Ocean Engineering 154 (2018) 94–105

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Ocean Engineering

journal homepage: www.elsevier.com/locate/oceaneng

Hydrodynamic modeling and experimental validation of acycloidal propeller

Atanu Halder *, Carolyn Walther, Moble Benedict

Department of Aerospace Engineering, Texas A&M University, College Station, TX, 77843, USA

A R T I C L E I N F O

Keywords:UUV (unmanned underwater vehicle)Cycloidal propellerDynamic virtual camberInflowWake

* Corresponding author.E-mail addresses: halder.atanu@tamu.edu (A. Halder),

https://doi.org/10.1016/j.oceaneng.2017.12.069Received 13 June 2017; Received in revised form 7 Nove

0029-8018/© 2018 Elsevier Ltd. All rights reserved.

A B S T R A C T

In this paper, a lower-order unsteady hydrodynamic model of a cycloidal propeller along with in-house experi-ments to validate the model is presented. Towards this, the hydrodynamics of a cycloidal propeller is investigatedthoroughly and various underlying physical phenomena such as dynamic virtual camber, effects of near and shedwake, leading-edge vortices are rigorously modeled. It is shown that the chord-wise variation of incidence ve-locity angle on cycloidal propeller blade is manifested as dynamic virtual camber, which depends on curvilinearflow geometry, pitch angle, pitch rate and inflow distribution. By including all these effects together, a gener-alized expression of additional lift due to virtual camber effect is developed. To capture the effects of near wake, anonlinear lifting line model is incorporated. Rapid pitching of rotor blades produces unsteady phenomena such asstrong leading-edge vortices and shed wakes. Polhamus leading-edge-suction analogy is applied to model leading-edge vortex. To capture the effects of shed wake, a method based on Theodorsen's approach is developed. Amodified Double Multiple Streamtube (D-MS) model is used for modeling the complex inflow characteristics of acycloidal propeller. The present hydrodynamic model is validated with measured time-history of forces obtainedfrom in-house experiments at low Reynolds numbers.

1. Introduction

In past few decades, unmanned underwater vehicles (UUVs) havebecome popular among researchers and scientists due to its wide range ofapplications and future possibilities. UUVs have huge potential foroceanographic studies such as deep water and shallow water marineexploration, as well as military applications such as maritime intelli-gence/surveillance/reconnaissance (ISR), mine countermeasures, anti-submarine warfare, inspection/identification, communication, payloaddelivery, information operations and time critical strike (U.S. Navy,2004). Oil and gas industry is also becoming interested in application ofUUVs for pipeline inspection and surveillance, sea-floor mapping etc.Most of the common UUVs use conventional screw propellers which aremechanically simple and moderately efficient in steady cruise (Blidberg,2001; Griffiths et al., 2004; Collar et al., 1994; Bandyopadhyay 2005).However, a UUV using a single fixed conventional propeller would sufferfrom low agility and would be highly susceptible to disturbances, espe-cially during station-keeping. To overcome this barrier many UUVs aredesigned with multiple propellers in staggered arrangements to generatethrust in different directions to improve control authority. Althoughthese types of configurations can offer superior maneuverability, many of

carolyn.walther@tamu.edu (C. Walt

mber 2017; Accepted 29 December 2

these additional propellers/motors become redundant in forward mo-tion, which adds to the weight and reduces efficiency (Roper et al.,2011). Thrust production on conventional propellers relies on steadyhydrodynamic mechanisms, and it is challenging to keep a steady flowattached to the hydrofoil surface, especially at low Reynolds numbers.Therefore, a conventional propeller blade can only operate over a narrowrange of angle of angle of attack if the flow has to stay attached. Thisleads to small stall margins, and the slightest perturbation could causethe flow to separate. On the other hand, in nature, fish fins and tailoperate in a highly unsteady hydrodynamic regime with large variationsin angle of attack and are more robust to perturbations. For this reason,bio-inspired and biomimetic propulsion systems, that exploits unsteadyhydrodynamics just like natural swimmers, have been investigated forthe last two decades, dating back to the RoboTuna developed at MITduring the mid-90s (Anderson, 1996; Read, 1999; Haugsdal, 2000;Flores, 2003; Polidoro, 2003). Although many of these bio-inspired ro-botic UUVs could be more efficient, they used complex linkage systemspowered with multiple servo motors or hydraulic/pneumatic actuators,which resulted in enormous mechanical complexity and weight whencompared to conventional propellers (Kato, 1998; Triantafyllou et al.,2004; Phillips et al., 2010; Zhou and Low, 2012; Palmre et al., 2013; Licht

her), benedict@tamu.edu (M. Benedict).

