TMR4220 - NAVAL HYDRODYNAMICS

Analysis of a propeller with lifting line theory

Leonard O. KiellandJonathan Schwartz

March 2018

CONTENTS

I Calculation of 2D lift coefficients 2

II Lifting line model without induced veloci-ties 2

III Simple model of induced velocities 3

IV Accurate model of induced velocities 4

V Cavitation 5

VI Simplifications 6

Appendix 7A task2.m . . . . . . . . . . . . . . . 7B task3.m . . . . . . . . . . . . . . . 7C task4.m . . . . . . . . . . . . . . . 9D percentdone_print.m . . . . . . . . 10E gammaderive.m . . . . . . . . . . 10F InductionFactors.m . . . . . . . . 10G SingularIntegration.m . . . . . . . 10

NOMENCLATURE

αeff Effective angle of attack [rad]β Geometric angle of attack [rad]βi Hydrodynamic pitch angle [rad]η0 Propeller efficiency [-]Γ(r) Circulation distribution [m2/s]ν Kinematic viscosity of water. Set to 1.19E-6

[m2/s]∂Γ/∂r Radial derivative of the circulation distribution

[m/s]φ Geometric pitch angle [rad]ρ Density of sea water. Set to 1025 [kg/m3]σ Cavitation number [-]c Chord length [m]CD Drag coefficient [-]CF Friction coefficient [-]CL Lift coefficient [-]CL Lift coefficient due to angle of attack [-]CLc Lift coefficient due to camber [-]D Diameter of propeller blades [m]damp Damping coefficient [-]dD Drag contribution [N/m]dK Tangential force contribution [N/m]dL Lift contribution [N/m]dQ Torque contribution [N]dT Thrust contribution [N/m]g Gravitational constant [m/s2]h Depth of propeller [m] (NOTE: h used for step-

size in code)

J Advance number [-]k Number of foil sections. Set to 100 for all

models.kQ Torque coefficient [-]KT Thrust coefficient [-]n Propeller rotative frequency [Hz]P Pitch [m]pa Atmospheric pressure [Pa]pv Vaporization pressure [Pa]Q Torque [N m]R Radius of propeller blades [m]r Radial distance to given foil section [m]r0 Fixed radial position when integrating [m]rh Radius of the propeller hub [m]Re Reynolds number [-]T Thrust [N]tol Convergence tolerance. Set to 10E-4 for all

models.UA Axial induced velocity [m/s]UT Tangential induced velocity [m/s]V Advance velocity [m/s]Vinf Total velocity [m/s]Vmax Max velocity when effect of camber and thick-

ness is included [m/s]x Non-dimensional radial distance to given foil

section, x = r/R [-]Z Blade number

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Abstract—A study on the influence on accuracy of liftingline theory, with and without the inclusion of inducedvelocities - using models with varying complexity. Theresults from the lifting line theory will be compared toexperimental data from the Wageningen B-series. It will beshown how the developed program can be used to check forpropeller cavitation and the validity of lifting line theorywill be discussed.

I. CALCULATION OF 2D LIFT COEFFICIENTS

Using linear foil theory, the effects of angle of attackand camber on the lift can be linearly superimposed.The effect of camber on lift is not flow dependantand is therefore only a condition of the specific foilsection geometry, this allows the calculation of the liftdue to camber to be done independently and then usedas an input to the lifting line code. Using the fastand quite accurate foil software XFoil, the different liftcoefficients due to camber for the specified foil sectionswere calculated.The procedure is as follows, the foil geometry isimported with help of the command "load<filepath>",then the command "oper" is used to launch the operatingmode. An angle of attack of zero degrees was appliedwith the command "alfa<angle>" to find the 2D liftcoefficient only due to the camber of the foil. Theresults were then saved as a .mat file to be used inthe lifting line code. In addition to the values obtainedby XFoil, the lift due to camber at the propeller huband blade tip, x = 0.167 and x = 1, were included aszero in the .mat file. The obtained values of the liftcoefficients following this procedure for the differentfoil sections can be found in table I.

x 0.2 0.3 0.4 0.5CLc 0.5523 0.5106 0.5558 0.5751

x 0.6 0.7 0.8 0.9CLc 0.5098 0.4105 0.2987 0.2123

TABLE ILIFT COEFFICIENT DUE TO CAMBER AT DIFFERENT FOIL SECTIONS.

II. LIFTING LINE MODEL WITHOUT INDUCEDVELOCITIES

The purpose of this exercise is to calculate a liftingline code for the propeller without including the effectof induced velocities. The torque coefficient KT , thrustcoefficient KQ and the efficiency η are then computedas a function of the given advance numbers J.The code works as follows, first the values of the

lift coefficients due to camber, the parameters of thegeometry and the values of the Wageningen B-seriesare loaded. Then the foil is discretized into a desirednumber of sections and an interpolation function is usedto calculate the chord length, thickness, geometric pitchangle and camber lift coefficient at these sections fromthe loaded data. Using the relation between Reynoldsnumber, incoming velocity V and rotative frequencyn specified in chapter 16 in the compendium [1],V and n were approximated at every given advancenumber. For each advance number the geometric angleof attack β, the effective angle of attack α, the total liftcoefficient CL and the velocity Vinf on each foil sectionare computed. These values are subsequently used tofind the total lift contribution of every foil section,which is decomposed into a thrust- and tangential-forcecomponent. To find the total thrust and torque, the dragforce on the foil sections must also be accounted for.Using eq. (13.33) in the compendium [1] to find thedrag coeffient, the drag force component on each foilsection is promptly calculated and then decomposedinto thrust- and tangential-force components. When thethrust and torque from the lift and drag are computed,the last step in the for loop is to compute the totaltorque and thrust. The total thrust component, dT ,is equal to the lift thrust minus the drag thrust. Thetotal torque component, dQ, is equal to the sum ofthe tangential force components multiplied with theirrespective distance from the center of the propeller. Atthis point the code has computed the torque and thethrust for each section. The next step is to integratethe total thrust and torque for the propeller, whichis done with the MATLAB trapz() function. Whenthe total thrust and torque for the different advancenumbers are calculated, the corresponding thrust- andtorque-coefficients and the propeller efficiency can befound using eq. 1. For more details on the procedure,see the MATLAB code in appendix A with comments.

KT =T

ρ · n2 ·D4(1a)

KQ =Q

ρ · n2 ·D5(1b)

η0 =J

2π· KT

KQ(1c)

2

Fig. 1. Thrust-coefficient KT , torque-coefficient KQ, and propellerefficiency η plotted against advance number J . Data obtained fromlifting line model predictions without induced velocities compared toWageningen B-series experimental data.

Figure 1 shows that while the simple model withoutinduced velocities gives results which generally followsthe same trend as the experimental data, the accuracy ispoor. Both the predicted lift and the predicted torque ismuch higher than in reality, which signals the need for amore precise model. The next exercise will include theinduced velocities in the calculations, which in theorywill reduce both predicted lift and predicted torque.

III. SIMPLE MODEL OF INDUCED VELOCITIES

To include the effect of induced velocities on the liftand torque, the simple model "Complete momentumtheory" layed out in chapter 13 in the compendium[1] was implemented in MATLAB. As the circulationdistribution is needed to calculate the induced velocitiesand the induced velocities are needed input when calcu-lating the circulation distribution, iteration is required.

