algorithm is applied to solve the different symmetric hydrofoil NACA ..... Cavity pressure (water vapour pressure). Pâ .... two-dimensional flow past an obstacle with a wake or cavity formation, as shown .... models; the first one is the simple closure model, shown in figure (1.10). .... become important under cavitating condition.
Zagazig University Faculty of Engineering Dept. of Math. & Eng. Phys.
Boundary Element Method and Its Applications of Non-Linear Analysis of Flow around Hydrofoils
By
Norhan Alaa El Dain Mohamed B.Sc. Civil Engineering, Faculty of Engineering, Zagazig University (1999) A Thesis submitted in Partial Fulfillment For M.Sc. Degree In Engineering Mathematics Faculty of Engineering Zagazig University Under the Supervision of Prof. Said G. Ahmed Prof. of Engineering Math. Dept. of Math. & Eng. Phys. Faculty of Engineering Zagazig University Egypt
Assoc Prof. Ahmed F. Abdel Gawad Assoc. Prof. of Mechanical Power Eng. Dept. of Mech. Power Eng. Faculty of Engineering Zagazig University Egypt 2007
ﱠﺣِﻴﻢ ٌﻟﺮ ﺣﻤﻦِ ا ٌﻟﺮ ٌِﻪﻠﻟﺍ ا ِﺴﻢ ﺑ ِ ﱠ َ
َح ٌﺷﺮ ﱢ أ َب َ ر َﻗﺎل ٌ ﻟﻰِ ِى َﺪر ﺻ ﻳﺴﱢﺮِ ِى َﻣﺮ ﻟﻰ أ و َ َ ٌ ﻣﻦ ة ﻘﺪ ﻋ ﻞ ﻠ ﺣ أ و ً ُ َ ﱢ َ ُ ِ ﻬﻮا ﻘ ﻔ ﻧﻰ ﻳ ﺎ ﺴ ﻟ َ ْ َ ﱢَ ُ َﻗﻮِ ﻟﻰ ﺻﺪق ﻪﻠﻟﺍ اﻟﻌﻈﻴﻢ
ُورةُ َطه أية ٢٥إلى (٢٨ )س َ
Dedicated to: My dear parents……… My dear husband Amr……… My dear sons Alaa & Mustafa…….. And …………. To my great family………….
Acknowledgments First for most and always, I bow my head in gratitude to Great Allah, the merciful, the compassionate who gave me the ability to complete this work. I would like to express my deepest appreciation and sincere gratitude to Prof. Dr. Said G. Ahmed, Prof. of Engineering Mathematics, Faculty of Engineering, Zagazig University, for suggesting the subject, grateful help, valuable advice and continuous encouragement throughout the performance of this work. My cordial thanks and deepest appreciation to Prof. Dr. Ahmed F. Abdel Gawad, Assoc. Prof. of Mechanical Engineering, Faculty of Engineering, Zagazig University, for his continous help during all stages of this work. Supervisors and myself want to introduce special thanks to Prof. Dr. Mohamed A. El Shaer, Prof. and Head of Department of Eng. Physics and Mathematics for his kind help. Grateful thanks are extended to the staff of the Department of Eng. Physics and Mathematics, for their help during the course of study.
Abstract Cavitation is one of the most important problems in hydrodynamic applications which causes a noticeable deterioration of the machine performance. Thus, accurate prediction of cavitation is very important in estimating the hydrodynamic performance of pumps, marine propellers and high speed hydrofoils. For this reason, substantial efforts have been taken by many researchers to develop capabilities to predict the extent of cavitation for various types of geometries. Several researchers have successfully analyzed the cavitation phenomenon and its application. They used different analytical and numerical methods to determine the shape and the size of the cavity and to know the velocities and/ or pressure along the boundaries. Due to the complexity of the cavitation problem and due to the difficulty to obtain analytical solutions many researchers prefer numerical methods. The most popular numerical technique for analyzing cavitation problems is the boundary element method (BEM). Boundary element method has gained its popularity from its simplified nature and many other reasons that will be encountered through the thesis.
The boundary element method was used herein as a mathematical tool to solve the cavity flow around hydrofoils. Two major difficulties meet the researcher in this field of study. These difficulties are the determination of the cavity free surface and the potential at the leading point. In the present thesis, these difficulties are solved by a new suggested technique and it gives excellent results. Also, the new technique saves the time and effort needed by previous techniques. The algorithm is applied to solve the different symmetric hydrofoil NACA sections with studying the effect of three different parameters. Excellent agreement was obtained with the available existing results.
Contents 1
2
INTRODUTION AND LITERATURE REVIEW 1.1 Introduction…………………………………………….............
2
1.2 The nature of cavitation phenomenon…………………..............
3
1.3 Cavitation models………………………………………............
8
1.3.1 Riabouchinsky model………………………….............
9
1.3.2 Open-wake model………………………………...........
10
1.3.3 Reentrant-jet model………………………....................
12
1.3.4 Cusped-wake model……………………………...........
13
1.3.5 Batchelor wake model………………………………….
13
1.3.6 Vortex models…………………………………............
13
1.3.7 Lavrentiev wake model……………………...................
14
1.3.8 Displacement-thickness model…………………..…......
14
1.3.9 Other models……………………………………..........
16
1.4 Survey of some analytical methods………………………….....
17
1.5 Survey of some numerical methods…………………………....
19
1.6 Background of boundary element technique……………...........
24
1.7 Objective of the work………………………………………….
30
1.8 Structure of the work…………………………………………...
31
MATHEMATICAL MODELLING AND FORMULATION 2.1 Introduction………………………………………………….....
34
2.2 Mathematical formulation…………………………………….…
34
2.3 Boundary conditions for infinite domain problems……..……...
37
2.3.1 Kinematic boundary condition………………………....
37
2.3.2 Dynamic boundary condition…………………………..
38
2.4 Riabouchinsky model…………………………………...……..
41
2.4.1 Riabouchinsky model for vertical flat plate....................
41
2.4.2 Riabouchinsky model for oblique flat plate....................
44
2.4.3 Riabouchinsky model for wedge……………………….
45
2.5 Hydrofoil cavity model…………………………………………
46
3 BOUNDARY ELEMENT METHODS AND NUMERICAL ALGORITHMS 3.1 Introduction…………………………………………………....
52
3.2 The divergence theorem……………………………………….
53
3.3 Green’s second identity……………………………….............
54
3.4 Discretisation techniques……………………………..............
63
3.4.1 Constant element……………………………………....
64
3.4.2 Linear element………………………………………....
66
3.5 Numerical algorithms………………………………………......
70
3.5.1 Aitchison algorithm…………………………………....
70
3.5.2 Wrobel algorithm…………………………….............
72
3.5.3 Yas’ko algorithm……………………………………..
73
3.5.4 Ingber algorithm………………………………………
74
3.5.5 Kinnas and Fine algorithm…………………….............
74
3.5.6 Ahmed algorithm……………………………………...
75
3.6 New (present) numerical algorithm……………………….…
77
3.6.1 Discretization process ………………………………..
78
4
RESULTS AND DISCUSSION 4.1 Introduction………………………………………………….....
81
4.2 Re-introduce of some previous results…………………..…...
82
4.3
4.2.1 Flow over vertical flat plate…………………………...
82
4.2.2 Flow over wedge………………………………...........
87
New results………………………………………………….....
88
4.3.1 NACA-16006 Hydrofoil………………………………
89
4.3.2 NACA-0012 Hydrofoil………………………………..
102
4.3.3 NACA-0015 Hydrofoil………………………………..
114
4.3.4 NACA-0025 Hydrofoil………………………………..
129
4.3.5 Comparisons between hydrofoil sections……………..
140
4.3.6 Comparison with others ……………………………..
144
5 CONCLUSIONS AND FUTURE WORK 5.1
Conclusions ……………………………………...……………..
148
5.2 Suggestions for future work …………………………………...
150
REFERENCES…………………………………………………....
151
List of Figures 1.1
Classifications of cavitation ……………………………………
5
1.2
Cavity photos…………………………………………………..
6
1.3
Riabouchinsky model………………………………………...…
10
1.4
Two-dimensional wake formation…………………………..….
11
1.5
Open-wake model……………………………………………...
12
1.6
Reentrant-jet model………………………………………….....
12
1.7
Cusped-wake model……………………………………….…..
13
1.8
The Displacement-thickness model………………………….....
15
1.9
Ingber and Hailey model……………………………….…….…
15
1.10
Simple closure model…………………………………….…..
16
1.11 Short plate termination model…………………………….…….
17
2.1
The lengths of the element i…………………………………....
39
2.2
The Riabouchinsky model for vertical flat plate………….….…
41
2.3-a The Riabouchinsky model for vertical flat plate with complete domain…………………………………………….…
42
2.3-b The Riabouchinsky model for vertical flat plate with half domain……………………………………………………..
43
2.3-c The Riabouchinsky model for vertical flat plate with quarter domain…………………………………………..............
44
2.4
The Riabouchinsky model for oblique flat plate……………......
45
2.5
The Riabouchinsky model for wedge…………………………..
45
2.6
Partially cavitation on the two-dimensional hydrofoil……….....
46
3.1
The unit normal vector……………………………………….....
54
3.2
Domain of the cavitation around hydrofoil…………………….
56
3.3
Domain without a singular point……………………………….
57
3.4
Domain with source point on the boundary…………………...
61
3.5
Geometry of constant element………………………...……....
64
3.6
Geometry of linear element……………………………………
69
3.7
General cavity flow model……………………………….........
70
3.8
Riabouchinsky model in a finite domain……………………....
70
3.9
Region of solution………………………………………….....
71
3.10 Yas'ko problem………………………………………………..
73
3.11 Ahmed domain problem………………………………..
76
4.1
Domain of flow problem over vertical flat plate…………….....
82
4.2
The final free surface for three cavity length…………………..
85
4.3
The final free surface for five cavity length…………………....
86
4.4
Domain of flow problem over wedge………………………….
87
4.5
Profiles of different tested NACA sections……………………
88
4.6-a Scaled hydrofoil section NACA-16006 at U 1.0 m / s ………
89
4.6-b Non-scaled hydrofoil section NACA-16006 at U 1.0 m / s …
90
4.7
Non-scaled hydrofoil section NACA-16006 at U 0.5 m / s …
91
4.8
Non-scaled hydrofoil section NACA-16006 at U 2.0 m / s ...
92
4.9
Non-scaled hydrofoil section NACA-16006 at U 3.0 m / s ...
93
4.10 Non-scaled hydrofoil section NACA-16006 at 0.852 …….
94
4.11 Non-scaled hydrofoil section NACA-16006 at 1.26933 …..
95
4.12 Non-scaled hydrofoil section NACA-16006 at = 3o…………
96
4.13 Non-scaled hydrofoil section NACA-16006 at = 5o………...
97
4.14 Effect of free-stream velocity NACA-16006 (Non-scaled)…….
98
4.15 Effect of cavitation number NACA-16006 (Non-scaled)……....
99
4.16 Effect of angle of attack NACA-16006 (Non-scaled)…..……..
100
4.17 Non-Scaled hydrofoil section NACA-0012 at U 0.5 m / s …
102
4.18 Scaled hydrofoil section NACA-0012 at U 1.0 m / s ……...
103
4.19 Scaled hydrofoil section NACA-0012 at U 2.0 m / s ……...
104
4.20 Scaled hydrofoil section NACA-0012 at U 4.0 m / s ……....
105
4.21 Scaled hydrofoil section NACA-0012 at 0.825 …………...
106
4.22 Scaled hydrofoil section NACA-0012 at 1.1247 …….........
107
4.23 Scaled hydrofoil section NACA-0012 at = 3o………………
108
4.24 Non-scaled hydrofoil section NACA-0012 at = 5o………….
109
4.25 Effect of free-stream velocity NACA-0012 (Scaled)…..………
110
4.26 Effect of cavitation number NACA-0012 (Scaled)…..………..
111
4.27 Effect of angle of attack NACA-0012 (Scaled)…………..........
112
4.28 Non-scaled hydrofoil section NACA-0015 atU 0.25 m / s ...
114
4.29 Non-scaled hydrofoil section NACA-0015 at U 0.5 m / s …
115
4.30 Scaled hydrofoil section NACA-0015 at U 1.0 m / s ………
116
4.31 Scaled hydrofoil section NACA-0015 at U 2.0 m / s ………
117
4.32 Scaled hydrofoil section NACA-0015 at 0.67585 ………..
118
4.33 Scaled hydrofoil section NACA-0015 at 0.825 …………..
119
4.34 Non-scaled hydrofoil section NACA-0015 at 0.9774 ……
120
4.35 Scaled hydrofoil section NACA-0015 at = 6o………………
121
4.36 Non-scaled hydrofoil section NACA-0015 at = 3o……........
122
4.37 Scaled hydrofoil section NACA-0015 at = 3.5o…………….
123
4.38 Effect of free-stream velocity NACA-0015 (Scaled)…............
124
4.39 Effect of cavitation number NACA-0015 (Scaled)...…….........
125
4.40 Effect of angle of attack with cavity number 0.67585 NACA-0015 (Scaled)……...………………………………….
126
4.41 Effect of angle of attack cavitation number 0.9774 NACA-0015 (Scaled)…..……………………………………...
127
4.42 Scaled hydrofoil section NACA-0025 at U 0.5 m / s ……...
129
4.43 Scaled hydrofoil section NACA-0025 at U 1.0 m / s ……...
130
4.44 Scaled hydrofoil section NACA-0025 at U 2.0 m / s ……...
131
4.45 Scaled hydrofoil section NACA-0025 at 0.67585 ………..
132
4.46 Scaled hydrofoil section NACA-0025 at 0.825 …………..
133
4.47 Scaled hydrofoil section NACA-0025 at 0.9774 …………
134
4.48 Scaled hydrofoil section NACA-0025 at = 6o ……………...
135
4.49 Effect of free-stream velocity NACA-0025 (Scaled)..…………
136
4.50 Effect of cavitation number NACA-0025 (Scaled)……..……..
137
4.51 Effect of angle of attack NACA-0025 (Scaled)…..…………...
138
4.52 Comparison between the results of Shimizu et al. [85] and the present results for NACA-0015 hydrofoil……………………...
145
List of Tables 4.1
The calculations of three different examples of a vertical flat plate…………...…………………..........
84
4.2 Results of NACA-16006……………………………...
101
4.3 Results of NACA-0012……………………………….
113
4.4 Results of NACA-0015……………………………….
128
4.5 Results of NACA-0025……………………………….
139
4.6 Effect of ……………….…………………………....
141
4.7 Effect of …………………………………………….
142
4.8 Effect of U ………………………………………….
143
4.9 Comparison between the present results and those of [85] ………………………………………
146
5.1 The ranges of the control parameter ………………..
149
Nomenclature BEM
Boundary element method.
C
Chord length of the hydrofoil.
h s L
Thickness at the cavity trailing edge.
L
Cavity length.
=l
Dimensionless cavity length.
nˆ g
Outward unit normal vector to the surface.
pv pc
Cavity pressure (water vapour pressure).
P
Free-stream pressure.
q
Total velocity vector.
qc
Total cavity velocity.
qi
Velocity of the element i at the cavity surface
r
Distance between the field and source points.
S
Arc length along the cavity surface.
S
Unit vector tangent to the cavity surface.
sL
Total arc length along the cavity surface.
U = U
Free-stream velocity.
Acceleration due to gravity.
Greek
Flow angle of attack.
Fluid density.
Bernoulli's constant.
Wedge angle.
Inclination angle of the plate.
Control parameter.
very small radius.
Total potential.
in
Inflow velocity potential.
Perturbation potential.
i
Incident potential (perturbation potential of the element i at the cavity surface).
A
Perturbation potential at point A.
Boundary of the domain.
Solution domain.
Cavitation number.
Field point.
Source point.
CHAPTER (1) INTRODUTION AND LITERRATURE REVIEW
Chapter (1)
Introduction and Literature Review
CHAPTER (1) INTRODUTION AND LITERATURE REVIEW 1.1 Introduction The solution of a differential equation in a certain domain and satisfying a certain condition is referred to as boundary-value problem. If one or more of the boundary is not known and moving with time, the problem then is referred to as moving-boundary problem. Furthermore, if the governing equation is time independent as well as the boundary condition, then the problem referred to as free-boundary problem. All these types have practical, industrial and engineering applications. Phase change problem is a direct example of moving-boundary problem, meanwhile, cavitation problem is a direct example of free-boundary problem. Cavitation is a challenging hydrodynamic phenomenon both in physical science and engineering applications. In physical science, cavitation flows involve many complicated and interdisciplinary physical phenomena in which both physicists and applied mathematicians are interested [1]. In engineering applications, cavitation plays a major role in surface seagoing vessel design and operation, as well as in hydraulic equipment.
