fluid and structural modeling of cavitating propeller...

Fluid and Structural Modeling of Cavitating Propeller Flows

Yin Lu Young, Princeton University, [email protected] A. Kinnas,The University of Texas at Austin, [email protected]

ABSTRACT

A three-dimensional low-order boundary elementmethod (BEM) is used for the hydrodynamic anal-ysis of propeller flows. The present method is ableto predict the time-dependent cavity planforms onfully-submerged and partially submerged propellers.An overview of the hydrodynamic formulation is pre-sented, as well as validation studies with experimen-tal measurements and observations. To account forhydroelastic effects, the present BEM is coupled witha 3-D transient finite element method (FEM). Anoverview of the formulation and numerical implemen-tation for the hydroelastic coupling is presented alongwith preliminary results.

1 Introduction

The motivation of this research is to develop a reliableand robust tool to predict the dynamic performanceof general cavitating propeller flows, with particu-lar emphasis on supercavitating and surface-piercingpropellers. Supercavitating propellers are often be-lieved to be the most fuel efficient propulsive devicefor high-speed vessels. They tend to have smaller vol-ume change and produce bubbles that collapse down-stream of the blade trailing edge, which result in re-duced noise and blade surface erosion. A surface-piercing propeller is a special type of supercavitatingpropeller which operates at partially submerged con-ditions. Surface-piercing propellers can be more effi-cient than fully submerged supercavitating propellersdue to the reduction of appendage drag since most ofthe propeller assembly (e.g. shafts, struts, hub, etc.)is elevated above the water surface.

Since the pioneering works of [21, 24, 22], bound-

ary element methods (BEMs) have been widely usedfor the analysis of complex flows around propellers.Most recently, a 3-D perturbation potential-basedBEM developed by [27, 15] has been further ex-tended to predict simultaneous face and back cav-itation on conventional fully submerged propellers[68], supercavitating propellers [69, 73], and surface-piercing propellers [69, 74] in non-axisymmetric in-flow. For cases without blade vibration, the devel-oped BEM has been shown to predict forces and cav-itation patterns that compared well with experimen-tal measurements. However, hydroelastic effects be-come important for large supercavitating and surface-piercing propellers operating at low advance coeffi-cients. Since the cross section of these types of pro-pellers usually has a sharp leading edge and a thicktrailing edge, bending and/or torsional oscillationsmay occur [44, 5, 38, 39, 40]. For surface-piercing pro-pellers, the problem is exagerrated because 1) higherpossibility of resonance due to the cyclic loading andunloading of the blades associated with the blade’sentry to and exit from the free surface, and 2) thethin leading edge tends to absorb most of the impactpressure during the blade entry phase. Thus, bladestrength, fatigue, and resonant vibration issues mustbe considered during the design and analysis of su-percavitating and surface-piercing propellers.

Previous Theoretical Work - Hydrody-namic Analysis

The first theoretical design method for supercavitat-ing propellers was developed by [50], and followedby [53, 10, 3, 66]. However, these methods werebased on 2-D studies, and required many approxima-tions and empirical corrections. Recently, more rig-

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Fifth International Symposium on Cavitation (CAV2003)

Osaka, Japan, November 1-4, 2003

orous methods were developed by [23, 59, 55]. Nev-ertheless, these methods were based on the optimiza-tion of 2-D cavitating blade sections to yield minimaldrag for a given lift and cavitation number. A 3-Dvortex-lattice method was developed by [31] to pre-dict the steady performance of supercavitating pro-pellers. Their model assumed the pressure over theseparated zone to be constant and equal to the va-por pressure. A variable length separated zone modelusing a similar vortex-lattice method was presentedin [30]. However, all of the above mentioned liftingsurface methods cannot capture accurately the flowdetails at the blade leading and trailing edge due tothe breakdown of linear cavity theory.

