CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

HYRODYNAMIC MASS OF BLUFF BODIES

WITH AND WITHOUT CAVITY

A thesis submitted in partial fulfillment of the requirements

For the Degree of Master of Science

In

Mechanical Engineering

by

Mohamed Elgabaili

August 2012

ii

The thesis of Mohamed Elgabaili is approved:

Dr. Larry Caretto Date

Dr. Abhijit Mukherjee Date

Professor, Hamid Johari, Chair Date

California State University, Northridge

iii

ACKNOWLEDGMENT

“It is He Who brought you forth from the wombs of your mothers when ye knew nothing;

and He gave you hearing and sight and intelligence and affections: that ye may give

thanks (to Allah)” [Quran, 16:78]. I thank Allah the most merciful and the most gracious

for his graces, blessings and everything. Then I would like to extend my greatest

gratitude and appreciation toward Professor Hamid Johari for his support, attention and

guidance to make this work pleasant for me. I would also like to thank my graduate thesis

committee member Dr. Larry Caretto and Dr. Abhijit Mukherjee for their constructive

input, my family for their support and encouragement.

iv

TABLE OF CONTENTS

Signature Page .................................................................................................................... ii

Acknowledgement ............................................................................................................. iii

List of Figures .................................................................................................................... vi

List of Tables .......................................................................................................................x

Nomenclature ............................................................................................................................... xii

Abstract ............................................................................................................................ xiv

1.Introduction ................................................................................................................................. 1

2.Theory ...............................................................................................................................4

2.1. Potential Flow ....................................................................................................................... 4

2.2. Concept of Hydrodynamic Mass ....................................................................................... 5

2.3. Relative kinetic Energy ....................................................................................................... 6

3. Numerical Method .................................................................................................................. 11

3.1. Overview ............................................................................................................................. 11

3.2. Defining the problem in COMSOL................................................................................. 11

3.2.1. The Physics Interface Setup ....................................................................................... 11

3.2.2. Global Definition ......................................................................................................... 14

3.2.3. Creating the geometry ................................................................................................. 15

3.2.4. Boundary Conditions ................................................................................................... 16

3.2.5. Meshing ......................................................................................................................... 32

4. Classical Models and Verification ........................................................................................ 36

4.1. Classical Models ................................................................................................................ 36

4.1.1. Circle (2D Cylinder) ................................................................................................... 36

4.1.2. 3D Cylinder Model ...................................................................................................... 38

4.1.3. Sphere Model ................................................................................................................... 21

v

4.1.4. Disk Model.................................................................................................................... 32

4.2. Refinements ........................................................................................................................ 34

4.2.1. Computation Domain Refinements ........................................................................... 24

4.2.2. Mesh Refinements ....................................................................................................... 28

4.3. Verifications ....................................................................................................................... 32

5. Bluff Bodies ............................................................................................................................. 33

5.1. Solid Cylinder ................................................................................................................... 33

5.1.1. Solid Cylinder Correlation .......................................................................................... 34

5.2. Cup ...................................................................................................................................... 37

5.2.1. Cup Results ................................................................................................................... 45

5.2.2. Hydrodynamic Mass Correlation for Cup ................................................................ 43

5.3. Parachute Canopies ........................................................................................................... 48

5.3.1. Canopy Models............................................................................................................. 48

5.3.2. Preparing Canopy Models in COMSOL ................................................................... 60

5.3.3. Potential Flow Results for Parachute Models .......................................................... 52

5.3.4. Correlation of Hydrodynamic Mass Parachutes Canopies ..................................... 58

6. Summary and Conclusion ...................................................................................................... 64

Reference ..................................................................................................................................... 67

Appendix A ................................................................................................................................. 68

Appendix B .................................................................................................................................. 80

Appendix C .................................................................................................................................. 75

Appendix D .................................................................................................................................. 83

vi

LIST OF FIGURES

Figure Title Page

3.1 The organization of COMSOL Desktop 13

3.2 Screen capture showing COMSOL space dimensions 12

3.3 Screen capture showing the COMSOL Multiphysics modules 12

3.4 A template of PDEs 13

3.5 COMSOL CAD tools 15

3.6 Potential flow streamlines around a cylinder 17

3.7 Inlet and outlet uniform-stream boundary conditions 18

3.8 Computational domain boundaries with zero normal velocity condition 18

3.9 Zero Flux setting window 21

3.10 Source/Flux setting window 23

3.11 Mesh toolbar with varieties of elements and features 22

3.12 Mesh setting window 23

3.13 Screen capture showing element size parameters 24

4.1 Streamlines over the cylinder 26

4.2 COMSOL flow model for 2D cylinder 27

4.3 Cylinder model 28

4.4 Parameters setting for the cylinder model 25

4.5 The cylinder Model in COMSOL 31

4.6 COMSOL model for sphere in potential flow 33

4.7 Schematic of the disk 32

4.8 COMSOL model of the disk 33

vii

4.9 The effect of ratio on the hydrodynamic mass coefficient of circle 35

4.10 Hydrodynamic mass coefficients for a 3D cylinder with different

values 36

4.11 The effect of domain size on hydrodynamic mass coefficient of a sphere 36

4.12 The effect of domain size on hydrodynamic mass coefficient of a disk 37

4.13 The effect of mesh type on the hydrodynamic mass coefficient of the

cylinder 38

4.14 Mesh refinements effect on the hydrodynamic mass coefficient of the

sphere 35

4.15 Mesh refinement effects on the hydrodynamic mass coefficient of the

disk 35

4.16 Predefined normal element size parameters 41

5.1 Solid cylinder aligned with the flow. Radius and length are indicated

by and . 43

5.2 Curve fitting to 45

5.3 Comparison of added mass values computed numerically with the

correlation in Eq. 5.3 46

5.4 Schematic views of cup 47

5.5 Various dimension settings that were applied to a cup model 48

5.6 Streamlines for a cup with 45

5.7 Streamlines for a cup with 51

5.8 Streamlines for a cup with 51

5.9 Velocity distributions on cup surfaces 52

5.10 Hydrodynamic mass of cup as a function of outer radius 54

5.11 Added mass of cup versus its length 54

5.12 The influence of cup‟s thickness on the added mass of cup 55

5.13 Prediction error of the correlation in Eq. 5.5 for various H/D. 57

viii

5.14 Prediction in error of the correlation in Eq. 5.5 with for various thicknesses

ratio‟ and fixed and 57

5.15 Canopy Shapes from [9] 59

5.16 Free stream flow direction on computational domain of Model 8 60

5.17 The tetrahedral mesh elements for Model 8 61

5.18 Slices of velocity potential for Model 1 52

5.19 Slices of velocity potential for Model 8 63

5.20 Streamlines on a y-z plane for Model 1. 64

5.21 Streamlines on a y-z plane for Model 8. 64

5.22 Distribution of velocity component in the y-direction for Model 1 65

5.23 Distribution of the velocity component in y-direction for Model 8 65

5.24 Distribution of the velocity component in x-direction for Model 1 66

5.25 Distribution of the velocity component in x-direction for Model 8 66

5.26 Distribution of the relative kinetic energy on a cut plane at the opening

of Model 1 67

5.27 Distribution of the relative kinetic energy on a cut plane at the opening

of Model 8 67

5.28 The Maximum projected area of Model 5 69

5.29 Comparison between hydrodynamic mass for disk, the solid canopies

and the correlation in Eq. 5.7 72

5.30 A Comparison between the correlations and numerically computed

added mass of parachute canopies 73

1. C Drawing views of Model 1 85

2. C Drawing views of Model 2 86

3. C Drawing views of Model 3 87

4. C Drawing views of Model 4 88

ix

5. C Drawing views of Model 5 89

6. C Drawing views of Model 6 90

7. C Drawing views of Model 7 91

8. C Drawing views of Model 8 92

x

LIST OF TABLES

Table Title Page

2.1 Hydrodynamic mass of symmetric bodies 8

4.1 A list of required parameters in the Model Builder 27

4.2 Normal mesh refinement parameters 43

4.3 A comparison of numerical and analytical solutions of hydrodynamic

mass coefficient 42

5.1 Different value of hydrodynamic mass coefficient for different size of cup 53

5.2 Maximum projected area, equivalent diameters and the calculated

hydrodynamic masses for each model 69

5.3 Hydrodynamic mass coefficient C, for parachute models 70

5.4 Performing of nonlinear regression on the first term on the right-side

of Eq. 5.6 71

5.5 Performing regression to find the value of parameter in Eq. 5.6. 72

1. A The verification of hydrodynamic mass coefficient, Cn of circle 78

2. A Verification of hydrodynamic coefficient of the classical cylinder

model 78

3. A Verification of hydrodynamic mass coefficient of the sphere 79

4. A Domain size effect on the hydrodynamic mass coefficient for disk 75

1. B Hydrodynamic mass results of solid cylinder aligned to the flow 80

2. B Spreadsheet of the regression solver to correlate data for hydrodynamic 82

mass of cup

1. C A list of dimensions for Model 1 85

2. C A list of dimensions for Model 2 86

3. C A list of dimensions for Model 3 87

4. C A list of dimensions for Model 4 88

xi

5. C A list of dimensions for Model 5 89

6. C A list of dimensions for Model 6 90

7. C A list of dimensions for Model 7 91

8. C A list of dimensions for Model 8 92

1. D Hydrodynamic mass versus domain sizes for Model 1 83

2. D Hydrodynamic mass versus domain sizes for Model 2 83

3. D Hydrodynamic mass versus domain sizes for Model 3 84

4. D Added mass versus domain sizes of Model 4 84

5. D Added mass versus domain sizes of Model 5 84

6. D Hydrodynamic mass versus domain size of Model 6 94

7. D Added mass versus domain size for Model 7 95

8. D Added mass versus domain size for Model 8 95

xii

NOMENCLATURE

Velocity vector

Velocity potential

Velocity component in x-direction

Velocity component in y-direction

Velocity component in z-direction

Dynamic force

Mass of the accelerated body

Hydrodynamic mass or the added mass

Numerically computed value of the added mass

Velocity of the accelerated body or the uniform stream velocity

Hydrodynamic mass coefficient

Numerically computed value of the hydrodynamic mass coefficient

Drag coefficient

Surface area of the body

Density of the fluid medium

Total Enclosed volume of the moving body

Enclosed volume of the body (volume of the cavity)

Relative velocity of the flow with respect to the accelerated body

Azimuthal angle

Velocity component in -direction

Velocity component in the radial direction

Relative velocity component in -direction

Relative velocity component in the radial direction

xiii

Radius

Height of cylinder, cup, and parachute canopies

Thickness

Depth

Normal unit vector

Side length of the computational domain

Outer radius

Inner radius

Maximum projected area of parachute canopy

Equivalent diameter of the maximum projected area

Equivalent radius of the maximum projected area

xiv

ABSTRACT

Hydrodynamic Mass of Bluff Bodies with and without Cavity

by

Mohamed, Elgabaili

Master of Science in Mechanical Engineering

The hydrodynamic mass is virtual mass that can be added to the mass of moving body

through fluid medium to account for the resistant due to pressure gradient. In this

research, potential flow about bluff bodies with and without cavity was computed

numerically to obtain the hydrodynamic mass. All the geometries are idealized, rigid, and

nonporous and consist of cylinders and cups facing the flow and eight models of round

parachute canopies during inflation. The numerical solution was carried out using finite

element solver, COMSOL Multiphysics. In order to ensure the validity of the results, the

solver was first used to find the solution for certain classical geometries that have

analytical expressions for added mass. From the numerical results for the added mass of

the cylinders, cups, and parachute canopies, correlations were developed to show the

influence of various geometrical parameters.

