Fundamentals of ship resistance and propulsion 2 (mt520)

mt520

July 2003

FundamentaJs of Ship Resistance and

Propulsion

Technical University Deift

G. Kui.per'

January 29, 2003

1710-K

3

Vakcode: mt520

Vaknaam: Weerstand en Voortstuwing 2

.

Het betreft een college

TUD studiepunten: 1,5

ECTS studiepiinten: 2,75

Sùbfaculteit der Werktuigbouwkiinde en Maritiéme Techniek

Docent: Koning Gans, dr.ir.H.J. de Tel: 015/27 81852

Tefwoorden:

Stronaingsmodel, Vereenvoudigen van de stromiigsmode1ien, Potentia1theorie, Stationaire vrije vloeistofoppervlaktheorie, Omstromingsberekeningen rondom profiel, Lifting line theorie, Schroefóntwerp

Curstisjiar:

3

Periode:

3/O/O/O

Coll.uren p/w:

3 Toetsvorm: schriftelijk

Tentamenperiode: 1,2

Voorkennis:

Weerstand en Voortstuwing i, stromingsleer

TJitgebreide beschrijving van bet onderwerp:

Overzicht van programmatuur _{voor het berekenen van de omstroming van}

romp en voortstuwer. Uitgangspunten en berekeningstechnieken.

Aan de orde komen het gebruik van bewegingsvergelijkingen,

potentiaalberekeningen, Navier Stokes oplossingen, linearisaties, invloed van viscositeit, grenslagen

C1'legemateriaal: collegedictaat

Referenties vanuit de literatuur: Principle of Naval Architecture Opmerkingen: tentaminering: Open boek tentamen

Leerdoelen:

- Het begrijpen van de Navier Stokes Vergelijkingen. Het kunnen afleiden

vaneenvoudige stromingsmodeilen4zoals Euler, grenslaag en

potentiaal-vergelijkingen) en weten welke fysica erachter steèkt. Het _kmrien

toepassen van de j niste randvoorwa arde bij de bovengenoemde

stromingsmodellen, het kunnen toepassenvan elementaire

potentiaai-oplossingen. Het kunnen afleiden van de vrijvloeistofoppervlakte stromingen.

Begrij pen van diverse numerieke stromingsmodellen_{en weten warop deze}

stromingsmodellen zijn gebaseerd en wanneer deze nmnerieke modellen mogen worden to egepast.

- Het begrijpen van de omstroming van pröfielen en bet kunnen uitrekenen van de locale snelheden en drukken op bet oppervlak van het profiel, met

behuip van profieltabellen. Kennismaken met de dragende lijn theorie.

Het kunnen reproduceren van géinduceerde sneiheden _{met behuip van deze}

Contents

i

2

Equations of Motion

1.1 The Continuity Equation

1.2 The Equations of Motion

1.2.1 Rotation and Deformation

1.2.2 Relation between Stresses and Strain

1.2.3 Navier-Stokes Equations

...

1.3. A Simple Example

1.4 The Euler Equations . . .

1.5 The Bernoulli Equation

...

1.6 Summary

Potential flow

2.1 Singularities in Potential Flow

2.1.1 Uniform Flow . . 2.1.2 Source

213 Vottex

16 23 25 13 14 18 19 20 22 24 27 28 28 29 30 3 2.1.4 Dipole (3D)

2.2 A Simple Example of Potential Flow

2.3 Forces on a Vortex

2.4 Panel Methods

2.4.1 The Lifting Problem

2.5 Summary

Boundary Layers

3.1 The non-dimensional Navier Stokes Equations

...

3.2 The Boundary Layer Equation

3.2.1 _{Scaling the Thickness of'the Boundary Layer}

3.3 Solutions of the Boundary Layer Equations: B1aius

41 32 33 36 37 40 43 44 44 47 47

4

3.4 _Turbulence

Flow Calculations without Waves

48

51

4.1 _{Potential Flow Calculations 54}

4.1.1 _{Panel 'Methods withòut Free Surface .} . . _.,

4.1.2 _{Assessment of Various Btilb Designs}

54

55

4.1.3 Knuckles and Bilge. Keels.. ₅₈

4.1.4 Assessment of the Aftbody ₅₉

4.2 Navier-Stokes Solutiòns ₆₂

5 Fiów Calculations with

a Free Sürface

₆₉

5.1 The Linearized Fee Surface Condition ₆₉

5.2 Kelvin Sources ₇₃

5.3 _{Applications of the Kelvin 'Sources 73}

5.3.1 The Michell Theory 73

5.4 Kelvin Sources for Catamaran Hulls, an Example 76

5.5 Dawson's Method ' 78

5.5.1 Applications of Dawson's Method ₇₉

56

_{General Considerations to Assess Programs}

. 82

'6

_{Profile Characteristics}

₈₉

6.1 The Pressure Distribution ₈₉

6.2 The Loading Distribution ₉₂

6.2. I _{The Lift Curve}

' 93

6.3 The 'Zero Lift Angle ' ' 95

6.4 The Leading Edge Suction Peak ₉₅

.6.4.1 The' Ideal Angle of Attack ,

6.42 Proffle Drag ' 96 97 6.5 Profile Series ' '99 6.5.1 Thickness Distributions ₉₉ 6.52 Caniber Distributions ' 100.

6.5.3 Derivation of the Local Pressure of a Proffle 103

6.5Á _{Considerations to Choose or Design a Profile 104}

7 Lifting Line Propeller Design

' 107'

7.1 Lifting Line Thèory _{. 110'}

7.1.1 _{Two-dinîensionai Lifting Lines.,.}

_..

₁₁₀ 7.1.2 _{Lifting Lines, in Three Dimensions (finite span) . 111}

January 29, 2003, Preface 7

7.1.3 _{Lifting Line Theory for a Propeller 113}

7.2 Optimum R.adiaJ Loading Distribution 115

7.3 induction Factors ₁₁₆

7.4 Propeller Design using the Induction Factors 117

7.4.1 _{Determiñatjon of the Inflow 118}

7.4.2 _{Determination of the Blade Sections 119}

7.4.3 Strength of the Propeller 122

7.4.4 Stresses due to Loading 122

7.4.5 _{Stresses due to Centrifugal Forces 123}

7.4.6 Approximate Methods 124

7.4.7 Lifting Surface Corrections ₁₂₆

7.4.8 Viscous Forces 131

A TABLES

i

B WOORDENLIJST

_vii

Prefâce.

This is an introductory course _{on ship resistance and propulsion for the}

Mar-itime. Technology Department of the Deift University of Technology. The text is written fôr students who have only basic knowldge of mathematics and fluid' dynamics Vector and tensor notation is therefore avoided. The propeller inflow is averaged in time and' space to an average uniform inflow,

and the propeller loading is consequently assumed to be steady. Variable

conditions will be adressed in a more advanced course

The intention of the course is to describe the models which_{are used. '1t}

does not 'contain the complete diagrams, data and 'Fonnulas necessary fôr the actual application of the method. These will, nowadays, often be contained

in a computer program. The use of computer _{programs in routine}

calcu-lations makes it even more necessary that the tiser undérstands the model which is used and the restrictions which are inherent to such a model.For an engineer it is risky 'to refer only to "a formula" without understanding the basic theory behind it. Relying only on computer programs, which may centin fudge factors or errors, can cause-problems.

it is essential for an 'engineer to be able to formulate rapidly _{a crude}

approximation of a problem and to grasp the main rariables involved. For_this and for proper use of complicated programs understanding the basic approach

is more important than the detailed' development of a theory. ]eaching the

basic theory 'is the aim of this course.

