Propeller Performance Analysis Using Lifting Line Theory. by Kevin M. Flood B.S., (1997) Hobart College M.A., (2002) Webster University

Transcription

1 Prpeller Perfrmance Analysis Using Lifting Line Thery by Kevin M. Fld B.S., (997) Hbart Cllege M.A., (00) Webster University Submitted t the Department f Mechanical Engineering in Partial Fulfillment f the Requirements fr the Degrees f Naval Engineer and Master f Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June K. M. Fld. All rights reserved The authr hereby grants t MIT permissin t reprduce and t distribute publicly paper and electrnic cpies f this thesis dcument in whle r in part in any medium nw knwn r hereafter created. Signature f Authr Department f Mechanical Engineering May 8, 009 Certified by Mark S. Welsh Prfessr f the Practice f Naval Cnstructin and Engineering Thesis Supervisr Certified by Richard W. Kimball Thesis Supervisr Accepted by David E. Hardt Chairman, Departmental Cmmittee n Graduate Students Department f Mechanical Engineering

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3 Abstract Prpeller Perfrmance Analysis Using Lifting Line Thery by Kevin M. Fld Submitted t the Department f Mechanical Engineering n May 8, 009 in Partial Fulfillment f the Requirements fr the Degrees f Naval Engineer and Master f Science in Mechanical Engineering Prpellers are typically ptimized t prvide the maximum thrust fr the minimum trque at a specific number f revlutins per minute (RPM) at a particular ship speed. This prcess allws ships t efficiently travel at their design speed. Hwever, it is useful t knw hw the prpeller perfrms during ff-design cnditins. This is especially true fr naval warships whse missins require them t perfrm at a wide range f speeds. Currently the Open-surce Prpeller Design and Analysis Prgram can design and analyze a prpeller nly at a given perating cnditin (i.e. a given prpeller RPM and thrust). If these values are varied, the prgram will design a new ptimal prpeller fr the given inputs. The purpse f this thesis is t take a prpeller that is designed fr a given case and analyze hw it will behave in ff-design cnditins. Prpeller perfrmance is analyzed using nn-dimensinal curves that depict thrust, trque, and efficiency as functins f the prpeller speed f advance. The first step in prducing the pen water diagram is t use lifting line thery t characterize the prpeller blades. The bund circulatin n the lifting line is a functin f the blade gemetry alng with the blade velcity (bth rtatinal and axial). Lerbs prvided a methd t evaluate the circulatin fr a given set f these cnditins. This thesis implements Lerbs methd using MATLAB cde t allw fr fast and accurate mdeling f circulatin distributins and induced velcities fr a wide range f perating cnditins. These values are then used t calculate the frces and efficiency f the prpeller. The prgram shws gd agreement with experimental data. Thesis Supervisr: Mark S. Welsh Title: Prfessr f the Practice f Naval Cnstructin and Engineering Thesis Supervisr: Richard W. Kimball 3

4 Acknwledgements The authr thanks the fllwing individuals fr all f their supprt and assistance with this thesis: Prfessr Rich Kimball fr all his wisdm and guidance during nt nly the thesis prcess but als during the tw classes he taught. His classes and teaching methd were amng the best experienced at MIT. CAPT Patrick Keenan fr the leadership and directin he prvided thrughut his time leading the curse N prgram. His curse was ne f the mst interesting the authr experienced at MIT. CAPT Mark Welsh fr the leadership and supprt he prvided in the hme stretch f my MIT experience. And mst f all, my family fr all f their supprt and understanding during my time at MIT. Missy, Margaret, Jsh, and I had a wild ride filled with fun and frustratin fr the past three years. 4

5 Table f Cntents Abstract... 3 Acknwledgements... 4 List f Figures... 7 List f Tables Intrductin and Recent Prpeller Design Advances Recent Advances Theretical Fundatin.... Velcity Field f Symmetrically Spaced Helical Vrtex Lines.... Velcity Field f Symmetrically Spaced Helical Vrtex Sheets Applicatin t Mderately Laded Free-Running Nn-Optimum Prpellers Determining the Circulatin and Induced Velcities Implementatin and Validatin MATLAB Implementatin f the Building Blcks Validatin f the Building Blcks with Lerbs Example Data MATLAB Implementatin fr Finding Circulatin, Induced Velcities, and the Resultant Frces Validatin f Circulatin, Induced Velcity, and Frces Using OpenPrp MATLAB Implementatin fr Determining Prpeller Perfrmance Characteristics at Off-Design Advance Ratis Validatin f Prpeller Perfrmance Characteristics at Off-Design Advance Ratis Cnclusins and Recmmendatins Cnclusins Recmmendatins fr future wrk

6 References... 4 Appendix A. MATLAB Cde A. Find_Lerb_Inductin_Factrs.m A. Calculate_Inductin_Furier_Cefficients.m A.3 Calculate_hm_factrs.m A.4 Calculate_Angle_f_Zer_Lift.m A.5 Calculate_Gm_and_new_AlphaI.m A.6 Frces.m A.7 Sme_Open_Water_Characteristics.m... 5 A.8 Open_Water_Characteristics.m Appendix B. User s Manual B. Cde t Prduce Perfrmance Curves f OpenPrp Default Design B. Cde t Prduce Perfrmance Curves f DTMB 49 Prpeller

7 List f Figures Figure : Vrtex Pattern Representing a Lifting Wing (3)... 9 Figure : Pitch Angle Relatinships t Velcities and Frces (4)... 0 Figure 3: Velcity Diagram f a Mderately Laded Prpeller (4)... Figure 4: Nn-Dimensinal Circulatin Cmparisn... 3 Figure 5: Nn-Dimensinal Induced Velcities Cmparisn... 3 Figure 6: Perfrmance Curves f a Generic Prpeller Figure 7: Rbustness Results fr the Initial Estimatin f α i Figure 8: OpenPrp Input Parameters fr the DTMB 49 Prpeller Figure 9: Pitch ver Diameter Rati fr the DTMB 49 Prpeller and OpenPrp Output Figure 0: OpenPrp Representatin f the DTMB 49 Prpeller Figure : DTMB 49 Prpeller Perfrmance Curves List f Tables Table : Input Data fr Lerbs' Example... 5 Table : Axial Inductin Factr Difference... 5 Table 3: Tangential Inductin Factr Difference... 6 Table 4: Axial Inductin Factr Furier Cefficient Difference... 6 Table 5: Tangential Inductin Factr Furier Cefficient Difference... 6 Table 6: Axial h m Factr Difference... 7 Table 7: Tangential h m Factr Difference... 7 Table 8: Thrust, Trque, and Efficiency Differences Table 9: Gemetry f the DTMB 49 Prpeller (4)