1

Chapter 1

Introduction

The concept of added mass or virtual mass has been the subject of interest for many

years for its significance in hydrodynamic analysis and fluid-structure interactions.

Having sufficient information regarding the added mass for a certain object leads to

knowledge of the inviscid hydrodynamic forces acting on that object. This kind of

information is necessary for enhancing existing designs or developing a new one.. Prior

work in fluid-structure interaction (FSI) investigated the added mass of symmetric

objects. The added mass‟s term was first introduced in 1828 by Friedrich Bessel. During

his experimental work on the motion of a pendulum in a fluid, Bessel discovered that

when the pendulum moved in fluid medium the motion had a longer time period than

when it moved in vacuum. The investigation of this phenomenon revealed that the

effective mass of the pendulum is greater than the real mass by a virtual mass due to

pressure resistances in the fluid medium. Since then many studies have been carried out

to find the added mass for a variety of symmetric geometries. For instant in 1843 a study

was carried out by George Stokes on an infinite cylinder moving with uniform

acceleration in infinite domain of fluid, and it revealed that the added mass or virtual

mass for the infinite cylinder is equal to the mass of the fluid it displaced. The common

means to find the added mass in the earliest studies focused on using applied

mathematical theories. In 1879 Sir Horace Lamb published his book Hydrodynamics;

this text includes theoretical solutions of added mass for ellipsoids. More recently, studies

in the 1960s further developed solutions for added mass of more complicated shapes. For

example Ibrahim [1] performed mathematical analysis using the method of images to find

the added mass of spherical shells that represent idealized forms of round parachute

2

canopies. Ibrahim obtained the potential flow about these asymmetrical shells by

intersecting two symmetrical spheres to form an asymmetrical lens, and then applied the

theory of superposition in ideal flow [1]. Such efforts continued to search for solution to

more complicated shapes using the superposition of symmetrical shapes, but the fact that

not all shapes can be presented with superposition prevented general solutions. Real

parachutes canopies have curved bulges on their sides, do not have a complete spherical

shape and their concavities are arbitrary.

Typically in fluid dynamic applications, the hydrodynamic force due to translation

motion of an object is decomposed into a viscous drag and potential flow force. The

hydrodynamic force decomposition presented by Morrison [13] as follows:

( )

(1.1)

where is the mass of the object, is the added mass based on the potential flow

approximation, is the instantaneous velocity of the object, is the drag coefficient

and is the surface area of the object. Afterward, Lighthill [11] developed the concept

which validates a decomposition of hydrodynamic forces into potential flow and vortex-

flow forces. The theory began from D‟Alembert theorem introducing that steady

incompressible potential flow with uniform stream has no force exerted on a body

exposed to it. In incompressible potential flow there is no single fluid element having

non-zero vorticity or dilation. It is thus deemed that in more realistic flow, any element

with non-zero vorticity or dilation may be considered as a source of the hydrodynamic

force and the potential flow force is only introduced when there is acceleration in the

flow. Based on this theory of hydrodynamic force decomposition, experimental works

were conducted to develop better understanding of fluid elements surrounding bluff

3

bodies. For instance, an expression for estimating the instantaneous hydrodynamic force

on mobile body in fluid medium was developed by Noca et al. [12], this expression

requires the knowledge of the hydrodynamic mass, based on the potential flow. Desabrais

[9] conducted experimental work on flexible parachute canopy during inflation; one of

the experiment‟s aims was to examine whether the primary source of the drag force is the

result of an unsteady potential flow (added mass force) or that associated with the

production of vorticity in the wake behind the canopy. The study showed that the

unsteady potential force contributed to no more than 10% of the total peak opening force.

This conclusion was based on assuming constant hydrodynamic mass coefficient for the

canopy during the inflation, since there was no information available regarding the

hydrodynamic mass of the canopy during different stages of the inflation. The primary

motivation for this research is to compute the hydrodynamic mass of bluff bodies with

arbitrary concavities such as parachute canopies. An extension of this objective is to

correlate the computed added mass with the geometric parameters so that such

correlations may be used for design purposes. To achieve these objectives, the velocity

potential was obtained numerically using COMSOL Multiphysics Solver (FEMLAB),

and then the added mass was calculated by integrating the relative kinetic energy of the

flow over the entire computational domain. To test the validity of numerical scheme, the

numerical approach was applied to the symmetrical classical geometries first, and the

computed mass was compared with the available theoretical values. This research

provides better estimates for the potential flow force as a result of the computed

hydrodynamic mass. The current CFD models are limited for the fully inflated parachute

models since the canopy geometry changes dynamically during inflation.

4

The body of this report is comprised of six chapters. Chapter 2 discusses potential flow

theory and presents the governing equation; some background information regarding the

physical concept of hydrodynamic mass is also provided in this chapter. Chapter 3

describes the numerical approach and the steps that were taken to set up the solver such

as defining the physical interface and boundary conditions. Chapter 4 explores the results

for classical geometries and compares the analytical solutions with the obtained

numerical ones. Various computational domains and mesh definitions were experimented

with in this chapter. Mesh refinement was carried out by adjusting the elements size.

Chapter 5 presents the results of the added mass analysis for solid cylinders aligned with

the flow, cups facing the flow, and several parachute models. Specific correlations of

added mass were also developed. Chapter 6 provides a summary of the findings and

possible topics for future work in this field.

5

Chapter 2

Theory

In this research potential flow theory was utilized to arrive at numerical solutions for

hydrodynamic mass of an arbitrary object. All other effects associated with boundary

layers and viscous wakes may be separately addressed.

2.1. Potential Flow

The potential flow is an ideal flow with irrotational motion and zero viscosity. In reality

such an ideal situation doesn‟t exist, but if there is no significant viscous stresses that

could cause any considerable rotational motion or separations in the flow pattern, the

flow can be deemed ideal [2]. Potential flow is dictated by geometry; the flow pattern

mainly depends on the shape of the body and any nearby interacting walls [2].

In potential flow, the term velocity potential is used to define a scalar function whose

gradient provides the velocity field [3].

(2.1)

or in term of each component of velocity:

,

,

. (2.2)

There is no rate of expansion in incompressible flow, thus the divergence of velocity

vector is equal to zero and this leads to velocity potential satisfying the Laplace equation

throughout the domain.

( ) (2.3)

6

2.2. Concept of Hydrodynamic Mass

In potential flow, there are no unbalanced shear forces and vorticity is zero. When a

body is accelerated in potential flow, it will act as if it is more massive by an amount

called the hydrodynamic mass or the added mass. As the body moves, it will have to

accelerate the surrounding fluid in addition to its own mass. In other words, the

hydrodynamic mass is the mass of the fluid surrounding the body which will be

accelerated with the body due to the pressure force [4]. The hydrodynamic mass is one of

the key parameters in studying the fluid dynamics around submerged vehicles; its

significance affects the dynamic forces. Therefore, the consideration of the added mass

is very important when applying Newton‟s second law to obtain the dynamic forces

under potential flow consideration.

( ) ( )

(2.4)

Here, ( ) is the velocity of the accelerated body, is the mass of the body and is

the hydrodynamic mass [3].

Traditionally, the hydrodynamic mass is written as

(2.5)

where is the hydrodynamic mass coefficient, is the density of the fluid and is the

enclosed volume of the accelerated body.

The added mass coefficient depends on the geometry of the object and the direction of

movement [3]. For a few symmetric geometries, the added mass can be found

analytically and these values are available in literature [5-7]. Because analytical solutions

7

for non-symmetric shapes do not exist in general, experimental or numerical methods

have to be pursued.

2.3. Relative Kinetic Energy

Usually the hydrodynamic mass is calculated by neglecting the frictional forces (ideal

flow), and therefore the forces stem from pressure gradient in the stream. There are

several ways to find the hydrodynamic mass; one is to equate the total integral of relative

kinetic energy of the fluid surrounding the body with the kinetic energy associated with

the hydrodynamic mass [3]. An example of this approach is given below, starting with

the basic equation (2.6).

∫

(2.6)

Here, is the relative flow velocity with respect to the instantaneous velocity of the

accelerated body, is the mass element. The integral is taken over the entire fluid

volume extending to infinity.

The procedures for implementing this method are to find the velocity field from the

equations that govern the potential flow and then solve Eq. (2.6) for the added mass. In

potential flow, the velocity potential satisfies the Laplace equation, as stated earlier in

Eq. (2.3).

For simple geometries the solution of Eq. (2.3) is available analytically in the literature.

For bodies with arbitrary shapes, Eq. (2.3) needs to be solved numerically. Then, the

velocity field is obtained by taking the gradient of velocity potential, as introduced earlier

in Eq. (2.1). Then relative velocity field is calculated by subtracting the body velocity

from the computed velocity field from Eq. (2.1) as following:

8

(2.7)

To illustrate this method in more detail, the hydrodynamic mass of a sphere moving with

velocity in potential flow is derived below [3].

1- The Solution of the Laplace equation of a sphere with radius R with boundary

conditions as and

on the sphere surface ( ) is [3]:

( )

(

)

(2.8)

Here is the azimuthal angle, and is the radial direction.

2- Taking the gradient of velocity potential gives the velocity components in

tangential and radial directions:

(

)

(2.9)

(

)

(2.10)

To find the relative velocity of the fluid to the body, we subtract the velocity component

of the body from both equations above. Hence, the result will be:

(

)

(2.11)

(

)

(2.12)

Summing the square of Eq. (2.11) and (2.12) we get the square of the relative velocity as

follow:

(

)

(2.13)

The mass increment for the flow in spherical coordinate is given by:

(2.14)

9

Substituting Eqns. (2.13) and (2.14) in Eq. (2.6), and then integrating the left side from

to

∫ ∫

(2.15)

Hence the hydrodynamic mass for a sphere accelerated in potential flow is

(2.16)

Comparing Eq. (2.16) with Eq. (2.5) reveals that the hydrodynamic mass for a sphere is

equal to one-half of the mass of a sphere having a density equal to fluid density. Thus,

one can relate with the hydrodynamic mass coefficient , the volume of the

accelerated body and the fluid density. The hydrodynamic mass coefficients for simple

symmetric shapes calculated theoretically are presented in Table 2.1.

Table 2.1 shows Hydrodynamic mass of symmetric bodies.

In the sphere case, the direction of movement doesn‟t have any effect on the value of

hydrodynamic mass due to symmetry. On the other hand, the direction of motion of a

cylinder has a significant influence on the hydrodynamic mass coefficient. For instance,

the hydrodynamic mass coefficient for cylinder moving in a direction normal to its axis is

Shape Direction of

motion

Hydrodynamic

mass (m’)

Hydrodynamic

mass coefficient

( )

Source

Sphere Horizontal

/Vertical

0.5 [3]

[7]

Cylinder with

length and R

radius

Normal to the

cylinder axis 1.0 [4]

Infinitely thin

Disk with radius

R

Along the disk

axis

[7]

10

equal to the mass of a cylinder of fluid density, and this value is different when the

cylinder moves along its axis. Numerical solutions for hydrodynamic mass coefficient of

finite length cylinders moving along their axis are developed in Chapter 5.

In one of the early studies by Lamb (1932), the technique of solving the velocity

potential to obtain the kinetic energy of the relative velocity field was used to find the

hydrodynamic mass coefficient for ellipsoids in harmonic motion. The circular disk

moving along its axis is quite interesting since the disk radius is the dominant dimension

in the hydrodynamic mass expression, see Table 2.1.