Structure of the

course

The prediction of resistance, propeller and propulsion characteristics are

January 29, 2003, Preface 9

extrapolation frm modl test results.

systematic or statistical data

flow ca1cuiations

This sequence is natural since model tests form the baths formany

sys-tematic data sets The development of computatiónaj fluids dynamics (CFD) has been rapid in the last. decade, so these methods have becóme a

consider-able help in the prediction of the behavior of ships at fullscale.

Model tests and computations are often cOmplementary, both having their

advantages and disadvantages. Model tests have the disadvantage ofpossible

scale effects, but have the advantage that complex flOw jihenomena

_canbe

simulated.

Calculations. have the advantage that the flow can be calculated in detail and that variations can be made rather easily. However, drastic simplifications

such as inviscid flow are used: in the calcuintión& An important _{aim of this,}

course is to explain the complementary role of cakulations and model tests.

Textbooks

This course is not intended to provide a full inventory of practical rhethods

of ship design or for the prediction of resistance and _{propulsion, fOr this the}

Principles of Naval Architecture [3t] is more suitable. The basis of the math-ematical description of marine hydrodynamics can be founìd in Newman's

book with the samè title [40] _{Reiated'specialized books are Lighthill's book}

on waves [32]and the books of Knapp [26] and Young [53] on cavitation. An introduction in basi aerodynamics with numerical solutions of potential flow problems can be found in Katz and Plòtkin [24]

The emphasis in this course is on the practical application of first _principles

to the prediction of the behavior of ships and propellers Insightm these first

principles increases understanding of the complex phenomena and forms a basis for intelligent problem solving.

10 _{C.Kuiper Resistance and Propulsion, January 29, 2003}

Additional data

The text includes some additional çhita, such as 'common formuis, wich is not

nessesarily relevent to the main idea. This information is. _{on]y given for the}

convenience of the reader using the material for his own purposes It is not

a part of' the text and does not add to the understanding of the problems.

important Fórrnulàs.

or statements

Some conclusions, fórmulas' and' definitions _{are important throughout the}

text. A 'box 'has been placed around this information to smilifr _recognition

and retrieval.

References

Since this course is, intended to provide an understanding of the basic ap-proach of a topic, 'a limited use of references has been made. In most cases users of this course will not have studied literature of the subjects in depth. No efforts have been made to refer to the most recent literature. The. refer-erces in the. Proceedings of the. International 'Towing Tank Conference [21]

or textbooks can be used to locate this information. Whennames are linked

to formulations or theories, thesê names are given with the year, but without the reference. E.g. the Betz 'condition for optimum efficiency is mentioned to date-baek-te-19-29 j-but-is-not-mentioned in the references. Full references are

only made when full sets of diagrams, data_{or formulas have been used. This}

makes' the list of references less dependent _{on the most recent publications.}

'Acknowledgements

Many students and colleagues from Marin have given comments, corrections and material fór this 'course. The help of Mr. Hoekstra, van' Gent, van

Wijngaarden, Holtrop, de Koriing-G'ans. and Mrs.. Raven is gratefully

ac-'knowl'edged. Special: help was. received 'from' SM.M.Bernaert. She has done the redactional work and has improved many topics The text will be

January 29, 2003, Preface 11 pages and on the title page Any comments can be helpful to improve it and will be very welcome.

Chapter 1

Equations of Motion

Objective:

To show the origin of the equations of motion The derivation of the formula 's is not impOrtant. The _{purpose is to show that the equations} of motion are the equivalent of Newtons second law, -to show the assumptions used to arrive at the equations (Hooke 's law, Newtonian fluidj. Also the con-cept of rotation is introduced.

The -equations of rnotiòn describe the relation between forces and motiòns

in the fluid. The equations are rewritten forms of Newtons'_{second 1aw:}

F(t). = mü _(1.1)

where F=force in Newtons, _{mmass in kg and x=position in m.}

The equations of motion therefore describe a relation, not a situation!

The situation, local velôcities and _{pressures (forces), can -only be found from}

a combination of the equations of motion and boundary conditions. The,

boundary conditions are situations in_{pressureor velocity at a certain position}

at a certain- time. When the boundary conditions_{are suficient, the velo:cities}

-and pressures at other times and: positions _{can then be cakulated using the}

equations of motion.

Example A simple and well known example is: the applicationof Newtons'

law directly to a bod' with -mass m. 'The equation of motion is eq. 1.1 and the boundary conditions are e;g.

14 _{G.Kuiper, Resistance and Propulsion 2, January 29, 2003}

y(0) = O x(0) = O

F(t)=f(t)

This makes it possible to calculate the velocity of thebody at an arbitrary time t from:

From the boundary conditions it is found that C = O. In this case the

equation of motion is integrated.

When the velocity ±(t) is given as a boundary condition instead of f(t) the forces can be derived directly from the équation of motion by using the time-derivative of the given velocity.

This example is meant to ilhistrate the character of the equations of

motion in a fluid. A very useful notation for three-dimensional flow fields

is vector notation. In the follöwing , however, the equations öf motion will

be derived without using this notatiön because it is not essential. To avoid

excessive long formulat ions. the derivation will be in_{one direction only and for}

incompressible flow. The purpose of this derivation is to give basicinsight

in the equations of motion and to show the concepts of _{stress-strain and}

of-rotation=deförmation--Abasic-insight--iii-the-equations of mttiorris-also necessary to understand simplifications which will be made later on, in order to make calculations feasible.

Before deriving these equations of motion another basic equation has to be formulated.

1.1

_{The Continuity Equation}

Nomenclature:

In flow calculations the velocity of particulars has .a direction and a mag-nitude. The components are mentioned u, y and w. These are the velocities

pt

January 29, 2003, Equations of Motion 15

in respectively in x-, y- and z-direction. These components are local

veloci-ties. There are also global velbcity components, they are expressed with the

capitals: U, V and W (respectively inx-, y-, and z-direction).

Sometimes pertuated velocities are considered. When this is the case, it

will be mentioned. Then, the pertuation components are respectively u, y

and w.

The continuity equation expresses that no mass is lost when there is no flow through a control surface. When a cube of fluid is observed at a cer-tain time its shape may change over, time, its deni'ty may change, but the amountS of mass is unchanged regardless the deformations or compression.

For an incompressible fluid this means that not only the mass, but also the

volume of such a cube remains unchanged. We will _{assume incompressible}

fluid here, because the line of reasoning remains the same as in compressible flow. Water can nearly always be considered as incompressible.

The formulation of the continuity equation is an example of the use of an arbitrary control volume which is fixed in time and position.

Figure 1.1: Control Volume Fixed in Time and Position

Consider a control volume as shown in Fig. 1.1 with sides dx, dy and dz. The velocity of the fluid entering the control volume from the left side

ADFG is now called u. (Note that this is the average velocity over the plane,

not a local velocity. This is useful because afterwards the size of the control

16 _{G.Kuiper, Resistance and Prvpulsion 2, January 29, 2003}

the local velocity in point G.). The velocity leaving he cóntrol volume at the

right side BCEH is then u+ dx. Partial derivatives are used here although

only the derivative in x-direction is relevant. Sirnilary, the velocity of the

fluid entering the control volume from the back is called _{y and that of the}

fluid entering from the bottom is called w.