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Notation

A Streamline Areac Chord LengthC Theodorsen's FunctionCd Drag coefficientCdle Drag coefficient with leading-edge vortexCdwole Drag coefficient without leading-edge vortexCl Lift coefficientClle Lift coefficient with leading-edge vortexClnc Non-circulatory lift-coefficientCl0 Additional lift coefficient due to virtual camberClquasy steady Quasi-steady lift coefficientClunsteady Unsteady lift coefficientClwole Lift coefficient without leading-edge vortexD DragFd Force on the blade along the direction of flow at

downstreamfd Force per unit area along the direction of flow at

downstreamFu Force on the blade along the direction of flow at upstreamfu Force per unit area along the direction of flow at upstreamK A constant that represents Lift-curve slope at small angle of

attackkf Reduced frequencyKv Non-dimensional coefficient due to extra vortex lift by

leading-edge-suctionL Lift_m Mass flow rateNb Number of blades in cycloidal propellerR Radius of cycloidal propellerR'ðxÞ Distance of local blade location (x) from center of cycloidal

propellerS Planform area of propeller blades Position of blade in propeller disc arcUT Component of resultant flow velocity along the direction of

bladeUP Component of resultant flow velocity along the

perpendicular direction of bladeVðxÞ Resultant flow speed at a particular chord-wise location (x)

on propeller blade

V!ðxÞ Resultant flow velocity at a particular chord-wise location

(x) on propeller blade

V!

bðxÞ Blade velocity at a particular chord-wise location (x) on

propeller bladeVd Downstream inflow velocityVd wake Downstream wake velocity

V!

iðxÞ Inflow velocity at a particular chord-wise location (x) onpropeller blade

viðxÞ Inflow speed at a particular chord-wise location (x) onpropeller blade

V!

pðxÞ Relative flow velocity at a particular chord-wise location (x)on propeller blade due to blade pitching

Vu Upstream inflow velocityVu wake Upstream wake velocityV∞ Free stream velocityX Chord-wise location from leading-edge along virtual chord

linex Chord-wise location from pitching axisx1 Position of leading-edge of virtual airfoil along actual

chord-lineY Vertical distance of virtually cambered airfoil from virtual

chord liney0 Span-wise location from mid spany1 Position of leading-edge of virtual airfoil normal to actual

chord-lineα Angle of attackαeff Effective angle of attackαi Induced angle of attackαx Angle of attack at a particular chord-wise location (x) on

propeller bladeβ Angle between inflow velocity and vertical axisΓ Circulationγd Inflow angle at downstream locationsγu Inflow angle at upstream locationsη Non-dimensional parameter to represent chord-wise

distanceηle Represents percentage of partial leading-edge-suctionθ Prescribed blade pitchρ Fluid densityσ SolidityΨ Azimuthal location of blade in a cycleϕ Angle between local tangent and resultant flow velocity

with respect to propeller bladeϕs Slope of virtual chord-line with respect to actual chord-lineΩ Rotational speed of rotor

Fig. 1. Cycloidal propeller based unmanned underwater vehicle (UUV).

A. Halder et al. Ocean Engineering 154 (2018) 94–105

et al., 2004; Triantafyllou et al., 2000). Some recent studies investigatedthe use of smart materials such as shape memory alloys to actuate themotion, thereby introducing other limitations such as bandwidth (Low,2011; Roper et al., 2011). In the present study, a cycloidal propulsionsystem based UUV (Fig. 1) is proposed. Cycloidal propellers exploit un-steady hydrodynamics for efficient thrust production while they alsooffer high maneuverability through instantaneous thrust vectoring.

As shown in Fig. 1, in the case of a cycloidal propeller, the blade spanis parallel to the axis of rotation of the propeller. As shown in Fig. 2, thepitch angle of each blade is cyclically altered in such way that the bladeexperiences positive geometric angles of attack at both the top (up-stream) and bottom (downstream) halves of the azimuth cycle resultingin a net thrust. The magnitude of the net thrust can be adjusted byvarying either the amplitude of blade pitch or the rotational speed; whiledirection of the net thrust vector can be altered by varying the phase ofcyclic blade pitch. Themain advantage of the cycloidal propulsion systemis that a complete 360� instantaneous thrust vectoring can be achieved by

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Fig. 2. Cycloidal propeller blade kinematics.

Fig. 3. Voith-Schneider propellers installed under the ship hull.