The difference in lift in this procedure compared tothe procedure specified in chapter 2 comes from thechange in effective angle of attack due to the inducedvelocities. The calculation of the effective angle of attacknow depends on the hydrodynamic pitch angle βi, whichis found by eq. 2. Note that the second equality onlyholds for the simple model used here, and not themodel applied in chapter 4. This equation also gives therelation between the two induced velocities and leads to aquadratic polynomial which is solved to obtain the axialinduced velocity UA. The tangential velocity is foundfrom Biot-Savart’s law, see eq. 3.

tan(βi) =V + UA/2

2πrn− UT /2=UT

UA(2)

UT (r) =Γ(r)

2πr(3)

Many of the initial steps of the procedure are the sameas in chapter 2, where the geometry and camber lift co-efficients are loaded and interpolated for the discretizedfoil sections. What differs is the specification of iterationparameters and the construction of an initial circulationdistribution. The code is set to iterate a maximum num-ber of steps called itermax, but will break the iterationsif a convergence criterion is met or exit the program ifNaN is detected. Each iteration the circulation is updatedusing a damping factor to ensure stability. This dampingfactor, inspired by the lifting line method outlined inAnderson 2017 [2], is given a value of damp = 0.05.The circulation is given as an array of values where therows indicate advance number and the columns indicatefoil section. To check the convergence of the circulationarray, the two-norm error seemed to be a logical choice.The two-norm error for a solution step is given in eq.4a, and is a method often used for convergence testingof a variable distribution in CFD. The iteration was setto break when the relation between the two-norm errorof the first step to the two-norm error of the current stepfell below a given tolerance, see eq. 4b.

||Γk+1 − Γk||2 =

∑i

∑j

(Γk+1 − Γk)2

0.5

(4a)

||Γ2 − Γ1||2||Γk+1 − Γk||2

≤ tol (4b)

The initial circulation was set to an elliptical distributionwith a maximum circulation of Γ0 = 5. Then, for everyiteration the new circulation at every foil section wasdetermined by following the procedure listed below.

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1) Calculate tangential and axial induced velocitiesand total velocity at each foil section.

2) Calculate hydrodynamic pitch angle and use it tofind effective angle of attack, αeff = φ− βi.

3) Find lift coefficient as the contribution of a camberpart and angle of attack part, CL = CLc+2παeff .

4) Find the "new" circulation using Γnew = 12Vinf ·

Z · c · CL.5) Udate the circulation using a damping factor,

Γinput = Γold +D · (Γnew − Γold).6) Check for convergence using eq. 4b.

After convergence is reached, the calculation of torqueand thrust is essentially the same as the one outlined inchapter 2. For more details on the procedure, see theMATLAB code in appendix B with comments.

Fig. 2. Thrust-coefficient KT , torque-coefficient KQ, and propellerefficiency η plotted against advance number J . Data obtained fromlifting line model predictions with simple model of induced velocitiescompared to Wageningen B-series experimental data.

Figure 2 shows that the accuracy of the predictedthrust- and torque-coefficients and especially the effi-

ciency is much better than the previous results. Whileboth the thrust- and torque-coefficients are slightly over-predicted, the efficiency is almost spot-on for loweradvance numbers. Whether the accuracy of the efficiencyis indicative of the method or if it’s a "lucky" hit becauseof the spread of thrust- and torque-coefficients is hard tosay.

IV. ACCURATE MODEL OF INDUCED VELOCITIES

Up to this point it has been assumed that the propellerconsists of an infinite number of blades. The aim of thisexercise is to use a more complete model to account forthe effect of a finite number of blades. According to S.Steen [1]:

"For a given mean circulation, the effect of afinite number of blades means an increase ininduced velocity at the blade"

An increase in induced velocity at the blade will lead toa lower effective angle of attack, and thus to a lowerprediction of torque and thrust on the propeller. Theeffect of finite blade numbers will in this exercise beimplemented through the induction factor method. Theinduced velocities are larger close to each blade thanthe value obtained from the complete momentum theory.How much larger depends on blade number and flowparameters, and while complicated, a direct calculationof the local induced velocity is possible. The aim of theinduction factor method is to simplify this direct cal-culation. Using Biot-Savart’s law, the induced velocitiesat the propeller using the induction factor method areexpressed as

UA,T (r0) =

∫ R

rh

ia,t(r0, βi)

2π· ∂Γ(r)

∂r· dr

r0 − r, (5)

where ia,t(r0, βi) is the axial or tangential inductionfactor which aims to take the effect of the helicalstructure of the trailing vortices into account. This al-ternate method of determining the induced velocitiesis the main difference between the induction factormethod and the complete momentum theory. The initialsteps of the procedure is therefore almost exactly thesame as in the previous exercise, except for the initialcirculation distribution. As the induction factor methodstruggles with instability, it was of importance to havea close initial circulation to ensure that the differenceover a solution step wasn’t too large. Therefore theinitial circulation was determined by interpolating fromthe circulation distribution obtained using the completemomentum theory in the previous section. Note that thecirculation from the complete momentum theory had tobe divided by the blade number, as in the induction factor

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method the circulation distribution is given for a singleblade, not the whole propeller. Then, for every iterationthe new circulation at every foil section was determinedby following the procedure listed below.

1) Calculate mean tangential and mean axial inducedvelocities using complete momentum theory ateach foil section. Calculate the tentative hydrody-namic pitch angle using these velocities.

2) Find the derivative of the circulation by finitedifferencing. See appendix E.

3) Correct the tangential and axial induced velocitiesby the induction factor method. Iterating overevery foil section,

a) find the induction factors over the runningvariable for the fixed point, then

b) calculate the tangential and axial inducedvelocities by using eq. 5. This requires asingular integration technique.

4) Calculate hydrodynamic pitch angle and use it tofind effective angle of attack, αeff = φ− βi.

5) Find lift coefficient as the contribution of a camberpart and angle of attack part, CL = CLc+2παeff .

6) Find the "new" circulation using Γnew = 12Vinf ·

c · CL.7) Udate the circulation using a damping factor,

Γinput = Γold + damp · (Γnew − Γold).8) Check for convergence using eq. 4b.

Again, after convergence is reached the calculation ofthrust and torque is almost the same as in the last twosections. The only difference is that the results given areonly the effects on a single propeller blade, and the finalthrust and torque is multiplied by the blade number toobtain results for the whole propeller. For more detailson the procedure, see the MATLAB code in appendix Cwith comments.Figure 3 shows that the induction factor gives fairlyaccurate results for both thrust and torque, while theefficiency is to a degree less accurate then for the com-plete momentum theory. Again, whether the efficiencyis indicative of the general accuracy of the method isnot clear and it would be reasonable to say that theinduction factor method gives more accurate results thanthe complete momentum theory.

V. CAVITATION

To check for cavitation risks, the lifting line code canbe extended to calculate the local velocity at each bladesection. To do this requires knowledge of the influenceof camber and thickness on the local velocity. For mostcambered lines and thicknesses, data from the handbookAbbot & Doenhoff (2010) [3] can be superimposed to

Fig. 3. Thrust-coefficient KT , torque-coefficient KQ, and propellerefficiency η plotted against advance number J . Data obtained fromlifting line model predictions with accurate model of induced velocitiescompared to Wageningen B-series experimental data.

determine this influence and establish the resulting localvelocity at each foil section. In the compendium [1]chapter 6.4, S. Steen explains that

"In order for the initial cavitation bubble notto grow into a real cavity, it is required that theexternal pressure on the surface of the cavitymust be larger than the water vapour pressure"

This relation in accordance with Bernoulli’s equationfrom a point where the flow is undisturbed up to the pointof the bubble, allows us to write out the requirement fornon-cavitation as:

pa + ρg(h− r) +1

2ρ(V 2

inf − V 2max) > pv. (6)

Rearranged, this becomes

σ =pa − pv + ρg(h− r)

12ρV

2inf

> −(

1− (Vmax

Vinf)2), (7)

5

where σ is the cavitation number.In principle, a cavitation check can be summarized inthe following three steps done on every foil section:

1) Calculate cavitation number σ using the total ve-locity from the lifting line code, see eq. 7.

2) Find the maximum velocity, Vmax, at the suctionside of each section by superimposing the effectsof camber and thickness. These effects can be de-termined from tabular data, e.g. Abbot & Doenhoff(2010) [3].