The total volume change dV in a time-step dt is now:

dV

_{= (dx)ddzdt + (dy)dxdzdt + (dz)dxdydt}

In incompresi'ble 'flow this volume change must be zero for any volume

and for any time-step, so that

Ou 0v 0w

(1.3)

Ox ôy ôz

This is the continuity equation for incompressible flów. This equation can be used in the formulation of the equations of motion.

1.2

_{The Equati'ons of Motion}

For the formulation of the equations of motion we consider the forces on a cube of fluid. These forces äre decomposed' in three directions, so that at each side of the cube there is one pressure force (indicated a) and two friction forces (indicatecLi-)

In Fig. 1.2 a cross section of a cube is shOwn with

at the bottom the

pressure force a and the fiction forceT fl x'- direction. The related forces at

the 'opposIte side of the cube are o- ± _{dz and y + Edz. These forces are}

always in opposite directions and the resulting forces on the fluid particle

can therefore be written as dz and Edz only.

in Fig. 1.3 the resulting forces acting on a cube of fluid are indicated.

Note that in this case this is not a cöntrol volume, but a material cube of

fluid at a certain time. The first index indicates the plane at which the force

is acting, the second index is the direction of the force, So _{is a friction}

force on a plane perpendicular to the x-axis in the direction of _{the z-axis.}

We can now förmulate the resulting, force in each directión on the fluid particle. For sake of simplicity this will only be done in the x-direction

Figure i2; Forces on a Cross Section of a Cúbe in a Fluid

Figure L3: Resulting Forces in x-direction

The resulting force in xdirection is

18 _{C.Kuiper, Resistance and Propulsion 2, January 29, 2003}

s

F

(&TXd

_{)dd + (dy)dxdz ± (dz)dxdy}

This force can now simply be used. ineq. 1.1, resulting in

F = pdxdydz (1.4)

where dxdydz is the volume of the element. The result of thissubstitution

du

ôa

a-t a 3y ôz (1.5)

In eq. 1.5 the relation between the fluid motion u. and the stresses in the fluid is given The stresses m the fluid are in turn related to the deformations and compressions in the fluid. For a further formulation of the equations of

motion the relation between the deformations (strain) _{in the fluid and the}

stresses on a fluid element is required.

1.2..i

Fotation and Deformation

The stresses in the fluid will now be related to the deformations of the fluid

particle. _{The cube, considered before, will be deformed .by the forces on}

its sides. For simplicity only two dimensions will be considered, so a cross

section of the fluid particle in the _{x-z plane as shown in Fig. 1.4.}

The originally rectangular shape of the fluid particle under consideration

will he deformed after a time At to _{an arbitrary cross section as given in}

Fig. 1.4. After the time-step:

B(ix, O) is in B'(Lx + x)

D(O, Ay) is in D'(Ly, Lìy +

C(x,

y) is in C'(tXx +

_{-I- Ay, Ly +}

_{x + y)}

C

January 29, 2003, Equations of Motion 19

A B A' X

Figure 1.4 Deformation of a Fluid Particle

the rotation of the diagonal AC, which is

-

). The value

-is called the rotation of the element.

the deformation of the element, which is writteñ as

₊

). This is

called the dilatation of the element.

The dilatation of the element causes a strain in the element.

1.2.2

_{Relation between Stresses and Sffn}

The relation between the dilatation of the element and the frictional stresses

is written as

0v Ou

=

+

(1.6)

Ou 0w

20 _{G.Kuiper, Resistance and Propulsion 2, January 29, 2003}

Note that the frictional stress in the x-direction contains also the velocity

gradients in the y-direétion. The parameterji is the dynamic viscosity of the

fluid.

Also when no dilatation occurs a deformation is possible, as shown in

Fig 1.5.ln this case the pressure gradient in z-directioncauses a defórmation.

In such a case the follOwing relation yields:

8cY ô ôu

=

Here p is the mean static pressure. This relation is not evident. It is. valid: for an isotropic fluid and the stress-strain relation uses the same dy-namic viscosity as the relation between viscous stresses and deformations.

y

D

(1.7)

A B A B'

Figure_L _{Deformation_of_a_Fluid_Particle_due_to the Pressure_Gradient}

Lbwgvgle}

The fluids for which the viscosity is a constant independent of the

flow-field are called Newtonian fluids _{Water is such a fluid. An example of a}

non-newtonian fluid is blood. In blood the viscosity increases with increas-ing velocity gradient. This is also the case with many polymer solutions.

Using the foregoing the equations of motoncan be formulated.

1.2.3

Navier-Stokes Equations

Eq. 1.5 can be rewritten using the foregoing as:

y

C,

January 29, 2003, Equatiöns of Motion 21

The total derivative of the velocity u can be written in its partial

deriva-tives as

=

+ u + v ± w so that the formulation of the equation

of motion in the x-direction yields:

Su Su Su Su

Sx Sy Sz (1.9)

Sp S2u S2u a2u

a Su

0v 5w

+ /L-(-+-±-)

_öxôx

Oy Sz

This equation is also called the Navier-Stokes equation, because _the

French mathematician Navier and the English mathematician Stokes

deivel-oped this equation independently in the middlle of the nineteenth_century.

In incompressible flow this equation simplifies because eq. 1.3 can be used, resulting in

au au au

au

p{- +u ±v-- +w} =

52u S2u _52u

52w

ôzS)

5v 5w ay ôz (1.8) (1.10)

The equations in the other two directions çan simply be found by changing the x,y and z and u,v,w variab1es and indices.

du o a

a

a

3v ôu

=

ô ôu 5w

ap 52u 52v 52u 82u

=

ap a2u a2u a2u

ô Su

22 _{G.Kuiper, Resistance and Propulsion 2, January 29, 2003}

1.3

_{A Simple Example}

The Navier-Stokes equations are too difficult to solve in general. As a simple

illustration the flow between two parallel wallscan be considered (Fig. 1.6).

In this case there is only motion in one direction, and only the equation of motion in that direction has to be solved. Moreovèr, the flow is steady and does not change in time

Figure L6: The Flow between Two Parallel Wails

When the flow is in x-direction the Navier-Stokes equation in x direction

has to be considered. Since the flow is steady the time-derivatives are zero.

Also the x-derivatives of the velocity arezero since the flow proffle remains

the saxne-aallst-at-ions--T-here-is-onIy-a-pressure gradientg in-x direction.

The pressure gradient is negative since thepressure decreases in x-direction.

Because the flow is two-dimensional there are no derivatives in z-direction and the y- and w-components of the flow are zero. This simplifies the N.S. equation to:

ap 02u

The solution of equation 1.11 is now straightforward:

au ₌

yap

+ cl

(1.12)

January 29, 2003, Equations of Motion 23

u =

+C1y+C2

2t ax

This is the integrated equation of motion. The constants can be deter-mined using boundary conditions. These conditions are:

s v=O

u(b) = u(b) =O

where b and b are the locations of the wall in y-direction. With these

boundary conditions it has been assured that the flow at the wall is parallel

to the wall and the condition of viscous flow has also been imposed: the

velocity of the fluid close to the wall is zero.