8 . Intrductin and Recent Prpeller Design Advances Prpellers are typically ptimized t prvide the maximum thrust fr the minimum trque at a specific number f revlutins per minute (RPM) at a particular ship speed. This prcess wrks very well fr merchant ships that travel acrss the cean at a relatively cnstant speed. There is a certain RPM at which their engines perate mst efficiently. If their prpeller can prduce the thrust t achieve the desired speed while turning at the prper rate, then the ship will arrive at its destinatin using the minimal amunt f fuel. The prcess is nt much different fr a Naval ship except that tactical situatins ften require a wider range f speeds t be used in rder t accmplish the missin. Therefre, even thugh a ship s prpeller can be ptimized fr a given set f cnditins, it is very useful t knw hw the prpeller perfrms at ff-design cnditins. The purpse f this thesis is t take a prpeller that is designed fr a given case and analyze hw it will behave at ff-design advance rati values. This will be dne by prducing the prpeller pen water diagram as described by Wud and Stapersma in (). Prpeller perfrmance can be expressed in fur nn-dimensinal parameters: The advance rati, J S. The thrust cefficient, K T. The trque cefficient, K Q. The pen water efficiency, η. The advance rati nn-dimensinalizes the ships velcity (V S ), which is the same as the velcity f advance (V a ) since there are n wake effects in an pen water analysis, using the prpeller tip speed as shwn in equatin (.). Thrust, T, and trque, Q, are als nn-dimensinalized using the prpeller s speed, n, and diameter, D, as well as the density f seawater, ρ, as shwn in equatins (.) and (.3) respectively. The pen water efficiency is expressed in terms f the abve parameters as shwn in equatin (.4). A prpeller pen water diagram is cnstructed such that K T, K Q, and η are all functins f J S. In keeping with cmmn practice, K Q is multiplied by ten in rder t better present the curves n ne set f axes. 8

9 VS J S = (.) nd K T T = (.) ρ n D 4 K Q Q = (.3) ρ n D 5 = KT J η π K Q S (.4) The first step in prducing the pen water diagram is t use lifting line thery t characterize the prpeller blades. In this thery the blade is replaced by a straight line. As described in Abbtt (), the circulatin, Γ, abut the blade assciated with lift is replaced by a vrtex filament. The vrtex filament lies alng the straight line; and, at each span-wise statin, the strength f the vrtex is prprtinal t the lcal intensity f lift. Accrding t Helmhltz s therem, a vrtex filament cannt terminate in the fluid. The variatin in vrtex strength is therefre assumed t result frm superpsitin f a number f hrseshe shaped vrtices, as shwn in Figure frm (3). The prtins f the vrtices lying alng the span are called the bund vrtices. The prtins f the vrtices extending dwnstream indefinitely are called the trailing r free vrtices. Circulatin Γ Γ Γ3 Γ4 Γ5 Γ4 Γ5 Γ3 Γ Γ Figure : Vrtex Pattern Representing a Lifting Wing (3) 9

10 The effect f the trailing vrtices crrespnding t a psitive lift is t induce a dwnward cmpnent f velcity at and behind the blade. This dwnward cmpnent is called the dwnwash. The magnitude f the dwnwash at any sectin alng the span is equal t the sum f the effects f all f the trailing vrtices alng the entire span. The effect f the dwnwash is t change the relative directin f the fluid stream ver the sectin. The sectin is assumed t have the same hydrdynamic characteristics with respect t the rtated stream as it had in the nrmal tw-dimensinal flw. The rtatin f the flw effectively changes the angle f attack. () The methd described by Lerbs in (4) is used t determine the circulatin and induced velcity distributins fr given values f J S using the hydrdynamic prperties f the fil sectins at specified radial distances alng the prpeller blade. With the circulatin and velcities calculated, functins built int the Open-surce Prpeller Design and Analysis Prgram are then used t determine the values f K T, K Q, and η. Finally, the whle prcess is repeated again fr anther value f J S until the entire diagram is prduced.. Recent Advances Open-surce Prpeller Design and Analysis Prgram (OpenPrp) is an pen surce MATLAB -based suite f prpeller numerical design tls. This prgram is an enhanced versin f the MIT Prpeller Vrtex Lattice Lifting Line Prgram (PVL) develped by Prfessr Justin Kerwin at MIT in 00. OpenPrp v.0, riginally titled MPVL, was written in 007 by Hsin-Lung Chung and Kate D Epagnier and is described in detail in (3) and (5). Tw f its main imprvements versus PVL are its intuitive graphical user interfaces (GUIs) and greatly imprved data visualizatin which includes graphic utput and three-dimensinal renderings. OpenPrp v.0 was written in 008 by Jhn Stubblefield and is described in detail in (6). Versin adds the ability t mdel an axially symmetric ducted-prpeller with n gap between the duct and the prpeller. OpenPrp v3.0 was written by Brenden Epps in 009 and brings the capability t design a turbine as well as a mre generalized ptimizatin rutine. 0

11 OpenPrp was designed t perfrm tw primary tasks: parametric analysis and single prpeller design. Bth tasks begin with a desired perating cnditin defined primarily by the required thrust, ship speed, and inflw prfile. The parametric analysis prduces efficiency diagrams fr all pssible cmbinatins f number f blades, prpeller speed, and prpeller diameter fr ranges and increments entered by the user. The efficiency diagrams are then used t determine the ptimum prpeller parameters fr the desired perating cnditins given any cnstraints (e.g. prpeller speed r diameter) specified by the user. The single prpeller design rutine prduces a cmplete prpeller design which is ptimized fr the desired perating cnditin and defined prpeller parameters (number f blades, prpeller speed, prpeller diameter, hub diameter, etc). If these values are varied, the prgram will design a new ptimal prpeller fr thse given inputs, but it will nt analyze the same prpeller at new cnditins. In 008 Christpher Petersn develped MATLAB executables that interface with a mdified versin f XFOIL fr determining the minimum pressure f a fil perating in an inviscid fluid which are described in detail in (7). XFOIL is an analysis and design system fr Lw Reynlds Number Airfils. XFOIL uses an inviscid linear-vrticity panel methd with a Karman-Tsien cmpressibility crrectin fr direct and mixed-inverse mdes. Surce distributins are superimpsed n the airfil and wake permitting mdeling f viscus layer influence n the ptential flw. Bth laminar and turbulent layers are treated with an e 9 -type amplificatin frmulatin determining the transitin pint. The bundary layer and transitin equatins are slved simultaneusly with the inviscid flw field by a glbal Newtn methd as described by Drela in (8). Petersn s cde creates minimum pressure envelpes, similar t thse published by Brckett (965). The mdified XFOIL and MATLAB interface was intended fr future interface with OpenPrp.