2
Chapter (1)
Introduction and Literature Review
Propellers, hydrofoil ships, hydraulic turbines and pumps may suffer from its consequences in many ways. All these devices encompass lifting surfaces and a liquid working medium. When cavitation occurs unexpectedly, severe decrease in performance is observed and in many cases, the machine itself suffers from damages. Systems that operate for long time intervals without maintenance, like a marine propeller, may be completely destroyed by such damages [2]. In addition, the presence of a cavitation region has a large effect on surface pressure distribution as well as the mass flow capability of internal flow devices operating in this environment. For these reasons, substantial efforts have been taken by many researchers to develop capabilities to predict the extent of cavitation for various types of geometries and flow situations [3].
1.2 The nature of cavitation phenomenon Cavitation is generally a two-phase, three-component flow. The two phases are most importantly involved, i.e., a liquid and its own vapor. However, in almost all real cases, at least a trace quantity of noncondensable gas such as air is also involved significantly in both bubble collapse and inception, but particularly inception. The non-liquid portion in general can be either in the form of quasi-fixed cavities or traveling bubbles. In quasi-fixed cavities (attached cavitation), the vapor region lies near a solid boundary, while traveling bubbles (vortex or
3
Chapter (1)
Introduction and Literature Review
traveling cavitation) refer to a bubble field completely surrounded by the parent liquid. To ease the research work, the quasi-fixed cavities are usually classified into three main groups. The first group is called the semi-fixed cavities (partially cavitation) that are usually found in cases of relatively low cavitation
P Pc number ( ) . The second group is known as fixed cavity (super1 2 U 2 cavitation) that is formed for very much lower cavitation number (σ). For sufficiently high values of σ, cavitation will never happen and this is known as the third group, see figure (1.1). Also, the cavitation photos of the three groups are shown in figure (1.2) [4]. The type of cavitation (partial or super) depends on the operating conditions (cavitation number, flow velocity, angle of attack, etc.) and the fluid properties as well as the profile of the hydrofoil section.
4
Chapter (1)
Introduction and Literature Review
Cavitation
Liquid
Non-liquid
Traveling bubbles
Quasifixed cavities
Semi-fixed cavities
Fixed cavities
Partially Cavitation
Super-cavitation
No cavities
Figure (1.1): Classifications of cavitation.
5
Chapter (1)
Introduction and Literature Review
(top is Partially Cavitation, middle is traveling bubbles, bottom is Super-cavitation) Figure (1.2): Cavity photos [4].
6
Chapter (1)
Introduction and Literature Review
In extreme cases, the flow regime will become that of super-cavitation, i.e., the cavity termination will be downstream of the cavitating body (hydrofoil, etc.). If the cavitation number, σ, is then raised from the super-cavitating condition, the cavity length will be reduced so that its termination point will move upstream, and then cavity attachment will occur on the body. A further increase in σ will cause the fixed cavity to disappear completely. However, after complete disappearance of the fixed cavity, there will still remain small cavitation bubbles in the region of minimum pressure. These generally will be entrained in the flow, and hence are traveling cavities. Even for those cavitation regimes dominated by a fixed or semi-fixed cavity, it is generally true that traveling bubbles exist in the interface region between the cavity and the main liquid flow. So, the collapse in the region of cavity termination of the traveling bubbles is presumed responsible for the cavitation damage observed in this region. The traveling bubbles begin to collapse under the influence of increasing external pressure. In some cases and to simplify the problem, the cavity was usually macroscopically treated as a single big bubble of finite length, which encompasses the whole region where cavitation occurs and micro-bubbles dominate. The pressure inside the cavity bubble is usually assumed to be constant [1].
7
Chapter (1)
Introduction and Literature Review
1.3 Cavitation models
The modeling of cavitating flows has taken two distinctly different approaches. The first approach is interface-tracking scheme, which presumes a distinct interface between liquid and vapor and solves governing equations on problems, which are characterized by a cavitated region of nearly fixed dimensions, and is not suitable for traveling cavitation problems. Typically, the interface is located by assuming the cavitated region is at a constant pressure equal to the local vapor pressure. Numerically, it is impossible to impose a constant pressure condition on the entire bubble because of the recovery of pressure at the end of the cavitation region. For this reason, interface-tracking scheme must employ a wake model to approximate the two-phase behavior in this complex area. Unfortunately, it is difficult to introduce physics into the wake model since the single-phase flow solver cannot address two-phase effects. Most authors agree that the choice of the wake model, within limits, has little influence to the force body of the cavity. Depending on the flow conditions and the extent of the wake, the interface tracking method can predict the overall behavior of cavitating flow fairly well. The second approach is a two-phase scheme, which solves the
governing equations in the entire liquid/vapor domain by introducing a pseudodensity, which can vary between liquid and vapor density extremes. This two-
8
Chapter (1)
Introduction and Literature Review
phase approach has the advantage that no special wake treatment is required but has the disadvantage in that the interface of gas and liquid is not determined and must be inferred from density values in neighboring cells. Also, it can treat traveling as well as attached cavitation. This technique can also be used in turbulent cavitation flows in which a bubble field (rather than a distinct interface) is present [3]. 1.3.1 Riabouchinsky model When a thin rigid plane is fixed in a fluid in motion, and normal to the direction of flow, it is observed that, near to its edges, the formation of a “free surface”, or surface of slip, which remains well defined for a certain distance from the plane. Experience showed that the pressure behind the plane is distributed in an approximately uniform manner over its surface, and is less than the pressure in the undisturbed current. Riabouchinsky [5] studied the case in which two planes are disposed, one behind the other, at right angles to the stream, and their edges connected by free surfaces. The velocity at the free surface is greater than the velocity of the stream. The two planes mutually attract each other, but the resultant pressure of the fluid on the system of the two planes is evidently zero, see figure (1.3).
9
Chapter (1)
Introduction and Literature Review
z'
k
C/
C
k/
z
Figure (1.3): Riabouchinsky model [5]. 1.3.2 Open-wake model This model was first introduced by Joukowsky [6] in a fundamental work which was apparently extended by McNown and Yih [7], Roshko [8], and Eppler for symmetric bodies in non-lifting flows. Wu [9] modified the model in order to extend it to lifting flows past arbitrary bodies. The wake model for the free-stream line theory is proposed to treat the two-dimensional flow past an obstacle with a wake or cavity formation, as shown figure (1.4) [10].
10
Chapter (1)
Introduction and Literature Review
Near-wake
Separation point
Laminar Farwake
Closure or Reattachment
Vortical or Turbulent Farwake
Figure (1.4): Two-dimensional wake formation [10]. In this model, the wake flow is approximately described, in the large, by an equivalent potential flow such that, along the wake boundary, the pressure first assumes a prescribed constant under-pressure in a region down stream of the separation points (called the near-wake) and then increases continuously from this under-pressure to the given free-stream value in an infinite wake strip of finite width (called the far-wake). Application of this wake model provides a rather smooth continuous transition of the hydrodynamic forces from the fully developed wake flow to the fully wetted flow as the wake disappears. When applied to the wake flow
11
Chapter (1)
Introduction and Literature Review
past an inclined flat plate, this model yields the exact solution in a closed form for the whole range of the wake under-pressure coefficient, see figure (1.5) [9]. c U P=P∞
A
P=Pc (α B
c'
I P increases from Pc to P∞ I
Figure (1.5): Open-wake model [9]. 1.3.3 Reentrant-jet model This model seems to have been first suggested by Prandtl and H. Wagner [11], later formulated by Kreisel [12] and independently by Efros [13]. In this model, the two dividing streamlines reverse direction, carrying a finite part of main stream into a jet, which flows through the cavity in an undetermined direction for lifting flows and is continued mathematically into a second Riemann sheet, see figure (1.6).
A
P=Pc
J
R B x
Figure (1.6): Reentrant- jet model [10].
12
Chapter (1)
Introduction and Literature Review
1.3.4 Cusped-wake model Brillouin [14] appears to have been the first to discuss the finite cavity, which is terminated with a cusp and especially the fact that the minimum pressure no longer falls on the cavity boundary. This model was applied to the circular cylinder by Southwell and Vaisey [15] and was taken as a realistic model for truncated airfoil by Lighthill [16], see figure (1.7) [10]. A I
o
c
I
B Figure (1.7): Cusped – wake model [10]. 1.3.5 Batchelor wake model With wakes at high Reynolds numbers in mind, Batchelor [17] proposed a closed wake model, which encloses a pair of stationary viscous, eddies of constant vorticity. Of particular concern was the uniform validity of possible wake models as the viscosity. An outstanding property of this model is that the resistance of the fluid on the body vanishes in this limit [10].
1.3.6 Vortex models Systems of stationary vortices are observed in the numerical calculations of wakes at low Reynolds number, for example, the early computations Thom [18] and Apelt [19] show a pair of stationary vortices behind a circular cylinder up to a Reynolds number of 40. For high Reynolds numbers, various
13
Chapter (1)
Introduction and Literature Review
kinds of twin vortices, such as the single-spiral and double-vortices, have been proposed as possible wake and cavity models by Tulin [20].
1.3.7 Lavrentiev wake model Lavrentiev [21] proposed a closed wake-bubble model, which terminates with a stagnation point at the closure and has a recalculating flow that encloses two separate regions of constant pressure. Its potential utility in the determination of viscous near-wake remains to be realized [10].
1.3.8 Displacement-thickness model This model was produced by Lemonnier and Rowe [22]. Their model aims to be flexible, simple and adjustable; flexible to allow it to be used under a variety of conditions, simple so that extrapolations to unsteady or threedimensional flows can be made and adjustable so that it can be made to fit experimental results. The model is suitable for complex flows such as: partial cavitation on a thin foil, cavitation behind a rounded body, consideration of the gravity effect in the case of an axisymmetrical body and simultaneous calculation of two cavities, see figure (1.8).
14
Chapter (1)
Introduction and Literature Review
L D
λL cavity
Potential flow Figure (1.8): The Displacement – thickness model [22]. Also, Ingber and Hailey [23] used a displacement thickness model to connect the cavity to the body in the reattachment zone. Their closure model is used not only to approximate the shape of the aft-portion of the cavity but also to determine the point of detachment that point, at which the streamline divided, the angle at which the dividing streamline makes with the attached flow and the cavity length, see figure (1.9). Cavity surface y
z U
α
blunt-ended body
blunt-ended body
Figure (1.9): Ingber and Hailey model [23].
15
Chapter (1)
Introduction and Literature Review
1.3.9 Other models Aside from the above mentioned models, Hopkinson considered the flow bounded by two plates and two free streamlines with interior sources and vortices. Artificial plates have been used to intercept free streamlines and to control the open wake width (Gadd) [24]. The one-side reentrant jet at the edge of a wedge or plate was treated by Cox and Clayden [25] and Cumberbatch [26]. Also, Wu and Chen [1] studied two different closure models; the first one is the simple closure model, shown in figure (1.10).
End point
Wake Foil
Cavity
Figure (1.10): Simple closure model [1]. The cavity shape and the tangential speed on the cavity surface are part of the solution to be determined. This is a purely computational model due to its simplicity in numerical implementation. In fact, from the mathematical point of view, a closed finite body with a constant pressure does not exist in exact potential flow theory. This kind of termination condition is contradictory to the constant pressure condition
16
Chapter (1)
Introduction and Literature Review
inside the cavity (or the constant tangential velocity condition on the cavity surface). The second model is the short plate termination model (or modified Riabouchinsky model). As shown in figure (1.11), the cavity ends at a plate normal to the incoming flow. In addition to the cavity shape and the tangential velocity, the height of the short plate is unknown and need be determined as the flow solution is sought.
Cavity surface
In Flow Wake Foil
Short plate θ=± π/2 Figure (1.11): Short plate termination model [1].
1.4 Survey of some analytical methods The analytical calculation of cavitating flows is based on two techniques: the small-perturbation theory proposed by Tulin, and the nonlinear theory that was originated from the work by Helmoltz [27], Kirchhoff
17
Chapter (1)
Introduction and Literature Review
[28] and Levi-civita [29]. Following the linear super cavitating flow theory, first introduced by Tulin [30, 31, 20]. Acosta [32] provided the first partial cavitation solution specifically for a flat plate hydrofoil. He also expressed the cavitation number explicitly in terms of the non-dimensional cavity length and the angle of attack. Of course, all these results are applicable only to flow at a small angle of attack as the assumptions of the linear theory imply. Geurst [33] employed a conformal transformation technique to derive an implicit theoretical relation among the angle of attack, cavitation number, and non-dimensional cavity length for a super-cavitating flow. The relation contains coefficients in integral form, which can be evaluated using a typical numerical quadrature procedure. Nevertheless, the cavity length can be readily obtained for a given cavitation number, and viseversa, provided that the incident angle of uniform flow is specified. Furthermore, he also studied the special case of a flat plate hydrofoil and obtained a simple analytical expression for these physical quantities. Davies [34] formulated the linearized cavitating hydrofoil problem in terms of singular integral equations with respect to unknown vorticity and source distributions, and inverted the resulting integral equations. Unfortunately, the final expressions for the vorticity and source distributions were coupled to each other. Persson [35] focused on a super-
18
Chapter (1)
Introduction and Literature Review
cavitating flat plate and inverted the equations to obtain analytical expressions for their distributions. Kinnas [36] extended the analytical inversion to super cavitating hydrofoils of arbitrary inversion to super-cavitating hydrofoils of arbitrary shape. He expressed the cavitation number, the vorticity and source distributions in terms of integrals of quantities, which depended only on the foil geometry and the cavity length [37]. The analytical calculation of cavitating flows using conformal transformation is difficult, and extension to three-dimensional flow is impossible outside the framework of perturbation methods. The most comprehensive analytical developments are probably those proposed by Furaya [38], whose non-linear theory dealt successively with the case of a super-cavitating section near a free surface, a super-cavitating foil near a free surface [39] and a partially or fully cavitating cascade [22]. Due to the complexity of the cavitation problem and due to the rare of analytical solution, many researchers prefer numerical methods.
1.5 Survey of some numerical methods Finite difference method proposed by Garadedian [40]. The solutions are obtained for cavities behind a disk and a sphere. He approached the axisymmetric case by successive corrections to the corresponding planar flow, each correction involving the solution of a linear mixed boundary value
19
Chapter (1)
Introduction and Literature Review
problem. In addition, Jeppson [41] successfully solved a number of plane and axisymmetric cavity flow problems and avoided the difficulty of dealing with the unknown location of the free surface by considering the velocity potential and stream function as independent variables, and the coordinates as dependent variables. Brennen [42] also used the finite difference method to obtain the solution for cavities behind a disk and a sphere, the Garadedian case, in different size of solid wall tunnel. Later, Mogel and Street [43] developed a numerical method for steady state cavity flows that provides a systematic correction of an initial, assumed free streamline position. The method used a finite difference representation with velocities as dependent variables and obtained a solution by successive over-relaxation. Boundary integral equation method was used in works of Amromin and Ivanov [44], Gyzevsky [45] and Kojouro [46]. In these works, the vortex surface distribution was used and resulting system of nonlinear algebraic equations was solved by the different numerical methods such as finite element method and boundary element method [47]. Aitchison [48] described a method for the numerical solution of planar and axisymmetric Riabouchinsky cavity flows. The problem is formulated as the minimization of a function over a variable region, which is solved numerically by the method of variable finite elements.
20
Chapter (1)
Introduction and Literature Review
However, several numerical difficulties are associated with domain numerical methods for cavitation problems, which limit their effectiveness. In particular, the cavitation problem is essentially an external flow problem causing discretization problems for either the finite difference method or the finite element method. Not only does the exterior domain need to be truncated, but also in many cases, a finer mesh is required near the separation point resulting in variable grid spacing. The most popular numerical technique for analyzing cavitation problems has been the boundary element method (BEM) because of the reduction of the gridding difficulties that the boundary element method affords. Several
researchers
have
successfully
analyzed
planar
and
axisymmetric cavitation problems using the boundary element method. Dagon [49], Aitchison and Karagearghis [50] who presented a boundary element method formulation for axisymmetric Riabouchinsky cavity flow problems, and Lemonnier and Rowe [22] who used the panel technique with minimization of a certain vector characterizing the discretion error which may become important under cavitating condition. Several practical examples were presented: partial cavitation on an isolated foil, cavitation behind a blunt ended body, and the problem of two cavities around an axisymmetrical body.
21
Chapter (1)
Introduction and Literature Review
Also, Worbel [51] presented a different boundary element method approach to the same problem that was presented by Aitchison and Karageorghis [50] in which it is demonstrated that the determination of the free surface-location is equivalent to the solution of a system of nonlinear equations. Ingber and Hailey [52] used the boundary element solution for cavitating flows about axisymmetric bodies both at zero and nonzero angle of attack. The nonzero angle of attack case results in a nonaxisymmetric cavity were presented. Kinnas and Fine [53] treated with the partially cavitating twodimensional hydrofoil problem using nonlinear theory by employing a low order potential based boundary element method. Also, they extended their work to treat the partially cavitating three-dimensional hydrofoil problems. Ahmed and Worbel [54] solved the planner cavity flow problem through the boundary integral method and a new algorithm depending on the correction of the free surface location iteratively. Also , Ahmed et al. [55] studied deeply cavitation for different and several practical cases of obstacle starting with vertical flat plate, oblique flat plate with a wide range of inclined angle from 0
л/2 and wedge of different angle of inclination.