A. Halder et al. Ocean Engineering 154 (2018) 94–105

simply phase-shifting the cyclic pitch which could potentially improvethe agility and disturbance rejection capability of a UUV propelled usingthis concept. This could also be more power efficient and significantlyfaster than swiveling a conventional screw-propeller. There have beenmany recent studies (Jarugumilli et al., 2013a, 2013b; Benedict et al2013a, 2013c, 2014a, 2014b; Zachary et al., 2013; Shrestha et al., 2012,2014; Hrishikeshavan et al., 2015; Jarugumilli et al., 2014) on cycloidalrotor systems for small-scale aerial vehicle applications, which showedthat an optimized cycloidal rotor can be more efficient than conventionalrotors (Benedict, 2010; Benedict et al., 2011a, 2013b). The fundamentalreason behind this is the decrease in induced power requirement due touniform load distribution along blade span, which also reduces rotorspeed required to generate a certain thrust. Also, unsteady fluid dynamicmechanisms such as leading-edge vortices improve performance ofcycloidal propulsion system, as revealed by Particle Image Velocimetry(PIV) studies (Benedict et al., 2010a, 2010b). Furthermore, due to dy-namic pitching at high amplitudes and reduced frequencies, cycloidalpropeller blades can potentially exploit dynamic stall to generate veryhigh lift coefficients, even twice the static values (Walther et al., 2017).

For underwater applications, cycloidal propellers (commonly calledVoith Schneider Propellers or VSPs, Fig. 3) have been historically used ontug-boats, mine-sweepers and other applications which require highthrust at low speeds, which could be easily vectored for tight maneu-vering, station-keeping, etc (Haberman and Harley, 1961; Haberman andCaster, 1962; Sparenberg and De Graaf, 1969; Ficken and Dickerson,1969; James, 1971; Zhu, 1981). Notably there have been only a limitednumber of studies on cycloidal propellers for marine applications, espe-cially at high advance ratios (Dickerson and Dobay, 1970; Bose and Lai,1989). A detailed literature survey on the research on cycloidal marinepropellers is provided in Bartels and Jürgens (2006).

It is significant to note that not many systematic scientific studieshave been conducted to unravel the complex unsteady fluid dynamics ofcycloidal propellers in order to fully exploit its potential. In the presentliterature, there are very limited experimental data as well as analyticalmodels that can predict performance of a cycloidal propeller, especiallyfor underwater propulsion. Moreover, all the previous analytical modelswere only validated with time-averaged performancemeasurements. Thefirst attempt to develop an aerodynamic model for cycloidal rotor wascarried out by Wheatley in 1930s (Wheatley, 1933; Wheatley andWindler, 1935). The model predictions were compared withtime-averaged wind-tunnel measurements; however, it showed poor

96

correlation with experimental data. Recently, more fluid-dynamicmodels were developed by McNabb (2001), Kim et al. (2004) andHwang et al. (2005, 2006). However, these studies were performed onlarge scale cycloidal rotors operating at higher Reynolds numbers andshowed good correlation with time-averaged thrust and power mea-surements. At MAV scales, CFD studies were carried out by Iosilevskii andLevy (2006), which showed good correlation with experimental resultsand also exposed some of the complex underlying fluid dynamic phe-nomena. Recently an aeroelastic model of cycloidal rotor was developedby Benedict et al. (2011b), which utilized a simplified aerodynamicmodel and mainly focused on understanding effects of flexible bladedeflections on cycloidal rotor performance. Recently Voith Schneiderpropellers (VSP) have attracted significant attention from researchers. InEsmailian et al. (2014), CFD simulations have been carried out to predictVSP performance. Some studies have been carried out to developsimplified models of VSP (Jürgens et al., 2007).

However, there have been very limited efforts to develop a lowerorder high fidelity hydrodynamic model that could be utilized for designand optimization of cycloidal propeller. Existing models lack detailedmodeling of the physics of cycloidal propellers and therefore, thesemodels are not capable of accurately predicting instantaneous bladefluid-dynamic forces. None of the previous studies on cycloidal propellersprovide information on the unsteady flow phenomena including dynamicvirtual camber/incidence and dynamic stall on the blades. Therefore, thepresent paper focuses on the development of a detailed high-fidelityunsteady hydrodynamic model, which is capable of predicting not onlythe time averaged forces, but also the time history of forces on a cycloidalpropeller blade. At the same time, the present model is computationallyinexpensive in the sense that it can be used for design optimization ofcycloidal propellers. To achieve this goal, key fluid-dynamic phenomenaon a cycloidal propeller are investigated, and rigorous modeling of vir-tual camber, effects of near and shed wakes, leading-edge vortices arecarried out. Moreover, the inflow model is improved by modifyingexisting Double Multiple Streamtube (D-MS) model. All these improve-ments enabled the model predictions to correlate reasonably well withthe time-history of forces obtained from in-house experimental data. Thelong-term goal of this project is to develop a high-fidelity hydroelasticcycloidal propeller model to be used in a computational design frame-work for next generation of cycloidal propeller based UUVs.

