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A Numerical and Experimental Investigation of Taylor Flow Instabilities in Narrow Gaps and Their Relationship to Turbulent Flow

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A NUMERICAL AND EXPERIMENTAL INVESTIGATION OF TAYLOR FLOW INSTABILITIES IN NARROW GAPS AND THEIR RELATIONSHIP TO TURBULENT FLOW IN BEARINGS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Dingfeng Deng August, 2007 A NUMERICAL AND EXPERIMENTAL INVESTIGATION OF TAYLOR FLOW INSTABILITIES IN NARROW GAPS AND THEIR RELATIONSHIP TO TURBULENT FLOW IN BEARINGS Dingfeng Deng Dissertation Approved: Accepted: _______________________________ _______________________________ Advisor Department Chair Dr. M. J. Braun Dr. C. Batur _______________________________ _______________________________ Committee Member Dean of the College Dr. J. Drummond Dr. G. K. Haritos _______________________________ _______________________________ Committee Member Dean of the Graduate School Dr. S. I. Hariharan Dr. G. R. Newkome _______________________________ _______________________________ Committee Member Date R. C. Hendricks _______________________________ Committee Member Dr. A. Povitsky _______________________________ Committee Member Dr. G. Young ii ABSTRACT The relationship between the onset of Taylor instability and appearance of what is commonly known as “turbulence” in narrow gaps between two cylinders is investigated. A question open to debate is whether the flow formations observed during Taylor instability regimes are, or are related to the actual “turbulence” as it is presently modeled in micro-scale clearance flows. This question is approached by considering the viscous fluid flow in narrow gaps between two cylinders with various eccentricity ratios. The computational engine is provided by CFD-ACE+, a commercial multi-physics software. The flow patterns, velocity profiles and torques on the outer cylinder are determined when the speed of the inner cylinder, clearance and eccentricity ratio are changed on a parametric basis. Calculations show that during the Taylor vortex regime velocity profiles in the radial direction are sinusoidal with pressure variations in the axial direction even for the case of the “long journal bearing” (L/D>2). For the concentric case ( ε = 0), both velocity and pressure profiles are axisymmetric and time-independent during the Taylor vortex regime. During the wavy vortex regime the radial velocities maintain their sinusoidal profiles, while pressure varies in both axial and circumferential directions. Both velocity and pressure profiles are non-axisymmetric and time-dependent. An order of magnitude analysis of the Navier-Stokes equation terms shows that the inertia, viscous, pressure and iii the Reynolds stress terms are equally significant during the transition regime (Taylor to wavy vortex regimes). Based on these findings, a new model for predicting the flow behavior in long and short journal bearing films in the transition regime is proposed. The new model indicates that the velocity profiles are sinusoidal and depend on the local Reynolds number (or Taylor number) and the position in the axial direction. Unlike the modified turbulent viscosity of the most accepted models (Constantinescu, Ng-Pan, Hirs and Gross et al.), the viscosity used in the new model is kept at its laminar value. A comparison is made between the results of this model and the four most accepted turbulence models mentioned above. Experimental torque measurements and flow visualization are performed for three kinds of oils with different viscosities. It is shown that in general there is a good agreement between the numerical and experimental torques except those in turbulent regime. The different flow patterns in the turbulent regime accounts for the difference between the numerical and experimental torques. Comparison between numerical and experimental flow patterns is also made and it shows that they match well in the Couette, Taylor and Wavy regimes. There is a discrepancy between numerical and experimental flow patterns in Pre-wavy regime. The reason is that the vorticity is strong enough to destroy the organized Taylor vortices but it is not strong enough to hold the particles in the center areas of the vortices during experiments in this regime. In general there is a good agreement between the numerical and experimental results including torque measurements and flow patterns. The new model for predicting the flow behavior in journal bearing films in the transition regime is justified. iv ACKNOWLEDGEMENTS I would like to express my appreciation and gratitude to my advisor, Dr. M.J. Braun, for his continuous and invaluable advisement. I am glad that I met Dr. Braun during my first semester of graduate school. Since then he has been my professional advisor, life mentor and personal friend. It is obvious that I have achieved my goals under the five years’ guidance of Dr. Braun. I would like to thank my committee members, Dr. Drummond, Dr. Hariharan, Mr. Hendrick, Dr. Povitsky and Dr. Young, for their comments and guidance during the course of my dissertation. I would also like to thank Dale Ertley for his work on building the installation for my experiments and Bob Shardy for his work on installing and trouble-shooting the CFD software package. Additionally I received unlimited support and courage from my friends and family. I would not have accomplished this without them. v TABLE OF CONTENTS Page LIST OF TABLES........................................................................................................xii LIST OF FIGURES. .....................................................................................................xiv NOMENCLATURE. ....................................................................................................xxiii CHAPTER I. THEORETICAL INTRODUCTION AND LITERATURE REVIEW........................1 1.1 Introduction.......................................................................................................1 1.2 Theories of Fluid Instability in Concentric Cylinders and Literature Review..............................................................................................4 1.2.1 Rayleigh’s Stability Criterion. ..............................................................4 1.2.2 Linear Theory of the Instability. ...........................................................8 1.2.3 Literature Review on Linear Theory.....................................................12 1.2.4 Weakly Nonlinear Theory.....................................................................15 1.2.5 Literature Review on Weakly Nonlinear Theory..................................18 1.2.6 Theory of Wavy Taylor Vortices and Literature Review. ....................20 1.3 Theories of Fluid Instability in Eccentric Cylinders and Literature Review..............................................................................................25 1.3.1 DiPrima’s Local Theory and Literature Review...................................25 1.3.2 DiPrima and Stuart’s Non-local Theory and Literature Review. .........31 1.4 Theories of Turbulence in Bearings and Literature Review.............................33 vi 1.4.1 Constantinescu’s Theory.......................................................................33 1.4.2 Ng-Pan Theory......................................................................................37 1.4.3 Bulk Flow Theory of Hirs.....................................................................39 1.4.4 Literature Review on Theories of Turbulence in Bearings...................42 II. SCOPE OF WORK. ....................................................................................................47 2.1 Numerical Simulations and Calculations of Long and Short Bearings. ...........47 2.2 The Nature of “Instability” & “Turbulence” in Small Gap Journal Bearings................................................................................................47 2.3 New Models for Transition Flow of Thin Films in Long and Short Bearings...................................................................................................48 2.4 Experimental Verification of the Numerical Results........................................49 III. NUMERICAL ALGORITHM...................................................................................50 3.1 General Introduction to CFD-ACE+.................................................................50 3.2 Numerical Methodology Adopted by CFD-ACE+...........................................50 3.2.1 Discretization. .......................................................................................51 3.2.2 Velocity-Pressure Coupling..................................................................55 3.2.3 Boundary Conditions. ...........................................................................57 3.2.4 Solution Methods..................................................................................58 3.3 Flow Module.....................................................................................................61 IV. NUMERICAL RESULTS (LONG BEARING)........................................................62 4.1 Introduction.......................................................................................................62 4.2 Geometry and Boundary Condition Applications.............................................63 4.2.1 Geometry...............................................................................................63 4.2.2 Boundary Condition Applications ........................................................67 vii 4.3 Convergence Criteria and Numerical Accuracy.
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