The first known method for the analysis of surface-piercing propellers was developed by [65]. They ap-plied a blade element method based on 2-D hydrofoiltheory, but ignored the effect of adjacent blades, cav-ities, and wake vortex sheets [43]. Later, [42] applieda lifting line approach that included the effect of im-mersion, but the propeller was assumed to be lightlyloaded such that no natural ventilation of the pro-peller and its vortex wake occur. Furuya developed alifting-line approach that included the effect of pro-peller ventilation in [16, 17]. The blades were reducedto a series of lifting lines, and the method was com-bined with a 2-D water entry-and-exit theory devel-oped by [60, 61] to determine thrust and torque coef-ficients. In general, the predicted thrust coefficientswere within acceptable range compared to measuredvalues presented in [19], but significant discrepancieswere observed for the torque coefficients. In 1991,[57] extended the conventional propeller theory givenin [33] to predict steady forces on surface-piercingpropellers by assuming that the steady sectional liftcoefficient is linearly related to the increment bladesection angle of attack. Later, an unsteady liftingsurface method was developed for the analysis of 3-Dfully ventilated thin foils [64] and 3-D fully venti-lated partially submerged propellers [62, 63]. Simi-lar to [16, 17], the negative image method was usedand the flow was assumed to separate from both theleading edge and trailing edge of the blade, form-ing on the suction side a cavity that vents to the at-mosphere. Comparisons with experimental measure-ments by [19] and numerical predictions by [16, 17]

were within reasonable agreement for one propellerin a limited range, but substantial discrepancies wereobserved for another propeller. The 3-D vortex-lattice lifting surface method developed by [31, 30] forthe analysis of supercavitating propellers was also ex-tended to treat surface-piercing propellers. However,the method performed all the calculations assumingthe propeller to be fully submerged, and then mul-tiplied the resulting forces with the propeller sub-mergence ratio. Thus, only an estimate of the meanforces can be obtained while the complicated phe-nomena of blades’ entry to, and exit from, the wa-ter surface were completely ignored. A 2-D time-marching BEM using the negative image method wasdeveloped by [48] for the analysis of the flow fieldaround a fully ventilated partially submerged hydro-foil, but it only considered the hydrofoil’s entry to,but not exit from, the water surface.

Previous Theoretical Work - Hydroe-lastic Analysis

Traditionally, the modified cantilever beam theory isused to determine blade stresses. The theory wasdeveloped by D.W. Taylor [51] before 1920, whichassumed the blade to be a cantilever beam loaded bythrust and torque distributed linearly over the radius[49]. Later, modifications were made to include theeffects of rake, skew, and centrifugal force [36, 49, 1].However, the beam theory cannot accurately predictthe stress distributions for complex blade geometries(e.g. highly skewed propellers, supercavitating pro-pellers, etc) due to its simplified assumptions. Later,more sophisticated theoretical models based on themeasured or calculated mode shapes and resonancefrequencies of propeller blades in air were developedby [52, 7]. In 1981, [38, 39] developed a linearizedtheory for the analysis of flow around a chordwiseflexible supercavitating hydrofoil when clamped atthe trailing edge and when supported elastically, butthe method was limited to 2-D. Recently, finite el-ement methods coupled with hydrodynamic modelshave also been employed for the analysis of dynamicblade stresses. These include the works of [18] usingthin shell elements, [2] using thick shell elements, and[32] using 3-D isoparametric brick elements. How-

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ever, all of these methods were developed for analysisof fully submerged, non-cavitating propellers. Mostrecently, [13, 14] applied a 3-D FEM using a combina-tion of 2-D semi-loof shell elements and 3-D brick ele-ments for the hydroelastic analysis of surface-piercingpropellers. However, due to the use of assumed in-stead of evaluated hydrodynamic load models, themethod cannot provide accurate description of thedynamic stresses.

Objective

The objective of this work is to develop a consis-tent, 3-D, coupled BEM-FEM model to determinethe time-dependent cavitation patterns and hydroe-lastic response of general cavitating propellers.

Hydrodynamic Formulation

x = xs

yys

z

ωt

zs

(x,y,z)r

θB

ω

midchord partialcavity on back side

wake

supercavityon face side

hub

non-axisymmetriceffective inflow wake

supercavity on back side

hub

blade

Figure 1: Propeller subjected to a general inflowwake. The propeller fixed (x, y, z) and ship fixed(xs, ys, zs) coordinate systems are also shown. From[73].