22
Chapter (1)
Introduction and Literature Review
Yas’ko [47] presented a new numerical iterative algorithm based on direct boundary element method, which is proposed for the solution of the steady cavitating. Kinnas et al. [56] summarized some applications of boundary element method to predict sheet or developed tip vortex cavitation on lifting bodies using the nonlinear cavity theory. They applied Dirichlet type boundary condition on cavity surface and Neumann type on wetted body surface for two-dimensional and three-dimensional hydrofoils and submerged marine propellers in non-uniform inflow. Vaz et al. [57] developed a new boundary element method in which the elements are located on the foil surface and the boundary conditions for the cavity surface have been reformulated based upon a Taylor expansion. Kinnas et al. [58] studied the steady fully wetted and cavitating flows over hydrofoil. The non-linear analysis of the inviscid cavitating hydrofoil flow is based on a potential-based boundary element method. Mohamed et al. [59, 60& 61] used the boundary element method to solve the cavitation problem around two-dimensional hydrofoil with various NACA section and a new algorithm is obtained to determine the location of the free surface.
23
Chapter (1)
Introduction and Literature Review
1.6 Background of boundary element technique Engineers and physical scientists have in recent years become very conversant with numerical technique of analysis. These techniques are based on the approximate solution of an equation or set of equations describing a physical problem. The first widely known approximate method was finite difference, which approximate the governing equation of the problem by defining a series of nodes at which the discrete version of the differential equation is satisfied. Another approximate method is the finite element method, which has attracted the attention of the analysts largely due to its property of dividing the continuum into a series of elements, which can be associated with physical parts. In addition, in finite elements the differential equation is satisfied in an average sense over a region or element [62]. The boundary element method is firmly established as an important alternative technique to the prevailing numerical methods of analysis in continuum mechanics because unlike the domain discretization techniques such as finite element and finite difference methods, only the boundary needs to be discretized, leading to considerable savings in computer storage and running time [63]. Also, the boundary element method can now be used especially when the better accuracy wanted or where the domain extends to infinity because of
24
Chapter (1)
Introduction and Literature Review
its important property that it only needs discretization of the surface rather than the volume. So, it requires less meshing than finite element analysis and thus it is comparatively faster in generating or refining the mesh [64]. This advantage is particularly important for designing as the process usually involves a series of modifications which are difficult to carry out using finite elements. Moreover, boundary element method doesn’t require a complete remeshing when design changes. This technique consists in transformation of the partial differential equation describing the behavior of the unknown inside and on the boundary of the domain into an integral equation relating only boundary values, and then finding out the numerical solution of this equation. If values at internal points are required, they are calculated afterwards from the boundary data [65]. Since all numerical approximations take place only at the boundaries, one reduces the dimensionality of the problem and a smaller system of equations obtained in comparison with those achieved through differential methods. Historically, the application of integral equation to formulate the fundamental boundary-value problems of potential theory dates back to 1903 when Fredholm [66] demonstrated the existence of solutions to such equations.
25
Chapter (1)
Introduction and Literature Review
Due to the difficulty of finding analytical solution, the use of integral equation has, to a great extent, been limited to theoretical investigations of existence and uniqueness to solutions of problems of mathematical physics. However, the advent of high-speed computers made it possible to implement discretization procedures analytically and enabled numerical solutions to be readily achieved. Fredholm integral equations followed from the representation of harmonic potentials by single-layer or double-layer potentials and set up the foundations of the so-called indirect boundary element method. Vector integral equations analogous to the Fredholm integral equations of potential theory were introduced by Kupradze in context of the theory of elasticity [62]. Integral equations for linear problems can alternatively be formulated through the application of Green’s third identity, which represents a harmonic function as the superposition of a single-layer and a double-layer potential. Taking the field point to the boundary, an integral equation relating only boundary values and normal derivatives of the harmonic function is obtained [62]. The Green’s function method of solving boundary-value problems is most directly applicable to elliptic partial differential equations. In fact, the concept of Green’s function grew out of a detailed study of such boundary-
26
Chapter (1)
Introduction and Literature Review
value problems, but the method can also be extended to solve parabolic and hyperbolic partial differential equations. Jaswon and Symm [67, 68] presented a numerical technique to solve Fredholm boundary integral equations. The technique consists of discretizing the boundary into a series of small segments (elements). Assuming that the source density remains constant within each segment, they obtained accurate solutions for simple two-dimensional Neumann (all conditions are natural) and Dirichlet (all conditions are essential) problems. They also proposed a more general numerical formulation for solving Cauchy boundary-value problems (also called mixed because the boundary conditions are essential on parts of the boundary and natural on other parts) through the application of Green’s third identity, which yields a boundary integral equation where boundary values and normal derivatives of the physical variable play the role of the fictitious source densities. Results using this formulation are reported by Symm, Jaswon and Ponter [69]. Efforts such as the pioneering work by Hess and Smith [70] remained as special cases rather than being interpreted as a way of generating a whole new method of solutions for general engineering problems. Nevertheless, Hess and Smith [70] developed many powerful programs for the solution of Laplace type boundary-value problems, which were applied to potential flow and arbitrary bodies, using what is now called the indirect boundary element
27
Chapter (1)
Introduction and Literature Review
technique. They extended their formulation to analyze the three-dimensional objects as well as two-dimensional and their codes are still popular in aerodynamics. The boundary element method is now used widely as an important technique in one of the most important application such as temperature diffusion, some types of fluid flow motion, flow in porous media, electrostatic and many other applications which their equation can be written as a function of a potential and governed by the Laplace, Poisson or Helmholtz equation in two-dimensional, axisymmetric or fully threedimensional problems. All these cases are potential problems and they can generally be efficiently and economically analyzed using boundary elements [62]. The boundary element method succeeds in solving many engineering application, specially the cavitation problem on the hydrofoils. This problem was treated by many researchers, for example, Widnall treated the case of supercavitating foil in unsteady flow using the surface-lifting method. Nishiyama and Miyamoto treated the problem of a supercavitating foil placed under a free surface [22]. Hough and Moran [71], and Plotkin [72] used the thin-foil approximation with linearized free surface condition. Tsen and Guilbaud studied the effect of the plan shape of a superventilated foil from both experimental and theoretical standpoints [22]. In
28
Chapter (1)
Introduction and Literature Review
(1979), Verron extended Tsen and Guilbaud’s method to the case of basevented foils. In his work the plane shape of the cavity is calculated for nonzero cavitation numbers [22]. Yeung and Bouger [73] dealt with thick-foil methods, which provided a precise representation of the flow near the hydrofoil surface. Yamaguchi and Kato presented methods well suited to the case of foils having a large relative thickness [22, 74]. Kinnas and Fine [75] proposed a nonlinear analysis of the flow around partially or super-cavitating hydrofoils using a potential-based panel method. They stated that their method is very promising as it converges to the final cavity shape quicker than the other numerical schemes that use surfacevorticity-velocity-based panel methods. Xie and He [76] conducted numerical investigations on submerged cavitating slender axisymmetric bodies with gravity effect. They used the boundary element method to solve the potential flow problem. They also developed a new iteration method named Adaptive Modified Newton iteration to determine the cavity shape. Bal
[74]
described
a
potential-based
panel
method
for
the
hydrodynamic analysis of two-dimensional hydrofoils moving beneath the free-surface with constant speed without considering cavitation.
29
Chapter (1)
Introduction and Literature Review
Berntsen et al. [77] presented numerical and experimental work to investigate the effect of cavitation on the dynamic features of hydraulic machinery. The numerical calculations were done with the commercial code fluent V. 5. Their results showed that cavitation induces large fluctuations in lift on hydrofoils. Dular et al. [78] made a study of visual and erosion effects of cavitation on simple single hydrofoil configurations in a cavitation tunnel. They used a two-dimensional hydrofoil with circular leading edge for the experiments. They established a relation between characteristics of cavitation structures and cavitation damage. Wu and Chen [1] studied the effects of different cavity closure models on the development sheet supercavitation of two-dimensional hydrofoil. The study was carried out under the assumption of potential flow for which a cavity closure model was needed. A potential-based boundary element method has been developed for this purpose. The models, which they employed, included the constant velocity model and the short plate termination model.
1.7 Objective of the work The present work deals with the partially cavitating flow over twodimensional hydrofoils. The objective of the present work is to overcome the main difficulties which usually appear during the determination of the free
30
Chapter (1)
Introduction and Literature Review
surface around the hydrofoils. These main difficulties are, the free-surface location is not known in advance and the potential at leading point is not known also. Thus, the boundary element method with a new proposed algorithm is used to overcome these difficulties.
1.8 Structure of the present work The contents of this work are presented in five chapters. Chapter one includes an introduction and a literature review which is consisted of two parts; part one concentrats on the cavitation problems and the different models that deal with this phenomenon by using the analytical or numerical methods; the second part concerned with the boundary element method and its application of the cavitation problems. In chapter two, the mathematical formulation of the cavitation problem is presented. This chapter discusses the boundary conditions for infinite domain problem, which are dealt with two different models, the Riabouchinsky model with three cases; the vertical flat plate, the oblique flat plate and the wedge. The second model is the hydrofoil cavity model and the boundary conditions which should be applied. Chapter three focuses on the boundary element formulation and its derivation to the flow potential problems. In chapter three, it is described in detail the different numerical algorithms which delt with the cavitation problems.
31
Chapter (1)
Introduction and Literature Review
In chapter four, the results of the numerical examples are presented. Four different NACA hydrofoil sections are solved with three different operating parameters. The effect of these parameters and the profile of NACA section are discussed in this chapter. Chapter five presents the conclusions about the solution of the cavitation problem on the hydrofoil and the results which are presented in the chapter four. Also, suggestions for future research work are included.
32
CHAPTER (2) MATHEMATICAL MODELLING AND FORMULATION
Chapter (2)
Mathematical modeling and Formulation
CHAPTER (2) MATHEMATICAL MODELLING AND FORMULATION 2.1 Introduction Cavitation around hydrofoils belong to free boundary problem in which a set of partial differential equations with associated boundary conditions must be solved within domain in which one or more of its boundaries moves freely. This boundary is called free surface, its location is not known in advance and should be found as a major part of the solution of the problem. In the present chapter, different models describing cavitation flow will be presented. In addition to the mathematical formulation, some important models describing the flow should also be introduced.
2.2 Mathematical formulation For the steady, irrotational, planar flow of an incompressible, inviscid fluid past an obstacle and infinite domain problems, the following equations apply: (1)
Continuity equation:
u v 0 x y
(2.1)
34
Chapter (2)
(2)
Mathematical modeling and Formulation
Irrotationality equation:
u v 0 y x (3) P
(2.2)
Bernoulli’s equation: 1 2 q gy 2
(2.3)
In equation (2.1), u and v are the horizontal and vertical components of the velocity whose magnitude is given by the following formula:
q u 2 v 2
(2.4)
Where: x&y are the horizontal and vertical coordinates of a fixed system. P
is the pressure.
is the fluid density.
g
is the acceleration due to gravity.
is Bernoulli’s constant.
If the effects of gravity are neglected, equation (2.3) takes the following form: P
1 2 q const . 2
(2.5)
The pressure is taken to be constant in the cavity. Denoting quantities referenced to the cavity or free-stream line by the subscript c, then:
35
Chapter (2)
P
Mathematical modeling and Formulation
1 1 U 2 Pc qc2 const . 2 2
(2.6)
Where:
P
is the flow pressure at infinity.
U
is the inflow velocity at infinity.
Pc
is the pressure in the cavity.
qc
is the velocity on the cavity free-surface.
The Prandtl cavitation number σ is defined as [43]: P P c 1 U 2
(2.7)
2
From (2.6) and (2.7), one can get the following formula:
qc2
U 2
1
(2.8)
The velocity potential Φ can be introduced such that: u
x
(2.9)
v
y
(2.10)
Substituting (2.9) and (2.10) into the continuity equation leads to:
2 x
2
2 y
2
0
(2.11)
36
Chapter (2)
Mathematical modeling and Formulation
It can be seen that the potential given by (2.9) and (2.10) identically satisfies the irrotationality equation. The total potential and perturbation potential are related as follows:
(x , y ) (x , y ) in (x , y )
(2.12)
Where, the inflow velocity potential in corresponds to the uniform inflow velocity of magnitude U at an angle of attack of the fluid with the obstacle [75] as follows: in (x , y ) U (x cos y sin )
(2.13)
In addition, the perturbation potential will satisfy Laplace’s equation in the flow domain outside the cavity and the obstacle: 2 0
(2.14)
2.3 Boundary conditions for infinite domain problems When the obstacle placed in an infinite length and infinite depth of a channel (an infinite domain), the flow separates of the obstacle and makes a cavity flow. Then, the boundary conditions must be known on the different surfaces to solve the cavity problem.
2.3.1 Kinematic boundary condition
According to this condition, the flow is required to be tangent to the wetted surface as well as to the cavity surface. The kinematic boundary condition is given by:
37
Chapter (2)
Mathematical modeling and Formulation
0 n
(2.15)
Where, n is the outward unit normal at any point on the wetted surface or the free-surface. Substituting from equation (2.12) into (2.15), one can get: in n n
(2.16)
Apply equation (2.16) on the wetted surfaces and the cavity surface, the right hand side of equation (2.16) can be written as: in U (n x cos n y sin ) n
(2.17)
Where, n x and n y are the components of n in the x and y direction, respectively, for each element of the free-surface [75]. Substituting from equation (2.16) into (2.17), then equation (2.16) takes the following form: U (n x cos n y sin ) U nˆ n
(2.18)
2.3.2 Dynamic boundary condition This condition is applied on the cavity surface. Simply, it means that the
pressure is required to be constant at each point of the free-surface and equals to Pc . Also, the magnitude of the total velocity on the cavity surface, qc , is found to be constant, where:
38
Chapter (2)
Mathematical modeling and Formulation
qc U 1
(2.19)
Then, the dynamic condition on the cavity surface is given by: qc s
(2.20)
In equation (2.20), s denotes the curvilinear length from the plate tip (A) on the cavity free-surface, see figure (2.1). y i
si s
wi
A zi x
Figure (2.1): The lengths of the element i.
Substituting of equation (2.12) into (2.20), then: in qc s s
(2.21)
Or, in qc s s
(2.22)
Integrate both sides with respect to s, one can get:
sAi qc s i in sAi
(2.23)
Or, in some details:
39
Chapter (2)
Mathematical modeling and Formulation
s i A qc s i in (s i ) in (A )
(2.24)
Where, s i is the curvilinear length from the plate tip (A) to the node of each element. From equation (2.13), the in over the curvilinear distance s i is: in (s i ) U (x s cos y s sin ) i i
(2.25)
Replacing both horizontal and vertical components of s i , then equation (2.24) takes the new form: in (s i ) U (z i cos w i sin )
(2.26)
Where:
zi
is the horizontal distance from the point (A) to the node of the element i.
wi
is the vertical distance from the point (A) to the node of the element i.
Therefore; the dynamic condition takes the following form:
s i A qc s i U (z i cos w i sin ) in (A )
(2.27)
But, in (A ) 0 , then:
s i A qc s i U (z i cos w i sin )
(2.28)
Also, equation (2.28) can be written as:
U (z i cos w i sin ) A qi i si Where [75]:
qi
is the velocity of the element i at the cavity surface.
40
(2.29)
Chapter (2)
Mathematical modeling and Formulation
i
is the perturbation potential of the element i at the cavity surface.
A
is the perturbation potential at point A. By equation (2.29), a complete mathematical formulation for cavitation
flow problem was introduced. In the next section, a survey of previous models
suggested by different researchers will be presented.
2.4 The Riabouchinsky model This model was proposed by Riabouchinsky [5]. The model assumed that an image plate can be placed in the flow at some point downstream from the original plate and that the flow geometry will be symmetric about the line midway between the plates. 2.4.1 The Riabouchinsky model for vertical flat plate
In this case, the domain over which Laplace's equation should be satisfied may be one of the following three cases; as shown in figure (2.2).
A
A´
A
A o
B
B´
(a) complete domain.
B
(b) half-domain.
(c) quarter-domain.
Figure (2.2): The Riabouchinsky model for vertical flat plate.
41
Chapter (2)
Mathematical modeling and Formulation
Without details, the governing equation with the associated boundary condition for the three cases of domain will be as follows:
1.
Complete domain
Laplace's equation given by (2.14) should be solved over the domain shown in figure (2.2-a) with the kinematic boundary condition given by equation (2.18) taking into consideration that the component n y equals zero, only on the flat plates. Over the cavity surfaces the dynamic boundary condition given by equation (2.29) will be applied, see figure (2.3-a).