3) Check the requirement for non-cavitation,(Vmax

Vinf

)2− 1 > σ. Typically a threshold is used

to be safe, e.g. > 0.8 · σ.As an alternative to tabular data one can use softwarelike XFoil to find

(Vmax

Vinf

)2− 1 = −CP at different foil

sections and interpolate the values into the lifting linecode, similar to how the lift coefficients due to camberwere treated. The requirement for non-cavitation remainsthe same.

VI. SIMPLIFICATIONS

Generally a lifting line model is only suited for foilswith aspect ratios that aren’t too small and with little tono skew. It works best for lightly loaded propellers, aslarge loads leads to effects which cannot be captured ac-curately with lifting line theory. Many of the limitationsof the lifting line model comes from the simplification ofcirculation distribution over the propeller blades. In thismethod, the circulation is assumed to be concentrated ata line going straight through the propeller blade, whichmeans that the chord-wise circulation distribution isneglected. A propeller with a skew or small aspect ratiowill generally have chord-wise circulation distribution,and thus the assumption brings with it large inaccuraciesin these cases. Lifting line models do not include interac-tion with propeller hub and similar objects, which limitsthe use to light propeller loading where these effectsare small. Furthermore, viscous effects like friction areapproximated only through empirical formula, whichmay give somewhat skewed results at high loads.For the propeller in this case the aspect ratio is quitehigh and there is no skew angle which indicates that thelifting line model can give quite accurate results. Thisis validated with our results in exercise 4, as the liftingline method with accurate model of induced velocitiesgave thrust- and torque-coefficients and efficiency whichagreed well with experimental data. As the advancenumber grows, the lifting line model results divergesfrom the experimental data, which is to be expectedas the lifting line model is less accurate with largerpropeller loading.

REFERENCES

[1] S. Steen, TMR4220 Naval Hydrodynamics - Foiland Propeller Theory. Akademika forlag, 2014.

[2] J. D. Anderson, “5.4 a numerical nonlinear lifting-line method,” in Fundamentals of aerodynamics.McGraw-Hill Education, 2017.

[3] I. H. Abbott and A. E. v. Doenhoff, Theory ofwing sections: including a summary of airfoil data.Dover, 2010.

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APPENDIX

A. task2.m

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%3 %−−−−−−−−−−−−− LIFTING LINE METHOD WITHOUT INDUCED VELOCITIES−−−−−−−−−−−−%4 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%5 % C a l c u l a t e s t o r q u e , t h r u s t and e f f i c i e n c y f o r a s p e c i f i e d p r o p e l l e r %6 % geomet ry and compares t h e r e s u l t s w i th e x p e r i m e n t a l d a t a . Th i s model %7 % does n o t i n c l u d e t h e e f f e c t o f i n d u c e d v e l o c i t i e s %8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%9

10 c l e a r a l l11 c l o s e a l l12 c l c1314 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%15 % Load l i f t c o e f f i e c i e n t s , Wagenigen d a t a and geomet ry d a t a %16 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%17 l o a d L i f t R a d i u s . mat ;18 wagB= dlmread ( ’wagB . t x t ’ ) ; % J , KT, KQ, e t a19 geomet ry = dlmread ( ’ Geometry . t x t ’ ) ; % r / R , chord /D, t /D, P /D2021 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%22 % De f i n e f low & p r o p e l l e r p a r a m e t e r s %23 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%24 Re =2 .4 E6 ;25 x0 = 0 . 7 ;26 cx0 = 0 . 2 2 5 ; % from geomet ry f i l e27 nu =1.19E−6; % from m e c h a n i c a l p r o p e r t i e s o f w a t e r28 rho =1025;29 D=1;30 Z=4;3132 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%33 % D i s c r e t i z e p r o p e l l e r i n t o f o i l s e c t i o n s %34 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%35 k =100;36 h =(1−0.167) / ( k−1) ;37 x = 0 . 1 6 7 : h : 1 ; % x= r / R38 r =x∗D/ 2 ;3940 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%41 % I n t e r p o l a t e f o r chord l e n g t h , t h i c k n e s s and p i t c h a n g l e a t s e c t i o n s %42 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%43 chord = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 2 ) , x , ’ p c h i p ’ ) ;44 t h i c k n e s s = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 3 ) , x , ’ p c h i p ’ ) ;45 PdivD = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 4 ) , x , ’ p c h i p ’ ) ;46 p h i = a t a n ( PdivD∗D. / ( 2∗ p i∗r ) ) ;4748 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%49 % I n i t i a l i z e v e c t o r o f advance numbers and f i n d c o r r e s p o n d i n g V and n %50 % from eq . i n c h a p t e r 16 ( compendium ) %51 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%52 J = [ 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 . 1 ] ;53 V= Re∗nu / cx0 ∗ 1 . / s q r t ( 1 + ( x0∗p i . / J ) . ^ 2 ) ;54 n= V . / ( J∗D) ;5556 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%57 % I n t e r p o l a t e l i f t c o e f f i c i e n t a t z e r o aoa from t h e XFoi l r e s u l t s %58 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%59 C_Lc = i n t e r p 1 ( L i f t R a d i u s ( : , 1 ) , L i f t R a d i u s ( : , 2 ) , x , ’ p c h i p ’ ) ;6061 %−−−−−−−−−−−−−−−−−−−−−%62 % I n i t i a l i z e m a t r i c e s %63 %−−−−−−−−−−−−−−−−−−−−−%64 b e t a = z e r o s ( l e n g t h ( J ) , l e n g t h ( x ) ) ; a l p h a = b e t a ; C_L= b e t a ; Vinf = b e t a ;6566 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%67 % C a l c u l a t e t h r u s t and t o r q u e on e v e r y f o i l s e c t i o n f o r %68 % e v e r y advance number %69 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%70 f o r i =1 : l e n g t h ( J )7172 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%73 % C a l c u l a t e a n g l e btw v e l o c i t y and f o i l and use i t t o f i n d e f f e c t i v e %74 % a n g l e o f a t t a c k . F ind t o t a l v e l o c i t y a t e v e r y f o i l s e c t i o n %75 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%76 b e t a ( i , : ) = a t a n ( J ( i ) . / ( p i∗x ) ) ;77 a l p h a ( i , : ) =phi−b e t a ( i , : ) ;78 Vinf ( i , : ) = s q r t ( (2∗ p i∗r∗n ( i ) ) . ^ 2 + V( i ) . ^ 2 ) ;7980 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%81 % Find l i f t c o e f f i e c i e n t a s t h e c o n t r i b u t i o n o f camber p a r t and %82 % a n g l e o f a t t a c k p a r t %83 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%84 C_L ( i , : ) = C_Lc ( : ) ’ +2∗p i∗a l p h a ( i , : ) ;8586 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%87 % Find l i f t c o n t r i b u t i o n a t e v e r y f o i l s e c t i o n , decompose i n t o t h r u s t %88 % and t a n g e n t i a l f o r c e %89 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%90 dL ( i , : ) = 0.5∗ rho∗Vinf ( i , : ) . ^ 2 .∗C_L ( i , : )∗Z.∗ chord ;91 dTi ( i , : ) =dL ( i , : ) .∗ cos ( b e t a ( i , : ) ) ;92 dKi ( i , : ) =dL ( i , : ) .∗ s i n ( b e t a ( i , : ) ) ;9394 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%95 % Find drag c o e f f i e n t by use o f eq . ( 1 3 . 3 3 ) i n compendium %96 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%97 Re_c ( i , : ) = Vinf ( i , : ) .∗ chord / nu ;98 C_F = 0 . 0 7 5 . / ( l og10 ( Re_c ( i , : ) )−2) . ^ 2 ; % from ITTC 57 ’