Fundamental Assumptions

For the hydrodynamic analysis, the propeller is as-sumed to be a rigid solid body which rotates at a

constant angular velocity (ω) in an unbounded fluid.The inflow is assumed to be constant over the ax-ial extent of the propeller. It represents the effectivewake of the ship or cavitation tunnel. The effectivewake models the vorticity in the inflow in absence ofthe propeller (also called the nominal wake) and thevorticity due to the propeller. It is determined viathe coupling of a potential method with an unsteadyEuler solver (which models the rotational part of theflow) [9, 26, 8].

The problem is solved with respect to a coordinatesystem that rotates with the blades. The resultingflow is assumed to be incompressible and inviscid.To correct for viscous effects, the viscous force is in-cluded by applying a uniform friction drag coefficienton the wetted portions of the blade and hub.

Only vapor- or gas-filled sheet cavities on the pro-peller blades are modeled. The sheet cavities are as-sumed to be constant pressure surfaces that grow andcollapse with time. The cavities are allowed to growon both sides of the blade surface and the detachmentlocations are searched for numerically.

Problem Definition

Consider a cavitating propeller subject to a generalinflow wake, ~qw(xs, ys, zs), as shown in Fig. 1. Theinflow wake is expressed in terms of the absolute (shipfixed) system of coordinates (xs, ys, zs). The inflowvelocity, ~qin, with respect to the blade fixed coordi-nates (x, y, z), can be expressed as the sum of theinflow wake velocity, ~qw, and the propeller’s angularvelocity ~ω, at a given location ~x:

~qin(x, y, z, t) = ~qw(x, r, θB − ωt) + ~ω × ~x (1)

where r =√

y2 + z2, θB = arctan(z/y), and ~x =(x, y, z).

The inflow (~qw) is assumed to be the effective wake,which is defined as the total (actual) velocity minusthe propeller induced velocity by the potential flowtheory. Thus, the perturbation potential, φ(x, y, z, t),can be expressed as follows:

~q(x, y, z, t) = ~qin(x, y, z, t) +∇φ(x, y, z, t) (2)

where φ satisfies the Laplace’s equation in the fluiddomain (i.e. ∇2φ = 0).

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The BEM formulation for flow around a cavitat-ing propeller subjected to a non-axisymmetric inflowis given in [28, 73]. In brief, Green’s third identity,Eqn. 3, is solved with respect to the propeller fixedcoordinates (x, y, z):

2πφp(t) =∫

SBD

[φq(t)

∂G(p; q)∂nq(t)

−G(p; q)∂φq(t)∂nq(t)

]dS

+∫

SW (t)

∆φw(rq, θq, t)∂G(p; q)∂nq(t)

dS (3)

where SBD = SB(t)∪SC(t)∪SH(t). SB , SC , SH , andST represent the wetted blade, cavitating, and hubsurfaces. The subscript q corresponds to the vari-able point in the integration. G(p; q) = 1/R(p; q) isthe Green’s function with R(p; q) being the distancebetween points p and q. ~nq is the unit vector nor-mal to the integration surface, pointing into the fluiddomain. ∆φ is the potential jump across the wakesurface, SW .

Boundary Conditions

• The flow is tangent to the wetted portions of theblade and hub surfaces.

∂φ

∂n= −~qin(x, y, z, t) · ~n (4)

• The pressure inside or on the cavity surface isconstant and equal to the prescribed pressure(Pc), i.e. Pc = Pv (vapor pressure) for vapor-filled cavities, and Pc = Patm (atmospheric pres-sure) for ventilated cavities. It can be shownthat the dynamic boundary condition rendersthe following expression for ∂φ

∂s [28]:

∂φ

∂s= −~qin · ~s + Vv cos ψ + sin ψ× (5)√

n2D2σn + |~qw|2 + ω2r2 − 2gys − 2∂φ

∂t− V 2

v

where Vv ≡ ∂φ∂v + ~qin · ~v; ~s and ~v are the local

unit vectors in the chord-wise and span-wise di-rection, respectively. ψ is the angle between ~sand ~v. σn ≡ (Po − Pc)/(0.5ρn2D2) is the cav-itation number. ρ is the fluid density and r is

the distance from the axis of rotation. Po is thepressure far upstream on the shaft axis; g is theacceleration of gravity and ys is the ship fixed co-ordinate, shown in Fig. 1. n = ω/2π and D arethe propeller rotational frequency and diameter,respectively.