KBC:
U nˆ n
U (z i cos w i sin ) A DBC: q i i si KBC:
U nˆ n
A
A'
B
B'
Inflow
Figure (2.3-a): The Riabouchinsky model for vertical flat plate
with complete domain.
42
Chapter (2)
2.
Mathematical modeling and Formulation
Half - domain
Laplace's equation given by equation (2.14) with the same boundary condition is solved and the only difference appears here is the boundary condition at the vertical centerline
0 , see figure (2.3-b). n
KBC:
U nˆ n
U (z i cos w i sin ) A DBC: q i i si KBC:
U nˆ n
0 n
A
Inflow B Figure (2.3-b): The Riabouchinsky model for vertical flat plate
with half domain. 3.
Quarter - domain
The governing equation is the same as the two previous cases except on the horizontal centerline
0 , see 0 and on the vertical centerline x y
figure (2.3-c).
43
Chapter (2)
Mathematical modeling and Formulation
KBC:
U nˆ n
U (z i cos w i sin ) A DBC: q i i si KBC:
U nˆ n
A
Inflow
o
0 x
0 y Figure (2.3-c): The Riabouchinsky model for vertical flat plate
with quarter domain.
2.4.2 The Riabouchinsky model for oblique flat plate
In this case, a major difference will appear, that is two free surfaces encountered called upper and lower free-surfaces. Their final shape depends mainly on the angle of inclination of the flow and other parameters. The domain may be one of the two cases shown in figure (2.4). The boundary condition should take angle of attack and inclination of the plate in the calculation and also the iterative scheme will be modified to track the two free-surfaces simultaneously.
44
Chapter (2)
Mathematical modeling and Formulation
A
A´
) (
A
)
B
) (
B'
B
(a) complete domain.
(b) half-domain.
Figure (2.4): The Riabouchinsky model for oblique flat plate. 2.4.3 The Riabouchinsky model for wedge
In this case, the domain consists of two parts. The first one is the wedge, called wetted body and the second one is the cavity bounded by the wedge and the similar free-surface, as shown in figure (2.5).
A
A'
A
A
B
B'
(a) complete domain.
o
B
(b) half-domain.
(c) quarter-domain.
Figure (2.5): The Riabouchinsky model for wedge.
The governing equation and boundary conditions for the three cases of the domain will be as follows:
45
Chapter (2)
Mathematical modeling and Formulation
Complete domain
Laplace's equation given by equation (2.14) was applied as a governing equation, with the kinematic boundary condition given by equation (2.18) was applied on the wedges and the cavity surfaces. Also, over the cavity surface the dynamic boundary condition given by equation (2.29) was applied. The information in the previous case of the Riabouchinsky model for vertical flat plate with half and quarter domain, respectively, will be taken into account if half or quarter domain of this case is used.
2.5 The Hydrofoil cavity model The partially cavitating hydrofoil with chord length c 1 subjected to a uniform inflow U at an angle of attack and with ambient pressure P is considered, as shown in figure (2.6).
Y
l
l
sc D
T
sf
n L
X
U
)α c
Figure (2.6): partially cavitation on the two–dimensional hydrofoil [53].
46
Chapter (2)
Mathematical modeling and Formulation
The cavity detaches from the foil surface at point D on the suction side of the foil and rejoins the foil at point L . The point D is kept fixed and unless otherwise mentioned, it will coincide with the foil leading edge. It is assumed that the fluid is inviscid, incompressible and that the flow is irrotational. The total velocity flow field q can be expressed in terms of either the total potential, , or the perturbation potential, , as follows [79]:
q U
(2.30)
The total and perturbation potentials are related as in equation (2.12), where the inflow velocity potential in corresponds to the uniform inflow in equation (2.13). The perturbation potential will satisfy Laplace’s equation (2.14) in the domain outside the hydrofoil and inside the cavity. Further boundary conditions should be applied as follows: Kinematic boundary condition
Over the wetted foil and the cavity free-surface, equation (2.18) should be applied. Dynamic boundary condition
The pressure is required to be constant on the cavity surface and equal to
Pc . Applying Bernoulli’s equation given by (2.3) and the definition of the cavitation number (2.7), then, the magnitude of the total velocity on the cavity surface, qc , is found to be constant as in equation (2.19).
47
Chapter (2)
Mathematical modeling and Formulation
Kutta condition This condition requires finite velocities at the foil trailing edge.
(2.31)
Condition at infinity At infinity, the perturbation velocities should go to zero.
0
(2.32)
Cavity termination model
A termination model must be applied at the end of the cavity. A pressure recovery termination model is employed, by which the velocity in a transition zone of length λl as shown in figure (2.6) departs from its constant value on the cavity between D and T according to a prescribed algebric law in the transition zone between T and L :
qtr U 1 (1 f (s f ))
(2.33)
Where f (s f ) is defined as follows [53]:
0 f (s f ) s s A ( f T )v s L sT
if s f sT
(2.34)
if sT s f s L
Where s f is the arc-length of the foil beneath the cavity, measured from the cavity leading edge, and A (0 A 1) and v (v 0) are arbitrary constants [53]. In addition to the above boundary conditions, it is assumed that the cavity height vanishes at its trailing edges, i.e;
48
Chapter (2)
Mathematical modeling and Formulation
h (s L ) 0
(2.35)
On the wetted part of the foil that the value of
is known and given by the n
kinematic boundary condition (2.18). A general expression for the velocity on the cavity including the transition zone is given by the following algebraic law: in qc (1 f (s f )) sc sc
(2.36)
Where, sc (see in figure (2.6)) is the arc length of the cavity. The expression for
on the cavity can be found by integrating (2.36):
(sc ) (0) in (sc ) in (0) qc
sc
(1 f
(s f )) dsc
(2.37)
0
Also, equation (2.36) can be re-written as:
(sc ) (0)
sc
( U S c qc (1 f (s f )) dsc
(2.38)
0
Where, (0) is the potential at the leading edge of the cavity. In the numerical scheme, (0) is expressed in terms of the unkown potentials on the wetted part of the foil in front of the cavity. Also, the cavity shape is updated by an amount
h (sc ) applied normal to the cavity surface, which is determined from integrating the following ordinary differential equation: qc (1 f (s f ))
dh U nˆ ds n
(2.39)
49
Chapter (2)
Mathematical modeling and Formulation
Then, for a given cavitation number, the thickness of the cavity, , can be written as follows [56]: sc 1 (sc ; ) h (sc ) (U nˆ )dsc n qc (1 f (s f )) 0
(2.40)
Where, h (sc ) is the cavity height and it represents the amount by which the cavity surface is updated. The cavity closure condition at the cavity end, , will be as follows [53]:
(l ; ) 0
(2.41)
In the next chapter, a complete boundary element formulation is developed. Then, some of the most important numerical algorithms are also introduced followed by a new algorithm developed to overcome the difficulty appears when estimating the potential (A ) .
50
CHAPTER (3) BOUNDARY ELEMENT METHODS AND NUMERICAL ALGORITHMS
Chapter (3)
Boundary Element Methods
CHAPTER (3) BOUNDARY ELEMENT METHODS AND NUMERICAL ALGORITHMS 3.1 Introduction Since 1970, the boundary element methods started solving a wide range of practical, engineering and industrial problems. Simply, the boundary element methods transfer the governing equation and its associated boundary conditions to a boundary integral equation. This equation is valid for the boundary in addition to the internal points of the domain. The basic idea to derive the integral equation is to minimize the error and dealing only with the boundaries. There are two directions when deriving the integral equation, the direct and indirect approaches. In the present thesis, the first approach will be used for the derivation of the integral equation. The direct approach of formulation consists of transforming the partial differential equation describing the behavior of the unknown inside and on the boundary of the domain into an integral equation relating only boundary values, and then finding out the numerical solution of the integral equation. Before deriving the boundary integral equation, it is important to make review of some basic theorems that play an important role in the derivation of the integral equation. It is sensible to start with the divergence theorem [80] in
52
Chapter (3)
Boundary Element Methods
vector calculus, which equates an integral over the volume V to an integral over the surface S.
3.2 The divergence theorem Define a vector quantity F in its components in the x , y and z directions as follows:
F Fx i Fy j Fz k
(3.1)
In which, i , j and k are the unit vectors in the directions x , y and z , respectively. The divergence theorem states that: F F x y Fz x y z V
dv F n dS S
53
(3.2)
Chapter (3)
Boundary Element Methods
n n
n z
y
n
Z
x
y x Figure (3.1): The unit normal vector. The vector n can be written as:
n nx i n y j nz k
(3.3)
The left hand side of equation (3.2) is called the divergence of F and written as F , therefore equation (3.2) can be re-written as follows:
F dv V
F n dS
(3.4)
S
3.3 Green’s second identity One of the most important mathematical tools that mostly used when deriving the boundary integral equation is the Green’s second identity. Consider two arbitrary functions and * continuous at all points of the volume V and both have values which depend on the position in the volume. Consider the following product [81]:
54
Chapter (3)
Boundary Element Methods
F *
(3.5)
Therefore, the divergence of the function F is given by: F 2 * *
(3.6)
Applying the divergence theorem to the product * , leads to:
2 * * V
dv *n dS
(3.7)
S
Again, apply the divergence theorem to * , leads to:
* 2 * V
dv * n dS
(3.8)
S
Subtract equation (3.7) from equation (3.8), leads to:
* 2 2 * V
dv * * n dS
(3.9)
S
Equation (3.9) is the starting point for the derivation of the boundary integral equation for potential flow applications.
Now, it is the time to start derivation of the boundary integral equation corresponding to the flow potential problems. Consider a domain , between the upper surface of the hydrofoil and the free-surface, see figure (3.2):
55
Chapter (3)
Boundary Element Methods
n
Cavity
S q
Hydrofoil
c SL
U l
Figure (3.2): Domain of the cavitation around hydrofoil. Over this domain, Laplace’s equation is well defined with different associate boundary conditions [82]. The starting point of the formulation is Green's second identity:
* * * 2 d n n d
2 *
(3.10)
In equation (3.10), is a bounded two-dimensional region and is its closed boundary curve. The functions and * must be differentiable at least to the orders that appear in the integrals, the later function is called the fundamental solution for Laplace’s equations and has two different forms according to the dimension of the problem. These forms are as follows:
*
1 ln r 2
*
1 4 r
For 2-D problems
(3.11)
For 3-D problems
(3.12)
56
Chapter (3)
Boundary Element Methods
It should be remembered that the function * x , y is the potential at a point
y called the field point due to a unit source at the point x called the source point. This function and its derivative should be continuous at least up to the second order everywhere over the domain of the problem except at the source point as shown in figure (3.3).
Figure (3.3): Domain without a singular point [62]. Removing the singular point from the domain by removing a circle centered at the source point and having a very small radius denoted by . Again, applying the Green’s second identity to the same domain but without the circle containing the singularity. Let the new domain is denoted by new , the application of the second Green’s identity over the new domain leads to:
57
Chapter (3)
Boundary Element Methods
* * * d 2 d n n new
2 *
* * d n n (3.13)
Also, it should be remembered that both and * satisfies Laplace’s equation over the new domain new . The next step of the derivation of the integral equation is to evaluate the
integral
* n d , taking into consideration that the potential inside
the integral is the potential at the field point y . Re-write this integral in the following form: * x , y y d y n
* x , y y x d y x n y
* x , y d y n y
(3.14) Evaluation of the integrals involving potential derivative The following procedure is used to evaluate the following integral:
* x , y d y I1 n y
(3.15)
* x , y is the normal derivative of the fundamental solution, The term n y i.e;
58
Chapter (3)
Boundary Element Methods
* x , y 1 r 2 r n y n y
(3.16)
In equation (3.16): r
is the distance between the source and the field point.
ny
is the outward unit normal ( inward the circle).
The term
r equals –1, therefore, equation (3.16) takes the following form: n y
* x , y 1 n y 2 r
(3.17)
Because the domain under consideration is a circle, let us now transform into the polar coordinates and taking the limit as the radius 0 , then the second integral in the right hand side of equation (3.14) will take the following form:
2 * x , y d y lim 1 d 1 lim 0 n y 0 2 0
(3.18)
The first integral in the right hand side of equation (3.14) equals zero due to the same reasons that taking into consideration when evaluating equation (3.18), therefore;
* x , y d y 0 y x n y
(3.19)
Substituting from equations (3.18) and (3.19) into equation (3.14), leads to:
59
Chapter (3)
Boundary Element Methods
* x , y d y x lim y n 0
(3.20)
Evaluation of the integrals involving potential This integral appears in the second integral in the right hand side of equation (3.13) as follows:
I2
y * x y , d y n
(3.21)
Carry out the same procedure as done when evaluating the integral involving the potential derivative, leads to the following:
2 * y 1 ln d 0 I * lim x , y d y lim n y 0 0 2 0
(3.22)
Back again to the first integral of the right hand side of equation (3.14), the limit of this integral over and due to the same procedure carried out as before is as follows:
* x , y y * d y x x , y lim y n y n y 0
(3.23)
Therefore, the following integral equation can be obtained:
y * x , y * d y x x , y y n y n y
60
(3.24)
Chapter (3)
Boundary Element Methods
The above procedure was the case when the source point was inside the domain of the problem. In what follows, the integral equation will be derived if the source point is moved to the boundary of the domain. Suppose that the source point x is moved to the boundary and then take the limit when x tends to on the boundary as shown in figure (3.4).
n
Figure (3.4): Domain with source point on the boundary [62]. Again, and as done before, the point should be removed by removing a semi-circle containing this point. Carry out the same procedure as done before, taking into consideration that the upper limit of integration in equation (3.18) becomes , leads to:
* x , y y * d y 1 lim y x , y 2 n y n y 0
(3.25)
Therefore, the following integral equation can be obtained:
y * , y 1 * d y , y y 2 n n y y 61
(3.26)
Chapter (3)
Boundary Element Methods
Equation (3.26) is valid for every point on the smooth curve of the boundary of the domain. The general form of equation (3.26) can be written as follows:
y * , y * c , y d y y d y n n y y
(3.27)
The terms in the right hand side of equation (3.27) can take a simple form as follows:
q*
* ,y n y (3.28)
q
y n y
Therefore, equation (3.28) can now take the following form:
c y q * , y d y * , y q y d y
(3.29)
The multiplier c takes one of the following values as follows:
c 0
If outside the domain
c 1
If inside the domain
c
2
On the domain
(3.30)
Where is the angle of the boundary at the point . It should be noted that at every source point, three different values are known; the multiplier c ,
* and q * . 62
Chapter (3)
Boundary Element Methods
Thus, the boundary integral equation gives the potential at any source point inside the domain or on the boundary as a function of the potential and normal flux density components around the boundary. The next stage shows how the integral equation may be expressed in a discrete form, leading to the boundary element method itself. Before dealing with the discretized form of the integral equation, it is important to re-write equation (3.29) in a discretized form as follows:
c
* y q , y d element
element
* , y q y d element
element
(3.31) It will be fair enough to say that up to equation (3.31) no approximation has been introduced. Also, it is important to remember that if all integrations can be carried out analytically, then an exact solution will then be obtained. Unfortunately, some of the integrals cannot be found analytically and should be obtained numerically; this will be explained well later in this chapter. Before dealing with the numerical integrations, a brief review of the discretization techniques should be introduced.
3.4 Discretization Techniques There are three different types of discretization, the type of any one depends on the distribution of both potential and potential derivative over the element. These types are, constant, linear and quadratic. The accuracy, nature
63
Chapter (3)
Boundary Element Methods
of the problem and the importance of the results are three parameters that control choosing the type of the elements used.
3.4.1 Constant element
In this case, both potential and potential derivative are assumed to be constant along the element and that element has only one node denoted by i , see figure (3.5).
Potential Or Potential derivative
Element length Node i
Figure (3.5): Geometry of constant element [62].
Suppose that the boundary is divided into N-constant elements, therefore, the discretized form of equation (3.31) will take the following form: N 1 i q *d j 2 j 1
N j *d j j 1
64
j q
(3.32)
Chapter (3)
Boundary Element Methods
Where i is any boundary node, which is located in the middle of the element and j is the element number. Referring to equation (3.32), one can get two different integrals: I1 q *d j
(3.33)
I 2 *d j
(3.34)
The integrals in equations (3.33) and (3.34) relate two nodes i and j where the fundamental solution is acting on the first. Carrying out these integrals on all elements of the boundary leads to the influence coefficients. These coefficients can be denoted as: ij
q *d j
(3.35)
G ij *d j
(3.36)
H
Substutiting from equations (3.35) and (3.36) into equation (3.32), leads to: N N 1 i ij j H G ij q j 2 j 1 j 1
(3.37)
It should be remembered that the node at which the fundamental solution acting varires over each element on the boundary, therefore the influence coefficient given by equation (3.36) can take another form as follows:
65
Chapter (3)
Boundary Element Methods
H ij ij H ij 1 H 2
i j (3.38) i j
Substuting equation (3.38) into equation (3.37), and re-arrange, leads to: N
H ij j
j 1
N
G ij q j
(3.39)
j 1
In simple matrix form, equation (3.39) can be re-written as: H GQ
(3.40)
Referrring to the the influence coefficients given by equations (3.35) and (3.36), one can meet four different cases that depend on the location of the source and field points. When the source point being the field point, then a singularity will be encountered, i.e; H ii and G ii , these integrals have special treatment, while
H ij and G ij being calculated using simple Gauss quadrature rules, for more details see [84].