11

Chapter 3

Numerical Method

3.1. Overview

In this project COMSOL Multiphysics Solver (FEMLAB) is used to solve the

hydrodynamic mass for a variety of geometries including the classical geometries such as

cylinder, sphere and disk, and eight geometries of a round parachute canopy during

inflation. COMSOL is a finite element solver that provides solutions for many partial

differential equations in two and three space dimensions and for stationary or time

dependent problems. The Laplace equation solver in COMSOL along with appropriate

boundary conditions was used to solve the velocity potential equation. After defining the

velocity potential numerically, the velocity field was calculated by taking the gradient of

the velocity potential. The final step was to define the hydrodynamic mass using the

relative kinetic energy for the accelerated flow surrounding the moving body as

expressed in Eq. (2.6). The hydrodynamic mass was calculated by the software. In order

to run the program many steps must be taken first, including choosing the physics mode,

creating the geometry, specifying boundary conditions and meshing.

3.2. Defining the problem in COMSOL

This section discusses the set up and the definitions of the problem such as defining

physical interfaces, geometric parameters and boundary conditions. Moreover, various

COMSOL features and their implementation are also discussed.

3.2.1 The Physics Interface Setup

The COMSOL Multiphysics provides number of predefined physics interfaces which

facilitate a quick model set-up for a variety of applications such as heat transfer, fluid

12

flow, structural mechanics, and electromagnetic applications [8]. The COMSOL Desktop

is organized conveniently as shown in Figure 3.1.

Figure 3.1. The organization of COMSOL Desktop

To specify the physics interface three steps has to be done as follow:

1- Specifying a space dimension: When COMSOL Desktop is launched, the first step

is to choose the type of space dimensions 2D or 3D. COMSOL offers a variety of

space dimensions ranging from 3D and 2D to 1D and 0D as show in Figure 3.2.

13

Figure 3.2. Screen capture showing COMSOL space dimensions.

2-Add Physics: Once the blue arrow at the top of Select Space Dimension bar is clicked

the Add Physics bar appears, replacing the Space bar.

Figure 3.3. Screen capture showing the COMSOL Multiphysics modules.

14

In this project the focus will be on the mathematics module. This mathematics module is

comprised of various formulations for PDEs and ODEs that in general represent equation

based modeling for several physics interfaces; the PDE Module is the subject of interest

in this research and COMSOL is prepared with flexibility and options to implement any

PDE system.

Figure 3.4. A template of PDEs.

The core equation in this project is the velocity potential equation, which satisfies the

Laplace equation and there are many means to implement this PDE in COMSOL. One

direct and easy way is to pick the Laplace Equation node from the Classical PDE

template as shown in Figure 3.4. Another path is to choose the Coefficient Form or the

General Form from the PDE template, and then implementing the Laplace equation

15

become a matter of matching coefficients. For example, the Coefficient Form given in

COMSOL is:

( ) (3.1)

In order to transform Eq. 3.1 to the Laplace equation set , and all the other

coefficients ( ) to zero. For more information regarding the physical

meaning of these coefficients, interested reader is referred to the COMSOL manual [8].

By clicking on the Laplace Equation node in the model wizard, the PDE system is

chosen and it will appear with default boundaries on the model builder toolbar.

3- Select study type: After the Classical PDE is selected, the study type template will

show up with various types of study such as stationary, time dependent, and eigenvalue.

In the present study the system is steady state. Therefore, stationary study was chosen.

3.2.2. Global Definition

In this part, parameters and variables relevant to geometric dimensions and flow

conditions are defined, and certain values are set. Some of these values may be reset later

to examine the effect of a particular parameter on the results. In the present study,

geometric parameters are defined for each case, and they vary from one case to another.

On the other hand, there are common parameters between all shapes in this study

particularly in the definitions of the boundary conditions like the value of uniform

freestream velocity , and flow properties such as density which are involved in our

general equation for calculating the hydrodynamic mass. For simplicity, the freestream

velocity and fluid density are set equal to 1 m/s and 1 kg/m,3 respectively.

16

3.2.3. Creating the geometry

Creating the geometry can be achieved through the CAD tools in COMSOL or by one

of COMSOL product interfaces such as CAD Import Module, LiveLink™ for

AutoCAD® , LiveLink™ for Creo™

Parametric , LiveLink™

for SolidWorks ,

LiveLink™ for Pro/ENGINEER and LiveLink™

for SpaceClaim. All these features allow

COMSOL to run an analysis for very complex shapes with high design quality. Right

clicking on the Geometry root node in the Model Builder template, the geometry toolbars

will appears with all the features and options as shown in Figure 3.5. In the present work,

the classical shapes will be modeled using solid modeling and boundary modeling in

COMSOL CAD tools interface. However, the parachute canopy geometries were

modeled and modified in SolidWorks, and then imported into COMSOL using CAD

Import Module feature.

Figure 3.5. COMSOL CAD tools.

17

COMSOL has extraordinary versatility in handling complicated designs with narrow

curved areas and sharp edges. These kinds of special tools and advanced features made it

possible for COMSOL to independently model the geometry and then run the analysis.

Moreover, in case the geometry is imported from Solidworks or other solid modeling

programs, COMSOL has the ability to remodel and repair any geometric errors

associated with the original imported model. It can be seen in the Geometry toolbar that

COMSOL offers solid modeling and boundary modeling, and both are used in our study

of hydrodynamic mass to build the geometries such as the cylinder, sphere, cup and the

computational domain. Boolean operation is applied to combine and form the final

domain. In every case, the geometry for the object and the surrounding medium were

combined together as one entity using the Difference feature in Boolean Operations. The

Difference feature will keep the interior boundaries between the object and the fluid

medium by subtracting the object from the flow medium we ensure that the equation of

interest (Laplace equation) is only implemented in the resultant volume which is only the

fluid volume without the solid object.

3.2.4. Boundary Conditions

Boundary conditions for potential flow differs from the ones for viscous flow; the

potential flow is inviscid flow and the no slip condition cannot be applied on the solid

surfaces since there is no shear stress acting against the flow on the surface. In an ideal

fluid, flow takes the shape of the body on the surface. Therefore, velocity component

normal to the surface is zero. This tendency for potential flow to smoothly follow the

shape of an object as shown in Figure 3.6 is the main effect for zero vorticity and the

ability to introduce the velocity potential.

18

Figure 3.6. Potential flow streamlines around a cylinder.

In order to solve the velocity potential equation ( ), two kind of boundary

conditions need to be implemented. The first type is quite common and known as

uniform-stream conditions, and the second is zero velocity normal to the boundary at the

body surface. In the current study, every case will be subjected to uniform flow with

velocity U far from the body. Thus, the boundary conditions (uniform freestream

conditions) are applied at the inlet and the outlet of the computational domain.

For instant if the uniform flow (U) is in x-direction normal to the inlet boundary plane,

the boundary condition at the inlet plane will be:

U (3.2)

the same condition is applied to the outlet boundary plane, away from the body, the

velocity will be uniform and equal to U.

19

Figure 3.7. Inlet and outlet uniform-stream boundary conditions.

The rest of the boundaries will be subjected to zero velocity normal to boundaries at both

the body and domain surfaces as depicted in Figure 3.8 below.

Figure 3.8. Computational domain boundaries with zero normal velocity condition.

20

The PDE interface in COMSOL Multiphysics has various types of boundary conditions

that can be modified to match different problems. By default once the PDE is chosen and

the type of study (stationary or time dependent) added, two conditions appear in the

Model Builder template under the PDE root node. The two boundary conditions are

Initial Values and Zero Flux boundary condition. The Initial values are defined for

domain entity and it is available in two forms, dependent variable initial value and the

initial time derivative of the dependent variable. The initial dependent variable serves as

an initial value or function for nonlinear and transient study but in our study it is set to

zero as the initial guess is for the stationary case. The initial time derivative is also set to

zero since the present study is stationary.

Second condition added by default is Zero Flux boundary condition; the flux here in the

current study is represented by the normal velocity to the boundaries. Hence this

boundary condition satisfies the zero normal velocity condition of potential flow.

The Zero Flux condition is given by:

( ) (3.3)

Here, is the unit normal vector on the chosen boundary. Eq. 3.3 is applied by default to

all boundaries on the body and domain unless other boundary conditions are assigned to

certain boundary surface. In that case, that boundary will be governed by the new

condition and listed as „„overridden‟‟ boundary in zero flux boundaries list as shown in

Figure 3.9.

21

Figure 3.9. Zero Flux setting window.

In this project, Source/flux boundary parameter in COMSOL is assigned to impose

uniform-stream conditions at both inlet and outlet of the domain. To bring these boundary

conditions into play, right click on the Laplace Equation root node in Model Builder. The

toolbar with various choices of conditions will appear and allow the choice of

Source/Flux. Immediately after clicking the Source/flux, the setting window for

Source/flux comes into view as exhibited in Figure 3.10. The boundary Flux/Source

was set equal to – (freestream velocity) at the inlet. Moreover, the Boundary

Absorption/Impedance Term was set equal to zero. The same thing is achieved at the

outlet boundary with positive sign for ( ) , where is the uniform stream

velocity and it is previously defined in the Parameters window.

22

Figure 3.10. Source/Flux setting window.

After all the previous setting were applied, the computational domain became fully

defined and the next step was to generate the computational grid or mesh by subdividing

the computational domain into small cells. Then GMRES (Generalized Minimum

Residual) iterative method was selected with Incomplete LU factorization to perform the

iteration to find velocity potential for each Model.

23

3.2.5. Meshing

In this part the generation of computational grid or mesh is discussed. Moreover, some

grid setting and types is explored. COMSOL has distinct types of mesh elements for 1D,

2D and 3D. In this project, the Triangular and Tetrahedral mesh elements were used most

frequently to discretize the computational domain. Finer elements are generated by

default in narrow and curved areas. Furthermore, these elements can be adjusted through

the mesh setting window. The first step is to choose the type of elements, more than one

types can be applied for different parts of the geometry. For example, sometime the

geometry has complex items that cannot be meshed with only one type of elements.

Figure 3.11. Mesh toolbar with varieties of elements and features.

Right clicking on the mesh root node, the mesh toolbars will shows up as depicted in

Figure 3.11. This toolbar will allow the choice of elements type. The determination of the

24

elements type depends on the geometry and type of problem. After choosing certain

elements, the mesh setting window will appear under the main setting window in

COMSOL Desktop.

Figure 3.12. mesh setting window.

Two choices are available for the sequence type, User-controlled mesh and physics-

controlled mesh. The physics controlled mesh is tied with the type of physics that was

chosen to define the problem in the Add physics section. In this project, the sequence

type was set to the User-controlled mesh since many meshing parameters were tested. In

addition, the size of elements can be adjusted manually by clicking on the size node in

the Model Builder section. The size setting window is shown in Figure 3.13.

25

Figure 3.13. Screen capture showing element size parameters.

The parameters shown above play major roles in determining the accuracy of the

solutions. Therefore different settings were applied to seek precise solutions. Finer

elements provide more accurate solutions. The maximum element growth rate specifies

the maximum rate at which the element can grow from an area with small elements to

another with larger elements. The value of the maximum element growth rate must be

greater or equal to one. For example, with a maximum element growth rate of 1.4, the

element size can grow by at most 40% from one element to another. Another significant

parameter is the Resolution of curvature; this value specifies the size of boundary

elements confined to curved areas. A smaller value of Resolution of curvature parameter

26

gives finer boundary elements. The last parameter is the Resolution of narrow regions;

this parameter governs the number of layers of elements that are generated in narrow

regions. Values close to one are preferred for accuracy when setting this parameter.

27

Chapter 4

Classical Models and Verification

In this Chapter, several classical geometries will be examined. The added mass

coefficients for these models were calculated numerically and compared with the

available theoretical values for verifications purposes. The classical geometries are very

simple and symmetric such as cylinder, sphere, and disk.