• The flow is tangent to the cavitating surfaces.As shown in [28], the kinematic boundary con-dition renders the following partial differentialequation for the cavity thickness, h, which is de-fined normal to the blade surface:

∂h

∂s[Vs − cos ψVv] +

∂h

∂v[Vv − cos ψVs]

= sin2ψ

(Vn − ∂h

∂t

)(6)

where Vs ≡ ∂φ∂s + ~qin · ~s and Vn ≡ ∂φ

∂n + ~qin · ~nare the tangential and normal component of thetotal velocity vector, respectively.

• The velocity at the blade trailing edge is finite.

• For fully submerged propellers, the cavity thick-ness at the cavity trailing edge must be equalto zero at all locations. For partially submergedpropellers, the cavity vents to the atmosphereand remains open at the free surface.

• For partially submerged propellers, the dynamicboundary condition requires the pressure every-where on the free surface to be constant andequal to the atmospheric pressure, and the kine-matic boundary condition requires the normalvelocities of the fluid and of the free surface tobe equal. In the present method, the linearizedform of the combined free surface kinematic anddynamic boundary conditions is applied:

∂2φ

∂t2(x, y, z, t) + g

∂φ

∂ys(x, y, z, t) = 0 (7)

at ys = −R + h

where h and R are the blade tip immersion andblade radius, respectively. ys is the vertical ship-fixed coordinate, also defined in Fig. 1.

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Assuming that the infinite Froude number con-dition (i.e. Fr = V/

√gD → ∞) applies, Eqn. 7

reduces to:

φ(x, y, z, t) = 0 at ys = −R + h (8)

The above equation implies that the negative im-age method can be used to account for the effectof the free surface. Consequently, only verticalmotions are allowed on the free surface.

Face And/Or Back Cavitation WithSearched Detachment

The present BEM is able to search for cavity detach-ments on both sides of the blade surface in steady orunsteady flow conditions via an iterative algorithm.First, the initial detachment lines at each time step(or blade angle) are obtained based on the fully wet-ted pressure distributions. The detachment lines arethen adjusted iteratively at every revolution until thefollowing Villat-Brillouin [6, 56] smooth detachmentcriterion is satisfied:

1. The cavity has non-negative thickness at its lead-ing edge, and

2. The pressure on the wetted portion of the bladeupstream of the cavity should be greater thanthe prescribed cavity pressure, Pc.

Convergence and validation studies of the cavitysearch algorithm are presented in [28, 37, 68, 67].

Supercavitating Propellers

The current BEM is also able to predict the perfor-mance of supercavitating propellers by modeling theseparated region behind the thick blade trailing edge.In the present method, the pressure within the sepa-rated region(also called the base pressure) is assumedto be uniform [47, 54] and equal to the prescribed cav-ity pressure [31]. The logic behind these assumptionare:

1. The base pressure should equal to the cavitypressure in the case of supercavitation.

2. The pressure change along the blade trailingedge should be smooth when one part of theblade is wetted or partially cavitating, and an-other is supercavitating.

3. Most supercavitating propellers operate in su-percavitating conditions.

Hence, the size and extent of the separated region canbe solved like an additional cavity bubble and are al-lowed to change with time. For supercavitating pro-pellers, the pressure acting on the thick blade trailingedge is included by multiplying the separated regionpressure acting normal to the blade trailing edge withthe corresponding trailing edge area. Details of thenumerical algorithm, as well as convergence and val-idation studies, are presented in [73, 71].

Surface-Piercing Propellers

For surface-piercing propellers, the vortical interac-tion between the propeller and the inflow is tem-porarily ignored (i.e. the effective wake is assumedto be the same as the nominal wake). To account forthe vortical interaction, the current BEM can be cou-pled with an unsteady Euler or Reynolds AveragedNavier-Stokes (RANS) solver which includes a freesurface model to determine the true effective wake.

Since the air loading of the propeller is small com-pared to the hydrodynamic load, Green’s formula(Eqn. 3) is only solved for the total number of sub-merged blade and wake panels. The values of φ and∂φ∂n are set equal to zero on the blade and wake panelsthat are above the free surface. The solution algo-rithm for partially submerged propellers is similar tothat for fully submerged supercavitating propellers,which are given in detail in [70, 67, 71, 74].