3.4.2 Linear element The first difference between the constant and linear element is the multiplier
c where in the constant element it takes
1 because the boundary is 2
smooth, but now it will be used as it is in the mathematics. Discretized equation (3.31) with N linear elements, then it takes the following form:
66
Chapter (3)
c i i
N
Boundary Element Methods
* q d
j 1
N
*qd
(3.41)
j 1
In this case, it is imposible to take the potential and the potential derivative out of the two integrals q *d and
*d because over each element these
values vary linearly, see figure (3.6). Therefore, the values of and
at n
any point over the element can be defined in terms of their nodal values and two linear interpolating functions 1 and 2 of the homogenous coordinate
such that: 1 1 n
2 1
2 1 n
2
T
(3.42) T
2 n
(3.43)
In which, the functions 1 and 2 are as follows: 1 0.5 1 (3.44) 2 0.5 1 * n d 1
1 * 1 d 2 hij n 2
67
1 hij2 2
(3.45)
Chapter (3)
Boundary Element Methods
1 1 n n * * d g1 g 2 d n 1 2 ij ij 2 2 n n
hij1
* 1 n d j j
hij2
(3.46)
* 2 n d j j (3.47)
g ij1
j
* d 1
g ij2
j
j
* d 2
j
Therefore, the final boundary integral equation takes the following form: j
H ij j
j 1
j
j
j 1
n
G ij
(3.48)
Finally, as done before for a constant case, the system given by equation (3.48), can now recast in a matrix form as follows:
H GQ
(3.49)
In the present and next chapters, all calculations will be performed using these two different types of elements, therefore, it is not necessarily to represent the third type.
68
Chapter (3)
Boundary Element Methods
Potential Or Potential derivative
Potential Or Potential derivative
Node i 1
Node i 1 Node
i
Element length
Figure (3.6): Geometry of linear element [62].
After this complete formulation of the boundary element method, remainning part of the present chapter introduces different numerical algorithms for cavitation problems. These algorithms will be introduced in some details in the next section.
69
Chapter (3)
Boundary Element Methods
3.5 Numerical algorithms 3.5.1 Aitchison algorithm Aitchison [48] presented in his paper a numerical solution for planar and axi-symmetric Riabouchinsky cavity flow. The problem was formulated as the minimization of a functional over a variable region, which was solved numerically by the variable finite elements. In the following figures, the general cavity model, followed by Riabouchinsky model in a finite domain then finally the region of the solution, respectively, are shown.
Figure (3.7): General cavity flow model [48].
Figure (3.8): Riabouchinsky model in a finite domain [48].
70
Chapter (3)
H
Boundary Element Methods
b d L
Figure (3.9): Region of solution [48]. It is well known from Birkoff and Zarantonello [83] that for a fixed cavity length, L and half height of the plate, there is only one value of the free stream line velocity, qc , which gives an analytical solution. From this fact, Aitchison derived the following algorithm, using the following formula for
qc , that is:
qc 2 p q 2
(3.50)
Aitchison algorithm can be summarized as follows: (1)
Solve the problem for a given value of qc , which is known to be too low.
(2)
Calculate the average pressure in each element p j
(3)
If any value of p j is negative, increase qc and repeat step (2), otherwise stop.
This means that, the main objective of this algorithm is to find lowest value of
qc for which the pressure is positive everywhere inside the flow.
71
Chapter (3)
Boundary Element Methods
3.5.2 Wrobel Algorithm In 1992, Wrobel [51] solved cavitation behined a plate placed perpendicular to the direction of the flow in a channel of finite width. Riabouchnisky model was used for modeling the problem, see figure (3.9). Wrobel’s algorithm can be summarized as follow: (1)
Assume an initial guess for the free-surface.
(2)
Use the boundary element method to find the potential derivative at each node of the free-surface.
(3)
1 Calculate Bernoulli’s constant at each node using i qc2 . 2 i
(4)
Evaluate the difference between Bernoulli’s constant at the tip of the plate and that value obtained in step (3).
(5)
Evaluate the vertical displacement from y i i y 3 . i
(6)
k 1 y k 1 yi i . Evaluate the following norm Norm k N yi
(7)
If the previous norm is smaller than a prescribed tolerance, stop, otherwise, guess new location using y ik 1 y ik y ik , then repeat the procedure again.
72
Chapter (3)
Boundary Element Methods
3.5.3 Yas’ko Algorithm In 1992, Yas’ko [84] solved plane and axi-symmetric flow and the cavity formed behined blunt-ended bodies. Yas’ko used Riabouchinsky model in his algorithm, see figure (3.10).
Free surface Wetted body cavity
Body
Z
ZA
Z
R
K
L
K
Figure (3.10): Yas’ko problem [84]. Yas’ko algorithm in steps (1)
Apply the boundary integral equation at each point on the wetted body and the free-surface and get N L number of unknowns.
(2)
Yas’ko derived an additional equation for the velocity of the freesurface.
(3)
By Gauss elimination method, he solved N L 1 equations for the unknowns. These unknowns are body, N
L
potential on the wetted
potential derivative on the free-surface and the velocity
of the free-surface.
73
Chapter (3)
(4)
Boundary Element Methods
He used the kinematic boundary condition to iterate for the shape of the cavity.
(5)
After iteration, he found the cavitation number as a solution parameter.
3.5.4 Ingber Algorithm In this algorithm [23], flow about submerged, fully cavitating bodies at zero and non-zero angle of attack were considered. A cavity closure model was used in the algorithm and making use of the boundary element method. The domain of the problem with the closure model is shown in figure (1.9). Ingber assumed that he could predict correctly the point of detachment and focused on determining the correct cavity length. To do that he proposed two iterations, the first named as inner and the other named outer. Inner iteration In this iteration, the point of detachment and the cavity length are given and assumed cavity shape. Then, he uses the direct boundary element method to satisfy either kinematics or dynamic condition. In steps, the inner iterations can be summarized as follows: (1)
Assume cavity length and cavity shape.
(2)
Apply direct boundary element, where the kinematics boundary condition is known on both wetted body and the free-surface.
Outer iteration In this iteration, an adjustment for the cavity length still carried out until the dynamics condition will be satisfied everywhere along the cavity surface.
3.5.5 Kinnas and Fine Algorithm In 1993, Kinnas and Fine. [53] suggested an algorithm for numerical nonlinear analysis for the flow around two- and three-dimensional partially
74
Chapter (3)
Boundary Element Methods
cavitating hydrofoil. In few steps their algorithm can be summarized as follows: (1)
Solve the cavity problem (first iteration) with fixed cavitation number o to find the corresponding cavity length, o .
(2)
Solve the cavity problem with fixed cavity length, o , for Kiterations and find the converged value of the cavitation number,
o/ .
(3)
Define a new cavitation number from, 1 o o o/ .
(4)
Solve the cavity problem (first iteration) with fixed cavitation number n and find the correct cavity length, n , n 1 .
(5)
Solve the cavity problem with fixed cavity length, n for Kiterations and find the converged value of the cavitation number,
n/ , n 1 . (6)
Apply Newton-Raphson iteration solution for n/ 0 and find a new cavitation number n 1 .
(7)
Repeat steps (4-6) until n/ within a prescribed tolerance.
3.5.6 Ahmed Algorithm Between 1994 and 1996, Ahmed et al. [55], solved the two-dimensional cavity flow problem, see figure (3.11), using the boundary element method. A new algorithm for that problem was derived and can be summarized in few steps as follows: (1)
Assume an initial guess for the free-surface position.
75
Chapter (3)
(2)
Boundary Element Methods
Apply the kinematics condition on both wetted and free-surface, then use the boundary element formulation to determine the potential on both surfaces.
(3)
Find qi
(4)
approximate si A x i si
velocity
at
each
node
using
.
Evaluate the error at each point i given by the difference between
q i and q A . (5)
Verify convergence, i.e., if the difference calculated in step (4) is less than a specified tolerance at all free-surface points; if this is not so, use y i y 2 q A q to move the free-surface nodes and i i repeat steps 2-5 until convergence.
K
Y A
B
X
Figure (3.11): Ahmed domain problem [55].
76
Chapter (3)
Boundary Element Methods
3.6 NEW (PRESENT) NUMERICAL ALGORITHM After the previous review on the most numerical algorithms that were developed to solve cavity flow problem. A new numerical algorithm based on the closure model and uses the boundary element method will be introduced herein and listed in steps as follows [59, 60& 61]:
(1)
Solve the BE-code over domain bounded by the upper surface of the hydrofoil and the free-surface as shown in figure (2.6). Over the hydrofoil surface, equation (2.18) is used as a boundary
condition. Over the free-surface, the potential in equation (2.38) is applied. For
simplicity, 0 [56], then, equation (2.38) takes the following formula s
s 0 U S U 1 ds
(3.51)
0
Assuming that in this step the 0 equals zero. (2)
Evaluate
M s 0 1 i 1 M
(3.52)
where is a control parameter.
77
Chapter (3)
(3)
Boundary Element Methods
Solve the BE-code once more over the same domain as in step (1) with equation (2.18) as a boundary condition over the hydrofoil and equation (3.51) using 0 obtained from step (2) as a boundary condition over the free-surface. This step leads to the potential . derivative at each node of the free-surface n i node
(4)
Evaluate the magnitude of the total free-surface velocity.
(5)
The cavity closure condition requires the thickness at the cavity trailing edge, h s L , to be zero. This is obtained by applying the following equation: S 1 L h s L U n qc 0 n i node
ds 0
(3.53)
In the first iteration, the panels representing the cavity are placed on the foil surface directly under the cavity. In the following iterations, the cavity shape is updated by an amount h s L .
3.6.1 Discretization Process The present algorithm used boundary element method with linear elements to get the required accuracy ( 0.0005 ) that appears in Eq. (2.41) as the cavity closure condition. This accuracy is satisfied using a number of elements that varies for different cavity lengths. However, for all cases, the linear elements that were used to discretize the domain have different lengths. 78
Chapter (3)
Boundary Element Methods
Cavitation phenomenon emerges at the leading edge of the hydrofoil (at the separation point). So, the number of elements (with short length) must be increased in this area in comparison with the trailing edge area of the cavity. For example, at the cavity length l=0.6, the total number of linear elements at both the cavity surface and the hydrofoil surface equals to 9. Thus, the total number of the linear elements used within the domain equal to 18. The length of the shortest element equals to 0.0125 of chord length (C). This shortest element was placed at the leading edge. The length of the longest element equals to 0.1 of chord length (C). This longest element was placed at the trailing edge of cavity. In the next chapter, results will be presented starting by previous results obtained by other researchers but using the present algorithm to check its validity, and then new results will be listed in more details.
79
CHAPTER (4) RESULTS AND DISCUSSION
Chapter (4)
results and Discussion
CHAPTER (4) RESULTS AND DISCUSSION 4.1 Introduction Cavitation problem is highly non-linear due to the inherent nonlinear conditions encountered in its mathematical formulation, therefore, analytical solutions are so difficult to obtain. Because of that many researchers directed their efforts to numerical solutions. Different numerical approaches had been done by the aid of different mathematical models, and some of these models had been introduced in chapter three of the present thesis. The first section of the present chapter starts by re-introducing previous results [55] for the flow problem past vertical, oblique flat plates and over a wedge of vertex angle , 0
2
. The purpose of re-introducing these results
is to gain skills of discretization techniques, techniques of nodes numbering and element, modification of results and any other parameters that appear when solving the problem using the boundary element methods. The second part of the present chapter introduces a numerical solution of cavitation problem around hydrofoil using the boundary element method with a new numerical algorithm developed for this purpose. In this algorithm, the problem is solved in a completely different approach, i.e., only the domain 81
Chapter (4)
results and Discussion
considered is that bounded by the upper surface of the hydrofoil and the freesurface, therefore the difficulty in evaluating the potential at the tip of the hydrofoil becomes easy rather than before. Different cases of NACA hydrofoils are tested using the proposed technique and give good results [59, 60& 61].
4.2 Re-introduce of some previous results 4.2.1 Flow over vertical flat plate The flow over vertical flat plate was studied before by different researchers, such as Yas’ko [47] and Ahmed et al. [55]. Fortunately, this problem has an analytical solution [5]. Therefore, re-introduced of some previous results and the comparison with the analytical solution will give a good impression about understanding the technique of the boundary element method. The configuration of the flow problem over vertical flat plate is shown in figure (4.1).
Free Surface
R H
Z Figure (4.1): Domain of flow problem over vertical flat plate [55].
82
Chapter (4)
results and Discussion
In figure (4.1): Z
Is the cavity length
H
Is the vertical plate length
R
Is the maximum position of the free-surface. Three different examples in [55] have been solved. The summary of
these examples with previous and present calculations is listed in table (4.1). The purpose of these results is to check the validity of the designed code and to handle different parameters that have a direct effect on the accuracy of the results. Another topic analyzed is the effect of the cavity length on the final shape of the free-surface. Two different cases are solved for three and five cavity length, respectively. These results are shown in figures (4.2) and (4.3), respectively.
83
Table (4.1) The calculations of three different examples of a vertical flat plate.
Analytical Yas’ko [5]
[47]
q
R H
Z R
Ahmed [55] Z
H
Analytical Yas’ko Z R
[5]
[47]
Given 15.568
15.85
0.778
31.798
31.74
1.589
245.60
243.3
12.28
Ahmed [55] R
H
Analytical Yas’ko Ahmed R H
[5]
3.4612
1.2042
1.2042 1.2069
4.7084
1.1402
1.1402 1.1422
12.258
1.0488
1.0488 1.0494
[47]
[55]
Calculated
0.05
15.568
3.4847
3.491
0.17306
31.798
4.7305
4.727
0.23542
245.60
12.201
12.07
0.61293
84
0.05
Chapter (4)
results and Discussion
0.40
0.30
0.20 Vertical flat plate Half cavity length = 1.5
A......Initial guess Initial guess B......1- iteration iteration (1) C......3- iteration D......5- iteration iteration (3) E......6- iteration
0.00
iteration (5)
-0.10
iteration (9)
-0.20
Figure (4.2): The final free surface for three cavity length.
85
3.0
2.7
2.4
2.1
x
1.8
1.5
1.2
0.9
0.6
-0.40
0.3
-0.30
0.0
Y
0.10
Chapter (4)
results and Discussion
1.20 Vertical flat plate Half cavity length = 2.5 Initial guess iteration (3)
0.80
iteration (5) iteration (7) Final iteration (9)
Y
0.40
0.00
-0.40
x Figure (4.3): The final free surface for five cavity lengths.
86
5.0
4.0
3.0
2.0
0.0
-1.20
1.0
-0.80
Chapter (4)
results and Discussion
4.2.2 Flow over wedge The second case that had been introduced was the flow past a wedge of a half angle , 0
2
. The domain of the problem is shown in figure
(4.4).
R
Z
Figure (4.4): Domain of flow problem over a wedge. For a wide range of calculations about this case, the reader should refer to [55].
87
Chapter (4)
results and Discussion
4.3 New results To test the validity of the present algorithm, previous results by Kinnas et al. [56] were achieved by the present algorithm with an execllent agreement for
NACA-16006. In Kinnas’s case study the cavitation number was 1.097 and the cavity length was found to be
L 0.4 . c
Thus, the computations were expanded to study the effect of different parameters has direct effect on the cavity shape and length. These parameters include; hydrofoil section, angle of attack, free-stream velocity, and cavitation number. Figure (4.5) shows the profiles of the different tested NACA sections (the definition of NACA sections consists of two main parts; the first part, for all NACA sections, is 00 which means that the profile is symmetric. The second part is (06, 12, 15 or 25) refers to the maxiumum thickness of the NACA section relative to the chord length of the hydrofoil). 0.5
Hydrofoil NACA-0025 Hydrofoil NACA-0015 Hydrofoil NACA-0012 Hydrofoil NACA-16006
0.4
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.5): Profiles of different tested NACA sections.
88
Chapter (4)
results and Discussion
4.3.1 NACA-16006 Hydrofoil Case (1)
In this case study, three parameters are prescribed and the present code is used to find the cavity length
L 0.4 . The three parameters are; the c
cavitation number 1.097 , the inflow velocity U 1.0 m / s and the angle of attack = 4o. The results of this case are shown in figures (4.6-a, b). It is clear that the cavity closure condition at the cavity end, , given in equation (2.41) is obtained at
L 0.4 . c
0.9 0.8
Hydrofoil l = 0.2 l = 0.3 l = 0.4 l = 0.5 l = 0.6 l = 0.7
0.7
y = Y/C
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
0
0.2
0.4
0.6
0.8
1
l = X/C
Figure (4.6-a): Scaled hydrofoil section NACA-16006 at U 1.0 m / s .
