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A STUDY ON VENTILATED HYDROFOIL SECTIONS USING FREE STREAMLINE THEORY AND CFD ANALYSIS Thesis submitted in partial fulfillment of the requirements for the degree in BACHELOR OF TECHNOLOGY in OCEAN ENGINEERING AND NAVAL ARCHITECTURE by Aproop Dheeraj (07 NA 1017) Under the guidance of Prof. Om Prakash Sha DEPARTMENT OF OCEAN ENGINEERING & NAVAL ARCHITECTURE INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR May, 2011
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Page 1: B.Tech Thesis

A STUDY ON VENTILATED HYDROFOIL SECTIONS USING FREE

STREAMLINE THEORY AND CFD ANALYSIS

Thesis submitted in partial fulfillment of the requirements for the degree in

BACHELOR OF TECHNOLOGY

in

OCEAN ENGINEERING AND NAVAL ARCHITECTURE

by

Aproop Dheeraj

(07 NA 1017)

Under the guidance of

Prof. Om Prakash Sha

DEPARTMENT OF OCEAN ENGINEERING & NAVAL ARCHITECTURE

INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR

May, 2011

Page 2: B.Tech Thesis

Department of Ocean Engineering &

Naval Architecture

Indian Institute of Technology

Kharagpur-721302

CERTIFICATE

This is to certify that the thesis entitled “A Study on Ventilated Hydrofoil Sections using Free

Streamline Theory and CFD Analysis” submitted by Aproop Dheeraj (Roll no. 07NA1017), in the

partial fulfillment for the requirement for the award of degree of Bachelor of technology in

Ocean Engineering & Naval Architecture by the Indian Institute of Technology, Kharagpur during

the academic session 2010-2011 is a bona fide record of the project work carried out by him

under my supervision and guidance. The approval does not necessarily endorse or accept every

statement made. Opinion expressed or conclusion drawn as recorded in the thesis. It only

signifies the acceptance of the thesis for the purpose for which it is submitted.

Date: ____________

Place: IIT Kharagpur Dr. O.P.Sha

Professor

Department of Ocean Engineering & Naval Architecture

Indian Institute of Technology

Kharagpur-721302

Page 3: B.Tech Thesis

ACKNOWLEDGEMENT

I would like to extend my sincere gratitude to Prof. O P Sha, my supervisor for the project, for

his support, patience and constant involvement in the project. I would also like to thank Mr.

Manu Korulla and Mr. Jagadeesh Kadiyam of NSTL, Vizag, for providing me with the requisite

data from experiments and their invaluable inputs at critical stages in the project that led to

significant progress.

Date: ____________

(Aproop Dheeraj)

Page 4: B.Tech Thesis

CONTENTS

1. Introduction

2. Free Streamline Theory Equations

3. Ventilated foil sections – experiment and CFD analysis

4. Extension of the theoretical model to a NACA-66 section

5. Correlation of theory with experimental and CFD results

6. Conclusions

Page 5: B.Tech Thesis

List of Symbols The symbols used in this thesis are generally explained when first introduced and their meaning will

normally be apparent from the context. The following list contains the main symbols used.

CL Lift coefficient

CD Drag coefficient

CM Moment coefficient

Cp Coefficient of pressure

Fr Froude number

g Acceleration due to gravity

pv Vapour Pressure

p Ambient Pressure

α Angle of attack

ν Kinematic viscosity

ρ Density

σ Cavitation number

Page 6: B.Tech Thesis

1. Introduction

Ever since sea-faring began, man has made continuous efforts to conquer the seas and establish his supremacy

over the mighty oceans. In this, one of the prime areas of focus has been to make waterborne craft faster. Ships

and boats have evolved from wind powered and paddle-oar systems to steam engines and turbines, nuclear

powered vessels and reciprocating diesel engines. But all of these focus on the propulsion of the vessel, to

overcome the resistance that is encountered while moving through water. Figure 1.1 depicts a typical drag (skin

friction)-velocity curve :