12 . Theretical Fundatin Each individual blade f a prpeller can be represented by its wn lifting line as described in Sectin. The dwnwash affects nt nly the blade being mdeled, but als every ther blade n the prpeller. Hwever, if the prpeller is symmetrical, these affects will cancel each ther ut. Therefre, the free vtices that are shed frm the lifting line will be identical and spaced at cnstant angles t each ther. The free vrtex lines shed frm the lifting line are nt acted n by frces. Their directins, accrding t wing thery, cincide with that f the resultant lifting system. This results in the free vrtex lines f a prpeller frming a general helical shape. When all f the free vrtex lines are cmbined, they frm a free vrtex sheet that is in the shape f a general helical surface. The shape f the helical sheets and their induced velcities are mutually dependant s certain assumptins abut the shape f the helical sheets are required. The methds described in this thesis are applicable t mderately laded prpellers, therefre the fllwing assumptins apply. The induced velcities are allwed t influence the shape f the free helical sheets. Hwever, the effects f centrifugal frces and wake cntractin n the shape f the sheets are neglected. The wrk f Betz, Lck, and Kramer shwed that fr a mderately laded prpeller with an ptimum circulatin distributin, the assumptin that the defrmatin f the helical sheets in the axial directin can be neglected prduced results in clse agreement with experimental data. The assumptin that the axial defrmatin can be neglected will be extended here t nn-ptimum circulatin distributins. Therefre, the vrtex sheets discussed in the upcming sectins are f general helical shape and are made up f cylindrical vrtex lines f a cnstant diameter and pitch angle in the axial directin. In additin, the curling up f the vrtex sheets a certain distance behind the prpeller caused by the sheet s self mtin is als disregarded. (4). Velcity Field f Symmetrically Spaced Helical Vrtex Lines There are tw pssible ways t determine the velcity field f symmetrically spaced helical vrtex lines, the integral by Bit-Savart and Laplace s differential equatin. Kawada in (9) used the Laplace equatin apprach t develp analytical expressins fr the velcity ptential. These

13 expressins prvide numerical results mre readily than the elabrate numerical integratin carried ut by Strscheletzky in (0) fllwing the Bit-Savart methd. (4) Kawada cnsiders a prpeller f g blades mdeled as a symmetric system f g helical tip vrtices cmbined with an axial hub vrtex which has the same strength as all f the tip vrtices cmbined. Kawada s ptential functin is nly justified when assuming that the vrtex system is infinitely lng in bth directins f the axis. By subtracting the ptential f the hub vrtex frm this expressin, the ptential fr an infinitely lng symmetrical system f g helical vrtices f cnstant radius, r, and cnstant pitch angle, β i, is btained. First, a cylindrical crdinate system (z, ψ, r) is defined where z equals zer at the prpeller and is psitive in the directin f flw, and ψ equals zer at the first lifting line. Then the ptential fr internal pints (r< r ) in the field is defined by equatin (.), and the ptential fr external pints (r>r ) is defined by equatin (.). (4) g g z r ng ng φ = Γ + Γ + i I ng r K ng r sin ngϑ (.) π k k n= k k g Γ m r ng ng φ = + e gψ π g K ng r I ng r sin ngϑ (.) π m= g k n= k k Fr the ptential equatins, Γ is defined as the infinitesimal circulatin f ne f the g vrtex lines which is related t the circulatin at ne f the lifting lines, Г, by equatin (.3). In additin, k and ϑ are defined in equatins (.4) and (.5), respectively. The functins I and K are mdified Bessel functins f the first and secnd kind. The prime symbl dentes the derivative with respect t the argument. (4) dγ Γ = dr (.3) dr k r ( β ) = tan (.4) k i z ϑ =ψ (.5) 3

14 4 In rder t find the induced velcities at a lifting line f the prpeller, equatins (.) and (.) can be used with ψ and z bth equal t zer. Hwever, these velcities wuld crrespnd t a vrtex system that extends frm z equals psitive infinity t negative infinity which is essentially a tw-dimensinal system. The vrtex system f an actual prpeller is three-dimensinal and the vrtex system extends nly frm z equals zer t psitive infinity. In rder t determine if velcities f the three-dimensinal system culd be determined frm the tw-dimensinal system, Lerbs (4) cnsidered the velcity-integrals by Bit-Savart fr the assumed case that bth r and β i are independent f ψ. It was cncluded frm these integrals that the effect f the axial and tangential cmpnents f the system between z equals negative infinity and the prpeller (z=0) is equal t the effect f the system between the prpeller and z equals psitive infinity fr pints n the lifting line. Therefre it fllws that the axial, a w, and tangential, t w, velcities induced at a lifting line f a prpeller are simply half the axial and tangential velcities induced frm the ptentials in equatins (.) and (.). These induced velcities are shwn fr bth the internal and external pints in equatins (.6) thrugh (.9). Γ = = = 4 n ng ng i ai r k ng K r k ng ni k r g k g z w π φ (.6) = Γ = = n ng ng e ae r k ng I r k ng nk k r g z w π φ (.7) = Γ = = n ng ng i ti r k ng K r k ng ni k r g r r w π ψ φ (.8) + Γ = = = 4 n ng ng e te r k ng I r k ng nk k r g k g r r w π ψ φ (.9) Fr the purpses f numerical integratin, the Bessel functins and their derivatives are estimated by Nichlsn s asympttic expansins (). The final results fr the induced velcities at ne f the lifting lines are shwn in equatins (.0) thrugh (.3). ( ) 4 B g k w ai + Γ = π (.0) 4 B g k w ae π Γ = (.)