The solution is C1 = O and C2

=

. This leads to the solution:

lap

2 2

u=--(b y)

The velocity distribution between the walls is thus arabo1ic.

1.4

The Euler Equations

Because-the Navier-Stokes-equations are very difficult-to solve, both analyt-ically and numeranalyt-ically, simplifications have been applied. The first

simpli-fication is the neglect of viscosity. This removes all terms with from the

Navier Stokes equations. The result is called the Euler-equation. From the

Euler equations some specific results can be derived . Although the Euler

Equations are 3D, now only the 2D-equations are considered. The stationary Euler-equation in x-direction is:

au

au

u + v =

_{ax ay}

_pax

This equation can be rewritten in terms of rotation as

24 _{G.Kuiper, Resistance and Propulsion , January 9, OO8} i ôp p 0x l&u2 5u

=

lau2

au

ia

2 2 =

(u+v)

This form can also be integrated in x-direction:

p 2 2 u V

-- + C = (u + y ) + j v( - )dx

(1.16)

p 2

_J

ôy ôx

Eq. 1.16 is a form of the Euler equation . These equations of motion

describe the flow in an inviscid, rotational flow. The Euler equations can

be used in flow regions where vortices, generated elsewhere, _{are present, but} where no new vorticity is generated. The path and behavior of the vorticity can be calculated by Euler solvers.

1.5

The Bernoulli Eqfuatiôn

The pressure can be expressed explicitly from the Euler equations.Thereare

two possibilities to do this:

along a streamline, where f {v(

-

) - u(.

-

)} ds = O

in irrotational flow where _Oy

-

_Ox = O

In both cases the integration leads to the Bernoulli-equation:

p+pV2 =c

(1.17)

When gravity is the only external force the constantcan be written as pgh+C, so that the Bernoulli equation takes the well known form

p+ pV +pgh= C

ôv Ox ax

au

0v +V(ay _ax (1.15) (1.18)

January 29, 2003, Equations of Motion 25 Note that in an inviscid and irrotational flow the Bernouffi equation is valid anywhere in the flow (it has the same constant everywhere in the flow)

In an inviscid but rotational flow the Bernoulli equation has the same constant along a streamline. Every streamline has its own constant, however.

1.6

Summary.

The equations of motion in the flow are a translation of the second law of

Newton:F orce = mass x acceleration When this law is applied to a particle

in the flow the Navier stokes equations evolve. In deriving the Navier Stokes equations use has been made of the law of Hooke, which relates the

deforma-tion of the fluid with the required stress. Also the fluid is characterized by a

single viscosity in all directions and for all deformations. The class of fluids with these characteristics are called Newtonian fluids.

The Navier Stokes equations can be simplified by neglecting the viscosity.

This results in the Euler equations. A further simplification can be made by

neglecting the rotation of the flow. In that case the equations of motion can be integrated and the Bermioulli equation evolves:

Chapter 2

Potential flow

Objective: To show the use of singularities in potential flow calculations.

The equations of motion relate the pressures in the fluid with the flow velocities.In an irrotational, inviscid flow the equation of motion can be writ-ten as the Bernouffi equation,, eq. 1. i& The continuity equation is given in

eq. 1.3. These two equations are the constitutive equations for an

irrota-tional, inviscid flow. That means that they _{are vaiid anywhere in the flow.}

A potential flow is now defined as a flow field which can be described by a scalar function, generally indicated as (x, y, z), which is the potential. The-velocities in the flow in any-direetion ar-e-the par-thu derivatives-in that direction of the potential:

u =

-Ox

V =

-ay

28 _{G.Kuiper, Resistance and Propulsion .2, Jamuar' 29, 2003}

The rotation of a potential flow e.g. in the x

_{- y}

plane can be wrjtten in

terms of the potential

@u avaa

aa

ay ôx Oyäx ôxOy

and this is always zero for a function _{which is twice differentiable. So}

a potential flöw is always irrotational

In such a flow the relation between the pressures and the velocities can be described by the Bernoulli equation.

'When the continuity equation is written in terms of the potential this results in

a2 52 32

a ay2 8z2

This equation is called the Laplace equation In order to describe a flow

field it is therefore necessary to find a solution of the Laplace equation.

The strength of the potential flow description is that it is possible _to

define a number of elementary solutions of the Laplace equation. Because

the solutions are superimposable, _{an arbitrary flow field can be expressed in}

terms of superpositions of these elementary solutions.

(2.1)

2.1

_{Singularities in Potential Flow}

2.1.1

Uniform Flow

The simplest solution of the Laplace equation is

=Ux

Its second derivative in x-direction is zero and the derivatives in y and z directions are also zero So this potential satisfies the Laplace equation. The meaning of this potential is a uniform steady flow in x-direction, since the

January 29, 2003, Potential Flow 29

2.1.2

Source

Another solution is the potential of a source. In two dimensional flow its potential is

=--1n(r)

(2.2)

r is the distance from the source with strength Q. It requires some alge-bra to show that this potential satisfies the Laplace equation.

The velocities in radial direction can be found from the derivative of the potential function to r, so

Q

Vr -2irr

There is only a velocity in radial direction because vo = O.

/\

Figure 2.1: Velocity Field of a Potential Source

This is called a source because the amount of fluid passing through a

circle at radius r is always Q. The flow field of a single source is shown in Fig. 2.1. VVhen the sign of the source is negative it is called a sink.

In three dimensions the potential of a source is

C.

4irr (2.3)

The velocities in radial direction are found from

a

30 _{C.Kuiper, Resistance and Propulsion 2, January 29, 2003}

Here the amount of fluid passing through a sphere around the source per unit time is always Q.

2.1.3

Vortex

The potential of a two-dimensional vortex is

2ir The velocities are now in tangential direction

10cI F

Ve = -- =

rôO 2irr

There is no velocity in radial direction since the potential ftmction is

independent of r. Here F is the vortex strength. The integral of the velocity around a contour in the flow is called the circulation of the vortex, defined as

F =

_j'vods

The flow field of a two-dimensional vortex is shown in Fig. 2.2.

The two-dimensional vortex can be considered as a vortex element per pendicular to the plane.

In three dimensions the same formulation 6f a vortex can be used. Only a vortex element is considered, and the velocity induced by such a vortex element is in a plane perpendicular to the vortex element, as shown in Fig. 2.3.

The velocity induced by the vortex element dl with strength F in an arbitrary point P can be written as

F sin O

dV= ds

4irr2

For an infinitely long straight vortex in 3 D the induced velocity is

r

2ira (2.5)

January 29, 2003, Potential Flow 31

Figure 2.2: Velocity Field of a Potential Vortex

Figure 2.3: Velocities Induced by a Vortex Line Element

where a is the distance perpendicular to the vortex line. This is a well

32 _{C.Kuiper, Resistance and PrOpulsion 2, January 29, 2003}

An isolated vortex element cannot exist in a potential flow, field, because in such a flow field no vorticity is generated. So a vortex element has to be part of a continuous vortex or it splits into two vortex elements with different

directions, but with the same total strength. A _{vortex can only end on a solid}

wall which is the boundary ofa potential flòw 'fie1d

2.1.4

Dipole (3D)

An elementary solution of the Laplace 'equation which is also often used is a combination of a source and a sink ('a negative source). The source and sink are infinitely close together, but the product of source/sink_{strength and} distance remains finite This product is the, dipole or doublet strength ji.