99 C_D = 2∗C_F.∗(1+2∗ t h i c k n e s s . / ( chord ) ) ;100101 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%102 % Find drag c o n t r i b u t i o n a t e v e r y f o i l s e c t i o n , decompose i n t o t h r u s t %103 % and t a n g e n t i a l f o r c e %104 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%105 dD ( i , : ) =0.5∗ rho∗Vinf ( i , : ) .^2∗Z.∗ chord .∗C_D ;106 dTd ( i , : ) =dD ( i , : ) .∗ s i n ( b e t a ( i , : ) ) ;107 dKd ( i , : ) =dD ( i , : ) .∗ cos ( b e t a ( i , : ) ) ;108109 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%110 % Find t o t a l t h r u s t and t o r q u e component a t e v e r y f o i l s e c t i o n %111 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%112 dT ( i , : ) =dTi ( i , : )−dTd ( i , : ) ;113 dQ ( i , : ) =( dKi ( i , : ) +dKd ( i , : ) ) .∗x∗D/ 2 ;114 end115116 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%117 % Numer ica l e r r o r s a t p r o p e l l e r ends a r e c o r r e c t e d %118 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%119 dT ( : , end ) =0;120 dQ ( : , end ) =0;121122 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%123 % I n t e g r a t e wi th MATLABs t r a p z ( ) i n t e g r a t i o n %124 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%125 T= t r a p z ( r , dT , 2 ) ;126 Q= t r a p z ( r , dQ , 2 ) ;127128 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%129 % C a l c u l a t e t h e t h r u s t and t o r q u e c o e f f i e n t %130 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%131 K_T= T ’ . / ( rho∗n .^2∗D^4) ;132 K_Q= Q ’ . / ( rho∗n .^2∗D^5) ;133134 %−−−−−−−−−−−−−−−−−−−−−−%135 % C a l c u l a t e e f f i c i e n c y %136 %−−−−−−−−−−−−−−−−−−−−−−%137 e t a 0 =J / (2∗ p i ) .∗ K_T . / K_Q;138139 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%140 % I n t e r p o l a t e t h r u s t , t o r q u e and e f f i e n c y from Wag . B f o r compar i son %141 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%142 K_Twag= i n t e r p 1 ( wagB ( : , 1 ) , wagB ( : , 2 ) , J ) ;143 K_Qwag= i n t e r p 1 ( wagB ( : , 1 ) , wagB ( : , 3 ) , J ) ;144 eta0wag = i n t e r p 1 ( wagB ( : , 1 ) , wagB ( : , 4 ) , J ) ;145146147 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%148 %−−−−−−−−−−−−−−−−−−−−−−−−−−PLOTTING−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%149 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%150 f i g u r e ( ’ r end ’ , ’ p a i n t e r s ’ , ’ pos ’ , [ 1 0 0 100 600 8 0 0 ] )151 ho ld on152 g r i d minor153 t i t l e ( ’ Th rus t , t o r q u e & e f f i c i e n c y w i t h o u t i n d u c e d v e l o c i t i e s ’ )154 xl im ( [ J ( 1 ) J ( end ) ] )155 yl im ( [ 0 1 . 2 5 ] )156 x l a b e l ( ’ J [−] ’ )157 y l a b e l ( ’K_T [−] 1 0 \ c d o t K_Q [−] \ e t a _ 0 [−] ’ )158159 p l o t ( J , K_T , ’ g ’ , wagB ( : , 1 ) , wagB ( : , 2 ) , ’g−−’ )160 p l o t ( J ,10∗K_Q, ’ b ’ , wagB ( : , 1 ) ,10∗wagB ( : , 3 ) , ’b−−’ )161 p l o t ( J , e t a0 , ’ r ’ , wagB ( : , 1 ) , wagB ( : , 4 ) , ’ r−−’ )162 l e g e n d ( ’K_T − l i f t i n g l i n e ’ , ’K_T − Wag . B ’ , ’ 1 0 \ c d o t K_Q − l i f t i n g l i n e ’ . . .163 , ’ 1 0 \ c d o t K_Q − Wag . B ’ , ’ \ e t a _ 0 − l i f t i n g l i n e ’ , ’ \ e t a _ 0 − Wag . B ’ )

B. task3.m

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%3 %−−−−−− LIFTING LINE METHOD WITH SIMPLE MODEL OF INDUCED VELOCITIES−−−−−−%4 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%5 % C a l c u l a t e s t o r q u e , t h r u s t and e f f i c i e n c y f o r a s p e c i f i e d p r o p e l l e r %6 % geomet ry and compares t h e r e s u l t s w i th e x p e r i m e n t a l d a t a . Th i s model %7 % i n c l u d e s t h e e f f e c t o f i n d u c e d v e l o c i t i e s t h r o u g h t h e c o m p l e t e %8 % momentum t h e o r y . %9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

1011 c l e a r a l l12 c l o s e a l l13 c l c1415 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%16 % Load l i f t c o e f f i e c i e n t s , Wagenigen d a t a and geomet ry d a t a %17 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%18 l o a d L i f t R a d i u s . mat19 wagB= dlmread ( ’wagB . t x t ’ ) ; % J , KT, KQ, e t a20 geomet ry = dlmread ( ’ Geometry . t x t ’ ) ; % r / R , chord /D, t /D, P /D2122 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%23 % S p e c i f y i t e r a t i o n p a r a m e t e r s %24 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%25 i t e r m a x =1000;26 t o l =10E−4;27 damp = 0 . 0 5 ;2829 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%30 % De f i ne f low & p r o p e l l e r p a r a m e t e r s %31 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%32 Re =2 .4 E6 ;33 x0 = 0 . 7 ;

7

34 cx0 = 0 . 2 2 5 ; % from geomet ry f i l e35 nu =1.19E−6; % from m e c h a n i c a l p r o p e r t i e s o f w a t e r36 rho =1025;37 D=1;38 PdivD =1;39 AEA0= 0 . 4 ;40 Z=4;41 span =D/2∗(1−0.167) ;4243 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%44 % D i s c r e t i z e p r o p e l l e r i n t o f o i l s e c t i o n s %45 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%46 k =100;47 h =(1−0.167) / ( k−1) ;48 x = 0 . 1 6 7 : h : 1 ; % x= r / R49 r =x∗D/ 2 ;5051 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%52 % I n t e r p o l a t e f o r chord l e n g t h , t h i c k n e s s and p i t c h a n g l e a t s e c t i o n s %53 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%54 chord = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 2 ) , x , ’ p c h i p ’ ) ;55 t h i c k n e s s = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 3 ) , x , ’ p c h i p ’ ) ;56 PdivD = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 4 ) , x , ’ p c h i p ’ ) ;57 p h i = a t a n ( PdivD∗D. / ( 2∗ p i∗r ) ) ;5859 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%60 % I n i t i a l i z e v e c t o r o f advance numbers and f i n d c o r r e s p o n d i n g V and n %61 % from eq . i n c h a p t e r 16 ( compendium ) %62 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%63 J = [ 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 . 1 ] ;64 V= Re∗nu / cx0 ∗ 1 . / s q r t ( 1 + ( x0∗p i . / J ) . ^ 2 ) ;65 n= V . / ( J∗D) ;6667 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%68 % I n t e r p o l a t e l i f t c o e f f i c i e n t a t z e r o aoa from t h e XFoi l r e s u l t s %69 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%70 C_Lc = i n t e r p 1 ( L i f t R a d i u s ( : , 1 ) , L i f t R a d i u s ( : , 2 ) , x , ’ p c h i p ’ ) ;7172 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%73 % I n i t i a l i z e m a t r i c e s & v e c t o r s %74 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%75 gamma= z e r o s ( l e n g t h ( J ) , l e n g t h ( x ) ) ;76 U_A=gamma ; U_T=gamma ; a l p h a =gamma ; Vinf =gamma ; C_L=gamma ; dTi=gamma ;77 dQi=gamma ; dL=gamma ; dT=gamma ; dTd=gamma ; dKd=gamma ; dQ=gamma ; dLi=gamma ;78 dQd=gamma ; gamma_new=gamma ; gamma_input=gamma ; b e t a _ i =gamma ;79 twonorm= z e r o s ( 1 , i t e r m a x ) ; Re_c= z e r o s ( 1 , l e n g t h ( x ) ) ;8081 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%82 % S e t i n i t i a l c i r c u l a t i o n t o an e l l i p t i c a l d i s t r i b u t i o n %83 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%84 gamma_0 =5;85 y=−span / 2 : h∗D/ 2 : span / 2 ;86 gamma_in i t =gamma_0∗s q r t ( 1−( y / ( span / 2 ) ) . ^ 2 ) ;87 f o r i =1 : l e n g t h ( J )88 gamma ( i , : ) = gamma_in i t ;89 end9091 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%92 % I t e r a t e t h e p r o c e d u r e u n t i l c o n v e r g e n c e i s s a t i s f i e d o r maximum a l l o w e d %93 % number o f i t e r a t i o n s i s r e a c h e d %94 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%95 f o r i t e r =1 : i t e r m a x96 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%97 % C a l c u l a t e t h e new c i r c u l a t i o n a t e v e r y f o i l s e c t i o n f o r %98 % e v e r y advance number %99 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