4.1. Classical Models

This section exhibits some geometric aspects for each model and shows brief presentation for

each model preparation into the Solver.

4.1.1. Circle (2D Cylinder)

This model is the only 2D model in this project,. The flow direction is shown along the

streamlines in Figure 4.1.

Figure 4.1. Streamlines over the 2D cylinder.

In order to build the model, some parameters need to be specified in the Parameters

section in the Model Builder. These parameters are shown in Table 4.1.

28

Table 4.1 A list of required parameters in the Model Builder.

The computational domain side‟s length as shown in Table 4.1 was the first

guess to start the run. Larger values were tried till consistent results were obtained. Figure

4.2 illustrates the 2D Cylinder inside the square computational domain with side length .

Figure 4.2. COMSOL flow model for 2D cylinder.

This model was created by using COMSOL CAD tools; square and circle were

implemented into COMSOL graphic window then the Difference parameter in Boolean

29

Operation was applied to subtract the chosen circle from the enclosed domain which in

this case is the square. Furthermore, assigned boundary conditions are enforced, and then

the model was discretized using triangular free elements.

4.1.2. 3D Cylinder Model

This model describes the potential flow over a cylinder in a direction normal to its axis.

Figure 4.3 shows the adapted geometry symbols.

Figure 4.3. Cylinder model.

Various flow and geometric parameters for this model are defined in the parameter

section as depicted in Figure 4.4. The radius is set to a constant value as shown below,

but the cylinder length is changed with domain size and set equal to domain‟s side

dimension to make sure that the entire cylinder length is covered in every trial.

30

Figure 4.4. Parameters setting for the cylinder model.

Figure 4.5 shows the computational domain with the cylinder model inside a cubic

block with a side‟s length L, where L had been defined previously in the parameter

section under the global definition in the Model Builder template. The uniform

freestream velocity was set to be in the direction parallel to the x-axis and normal to the

cylinder axis as demonstrated in Figure 4.5. The same boundary conditions, previously

explained in details in chapter 3, were applied to this model to solve the Laplace

equation. The solution was computed for a variety of domain volumes. The discretization

of the computational domain was done with finer tetrahedral mesh elements. In order to

ensure consistency of the results, the model was operated for different domain sizes and

mesh element settings. The results are exhibited and discussed in details in section 4.2.

31

Figure 4.5. The cylinder Model in COMSOL.

4.1.3. Sphere Model

The reason behind modeling this geometry is for verification purposes since the

analytical solutions can be compared with the numerically computed results. Creating the

model in COMSOL was done using COMSOL CAD tools in the Geometry toolbar, a

sphere with radius R equal to 0.2 meter was created first, and then enveloped with a cubic

block to create the surrounding domain, and finally Difference Boolean operations was

applied to subtract the sphere from the block and keep the interior boundaries. Boundary

32

conditions were assigned and flow parameters similar to the previous case were set. After

defining the boundary conditions on every surface, the computational domain was

meshed with free tetrahedral elements. The size of the elements was adjusted and

refined various times to arrive at precise results. Figure 4.6 shows the computational

domain in COMSOL graphic window, the sphere centered at the center of the cubic block

and the flow direction is parallel to x-axis.

Figure 4.6. COMSOL model for sphere in potential flow.

33

4.1.4. Disk Model

Developing this model stemmed from the need to develop correlation for the

hydrodynamic mass of a cylinder moving along its axis. A disk is essentially an infinitely

thin disk. The comparison between numerically calculated values for the hydrodynamic

mass of the disk with analytical values are explored and discussed in details in the

following section. Similar to the past models, different domain sizes were tried till

consistent results were obtained. The model consists of a disk with radius and

thickness enclosed by a cubic block of length . Difference function in

COMSOL Boolean operations was used to subtract the solid part from the block and then

the Laplace equation interface in COMSOL was applied to the resultant computational

domain. The disk was positioned in the center of the domain facing the flow. Furthermore

this model was modified to represent the model of Cylinder moving along its axis in

potential flow by setting in the parameters section, where is the length of the

cylinder. Schematic view of the disk is shown in Figure 4.7.

Figure 4.7. Schematic of the disk.

34

Uniform freestream velocity boundary conditions were set on the inlet and outlet surfaces

of the computational domain as indicated with green color in Figure 4.8, the rest of

surfaces are enforced to be zero flux boundary conditions. The computational domain

was discretized using tetrahedral elements and resolution factors at curvature and narrow

were set to reasonable values of 0.3 and 0.8, respectively.

Figure 4.8. COMSOL model of the disk.

35

4.2. Refinements

In this section refinements of domain and mesh size were preformed on each of

previous classical models to examine the effects of these parameters on the

hydrodynamic mass coefficient.

4.2.1. Computation Domain Refinements

A suitable computation domain size is very important in order to accomplish reliable

results for the added mass, i.e. independent from the domain size. Hence, the

computational domain size was modified many times until consistent values of added

mass coefficients were found. Integration of the relative kinetic energy of the flow is

preformed all over the computational domain, in order to compute the hydrodynamic

mass, it is crucial to have a computational domain size that reaches to the point where the

pattern of the flow is not affected by the examined object. Here the change of

computational domain is expressed as ratio of the side length, , to the radius, . For

each case of the classical geometries the investigations were done by detecting the

change in hydrodynamic mass coefficient with the increase in the ratio L/R.

The verifications were performed with predefined finer meshes where COMSOL

automatically assigns finer mesh sizes according to the computational domain size.

Investigation of mesh parameter effects are discussed in the next section.

For the circle the results of added mass coefficient start to approach a value of 1.001

approximately at

as shown in Figure 4.9. To carry out the investigation of

domain size effect on the added mass of the circle model, 15 runs were preformed for

36

domain lengths ranging between 3 to 17 meters. For more details, reader is referred to

Table (1.A) in Appendix A.

.

Figure 4.9. The effect of L/R ratio on the hydrodynamic mass coefficient of circle.

The 3D cylinder model is similar to the circle model and the only difference is the third

dimension which is the length of the cylinder, H. Since the classical model is an infinite

cylinder moving in infinite fluid in transverse direction, the representation of this model

in COMSOL is done by setting the length of the cylinder, H, equal to the side length of

the computational domain, L. The results for 3D cylinder also give constant values of

hydrodynamic mass coefficient equal to 1.001 when is larger than 60 as shown in

Figure 4.10. The investigations were accomplished by running the solver for 13 different

domain side lengths which vary from 1 to 18 meters. The results are shown in Appendix

A, Table (2.A).

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

0 20 40 60 80 100

Cn

L/R

37

Figure 4.10. Hydrodynamic mass coefficients for a 3D cylinder with different

values.

In the sphere model the convergence of the results happened at smaller values of as

depicted in Figure 4.11. Constant values of the hydrodynamic mass coefficient equal to

0.5013 were detected first at .

Figure 4.11. The effect of domain size on hydrodynamic mass coefficient of a

sphere.

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

1.05

0 20 40 60 80 100

Cn

L/R

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0 5 10 15 20 25 30

Cn

L/R

38

For the disk the results approached a constant value at which is considerably

small then the cylinder case.

Figure 4.12. The effect of domain size on hydrodynamic mass coefficient of a disk.

After exploring all the results for the added mass coefficients of classical models, it can

be concluded that the domain sizes that provide consistent results, crucially depends on

geometrical aspects of the objects.

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

0 5 10 15 20 25

Cn

L/R

39

4.2.2 Mesh Refinements

In this part a comparison between different mesh sizes was performed to investigate the

effect on the hydrodynamic mass coefficient. The investigation was done using the three

classical models cylinder, sphere and disk with the different mesh settings: extra fine,

finer, fine, normal, coarse, coarser, and extra coarse. The solutions were obtained at

ratio that gave consistent results for each model. Moreover, some element size parameters

were examined individually, using one of the classical models to see their influence on

the results for the hydrodynamic mass coefficient. The element size parameters are

explained in detail in Chapter 3 and shown in Figure 3.13. Numbers of 1, 2, 3, 4, 5, 6 and

7 were chosen to represent the mesh types: extra fine, finer, fine, normal, coarse, coarser,

and extra coarse respectively on the horizontal axis in the Figures 4.13, 4.14 and 4.15

below.

Figure 4.13. The effect of mesh type on the hydrodynamic mass coefficient of the

cylinder.

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0 1 2 3 4 5 6 7 8

Cn

( C

ylin

de

r)

Mesh type

40

Figure 4.14 Mesh refinements effect on the hydrodynamic mass coefficient of the

sphere.

Figure 4.15. Mesh refinement effects on the hydrodynamic mass coefficient of the

disk.

0.46

0.465

0.47

0.475

0.48

0.485

0.49

0.495

0.5

0.505

0 1 2 3 4 5 6 7 8

Cn

( S

ph

ere

)

Mesh type

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0 1 2 3 4 5 6 7 8

Cn

( D

isk)

Mesh type

41

All the results above indicate very good convergence of n at extra fine, finer, and fine

meshes which correspond to 1, 2, and 3 on the horizontal axis of the above plots.

Subsequently the results start to deviate from the exact solutions as the grid sizes get

coarser.

To save time, another approach can be pursued using a coarser mesh, such as normal

mesh, with customized element size parameters. The predefined normal grid parameters

are shown in Figure 4.16. In this approach the maximum size and the minimum element

size were kept fixed and equal to the same values that are given by default in the normal

mesh, since these two factors have a major effect on the solution time. Furthermore,

different selections were chosen for the rest of the parameters to enhance the solution and

make it closer to the one that had been obtained with the extra fine mesh.

Figure 4.16. Predefined normal element size parameters.

42

Table 4.2 displays various changes on the maximum growth rate, the resolution of

curvatures and narrow regions on the cylinder model to obtain accurate results for the

hydrodynamic mass coefficient using the normal mesh. These parameters were modified

in a way that would create finer element close to the cylinder surface and then a gradual

growth in size in neighboring elements as the distance gets farther from the cylinder

surface.

Table 4.2 Normal mesh refinement parameters for cylinder

predefined

mesh

the

maximum

growth

rate

resolution

of

curvature

resolution

of narrow

region

m’(kg)

65 normal 1.5 0.6 0.5 1.595 0.977

65 normal 1.3 0.5 0.6 1.605 0.983

65 normal 1.3 0.4 0.7 1.605 0.983

65 normal 1.2 0.4 0.7 1.607 0.984

65 normal 1.1 0.4 0.7 1.61 0.985

65 normal 1.1 0.3 0.8 1.62 0.992

65 normal 1.2 0.2 0.9 1.62 0.992

It is obvious through observing the results in Table 4.2 that the cylinder hydrodynamic

mass coefficient is falling closer to the value of when the maximum growth

approaches 1.1 for predefined normal mesh with curvature resolution factor falling closer

to the value of 0.2 and resolution of narrow regions gets closer to 1. Despite the short

computing time of the modified normal mesh, it is still not as accurate results as the

finer mesh setting. Finer meshes were specifically used in this project, since they were

43

proven to give the same results that are provided by the extra fine mesh as indicated in

Figures 4.13, 4.14 and 4.15.

4.3. Verifications

In this section the hydrodynamic mass coefficients that had been previously developed

theoretically for classical geometries is compared to the ones that were obtained

numerically using COMSOL. Table 4.3 shows a comparison between numerical and

analytical values of hydrodynamic mass coefficients; and represents the analytical

and numerically computed hydrodynamic mass coefficients, respectively. The

numerically values were obtained under the following conditions: finer mesh size with

resolution of curvatures set equal to 0.3, resolution of narrow regions set equal to 0.8 and

maximum growth rate 1.2.