Consider a source of strength Q and a sink of strength -Q at a distance i from each other. The potential in an arbitrary point P is

47r \r1 r2

(2.6)

The factor between brackets can be written as

r2 r1

r1 T2

When i - O the product r2r1 -* r2 and the distance _{r2 - r1 - i.cosû, where}

O is the. angle between the axis of the dipole and the _{arbitrary point P. When}

Ql -*

ji

the potential of the dipole can be writtenas

jicos(0)

3D= ₂

4irr

Analogue as in 3D, a dipole in 2D can be derived, the result is:

jicos (9')

2D =

2irr

The velocity field belonging to a dipole with its axis in the x-direction is shown in Fig. 2.4.

The velocity field 'indicates that dipoles _{can be written as vortices and}

vice-versa. In fact it can be shown that a vortex can be written as a deriva-tive of the dipole. So a velocity field containing dipoles is equivalent with a

(2.7)

January 29, 2003, Potential Flow 33

Figure 2.4: Velocity Field of a Doublet or Dipole velocity field with vortices.

Note that a dipole has a direction. This direction is the axis of the position

of source and ink. The potential ofa dipole is in fact the derivative of the potential of a source, so

(dzpole) _{a_(source) (2.9)}

This relation will be used in panel methods (see below).

The velocity of a source or vortex is infinite at the location of the _source

or vortex itself. The mentióned potentials are therefore called singularities.

Note that a single singularity describes the fow field everywhere in the flow.

2.2 A Simple Example of Potential Flow

The use of singu]arities to describe _{a flow field is now illustrated by the}

combination of a uniform flow with a single dipole (in 2D). A uniform flow in

x-direction has a potential function 1 Ux. When a dipole with a strength

N3, in x direction is put in a unifòrm flow the potentials can be added, so:

(stream) = (parale1) ± 1(dipole) = Ux N3, cosO

where r2 = x2 + y2 and x r cos O. The velocity in radial direction is then

found from

34 G.Kuiper, Resistance and Propulsion 2, January 29, 2003

N

vr==UcosO(1+

2irUr2

For r -' oo this reduces to

Vr = LT cos O, which is the undisturbed uniform flow in x-direction.

There is another special location, where

2_

N

r

2irU

At that location the radial velocity is exactly zero, so there is only tan-gential velocity. When this location is considered as the boundary between

inner and outer flow it turns out that the combination of a dipole and a

uniform flow represents the flow around a cylinder!.

Figure 2.5: Flow around a Dipole or Doublet in Uniform Flow

Note that the tangential velocity along the cylinder is not zero, as it

would be in real fluid. This is because the solution is in potential flow, which is inviscid.

The tangential velocity along the cylinder is also be derived from the

potential function:

(2.11)

vo =

= U sin 0(1

January 29, 2003, Potential Flow 35

At the location of the cylinder R2

_{= -, since there is only a tangential}

velocity, so

vo = 2UsinO

So there is a sinusoidal distribution of the velocity on the cilinder. The maximam velocity at O = ir/2 is twice the incoming velocity U. Because this is in a potential flow (irrotational) Bernoulli's law _{can be applied to find the} pressure from the velocities. The inviscid pressure distribution can thus be found. At the top and bottom position (O = ir/2) the velocity is maximum

and the pressure is consequently minimum, the minimum _{pressure is found}

from: Po + 1/2pU2= prnjn + 1/2p{2U]2 Or PminPO

-3

1/2pU2

-where Po iS the pressure in the undisturbed flow. The value_{-3 is called}

the pressure coefficient.

This example shows that a single dipole can describe the flow around a cylinder. Combinations of singularities in the flow can describe almost any flow patterns. When the flow around an arbitrary body has to be calculated the potential flow theory is used to locate singularities in the flow and to

determine the strength of these singularities in such _{a way that the body}

contour is exactly a streamline.

Because the pressure distribution in our example of the cylinder is sym-metrical in x-direction there is no resulting dragforce on the cylinder. This is a general feature of bodies in a potential flow (Paradox of d'Alembert).

36 _{G.Kuiper, Resistance and Propulsion 2, January 29, 2003}

The resistance of a body in

a parallel potential flow is zero.

This is due to the absence of the boundary layer, which causes not only the absence of a friction force along the surface, but also the absence of

pres-sure drag.

2.3

Forces on a Vortex

It is easy to add a circnlation around the cylinder by adding a vortex with strength F in the center of the cylinder. The potential of the flow is similar to eq. 2.10 with the potential of the vortex added. The radial velocity in eq. 2.11

does not change, because the radial derivative of the vortex potential iszero.

The tangential velocity component changes from eq. 2.12 into:

N F

vo=UsinO(1

)+-2'ïrUR2 2irr (2.13)

The velocity distribution now becomes asymmetrical.

At the top of the

cylinder with 6 = the velocity is

N

F

= U(1

_{- 27rUR2)} + tebottoinoLthecyiinder the velocity is

N F

= +U(i

_{- 2irUR2)}

+

The pressure distribution, which can again be found by application of Bernoulli's law, also is asymmetrical. Integration of the pressure over the cylinder gives after some algebra the resulting force pUF in the direction perpendicular to the flow.

So a vortex does give a force, perpendicular to the flow. The relation

between the side force L on a vortex and the incoming flow U is called the law of Kutta-Joukowsky:

L = pUF

January 29, 2003, Potential Flow 37

Vortices are so important because they can represent a lift force in the fluid. This will be used for the description of airfoil wings or propeller blades. An example of the application of lift on a single vortex is the Flettner rotor

The rotating cylinders generate a circulation around the cylinder due to frictional forces in the boundary layer close to the wall. In wind the result is a side force perpendicular to the wind.

2.4

Panel Methods

Without proof it is stated here that any potential flow field may be writtenas

a surface integral over the boundary S of a source distribution and a dipole distribution on S. (See e.g [24]). The dipole distribution has its direction normal to S. Note that the boundary surface S is the complete boundary of the flow, as shown in Fig. 2.6.

38 _{.G.Kuiper1, Resistance and' Propulsion , January 9, 003}

Figure 2.7 Panelling of an Axbitrary Body

When the outer surface goes to 'iifinity the surface S can be considered.

as the surface of the body and the cut from the body to hifinity. When this cut has no thiciness only dipoles are present there.. When. there is no lift on

the body a source distributibn over the surface of the body remains.

For a short indication Of panel methods it is sufficient to consider 'a source

distribution on a body. This situation is different from_{the examplé of a}

'cilin-der, because in that example the dipolewas in the. center_oLthecyJinder and

not on the surface.

The boundary condition On the. .body is that the normal velocity on the body is zero, so

On (2.14)

This condition is valid everywhere_{on, the surface of the body.}

The panel method is a :d:i,scretization of this boundary _{problem.' The}

surface' of the body is approximated by flat elements. At _{each element a}

January 29, 2003, Potential Flow 39

So the souice distribution on the body is also discretised. The boundary

condition eq., 2.14 is now applied _{on one point of the element. This point is} the control point, where the boundary condition is fulfilled.