100 f o r i =1 : l e n g t h ( J )101102 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%103 % C a l c u l a t e t a n g e n t i a l and a x i a l i n d u c e d v e l o c i t i e s and t o t a l %104 % v e l o c i t y a t e v e r y f o i l s e c t i o n %105 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%106 U_T ( i , : ) =gamma ( i , : ) . / ( 2∗ p i∗r ) ; % eq . ( 1 3 . 1 9 )107 f o r j =1 : l e n g t h ( x )108 a = 0 . 5 ; b=V( i ) ; c=−U_T( i , j ) .∗(2∗ p i .∗x ( j )∗(D/ 2 )∗n ( i )−U_T ( i , j ) / 2 ) ;109 U_A( i , j ) = (−b+ s q r t ( b^2−4∗a∗c ) ) / (2∗ a ) ; % eq . ( 1 3 . 2 8 )110 end111 Vinf ( i , : ) = s q r t ( ( V( i ) +0.5∗U_A( i , : ) ) . ^ 2 + . . .112 ( 2∗p i∗r∗n ( i )−0.5∗U_T( i , : ) ) . ^ 2 ) ;113114 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%115 % C a l c u l a t e hydrodynamic p i t c h a n g l e and use i t t o f i n d e f f e c t i v e %116 % a n g l e o f a t t a c k117 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%118 b e t a _ i ( i , 2 : end−1)= a t a n ( U_T( i , 2 : end−1) . / U_A( i , 2 : end−1)) ;119 a l p h a ( i , 2 : end−1) = p h i ( 2 : end−1)−b e t a _ i ( i , 2 : end−1) ;120121 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%122 % Find l i f t c o e f f i c i e n t a s t h e c o n t r i b u t i o n o f camber p a r t and %123 % a n g l e o f a t t a c k p a r t %124 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%125 C_L ( i , : ) = C_Lc + 2∗p i∗a l p h a ( i , : ) ;126127 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%128 % Update t h e new v a l u e o f gamma u s i n g a damping f a c t o r %129 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%130 gamma_new ( i , : ) =0.5∗ Vinf ( i , : )∗Z.∗ chord .∗C_L ( i , : ) ;131 gamma_input ( i , : ) =gamma ( i , : ) +damp∗(gamma_new ( i , : )−gamma ( i , : ) ) ;132 end133134 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%135 % C a l c u l a t e t h e two−norm e r r o r o f t h e i t e r a t i o n %136 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

137 twonorm ( i t e r ) = ( sum ( sum ( abs ( gamma_input−gamma ) . ^ 2 ) ) ) ^ 0 . 5 ;138139 %−−−−−−−−−−−−−−−−−−−−−−−−−−%140 % Updates t h e d i s t r i b u t i o n %141 %−−−−−−−−−−−−−−−−−−−−−−−−−−%142 gamma=gamma_input ;143144 %−−−−−−−−−−−−−−−−−−−−−−−−−−−%145 % P r i n t s p e r c e n t c o m p l e t i o n %146 %−−−−−−−−−−−−−−−−−−−−−−−−−−−%147 p e r c e n t d o n e _ p r i n t ( twonorm ( i t e r ) / twonorm ( 1 ) , t o l ) ;148149 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%150 % Checks i f c o n v e r g e n c e i s met , i f yes t h e n b r e a k t h e i t e r a t i o n s %151 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%152 i f twonorm ( i t e r ) / twonorm ( 1 ) < t o l153 c l c154 f p r i n t f ( ’ Convergence c r i t e r i o n met a f t e r %d i t e r a t i o n s . \ n ’ , i t e r ) ;155 b r e a k156 end157158 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%159 % Checks i f t h e r e i s any NaN i n gamma , i f yes t h e n c a n c e l s program %160 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%161 i f i s n a n ( sum ( sum ( gamma ) ) ) ==1162 c l c163 f p r i n t f ( ’NaN d e t e c t e d ! C a n c e l l i n g program . . . \ n ’ )164 r e t u r n165 end166167 end168169 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%170 % C a l c u l a t e t h r u s t and t o r q u e on e v e r y f o i l s e c t i o n f o r %171 % e v e r y advance number %172 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%173 f o r i =1 : l e n g t h ( J )174 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%175 % Find t h r u s t and t o r q u e c o n t r i b u t i o n b e c a u s e o f l i f t a t e v e r y f o i l %176 % s e c t i o n %177 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%178 dTi ( i , : ) = rho∗gamma ( i , : ) .∗(2∗ p i∗r∗n ( i )−U_T( i , : ) / 2 ) ; % eq . ( 1 3 . 3 7 )179 dQi ( i , : ) = rho∗gamma ( i , : ) .∗ (V( i ) +U_A( i , : ) / 2 ) .∗ r ; % eq . ( 1 3 . 3 8 )180181 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%182 % Find drag c o e f f i e n t by use o f eq . ( 1 3 . 3 3 ) i n compendium %183 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%184 Re_c ( i , : ) = Vinf ( i , : ) .∗ chord / nu ;185 C_F = 0 . 0 7 5 . / ( l og10 ( Re_c ( i , : ) )−2) . ^ 2 ; % from ITTC 57 ’186 C_D = 2∗C_F.∗(1+2∗ t h i c k n e s s . / ( chord ) ) ;187188 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%189 % Find t h r u s t and t o r q u e c o n t r i b u t i o n b e c a u s e o f d rag a t e v e r y f o i l %190 % s e c t i o n %191 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%192 dTd ( i , : ) =0.5∗ rho∗Vinf ( i , : ) . ^ 2 .∗ chord∗C_D( i )∗Z.∗ s i n ( b e t a _ i ( i , : ) ) ;193 dQd ( i , : ) =0.5∗ rho∗Vinf ( i , : ) . ^ 2 .∗ chord∗C_D( i )∗Z.∗ r .∗ cos ( b e t a _ i ( i , : ) ) ;194195 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%196 % Find t o t a l t h r u s t and t o r q u e component a t e v e r y f o i l s e c t i o n %197 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%198 dT ( i , : ) =dTi ( i , : )−dTd ( i , : ) ;199 dQ ( i , : ) =dQi ( i , : ) +dQd ( i , : ) ;200 end201202 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%203 % Numer ica l e r r o r s a t p r o p e l l e r ends a r e c o r r e c t e d %204 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%205 dT ( : , end ) =0;206 dQ ( : , end ) =0;207208 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%209 % I n t e g r a t e wi th MATLABs t r a p z ( ) i n t e g r a t i o n %210 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%211 T= t r a p z ( r , dT , 2 ) ;212 Q= t r a p z ( r , dQ , 2 ) ;213214 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%215 % C a l c u l a t e t h e t h r u s t and t o r q u e c o e f f i e n t %216 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%217 K_T= T ’ . / ( rho∗n .^2∗D^4) ;218 K_Q= Q ’ . / ( rho∗n .^2∗D^5) ;219220 %−−−−−−−−−−−−−−−−−−−−−−%221 % C a l c u l a t e e f f i c i e n c y %222 %−−−−−−−−−−−−−−−−−−−−−−%223 e t a 0 =J / (2∗ p i ) .∗ K_T . / K_Q;224225 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%226 % Save t h e c i r c u l a t i o n d i s t r i b u t i o n f o r use i n t a s k 4 %227 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%228 gamma_dis t ( : , 1 ) =x ’ ;229 gamma_dis t ( : , 2 : l e n g t h ( J ) +1)=gamma ’ ;230 save ( ’ gamma_dis t ’ , ’ gamma_dis t ’ )231232233 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%234 %−−−−−−−−−−−−−−−−−−−−−−−−−−PLOTTING−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%235 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%236 f i g u r e ( ’ r end ’ , ’ p a i n t e r s ’ , ’ pos ’ , [ 1 0 0 100 600 8 0 0 ] )237 ho ld on238 g r i d minor239 t i t l e ( [ ’ Th rus t , t o r q u e & e f f i c i e n c y wi th s i m p l e model o f ’ . . .