15 w ti gγ = B (.) r 4 π gγ w te = + r 4π ( B ) (.3) where:.5 + y ( ) y B, = m ln + (.4) ga,. 5, + ga y e g + y e A, = ± ( ) ( )( ) + y + + y + y + y ln ( + y + )( + y ) y m (.5) r = = (.6) k tan( β i ) r x y = = (.7) k tan x ( β ) i Unfrtunately, numerical accuracy f the induced velcities is degraded when the reference radius appraches the vrtex radius because the expressins g t infinity. This prblem can be avided by using the inductin factrs frm (9) as defined in equatins (.8) and (.9). The inductin factrs are the induced velcity cmpnents nn-dimensinalized by the velcity induced at r by a straight ptential vrtex extending frm z equals zer t psitive infinity f strength Γ situated at r. Bth the induced velcities and the nn-dimensinalizing velcity becme infinite in the same rder as r appraches r. Therefre, the inductin factrs remain finite as the radii apprach each ther. Cmbining the induced velcity equatins int the inductin factr equatins yields the final inductin factrs as shwn in equatins (.0) thrugh (.3). i a w a = Γ (.8) 4 π ( r r ) i t w t = Γ (.9) 4 π ( r r ) x x i = g ai x tan( β ) x + i ( B ) (.0) 5

16 i ae = g x x tan x B ( β ) x i (.) i ti x = g B (.) x i x = g ( B ) (.3) x te + It is interesting t nte that the inductin factrs are functins f gemetric prperties (i.e. the relative psitin f the reference pint and where the vrtex is shed as well as the angle at which it is shed) and are nt dependant n the circulatin. Als f nte is that the abve expressins apply t nly ne lifting line. In general, there will als be induced velcities at each lifting line (i.e. each blade) frm all ther lifting lines. Hwever, if the prpeller is symmetric, these effects will cancel each ther ut. (4) In rder t evaluate the accuracy f the assumptins leading t the calculatin f the inductin factrs, Lerbs (4) cmpared the calculated values t thse calculated using direct numerical integratin f the Bit-Savart integral by Strscheletzky (0) in the case f g equals 3. He fund the methd f Strscheletzky prduced higher values than the methd explained abve, but nly when i ti is smaller than Differences when the inductin factr is s small have a negligible effect n the verall final values.. Velcity Field f Symmetrically Spaced Helical Vrtex Sheets Sectin. develped the velcity field fr g helical vrtex lines. When multiple vrtex lines are shed frm the same lifting line, a vrtex sheet is frmed. The velcity cmpnents which are prduced by the vrtex sheets are btained by integrating the effects f the vrtex lines ver the span f the lifting line. By integrating ver the free vrtices and using equatins (.3), (.8), and (.9), expressins fr the induced velcities (nn-dimensinalized with the speed f advance) at statin x f the lifting line are shwn in equatins (.4) and (.5). These equatins are in terms f the nn-dimensinal circulatin (G) as defined in equatin (.6). (4) w V a S = x h dg dx ( x x ) i a dx (.4) 6

17 w V t S = x h dg dx ( x x ) i dx t (.5) Γ G = (.6) π DV S Expanding n the methd used by Glauert () in airfil thery, the values f the integrals in equatins (.4) and (.5) can be determined. This requires a new variable, φ, as defined in equatin (.7) which is zer when x equals the hub radius, x h, and π when x is at the blade tip. x = ( + x h ) ( x h ) cs( ϕ) (.7) Fr the purpse f analysis, G is estimated t g t zer at the hub and at the tip. Fr a real prpeller, the hub des carry sme circulatin as described in (). Therefre, this assumptin is nt always valid, but it des prvide a gd first rder apprximatin. This assumptin als des nt hld true fr prpellers with a zer-gap duct, but that is beynd the scpe f this thesis. Circulatin is cntinuus alng the span f the blade, therefre G can be written as the Furier series shwn in equatin (.8). The cntinuity f G als ensures that single vrtices f finite strength d nt ccur and that the series fr G, when placed int equatins (.4) and (.5), gives the cmplete induced velcity cmpnents. (4) m= ( ) G = G m sin mϕ (.8) In additin t the pitch angle and the number f blades, the inductin factrs frm Sectin. als depend n φ and φ. The inductin factrs, with respect t φ, can be written as an even Furier series as described by Schubert (3). This is shwn in equatin (.9). (, ϕ ) = I ( ϕ)cs( nϕ ) i ϕ (.9) n= 0 n It fllws that the expressins fr the nn-dimensinal induced velcities becme as shwn in equatins (.30) and (.3) where h is defined in equatin (.3). ( h will be defined later.) t m a m w V a S = x h m= mg 7 m h a m ( ) ϕ (.30)

18 h t m ( ϕ) = = π 0 w V t S = x h m= mg ( ϕ, ϕ ) cs( mϕ ) dϕ ( ϕ ) cs( ϕ) π t cs( ( m + n) ϕ ) I n ( ϕ) cs( ϕ ) cs( ϕ) it cs 0 m h t m dϕ + ( ) ϕ (.3) π 0 cs cs (( m n) ϕ ) ( ϕ ) cs( ϕ) dϕ (.3) The tw integrals inside the summatin f equatin (.3) can be slved by fllwing Glauert s methd (). Fr values f m greater than n, the summatin becmes equatin (.33), and fr values f m less than n, the summatin becmes equatin (.34). With these substitutins made, the final frm f equatin (.36). t h m becmes equatin (.35). A similar prcess is fllwed fr π sin ( ϕ) sin m t ( mϕ ) I n ( ϕ) cs( nϕ) n= 0 a h m, resulting in (.33) π sin ( ϕ) cs t ( mϕ) I n ( ϕ) sin( nϕ) n= m+ (.34) h h t m a m m π t t ( ϕ ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin mϕ I n ϕ cs nϕ + cs mϕ I n ϕ sin nϕ (.35) sin ϕ n= 0 n= m+ m π a a ( ϕ ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin mϕ I n ϕ cs nϕ + cs mϕ I n ϕ sin nϕ sin (.36) ϕ n= 0 n= m+ Equatins (.35) and (.36) becme indefinite at the hub (φ = 0º) and at the blade tip (φ = 80º). L Hspital s rule prvides values fr the functins at these pints as shwn in equatins (.37) and (.38). m t, a t, a t, a hm ( 0 ) = π m I n ( 0 ) + ni n ( 0 ) (.37) n= 0 n= m+ m t, a t, a t, a hm ( 80 ) = π cs( m 80 ) m I n cs( n 80 ) + ni n cs( n 80 ) (.38) n= 0 n= m+ 8