Table 4.3. A comparison of numerical and analytical solutions of hydrodynamic

mass coefficient.

Classical Models

Flow Direction Hydrodynamic mass formula

Error %

Circle normal to circle axis 1.0 1.001 0.1

Cylinder normal to cylinder axis

1.0 1.001 0.1

Sphere Horizontal/ Vertical

0.5 0.5013

0.26

Disk Along the disk axis 0.85 0.859 1.058824

The values based on the numerical solution have errors between 0.1 and 1.06% which

are quite acceptable. Therefore, the same conditions in term of mesh size and parameters

setting were applied to solve the arbitrary shapes as illustrated in Chapter 5. The

validation using the classical shapes provides the reliability of the numerical method for

its ability to produce accurate results for an arbitrary shape.

44

Chapter 5

Bluff Bodies

The main focus of this project was on developing correlations for hydrodynamic mass

of non-classical models from the numerical data that were computed through COMSOL.

The first non-classical geometry examined was a solid cylinder facing the flow, followed

by cup geometries, and eventually eight round parachute canopy models.

5.1. Solid Cylinder

Unlike the classical model of cylinder, in this model the direction of the flow is aligned

with the cylinder axis as shown in Figure 5.1.

Figure 5.1. Solid cylinder aligned with the flow. Radius and length are indicated by

and .

45

To correlate the hydrodynamic mass with the cylinder‟s geometric dimensions and ,

the hydrodynamic mass was obtained numerically for various dimensions of the solid

cylinder using COMSOL. Table (1.B) in appendix B shows the hydrodynamic mass

values obtained numerically for different values of ratio . At first, a very small

length m and a cylinder diameter ( ) ( ) were

examined. Afterward, the ratio was increased gradually in each run to reach a

maximum value in which is equal to four times . All the results were obtained with

the same flow conditions. In addition, tetrahedral finer meshes were used to grid the

computational domain.

5.1.2. Solid Cylinder Correlation

In this section the hydrodynamic mass data in Table (1.B) were utilized to find a

correlation for the hydrodynamic mass of solid cylinder. The correlation was obtained

using the regression solver in Excel. At First the hydrodynamic mass of solid cylinder

was set equal to factor (the hydrodynamic mass coefficient) multiplied by the volume

of the cylinder as presented in Eq. 5.1. This approach is the same as that for the classical

cylinder. In the classical model is equal to constant value of 1. On the other hand, the

numerical solution for the solid cylinder model showed that varies with the cylinder

dimensions.

(5.1)

In contrast to the classical model, values were not constant when cylinder‟s dimensions

changed. However, through investigations of the results which are exhibited in Table

(1.B), the following relationship between and was found: for any value of

46

and that produced the same value of there is constant value of . This link which

bounds with led to the development of a correlation between and using

the numerical values of hydrodynamic mass . Figure 5.2 depicts the resultant relation

between and for the solid cylinder on logarithmic scale.

Figure 5.2. Curve fitting to .

The curve fitting in Figure 5.2 has a residual squared value of for the

following curve fit:

(

)

(5.2)

Hence, substituting from Eq. 5.2 in Eq. 5.1 led to the following correlation:

(5.3)

To validate the correlation for the solid cylinder which is given in Eq. 5.3, a comparison

between the added mass values obtained through the correlation and the numerical values

is performed in Figure 5.3. The comparison was applied for ranging between

. Through observations of the two curves in Figure 5.3, apparently

C = 0.566 (H/D)-0.927 R² = 0.9999

0.1

1

10

100

0.01 0.1 1 10

C

H/D

47

both curves are almost identical, despite the presence of small errors which don‟t exceed

3% in the worst case.

Figure 5.3. Comparison of added mass values computed numerically with the

correlation in Eq. 5.3.

Furthermore, this developed correlation for aligned cylinder may be extended to the

disk case when is replaced with the disk‟s thickness. The disk has an added mass of

in contrast to the Eq.5.3. Thus, for a disk of and thickness

the analytical solution is whereas the correlation

results in for a fluid density of . The added mass result obtained for

disk using the aligned cylinder correlation is less than 2% from the analytical value.

Hence, the correlation can be reliable for both aligned cylinder and realistic disk, except

when H goes to the zero limits.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.5 1 1.5 2 2.5 3 3.5 4

Ad

de

d m

ass

of

alig

ne

d c

ylin

de

r (

kg)

H/D

Numerical

Eq. 5.3

48

5.2. Cup

Other interesting geometries that are used widely in hydrodynamic application are bluff

bodies with concavity such as cups facing flow. Unlike all previous solid geometries, the

movement of cup in fluid medium in the direction of the open face of the cup has

different aspects of hydrodynamic mass. Even though cup shapes are part of many

hydrodynamic applications, it is still unclear how its added mass can be related to its

geometric dimensions. Thus, in this section a cup model was designed to find the added

mass using the numerical method previously discussed. The hydrodynamic mass was

obtained numerically for a variety of cup length H, outer radius and thicknesses and

the results were correlated with these three geometric parameters. A schematic drawing

of the cup is presented in Figure 5.4.

Figure 5.4. Schematic views of cup.

49

Establishing the cup shape was done using COMSOL CAD Tools. In the model tree

under the geometry root node, two cylinders with different size were chosen first, and

then the small one was subtracted from the larger cylinder to create the void inside the

large cylinder which created the cup. After creating the cup, a cubic block of length

was built to surround the cup and establish the computational domain. The final step in

building the model geometry was to apply Difference operation to subtract the cup from

the cubic block. Some parameters had to be defined in the parameters section before the

geometry creation step took place, such as the dimensions of the two cylinders. Other

parameters such as, uniform velocity and flow density were specified as well. All the

results were calculated using the finer tetrahedral mesh. Figure 5.5 shows the settings

parameter for one of the cases that was computed.

Figure 5.5. Various dimension settings that were applied to a cup model.

50

5.2.1 Cup Results

In this section the streamlines and the velocity field were explored to observe the pattern

of the flow around the cup model and to develop better understanding of the added mass

that is accelerated by the cup movement. Three examples of the streamlines were taken

for a cup with different are shown in Figs 5.6, 5.7, and 5.8.

Figure 5.6. Streamlines for a cup with .

51

Figure 5.7. Streamlines for a cup with .

Figure 5.8. Streamlines for a cup with .

52

Figure 5.9 shows the distribution of velocity magnitude on the cup surfaces for

and the uniform freestream flow boundary condition of at the inlet of the

computational domain. The inlet was chosen to be facing the open side of the cup and

perpendicular to the x axis.

Figure 5.9. Velocity distributions on cup surfaces.

In Figure 5.9 regions of high velocity are observed on edges facing the flow direction

which gives the impression of high flow acceleration close to these edges. Furthermore,

the flow inside the cup seems to have the tendency of having the cup velocity which in

53

this case is zero; the cup is fixed and the flow approaches it. Therefore, the hydrodynamic

mass of the cup is expected to be larger than the one for the aligned cylinder.

5.2.2. Hydrodynamic Mass Correlation for Cup

When the traditional form of hydrodynamic mass ( ) is applied for the Cup

case, the results showed that is not constant value when the enclosed volume is

changing. Table 5.1 shows different values of with different cup‟s size. Hence, form of

correlation is needed to include solutions of hydrodynamic mass for different sizes of the

cup. In order to correlate the hydrodynamic mass of cup with the geometric parameters, it

is very important to examine the effect of , and first before going further. The

results in Figure 5.10 are plotted on logarithmic scale to show the hydrodynamic mass of

cup for different values of cup radius while keeping height and thickness values fixed

at and respectively.

Table 5.1. Different value of hydrodynamic mass coefficient for different size of cup.

( ) ( )

0.1 0.2 0.25 0.01 0.0355 0.013 2.825

0.56 0.2 1.4 0.01 0.091 0.07 1.3

0.52 0.2 1.3 0.01 0.088 0.065 1.33

0.1 0.5 0.1 0.01 0.435 0.078 5.53

0.5 0.5 0.5 0.01 0.785 0.393 2

0.5 0.1 2.5 0.01 0.016 0.0157 1.03

54

Figure 5.10. Hydrodynamic mass of cup as a function of outer radius.

Another investigation was done on the length as shown in Figure 5.11 where the

hydrodynamic mass of cup was calculated for a variety of lengths ranging between

with keeping the outer radius and the thickness fixed at

and respectively.

Figure 5.11. Added mass of cup versus its length.

0.01

0.1

1

10

0.1 1

hyd

rod

ynam

ic m

ass

of

cup

(kg

)

Ro (meter)

0.01

0.1

1

0.01 0.1 1 10

Ad

de

d m

ass

of

cup

(kg

)

H (m)

55

By observing Figure 5.10 it is very obvious that hydrodynamic mass of cup increases

with increasing cup radius with for a fixed thickness so that the inner radius increases as

well as the outer. The same thing can be concluded for the effect of cup‟s length on the

added mass, as seen in Figure 5.11. The resultant curve in Figure 5.11 indicates that

added mass is a nonlinear function of the cup length . Unlike the aligned cylinder case

another parameter that plays a major effect on the added mass of cup is the thickness .

Therefore, added mass was calculated for different cup thicknesses in the range of

with and fixed at and , respectively, and then

compared to the added mass of aligned cylinder has the same radius and length. The

results are depicted in Figure 5.12. On the horizontal axis the ratio of is chosen to

detect the influence of the thickness as fraction of the radius on the cup added mass as

fraction of the aligned cylinder added mass.

Figure 5.12.The influence of cup’s thickness on the added mass of cup.

The curve in Figure 5.12 gives clear indication that the added mass decreases when the

thickness of cup increases. The larger the thickness, the smaller the cavity inside the cup

1, 1

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2

m'c

up/m

'alig

ne

d

t/Ro

56

and the added mass approaches the value for the aligned cylinder, as the thickness

reaches values approximately 80% to 100% of outer radius. Therefore, the best way to

correlate the added mass of cup is to combine the added mass of aligned cylinder to the

added mass created due to the cavity inside the cylinder. The correlation for the added

mass of solid cylinder was found in the previous section but the challenge here is how to

predict the added mass generated by the cavity. To solve this dilemma the added mass of

the cavity was assumed to be proportional to the mass of the fluid inside the cavity as

follows:

( ) ( ) (5.4)

In order to find the values of , and , the cup‟s hydrodynamic mass was found

numerically for different dimensions values. Subsequently, these data were

organized into a spreadsheet as described in Table (2.B) Appendix B. Nonlinear

regression was preformed to estimate the values of and . The best fit gives values

for , and . Thus rewriting Eq. 5.4 with the values of , and

gives the following correlations:

( ) ( ) (5.5)

Comparison of the above equation to the data in Table (2.B) reveals that the accuracy of

the correlation varies with different cup sizes. In general, the accuracy of Eq. 5.5 in

predicting the added mass of cup mainly depends on the ratio. Figure 5.13 below

shows the error and how it varies with different values. For values ranging

between the errors do not exceed 2%. An overall assessment of the

correlation‟s accuracy yields the total squared residual of for

57

. Equation 5.5 was further investigated in term of how the error is affected by

the thickness ratio . Figure 5.14 shows small variance in error with the values.

The error ranges between a minimum value equal of 0.57% and a maximum value equal

of 1.03 %. The effect of ratio may be neglected since its variation results in small

errors in Eq. 5.5. Thus, the only restriction for using the correlation presented in Eq. 5.5

is the ratio

Figure 5.13. Prediction error of the correlation in Eq. 5.5 for various H/D.

Figure 5.14. Prediction error of the correlation in Eq. 5.5 for various thickness ratio

values, fixed and .

