flow and free-surface flows using VOF methodology

“…For both of these cases the internal liquid circulation induced by the motion of droplet is fluctuating between droplet front and rear face (streamlines in Figure 13). The recirculation behind the droplet is well predicted, while its dimension match the ones given by Feng and Michaelides [55] (missing exact number). Overall HRIC+HSHARP delivers the best results, with a total of 1.36% relative error in terminal velocity prediction (in 700ms), which for the specific grid density (40 cells in Radius) is considered to be as a satisfactory solution.…”

“…In that cases HRIC with HSHARP algorithm is implemented using three different dense grids, which unfortunately are not sufficient enough to capture in an efficient way the boundary layer formed on the top of the droplet. Feng and Michaelides [55] mention that a grid density of 500 cells in droplet radius is enough to resolve with accuracy this formed boundary layer showing that the streamlines are 'hugging' the droplet. Therefore given the numerical grid densities we use, the predicted "jump" of streamlines on the top right of the droplet can be probably considered as artificial and resembles that of a flow separation over an airfoil.…”

Forced convection heat transfer from a sphere to non‐Newtonian power law fluids

“…In brief, it consists of the meshing of 0.01D near the sphere having 100 control volumes along the halfsphere surface. Although the boundary layer thickness changes with the power law index (n), the grid used in the present work is fine enough to resolve the boundary layer effects 34 for the entire range of parameters used. The grid used in the present work is much finer than that used in other previous studies (see, for example, Tripathi et al, 35 Tripathi and Chhabra, 36 and Graham and Jones 37 ).…”

Modelling of cavitation in diesel injector nozzles

“…The Basset force has not been considered here following the finding of Meyer et al (1992). For the drag force coefficient the approach of Feng & Michaelides (2001) has been followed; i.e. for Reynolds numbers in the range of 0 6 Re B 6 5 we have…”