Application of a Projection Method for Simulating Flow of a Shear-Thinning Fluid

Masoud Jabbari,Evan Mitsoulis,Jesper Henri Hattel,James Mcdonough

doi:10.3390/fluids4030124

Abstract

In this paper, a first-order projection method is used to solve the Navier–Stokes equations numerically for a time-dependent incompressible fluid inside a three-dimensional (3-D) lid-driven cavity. The flow structure in a cavity of aspect ratio δ = 1 and Reynolds numbers ( 100 , 400 , 1000 ) is compared with existing results to validate the code. We then apply the developed code to flow of a generalised Newtonian fluid with the well-known Ostwald–de Waele power-law model. Results show that, by decreasing n (further deviation from Newtonian behaviour) from 1 to 0.9, the peak values of the velocity decrease while the centre of the main vortex moves towards the upper right corner of the cavity. However, for n = 0.5 , the behaviour is reversed and the main vortex shifts back towards the centre of the cavity. We moreover demonstrate that, for the deeper cavities, δ = 2 , 4 , as the shear-thinning parameter n decreased the top-main vortex expands towards the bottom surface, and correspondingly the secondary flow becomes less pronounced in the plane perpendicular to the cavity lid.

Highlights

Flow inside a cubic cavity with a top lid moving at a constant velocity has been the centre of many fundamental fluid flow studies [1,2,3] as well as practical applications [4,5,6]
Steady two-dimensional (2-D) lid-driven cavity (LDC) flow has been studied in literature at different Reynolds number [12,16]—Re = Ulid L/ν, where Ulid is the lid velocity, L the cavity length and ν the kinematic viscosity of the fluid
It is seen that the comparison is excellent at all Reynolds numbers considered here

Summary

Introduction

Flow inside a cubic cavity with a top lid moving at a constant velocity has been the centre of many fundamental fluid flow studies [1,2,3] as well as practical applications [4,5,6]. Steady two-dimensional (2-D) LDC flow has been studied in literature at different Reynolds number [12,16]—Re = Ulid L/ν, where Ulid is the lid velocity, L the cavity length and ν the kinematic viscosity of the fluid. Albensoeder et al [17] reported that the fluid bifurcates from steady to periodic, and eventually to turbulent flow when it goes above the critical Reynolds number, Recr ∼ O 103. This is, dependent on cavity aspect ratios, i.e., length-to-width, λ = L/W, and depth-to-width, δ = H/L [18,19]

Methods

Results

Conclusion