Frictional drag as a percentage of total resistance vs speed

The contributory factors to the total drag are the skin friction and wave resistance, amongst other losses (eddies,

vortex shedding etc). Depending on the type of vessel and the operating speed, different resistance components

have different degrees of contributions. The definition of “High Speed” in the case of waterborne vessels is

ambiguous, as it depends not on the absolute velocity, but the velocity in comparison to the size of the vessel. This

leads to the definition of the dimensionless “Froude number” (Fn):

Page 7: B.Tech Thesis

Where, ‘V’ is the velocity of the craft, ‘L’ its length and ‘g’ the acceleration due to gravity. In cases like SWATH hulls

where the definition of length is not clear, an alternative definition of the Froude number is used:

Where, is the displacement of the vessel. For high speed craft, the operation range is typically at Efforts to

reduce drag have resulted in modification of the hull form, and vary from planing craft and multihulls to hydrofoil

ships and hovercraft. Again, the idea behind most of these concepts to either lift the hull out of the water, thereby

reducing wetted surface area and thus the skin friction component of resistance, or the wave making component.

Favorable wave resistance characteristics are also observed simultaneously in some cases, like hydrofoil ships.

High speed craft typically operate at , and they fully or partially employ hydrodynamic lift. Apart from

reducing frictional losses, they also contribute to stabilizing the ship’s motion in waves. Hydrofoils may also be

used in conjunction with other designs, like multihulls and air cushion vehicles (hovercraft). Generation of lift is

basically a consequence of conservation of Energy from Bernoulli’s theorem, where the change in local velocities

creates a pressure difference, and thus a lifting force. It is however most effective at higher speeds and the lift

varies in proportion to the velocity squared. So two phases are observed in the operation of hydrofoil craft, where

at lower speeds there is not much lift, and a “take off” speed at which the hull lifts out of the water and

subsequently is a foil – operation mode. Prior to take off, as the speed increases, the trim also increases and the

center of gravity gradually rises. The maximum “hump drag” occurs at just around the take off speed, and on

further increase a speed it reduces due to reduction in wetted surface area, along with reduction in the trim angle.

However, at higher speeds, the skin friction increases again at higher speeds relatively and causes a rise in

resistance. The effective power however, continuously increases with velocity.

Foils may be used in various configurations, depending on the distribution of lift over the forward and aft foil

systems, the submergence of the foil system, the type of foil section used and the way the craft is stabilized.

Hydrofoils may be surface piercing or fully submerged. Surface piercing hydrofoils have the advantage of lift

control by means of lifting area variation so that these types of foils are inherently stable and do not necessarily

require additional lift control mechanisms for safe operations. Fully, but shallowly submerged hydrofoils obtain

some lift control through the free surface effect which reduces lift as the hydrofoil approaches the water surface,

but this is not a very reliable design approach for lift control as this effect is observed in a small range of immersion

depths. These types of hydrofoils are used in inland waterways only, as the foils are vulnerable to broaching in

waves. Fully, relatively deeply submerged hydrofoils do need a lift control system since their lifting area does not

vary with submergence. Trailing edge flaps linked to a digital ride control system are nowadays commonly used for

lift control, not only for operation in waves but also to assist take-off and to enable maneuvering. Due to the lower

wave excitation and high effectiveness of digital ride control systems coupled to trailing edge flaps, the sea keeping

behavior of deeply submerged hydrofoils is superior to that of surface piercing hydrofoils.

Virtually all craft today employ sub cavitating sections, which have an upper speed limit of 40 to 45 knots. Higher

spends require vented and super cavitation sections and have been employed in experimental and prototype craft.