89
Chapter (4)
results and Discussion
0.05 0.04 0.03
y = Y/C
0.02 0.01 Hydrofoil l = 0.2 l = 0.3 l = 0.4 l = 0.5 l = 0.6 l = 0.7
0
-0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.6-b): Non-scaled hydrofoil section NACA-16006 at U 1.0 m / s .
90
Chapter (4)
results and Discussion
Case (2)
In this case, the present code is used with the same three parameters which were used in the previous case. There are two parameters have the same value as the previous case, the cavitation number 1.097 and the angle of attack
= 4o. However, the inflow velocity,
U,
changed
to
becomeU 0.5 m / s . The results of this case are shown in figure (4.7). It is clear that the cavity length
L 0.425 achieved the cavity closure condition at the cavity end, c
given in equation (2.41).
0.06 hydrofoil l=0.4 l=0.425 l=0.6 l=0.7
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.7): Non-scaled hydrofoil section NACA-16006 at U 0.5 m / s .
91
,
Chapter (4)
results and Discussion
Case (3)
As the previous two cases, the parameters which have the same values are, the cavitation number 1.097 and the angle of attack = 4o. The inflow velocity U takes a new value, U 2.0 m / s . The results of this case are shown in figure (4.8). The present code finds the cavity length
L 0.3 that achieved c
the cavity closure condition at the cavity end, , given in equation (2.41).
0.06 hydrofoil l=0.2 l=0.3 l=0.4 l=0.5
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.8): Non-scaled hydrofoil section NACA-16006 at U 2.0 m / s .
92
Chapter (4)
results and Discussion
Case (4)
In this case study, the inflow velocity, U , takes a new value, U 3.0 m / s and the other parameters still have the same values as the previous
cases. In figure (4.9), all results of this case are shown and it is clear that the cavity length,
L 0.25 achieved the closure condition of equation (2.41). c
0.06 hydrofoil l=0.2 l=0.25 l=0.4 l=0.5
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.9): Non-scaled hydrofoil section NACA-16006 at U 3.0 m / s .
93
Chapter (4)
results and Discussion
Case (5)
In this case, the variable parameter is the cavitation number and the inflow velocity U has a prescribed value. The present code is used with the three parameters, the cavitation number 0.852 , the angle of attack = 4o
and the inflow velocity U 1.0 m / s to find the cavity length . The cavity length, , which achieved the cavity closure condition at the cavity end, , given in equation (2.41) is obtained at
L 0.47 . The results of this case are c
shown in figure (4.10).
0.06 hydrofoil l=0.4 l=0.47 l=0.6 l=0.7
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.10): Non-scaled hydrofoil section NACA-16006 at 0.852 .
94
Chapter (4)
results and Discussion
Case (6)
As the previous case, the two parameters which have the same values are, the angle of attack = 4o and the inflow velocity U 1.0 m / s . However, the cavitation number takes a new value, 1.26933 . The cavity length that achieved the closure condition in equation (2.41) is
L 0.225 . The results of c
this case are shown in figure (4.11).
0.06 hydrofoil l=0.2 l=0.225 l=0.4 l=0.5
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.11): Non-scaled hydrofoil section NACA-16006 at 1.26933 .
95
Chapter (4)
results and Discussion
Case (7)
In the present case, the angle of attack, , is studied as a variable parameter. The other two parameters, the cavitation number 1.097 and the inflow velocity U 1.0 m / s are fixed at the shown values. The present code is used with an angle of attack = 3o and gives the cavity length
L 0.34 c
which achieved the closure condition in equation (2.41). In figure (4.12) the results of this case are shown at different lengths.
0.06 hydrofoil l=0.3 l=0.34 l=0.5 l=0.6
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.12): Non-scaled hydrofoil section NACA-16006 at = 3o.
96
Chapter (4)
results and Discussion
Case (8)
In this case the same two parameters, the cavitation number 1.097 and the inflow velocity U 1.0 are used. However, the angle of attack, , has a new value, = 5o. It is clear that the cavity length
L 0.44 achieved the c
cavity closure condition at the cavity end, , given in equation (2.41). The results of this case are shown in figure (4.13).
0.06 hydrofoil l=0.4 l=0.44 l=0.6 l=0.7
0.05 0.04 0.03
y=Y/c
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.13): Non-scaled hydrofoil section NACA-16006 at = 5o.
97
Chapter (4)
results and Discussion
Figure (4.14) illustrates the variation of cavitation shape with free-stream velocity (U). The cavity size decreases as U increases. The cavity shape is almost the same for 3 and 4 m/s.
0.05 0.04 0.03
y = Y/C
0.02 0.01
Hydrofoil U = 0.5 m/s - l = U = 1.0 m/s - l = U = 2.0 m/s - l = U = 3.0 m/s - l =
0
-0.01
0.425 0.4 0.3 0.25
-0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.14): Effect of free-stream velocity, NACA-16006.
98
Chapter (4)
results and Discussion
Figure (4.15) shows the effect of cavitation number on the cavity shape and length at = 4o and U = 1.0 m/s. The cavitation length, l, decreases as increases.
0.05 0.04 0.03
y = Y/C
0.02 0.01 Hydrofoil Cav. Num. = 0.852 - l = 0.47 Cav. Num. = 1.097 - l = 0.4 Cav. Num. = 1.269 - l = 0.225
0
-0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.15): Effect of cavitation number, NACA-16006.
99
Chapter (4)
results and Discussion
The effect of the angle of attack () is shown in figure (4.16) for = 1.097 and U = 1.0 m/s. The height and length of cavitation increase with .
0.05 0.04 0.03
y = Y/C
0.02 0.01
Hydrofoil Alpha = 3.0 - l = 0.34 Alpha = 4.0 - l = 0.4 Alpha = 5.0 - l = 0.44
0
-0.01 -0.02 -0.03 -0.04 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.16): Effect of angle of attack, NACA-16006. From the previous results, it is shown that increase of the free stream velocity makes the cavity length decreased and the same effect occurs when cavitation number increased. Otherwise, increased of the angle of attack makes the cavity length increased.
100
Chapter (4)
results and Discussion
The summery of the previous results are listed in the table (4.2). Table (4.2) Results of NACA-16006.
Free-Stream Velocity (U) m/s
Cavitation Number (σ)
Angle of Attack ()
Cavitation Length (l)
(1)
1.0
1.097
4o
0.4
(2)
0.5
1.097
4o
0.425
(3)
2.0
1.097
4o
0.3
(4)
3.0
1.097
4o
0.25
(5)
1.0
0.852
4o
0.47
(6)
1.0
1.26933
4o
0.225
(7)
1.0
1.097
3o
0.34
(8)
1.0
1.097
5o
0.44
Case
101
Chapter (4)
results and Discussion
4.3.2 NACA-0012 Hydrofoil Case (1)
In this case study, the present code is used to solve cavitation for the hydrofoil section NACA 0012 with the three different parameters which were studied with the previous hydrofoil NACA section. The new values of the three parameters are, the cavitation number 0.9774 , the angle of attack = 4o and L
the inflow velocity U 0.5 m / s . The cavity length 0.3 achieved the c cavity closure condition at the cavity end, , given in equation (2.41). Figure (4.17) shows the results of this case.
0.2 hydrofoil l=0.2 l=0.3 l=0.4 l=0.5
0.15
y=Y/c
0.1
0.05
0
-0.05
-0.1 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.17): Scaled hydrofoil section NACA-0012 at U 0.5 m / s .
102
Chapter (4)
results and Discussion
Case (2)
In this case, the present code is used with the same three parameters which were used in the previous case. There are two parameters having the same value as the previous case, the cavitation number 0.9774 and the angle of attack = 4o. However, the inflow velocity changed to becomeU 1.0 m / s . The results of this case are shown in figure (4.18). It is clear that the cavity length
L 0.3496 achieved the cavity closure condition at the cavity c
end, , given in equation (2.41).
0.3 hydrofoil l=0.3 l=0.3496 l=0.5 l=0.6
0.2
y=Y/c
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.18): Scaled hydrofoil section NACA-0012 at U 1.0 m / s .
103
Chapter (4)
results and Discussion
Case (3)
As the previous two cases, the parameters which have the same values are, the cavitation number 0.9774 and the angle of attack = 4o. The inflow velocity takes a new value U 2.0 . The results of this case are shown in figure (4.19). The present code finds the cavity length
L 0.5 which c
achieved the cavity closure condition at the cavity end, , given in equation (2.41).
0.4 hydrofoil l=0.4 l=0.5 l=0.6 l=0.7
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.19): Scaled hydrofoil section NACA-0012 at U 2.0 m / s .
104
Chapter (4)
results and Discussion
Case (4)
In this case study, the inflow velocity takes a new value, U 4.0 m / s and the other parameters still have the same values as the previous cases. In figure (4.20), all results of this case are shown and it is clear that the cavity length, L 0.616 achieved the same condition in equation (2.41). c
0.4 hydrofoil l=0.5 l=0.6 l=0.616
0.3
0.2
y=Y/c
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.20): Scaled hydrofoil section NACA-0012 at U 4.0 m / s .
105
Chapter (4)
results and Discussion
Case (5)
In this case study, three different parameters are prescribed and the
present code is used to find the cavity length
L 0.443 . The three c
parameters are, the cavitation number 0.825 , the inflow velocity U 1.0 m / s and the angle of attack
= 4o. The results of this case are shown
in figure (4.21). It is clear that the cavity closure condition at the cavity end, , given in equation (2.41) is obtained, in all cases shown, at
L 0.443 . c
0.4 hydrofoil l=0.4 l=0.443 l=0.6 l=0.7
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.21): Scaled hydrofoil section NACA-0012 at 0.825 .
106
Chapter (4)
results and Discussion
Case (6)
As the previous case, the two parameters which have the same values are, the angle of attack = 4o and the inflow velocityU 1.0 m / s . However, the cavitation number takes a new value 1.1247 . The cavity length, , that achieved the closure condition in equation (2.41) is
L 0.244 . The results of c
this case are shown in figure (4.22).
0.4 hydrofoil l=0.2 l=0.244 l=0.4 l=0.5
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.22): Scaled hydrofoil section NACA-0012 at 1.1247 .
107
Chapter (4)
results and Discussion
Case (7)
In the present case, the angle of attack, , is studied as a variable parameter. The other two parameters, the cavitation number 0.9774 and the inflow velocity U 1.0 m / s are prescribed at the mentioned value. The present code is used with an angle of attack = 3o and gives the cavity length
L 0.247 which achieved the closure condition in equation (2.41). In figure c
(4.23), the results of this case are shown at different lengths.
0.4 hydrofoil l=0.2 l=0.247 l=0.4 l=0.5
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.23): Scaled hydrofoil section NACA-0012 at = 3o.
108
Chapter (4)
results and Discussion
Case (8)
In this case, the same two parameters, the cavitation number 0.9774 and the inflow velocity U 1.0 m / s are used. However, the angle of attack, , has a new value = 5o. It is cleared that the cavity length
L 0.6 achieved c
the cavity closure condition at the cavity end, , given in equation (2.41). The results of this case are shown in figure (4.24).
0.14 hydrofoil l=0.4 l=0.5 l=0.6 l=0.7
0.12 0.1 0.08
y=Y/c
0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.24): Non-scaled hydrofoil section NACA-0012 at = 5o.
109
Chapter (4)
results and Discussion
Figure (4.25) illustrates the variations in cavitation shape with free-stream velocity (U). The cavity length increases with U .
0.4
Hydrofoil U = 0.5 m/s - l = 0.3 U = 1.0 m/s - l = 0.35 U = 2.0 m/s - l = 0.5 U = 4.0 m/s - l = 0.62
0.3
y = Y/C
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.25): Effect of free-stream velocity, NACA-0012.
110
Chapter (4)
results and Discussion
The effect of cavitation number on the cavity shape and length at = 4o and U = 1.0 m/s is shown in figure (4.26). The cavitation length decreases as increases.
0.4
0.3 Hydrofoil Cav.Num.=0.825-l=0.442 Cav.Num.=0.9774-l=0.35 Cav.Num.=1.1247-l=0.244
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.26): Effect of cavitation number, NACA-0012.
111
Chapter (4)
results and Discussion
Figure (4.27) shows the effect of the angle of attack () for = 0.977 and U = 1.0 m/s. As expected, the height and length of cavitation increase with .
0.4
Hydrofoil Alpha = 3.0 deg. - l = 0.247 Alpha = 4.0 deg. - l = 0.350 Alpha = 5.0 deg. - l = 0.600
0.3
y = Y/C
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.27): Effect of angle of attack, NACA-0012. From the previous results, it is shown that increase of the free stream velocity makes the cavity length increased and the same effect occurs when angle of attack increased. Otherwise, increased of the cavitation number makes the cavity length decreased.
112
Chapter (4)
results and Discussion
The summery of the previous results are listed in the table (4.3). Table (4.3) Results of NACA-0012.
Free-Stream Velocity (U) m/s
Cavitation Number (σ)
Angle of Attack ()
Cavitation Length (l)
(1)
0.5
0.9774
4o
0.3
(2)
1.0
0.9774
4o
0.3496
(3)
2.0
0.9774
4o
0.5
(4)
4.0
0.9774
4o
0.616
(5)
1.0
0.825
4o
0.443
(6)
1.0
1.1247
4o
0.244
(7)
1.0
0.9774
3o
0.247
(8)
1.0
0.9774
5o
0.6
Case
113
Chapter (4)
results and Discussion
4.3.3 NACA-0015 Hydrofoil Case (1)
In this case study, the present code is used to solve the hydrofoil section NACA 0015 with the three different parameters which were studied in the previous hydrofoil NACA section. The new values of the three parameters are, the cavitation number 0.9774 , the angle of attack = 5o and the inflow L
velocity U 0.25 m / s . The cavity length 0.251 achieved the cavity c closure condition at the cavity end, , given in equation (2.41). Figure (4.28) shows the results of this case.
0.3 hydrofoil l=0.2 l=0.251 l=0.4 l=0.5
0.25 0.2 0.15
y=Y/c
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.28): Non-scaled hydrofoil section NACA-0015 at U 0.25 m / s .
114
Chapter (4)
results and Discussion
Case (2)
In this case, the present code is used with the same three parameters which used in the previous case. There are two parameters that have the same value as the previous case, the cavitation number 0.9774 and the angle of attack = 5o. However, the inflow velocity, U , changed to become U 0.5 m / s .
The results of this case are shown in figure (4.29). It is clear that the cavity length
L 0.3 achieved the cavity closure condition at the cavity end, c
, given in equation (2.41).
0.3 hydrofoil l=0.2 l=0.3 l=0.4 l=0.5
0.25 0.2 0.15
y=Y/c
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.29): Non-scaled hydrofoil section NACA-0015 at U 0.5 m / s .
115
Chapter (4)
results and Discussion
Case (3) As the previous two cases, the parameters which have the same values are, the cavitation number 0.9774 and the angle of attack = 5o. The inflow velocity takes a new value U 1.0 m / s . The results of this case are shown in figure (4.30). The present code finds the cavity length
L 0.4667 c
which achieved the cavity closure condition at the cavity end, , given in equation (2.41).
0.6
0.5
0.4 hydrofoil l=0.4 l=0.4667 l=0.6 l=0.7
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.30): Non-scaled hydrofoil section NACA-0015 at U 1.0 m / s .
116
Chapter (4)
results and Discussion
Case (4)
In this case study, the inflow velocity takes a new value U 2.0 m / s and the other parameters still have the same values in the previous cases. In figure (4.31), all results of this case are shown and it is clear that the cavity length, L 0.5 achieved the closure condition of equation (2.41). c
0.6
0.5
0.4 hydrofoil l=0.4 l=0.5 l=0.6 l=0.7
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.31): Scaled hydrofoil section NACA-0015 at U 2.0 m / s .
117
Chapter (4)
results and Discussion
Case (5)
In this case study, three different parameters are prescribed and the present code is used to find the cavity length,
L 0.5933 . The three c
parameters are, the cavitation number 0.67585 , the inflow velocity U 1.0 m / s and the angle of attack
= 4o. The results of this case are shown
in figure (4.32). It is clear that the cavity closure condition at the cavity end, , given in equation (2.41) is obtained at
L 0.5933 . c
0.5
0.4
0.3 hydrofoil l=0.4 l=0.5 l=0.5933 l=0.7
y=Y/c
0.2
0.1
0
-0.1
-0.2
-0.3 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.32): Scaled hydrofoil section NACA-0015 at 0.67585 .