0

0.5

1

1.5

2

2.5

3

0 0.8 1.6 2.4 3.2 4

Ab

solu

te v

alu

es

of

Erro

r %

H/D

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Ab

sollu

te v

alu

es

of

Ero

rr%

t/Ro

58

5.3. Parachute Canopies

In this section 8 models of round parachute canopies were built in Solidworks, and then

imported into COMSOL to find their hydrodynamic mass. Subsequently, added mass

values were correlated with certain geometric parameters.

5.3.1. Canopy Models

The eight models represent various stages of a parachute canopy during inflation. In the

experimental research conducted by Desabrais [9], the opening process of a flexible 30.5

cm constructed diameter canopy was conducted in a water tunnel to measure the velocity

field. Furthermore, digital photographs of the opening process were taken for during

various stages of inflation. Eight stages from these figures were deemed adequate to

represent the entire inflation process as shown in Figure 5.15. These shapes were utilized

to approximate the 8 solid models. The geometric aspects of each model are presented in

detail in Appendix C in which all the geometric dimensions are defined. Shape 1

represents the parachute model after initial release when the canopy is in the initial

“sock” shape. Shape 2 presents the canopy 1.5 second after release. Shape 3 represents

the canopy at 2.1 seconds after release. Shape 4 was captured at 2.5 second after the

initial release. Shape 5 is at the instant when maximum force is experienced. Shape 6

represents the canopy when the enclosed volume reaches a maximum value. Shape 7

represents the over-inflation stage. Shape 8 is from an image taken 4.2 seconds after

initial release and it represents the approximate steady state shape. The captured images

of these shapes can be seen in the Figure 5.15.

59

Figure 5.15: Canopy Shapes from [9]

Shape 1 Shape 2

Shape 8 Shape 7

Shape 6 Shape 5

Shape 4 Shape 3

60

5.3.2 Preparing Canopy Models in COMSOL

After the images in Figure 5.15 were created as solid models in Solidworks, the solid

models of the parachute canopy were imported into COMSOL. A cubic block of side

length was chosen to envelop each model to form the surrounding domain. Afterwards,

the same boundary conditions as the ones discussed in Chapter 3 were assigned to each

model to find the added mass. The hydrodynamic mass for each model was calculated for

a range of domain sizes from side length of 10 m until the length at which consistent

results appeared. More details on the hydrodynamic mass results with different sizes of

domain are presented in Appendix D. In Figure 5.16, Model 8 was chosen to illustrate

the freestream direction that was applied to the parachute models, and the position of the

parachute inside the computational domain.

Figure 5.16. Freestream flow direction in the computational domain and model 8.

61

Similarly, freestream flow with uniform velocity and density

was assigned to the inlet and outlet boundaries which are normal to y-axis as shown in

Figure 5.16 above. Tetrahedral elements were used to mesh the computational domain. In

addition, a finer element size was applied with parameters setting as follows: maximum

element growth rate equal to 1.3, resolution of curvatures equal to 0.3 and resolution of

narrow region equal to 0.8. Figure 5.17 below shows the tetrahedral grid pattern for

Model 8.

Figure 5.17. The tetrahedral mesh elements for Model 8.

After discretizing the computational domain, iterative method described in chapter 3 was

used to solve the velocity potential.

62

5.3.3 Potential Flow Results for Parachute Models

Some features of the flow field of parachute canopies such as velocity potential and

velocity field are exhibited in this section. The results shown are for Model 1 and Model

8 since their shapes are quite different. Model 1 is after the initial release and Model 8

represents the fully inflated parachute. The comparison between velocity potential of two

cases are shown in Figs. 5.18 and 5.19.

Figure 5.18. Slices of velocity potential for Model 1.

63

Figure 5.19 Slices of velocity potential of Model 8

Figs. 5.20 and 5.21 shows the streamlines on y-z cut plane. The velocity component in

the freestream direction (y-direction) of Model 1 and 8 are exhibited in Figs. 5.22 and

5.23. The results are shown on y-z cut plane through the center of the canopy as shown

below. The velocity magnitude gets larger at canopies edges that are facing the flow.

Furthermore, Figs 5.24 and 5.25 display the distribution of the velocity component in x-

direction on a x-z cut plane for Model 1 and Model 8.

64

Figure 5.20. Streamlines on a y-z plane for Model 1.

Figure 5.21. Streamlines on a y-z plane for Model 8.

65

Figure 5.22. Distribution of the velocity component in the y-direction for Model 1.

Figure 5.23. Distribution of the velocity component in y-direction for Model 8.

66

.

Figure 5.24. Distribution of the velocity component in x-direction for Model 1.

Figure 5.25. Distribution of the velocity component in x-direction Model 8.

67

Figure 5.26. Distribution of the relative Velocity on a cut plane at the opening of

Model 1.

Figure 5.27. Distribution of the relative velocity on a cut plane at the opening of

Model 8.

68

Figs 5.26 and 5.27 show the relative kinetic energy of the flow at the parachute canopy

skirt. The results in both figures show the higher relative kinetic energy in regions close

to the parachute body.

5.3.4. Correlation of Hydrodynamic Mass Parachutes Canopies

This section is devoted to finding a suitable correlation that relates the hydrodynamic

mass of the eight different models of parachute canopy during inflation with their

respective geometric parameters. Since these parachute models have distinct shapes

especially when comparing the first two models with the other six models, it was

necessary to find common ground in term of defining the dimensions for which the

hydrodynamic mass can be correlated. The presence of circular bulges in the last six

shapes requires a distinct definition for the diameter that applies equally to the six models

as well as the first two models. Therefore, an equivalent diameter was defined for the last

six models using the maximum projected area for each model. Figure 5.28 shows the

maximum projected area for model 5 which is red colored. This maximum projected area

was calculated for each Model using the COMSOL measurement tools, and then set equal

to an area of circle to find the equivalent diameter. Table 5.2 shows values of the

maximum projected area of each model and the equivalent diameter.

69

.

Figure 5.28. The maximum projected area of Model 5.

The computed hydrodynamic mass ( ) of the eight models is presented in Table 5.2 in

addition to the other related geometric dimensions such as the height , and the

equivalent radius .

Table 5.2. Maximum projected area, equivalent diameters and the calculated

hydrodynamic mass for each model.

Model No. ( ) ( ) ( ) ( )

( )

Model 1 1.26 1.2668 0.6334 4.995 3.3526

Model 2 15.76 4.48 2.24 4.565 48.7

Model 3 22.9 5.4 2.7 4.159 124.96

Model 4 37.05 6.868 3.434 3.248 189

Model 5 53.47 8.24 4.12 3.19 277.2

Model 6 56.24 8.4 4.2 2.434 267.9

Model 7 62.6 8.928 4.464 1.76227 296.34

Model 8 49.9 8.2 4.1 3.276 280.14

70

First, the relationship between the calculated added mass and the mass of the enclosed

volume for each model was examined. The approach stemmed from the fact that the

hydrodynamic mass for classical geometries is equal to a constant coefficient of order

one multiplied by the fluid mass of the accelerated object as described in Eq. 2.5.

Therefore, the enclosed volume of each model was calculated using COMSOL

measurement tools, and then the factor in Eq. 2.5 was calculated by dividing the

hydrodynamic mass by the enclosed mass. The results are shown in Table 5.3.

Table 5.3. Hydrodynamic mass coefficient C for parachute models.

Model ( ) ( ) ( )

( ) ( )

1 0.6334 4.995 3.72 3.3526 3.72 0.901237

2 2.24 4.565 29.98 48.7 29.98 1.624416

3 2.7 4.159 76.8 124.96 76.8 1.627083

4 3.434 3.248 93.82 189 93.82 2.014496

5 4.126 3.194 118 277.2 118 2.349153

6 4.2 2.434 92.61 267.9 92.61 2.892776

7 4.464 1.76227 82.67 296.34 82.67 3.584614

8 4.1 3.276 124.9 280.14 124.9 2.242914

Unlike the classical geometries, value for the parachute models is not constant, and

as exhibited above varies for different models. Therefore, it seems inapplicable to

develop correlation for the hydrodynamic mass of parachute canopies based on the

traditional method of Eq. 2.5. However, different ways were pursued using the nonlinear

regression solver in Excel to find a good correlation that can tie the dimensions , and

to the values of the hydrodynamic mass ( ).

71

Similar to the cup hydrodynamic mass, the hydrodynamic mass of a parachute canopy

during various stages of inflation may be written as the hydrodynamic mass of the solid

model of the parachute canopy without cavity plus the mass of fluid enclosed by the

canopy.

(5.6)

The first term on the right-side of Eq. 5.6 is the hydrodynamic mass of the enclosed

models (solid model) of the parachute canopies. Therefore, the cavity inside each model

was filled with solid to create the solid models of the canopies without cavity, and then

the hydrodynamic mass for each solid model was computed. Furthermore, the results of

the computed added mass ( ) were correlated using nonlinear regression to find the

values of , and . The results are shown in Table 5.4

Table 5.4. Performing of nonlinear regression on the first term on the right-side of

Eq.5.6.

Solid Models # ( ) ( ) ( )

0.82

1 0.633 4.995 0.56 0.65 0.01 3

2 2.24 4.565 22.8 29.06 39.25 0

3 2.7 4.159 55.28 50.90 19.18

4 3.434 3.248 107.6 104.72 8.29

5 4.12 3.19 177 180.85 14.83

6 4.2 2.434 193.75 191.59 4.65

7 4.464 1.76 234.3 230.04 18.15

8 4.1 3.276 172.4 178.23 34

∑ = 138.35

Substituting the values of , , and from Table 5.4 leads to a correlation of

hydrodynamic mass for the solid model of the parachute canopies (enclosed models) as

follows:

(5.7)

72

The resultant correlation in Eq. 5.7 for the solid parachute canopies without cavity is very

similar to the analytical solution of the disk (

). Figure 5.29 shows comparison

between the hydrodynamic mass of disk, numerical results of the solid models of the

parachute canopies (without cavity), and the correlation in Eq. 5.7.

Figure 5.29. Comparison between hydrodynamic mass for disk, the solid canopies

and the correlation in Eq.5.7.

Finally, Eq. 5.7 is added to the mass of the fluid enclosed by the canopy to find the

correlation of the hydrodynamic mass of the parachute canopies (with cavity). Regression

was performed in Table 5.5 to find the value of the parameter in Eq. 5.6.

Table 5.5. Performing regression to find the value of parameter in Eq. 5.6.

Model# ( ) ( ) ( ) ( ) ( ) Residual K

1 0.63 4.995 3.3526 2.8 3.20 0.022 0.91

2 2.24 4.565 48.7 27.14 53.75 25.465

3 2.7 4.159 124.96 71.66 116.07 79.069

4 3.434 3.248 189 86.26 183.165 34.051

5 4.12 3.19 277.2 110.3 281.158 15.66

6 4.2 2.434 267.9 85.08 268.964 1.13

7 4.464 1.76 296.34 72.17 295.67 0.45

8 4.1 3.276 280.14 117 284.63 20.16

∑

0

50

100

150

200

250

0 1 2 3 4 5

Ad

de

d m

ass

(kg)

Req (m)

Eq. 5.7

Numercal

Disk

73

Substituting the value of from Table 5.5 in Eq.5.6 leads to the correlation of

hydrodynamic mass of the parachute canopy during inflation as follows:

(5.8)

The accuracy of this correlation is investigated in Figure 5.30; the numerically computed

values are presented on the horizontal axis and the correlation added mass values are on

the vertical axis. Furthermore, the straight line in Figure 5.30 is plotted with slope equal

to 1 to measure the deviation in the added mass results.