Cavitation

Page 8: B.Tech Thesis

Cavitation is a phase change observed in high speed flows wherein the local absolute pressure reaches the vicinity

of the vapor pressure at the ambient temperature. It is signified by what is known as the cavitation number,

defined as

Physical cavities usually have finite length and positive cavitation number. Some mathematical formulations have

been suggested previously to investigate the possibility for obtaining the solution of steady cavity flows of ideal

fluid satisfying the following conditions:

(i) The cavitation number is greater than zero

(ii) The cavity pressure is uniform throughout the cavity

(iii) The pressure of the fluid is nowhere less than the cavity pressure

(iv) The pressure is continuous across the cavity boundary (no pressure jump)

Cavitation occurs in three stages, according to the degree of development of the cavity:

(i) Initial cavitation, which is characterized by the noise of collapsing bubbles. It is harmful as bubble

implosion can cause cracking and shock in moving parts like propellers and rudders and damage them,

hampering performance.

(ii) Partial cavitation, in which the cavity sustains over a part of the cavitating body and has a pulsating nature

(iii) Fully developed cavitation, in which the cavity dimensions exceed that of the body. It is observed for high

velocities (≥ 70 m/s). It can also be observed for considerably small velocities at penetration of bodies

through the free water surface or at horizontal motion in the conditions of crossing the free boundary. In

this case the supercavity is filled by atmospheric air and is related to artificial or ventilated supercavities

forming at the lower speeds.

Page 9: B.Tech Thesis

2. Free Streamline Theory Equations

Many models have been developed on potential flow theory, where the cavity is treated as a single bubble with

constant internal pressure. We bring our attention to three main models, all based around the classical Helmoholtz

boundary theory, which says that the cavity is of infinite length.

(i) Riabouchinksy model: An ‘image’ plate is introduced downstream of the plate to model the cavity. The

coefficient of drag is given by

CD(σ)={1+σ+[8(π+4) ]-1σ 2}CD(0) ≈(1+σ)CD(0)

Where CD(0)=0.88 is the classical Helmholtz solution for σ=0

The free streamlines run from A to A’ and the flow is assumed to be irrotational everywhere.

(ii) Re-entrant jet model: The free streamlines reverse direction at the rear of the cavity to form a jet which

flows upstream in the cavity and it supposed to not impinge on the real body but is removed

mathematical by allowing it to flow on a second Riemann sheet in the physical plane to infinity. The

point infinity acts as a doublet plus source, while the jet represents a sink.

(iii) Roshko’s dissipation model: In this model, the flow past a bluff body is divided into two parts. Near the

body it is described by the free streamline theory to allow a possible adjusment of the under

pressure. The flow farther downstream is described by an equivalent potential flow so that its

pressure increases constinuously to the free stream value as it approaches infinity. The actual

dissipative wake flow is thus represented.

Page 10: B.Tech Thesis

Roshko’s model is simpler in many respects and gives more accurate results, although it involves more complex

manipulations. All theories agree and give the same value for the coefficient of drag in case of pure drag problems

(given by the equation previously).

The main assumptions in employing the Roshko’s model are:

(i) Sharp leading and trailing edges of the foil

(ii) Separation of the streamlines occurs at the leading and trailing edges

(iii) The foil thickness is assumed to be zero (since it does not affect the flow once the cavity is fully

developed)

(iv) Totally dry (ventilated) upper surface of the foil

Using some conformal mappings and complex potential flow equations, we obtain for general profiles (as

illustrated in the diagram) , the coeffiecients of drag, lift and moment (for small cavitation numbers, typically <

1.5) as:

Radius of curvature definitions

Page 11: B.Tech Thesis

1) Ai

2) Bii

2 3/ 2

1

2

2

(1 ) 1)

A

A

A

iii Radius of curvature at A RKd y

dx

2 3/ 2

2

2

2

(1 ) 1)

B

B

B

iv Radius of curvature at B RKd y

dx

22

21 2 12(1 )[sin ]

6 6 2D

AC

J

22

1 22(1 )

6 6LC

J

2

1{ sin cos cos2

AA

231 1 1

2 34

1

cos 5(sin ) sin (sin )( )

4 (sin / 2) 2 4 2 4}

A A AA A

A

0

2

2

4(1 ) ( cos sin )

3M L DC K C C

J

1 2 3

8 84 sin ( sin ) cos sin

3 2 15J A A A

For small angles: 24 sin( )J .