Consider the velocity, induced by an arbitrary source panel in one contröt point P1, as: sketched in Fig. 2.7. The potential in P1 of_{an arbitrary panel is}

fUpa,e

dS J panel 4irr

where dS is a part of the paneL This integral is the influence function of a panel on a control point The nain variable is the distance r the control pomt and an arbitrary point on the panel S The potential m P is the sum

of the influence functions of ali panels and the normal velocity is _{the normal} derivative of this sum.

A singularity occurs in the influence function when P1 is located in the panel

which is considered, because in P1 the radius i is zero. This leads to a

sin-gular integrali over the panel with P1. Fortunately this integral has _{a finite}

sölùtion and can be analytically formulated.

When r is relatively large (say afe _{times the panelsize), the integration}

can be omitted because the panel distribution can be consideredas a single

source with strength cr8. The majority f_{the panels can thus be treated: as}

discrete sources,

The normal velocity induced by all panels in the contröl point P1 can thus be formulated as the sum of the influence functions _{of ail panels Application} of the boundary condition of zero normal velocity results in _{a set of N linear} equations, where N is the number of panels. A matrix of N x N results and this matrix can be solved.

There are different methods to formulate the problem, but the basic

assump-tions are the same. Refinements are possible by using curved panels instead

of flat ones and by using linear or quadratic formulations for _{the source}

dis-tribution on the panels. For a short review see [1:5]. An introduction in the formulation of the problem is given eg in [24].

40 _{C.Kuiper, Resistance and Propulsion 2, January 29, 2003}

2.4.1

The Lifting Pröblem

In the. foregoing description it _{was assumed that there was no lift on the}

body. When lift is present two important additions have to be made.

The first one is that there ae not 'only singularities on thebody, but also in the wake, which is expressed by the cut S in Fig. 2.6. Consequently the wake has to be panelled too. This can be a problem because the position of the wake is not always known. In case of a flat wing it can be approximated by positioning it in the plane of the wing, which is only true when the dipole strength or vortex strength is smalL This wake model therefore introduces

again sorne linearization. On a propeller the problem is more complicated

because the wake has: a complicated shape. This shape can be determined

only by the condition that the wake is in the direction of the (induced) _flow.

The induced flow is again dependent on the dipole strength, and this dilemma

is characteristic, for the propeller problem. Simplified wake_{models will be}

treated in the sectiOns on propeller design and analysis.

On the body and in the wake a dipole distribution is added to the

prob-lem. Every panel has not only a (uniform) source strength o, but also a

(umform) dipole strength ¡i of unknown magmtude The influence functions

of the panels become more complicatedas a result. The general shape of the

influence functhin is 'given in eq. 2.15.

I

Ii

1ia1\

J-a+--jdS

(2.15)

panel n

P1 =

In this equation the factor has been absorbed in the strength of the

sin-gularities.

The boundary condition remains the same, but an additional boundary

condition is necessary because there are additional unknown dipole _strengths

in the influence functions. A solution can be found by prescribing as an

adcli-tiönal condition that the floW at the aft stagnatiòn point remains finite it is

easier to consider a wing, where the flow velocity at the trailing_{edge has to}

remain flnite This condition is càJled the Kuttci conditIOn and it determines the strength of the dipoles or vörtiçes in' the wake. It adds one equation to the matrix of influence functions.

LST1NG STIP OF PANEU OUND VOTICITY N-LINES TRAILING EbGE

January 29, 2003, Potential Flow 41 A common problem of a panel method with lift is the flow around an airplane, as shown in Fig. 2.8.

-4-tRAIUNG EDGE

SEGMENT

Figure 2.8: Panels at an Airplane Sürface (From Hess)

The lift on the wings is represented by dipoles over the wings and the wake. The fuselage is covered with sources only.

Although the-for

often not so easy. When the panels are chosen wrongiy or inconsistently, the matrix of iufluence-,fanctions becomes ilbconditioned and the solutIon is no longer reliable.

2.5

Summary

The velocities in an incompressible, irrótational flow can be described by the Laplace equation and the Bernoulli equation. The latter equàtion relates the pressures with the velocities:

42 _{G.Kuiper, Resistance and Propulsion 2, January 29, 2008}

These flows are called potential flow fields.

Potentizl flow fields can be described with singularities. These are poten-tials which are sirigular(infinite) in the core. Typical singularities are vortices and sources. A body in the flow can be described by a closed streamline, which is generated by a singularity distribution in the flow. In a potential flOw no force is exerted on a body without circulation. (Paradox of d'Alembert). When circulation is present the force is perpendicular to the flow and its magnitude

is

F=pVF

where F is the circulation around the body contour.

Single vortIces have the property that they cann'ot begin or end in the flow. They are either closed or are connected to a solid vial 1.

The velocity induced by a vortex in the flow can be found from its potential. For a straight line vortex element this velocity is

given by the law of Biot and Savart

Fsinü

dv=

ds

4irr2

The flow around an aTbitrary three dimensional body can be described with a surface distributïon of singularities. When no circulation, is present a surface distribution of sources (and sinks,) is enough. When circulation is present also vortices or dipoles have to be distributed over the surface and

Chapter 3

Boundary Layers

'Objective: A basic approach of boundary layer theory and the definition of a turbulence model

Consider the flow over a two-dirnensïonaJ body in the x-y plane. The mean flow is in the x- direction.The flow can be considered inviscid, except in a region close to the body, where a strong velocity gradient is present.

In this boundary layer region the equations of motions e the Navier-Stokes

equations When the flow remains attached the viscous region is thm relative

to the chord f the profile and this thin layer is called 'the boundary layer.

A two-dimensional body is. considered, so the two-dimensional Navier-Stokes equations apply. These equations are:

ôu au

u F'

V-ôx ay 0v 0v

u +v-

_Ox Oy Ou 0v Ox ay

In these equations the dynamic viscosity a has 'been replaced by the kine-matic viscosfty u, which 'is found from' ii =

lap

82u ô2u

lop

02v 02v

-

± V('

+

Oect.

(3.1)

44 G.Kuiper Resistance and Propulsion 2, January 29, 2003

3.1

The non-dimensional Navier Stokes

Equa-tions

For further use these equations are first non-dimensionalized by replacing

/ u by u =

xbyx' =

ybyy'

pbyp' =

where U is the incoming undisturbed flow velocity and I is a relevant length of the body. For one term this is döne as follows:

=

(3.2)

,Du' U2

=

T)

T11s can be done for all terms in eq. 3d, which leads to the non-dimensional equivalent of the Navier-Stokes equation:

Du Du i 87) u A2, A2,

U-

-4-'ii-Dx

Dy =

p3xUlDxDy2)

(3.3) Dv Dv

= ---+{----+--

lap

y (D2v

D2v Dx Dy p Dy Ut \ Dz2 Dy2 Du Dv

±-

Dx Dy

In eq. 3.3 all primes are deleted, but all velocities and distances are now non- dimensional.The quantity u/UI is the inverse of the Reynolds number.