8

240 ’ i n d u c e d v e l o c i t i e s ’ ] )241 xl im ( [ J ( 1 ) J ( end ) ] )242 yl im ( [ 0 1 . 0 ] )243 x l a b e l ( ’ J [−] ’ )244 y l a b e l ( ’K_T [−] 1 0 \ c d o t K_Q [−] \ e t a _ 0 [−] ’ )245246 p l o t ( J , K_T , ’ g ’ , wagB ( : , 1 ) , wagB ( : , 2 ) , ’g−−’ )247 p l o t ( J ,10∗K_Q, ’ b ’ , wagB ( : , 1 ) ,10∗wagB ( : , 3 ) , ’b−−’ )248 p l o t ( J , e t a0 , ’ r ’ , wagB ( : , 1 ) , wagB ( : , 4 ) , ’ r−−’ )249 l e g e n d ( ’K_T − l i f t i n g l i n e ’ , ’K_T − Wag . B ’ , ’ 1 0 \ c d o t K_Q − l i f t i n g l i n e ’ . . .250 , ’ 1 0 \ c d o t K_Q − Wag . B ’ , ’ \ e t a _ 0 − l i f t i n g l i n e ’ , ’ \ e t a _ 0 − Wag . B ’ )

C. task4.m

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%3 %−LIFTING LINE METHOD WITH INDUCTION FACTOR METHOD FOR INDUCED VELOCITIES−%4 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%5 % C a l c u l a t e s t o r q u e , t h r u s t and e f f i c i e n c y f o r a s p e c i f i e d p r o p e l l e r %6 % geomet ry and compares t h e r e s u l t s w i th e x p e r i m e n t a l d a t a . Th i s model %7 % i n c l u d e s t h e e f f e c t o f i n d u c e d v e l o c i t i e s t h r o u g h t h e i n d u c t i o n %8 % f a c t o r method . %9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

1011 c l e a r a l l12 c l o s e a l l13 c l c1415 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%16 % Load l i f t c o e f f i e c i e n t s , c i r c u l a t i o n d i s t r i b u t i o n from t a s k 4 , %17 % Wagenigen d a t a and geomet ry d a t a %18 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%19 l o a d L i f t R a d i u s . mat ;20 l o a d ( ’ gamma_dis t . mat ’ ) ;21 wagB= dlmread ( ’wagB . t x t ’ ) ; % J , KT, KQ, e t a22 geomet ry = dlmread ( ’ Geometry . t x t ’ ) ; % r / R , chord /D, t /D, P /D2324 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%25 % S p e c i f y i t e r a t i o n p a r a m e t e r s %26 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%27 i t e r m a x =1000;28 t o l =10E−4;29 damp = 0 . 0 5 ;3031 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%32 % De f i n e f low & p r o p e l l e r p a r a m e t e r s %33 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%34 Re =2 .4 E6 ;35 x0 = 0 . 7 ;36 cx0 = 0 . 2 2 5 ; % from geomet ry f i l e37 nu =1.19E−6; % from m e c h a n i c a l p r o p e r t i e s o f w a t e r38 rho =1025;39 D=1;40 Z=4;41 span =D/2∗(1−0.167) ;4243 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%44 % D i s c r e t i z e p r o p e l l e r i n t o f o i l s e c t i o n s %45 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%46 k =100;47 h =(1−0.167) / ( k−1) ;48 x = 0 . 1 6 7 : h : 1 ; % x= r / R49 r =x∗D/ 2 ;5051 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%52 % I n t e r p o l a t e f o r chord l e n g t h , t h i c k n e s s and p i t c h a n g l e a t s e c t i o n s %53 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%54 chord = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 2 ) , x , ’ p c h i p ’ ) ;55 t h i c k n e s s = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 3 ) , x , ’ p c h i p ’ ) ;56 PdivD = i n t e r p 1 ( geomet ry ( : , 1 ) , geomet ry ( : , 4 ) , x , ’ p c h i p ’ ) ;57 p h i = a t a n ( PdivD∗D. / ( 2∗ p i∗r ) ) ;5859 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%60 % I n i t i a l i z e v e c t o r o f advance numbers and f i n d c o r r e s p o n d i n g V and n %61 % from eq . i n c h a p t e r 16 ( compendium ) %62 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%63 J = [ 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 . 1 ] ;64 V= Re∗nu / cx0 ∗ 1 . / s q r t ( 1 + ( x0∗p i . / J ) . ^ 2 ) ;65 n= V . / ( J∗D) ;6667 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%68 % I n t e r p o l a t e l i f t c o e f f i c i e n t a t z e r o aoa from t h e XFoi l r e s u l t s %69 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%70 C_Lc = i n t e r p 1 ( L i f t R a d i u s ( : , 1 ) , L i f t R a d i u s ( : , 2 ) , x , ’ p c h i p ’ ) ;7172 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%73 % I n i t i a l i z e m a t r i c e s & v e c t o r s %74 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%75 gamma= z e r o s ( l e n g t h ( J ) , l e n g t h ( x ) ) ;76 U_A=gamma ; U_T=gamma ; a l p h a =gamma ; Vinf =gamma ; C_L=gamma ; dTi=gamma ;77 gamma_new=gamma ; gamma_input=gamma ;78 dQi=gamma ; dL=gamma ; dT=gamma ; dTd=gamma ; dKd=gamma ; dQ=gamma ; dLi=gamma ;79 dQd=gamma ; d_gamma=gamma ; b e t a _ i =gamma ; i _ a = z e r o s ( 1 , l e n g t h ( x ) ) ; i _ t = i _ a ;80 twonorm= z e r o s ( 1 , i t e r m a x ) ; Re_c=gamma ;8182 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%83 % I n t e r p o l a t e c i r c u l a t i o n from t h e c i r c u l a t i o n d i s t . from t a s k 3 %84 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%85 f o r i =1 : l e n g t h ( J )86 gamma ( i , : ) = i n t e r p 1 ( gamma_dis t ( : , 1 ) , gamma_dis t ( : , i +1) , x ) / Z ;87 end

8889 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%90 % I t e r a t e t h e p r o c e d u r e u n t i l c o n v e r g e n c e i s s a t i s f i e d o r maximum a l l o w e d %91 % number o f i t e r a t i o n s i s r e a c h e d %92 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%93 f o r i t e r =1 : i t e r m a x94 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%95 % C a l c u l a t e t h e new c i r c u l a t i o n a t e v e r y f o i l s e c t i o n f o r %96 % e v e r y advance number %97 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%98 f o r i =1 : l e n g t h ( J )99