2. Methodology

Large amplitude, high reduced frequency pitchingmotion of cycloidalpropeller blades results in highly complex unsteady hydrodynamicscharacterized by strong leading-edge vortex formation, nonlinear dy-namic virtual camber effects, shed wake and blade-wake interactions. Foraccurate prediction of this complex physical system, detailed CFDmodeling is necessary which is out of scope for the present application,where the objective is to develop a design code for a cycloidal propeller.Therefore, a high fidelity lower-order model has been developed whichcan predict instantaneous blade hydrodynamic forces and propeller

Fig. 4. Flow-chart of hydrodynamic model.

Fig. 5. Negative virtual camber effect due to curvilinear flow.

Fig. 6. Chord-wise variation of flow velocity due to inflow distribution.

A. Halder et al. Ocean Engineering 154 (2018) 94–105

performance (cycle-averaged thrust and power) with sufficient accuracy.A flow-chart of the proposed hydrodynamic model is shown in Fig. 4.

The first step in the proposed hydrodynamic model is to compute themagnitude and incident angle of the resultant flow velocity at each localchord-wise location of propeller blade. For this purpose, blade speed isobtained from prescribed kinematics and flow speed is obtained frominflow information. Chord-wise variation of incident flow angle is man-ifested as virtual camber effect which is discussed in detail in the sub-sequent sections. Once sectional angle of attack is obtained, sectionalquasi-steady fluid-dynamic forces are computed using thin-airfoil the-ory. In the next step, several models are developed to incorporate effectsof near and shed wakes and leading-edge vortices. Using these models,unsteady fluid-dynamic forces on propeller blades are obtained. Aniterative procedure is used to calculate inflow and circulation, which arecomputed from the obtained blade forces and iterated until both circu-lation and inflows are converged.

Fig. 7. Positive virtual camber effect due to pitch rate.

2.1. Dynamic virtual camber

The cycloidal propeller blades experience a unique phenomenonknown as virtual camber. The virtual camber effect occurs due to thechord-wise variation of the incident velocity angle (or angle of attack) onthe hydrofoil. This effect is very predominant in cycloidal propellersbecause the flow over a cycloidal propeller blade is characterized by apitching hydrofoil in a curvilinear flow in the presence of inflow thatvaries along with azimuth. A hydrofoil in a curvilinear flow experiencesdifferent flow velocity magnitude and direction along the chord due togeometry and the curvilinear nature of the flow; this manifests as aneffective camber and incidence. Fig. 5 shows how curvilinear flow ge-ometry creates a negative virtual camber effect for a blade at 0� pitchangle. Therefore, a symmetric blade immersed in a curvilinear flow willbehave like a cambered blade in a rectilinear flow as shown in Fig. 5. Thisphenomenon is more significant for cycloidal propellers with a largechord-to-radius ratio (c/R).

Flow over a cycloidal propeller is not purely curvilinear when it isproducing thrust because of the induced inflow velocity, which also ef-fects virtual camber and incidence (Fig. 6).

Additionally, the blade pitch angle and pitch rate also affects thechord-wise velocity distribution and therefore, virtual camber. Fig. 7shows pitch rate causing chord-wise variation of flow incidence anglewhich is manifested as a positive virtual camber effect. An opposite pitchrate (nose-down pitch) would cause a negative virtual camber effect.

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Since blade pitch and pitch rate vary with azimuthal location, cor-responding virtual camber also changes with azimuth making it dynamicvirtual camber. To consider all these effects together, a generalizedexpression has been derived to represent the variation of local angle ofattack (αx) along blade chord.

All the velocity components of fluid relative to propeller blade isshown in Fig. 8. All the velocity components are computed along

tangential (bi) and normal (bj) to propeller hydrofoil chord. Blade velocity

(V!

b) at chord-wise location x from pitching axis, obtained from kine-matics of propeller can be expressed as Eq. (1).

V!

bðxÞ ¼ ΩR'cosðθ � ΔψÞbi �ΩR'sinðθ � ΔψÞbj (1)

In above equation, θ is geometric pitch angle of cycloidal propellerblade, ψ is azimuth location of blade in a cycle, Ω is rotational speed ofrotor, R'ðxÞ is distance of local blade location (x) from center of cycloidalpropeller. Using geometrical relations, blade velocity can be rewritten asEq. (2).