The detachment locations of the ventilated cav-ities are searched for iteratively by employing thesmooth detachment condition explained above. How-ever, during the exit phase (i.e. when part of theblade is departing the free surface), the ventilatedcavities are required to detach at or aft of the inter-section between the blade section and the free surface.In the current implementation, the ventilated cavitieson the pressure side of the blade are always assumedto detach from the blade trailing edge. However, it

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is possible to use the current algorithm to search forventilated cavity detachment locations on the pres-sure side of the blade. In addition, the current algo-rithm can also be modified to treat surface-piercingpropellers operating in base-cavitating conditions byadjusting the value of the cavity pressure in the dy-namic boundary condition.

Hydroelastic Formulation

Fundamental Assumptions

For the hydroelastic analysis, the propeller is mod-eled as a deformable solid body rotating in non-uniform wake. The fluid-structure interaction is mod-eled by coupling the current 3-D BEM with a 3-D transient FEM developed by Professor Prevost ofPrinceton University [46] over the lat twenty years.The field decompositions follows the work of [58, 32],but additional simplifications are made and the prob-lem is solved in the time domain using a Newmarktime integration scheme similar to that presented in[45]. Details of the formulation is presented in [72],and are summarized below.

Using linear decomposition, the blade position vec-tor ~χ can be expressed as:

~χ = ~x + ~δ(~r, t) (9)

where ~x and ~δ denote the position vector associatedwith rigid blade motion and vibratory blade motion,respectively.

Similar to the hydrodynamic formulation for rigidblades, the perturbation flow field is assumed to beincompressible, inviscid, and irrotational. Thus, theperturbation velocity can be represented as the gradi-ent of the perturbation potential Φ, where ∇2Φ = 0.Assuming linearity, Φ can also be decomposed intotwo parts:

Φ(~χ, t) = φ(~x, t) + ϕ(~χ, t) (10)

where φ(~x, t) denotes the potential due to rigid bladesrotating in non-uniform wake, and ϕ(~χ, t) denotes thepotential due to the vibrating blades in uniform wake.

Applying Taylor’s expansion and ignoring higherorder terms, the perturbation velocity can be simpli-fied as follows [72]:

∇Φ = ∇φ +∇ϕ (11)

Accordingly, the total velocity ~v can be written as:

~v(~χ, t) = ~q(~x, t) +∇ϕ(~χ, t) (12)

where ~q(~x, t) = ~qin(~x, t) + ∇φ(~x, t) (i.e. Eqn. 2) isthe fluid velocity due to rigid blades rotating in non-uniform wake; ∇ϕ(~χ, t) is the fluid velocity due tovibrating blades in uniform wake.

Grouping all the source terms on the right-hand-side, Green’s third identity yields the following equa-tion for ϕ [72]:

{ϕ} = [Cv]{

∂ϕ

∂n

}(13)

where [Cv] is the product of the dipole and sourceinfluence coefficient matrices.

Ignoring the higher order terms, the perturbationvelocity normal to the vibrating blade surface, ∂ϕ

∂n ,can be related to the normal component of the solidbody velocity at the element centroids [72]:

{∂ϕ

∂n

}=

{∂~δ

∂t· ~n

}(14)

Defining [T ] as the transformation matrix whichrelates the normal velocities at the element centroidsto the element nodal velocities:

{∂~δ

∂t· ~n} = [T ]{u} (15)

The following expression for ϕ can be obtained:

{ϕ} = [Cv][T ]{u} (16)

The unknown nodal velocities of the solid blade el-ements are determined by solving the dynamic equi-librium equation of motion:

[M ]{u}+ [B]{u}+ [K]{u} = {F}+ {f} (17)

where [M], [B], and [K] are the structural mass,damping, and stiffness matrices, respectively. {u},

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{u}, and {u} are the nodal acceleration, velocity, anddisplacement vectors, respectively. {F} is the nodalforce vector due to rigid blades rotating in nonuni-form inflow, and {f} is the nodal force vector due toblade vibration:

{F} = −∫

[N ]T {∆P}dS (18)

{f} = −∫

[N ]T {∆Pvib}dS (19)

where [N ] is the shape function for transforming thesurface tractions (pressure acting normal to the ele-ments) to consistent nodal forces. ∆P is the pres-sure distribution if the blades were perfectly rigid,and ∆Pvib is the pressure distribution due to bladevibration:

