International Journal of Rotating Machinery

Volume 2016 (2016), Article ID 8584067, 11 pages

http://dx.doi.org/10.1155/2016/8584067

## The Three-Dimensional Velocity Distribution of Wide Gap Taylor-Couette Flow Modelled by CFD

Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK

Received 9 October 2015; Revised 1 February 2016; Accepted 2 February 2016

Academic Editor: Ryoichi Samuel Amano

Copyright © 2016 David Shina Adebayo and Aldo Rona. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A numerical investigation is conducted for the flow between two concentric cylinders with a wide gap, relevant to bearing chamber applications. This wide gap configuration has received comparatively less attention than narrow gap journal bearing type geometries. The flow in the gap between an inner rotating cylinder and an outer stationary cylinder has been modelled as an incompressible flow using an implicit finite volume RANS scheme with the realisable model. The model flow is above the critical Taylor number at which axisymmetric counterrotating Taylor vortices are formed. The tangential velocity profiles at all axial locations are different from typical journal bearing applications, where the velocity profiles are quasilinear. The predicted results led to two significant findings of impact in rotating machinery operations. Firstly, the axial variation of the tangential velocity gradient induces an axially varying shear stress, resulting in local bands of enhanced work input to the working fluid. This is likely to cause unwanted heat transfer on the surface in high torque turbomachinery applications. Secondly, the radial inflow at the axial end-wall boundaries is likely to promote the transport of debris to the junction between the end-collar and the rotating cylinder, causing the build-up of fouling in the seal.

#### 1. Introduction

The understanding of the flow in the gap between concentric independently rotating cylinders is both of scientific and of practical interest for many engineering applications in rotating machinery. Specific examples include the lubricating flow between rotating shafts of turbopumps in rocket engines and of multispool turbofan engines and in the bearing housing of high [1] and low [2] bypass aircraft engines. Other areas of application are found in the bearing chambers of internal combustion aero-engines, rotating tube in tube heat exchangers, and the submerged pumps for water wells.

Lubrication is very important in turbomachineries where the inner cylinder (shaft) rotates and the outer cylinder (journal bearing) is stationary. In this application, the clearance is typically small enough, the lubricant is viscous enough, and the speeds are slow enough so that the flow is laminar. The flow is eccentric because the radial loading on the shaft reduces the bearing clearance on one side. With a properly designed bearing, the shaft, while turning, will not contact the bearing because the viscous shear force between the shaft and the lubricant carries the lubricant into this space. At high shaft speeds and high shaft loads, the laminar flow becomes first axially nonuniform and then nonaxisymmetric [3–9]. This is because the forces arising from viscosity are insufficient to overcome those associated with the fluid inertia. This transition increases the shaft torque significantly so that ball and or roller bearings are used in place of a journal bearing at these higher rotational speeds.

Many experimental investigations and numerical simulations have been conducted to understand the complexities of this flow. This activity dates back to 1888 and 1890, when Mallock [3, 4] and Couette [5] conducted independent experiments using concentric rotating cylinders. More recently, Liao et al. [8] conducted numerical simulations that reproduced three regimes of the Couette-Taylor system, namely, the steady circular Couette flow, the steady axisymmetric Taylor vortex flow, and the periodic spiral vortex flow. They validated their computational results using the experimental observations of Andereck et al. [9]. They concluded that this system exhibits a rich diversity of steady and chaotic flow patterns that are complex in nature and may arise as a result of small perturbations. These characteristics are typical expressions of hydrodynamic instabilities in the flow. Czarny et al. [10] performed a direct numerical simulation, using a three-dimensional spectral method, of a small axial length to diameter ratio annular flow driven by counterrotating cylinders. The numerical model predicted two different flow regimes, wavy vortices and interpenetrating spirals.

The flow enclosed between rotating coaxial cylinders is often characterised with respect to the Taylor number, , which expresses in nondimensional form the importance of the centripetal acceleration in a rotating flow relative to the viscous forces. In this study, where only the inner cylinder is rotating, the Taylor number Ta is defined aswhere is the radius ratio, and are the radii of the inner and the outer cylinders respectively, is the gap width, is the rotational speed of the inner cylinder, and is the fluid kinematic viscosity.

Many aspects of the flow developing between coaxial rotating cylinders are yet to be fully detailed. One of the advantages of 3D simulations over traditional experiments is the ability to investigate the salient features of the flow across the entire annulus on meridional, axial, and cascade planes. Whilst a 3D model is more demanding both in terms of its development time and of the computational resources, it has the potential to resolve the time-averaged three-dimensional motion of the localised flow disturbances induced by the rotation of the inner cylinder.

Adebayo and Rona [11, 12] measured by PIV the in-plane velocity between rotating cylinders at wide gap. These measurements were limited to the meridional plane where the PIV gave direct measurements of the in-plane velocity. Computational fluid dynamic (CFD) can overcome this limitation by estimating the full 3D velocity field as part of the flow field solution. The CFD model therefore enables quantifying and qualifying the important flow fields beyond the current limitations of conventional nonintrusive optics-based measurement techniques.

In this study, CFD is used to predict the flow pattern and examine in detail the velocity distributions both in the meridional and in the axial planes in a moderately wide gap setup. The accuracy of the predicted result is validated by comparing the velocity profiles from the CFD simulations to the PIV measurements by Adebayo and Rona [11, 12] in the meridional plane. Conclusions are drawn on the significant implication of these findings for high torque turbomachinery applications.

#### 2. Computation Domain and Flow Conditions

##### 2.1. Geometry

A three-dimensional (3D) numerical model is used to examine the velocity field flow in more than one plane. The model geometry is defined with respect to the coordinates shown in Figure 1. The cylinders are coaxial with the axis coinciding with the -direction of the cylindrical reference system . Two different coaxial assemblies, summarised in Table 1, are considered to allow a parametric study of the flow pattern in the annular region of the coaxial cylinders. The rotating speed of the inner cylinder is held constant at 52.36 rad/s in all test cases. The geometries modelled in this study were created using commercial CFD software GAMBIT 2.4.6.