Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Abstract

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C d ≡ F/(ηU) = k 0 + k 2 Re 2 + .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K(U/a) n−1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k 0 is the drag coefficient for creeping flows. The proportionality constant k 2 depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k 2 on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 1. Creeping flows

“…The variation of the drag coefficient with n depends on the choice of viscosity scale in the drag coefficient, (14). We have used here simply the averaged fluid velocity and the cylinder radius in the viscosity scale, i.e., η = η a ≡ K(U/a) n−1 .…”

“…The main results of each simulation are the cell-averaged fluid velocity components, V i (cf. (14) below). Convergence of this quantity was tested for the case in which the pressure drop over the unit cell was specified to be the same in each direction, such that V 2 should be equal to V 1 .…”

“…Flows at finite Reynolds numbers are the subject of a companion paper [14] and we shall restrict ourselves here to low-Reynolds-number flows. The Reynolds number becomes independent of the velocity at n = 2, but we shall consider values of n not larger than 1·5.…”

“…The effects of inertia are the subject of a companion paper [14]. Previous work on these flows is very limited.…”

“…Knowing the velocity variances is also useful for other purposes. The drag coefficient can be expressed in terms of the cell-averaged fluid velocity (see (14) below) and any simple scaling for the drag coefficient must therefore be corroborated by even simpler results for the velocity variances. Also, Hill, Koch and Ladd [22] showed recently that for Newtonian fluids the added mass coefficient of a sphere in an array can be related with good approximation to the steady-state velocity variances (the corresponding problem for cylinders is addressed in Section 8).…”