22 21 12 1 cos sin

2 2 4 2 2 2B AK K

22 2

1

9 1 12 1 cos sin8 2 2 4 4 2 2

B AA K K

Page 12: B.Tech Thesis

2 2

2

1cos sin

2 2 2B AA K K

22 2

3

1 1 12 1 cos sin8 2 2 4 4 2 2

B AA K K

The above result shows that ( - ), 1A , 2A and 3A are all of 2( , )O for all ; and in particular, ( -

), 2A and 3A reduce to 2 4( , )O ) for close to /2. Moreover, the fact that 3A is much smaller

.than 1A indicates a good accuracy of the expansion given by Eq. (3.13). If the above quantities are used to check

the curvature and slope of the solid boundary at some other points, say at / 2ie one can easily find that the

agreement is within a factor at most of 2( , )O . It should also be remarked here that if more terms were taken

in the expansion then the first three coefficients 1A , 2A , 3A would differ slightly from the above value. However,

it can be verified that the “improvement” of the solution by taking terms more than three is actually

unappreciable.

For segmental sections, the above equations will simplify to:

2 2212

(1 )[sin ]6 6 2

D

AC

J

2 2

21

2(1 ){[sin cos cos ]

6 6 2L

AC A

J

231 1 1

2 34

1

cos 5[(sin ) sin (sin )( )]}

4 (sin / 2) 2 4 2 4

A A AA A

A

0

2

2

4(1 ) ( cos sin )

3M L DC K C C

J

Where,2

2 2

2 2 1/ 2

1{ cos sin }

2 4( ) 4 sin [( ) ] 2 2

2 2 1/ 2( ) ] ( )

2

1 2 2 1/ 2

sin 9 (1 cos ) 9{1 } {1 }

8(4 sin ) 16 9[( ) ] 32( )A

2 2

2 2 2 1/ 2

1

{ cos sin }4 ( sin ) [( ) ] 2 2

AA

3 1 2 2 1/ 2

1 2 (1 cos )( ) { sin }

9 9(4 sin ) [( ) ]A A

Page 13: B.Tech Thesis

Definition of curvature for circular arc

Page 14: B.Tech Thesis

3. Ventilated Foil Sections

As discussed earlier, the value of the lift generated varies as the square of the velocity. Post take-off, with

increasing velocity, the hull rises higher and higher out of the water, and the foils move closer to the surface, and

consequently some of the pressure gradient is lost, causing a reduction in the lift. The craft then goes to a lower

draft where the hull’s position relative to the water surface can be maintained, and the process is repeated in a

‘bobbing’ kind of motion. It is thus desirable to control the lift beyond a certain velocity so that there is better ride

control. Usually, this is done either by varying the angle of attack of the foil by a control mechanism, or to provide

a trailing edge flap that effectively produces a camber in the foil section. Such methods are usually very

complicated in terms of instrumentation and expensive. For the purpose of lift dumping, and reduction in drag, an

alternative method has been proposed. Air is injected over the surface of the foil to create a cavity on the suction

side. This does away with moving parts designed for flap systems and changing the angle of attack, at the same

time reducing drag and providing ride control. The mechanism has been illustrated below

Lift dumping through ventilation of foils

Towing tank experiments have been performed at the Naval Science and Technological Laboratory (NSTL),

Vizag for NACA 66 airfoil sections at velocities between 3-6.5 ms-1 and small angles of attack (up to 8

degrees). The position of the slit for outlet of air was tried on the top surface, bottom surface and the

nose of the foil. However, it must be noted that the results so obtained are 2 D data, i.e. the foil is

assumed to have an infinite span (and consequently there is no bleeding of pressure or end effects).