19 The abve equatins allw the induced velcity cmpnents t be related t the circulatin distributin and the inductin factrs. The inductin factrs are knwn frm equatins (.0) thrugh (.3). Therefre the induced velcity fr any circulatin distributin that can be represented by equatin (.8) can be calculated, r vice versa. Hwever, it will be necessary t use successive iteratins and apprximatins since the inductin factrs depend n the pitch angles f the sheets which, in turn, depend n the induced velcities. (4).3 Applicatin t Mderately Laded Free-Running Nn-Optimum Prpellers The vrtex sheets described in Sectin. are made up f cylindrical vrtex lines whse pitch des nt change in the axial directin. The next step is t determine the cnditins impsed n the variatins f the pitch f the vrtex lines in the radial directin such that the entire system accurately mdels the vrtex system f an actual prpeller. The frce and the flw generated at a lifting line by the vrtex sheets f Sectin. can be related using an energy balance f the prpeller flw. The input pwer equals the useful pwer plus the increase in kinetic energy within the vlume that passes thrugh the slipstream in a single unit f time. Input pwer is als made up f an external input plus the wrk dne by the centrifugal pressure n the vlume f water in a single unit f time. As stated earlier, fr a mderately laded prpeller, it is acceptable t neglect the effects f the centrifugal pressure. Therefre, the energy balance takes the frm f equatin (.39) where the integral is taken ver a disc with the same diameter as the prpeller and ΔE is the kinetic energy f the induced flw within the vlume f fluid that passes a pint in the ultimate wake in ne unit f time. (4) A ( r dq vdt ) = ΔE ω (.39) At the lifting line, the left hand side f equatin (.39) can be expressed in terms f the relative flw s pitch by substituting dq and dt with their respective expressins frm the Kutta- Jukwski law. The Kutta-Jukwski law states that the frce at the lifting line is perpendicular t the resultant incming flw. Using the nmenclature f Figure, the equivalent f the left hand side is shwn in equatin (.40). (4) 9

20 Figure : Pitch Angle Relatinships t Velcities and Frces (4) A ( rωdq vdt ) = ρ ( gγ)( v + w ) R rh a w tan a ( β ) i + w t dr (.40) Using Green s therem, the kinetic energy f the flw which is included in a certain vlume can be represented by an integral ver the surface f the vlume. Applying this t the right hand side f equatin (.39) allws it t be expressed in terms f the pitch f the vrtex lines as shwn in equatin (.4). (4) R E = ρ rh R = ρ rh ( gγ)( v + w ) a ( gγ)( v + w ) a sin w w tan n ( β ) i a ( β ) i dr + wt dr (.4) Since the left hand side must equal the right hand side, the integrals in equatins (.40) and (.4) must be the same. Therefre β ( r) must equal β ( r) i must leave the lifting line at a pitch angle equal t the angle f the resultant inflw. The angle f the resultant flw is dependent n the accuracy f the apprximatins f the induced flw distributins, w a (x) and w t (x). The level f accuracy f the flws caused by the vrtex sheets in Sectin. is ascertained by cmparing them t a knwn slutin f the flw past a prpeller. 0 i. In ther wrds, the vrtex lines

21 One slutin fr cmparisn is that f an ptimum prpeller fr which the Betz cnditin, defined in equatin (.4), hlds. (4) v ωr wt ωr v + w a = cnstant (.4) Equatin (.43) shws the expansin f bth the numeratr and denminatr f the Betz cnditin. This shws that the Betz cnditin is met if the induced velcity terms f rder tw r greater can be neglected. Frm this it is expected that fr an arbitrary circulatin distributin, the induced velcity at a lifting line is sufficiently estimated frm the vrtex sheets f Sectin. when higher rder terms f the axial and tangential induced velcities are small cmpared t v and ωr, respectively. T this degree, the difference in the shape f the actual and mdeled vrtex sheets is insignificant. (4) ωr w v + w t a = = ( ωr w ) t wt wt ωr ω r w w a a ( v + w ) a v v... (.43).4 Determining the Circulatin and Induced Velcities Given that the gemetry and plar curves f a prpeller are knwn, it is pssible t estimate the circulatin and induced velcity distributins fr a given advance rati. In rder t ease the calculatins, the advance cefficient will be used in lieu f the advance rati. The advance cefficient is defined as the advance rati divided by pi. The first step is t use the Kutta-Jukwski law and the nmenclature f Figure 3 frm (4) t derive the relatinship between the lift cefficient and the circulatin at any radius alng the blade as shwn in equatin (.44). The slidity f the prpeller, s, is defined in equatin (.45) where l is the chrd length at a specified radius.

22 Figure 3: Velcity Diagram f a Mderately Laded Prpeller (4) ggv C L s = V (.44) gl s = πd (.45) Fr a small angle f attack α, the lift curve slpe is apprximately cnstant. Therefre, the lift cefficient can be fund as shwn in equatin (.46) which als shws the rearrangement f the different angular values base n Figure 3. Figure 3 can als be used t determine expressins fr bth the nn-dimensinal inflw velcity and the tangent f (β+α i ) as functins f the advance cefficient which are shwn in equatins (.47) and (.48), respectively. C L dcl = dα dcl ( α + α ) = [( ψ + α ) ( β + α )] V v dα ( β + α ) i i (.46) x wt = λ v (.47) cs

23 w + a ( ) v tan β + α = i (.48) x wt λ v Cmbining equatins (.44), (.46), and (.47) results in equatin (.49). x wt λ v dcl s dα dcl ( ψ + α β ) s ( α ) = gg cs( β + α ) dα i (.49) i The next step is t substitute the expressins fr w a, w t, and G (defined abve in equatins (.4), (.5), and (.8) ) int equatins (.49) and (.48). The results f this substitutin are shwn in equatins (.50) and (.5). dcl x s dα λ [( ψ + α β ) α ] G g sin =... t mh dcl ( mϕ ) cs( β + α ) + [( + ) ] m s ψ α β α m i m= i x h dα i (.50) tan ( + α ) + mg h a m m xh m= β i = (.5) x t mgmhm λ xh m= The final step in finding an apprximatin fr the circulatin distributin and the induced velcities at the lifting line is t satisfy equatins (.50) and (.5) at m statins alng the blade. Successive iteratins becme necessary by starting with a reasnable guess fr the distributin f α i and slving equatin (.50) fr the Furier cefficients, G m. These cefficients are then used in equatin (.5) t determine a new α i distributin. The average f the new and the ld α i distributins are taken as the starting distributin fr the next rund f iteratin. This prcess is repeated until the new and the ld α i distributins differ by nly a small amunt. A difference f less than ± 0. is cnsidered sufficient accuracy. (4) 3