100 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%101 % C a l c u l a t e mean t a n g e n t i a l and a x i a l i n d u c e d v e l o c i t i e s u s i n g %102 % c o m p l e t e momentum t h e o r y a t e v e r y f o i l s e c t i o n . %103 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%104 U_T ( i , : ) =Z∗gamma ( i , : ) . / ( 2∗ p i∗r ) ; % eq . ( 1 3 . 4 3 )105 f o r j =1 : l e n g t h ( x )106 a = 0 . 5 ; b=V( i ) ; c=−U_T( i , j ) .∗(2∗ p i .∗ r ( j )∗n ( i )−U_T( i , j ) / 2 ) ;107 U_A( i , j ) = (−b+ s q r t ( b^2−4∗a∗c ) ) / (2∗ a ) ; % eq . ( 1 3 . 2 8 )108 end109110 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%111 % C a l c u l a t e t e n t a t i v e hydrodynamic p i t c h a n g l e and f i n d t h e %112 % d e r i v a t i v e o f t h e c i r c . d i s t . ove r t h e p r o p e l l e r b l a d e %113 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%114 b e t a _ i ( i , 1 : end ) = a t a n ( ( V( i ) +0.5∗U_A( i , : ) ) . . .115 . / ( 2∗p i∗r∗n ( i )−0.5∗U_T ( i , : ) ) ) ;116 d_gamma ( i , : ) =gammaderive ( h∗D/ 2 , gamma ( i , : ) ) ;117118 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%119 % C o r r e c t t h e t a n g e n t i a l and a x i a l i n d u c e d v e l o c i t i e s wi th t h e %120 % i n d u c t i o n f a c t o r method %121 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%122 f o r j _ f i x e d =2: l e n g t h ( x )−1123 r0 = r ( j _ f i x e d ) ;124125 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%126 % Find i n d u c t i o n f a c t o r s ove r t h e r u n n i n g v a r i a b l e %127 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%128 f o r j =1 : l e n g t h ( x )129 [ i _ a ( j ) , i _ t ( j ) ] = I n d u c t i o n F a c t o r s ( r ( j ) , . . .130 r0 , b e t a _ i ( i , j ) ,Z ) ;131 end132133 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%134 % C a l c u l a t e t h e t a n g e n t i a l and a x i a l i n d u c e d v e l o c i t i e s by %135 % i n t e g r a t i o n o f Bio t−S a v a r t s law wi th i n d u c t i o n f a c t o r s . %136 % Thi s r e q u i r e s a s i n g u l a r i n t e g r a t i o n t e c h n i q u e %137 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%138 U_T ( i , j _ f i x e d ) = S i n g u l a r I n t e g r a t i o n ( r , i _ t .∗d_gamma ( i , : ) / (2∗ p i )

, r0 ) ;139 U_A( i , j _ f i x e d ) = S i n g u l a r I n t e g r a t i o n ( r , i _ a .∗d_gamma ( i , : ) / (2∗ p i )

, r0 ) ;140 end141142 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%143 % C a l c u l a t e hydrodynamic p i t c h a n g l e and use i t t o f i n d e f f e c t i v e %144 % a n g l e o f a t t a c k145 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%146 b e t a _ i ( i , 1 : end ) = a t a n ( ( V( i ) +0.5∗U_A( i , : ) ) . / ( 2∗ p i .∗ r∗n ( i )−0.5∗U_T( i

, : ) ) ) ;147 a l p h a ( i , 2 : end−1) = p h i ( 2 : end−1)−b e t a _ i ( i , 2 : end−1) ;148149 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%150 % Find l i f t c o e f f i e c i e n t a s t h e c o n t r i b u t i o n o f camber p a r t and %151 % a n g l e o f a t t a c k p a r t . C a l c u l a t e t o t a l v e l o c i t y a t f o i l s e c t i o n s %

%152 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%153 C_L ( i , : ) = C_Lc + 2∗p i∗a l p h a ( i , : ) ;154 Vinf ( i , : ) = s q r t ( ( V( i ) +0.5∗U_A( i , : ) ) . ^ 2 + . . .155 ( 2∗p i∗r∗n ( i )−0.5∗U_T( i , : ) ) . ^ 2 ) ;156157 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%158 % Update t h e new v a l u e o f gamma u s i n g a damping f a c t o r %159 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%160 gamma_new ( i , : ) =0.5∗ Vinf ( i , : ) .∗ chord .∗C_L ( i , : ) ;161 gamma_input ( i , : ) =gamma ( i , : ) +damp∗(gamma_new ( i , : )−gamma ( i , : ) ) ;162 end163164 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%165 % C a l c u l a t e t h e two−norm e r r o r o f t h e i t e r a t i o n %166 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%167 twonorm ( i t e r ) = ( sum ( sum ( abs ( gamma_input−gamma ) . ^ 2 ) ) ) ^ 0 . 5 ;168169 %−−−−−−−−−−−−−−−−−−−−−−−−−−%170 % Updates t h e d i s t r i b u t i o n %171 %−−−−−−−−−−−−−−−−−−−−−−−−−−%172 gamma=gamma_input ;173174 %−−−−−−−−−−−−−−−−−−−−−−−−−−−%175 % P r i n t s p e r c e n t c o m p l e t i o n %176 %−−−−−−−−−−−−−−−−−−−−−−−−−−−%177 p e r c e n t d o n e _ p r i n t ( twonorm ( i t e r ) / twonorm ( 1 ) , t o l ) ;178179 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%180 % Checks i f c o n v e r g e n c e i s met , i f yes t h e n b r e a k t h e i t e r a t i o n s %181 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%182 i f twonorm ( i t e r ) / twonorm ( 1 ) < t o l183 c l c184 f p r i n t f ( ’ Convergence c r i t e r i o n met a f t e r %d i t e r a t i o n s . \ n ’ , i t e r ) ;185 b r e a k186 end

9

187188 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%189 % Checks i f t h e r e i s any NaN i n gamma , i f yes t h e n c a n c e l s program %190 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%191 i f i s n a n ( sum ( sum ( gamma ) ) ) ==1192 c l c193 f p r i n t f ( ’NaN d e t e c t e d ! C a n c e l l i n g program . . . \ n ’ )194 r e t u r n195 end196 end197198 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%199 % C a l c u l a t e t h r u s t and t o r q u e on e v e r y f o i l s e c t i o n f o r %200 % e v e r y advance number %201 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%202 f o r i =1 : l e n g t h ( J )203 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%204 % Find t h r u s t and t o r q u e c o n t r i b u t i o n b e c a u s e o f l i f t a t e v e r y f o i l %205 % s e c t i o n %206 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%207 dTi ( i , : ) = rho∗gamma ( i , : ) .∗(2∗ p i∗r∗n ( i )−U_T( i , : ) / 2 ) ;208 dQi ( i , : ) = rho∗gamma ( i , : ) .∗ (V( i ) +U_A( i , : ) / 2 ) .∗ r ;209210 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%211 % Find drag c o e f f i e n t by use o f eq . ( 1 3 . 3 3 ) i n compendium %212 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%213 Re_c ( i , : ) = Vinf ( i , : ) .∗ chord / nu ;214 C_F = 0 . 0 7 5 . / ( log10 ( Re_c ( i , : ) )−2) . ^ 2 ; % from ITTC 57 ’215 C_D = 2∗C_F.∗(1+2∗ t h i c k n e s s . / ( chord ) ) ;216217 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%218 % Find t h r u s t and t o r q u e c o n t r i b u t i o n b e c a u s e o f d rag a t e v e r y f o i l %219 % s e c t i o n %220 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%221 dTd ( i , : ) =0.5∗ rho∗Vinf ( i , : ) . ^ 2 .∗ chord∗C_D( i ) .∗ s i n ( b e t a _ i ( i , : ) ) ;222 dQd ( i , : ) =0.5∗ rho∗Vinf ( i , : ) . ^ 2 .∗ chord∗C_D( i ) .∗ r .∗ cos ( b e t a _ i ( i , : ) ) ;223224 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%225 % Find t o t a l t h r u s t and t o r q u e component a t e v e r y f o i l s e c t i o n %226 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%227 dT ( i , : ) =dTi ( i , : )−dTd ( i , : ) ;228 dQ ( i , : ) =dQi ( i , : ) +dQd ( i , : ) ;229 end230231 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%232 % Numer ica l e r r o r s a t p r o p e l l e r ends a r e c o r r e c t e d %233 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%234 dT ( : , end ) =0;235 dQ ( : , end ) =0;236237 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%238 % I n t e g r a t e wi th MATLABs t r a p z ( ) i n t e g r a t i o n %239 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%240 T=Z∗ t r a p z ( r , dT , 2 ) ;241 Q=Z∗ t r a p z ( r , dQ , 2 ) ;242243 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%244 % C a l c u l a t e t h e t h r u s t and t o r q u e c o e f f i e n t %245 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%246 K_T= T ’ . / ( rho∗n .^2∗D^4) ;247 K_Q= Q ’ . / ( rho∗n .^2∗D^5) ;248249 %−−−−−−−−−−−−−−−−−−−−−−%250 % C a l c u l a t e e f f i c i e n c y %251 %−−−−−−−−−−−−−−−−−−−−−−%252 e t a 0 =J / (2∗ p i ) .∗ K_T . / K_Q;253254255 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%256 %−−−−−−−−−−−−−−−−−−−−−−−−−−PLOTTING−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%257 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%258 f i g u r e ( ’ r end ’ , ’ p a i n t e r s ’ , ’ pos ’ , [ 1 0 0 100 600 8 0 0 ] )259 ho ld on260 g r i d minor261 t i t l e ( ’ Th rus t , t o r q u e & e f f i c i e n c y wi th i n d u c e d v e l o c i t i e s & i n d u c t i o n