∆P = ρ

[12|~qw|2 +

12ω2r2 − gY − ∂φ

∂t− 1

2|~q|2

](20)

∆Pvib = ρ

[−∂ϕ

∂t− ~q · ∇ϕ

](21)

Since {∆P} is the pressure vector due to the non-vibrating blades, it can be computed separately viaEqn. 20 using the procedure explained in the ”Hydro-dynamic Formulation” Section. On the other hand,{∆Pv} is the pressure vector due to blade vibration,which depends on the unknown structural displace-ments. Following Eqn. 16, and recognizing that [Cv]and [T ] are only functions of the undeformed bladegeometry, the partial time derivative of ϕ can be com-puted as follows:

{∂ϕ

∂t

}= [Cv][T ]{u} (22)

Thus, Eqn. 21 can be rewritten as follows:

{∆Pv} = −ρ[Cv][T ]{u} − ρ[QD][Cv][T ]{u} (23)

where [QD] is the matrix operator representing ~q ·∇.In the FEM model, the Newmark update formulas

[41] are used for the time integration, and a predictor-multicorrector scheme is used to solve the coupledequations of motion, Eqn. 17. Details of the iterativesolution procedure are given in [72].

Validation Studies - Hydrody-namic Analysis

Extensive numerical and experimental validationstudies for the hydrodynamic analysis of cavitat-ing flows around fully submerged and partially sub-merged propellers are given in [68, 69, 73, 71, 74,29]. As examples, comparisons between experimen-tal measurements and numerical predictions of thehydrodynamic performance of a highly skewed fullysubmerged propeller (Propeller 4383), a supercavitat-ing fully submerged propeller (Propeller 3768), andtwo surface-piercing propellers (Propeller 4407 andM841B) are shown in this section.

Propeller 4383

To validate the treatment of fully submerged pro-pellers, numerical predictions are compared with ex-perimental measurements for propeller 4383, whichis a 5-bladed fully submerged propeller with a highskew angle of 72o. The propeller geometry is givenin [4] and [11]. Cavitation tests for propeller 4383were conducted in a 24-in cavitation tunnel at NavalShip Research and Development Center (NSRDC)[4]. Comparisons of predicted and measured thrust(KT = T

ρn2D4 ) and torque (KQ = Qρn2D5 ) coeffi-

cients as a function of advance ratio (JA = VA

nD )and cavitation number (σv = (Po − Pv)/(0.5ρV 2

A))are shown in Fig. 2 along with the predicted cavi-tation patterns for σv = 3.0. The symbol VA de-notes the advance speed of the propeller in open wa-ter. It should be noted that near the design con-dition (JA = VA/nD = 0.889), bubble cavitationwas observed in the experiment. However, the cur-rent method treats the cavity on the propeller bladesstrictly as sheet cavitation. In order to simulate theeffect of the bubble cavities, the authors incorporateda semi-empirical adjustment that alters the cavity de-tachment algorithm by requiring the height of thecavity near its leading edge to be greater than orequal to a prescribed tolerance. The purpose of thisadjustment is to avoid very thin sheet cavitation, andto allow for the possibility of pressure being slightlyless than the vapor pressure immediately in front of

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the cavity due to viscous flow and surface tension ef-fects.

σv=3.0, JA=0.5 σv=3.0, JA=0.75σv=3.0, JA=0.6

JA

KT,1

0KQ

0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

0.2

0.4

0.6

0.8Experiment (wetted)Experiment (cavitating)BEM

σv=3.0

σv=5.0

σv=5.0

σv=1.0

σv=3.0

σv=1.0

Propeller 4383, skew=72o

KT

10KQ

Figure 2: Comparison of the predicted and measuredcavitating performance for Propeller 4383 (72o skew).