24 3. Implementatin and Validatin Given the basic gemetry and plar curves f a prpeller, alng with an educated guess fr the hydrdynamic pitch angle f the resultant inflw, the theries f Sectin will prduce the circulatin and induced velcity distributins at the lifting line apprximatin f the prpeller blade fr a given advance rati. This data can then be used with the Frces.m functin defined in OpenPrp t determine the thrust, trque, and efficiency infrmatin fr the prpeller. 3. MATLAB Implementatin f the Building Blcks In rder t implement the theries f Sectin as a fast and reliable prpeller analysis tl, the equatins were cded using MATLAB. Althugh the nly equatins that are necessary fr the analysis are equatins (.50) and (.5), the majr required inputs were cded as separate functins in rder t prvide greater understanding t the reader as well as greater ease in trubleshting the cde. All f the MATLAB functins described in Sectin 3 can be fund in Appendix A. The first functin in the prcess is named Find_Lerbs_Inductin_Factrs.m. This functin takes as input the nn-dimensinal radial crdinate f a cntrl pint and a vrtex shedding pint in additin t the number f blades f the prpeller and the pitch distributin f the vrtex sheets. Then, using equatins (.0) thrugh (.3), it calculates the axial and tangential inductin factrs. The functin can nly d this prcess fr a single set f inputs, therefre it must be called multiple times in a lp t determine the inductin factrs at all cntrl pints caused by the entire vrtex system. The next functin is named Calculate_Inductin_Furier_Cefficients.m. This functin als takes as input the nn-dimensinal radial crdinates f cntrl pints and vrtex shedding pints with the axial and tangential inductin factrs determined in the previus functin. It returns bth the axial and tangential Furier cefficients defined in equatin (.9). The next functin is named Calculate_hm_factrs.m. This functin takes the inductin Furier cefficients and the nn-dimensinal radial crdinates f the cntrl pints as inputs in rder t 4

25 return the factrs a h m and equatins (.35) thrugh (.38). t h m frm the induced velcity equatins as they are defined in 3. Validatin f the Building Blcks with Lerbs Example Data The functins in Sectin 3. prvide the basic building blcks needed t slve fr the unknwn circulatin distributin and induced velcities. In rder t ensure that the MATLAB functins did nt cntain errrs, their utputs were cmpared t the data frm the example in Part Sectin A- f the Lerbs paper (4). The example cnsists f a 4-bladed prpeller with a zer hub radius at an advance cefficient f 0.. The specific sectin data is shwn in Table. ϕ ( ) x β ( ) Table : Input Data fr Lerbs' Example i Table and Table 3 shw the difference in the axial and tangential inductin factrs calculated by the Find_Lerbs_Inductin_Factrs.m functin and thse presented by Lerbs in (4). The minr differences are expected since Lerbs read the inductin factrs ff f a graph and the functin calculates them frm equatins. Radial Lcatin f Shed Vrtices (deg) Radial Lcatin f Cntrl Pints (deg) Table : Axial Inductin Factr Difference 5

26 Radial Lcatin f Shed Vrtices (deg) Radial Lcatin f Cntrl Pints (deg) Table 3: Tangential Inductin Factr Difference Table 4 and Table 5 shw the difference in the axial and tangential inductin factrs Furier cefficients calculated by the Calculate_Inductin_Furier_Cefficients.m functin and thse presented by Lerbs in (4). The ccasinal minr differences are in the third decimal place which is the extent f the accuracy f the data in (4). Radial Lcatin f Cntrl Pints (deg) Inductin Factr Furier Cefficients Table 4: Axial Inductin Factr Furier Cefficient Difference Radial Lcatin f Cntrl Pints (deg) Inductin Factr Furier Cefficients Table 5: Tangential Inductin Factr Furier Cefficient Difference Table 6 and Table 7 shw the difference in the factrs a h m and t h m frm the induced velcity equatins as calculated by the Calculate_hm_factrs.m functin and thse presented by Lerbs in (4). The ccasinal minr differences are typically in the third decimal place which is the extent f the accuracy f the data in (4). 6

27 Radial Lcatin f Cntrl Pints (deg) Inductin Factr Furier Cefficients Table 6: Axial h m Factr Difference Radial Lcatin f Cntrl Pints (deg) Inductin Factr Furier Cefficients Table 7: Tangential h m Factr Difference The abve results shw that the building blck functins have sufficient accuracy t be used in the calculatin f circulatin and induced velcities. 3.3 MATLAB Implementatin fr Finding Circulatin, Induced Velcities, and the Resultant Frces With the building blcks in place, the next step is t use them t slve equatins (.50) and (.5). This is dne using the Calculate_Gm_and_new_AlphaI.m functin. This functin returns estimatins f the Furier cefficients fr the nn-dimensinal circulatin functin shwn in equatin (.8), the unknwn angle α i frm equatin (.5), and the cmpnents f the induced velcity frm equatins (.30) and (.3) all f which are based n the initial estimate f α i. The inputs required by the functin are gruped int tw main categries, the prpeller gemetry and its perating cnditins. The imprtant aspects f prpeller gemetry include the slidity f the prpeller as defined in equatin (.45) as well as the pitch angle and the fil sectin data at specified nn-dimensinal radial crdinates. The first required value f fil data is the slpe f the lift curve as a functin f angle f attack. Appendix IV f Abbtt () prvides the plar plts f many NACA 4, 5, 6, and 7 series airfils. If the prpeller blade is made up f ne f these fils, then the slpe f the 7

28 curve can be determined graphically. Hwever, this is nt cnvenient when autmating the prcess with MATLAB. Wing thery states that the theretical slpe f the lift curve is π radians (). Experimental results have shwn clse agreement with this value, therefre it will be used fr this thesis. The next required piece f data fr the fil sectins is the angle f attack that prduces n lift (als called the angle f zer lift). Abbtt () prvides many ways t determine this angle. If the prpeller blade is made up f fil sectin cntained in Appendix IV f (), the angle f zer lift can be graphically determined frm the plar plts. Again, this is nt cnvenient fr the cmputing envirnment. If the mean line distributin is knwn, Abbtt utlines a methd derived by Munk t estimate the angle f zer lift with equatin (3.). It requires that the rdinates f the mean line as a percent f the chrd, y thrugh y 5, are knwn at specific chrdwise percentages, x thrugh x 5, defined belw. The cnstants k thrugh k 5 are defined such that the angle f zer lift is returned in degrees. α (3.) = k y + k y + k3 y3 + k4 y4 + k5 y5 where : k = 5.4 k = k k k = = = and x x x x x = = = = = The functin Calculate_Angle_f_Zer_Lift.m apprximates the angle f zer lift fr a NACA a=0.8 mean line fil. The functin is easily adaptable t ther mean lines by entering new x-y crdinates. The crdinates f many mean lines are listed in Abbtt s Appendix II (). The functin takes as input the maximum chamber rati and scales the mean line rdinates frm the design cnditins t the desired cnditins. It then uses equatin (3.) t calculate and return the angle f zer lift in degrees. The final inputs fr the Calculate_Gm_and_new_AlphaI.m functin are the perating cnditins. They cnsist f the advance cefficient, angle f flw, the factrs a h m and t h m, and the estimatin 8