f a c t o r s ’ )262 xl im ( [ J ( 1 ) J ( end ) ] )263 yl im ( [ 0 1 . 0 ] )264 x l a b e l ( ’ J [−] ’ )265 y l a b e l ( ’K_T [−] 1 0 \ c d o t K_Q [−] \ e t a _ 0 [−] ’ )266267 p l o t ( J , K_T , ’ g ’ , wagB ( : , 1 ) , wagB ( : , 2 ) , ’g−−’ )268 p l o t ( J ,10∗K_Q, ’ b ’ , wagB ( : , 1 ) ,10∗wagB ( : , 3 ) , ’b−−’ )269 p l o t ( J , e t a0 , ’ r ’ , wagB ( : , 1 ) , wagB ( : , 4 ) , ’ r−−’ )270 l e g e n d ( ’K_T − l i f t i n g l i n e ’ , ’K_T − Wag . B ’ , ’ 1 0 \ c d o t K_Q − l i f t i n g l i n e ’ . . .271 , ’ 1 0 \ c d o t K_Q − Wag . B ’ , ’ \ e t a _ 0 − l i f t i n g l i n e ’ , ’ \ e t a _ 0 − Wag . B ’ )

D. percentdone_print.m

1 f u n c t i o n [ ] = p e r c e n t d o n e _ p r i n t ( num , num_fin )2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3 %−−−−−−−−PRINTS THE PERCENTAGE COMPLETED TOWARDS CONVERGENCE GOAL−−−−−−−−−%4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%56 f r a c =num_fin / num ;7 c l c8 f p r i n t f ( ’ %6.2 f%% comple t ed \ n ’ , f r a c ∗100)9

1011 end

E. gammaderive.m

1 f u n c t i o n [ d_gamma ] = gammaderive ( h , gamma )2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3 %−−−−−CALCULATES THE DERIVATIVE OF GAMMA BASED ON FINITE DIFFERENCING−−−−−%4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%56 k= l e n g t h ( gamma ) ;78 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%9 % C a l c u l a t e f i r s t and l a s t d e r i v a t i v e wi th backwards and f o r w a r d s %

10 % d i f f e r e n c i n g %11 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%12 d_gamma ( 1 )=(−gamma ( 3 ) +4∗gamma ( 2 ) −3∗...13 gamma ( 1 ) ) / (2∗ h ) ;14 d_gamma ( k ) =(gamma ( k−2)−4∗gamma ( k−1)+ 3∗ . . .15 gamma ( k ) ) / (2∗ h ) ;1617 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%18 % C a l c u l a t e i n n e r d e r i v a t i v e s wi th c e n t r a l d i f f e r e n c i n g %19 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%20 f o r i = 2 : k−121 d_gamma ( i )=(−gamma ( i−1)+gamma ( i +1) ) / . . .22 (2∗h ) ;23 end2425 end

F. InductionFactors.m

1 f u n c t i o n [ i_A , i_T ] = I n d u c t i o n F a c t o r s ( r , r0 , b e t a _ i , Z )2 % Thi s f u n c t i o n c a l c u l a t e s t h e i n d u c t i o n f a c t o r s a s a s p e c i f i e d i n "

C a l c u l a t i o n o f Pe r fo rmance and C a v i t a t i o n C h a r a c t e r i s t i c s o fP r o p e l l e r s " by Van Oossanen .

3 % In t h e i n p u t v a r i a b l e s , r i s t h e r u n n i n g p o i n t , w h i l e r0 i s t h ef i x e d p o i n t . In t h e o r i g i n a l r e f e r e n c e , x i s t h e f i x e d p o i n tand x0 i s t h e r u n n i n g p o i n t .

4 % Thi s i s t h e r e a s o n t h a t x_x0 = r0 / r , r a t h e r t h a n x_x0 = r / r0 . Thev a r i a b l e names a r e k e p t a s c l o s e t o t h e o r i g i n a l r e f e r e n c e as

p o s s i b l e so t h a t i t i s5 % e a s i e r t o compare them d i r e c t l y67 x_x0 = r0 / r ;8 x0_x = r / r0 ;9

10 i f x_x0 == 111 i_A = cos ( b e t a _ i ) ;12 i_T = s i n ( b e t a _ i ) ;13 e l s e14 p = 1+( x_x0 ^ 2 / t a n ( b e t a _ i ) ^2 ) ;15 u = exp ( Z∗( l o g ( ( p^0.5−1)∗ (1 / s i n ( b e t a _ i )−1)^(−1)∗x0_x ) +p

^0.5−1/ s i n ( b e t a _ i ) ) ) ;16 g = ( s i n ( b e t a _ i ) ) ^3∗ (2+9 / ( t a n ( b e t a _ i ) ^2 ) ) +(3∗p−5)∗p ^(−3/2) ;17 f = s i n ( b e t a _ i ) ^(−0.5)∗p ^(−0.25) ;1819 i f x_x0 > 120 A = f ∗ ( 1 / ( u−1)−(g / (24∗Z ) )∗l o g ( u / ( u−1)) ) ;2122 i_A = ( x_x0−1)∗Z∗A/ t a n ( b e t a _ i ) ;23 i_T = (1−x0_x )∗Z∗(1+A) ;24 e l s e25 B = f∗(u/(1−u ) +( g / (24∗Z ) )∗l o g (1/(1−u ) ) ) ;2627 i_A = (1−x_x0 )∗Z∗(1+B) / t a n ( b e t a _ i ) ;28 i_T = ( x0_x−1)∗Z∗B ;29 end30 end31 end

G. SingularIntegration.m

1 f u n c t i o n J = S i n g u l a r I n t e g r a t i o n ( x , f , x0 )2 % Thi s f u n c t i o n computes t h e f o l l o w i n g i n t e g r a l : J = i n t f ( x ) / ( x0 −

x ) dx3 % The i n t e g r a t i o n u s e s a method t h a t s c a l l e d s i n g u l a r i t y s u b t r a c t i o n

.45 n = l e n g t h ( x ) ;6 f0 = i n t e r p 1 ( x , f , x0 ) ;78 in t eg rand_num = z e r o s ( 1 , n ) ;9

10 f o r i = 1 : n11 i f ( x ( i ) − x0 ) ~= 012 in t eg rand_num ( i ) = f ( i ) / ( x0 − x ( i ) ) − f0 / ( x0 − x ( i ) )

;13 end14 end1516 J = t r a p z ( x , i n t eg rand_num ) ;1718 J = J + f0∗(−l o g ( x ( n ) − x0 ) + l o g ( x0 − x ( 1 ) ) ) ;19 end

10