Figure 5.30. A comparison between the correlation in Eq. 5.7 and numerically

computed added mass of parachute canopies.

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

Co

rre

lati

on

ad

de

d m

ass

(kg)

Computed added mass (kg)

74

Chapter 6

Summary and Conclusion

The significance of hydrodynamic mass, and its effect on the dynamic behavior of

aerodynamic devices such as parachutes, has raised continuous demands for more

information on the hydrodynamic mass. Thus, the added mass of bluff bodies such as

aligned solid cylinder, cup, and parachute canopies, have been evaluated numerically by

means of a finite elements solver, COMSOL, and the solutions were validated by using

the solver to find the numerical solution of classical geometries. The comparison between

the added mass coefficient that had been obtain theoretically and the ones computed

numerically using the solver revealed very accurate results with very small errors which

did not exceed 1.06% in the worst case scenario. To enhance the accuracy of the

numerical solution, the classical models were tested with different element sizes and

types. Overall, the finer mesh size with customized parameters such as resolution of

curvatures and narrow regions was deemed the best choice to accomplish accurate

results.

The resultant correlation of added mass for aligned cylinder shows domination of the

cylinder radius with small effect to account for the length. The resultant correlation is

quite distinct from the hydrodynamic mass for the classical cylinder model due to the

difference in the flow direction. The accuracy of the correlation was estimated to give fit

with square residual . Furthermore the correlation can include the disk

case and gives satisfying results.

In the cup model the observation of the results showed that the added mass of the cup

can be presented as a combination of the added mass of aligned cylinder and the portion

75

of the fluid mass inside the cup. Therefore, the results for added mass of cup were

correlated according to this combination and the correlation was found to be quite good

for cup dimensions within .

The results for added mass of parachute models during inflations showed significant

changes in hydrodynamic mass values during the inflation. The added mass for the first

model was found equal to and for the last fully inflated model is equal

to . This huge variation in hydrodynamic mass refers to the tremendous change

in geometry during the inflation. In observing the change in parachute dimensions during

the inflation, it can be concluded that there is trade-off between the equivalent radius

( ) and the height ( ). The results showed that the radius has significant influence on

the hydrodynamic mass of parachute canopy. The hydrodynamic mass of a parachute

canopy during various stages of inflation may be written as the hydrodynamic mass of a

disk having the same area as the projected area of the canopy plus the mass of fluid

enclosed by the canopy.

Previous study on rigid spherical cup by Ibrahim (1965) had shown the effect of the

concavity parameter on the added mass. The possible suggested works in the future is to

combine Ibrahim (1965) analytical solutions with the work reported here to refine

solution that can give better presentation of hydrodynamic mass of the parachute canopy

models. Another suggestion is to develop solution that account for porosity of parachutes,

and cavities and wakes in flow. Moreover, prospective study could pursue solutions of

hydrodynamic mass for different shapes of parachutes such as rectangular parachutes, or

other design of parachutes such as the ones with hole at their apex. Furthermore the

results of this work can be utilized in future experimental works in flow about parachute

76

canopies to provide accurate results of the contribution of potential force in the total

opening force of the canopy.

77

References

[1] Ibrahim, S.K., “Potential Flow field and Added Mass of the Idealized Hemispherical

Parachute,” Journal of Aircraft, Vol. 4, No. 2, Mar.-Apr., 1967, pp. 96-100.

[2] Panton. R, “ Incompressible Flow,” John Wiley & Sons, Inc. New Jersey, Third

edition, 2005.

[3] White, Frank M. “Fluid Mechanics,” McGraw-Hill, Boston, MA. Fifth edition, 2003.

[4] B. Mutlu Sumer, Jorgen Fredsoe, "Hydrodynamics Around Cylindrical Structures

(Advanced Series on Ocean Engineering)". World Scientific, Singapore; River Edge, N.J.

1997.

[5] J. E. Slater, “A review of hydrodynamic added mass inertia of vibrating

Submerged structures,” Defense Res. Establishment Atlantic, Canada,

Tech. Memorandum, 1984.

[6] H. Chung and S. S. Chen, “Hydrodynamic mass,” DOE Tech. Rep.

CONF-840647-9, 1984.

[7] H. Lamb, “Hydrodynamics,” Dover Publications, New York, sixth edition, 1932.

[8] http://www.comsol.com/ ( Retrieved April, 2011)

[9] Desabrais, K.J., “Velocity Field Measurements in the Near Wake of a Parachute

Canopy,” Ph.D. thesis, Mechanical Engineering Department, Worcester

Polytechnic Institute, Worcester, MA, April 2002.

[10] Patton. Kirk. T, “Tables of Hydrodynamic Mass Factors for Translational Motion,‟‟

Underwater Technology Division, Chicago, Ill, ASME, 1965.

[11] Lighthill, J., “Fundamentals Concerning Wave Loading on Offshore Structures,”

Journal of Fluid Mechanics, Vol. 173, 1986, pp. 667-681,

[12] Noca, F., Shiels, D., and Jeon, D., “A Comparison of Methods for Evaluating

TimeDependent Fluid Dynamic Forces on Bodies, Using Only Velocity Fields and

Their Derivatives,” Journal of Fluids and Structures, Vol. 13, 1999, pp. 551-578.

[13] Morison J R, O'Brien M D, Johnson J W, and Schaaf S A,” The force exerted by

surface waves on piles”. Petrol Trans AIME. Vol. 189,1950.

78

Appendix A

Table (1.A). The verification of hydrodynamic mass coefficient of circle.

L (domain length) (m) R (radius of circle) (m) domain area L/R ( )

3 0.2 9 15 0.1296 1.031

4 0.2 16 20 0.1278 1.017

5 0.2 25 25 0.1271 1.011

6 0.2 36 30 0.1266 1.007

7 0.2 49 35 0.1264 1.005

8 0.2 64 40 0.1262 1.004

9 0.2 81 45 0.1261 1.003

10 0.2 100 50 0.126 1.002

11 0.2 121 55 0.1259 1.002

12 0.2 144 60 0.1259 1.002

13 0.2 169 65 0.1258 1.001

14 0.2 196 70 0.1258 1.001

15 0.2 225 75 0.1258 1.001

16 0.2 256 80 0.1258 1.001

17 0.2 289 85 0.1258 1.001

Table (2.A). The verification of hydrodynamic coefficient of the classical cylinder

model.

L (Domain length) (m) R (cylinder radius) (m) ( ) H ( ) L/R

1 0.2 0.1657 1 0.1256 5 1.319

2 0.2 0.2691 2 0.2512 10 1.071

3 0.2 0.3884 3 0.3768 15 1.030

4 0.2 0.5109 4 0.5024 20 1.016

5 0.2 0.6347 5 0.628 25 1.0101

6 0.2 0.759 6 0.7536 30 1.007

7 0.2 0.8837 7 0.8792 35 1.005

8 0.2 1.0091 8 1.0048 40 1.004

10 0.2 1.2594 10 1.256 50 1.0022

12 0.2 1.51 12 1.5072 60 1.0014

14 0.2 1.7611 14 1.7584 70 1.001

16 0.2 2.013 16 2.0096 80 1.001

18 0.2 2.26421 18 2.2608 90 1.001

79

Table (3.A) Domain verifications on the hydrodynamic mass coefficient of sphere

L (Domain length) (m) R (sphere radius) (m) ( ) L/R

1 0.2 0.0188 5 0.561

2 0.2 0.017 10 0.507

3 0.2 0.0168 15 0.501

4 0.2 0.0168 20 0.501

5 0.2 0.0168 25 0.501

Table (4.A) Domain size effect on the hydrodynamic mass coefficient for disk

L (domain length) (m) R (radius ) (m) t (thickness) (m) ( ) L/R

0.5 0.2 0.001 0.0256 2.5 1.0186

1 0.2 0.001 0.0219 5 0.8714

2 0.2 0.001 0.0216 10 0.859

3 0.2 0.001 0.0216 15 0.859

4 0.2 0.001 0.0216 20 0.859

80

Appendix B

Table (1.B) Hydrodynamic mass results of solid cylinder aligned to the flow

L (domain length) (m) R (m) H(m) ( ) H/D C

7 0.2 0.01 0.0224 0.025 17.83

7 0.25 0.0125 0.0438 0.025 17.85

7 0.2 0.012 0.0226 0.03 14.99

7 0.2 0.02 0.023 0.05 9.151

7 0.2 0.028 0.0233 0.07 6.622

7 0.2 0.036 0.0236 0.09 5.22

7 0.2 0.044 0.0238 0.11 4.30

7 0.2 0.052 0.0241 0.13 3.7

7 0.2 0.06 0.0243 0.15 3.22

7 0.2 0.068 0.0246 0.17 2.88

7 0.2 0.076 0.025 0.19 2.62

7 0.2 0.084 0.0252 0.21 2.38

8 0.2 0.092 0.0253 0.23 2.19

8 0.2 0.1 0.0257 0.25 2.045

7 0.2 0.1 0.0257 0.25 2.045

7 0.4 0.2 0.206 0.25 2.049

9 1 0.5 3.239 0.25 2.062

7 0.2 0.12 0.0261 0.3 1.731

9 0.5 0.3 0.4084 0.3 1.733

9 1 0.6 3.27 0.3 1.735

7 0.2 0.14 0.0265 0.35 1.506

9 0.5 0.35 0.4141 0.35 1.506

7 0.2 0.16 0.0266 0.4 1.323

8 0.25 0.2 0.0521 0.4 1.327

8 0.2 0.18 0.0271 0.45 1.198

8 0.5 0.45 0.4236 0.45 1.198

8 0.2 0.2 0.0272 0.5 1.0823

8 0.5 0.5 0.4254 0.5 1.0833

8 0.2 0.22 0.0274 0.55 0.991

9 0.5 0.55 0.4286 0.55 0.992

8 0.2 0.24 0.0278 0.6 0.922

8 0.25 0.3 0.0543 0.6 0.922

8 0.2 0.26 0.0279 0.65 0.854

10 0.5 0.65 0.4367 0.65 0.855

8 0.2 0.28 0.028 0.7 0.796

81

10 0.5 0.7 0.4377 0.7 0.796

8 0.2 0.32 0.0282 0.8 0.702

10 0.5 0.8 0.4427 0.8 0.704

8 0.2 0.36 0.0287 0.9 0.635

10 0.5 0.9 0.4492 0.9 0.635

8 0.2 0.4 0.0288 1 0.573

9 0.25 0.5 0.0562 1 0.573

8 0.2 0.44 0.029 1.1 0.524

12 0.5 1.1 0.4536 1.1 0.525

8 0.2 0.48 0.0291 1.2 0.482

12 0.4 0.96 0.2333 1.2 0.483

8 0.2 0.52 0.0293 1.3 0.448

9 0.25 0.65 0.0573 1.3 0.449

8 0.2 0.56 0.0295 1.4 0.419

10 0.25 0.7 0.0576 1.4 0.419

10 0.2 0.6 0.0296 1.5 0.393

10 0.3 0.9 0.1001 1.5 0.393

10 0.2 0.64 0.0297 1.6 0.369

10 0.3 0.96 0.1003 1.6 0.369

10 0.2 0.68 0.0298 1.7 0.349

10 0.3 1.02 0.1008 1.7 0.349

10 0.2 0.72 0.0299 1.8 0.330

10 0.3 1.08 0.101 1.8 0.331

10 0.2 0.76 0.03 1.9 0.314

10 0.3 1.14 0.1014 1.9 0.315

10 0.2 0.8 0.0301 2 0.299

10 0.3 1.2 0.1017 2 0.299

10 0.2 0.88 0.0302 2.2 0.273

10 0.2 0.96 0.0304 2.4 0.252

10 0.2 1.04 0.0305 2.6 0.233

10 0.2 1.12 0.0306 2.8 0.217

11 0.2 1.2 0.0307 3 0.204

11 0.2 1.28 0.0308 3.2 0.191

11 0.2 1.36 0.0309 3.4 0.181

12 0.2 1.44 0.031 3.6 0.171

12 0.2 1.52 0.031 3.8 0.162

12 0.2 1.6 0.0311 4 0.155

82

in Table (2.B) is defined as follow:

( ) ( )

Table (2.B) spreadsheet of the regression solver to correlate data for

hydrodynamic mass of cup

H(m) Ro(m) H/D t (m) [

-m']^2 K

0.02 0.2 0.05 0.01 0.0238 0.023897368 9.48053E-09 0.97

0.022 0.2 0.055 0.01 0.02415 0.024267117 1.37165E-08 m

0.024 0.2 0.06 0.01 0.0245 0.024624766 1.55665E-08 2

0.026 0.2 0.065 0.01 0.02485 0.0249722 1.49328E-08 n

0.028 0.2 0.07 0.01 0.02518 0.025310898 1.71343E-08 1.0

0.036 0.2 0.09 0.01 0.0264 0.026598901 3.95616E-08

0.044 0.2 0.11 0.01 0.0276 0.027811523 4.47419E-08

0.052 0.2 0.13 0.01 0.0287 0.028973916 7.50299E-08

0.06 0.2 0.15 0.01 0.02985 0.030100502 6.27511E-08

0.068 0.2 0.17 0.01 0.03095 0.03120032 6.26601E-08

0.076 0.2 0.19 0.01 0.03215 0.03227941 1.67471E-08

0.084 0.2 0.21 0.01 0.03325 0.033342007 8.46534E-09

0.092 0.2 0.23 0.01 0.03435 0.034391192 1.69681E-09

0.1 0.2 0.25 0.01 0.0355 0.035429278 5.00163E-09

0.12 0.2 0.3 0.01 0.0381 0.037986686 1.28401E-08

0.14 0.2 0.35 0.01 0.0407 0.040504 3.84159E-08

0.16 0.2 0.4 0.01 0.0433 0.042992661 9.44571E-08

0.18 0.2 0.45 0.01 0.0458 0.04546001 1.15593E-07

0.2 0.2 0.5 0.01 0.0484 0.047911026 2.39096E-07

0.22 0.2 0.55 0.01 0.05085 0.050349231 2.5077E-07

0.24 0.2 0.6 0.01 0.0533 0.052777203 2.73316E-07

0.26 0.2 0.65 0.01 0.05573 0.055196881 2.84216E-07

0.28 0.2 0.7 0.01 0.0582 0.057609754 3.4839E-07

0.3 0.2 0.75 0.01 0.0606 0.060016991 3.39899E-07

0.32 0.2 0.8 0.01 0.063 0.062419522 3.36955E-07

0.34 0.2 0.85 0.01 0.0654 0.064818097 3.38611E-07

0.36 0.2 0.9 0.01 0.0679 0.06721333 4.71515E-07

0.38 0.2 0.95 0.01 0.0702 0.069605729 3.53158E-07

0.4 0.2 1 0.01 0.0726 0.071995717 3.65158E-07

0.44 0.2 1.1 0.01 0.07735 0.076769825 3.36603E-07

0.48 0.2 1.2 0.01 0.0821 0.081537906 3.1595E-07

0.52 0.2 1.3 0.01 0.0867 0.086301612 1.58713E-07

0.56 0.2 1.4 0.01 0.09145 0.091062184 1.50401E-07

0.6 0.2 1.5 0.01 0.0961 0.095820571 7.80807E-08

83

0.64 0.2 1.6 0.01 0.1008 0.100577507 4.95033E-08

0.68 0.2 1.7 0.01 0.1054 0.105333568 4.41321E-09

0.72 0.2 1.8 0.01 0.11 0.110089212 7.95879E-09

0.76 0.2 1.9 0.01 0.1147 0.114844804 2.09682E-08

0.8 0.2 2 0.01 0.1193 0.119600639 9.03836E-08

0.88 0.2 2.2 0.01 0.1285 0.129113944 3.76927E-07

0.96 0.2 2.4 0.01 0.1377 0.138630553 8.65928E-07

1.04 0.2 2.6 0.01 0.14692 0.148151418 1.51639E-06

1.12 0.2 2.8 0.01 0.15613 0.157677179 2.39376E-06

1.2 0.2 3 0.01 0.1653 0.16720826 3.64145E-06

1.28 0.2 3.2 0.01 0.1744 0.176744937 5.49873E-06

1.36 0.2 3.4 0.01 0.1836 0.186287381 7.22202E-06

1.44 0.2 3.6 0.01 0.1927 0.19583569 9.83255E-06

1.52 0.2 3.8 0.01 0.2019 0.205389905 1.21794E-05

1.6 0.2 4 0.01 0.211 0.214950028 1.56027E-05

0.1 0.1 0.5 0.01 0.00563 0.005598632 9.83929E-10

0.5 0.5 0.5 0.01 0.7853 0.780459078 2.34345E-05

0.3 0.3 0.5 0.01 0.1666 0.165468794 1.27963E-06

0.04 0.2 0.1 0.01 0.027 0.027212713 4.52468E-08

0.1 0.5 0.1 0.01 0.4345 0.439064659 2.08361E-05

1 0.2 2.5 0.01 0.1423 0.143390408 1.18899E-06

0.5 0.1 2.5 0.01 0.0162 0.016361643 2.61284E-08

1.2 0.2 3 0.01 0.1653 0.16720826 3.64145E-06

0.6 0.1 3 0.01 0.0188 0.019041731 5.8434E-08

0.5 0.2 1.25 0.01 0.0844 0.08392022 2.30189E-07

0.5 0.2 1.25 0.02 0.0778 0.077315324 2.34911E-07

0.5 0.2 1.25 0.03 0.0717 0.071225685 2.24975E-07

0.5 0.2 1.25 0.04 0.066 0.065633091 1.34622E-07

0.5 0.2 1.25 0.05 0.0609 0.060519332 1.44908E-07

0.5 0.2 1.25 0.06 0.0563 0.055866204 1.88179E-07

0.5 0.2 1.25 0.1 0.0418 0.041496148 9.23259E-08

0.5 0.2 1.25 0.15 0.032 0.031717227 7.99608E-08

0.5 0.2 1.25 0.17 0.0302 0.02988777 9.74875E-08

0.5 0.2 1.25 0.19 0.0293 0.02903457 7.04531E-08

0.5 0.2 1.25 0.16 0.0309 0.030671708 5.21174E-08

0.5 0.2 1.25 0.18 0.0297 0.029347787 1.24054E-07

0.5 0.05 5 0.01 0.0029 0.00302859 1.65355E-08

0.5 0.1 2.5 0.01 0.0162 0.016361643 2.61284E-08

0.5 0.15 1.666667 0.01 0.0426 0.042535625 4.14409E-09

0.5 0.2 1.25 0.01 0.0844 0.08392022 2.30189E-07

0.5 0.25 1 0.01 0.144 0.142850181 1.32208E-06

84

0.5 0.3 0.833333 0.01 0.2235 0.221629324 3.49943E-06

0.5 0.35 0.714286 0.01 0.3253 0.322535011 7.64517E-06

0.5 0.4 0.625 0.01 0.4518 0.447821776 1.58263E-05

0.5 0.45 0.555556 0.01 0.6048 0.599724181 2.57639E-05

0.5 0.5 0.5 0.01 0.7853 0.780459078 2.34345E-05

0.5 0.55 0.454545 0.01 0.9989 0.992227443 4.4523E-05

0.5 0.6 0.416667 0.01 1.244 1.237215878 4.60243E-05

0.5 1 0.25 0.01 4.6449 4.651538312 4.40672E-05

Residual 0.000329161

85

Appendix C

Front view

Bottom view Top view

Figure (1.C). Drawing views of Model 1.

Table (1.C). A list of dimensions of Model 1.

Name Value symbol Unit

86

Front view

Top view Bottom view

Figure (2.C). Drawing of Model 2

Table (2.C). A list of dimensions of Model 2

Name Value symbol Unit

87

Front view

Top view Bottom view

Figure (3.C) Drawing of Model 3.

Table (3.C). A list of Model 3 dimensions.

Name Value symbol Unit

2.724

88

Front view

Top view Bottom view

Figure (4.C). Drawing of Model 4.

Table (4.C). A list of dimensions of Model 4

Name Value symbol Unit

3.44

89

Front view

Top view Bottom view

Figure (5.C). Drawing of Model 5.

:

Table (5.C). A list of dimensions of Model 5

Name Value Symbol Unit

4.248

90

Front view

Top view Bottom view

Figure (6.C). Drawing of Model 6.

Table (6.C). A list of dimensions for Model 6.

Name Value Symbol Unit

4.354

91

Front view

Top view Bottom view

Figure (7.C) Drawing of Model 7.

Table (7.C). A list of dimensions for Model 7.

Name Value Symbol Unit

4.587

92

Front view

Top view Bottom view

Figure (8.C). Drawing of Model 8.

Table (8.C). A list of dimensions for Model 8.

Name Value Symbol Unit

4.0898

93

Appendix D

All the results below were found using finer meshes with element size parameters 1.4,

0.2 and 0.8 for maximum element growth rate, resolution of curvatures and resolution of

narrow region respectively.

Table (1.D). Hydrodynamic mass versus domain sizes for Model 1.

( ) ( ) Domain side length (m) ( )

4.995 0.6334 10 3.3662

12 3.3591

15 3.3558

18 3.3551

20 3.3539

22 3.3526

24 3.3526

26 3.3526

Table (2.D). Hydrodynamic mass versus domain sizes for Model 2.

( ) ( ) Domain side length (m) ( )

4.56588 2.244 14 49.4615

16 49.202

18 49.0436

20 48.9435

22 48.8811

24 48.8484

26 48.8169

28 48.7933

30 48.7721

32 48.7588

34 48.7506

36 48.7468

38 48.7406

40 48.7348

42 48.7299

44 48.7213

50 48.7

55 48.7

60 48.7

94

Table (3.D). Hydrodynamic mass versus domain sizes of Model 3.

Table (4.D). Added mass versus domain sizes of Model 4.

( ) ( ) ( ) Domain h side length (m) ( )

3.248 3.43456 2.953 44 189.2059

50 189

55 189

60 189

Table (5.D). Added mass versus domain size of Model 5.

( ) ( ) ( ) Domain side length (m) ( )

3.194 4.248 3.781 44 277.7

50 277.47

55 277.35

60 277.2

64 277.2

Table (6.D). Hydrodynamic mass versus domain size of Model 6.

( ) ( ) ( ) Domain side length (m) ( )

2.434 4.354 3.979 44 268.23

50 268..15

55 268.0358

60 267.9

64 267.9

( ) ( ) ( ) Domain side length (m) ( )

4.159 2.7242 2.594 10 143.1795

20 126.5762

30 125.3545

40 125

50 124.96

55 124.8

60 124.79

64 124.79

95

Table (7.D). Added mass versus domain size for Model 7.

( ) ( ) ( ) Domain side length (m) ( )

1.76227 4.587 4.116 44 296.89

50 296.58

60 296.398

64 296.34

70 296.34

Table (8.D). Added mass versus domain size for Model 8.

( ) ( ) ( ) Domain side length (m) ( )

3.276 4.1 3.7295 44 280.672

50 280.41

55 280.277

60 280.19

64 280.14

70 280.14