29 f α i. The prpeller s rtatinal and axial speeds dictate the advance cefficient (as described in Sectin.4) as well as the angle f flw, β, as shwn in Figure 3. The factrs a h m and t h m are defined in equatins (.35) and (.36). The initial guess f the angle α i can be btained frm OpenPrp. With all f the inputs determined, the Calculate_Gm_and_new_AlphaI.m functin starts with equatin (.50). The left-hand side f the equatin is calculated fr every cntrl pint and the values are stred in a rw vectr f k elements, where k is the number f cntrl pints. Next, everything inside the curly brackets n the right-hand side f the equatin is calculated at all k cntrl pints using the values t k fr m thus creating a k by k matrix. The cmplete righthand side f equatin (.50) is the sum f the nn-dimensinal circulatin Furier cefficients (G m ) times everything inside the curly brackets. In matrix ntatin the sum f the prducts is simply the matrix multiplicatin f the curly bracket matrix (k by k) with G m (k by ). The builtin MATLAB functin linslve is then used t slve fr the unknwn G m clumn vectr. Nw that the cefficients fr the circulatin series (based n the estimate f α i ) are knwn, equatin (.5) is used t calculate a new estimate fr the distributins f α i and subsequently β i. Next, β i is used as the vrtex sheet pitch angle distributin. This leads t new inductin factrs with their wn Furier cefficients. These new cefficients are used t calculate new a h m and factrs fr input again int the Calculate_Gm_and_new_AlphaI.m functin. The values f the circulatin series cefficients are als used t estimate the axial and tangential velcities in equatins (.30) and (.3). This prcess is repeated until the riginal and new estimate f α i agree t within ± 0. r a predetermined number f iteratins are perfrmed. t h m Once there is cnvergence n the value f α i, the Frces.m functin frm OpenPrp is used. This functin is shwn in Appendix A fr cmpleteness, but it was nt altered frm its mst recent release. The frces f nte fr prpeller analysis, as explained in Sectin, are the thrust cefficient, the trque cefficient, and the efficiency. 9

30 3.4 Validatin f Circulatin, Induced Velcity, and Frces Using OpenPrp OpenPrp was chsen t design a prpeller because it has been prven t prvide accurate circulatin and induced velcity distributins fr a given design pint. OpenPrp als designs the blade shape and pitch angle based n wing thery. Therefre it shuld be pssible t use all f the gemetric data and the Lerbs methd (4) t arrive back at the same circulatin and induced velcity distributins. Lerbs mdeling f the circulatin as a Furier sine series necessitates that the circulatin ges t zer at bth the hub and blade tip. Fr this reasn, in designing the prpeller with the Single Prpeller Design ptin in OpenPrp, the Hub Image Flag and Ducted prpeller ptins were bth left unchecked. Thrughut the validatin prcess, errrs were discvered within the OpenPrp cde that affected the way the blades were designed but nt the calculatin f the circulatin and induced velcity distributins. The first errr nted was in the calculatin f Vstar just prir t generating the OpenPrp_Perfrmance.txt reprting file. This value was suppsed t be the magnitude f the resultant inflw t the lifting line at each cntrl pint nn-dimensinalized by the speed f the ship. Hwever, the ωr term had units f length ver time. This prblem was slved by dividing the term by the speed f the ship. This errr als cascaded int the circulatin, Gamma, and the lift cefficient, Cl, reprted in OpenPrp_Perfrmance.txt file. This lift cefficient was used by the Gemetry.m functin t scale the chamber f the mean line. The next errr detected als affects the scaling f the chamber f the mean line. The Single Prpeller Design graphical user interface allws the designer t enter the maximum chamber divided by chrd length fr multiple radial psitins alng the blade. This infrmatin in turn is multiplied by the lift cefficient in the Gemetry.m functin t set a new maximum chamber rati. Accrding t Abbtt (), the maximum chamber rati can be scaled by a cnstant factr t prduce the desired lift; hwever, the factr must be as shwn in equatin (3.). In general, if the designer inputs an arbitrary maximum chamber distributin, the crrespnding lift cefficient is unknwn and therefre the scaling factr cannt be determined. This prblem was slved fr this validatin run by inputting the maximum chamber rati fr the NACA a=0.8 mean line frm Appendix II f () int the graphical user interface. This data crrespnds t a lift cefficient f.0. Thus the scaling factr is simply the desired lift cefficient. 30

31 CL Desired Scaling Factr = (3.) CL Original The final errr detected was with the calculatin f the pitch angle in OpenPrp. The prgram simply tk the Ideal Angle f Attack inputted in the graphical user interface and added it t the hydrdynamic pitch angle, β i, that was calculated as part f the ptimizatin rutine. As described in Abbtt (), if the chamber is scaled by a cnstant factr t achieve a desired lift, the ideal angle f attack must als be scaled by the same factr. This prblem was slved by multiplying the ideal angle f attack by the factr in equatin (3.) befre determining the pitch angle f the fil sectin. OpenPrp was used t design a single prpeller using all f the default settings except fr the mdificatins nted abve. The utputs f that design prcess were then used as inputs t the Lerbs methd fr determining the circulatin and induced velcity distributins. Figure 4 shws bth the OpenPrp and Lerbs methd nn-dimensinal circulatin distributins. Figure 5 shw the induced velcity cmpnents fr bth the OpenPrp and Lerbs methd. Bth figures shw clse agreement in the values. The largest differences ccur at the blade rt. It is pssible this difference ccurs because the Lerbs methd frces the circulatin t be zer at the hub. Even thugh the ptin t eliminate hub effects in OpenPrp was chsen, the OpenPrp circulatin des nt apprach zer as quickly as the Lerbs methd. With accurate estimates f the circulatin and velcities, the next step is t calculate the resultant frces and cmpare them t the frces calculated in OpenPrp. The values f the thrust cefficient, the trque cefficient, and the efficiency frm using bth the Lerbs methd and OpenPrp are shwn in Table 8. The maximum difference between the values is nly apprximately.%. It is cncluded that the minr differences in the velcity and circulatin values have a negligible effect n the frces prduced by the prpeller. 3