Propeller 3768

To validate the treatment of supercavitating pro-pellers, the predicted force coefficients are comparedwith experimental measurements presented in [20] forpropeller 3768, which is shown in Fig. 3. Compar-isons of the predicted and measured thrust (KT ),torque (KQ), and efficiency (η) for σv = 0.6 and 1.68are shown in Figs. 3 and 4, respectively. As shownin the figures, the extent and volume of the cavitydecreased with increasing advance coefficient JA. Inaddition, the cavity changed from leading edge su-percavity on the back side (JA = 0.5) to a midchordsupercavity on the back side and a leading edge par-tial cavity on the face side (JA = 0.9). In general, thenumerical predictions compared well with experimen-tal measurements, and the predicted cavity shapesalso seemed reasonable. However, discrepancies be-tween numerical predictions and experimental mea-surements can be observed for higher advance coef-ficients because the blades start to become mostlywetted. In the partially cavitating and fully wettedflow regime, the pressure at the blade trailing edge

is probably different from the vapor pressure, thusviolating the assumptions in the present algorithm.

JA=0.5 JA=0.7

X Y

Z

JA

KT,1

0KQ,η

0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ExperimentBEM

σv=0.6

KT

KQ

η

JA=0.9

facecavity

backcavity

Figure 3: Top: Comparison of the predicted andversus measured cavitating performance for propeller3768. Bottom: Corresponding predicted cavity plan-forms. σv = 0.6.

Propeller 4407

To validate the treatment of surface-piercing pro-pellers, numerical predictions are compared with ex-perimental measurements for propeller model 4407.The propeller has 8 blades, tip skew angle of 45 de-grees, and zero rake. The BEM representation of pro-peller 4407 is shown in Fig. 5. The test was conductedat the Naval Ship Research and Development Cen-ter (NSRDC) with a specially designed dynamometerand instrumentation system. Details of the test setupand conditions are given in [12]. Comparisons of thepredicted and measured dynamic axial force Fx perblade as a function of the rotation angle measuredfrom the instant of blade entry is shown in Fig. 5.In [12], the report mentioned that the blade entryangle varied as much as 20 degrees during the testrun due to increase in free surface elevation as a re-sult of jet sprays and cavity displacement effects. Toaccount for the rise in free surface, the blade tip im-

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X Y

Z

JA=0.5 JA=0.7

JA

KT,1

0KQ,η

0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ExperimentBEM

σv=1.68

KT

10KQ

η

JA=0.9

facecavity

backcavity

Figure 4: Top: Comparison of the predicted andversus measured cavitating performance for propeller3768. Bottom: Corresponding predicted cavity plan-forms. σv = 1.68.

mersion angle (h/D) was increased by 9% (approxi-mately equates to a change of 10 degrees in the bladeentry angle) in all the numerical calculations. In gen-eral, the numerical predictions compared reasonablywell with experimental measurements.

Propeller M841B

To further validate the treatment of surface-piercingpropellers, numerical predictions for propeller model841-B are compared with experimental measure-ments collected by [43] at the free-surface cavita-tion tunnel at KaMeWa of Sweden. The propellerhas 4 blades with non-zero rake and skew. The ge-ometry of the propeller and the axial velocity dis-tribution in the propeller plane is shown in Fig. 6.Details of the experiments are given in [43]. Thepredicted and measured dynamic axial force coeffi-cients per blade, KFx = Fx/ρn2D4, as a function ofthe blade angle defined from the top vertical posi-tion is shown in Fig. 7. It should be noted that thestatic blade tip immersion ratio, h/D = 0.33, is usedin the numerical calculations. In general, the nu-merical predictions compared well with experimental

angle measured from the instant of entry (degrees)A

xial

forc

epe

rbl

ade,

Fx

(lbs

)0 60 120 180 240

-10

-5

0

5

BEM: JA=0.69BEM: JA=1.01EXP: JA=0.69EXP: JA=1.01

Propeller 4407, h/D=30%

Figure 5: Comparison of the predicted and measuredaxial force per blade in blade-fixed coordinates as afunction of the rotation angle defined from the instantof blade entry. Propeller 4407. h/D = 0.30.

measurements, particularly at higher advance coef-ficients. However, discrepancies between numericalpredictions and experimental measurements can beobserved due to 1) increase in free surface elevation asa result of jet sprays and cavity displacement effects,and 2) resonant blade vibration (which is evident viathe ”humps”, amplified fluctuations superimposed onthe basic load, shown in the experimental measure-ments. Additional comparisons of the predicted andmeasured dynamic and time-average performance, aswell as ventilation patterns, for propeller M841B arepresented in [74].