Additionally, strut effects have to be considered, for which data was taken over 75% and 100% of the foil

and multiplied by factors to obtain drag and lift coefficients. Another set of data was obtained from wind

tunnel testing of the foil, which takes into consideration only the effects of geometry of the foil (Abott

curve). Experimental results were compared with those obtained by CFD analysis in Fluent, with modeling

of the slit at various positions on the foil. Consistency was found for 3.33 ms-1, but difficulties arose in

Page 15: B.Tech Thesis

both experiment and CFD results of higher velocities. The corresponding results have been plotted in the

below:

Coefficient of lift (experiment) vs. angle of attack for different slit positions

Coefficient of drag (experiment) vs. angle of attack for different slit positions

Page 16: B.Tech Thesis

Coefficient of lift (CFD) vs. angle of attack for different slit positions

Coefficient of drag (CFD) vs. angle of attack for different slit positions

Page 17: B.Tech Thesis

4. Extension of the theoretical model to a NACA-66 section

In order to extend the free streamline theory equations developed by Wu to a general airfoil section, such as the

NACA-66 (used widely in Naval industries), it is necessary to approximate it either to a symmetric (circular arc) or

asymmetric general section, to calculate the values of A1,A2,A3 and Beta, and thus the lift and drag coefficients.

The first approach that was followed was to use the co-ordinates of the foil underside to calculate the first and

second derivatives and substitute and find the values of phi1 and phi2.

Lower surface of a NACA-66 section

First derivatives obtained from coordinates

Page 18: B.Tech Thesis

Second derivatives obtained from coordinates

It can be observed that the nature of variation of the slopes and curvatures by this method is rather

erratic and thus seems to be an unreliable approach. Next, a trend line was fitted through the curve to

obtain a polynomial equation of the surface:

Curve fitting through coordinates

By this method, the slopes and curvatures were calculated at the leading and trailing edges and were

obtained as:

Φ1=.346

Φ2=.064

Ka= -0.00337597

Kb= 0.000676

Substituting these values into the general case equations, the lift and drag curves were obtained, but

again it was observed for non zero cavitation numbers that for small angles in the range of 0-5 degrees

there were sudden spikes in the values of lift (as illustrated), which were unacceptable. It was observed

that the large variation to the order of thousands came due to changes in beta to the order of 0.1, and

beta was directly dependent on phi1. This method was also thus discarded.

Page 19: B.Tech Thesis

Lift coefficient trend obtained by using co-ordinates/curve fitting

Since it was apparent that it was desirable to have small leading and trailing edge angles, a segmental

approximation for the foil surface seemed to be a feasible option. Arcs were fitted through the mid chord

and maximum chord ordinates in AutoCAD. The value of the included angle was 27 degrees and 30

degrees respectively for each case.

Segmental approximation using AutoCAD

The value of the curvature was then substituted into the special case equations and the lift and drag

curves were obtained. There wasn’t any major difference between the values of the maximum chord and

the mid chord approximations. The curves showed a good trend at small as well as large angles of attack.

The cavitation number was increased up to 1.5, around which the lift curves started to show a decreasing

trend which is logically incorrect, validating Wu’s assumption of small cavitating numbers, corresponding

to larger sustained cavities and opposed to partial cavitation.

Page 20: B.Tech Thesis

Coefficient of lift vs. angle of attack for maximum chord segmental approximation

Coefficient of drag vs. angle of attack for maximum chord segmental approximation

Comparison of the lift coefficient values for different approximations

Page 21: B.Tech Thesis

Comparison of the drag coefficient values for different approximations

Lift coefficient -segmental approximation for large angles of attack

Drag coefficient – segmental approximation for large angles of attack

Page 22: B.Tech Thesis

It needs to be pointed out that the values obtained for the lift and drag coefficients using the segmental

approximation lie in the same range as those obtained for the actual section taken in the general case,

barring the problematic region lying between angles of attack of 0 and 5 degrees. Even in this region,

however, the curve obtained matched closely with the extrapolated values for the actual section, i.e. it

followed the trend that should have been expected, which validates the segmental approximation thus

made and any errors that it could have entailed.

5. Correlation of theory with experimental and CFD results

In order to compare the experimental and theoretical results, first a relationship needs to be established between

the velocity and the cavitation number, or, in the case of ventilated hydrofoils, the ventilation number.

