Finite Element Analysis and Program for Stokes Fluid Flow

152 Pages·2016·4.96 MB·English

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Theses
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Jeffrey Steele
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Steele, Jeffrey, "Finite Element Analysis and Program for Stokes Fluid Flow" (1977). Thesis. Rochester
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FINITE ELEMENT ANALYS IS AND PROGP-AH
FOR STOKES FLUID FLOW
by
Jeffrey M. Steele
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Mechanical Engineering
Approved by: Prof. Martin P. Sherman
---------------------------
Illegible Signature
Prof. ---------N-e-vi-lle- -F.- R-i-eg-e-r- -------
Prof. Illegible Signature
--~(~D-e-p-a-r~t-m-e-n-t~H-e-a-d~)------
DEPARTMENT OF MECHANICAL ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
August, 1977
I
Y
ACKNOWLEDGEMENTS
The author would like to express his appreciation
to those who have assisted him in the development of the
Stokes flow finite element program.
To professor Neville P. Rieger, his thesis advisor,
for his guidance and interest during this project.
To Dr. Lowell Wilcox, Ron Auble and Ken Arnold of
the Gleason Works for their assistance in the element
derivation and programming.
To Paul G. Warner for the use of his notes on his
potential flow finite element program and automatic
mesh generation routine.
ABSTRACT
The theory and computer program for finite element
analysis of Stokes (creeping) fluid flow is presented.
The element is an eight node, two dimensional isoparametric
quadrillateral. The element is formulated directly
from the Stokes equations of motion, a special case
of the Navier-Stokes equations where inertia terms are
dropped, using the method of weighted residuals with
Galerkin's criterion applied. Velocities in two directions
are solved for at all eight nodes while pressure is
solved simultaneously at the four corner nodes only.
Several test cases have been run with the program.
Very good accuracy was achieved in modeling velocity
profiles for Couette and Poiseulle . flow and for pressure
distribution of two lubrication models. In these four
cases error was less than l\%.
Limitations have been pointed out for the types of
flow which may be successfully modeled. Flow between
two concentric cylinders had an error of 31% between the
finite element and exact solutions.
Cascades of cylinders and turbine blades are modeled
by modeling only one cylinder or blade. The cascade of
turbine blades serves to illustrate the potential uses
of this technique.
ii
TABLE OF CONTENTS
PAGE
List of Figures
v
List of Tables
viii
Nomenclature
ix
1.0 INTRODUCTION
2
2.0 LITERATURE SURVEY AND PRIOR WORK k
2.1 Stream and Potential Function k
2.2 Stream Function - Stokes Flow 11
2.3 Direct Formulation - Stokes Flow Ik
*
3.0 NAVIER STOKES EQUATIONS 17
4.0 DERIVATION OF THE STOKES FLOW ELEMENT 26
4.1 Discretization of the Domain 27
4.2 Two Derivation Approaches 29
4.3 Interpolation Functions 31
4.4 Stokes Equations in Finite Element 39
Matrix Form
5.0 ASSEMBLY OF ELEMENTS 1*6
6.0 SOLUTION OF NODAL VARIABLES 52
7.0 MODELING SS
8.0 EXAMPLE PROBLEMS 66
8.1 Couette Flow 69
8.2 Poiseulle Flow 7k
8.3 Flow Between Concentric Cylinders 80
8.4 Plane Slider 85
8.5 Stepped Slider 91
8.6 Flow Around a Cylinder 95
8.7 Flow Through Cascade of 106
Turbine Blades
1X1
PAGE
9.0 CONCLUSIONS u$
10.0 RECOMENDATIONS 116
H-O REFERENCES 117
12- APPENDIX A - INPUT INSTRUCTIONS 119
J-3.0 APPENDIX B - PROGRAM LISTING 122
ir
LIST OF FIGURES
Figure Title Page
2.1 Potential and Stream Function 7
Element
2.2 Martin's Flow Around A Cylinder g
2.3 Cascade of Joukowski Airfoils 12
2.4 Potential Flow Through Cascade 13
of Joukowski Airfoils
3.1 Forces on Fluid Particle 20
4.1 Finite Element Discretization 28
4.2 Shape Function Mapping 32
4.3 Shape-Interpolation-Weighting 33
Functions
4.4 Linear Element 35
4.5 Quadratic Element 36
5.1 Degrees of Freedom I4.7
7.1 Modeling Example 58
7.2 Nodal Loadings - Single Element 62
7.3 Nodal Loading - Multiple Elements 63
7.4 Nodal Relations 65
Figure Title Page
3,.1 Couette Flow - Model 70
8,.2 Couette Flow - Results 73
8..3 Poiseulle Flow - Model (5
8,.4 Poiseulle Flow - Finite Element 76
Model
8,.5 Poiseulle Flow - Results 79
8,.6 Flow Between Concentric Cylinders 8l
Model
8.7 Flow Between Concentric Cylinders 8I4.
-
Results
8.8 Plane Slider - Model 86
8.9 Plane Slider - Pressure Results 89
8.10 Plane Slider - Velocity Results 90
8.11 Stepped Slider - Model 92
8.12 Stepped Slider - Pressure Results 9i|.
3.13 Flow Around Cylinder 97
8.14 Flow Around Cylinder - No Circ. - 99
Potential Flow
8.15 Flow Around Cylinder - No. Circ. - 100
Stokes Flow
8.16 Flow Around Cylinder - Circ. - ioi
Stokes Flow
8.17 Flow Around Cylinder - Circ. = 2Tf -102
Stokes Flow
8.18 Flow Around Cylinder - Pressure 103
Distribution - Stokes Flow
vi
Figure Title Page
8.19 Cascade of Cylinders - Potential loij.
Flow
8,,20 Cascade of Cylinders - Stokes Flow 105
8,,21 Cascade of Turbine Blades - Stokes 107
Flow Model
8,.22 Cascade of Turbine Blades - Potential
Flow Model 108
8,.23 Cascade of Turbine Blades - Stokes no
Flow Results
8,.24 Cascade of Turbine Blades - Potential
Flow Results in
8,.25 Cascade of Turbine Blades - Stokes 112
Flow Pressure Distribution
8. Cascade of Turbine Blades Stokes 113
.26
Flow Composite Results
8 Cascade of Turbine Blades Potential i ii)
.27
Flow Composite Results
vii
LIST OF TABLES
Table Title Page
2.1 Types of Fluid Flow 5
4.1 Linear Shape Functions 35
4.2 Quadratic Shape Functions 36
8.1 Summary of Test Case Results 67
viii