118
Chapter (4)
results and Discussion
Case (6)
As the previous case, the two parameters which have the same values are, the angle of attack = 4o and the inflow velocity U 1.0 m / s . However, the cavitation number takes a new value 0.825 . The cavity length, , that achieved the condition in equation (2.41) is
L 0.415 . The results of this c
case are shown in figure (4.33).
0.5
0.4
y=Y/c
0.3
hydrofoil l=0.4 l=0.415 l=0.6 l=0.7
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.33): Scaled hydrofoil section NACA-0015 at 0.825 .
119
Chapter (4)
results and Discussion
Case (7)
In this case, the cavitation number takes a value 0.9774 . The two parameters, the angle of attack, , and the inflow velocity, U , still have the same values. The cavity length that achieved the condition in equation (2.41) is L 0.2333 . The results of this case are shown in figure (4.34). c
0.2 hydrofoil l=0.2 l=0.2333 l=0.4 l=0.5
0.15
0.1
0.05
y=Y/c
0
-0.05
-0.1
-0.15
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.34): Non-scaled hydrofoil section NACA-0015 at 0.9774 .
120
Chapter (4)
results and Discussion
Case (8)
In the present case, the angle of attack, , is studied as a variable parameter. In this case, the other two parameters are the cavitation number
0.9774 and the inflow velocity U 1.0 m / s . The present code is used with an angle of attack = 6o and gives the cavity length
L 0.6 which achieved c
the closure condition in equation (2.41). In figure (4.35), the results of this case are shown for different lengths.
0.6
hydrofoil l=0.4 l=0.5 l=0.6 l=0.7
0.5
0.4
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.35): Scaled hydrofoil section NACA-0015 at = 6o.
121
Chapter (4)
results and Discussion
Case (9)
In this case, the same two parameters are prescribed, the cavitation number has another value 0.67585 and the inflow velocity U 1.0 m / s . However, the angle of attack has a new value = 3o. It is clear that the cavity length
L 0.315 achieved the cavity closure condition at the cavity end, c
given in equation (2.41). The results of this case are shown in figure (4.36).
0.3 hydrofoil l=0.3 l=0.315 l=0.5 l=0.6
0.25 0.2 0.15
y=Y/c
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.36): Non-scaled hydrofoil section NACA-0015 at = 3o.
122
,
Chapter (4)
results and Discussion
Case (10)
In this case study, the previous case is applied but with a new value to the angle of attack = 3.5o and the other parameters are prescribed with the same values. The present code is used to find the cavity length, . It is clear that the cavity closure condition at the cavity end, , given in equation (2.41) is obtained at
L 0.478 . The results of this case are shown in figure (4.37). c
0.6
0.5
0.4
hydrofoil l=0.4 l=0.478 l=0.6 l=0.7
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.37): Scaled hydrofoil section NACA-0015 at = 3.5o.
123
Chapter (4)
results and Discussion
Figure (4.38) explains the changes of cavitation shape and length with freestream velocity (from U = 0.25 to 2.0 m/s). The cavity height is slightly affected by the increase of U. However, the cavity length increases with freestream velocity.
0.5 Hydrofoil U = 0.25 m/s - l = 0.25 U = 0.5 m/s - l = 0.3 U = 1.0 m/s - l = 0.47 U = 2.0 m/s - l = 0.5
0.4
y = Y/C
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.38): Effect of free-stream velocity, NACA-0015.
124
Chapter (4)
results and Discussion
Figure (4.39) shows the effect of cavitation number on the cavity shape and length at = 4o and U = 1.0 m/s. The cavitation height and length decrease as increases.
0.5 Hydrofoil Cav. Num. = 0.676 - l = 0.593 Cav. Num. = 0.825 - l = 0.415 Cav. Num. = 0.977 - l = 0.233
0.4
y = Y/C
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.39): Effect of cavitation number, NACA-0015.
125
Chapter (4)
results and Discussion
Figures (4.40) and (4.41) illustrate the effect of the angle of attack () for = 0.67585 and 0.9774, respectively. A range of (from 3o to 6o) was tested for U = 1.0 m/s. Again, the height and length of cavitation increase with .
0.5 Hydrofoil Alpha = 3.0 deg. - l = 0.32 Alpha = 3.5 deg. - l = 0.48 Alpha = 4.0 deg. - l = 0.59
0.4
y = Y/C
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.40): Effect of angle of attack with cavity number = 0.67585, NACA-0015.
126
Chapter (4)
results and Discussion
0.5 Hydrofoil Alpha = 4.0 deg. - l = 0.23 Alpha = 5.0 deg. - l = 0.47 Alpha = 6.0 deg. - l = 0.60
0.4
y = Y/C
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.41): Effect of angle of attack cavitation number = 0.9774, NACA-0015. From the previous results, it is shown that increase of the free stream velocity makes the cavity length increased and the same effect occurs when angle of attack increased. Otherwise, increased of the cavitation number makes the cavity length decreased.
127
Chapter (4)
results and Discussion
The summery of the previous results are listed in the table (4.4). Table (4.4) Results of NACA-0015.
Free-Stream Velocity (U) m/s
Cavitation Number (σ)
Angle of Attack ()
Cavitation Length (l)
(1)
0.25
0.9774
5o
0.251
(2)
0.5
0.9774
5o
0.3
(3)
1.0
0.9774
5o
0.4667
(4)
2.0
0.9774
5o
0.5
(5)
1.0
0.67585
4o
0.593 3
(6)
1.0
0.825
4o
0.415
(7)
1.0
0.9774
4o
0.2333
(8)
1.0
0.9774
6o
0.6
(9)
1.0
0.67585
3o
0.315
(10)
1.0
0.67585
3.5o
0.478
Case
128
Chapter (4)
results and Discussion
4.3.4 NACA-0025 Hydrofoil Case (1)
In this case study, the present code is used to solve the cavitation of the hydrofoil section NACA 0025 with the three different parameters which were studied for the previous hydrofoil NACA section. The new values of the three parameters are, the cavitation number 0.9774 , the angle of attack = 5o L
and the inflow velocity U 0.5 m / s . The cavity length 0.25 achieved c the cavity closure condition at the cavity end, , given in equation (2.41). Figure (4.42) shows the results of this case.
0.4
hydrofoil l=0.2 l=0.25 l=0.4 l=0.5
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.42): Scaled hydrofoil section NACA-0025 at U 0.5 m / s .
129
Chapter (4)
results and Discussion
Case (2)
In this case, the present code is used with the same three parameters which used in the previous case. There are two parameters that have the same value as the previous case, the cavitation number 0.9774 and the angle of attack = 5o. However, the inflow velocity changed to become U 1.0 m / s . The results of this case are shown in figure (4.43). It is clear that the cavity length
L 0.419 achieved the cavity closure condition at the cavity c
end, , given in equation (2.41).
0.4
hydrofoil l=0.3 l=0.4 l=0.419 l=0.6
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.43): Scaled hydrofoil section NACA-0025 at U 1.0 m / s .
130
Chapter (4)
results and Discussion
Case (3)
As the previous two cases, the parameters which have the same values are, the cavitation number 0.9774 and the angle of attack = 5o. The inflow velocity takes a new value U 2.0 m / s . . The results of this case are shown in figure (4.44). The present code finds the cavity length
L 0.427 c
which achieved the cavity closure condition at the cavity end, , given in equation (2.41).
0.4
hydrofoil l=0.3 l=0.4 l=0.427 l=0.6
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.44): Scaled hydrofoil section NACA-0025 at U 2.0 m / s .
131
Chapter (4)
results and Discussion
Case (4)
In this case study, three different parameters are prescribed and the present code is used to find the cavity length
L 0.278 . The three parameters c
are, the cavitation number 0.67585 , the inflow velocity U 1.0 m / s and the angle of attack = 4o. The results of this case are shown in figure (4.45). It is clear that the cavity closure condition at the cavity end, , given in equation (2.41) is obtained at
L 0.278 . c
0.4
hydrofoil l=0.2 l=0.278 l=0.4 l=0.5
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.45): Scaled hydrofoil section NACA-0025 at 0.67585 .
132
Chapter (4)
results and Discussion
Case (5)
As the previous case, the two parameters which have the same values are, the angle of attack = 4o and the inflow velocityU 1.0 m / s . However, the cavitation number takes a new value, 0.825 . The cavity length, , that L 0.238 . The results of c
achieved the closure condition in equation (2.41) is this case are shown in figure (4.46).
0.4
hydrofoil l=0.2 l=0.238 l=0.4 l=0.5
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.46): Scaled hydrofoil section NACA-0025 at 0.825 .
133
Chapter (4)
results and Discussion
Case (6)
In this case, the cavitation number takes a value 0.9774 . The two parameters, the angle of attack, , and the inflow velocity, U , still have the same values. The cavity length, , that achieved the condition in equation (2.41) L 0.212 . The results of this case are shown in figure (4.47). c
0.4
hydrofoil l=0.2 l=0.212 l=0.4 l=0.5
0.3
0.2
y=Y/c
is
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.47): Scaled hydrofoil section NACA-0025 at 0.9774 .
134
Chapter (4)
results and Discussion
Case (7)
In this case, the angle of attack, , is studied as a variable parameter. The other two parameters, the cavitation number 0.9774 and the inflow velocity U 1.0 m / s are prescribed at the shown value. The present code is used with
an angle of attack = 6o and gives the cavity length
L 0.5 which achieved c
the closure condition in equation (2.41). In figure (4.48) the results of this case are shown at different lengths.
0.4
hydrofoil l=0.3 l=0.4 l=0.5 l=0.6
0.3
y=Y/c
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.48): Scaled hydrofoil section NACA-0025 at = 6o.
135
Chapter (4)
results and Discussion
Figure (4.49) explains the changes of cavitation shape and length with freestream velocity (from U = 0.5 to 2.0 m/s). The cavity height is slightly affected by the increase of U. However, the cavity length increases with free-stream velocity.
0.5
Hydrofoil U=0.5 m/s - l=0.25 U=1.0 m/s - l=0.419 U=2.0 m/s - l=0.427
0.4
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.49): Effect of free-stream velocity, NACA-0025.
136
Chapter (4)
results and Discussion
Figure (4.50) shows the effect of cavitation number on the cavity shape and length at = 4o and U = 1.0 m/s. The cavitation height and length decrease as increases.
0.5
Hydrofoil Cav.Num.=0.676 - l=0.278 Cav.Num.=0.825 - l=0.238 Cav.Num.=0.977 - l=0.212
0.4
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.50): Effect of cavitation number, NACA-0025.
137
Chapter (4)
results and Discussion
Figure (4.51) illustrate the effect of the angle of attack () for = 0.9774. A range of (from 4o to 6o) was tested for U = 1.0 m/s. Again, the height and length of cavitation increase with .
0.5
Hydrofoil Alpha=4.0 deg. - l=0.212 Alpha=5.0 deg. - l=0.419 Alpha=6.0 deg. - l=0.5
0.4
y=Y/c
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l=X/c
Figure (4.51): Effect of angle of attack, NACA-0025. From the previous results, it is shown that increase of the free stream velocity makes the cavity length increased and the same effect occurs when angle of attack increased. Otherwise, increased of the cavitation number makes the cavity length decreased.
138
Chapter (4)
results and Discussion
The summery of the previous results are listed in the table (4.5). Table (4.5) Results of NACA-0025.
Free-Stream Velocity (U) m/s
Cavitation Number (σ)
Angle of Attack ()
Cavitation Length (l)
(1)
0.5
0.9774
5o
0.25
(2)
1.0
0.9774
5o
0.419
(3)
2.0
0.9774
5o
0.427
(4)
1.0
0.67585
4o
0.278
(5)
1.0
0.825
4o
0.238
(6)
1.0
0.9774
4o
0.212
(7)
1.0
0.9774
6o
0.5
Case
139
Chapter (4)
results and Discussion
4.3.5 Comparisons between hydrofoil sections To identify the effect of the hydrofoil section (maximum thickness), comparisons should be made between their results at the same operating conditions. A summary of the results are shown in tables (4.6), (4.7) and (4.8) for the effect of cavitation number (), angle of attack () and the free-stream velocity (U), respectively. Considering figures (4.15), (4.26), (4.39) and (4.50), it is clear that the cavitation length decreases with the hydrofoil thickness at the same value of . However, it seems that the height increases with the hydrofoil thickness. For all NACA sections, it is clear that the cavitation length decreases with increasing of the cavitation number as shown in table (4.6).
140
Chapter (4)
results and Discussion
Table (4.6) Effect of .
NACA Section
NACA-16006
NACA-0012
NACA-0015
NACA-0025
Angle of Attack (α)
4.0˚
4.0˚
4.0˚
4.0˚
Free-stream Velocity (U) m/s
1.0
1.0
1.0
1.0
141
Cavitation Number (σ)
Cavitation Length (l)
0.852
0.47
1.097
0.4
1.269
0.225
0.825
0.442
0.977
0.35
1.125
0.244
0.676
0.593
0.825
0.415
0.977
0.233
0.676
0.278
0.825
0.238
0.977
0.212
Chapter (4)
results and Discussion
Figures (4.16), (4.27), (4.41) and (4.51) show that similar behavior as the behavior of the cavitation number is noticed for the cavitation length. Generally, the cavitation length varies with the hydrofoil thickness at the same angle of attack (e.g. = 5o). However, for all NACA sections, it is clear that the cavitation length increases with increasing of the angle of attack as shown in table (4.7). Table (4.7) Effect of .
NACA Section
NACA-16006
NACA-0012
Cavitation Number (σ)
1.097
0.977
0.977
Free-stream Velocity (U) m/s
1.0
1.0
1.0
NACA-0015 0.677
NACA -0025
0.977
1.0
1.0
142
Angle of Attack ()
Cavitation Length (l)
3.0˚
0.34
4.0˚
0.4
5.0˚
0.44
3.0˚
0.247
4.0˚
0.35
5.0˚
0.60
4.0˚
0.23
5.0˚
0.47
6.0˚
0.60
3.0˚
0.32
3.5˚
0.48
4.0˚
0.59
4.0˚
0.212
5.0˚
0.419
6.0˚
0.5
Chapter (4)
results and Discussion
Figures (4.14), (4.25), (4.38) and (4.49) illustrate that the cavitation height increases with hydrofoil thickness at the same free-stream velocity (e.g. U = 1.0 m/s). However, the cavitation length decreases with hydrofoil
thickness. For all NACA sections, it is clear that the cavitation length increases with increasing of the free-stream velocity except the NACA 16006, as shown in table (4.8). Table (4.8) Effect of U.
NACA Section
NACA-16006
NACA-0012
NACA-0015
NACA-0025
Angle of Attack (α)
4.0˚
4.0˚
5.0˚
5.0˚
Cavitation Number (σ)
1.097
0.977
0.977
0.977
143
Free-stream Velocity (U) m/s
Cavity Length (l)
0.5
0.425
1.0
0.4
2.0
0.3
3.0
0.25
0.5
0.3
1.0
0.35
2.0
0.5
4.0
0.62
0.25
0.25
0.5
0.3
1.0
0.47
2.0
0.5
0.5
0.25
1.0
0.419
2.0
0.427
Chapter (4)
results and Discussion
4.3.6 COMPARISON WITH OTHERS To emphasis on the validity and importance of the present boundary element algorithm, a comparison is made between the present results and those of Shimizu et al. [85]. They used the large eddy simulation (LES) method, which is a well-known computational fluid dynamics technique for real viscous turbulent flows, to predict cavitating flow around a two-dimensional NACA0015 hydrofoil. They performed the computations at Re = 1.2106 and = 8o. Their computational mesh consisted of 1.2 million finite elements. Figure (4.52) shows the comparison between the results of Shimizu et al. [85] (Fig. 4.52-a) and the present results (Fig. 4.52-b) for the NACA-0015 hydrofoil. The present results were carried out at a corresponding Reynolds number of 1.2106 for real viscous flow. Reynolds number (Re) is calculated based on the free-stream velocity (U) and chord length (c). Thus, Re = (U c)/ and is the fluid kinematic viscosity. As can be seen, the present results compare well to those of [85] when concerning the dimensionless cavitation length (l). Table (4.9) shows a comparison between the present results and those of [85] for the dimensionless cavitation length (l). The maximum difference between the present results and those of [85] is 4% (This value of 4% is relative to the hydrofoil chord length, C). The main advantages of the present algorithm are its relative simplicity as
144
Chapter (4)
results and Discussion
well as its small software, hardware and run-time requirements in comparison with the much more complicated real viscous flow codes.
(top = 1.6, middle = 1.15, bottom = 0.9)
Figure (4.52-a): Results of Shimizu et al. [85], = 8o. 0.5 Hydrofoil Cav. Num. =0.90 - l = 0.6 Cav. Num. =1.15 - l = 0.3 Cav. Num. =1.60 - l = 0.2
0.4
y = Y/C
0.3
0.2
0.1
0
-0.1
-0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
l = X/C
Figure (4.52-b): Present results, = 8o.