32 Lerbs Methd OpenPrp Results Nn-Dimensinal Circulatin (G) Nn-Dimensinal Radial Crdinate Figure 4: Nn-Dimensinal Circulatin Cmparisn Nn-Dimensinal Induced Velcity Cmpnents Lerbs Methd (Axial) OpenPrp Results (Axial) Lerbs Methd (Tangential) OpenPrp Results (Tangential) Nn-Dimensinal Radial Crdinate Figure 5: Nn-Dimensinal Induced Velcities Cmparisn 3

33 OpenPrp Results Lerbs Methd Percent Difference (%) K T *K Q Efficiency Table 8: Thrust, Trque, and Efficiency Differences 3.5 MATLAB Implementatin fr Determining Prpeller Perfrmance Characteristics at Off-Design Advance Ratis The Sme_Open_Water_Characteristics.m functin fllws essentially the same prcess utlined in Sectin 3.3. The nly exceptin is that instead f nly lking at just ne advance rati value, the functin lps thrugh the prcess fr a range f advance rati values. Fr each advance rati, the cnverged values f α i and β i are used as the initial estimates fr the next advance rati. The inputs t the functin are gruped int the fllwing three main categries: items cntained in the design structure defined in OpenPrp v3.0, items that are defined by the prpeller s gemetry but are nt part f the design structure, and the advance rati range f interest. Many f the parameters frm the design structure are defined at a set f cntrl pints; therefre, ne f the first pieces f infrmatin that is required is the nn-dimensinal radial crdinate f thse cntrl pints. The chrd-ver-diameter rati, the hydrdynamic pitch angle, the (nn-induced) nn-dimensinal velcity cmpnents, and the drag f the fil sectin must all be defined at the cntrl pint lcatins. Other prpeller gemetry data frm the design structure that is required includes the hub radius, the number f blades, and the radius f the vrtex that is shed frm the hub. The last parameters required frm the design structure are the size f the radial increment used t apprximate the frce integratins, the vlumetric mean inflw velcity, the hub image flag, and the duct thrust cefficient. Althugh the scpe f this thesis des nt include ducted prpellers, a value f zer thrust must be entered in rder t keep the Frces.m functin unchanged frm OpenPrp. 33

34 The parameters that are nt part f the design structure but are defined at the cntrl pints are the pitch ver diameter and the maximum chamber rati distributins. The diameter f the prpeller is als required. The last required input is the advance rati range ver which the prpeller is t be analyzed. The value f α i is easily btained frm the infrmatin cntained in the design structure. OpenPrp prvides this infrmatin at the design pint f the prpeller. Therefre, the Open_Water_Characteristics.m functin starts at the design advance rati and then wrks t bth extremes f the advance rati range f interest with a step size f 0.0. The Open_Water_Characteristics.m functin calls the Sme_Open_Water_Characteristics.m functin first fr the range f advance ratis less than the design value and then again fr the range abve the design value. After calculating the frces and efficiencies ver the entire range, the functin checks t see if the efficiency ever becmes negative. This pint is where the prpeller is wind-milling and n lnger prviding any useful thrust t the ship. The functin then returns the J S, K T, K Q, and efficiency values fr pints befre the wind-milling pint. 3.6 Validatin f Prpeller Perfrmance Characteristics at Off-Design Advance Ratis The same prpeller designed by OpenPrp in Sectin 3.4 was used t validate the prpeller perfrmance MATLAB cde described in Sectin 3.5. The result is shwn in Figure 6. In rder t determine the rbustness f the prcess described in Sectin 3.5, the cde was altered s that it wuld nt nly start at the design pint and wrk tward the endpints, but t als start at the lwest value in the advance rati range and wrk up t the highest and vice versa. In bth cases, the initial guess f α i (crrespnding t the design pint) was used. The results are shwn in Figure 7. There is n significant difference in the results f the three methds; therefre, the methd is nt very sensitive t the initial guess f α i. 34

35 K T, 0*K Q, η Advance Rati (J s ) Thrust Cefficient (K T ) 0*Trque Cefficient (K Q ) Efficiency (η) Figure 6: Perfrmance Curves f a Generic Prpeller 0.5 K T Advance Rati (J s ) 0*KQ Advance Rati (J s ) η Advance Rati (J s ) Slid Line = Start at Design J s, Square = Start at Lw J s, + = Start at High J s Figure 7: Rbustness Results fr the Initial Estimatin f α i 35

36 It is imprtant t nte that the general shape f the curves in Figure 6 is similar t the examples in Wud (). Hwever, there is n data fr this particular prpeller t validate that the values are accurate. Therefre, the numerical accuracy f the prgram is validated using the DTMB 49 prpeller. The DTMB 49 prpeller was chsen fr the validatin due t its relatively simple gemetry (it has n rake r skew) and the extensive amunt f experimental data available fr it. The gemetric characteristics f the DTMB 49 prpeller as reprted by Black (4) are shwn in Table 9. Diameter, D =.00 ft (.305m) Rtatin: Right Hand Number f Blades: 3 Hub Diameter Rati: 0.0 Skew θ s, Rake: Nne Design Advance Cefficient, J: Sectin Thickness Frm: NACA66 (DTMB Mdified) Sectin Meanline: NACA a=0.8 Design Thrust Cefficient, K T : 0.50 r/r c/d P/D θ s i T /D t M /c f M /c Table 9: Gemetry f the DTMB 49 Prpeller (4) This data frm Table 9 was entered int OpenPrp in the Single Prpeller Design GUI shwn in Figure 8. The prpeller prduced by OpenPrp with these inputs is shwn in Figure 0. The thickness frm f the DTMB 49 prpeller is the NACA 66 (DTMB mdified). OpenPrp des nt currently design prpellers with this thickness frm s the NACA 65A00 frm was used instead. Neither OpenPrp nr any f the MATLAB functins described abve use the thickness f the blade in any calculatins. Therefre, the difference in the thickness frms is nt 36

37 cnsidered a prblem fr the validatin f the perfrmance curves. The prpeller prduced by OpenPrp als has a different pitch ver diameter distributin than the actual DTMB 49 prpeller as shwn in Figure 9. The actual distributin was inputted int the Open_Water_Characteristics.m functin fr validating the results. Figure 8: OpenPrp Input Parameters fr the DTMB 49 Prpeller 37