3.2

The Boundary Layer Equation

Terms which are small when the boundary layer is thin will now be neglected. The boundary layer with thickness 6 is thin when q «i. Because the profile

=0

January 29, 2003, Boundary Layers 45 is assumed to be thin also the vertical velocity y will be small. This leads to the following orders of magnitude of the non-dimensional quantities:

u=ø(1)

x=ø(1)

vø(6)

.y=ø(S)

This makes it possible to identify terms of order i5. In x-direction eq. 3.3

is

Ou Ou lOp i

ôu

02u

The terms of this equation have the following order. of magnitude:

1

i

lOp

i

I i

F1- +8=

_-

±(- ±

The last term between the brackets can only be of order i when R

-which is the case for iarge Reynolds numbers. Neglecting the terms of order 8 the boundary layer equation is found:

Ou Ou

löp__i Ou

(3.4)

It is important to remember that the Reynolds number in this case has to be of the order 1/82. So when the boundary layer thickness is of the order of 1 percent of the length of the body, the Reynolds number based on the

length of the body has to be of the order of iü. Still these equations are only valid for laminar boundary layers.

In y-direction a similar procedure can be followed. The Navier-Stokes equation in that direction is

46 _{CKuiper, Resistance and Propulsion 2, January 29, 2003}

which has terms with the following order of magnitude

6 6.

ia

26

6

All terms except the pressure term are now of order ô or smaller, so the

equation of motion in y-direction reduces to

ay

This means that the pressure in the boundary layer is constant over the

boundary layer thickness. This is an important conclusion because _{now the}

pressure. outside the boundary layer can be identified with the wall pressure.

Outside the boundar layer the flow is modeled kike a imdisturbed

poteri-tial fiw and the velocitycomponents are respectively U and y. The stream-line at the edge of the boundary layer is along the profile of the boundary

layer.

So there is a relation between the paralll and

cross components,

namely that the direction of the streamline: is equal to the tangent of the

proffle of the boundary layer.

- = - or y = U

_u

Ox Ox

Because the boundary layer is small, the velocity component y is small.

using BernoiiilWs equation, the square of y is very smaJrd can be negld.

Thsi way, the pressure is only dependent of the velocity _{component in}

x-direction:

au

Op

ox ox

This can be. integrated in x-direction, which leads to

p+,pU2

_{= C}

So outside the boundary layer the BernouJli equation is valid again. The

January 29, 2003, Boundary Layers 47

3.2.1

Scaling the Thickness of the Boundary 'Layer

in the foregoing, derivation of the bóundary layer equations the assumption

was made that R (X 1/i2. Inversely this means that ' oc (The prime

means the nondimensiona1 boundary layer thickness f) When a model is

now towed at scale ratio 25 at the full scale 'Froude number -, the ratio

between R, and Rnm is 125,, where the index m stands for ml and s for

ship. This means that

=Vi

So the boundary layer at the model is relatively more than ten. times thicker than at the ship. Relatively, because the boundary layer is taken

relative to the size of the ship This causes the scale effects on the wake distribution and on the resistance.

3.3

Solutions of the Boundary Layer

Equa-tions: Biasius

The 'boundary layer equation eq. 3.4 can be. integrated when it is assumed that the velocity profile at every x-position has a similar shape

J (3.5)

Such a solution' is called a similarity solution. The specific friction at the wall is found from

Ou

Tw

0Y

y0

Blasius assumed' a simple similarity solution by taking the velocity distribu-tion in the boundary layer as

U -

"5

This made it possible to integrate the boundary layer equation in combination with the no slip condition at the wall. The resistance D of a flat plate with

48 _{G.Kuiper, Resistance and Propulsion 2, January 29, 2003}

length i and width b canbe found from the integration of the specific friction

coefficient 'r iver the length of the plate at both sides: D

=

2bf

r(s)ds

e

The solution of Blasius for a flat plate is

1.328 cf

=

where D

=

:pU2

3.4

Turbulence

The foregoing boundary layer equations are valid för laminar flow. The vis-cosity used in the equations i the kinematic visvis-cosity, a property of the fluid.. Simultaneously a high Reynolds number is assumed, so the Blasius solution

is only valid for R iO4 and until the boundary layer becomes turbu1nt at

R7., io

Thrbulence is the occurrence of random motions additional to the mean velocity. So the instantaneous velocity of a fluid particle can be described by the mean velocity and a turbulent component:

u

=

u' with O

y = + y' with = O

p + p' with O

To illustrate the effect of turbulence the N:avierStokes equation in x-direction (eq. 3.1) is considered:

Ou Ou Op 02u 02u.

p(u- + v-h)

-- ± i(- +

Substituting the instantaneous velocity in this equation and taking the time average results in a new formulation, the term p can be written as

i (2 + 2iu' + u'2)

-

2 ax

i ôz i au'2 an

i a'

= --+ =+---

_2ax

_2ax

_ax

_20x

The turbulent equation of motion in x-direction becomes in this way:

a

a

p(u +v)

-ax ay

January 29, 2003, Boundary Layers 49 The latter is true because the time-average of the turbulent pressure com-ponent is zero. So this pressure term is not changed by turbu1ence This is true for all right hand terms, but not for the left hand terms. E.g. the term

u can be written as au 13u2

2a

a2i

a2u i

ai

aJI.

-

_-ä--

_ay (3.7) (3.8)

The extra terms require further attention, because they describe the éffect

of tiirbulence The term 2 will be small relative to the derivative

in y direction, because the variations of the velocities normal to the. wall (in y-direction) are much larger than those in flow direction. (See also the consideration of a thin boundary) To assess the effects of turbulence only the

second term is sufficient. Similarly the term will be small relative

These term can therefore alsobeneg1ected.

When it is assumed that the term can be written as

a

AT

ay it possible to. rewrite eq. 3.8 as:

/ ô

au\

ai

a2u 52u

p(+ I

_{\ ax}

_ayj

=----+,i--+A--

_{ax ay2}

ay2

(3.9)

(3.10) When written in this form is is clear that the turbulence can be taken as an additional viscosity, also called the eddy viscosity . This is of course due

50 GKuiper, Resistance and Propulsion , January 9, 003

to the assumption of eq. 3.9. This assurnption is called the turbulence model,

which in this case relates the turbulent viscosity in a !jfl fashion rith thê

velocity gradient in. the bomdary layer. Other turbulent mode1 lave been formulated. No final turbulence model that .can describe the phenomena in

rai flow has been formulated'yet.

The eddy. viscosity is generally much greater than the dyflamic viscosity, so AT is greater than p. The determination of A is therefore important. A well-known model is that of Prandtl, who assumed that

pl

where i is the average distance of the turbulent motion in the, boundary

layer. The model. is therefore called the mixing length model. The problem

is of courseto determine this length. In practice there is not such a single

length scalè, but this model' gives a conceptual physical background to the turbulence model of eq 310

Chapter 4

Flow Calculations without.

Wave

s

Objective:

Review of the use Qf calculations for resistance and flow No extensive mathematical formulations are used, the purpose is to be able to use the available programs intelligently.

A complete description of the flow around an arbitrary body is given by the Nàvier-Stokes equations, provided that at high Reynolds numbers a so-lution can be found for the formulation of the eddy viscosity, the ficticious viscosity which is caused by the turbulence in the flow. The flow around a ship hull can be found in priciple by applying the no slip condition at the 'wall and by applying the appropriate boundary conditions at the free sur-face. The free surface conditions are twofold. The first is the kinematical free surface condition : the fluid. particles at the free surface have to remain

at that surface The second is the dynamical free surface condition : the pressure at the free surface is always equal to the atmospheric pressure. In principle this problem is well fórmulated' and using a volume discretization of the fluid it can be solved.