It should be noted that blade ringing and signif-icant blade force fluctuations were observed duringexperiments for propeller 4407 and M841B [12, 43].The blade loading history for both propellers alsoindicated the development of very high stresses atthe instant of impact. Similar observations were alsomentioned in other experimental studies of surface-piercing propellers presented in [35, 13, 14]. Thisimplies that fatigue and blade strength are seriousconcerns for surface-piercing propellers.

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X

Y

Z

h/D=0.33

Vx

y/R

0 0.25 0.5 0.75 1

0

0.2

0.4

0.6

0.8

1

flat plate

free surface

velocity distribution in propeller plane

Figure 6: Geometry of propeller M841B and axialvelocity distribution at the propeller plane. From[74].

Validation Studies - Hydroelas-tic Analysis

Complete implementations of the hydroelastic cou-pling is still underway. To verify the accuracy ofthe current FEM in predicting the vibratory char-acteristics of propeller blades, validation studies areconducted for a series of twisted cantilever plates pre-sented in [34, 25]. The twisted cantilever plates areselected because there exists an extensive collectionof experimental [34] and theoretical [25] results dueto a join government/industry/university initiative.The test specimens were precision machined and care-fully tested at two separate laboratories [34, 25]. Thegeometry of six of the twisted plates are shown inFig. 8. A is the span length, B is the chord length,H is the thickness, and φ is the twist angle in de-grees. In the FEM model, 50 (spanwise) x 20 (chord-wise) x 3 (thickness) 8-noded solid brick elementswere used. Comparisons of the predicted and mea-sured frequency parameter, λ = ωA2/

√ρsH/D (ω is

the vibrational frequency and ρs is the solid density),for the first two modes (in air) are shown in Fig. 9.In general, the numerical predictions compared wellwith experimental measurements. However, addi-tional validation studies are needed for more realisticpropeller geometries in air and in water. Experimen-tal validations studies of the dynamic performance offully submerged and partially submerged propellerswith and without hydroelastic effects are also needed.

JA=0.8, Fr=6

blade angle

KF

x

0 90 180 270 360

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

KFx (BEM)KFx (EXP)

KFx

JA=1.2, Fr=6

blade angleK

Fx

0 90 180 270 360

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

KFx (BEM)KFx (EXP)

KFx

Figure 7: Comparison of the predicted and measuredaxial force coefficient per blade in blade-fixed coordi-nates as a function of the blade angle defined from thetop vertical position. Propeller M841B. h/D = 0.33.

2 Conclusions

The developed 3-D BEM is able to predict the time-dependent cavity planforms and resulting hydrody-namic forces on fully-submerged and partially sub-merged propellers. An overview of the formulationand numerical implementation for the hydroelasticcoupling is presented along with preliminary results.Current efforts include:

• Fully implement the hydroelastic coupling al-gorithms to study the effect of dynamic bladestresses and deformations.

• Systematically examine the behavior of the cou-pled model at higher modal frequencies.

• Perform additional experimental and numericalvalidation studies.

ACKNOWLEDGMENT

Support for this research was provided by Phase III ofthe “Consortium on Cavitation Performance of High

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Y

X

Z

φ=30o, A/B=1, B/H=20

Y

X

Z

φ=60o, A/B=1, B/H=20

Y

X

Z

φ=0o, A/B=1, B/H=20

Y

X

Z

φ=60o, A/B=1, B/H=5

Y

X

Z

φ=0o, A/B=1, B/H=5

Y

X

Z

φ=30o, A/B=1, B/H=5

Figure 8: FEM representation of the twisted can-tilever blades. A is the span length, B is the chordlength, H is the thickness, and φ is the twist angle indegrees.

Speed Propulsors” with the following members: ABVolvo Penta, American Bureau of Shipping, El PardoModel Basin, Hyundai Maritime Research Institute,Kamewa AB, Michigan Wheel Corporation, NavalSurface Warfare Center Carderock Division, Office ofNaval Research (Contract N000140110225), UlsteinPropeller AS, VA Tech Escher Wyss GMBH, andWartsila Propulsion.

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5 EXP: A/B=1, B/H=20EXP: A/B=1, B/H=5FEM: A/B=1, B/H=20FEM: A/B=1, B/H=5

φ

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15

20

25 EXP: A/B=1, B/H=20EXP: A/B=1, B/H=5FEM: A/B=1, B/H=20FEM: A/B=1, B/H=5

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