Figure (4.52): Comparison between the results of Shimizu et al. [85] and the
present results for NACA-0015 hydrofoil. 145
Chapter (4)
results and Discussion
Table (4.9) Comparison between the present results and those of [85]. Case
1 2 3
0.90 1.15 1.60
Dimensionless cavitation length (l) Present model 0.60 0.30 0.20
146
Dimensionless cavitation length (l) Results of Shimizu et al. [85] 0.63 0.34 0.19
CHAPTER (5) CONCLUSIONS AND FUTURE WORK
Chapter (5)
conclusions
CHAPTER (5) CONCLUSIONS AND FUTURE WORK In the present work, the boundary element method is used with a new numerical algorithm to predict the cavitation around two-dimensional hydrofoils. The main difficulty encountered when predicting the cavitation around the hydrofoil, is the determination of The potential at the leading edge of the hydrofoil. The present algorithm overcomes this difficulty and found the control parameter for various NACA sections. In addition to the previous difficulty, three main parameters and their direct effect on the cavitation are also investigated. The present algorithm was first tested on some existing results and an execllent agreement was obtained, then more computations and results were performed. 5.1
Conclusions
Based on the results of the previous chapter, the following points can be stated: (1)
The present approach compares very well to the results of others and those of real flow simulations.
(2)
The proposed treatment of potential function at the inception of cavitation enhanced considerably the performance of the model.
(3)
Generally, the cavitation height and length are affected by the
148
Chapter (5)
conclusions
maximum thickness of the hydrofoil. (4)
The shape, height and length of the cavity depends on the cavitation number , angle of attack and free-stream velocity U .
(5)
For all tested sections, the height and length of cavitation increase with angle of attack .
(6)
For all sections, the cavitation length decreases as increases.
(7)
Except for NACA16006, the cavity length increases with the free-stream velocity U .
(8)
The values of the control parameter is found within the following ranges, see table (5.1): Table (5.1) The ranges of the control parameter .
Control parameter
Type of NACA
Range
NACA-16006
2.9 3.63
NACA-0012
1.05 3.60
NACA-0015
1.4 3.56
NACA-0025
1.015 2.243
149
Chapter (5)
conclusions
5.2 Suggestions for future work Cavitation phenomenon is very important and complicated. Many parameters that affect cavitation are studied in the present thesis. Others need more investigation. Thus, points of future work may be stated as: 1- Three-dimensional investigations of hydrofoil flow. 2- Expanding investigation to unsymmetrical hydrofoil sections. 3- Investigating the effect of other parameters on the free-surface of the cavity such as time, depth of water, temperature. 4- The viscosity may be also included for real flow solution using other schemes.
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بسم ﷲ الرحمن الرحيم اﻟﻬدف ﻣن اﻟﺑﺣث: ﻳﻬدف اﻟﺑﺣث اﻟﻲ دراﺳﺔ ظﺎﻫرة ﺗﻛوﻳن ﺗﺟوﻳف )اﻟﺗﻛﻬف( ﺣوﻝ اﻟرواﻓﻊ اﻟﻣﺎﺋﻳﺔ ﻣن ﻣﻧظور رﻳﺎﺿـﻲ ﻳﺗﻣﺛــﻝ ﻓــﻲ اﻟﺗﺣﻠﻳــﻝ اﻟرﻳﺎﺿـﻲ ﻟﻣﺗﻐﻳـرات ﺗﻠــك اﻟظــﺎﻫرة و ﻣﺳــﺗﺧدﻣﺎ طرﻳﻘــﺔ اﻟﻌﻧﺎﺻــر اﻟﺣدﻳـﺔ ﻛطرﻳﻘ ــﺔ ﻣﺗﻘدﻣ ــﺔ ﻓ ــﻲ اﻟﺗﺣﻠﻳ ــﻝ اﻟرﻳﺎﺿ ــﻲ ﻛ ــذﻟك ﻳﻬ ــدف اﻟﺑﺣ ــث اﻟ ــﻲ إﺳ ــﺗﻧﺗﺎج اﺳ ــﻠوب ﻏﻳ ــر ﺗﻘﻠﻳ ــدي ﻟﻠﺗﻌﺎﻣﻝ ﻣﻊ ﻫذة اﻟظﺎﻫرة ﻳوﻓر وﻗت اﻟﺣﻝ وﻛذﻟك ﻳﻌطﻲ ﻧﺗﺎﺋﺞ دﻗﻳﻘﺔ ﻓﻲ ﻣﺣﺎوﻟـﺔ ﻻﺿـﺎﻓﺔ ﺟدﻳـد ﻓـﻲ ﻫذا اﻟﻣﺟﺎﻝ .
ﻣﻠﺧص اﻟﺑﺣث: ﺗﻌﺗﺑــر ظــﺎﻫرة ﺗﻛــوﻳن ﺗﺟوﻳــف )اﻟﺗﻛﻬــف( ﺣــوﻝ اﻟرواﻓــﻊ اﻟﻣﺎﺋﻳــﺔ ﻣــن أﻫــم اﻟﺗطﺑﻳﻘــﺎت اﻟﻌﻣﻠﻳــﺔ ﻟﻣﺳﺎﺋﻝ دﻳﻧﺎﻣﻳﻛﺎ اﻟﻣواﺋﻊ .و ذﻟك ﺑﺳﺑب اﻟﺗﻠﻔﻳﺎت اﻟﺗﻰ ﺗﺣدث ﻧﺗﻳﺟﺔ ﻫذة اﻟظـﺎﻫرة و اﻟﺗـﻰ ﺗـؤدى اﻟـﻰ اﻧﺧﻔـ ــﺎض أداء اﻟﻌدﻳـ ــد ﻣـ ــن اﻷﺟﻬ ـ ـزة اﻟﻣﻳﻛﺎﻧﻳﻛﻳـ ــﺔ ﻣﺛـ ــﻝ اﻟرﻓـ ــﺎص ،أﺳـ ــطﺢ اﻟرواﻓـ ــﻊ اﻟﻣﺎﺋﻳـ ــﺔ ﻟﻠﺳـ ــﻔن، ﻧظر ﻷﻫﻣﻳﺔ ﻫذﻩ اﻟظﺎﻫرة ﻓـﻰ اﻟﺗطﺑﻳﻘـﺎت اﻟﻬﻧدﺳـﻳﺔ ﻓﻘـد ﺗـم د ارﺳـﺗﻬﺎ ﻣـن اﻟﺗورﺑﻳﻧﺎت و اﻟﻣﺿﺧﺎت .و اً ﺧﻼﻝ ﻫذﻩ اﻟرﺳﺎﻟﺔ و اﻟﺗﻲ ﺗﻘﻊ ﻓﻲ ﺧﻣﺳﺔ أﺑواب ﻛﺎﻵﺗﻲ:
اﻟﺑﺎب اﻷوﻝ ﻋﺑــﺎرة ﻋــن ﻣراﺟﻌــﺔ ﺗﺎرﻳﺧﻳــﺔ ﻣﻘﺳــﻣﺔ اﻟــﻰ ﺟ ـزﺋﻳن رﺋﻳﺳ ـﻳﻳن .اﻟﺟــزء اﻷوﻝ ﻳﺳــﺗﻌرض ﻛﻳﻔﻳــﺔ ﺣدوث اﻟظﺎﻫرة و أﺳﺑﺎﺑﻬﺎ ﺛم ﻳﺗﻧﺎوﻝ اﻟدراﺳﺔ اﻟﻧظرﻳﺔ و اﻟﺗﺟرﻳﺑﻳﺔ ﻟﻣﻌﺎﻟﺟﺗﻬﺎ ﻟﻌدﻳد ﻣـن اﻟﺑـﺎﺣﺛﻳن ﻣـن ﺧﻼﻝ إﺳﺗﺧدام ﻧﻣﺎذج ﻣﺧﺗﻠﻔﺔ و طرق ﺣﻝ ﻣﺗﻌددة ﺳواء ﻛﺎﻧت ﺗﺣﻠﻳﻠﻳﺔ أو ﻋددﻳﺔ .أﻣﺎ اﻟﺟـزء اﻟﺛـﺎﻧﻰ ﻓﻳﻌرض دراﺳﺔ ﺗﻔﺻﻳﻠﻳﺔ ﻷﺣدى طرق ﻟطرﻳﻘـﺔ اﻟﻌﻧﺎﺻـر اﻟﺣدﻳـﺔ و ﻣﻣﻳزاﺗﻬـﺎ ﻛﺄﺣـدى اﻟطـرق اﻟﻌددﻳـﺔ اﻟﻣﺳﺗﺧدﻣﺔ ﻓﻰ ﺣﻝ ﻣﺛﻝ ﻫذة اﻟﻧوﻋﻳﺔ ﻣن اﻟﺗطﺑﻳﻘﺎت اﻟﻬﻧدﺳﻳﺔ.
اﻟﺑﺎب اﻟﺛﺎﻧﻰ ﻋﺑﺎرة ﻋن ﺟزﺋﻳن أﺳﺎﺳﻳن اﻟﺟزء اﻷوﻝ ﻳﺗﻣﺛﻝ ﻓﻲ اﻟﻧﻣذﺟـﺔ اﻟرﻳﺎﺿـﻳﺔ ﻟﻠﺳـرﻳﺎن ﻓـوق اﻟرواﻓـﻊ ﻣﺗﻣــﺛﻼً ﻓــﻲ اﻟﻣﻌــﺎدﻻت اﻷﺳﺎﺳــﻳﺔ اﻟﻣﻛوﻧــﺔ ﻟﻠظــﺎﻫرة و ﻛــذﻟك اﻟﺷــروط اﻟﺗــﻲ ﻳﺟــب ﺗﺣﻘﻳﻘﻬــﺎ ﻟﻠوﺻــوﻝ ﻟﺣــﻝ ﺗﻠــك اﻟظــﺎﻫرة .اﻟﺟــزء اﻟﺛــﺎﻧﻲ ﻳﺗﻣﺛــﻝ ﻓــﻲ ﻓﺣـص ﺷــﺎﻣﻝ و ﺗﻔﺻــﻳﻠﻰ ﻟﻠﻧﻣــﺎذج اﻟﺗــﻲ ﺗــم إﺳــﺗﺧداﻣﻬﺎ ﻣــن ﻗﺑــﻝ ﻓــﻲ ﻧﻣذﺟــﺔ ﻣﺳــﺄﻟﺔ اﻟﺳ ـرﻳﺎن ﺑــدءاً ﺑﺎﻟﺳ ـرﻳﺎن ﻋﺑــر ﻋواﺋــق أرﺳــﻳﺔ و ﻣﺎﺋﻠــﺔ ﺑزواﻳــﺎ ﻣﺧﺗﻠﻔ ـﺔ ﺛــم اﻟﺳرﻳﺎن ﻋﺑر اﻟرواﻓﻊ و ﻫﻰ ﻣﺣﻝ اﻟدراﺳﺔ اﻷﺳﺎﺳﻳﺔ ﻟﻠرﺳﺎﻟﺔ.
اﻟﺑﺎب اﻟﺛﺎﻟث ﻳﻘــوم ﻓﻳــﻪ اﻟﺑﺎﺣــث ﺑﻌــرض ﺗﻔﺻــﻳﻠﻲ و إﺳــﺗﻧﺗﺎج ﻛــﻝ اﻟﻣﻌــﺎدﻻت و اﻟﺗﻛــﺎﻣﻼت ﻷﺣــد أﺷــﻬر اﻟطــرق اﻟﻌددﻳــﺔ و ﻫــﻰ طــرق اﻟﻌﻧﺎﺻــر اﻟﺣدﻳــﺔ ﻣﺳــﺗﺧدﻣﺎً إﺣــدى اﻟطــرق ﻓﻳﻬــﺎ .ﺑﺎﻹﺿــﺎﻓﺔ اﻟــﻰ ﺗﻘــدﻳم
اﻟﻌدﻳد ﻣن اﻟﺧوارزﻣﺎت اﻟﺳﺎﺑﻘﺔ و اﻟﻣﺳﺗﺧدﻣﺔ ﻟﺣﻝ ﻣﺳـﺄﻟﺔ ﺗﻛـوﻳن ﺗﺟوﻳـف )اﻟﺗﻛﻬـف( ﺛـم ﻳـﺗم ﻋـرض اﻟﺧوارزم اﻟﺧﺎص ﺑﺎﻟﺑﺎﺣث و اﻟذى ﺗﻐﻠب ﺑﻪ ﻋﻠﻰ ﻣﺷﻛﻠﺔ إﻳﺟﺎد ﻗﻳﻣﺔ اﻟﺟﻬد اﻟﻣﺿـطرب ﻋﻧـد ﻣﻘدﻣـﺔ ﺳطﺢ اﻟراﻓﻊ اﻟﻣﺎﺋﻰ و ﻫﻰ إﺣدى اﻟﻣﺷﻛﻼت اﻷﺳﺎﺳﻳﺔ ﻓﻰ ﻫذا اﻟﻧوع ﻣن اﻟﺗطﺑﻳﻘﺎت.
اﻟﺑﺎب اﻟراﺑﻊ ﻳﻘــدم اﻟﻧﺗــﺎﺋﺞ اﻟﻌددﻳــﺔ اﻟﺗــﻰ ﺗوﺻــﻝ اﻟﻳﻬــﺎ اﻟﺑﺎﺣــث ﻣــن ﺧــﻼﻝ د ارﺳــﺔ أرﺑﻌ ـﺔ ﻧﻣــﺎذج ﻣﺧﺗﻠﻔــﺔ ﻷﺳطﺢ رواﻓﻊ ﻣﺎﺋﻳﺔ ﻓﻰ ظﻝ ﺗﺄﺛﻳر ﺛﻼث ﻋواﻣﻝ أﺧرى وﻫﻰ :ﺳـرﻋﺔ اﻟﻣـﺎﺋﻊ ،زاوﻳـﺔ إﺻـطدام اﻟﻣـﺎﺋﻊ ﻣــﻊ ﺳــطﺢ اﻟ ارﻓــﻊ اﻟﻣــﺎﺋﻰ و درﺟــﺔ ﺗﻛــون اﻟﺗﺟوﻳــف )اﻟﺗﻛﻬــف( .و ﻛﺎﻧــت اﻟﻧﺗــﺎﺋﺞ ﻣﻘﺑوﻟــﺔ ﺟــداً ﻣﻘﺎرﻧــﺔ ﺑﻧﺗﺎﺋﺞ ﺑﺎﺣﺛﻳن ﺳﺎﺑﻘﻳن.
اﻟﺑﺎب اﻟﺧﺎﻣس ﻳﻌــرض ﻓﻳــﺔ اﻟﺑﺎﺣــث اﻟﻧﺗــﺎﺋﺞ اﻟﻌﺎﻣــﺔ اﻟﺗــﻰ ﺗوﺻــﻝ اﻟﻳﻬــﺎ ﻣــن ﺧــﻼﻝ د ارﺳــﺔ اﻟظــﺎﻫرة ﻣﺣــﻝ اﻟﺑﺣــث و اﻟﻌواﻣــﻝ اﻟﻣﺧﺗﻠﻔــﺔ اﻟﺗــﻲ ﺗــؤﺛر ﻓــﻲ دﻗــﺔ اﻟﻧﺗــﺎﺋﺞ .ﺑﺎﻻﺿــﺎﻓﺔ إﻟــﻰ ﻣﻘﺗرﺣــﺎت ﻹﺳــﺗﻛﻣﺎﻝ ﻫــذﻩ اﻟدراﺳﺔ ﻓﻰ دراﺳﺎت ﻣﺳﺗﻘﺑﻠﻳﺔ.
اﻟﻠﻬﻢ أﻧﻔﻌﻨﻰ ﺑﻤﺎ ﻋﻠﻤﺘﻨﻰ و ﻋﻠﻤﻨﻰ ﻣﺎ ﻳﻨﻔﻌﻨﻰ و زدﻧﻰ ﻋﻠﻤﺂ
جامعة الزقازيق كلية الھندسة قسم الرياضيات و الفيزياء الھندسية
طريقة العناصر الحدية و تطبيقاتھا فى التحليل الغير خطى للسريان فوق الروافع مقدمة من المھندسة
نورھان عالء الدين محمد بكالوريوس الھندسة المدنية-كلية الھندسة جامعة الزقازيق )(١٩٩٩ البحث مقدم للحصول على درجة الماجستير فى الرياضيات الھندسية كلية الھندسة-جامعة الزقازيق تحت أشراف أ.د /سعيد جميل أحمد أستاذ الرياضيات الھندسية قسم الرياضيات و الفيزياء الھندسية كلية الھندسة – جامعة الزقازيق جمھورية مصر العربية
أ.د.م /أحمد فاروق عبد الجواد أستاذ مساعد ھندسة القوى الميكانيكية قسم ھندسة القوى الميكانيكية كلية الھندسة – جامعة الزقازيق جمھورية مصر العربية
جامعة الزقازيق كلية الھندسة ٢٠٠٧