We have,

If we substitute the value of Pv, the vapor pressure with the cavity air pressure, the following values are obtained

(for 3.33 ms-1) under different ventilating conditions:

No Ventilation

velocity angle Pc Pl sig sig

alternate

3.33 0 98366.31 108682.5 1.860634436 0.09927937

5 0 94668.69 108682.5 1.12110492 0.13076405

6.5 0 90056.51 108682.5 0.881703858 0.16752781

Ventilation (T)

velocity angle Pc Pl sig sig

alternate

3.33 0 98808.82 108682.5 1.780822552 0.09461785

3.33 2 97886.84 108682.5 1.94711057 0.10437515

3.33 4 97153.32 108682.5 2.079409608 0.11226323

3.33 6 96750.28 108682.5 2.15210221 0.11664553

Ventilation (F)

velocity angle Pc Pl sig sig

alternate Avg sig

Page 23: B.Tech Thesis

3.33 0 98151.92 108682.5 1.899300363 0.10155202 1.88247133

3.33 4 97446.83 108682.5 2.026471715 0.10909342 Avg sig alt

3.33 6 96349.89 108682.5 2.224316713 0.12103335 0.11502045

Ventilation (B)

velocity angle Pc Pl sig sig

alternate

3.33 0 98178.46 108682.5 1.894514641 0.10127022

3.33 2 97184.33 108682.5 2.073816844 0.11192749

3.33 4 96781.2 108682.5 2.146524957 0.11630808

3.33 6 96114.53 108682.5 2.266765288 0.12362869

Correlation between sigma and velocity for different ventilation conditions

Since the cavity, as such is sustained due to the presence of air, an alternate number may be defined with the

additional term of air pressure in the denominator.

Uair is the velocity of air at the slit and is typically around 70 ms-1, and the value is equal to the air

pressure, which has to be greater than the stagnation point pressure. But by convention, cavitation number is

defined for the lowest pressure on the surface of the foil, and the corresponding value has been taken as Pc for

different cases. The above definition has been thus defined as the ratio of the total static pressure to the total

dynamic pressure.

The cavitation number is shown to follow an increasing trend with the angle of attack for different ventilation

conditions (F – Front, T – top, B – bottom). An average value of 1.88 is obtained for the first case (standard

definition) and a value of about .12 is obtained for the modified definition. The corresponding drag and lift curves

have been plotted against experimental and cfd values:

Page 24: B.Tech Thesis

Lift coefficient - comparison of theory against experiment and CFD results

Page 25: B.Tech Thesis

Drag coefficient - comparison of theory against experiment and CFD results

Page 26: B.Tech Thesis

6. Conclusions The value of the equivalent of the cavitation number for ventilated foils by the standard definition is

found to be too large to be adequately modeled using Wu’s theory. It is suggested that either the theory

be extended to include more terms in its expansion (since it neglects terms of order greater than ).

Also, reliable experimental data should be generated at higher angles of attack and higher velocities,

where at least a transition from fully ventilated to partially cavitating flows may be observed for better

usage of the theory. The geometry of the foil seems to have a limited effect on the variation in the lift and

drag coefficient values as shown by the comparison of segmental and general case results.

Page 27: B.Tech Thesis

REFERENCES

[1.] Roshko, A. (1954). A new hodograph for free streamline theory. NACA TN 3168

[2.] Wu, T.Y. (1956). A free streamline theory for two-dimensional fully cavitated hydrofoils. J. Math. Phys., 35, 236–

265.

[3.] Wu, T.Y. (1957). A linearized theory for nonsteady cavity flows. Calif. Inst. of Tech. Eng. Div. Rep. No. 85-6.

[4.] Wu, T.Y. (1962). A wake model for free streamline flow theory, Part 1. Fully and partially developed wake flows

and cavity flows past an oblique flat plate. J. Fluid Mech., 13, 161–181.

[5.] Wu, T.Y. (1972). Cavity and wake flows. Ann. Rev. Fluid Mech., 4, 243–284.


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