The solution of this problem is still not really feasible. One reason is that the viscous phenomena require a very fine grid because the sçale at which energy is dissipated is very small. On the other hand the scale of the waves at the free surface. is large. In principle a large flow region with a very fine

52 _{GKuiper, Resistance and Propulsion 2, January 29, 2003}

volume grid has to be used for the solution and this is still beyond the present computing capacity. Another reason is that the free surface is a part of the solution. It is not known beforehand where the, free surface boundary condi-tion has to be applied.

Programs using either volume discretizations or surface panelling there-fore contain essential simplifications. Only when the implications of these simplificaticns are understood a proper use can be made Of Computational Fluid Dynamics (CFD).

The first and basic simplification is that the regions of viscous flow and

the regions of potential flow with a free surface are separated, This is an

assumption comparable with the Froude hypothesis, which separates viscous and residuary resistance.

By far the most important 'calculation 'retbods are those which calcu- " '

late the inviscid outer flow. The presence of a thin boundary layer makes it possible to regard the flow outside the boundary layer as a potential flòw

and the pressure at the wall i equal to the pressure at the outside of the thin boundary layer. In that case the flow around a ship hull can be

re-gardad as a potential flow, because the viscosity can be neglected and as a

result no rotation is generated. The uthforrn inflow in front of the ship is

albo irrcitational, so the fiöw around: the 'ship can be described by the Laplace equation. The boundary conditions are the free' surface conditions 'and tan-gential flow along the outside of the boundary 'layer. Since the boundary layer is considered thin this can be approximated by tangential flow along the hull.

The viscous region can in principle 'be calculated using 'boundary layer equations, because the boundary is considered to be thin. Efforts have been made to use two-dimensional boundary layer equations along streamlines The boundary layer is' considered two-dimensional when no cross flow per-pendicular to the stre rnkne occurs. The streamlines and the pressure distri-bution along 'the streamlines have to be found from the potential calculations of the outer flow. The results of such calculations give a distribution of lo-cal friction coefficients along the streamline integration of the longitu'di:nal component over the hull gives the resistance.

aver-G.Kuiper January 29, 2OO, Flow calculations 53

UMIT1NG STREAMLINES

Figure t L Longitudinal, separation at the bow

t

t i

F

age ship hull the effects Of cross-flow and separation are too 'importarït to be ignored, and the presence of a thick boundary layer in the stern region has too much influence on the rsistance Cross-flöw is too important to be ignored. Three-dimensional boundary layer calculations are complicated and they blow up in regions of separation, where vortices leave the hull. In those regions the basic assumptions of boundary layer flow are violated. Until now no reliable calculatiOn methods are available to calctilä.tê the resistance. Ef-forts to db this are madè using the full Naviér-Stokes solutions, as will be discused later in this chapter.

The separation of the viscous and the inviscid regions neglcts the interL action between the viscous flow region and the potential flow region. The

54 G.Kuiper, Resistance and Propulsion 2, January 29, 2003

main region where problems occur is the wake, where the flow is 'highly ro-tational and viscosity cannot be neglected. Still it is a part of the potential

flow region. But also elsewhere on the hull where any type of separation

occurs, such as at the bilges in the forebody (see Fig. 4.1) the assumptions are viohited. Also when the boundary layer becomes very thick, as in some regions in the aftbody, the solution of the potential theory will be inaccurate. Keeping in mind these restrictions the potential flOw calculations can be used intelligently to optimize hull. forms or to calculate flow patterns which are difficult to measure at modél scale.

4.1

Potential Flow Calculations

4.1.1

Panel Methods without

ee Surface

The simplest category of calculations of the flow arouiid the ships hull are the potential flow calculations without a free surface. The undisturbed free surface is assumed to be unchanged by the flow around the hull. in practice this is realised by mirroring the ship hull at the undisturbed: free surface. This socalled: double hull is used for flOw calculations in an unbounded fluid. Because the undisturbed waterline is a plane of symmetry, the velocities nor-mal to that plane will always be zero. A drawback is, of course, :that twice the amount of panels have to be used to cover the wholé double body.

When no lift is present the singularity distribution used is a source

dis-tribution. The sources are distributed on the discretized hull surface. The

elements of the hull surface are called panels. As boundary conditions the condition of tangential flow at the ships hull, is used. This condition is ap-plied at the center of the panels, the so-calléd control points.

The simplest panel shape is a quadri1ateral flat plane , on which a uniform

distribution of sources with constant strength is placed Tins implies that

the distribution of the sources is a step-function instead of a continuous dis-tribution. This approach, first applied by Hess and Smith [14], approximated the shape of a ships double hull with a large number (N) of such flat panels. On each panel a umform source/srnk distribution is placed with an unknown

source strength Q. The potential function (x, y, z') can then be written

as the sum of the potential functions of all elements. The velocity at an

C.Kuiper January 29, 2003, Flow Calculations 55 The influence function of the own panel is singular,, but the singularity 'can be integrated over'the.panel.So after some'mathematics the velocity in an ar-bitrary control point can be expressed as a linear function of the N unknown source strengths Qi..

Next the boundary condition of tangential flow is applied at eaçh control point. This results in N boundary conditions 'for the N panels. This system can be solved, resulting in the strength Q at every panel. The velocities at every position at the ships hull and around the ship' can then be found 'from

the derivative of the potential function 1(x, y, z) in the desired direction

Note that the panel size limits the accuracy of the derivative, because the' derivative has to be determined from the difference between two panels. The pressures can be derived from the velOcities using Bernoulli's law.

It is. important to realize the consequences of the simplifications made. First it is potential' flow, so the fluid is inviscid'. That means that necèssar-uy the resistänce is zero (Paradox of d'Alembert) and .that flow separation

does not occur. Just as in the case of a cylinder .the pressure recovery in

the aftbody 5 complete. The water surface is also undisturbed, so the wave resistance is also zero.

How can' the results of such calculations be used? They can be used to assess qualitatively the relative merit of various alternatives and therefore to optimize the hull before model tests or further calculations are carried out. They can also' be used to indicate', improvements, because the calculations provide data such as pressure 'and velocity distributions, which are not mea-sured Examples are giveu below.

4.1.2

Assessment of Various Bulb Designs

in Fig. 4.2 three forebodies are shown of a full bulk carrier. [23] For these

designs panel calculations were carried' out, The results are shown in Fig. 4.3 The dotted lines in 'this Figure are lines Of equal pressure coefficient. Since the pressure coefficient 'Cp is dèflned' as

i2, a

low (highly negative) value of C means a low pressure on the hull.

A bulbous bow is 'used to minimize wave resistance, but excessive flow separation has. to be avoided as well, since this increases the residuary resis-tance' again. 'The risk of flow separation is largest whèn the.pressure gradients

56 G. Kuiper, Resistance and Propulsion 2, January 29, 2003

SHIP Pi

SHIP P2

Figure 4 2 TwoBulb Configurations for a Bulk_Carrier

are large. In this case, the pressure gradients are largest when the minimum pressure is low. From the calculations this mmi mum pressure occurs at frame

19 near the bottom. The location does not vary significantly between the

alternatives. The fórebody with the smallest risk of flow separation is the one with the smallest perturbation of the flow, so the :011e with the highest minimum pressure coefficient. Ship Pl has a minimum pressure coefficient of -0.35, which is lower than that of ship P2. This indicates that ship. Pl has a potentially higher residuary resistance.