V!

bðxÞ ¼ ΩR cos θ bi �ΩðR sin θ � xÞbj (2)

In the above equation, R is radius of cycloidal propeller. Flow velocity

Fig. 8. Velocity components at a local chord location on cycloidal propel-ler blade.

Fig. 9. Co-ordinate transformation from physical airfoil co-ordinate to virtual

A. Halder et al. Ocean Engineering 154 (2018) 94–105

(V!

p) relative to propeller blade due to blade pitching is given as

V!

pðxÞ ¼ x _θbj (3)

The inflow speed is represented as viðxÞ acting on propeller blade at

an angle βðxÞ w.r.t vertical axis. Thus, inflow velocity (V!

i) can beexpressed as

V!

iðxÞ ¼ ��vi sinðβ þ θÞbi þ vi cosðβ þ θÞbj� (4)

Considering all the above effects, the net relative flow velocity (V!)

with respect to propeller blade is calculated (Eq. (5)).

V!ðxÞ ¼ V

!pðxÞ þ V

!iðxÞ � V

!bðxÞ (5)

Substituting, Eqs. (2)–(4) into Eq. (5), the net flow velocity can beexpressed as.

V!ðxÞ ¼ �UTbi � UPbj (6)

where,

UT ¼ ΩR cos θ þ vi sinðβ þ θÞ (7)

UP ¼ x�Ω� _θ

�þ vi cosðβ þ θÞ �ΩR sin θ (8)

This information all together gives net flow speed ( VðxÞ) and incidentangle of attack (αxðxÞ) at each local chord-wise location (Eqs. (9) and(10))

VðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2

T þ U2P

q(9)

αxðxÞ ¼ �tan�1

x�Ω� _θ

�þ vi cosðβ � θÞ �ΩR sin θ

ΩR cos θ þ vi sinðβ þ θÞ

!(10)

In the above expression, vi is assumed to be zero for first iteration andupdated using inflow model (modified D-MS model, discussed later) for

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subsequent iterations. For a cambered airfoil in straight flow, the localangle of attack varies along chord according to following equation (Eq.(11)).

αxðxÞ ¼ θ � dydx

ðxÞ (11)

By comparing, Eq. (10) and Eq. (11), expression for virtual camberline of cycloidal propeller blade is obtained (Eq. (12)).

dydx

¼ θ þ tan�1

x�Ω� _θ

�þ vi cosðβ � θÞ �ΩR sin θ

ΩR cos θ þ vi sinðβ þ θÞ

!(12)

Modified thin airfoil theory has been used to calculate quasi-steadylift from the chord-wise variation of incidence. For this purpose, airfoilco-ordinates are transformed along chord line. Fig. 9 shows co-ordinatetransformation from actual chord-line (x, y) to virtual chord-line (X, Y)and Equations (13) and (14) provide expressions for this co-ordinatetransformation.

X ¼ ðy� y1Þsin ϕs þ ðx� x1Þcos ϕs (13)

Y ¼ ðy� y1Þcos ϕs � ðx� x1Þsin ϕs (14)

Since, the rotor blades in a cycloidal propeller goes through highangles of attack, small angle of attack assumption is relaxed while usingthin airfoil theory and a more general expression (Eq. (15)) is used.

Clquasi steady ¼ K sin αþ Cl0 (15)

In the above equation, K is a constant that represents Lift-curve slopeat small angle of attack, Cl0 is additional lift due to the effects of virtualcamber. Cl0 is a complex function of incidence angle given by Eq. (16). Inthis equation, dY

dX is obtained from Eqs. (12)–(14). dYdX represents slope of

virtual camber line in transformed co-ordinate X-Y. Using numericalintegration procedure, Cl0 is computed. In Eq. (16), non-dimensionalparameter η is used ( x ¼ c

2 ð1� cos ηÞ ) for integration purposes.

Cl0 ¼ 2 cos α∫ π0

dYdX

ðcos η� 1Þdη (16)

2.2. Nonlinear lifting line model

A nonlinear lifting line theory is implemented numerically to incor-porate the effects of near wake and tip vortices (finite-span effect). In thiscase, a linear geometry of trailing vortex lines is assumed. For this pur-pose, the wing is divided into number of span-wise elements. Nodal pointis set to center of each element. Induced angle of attack at each nodalpoint is obtained using following formula (Eq. (17)).

αiðy0Þ ¼ 14πV∞

∫ b=2�b=2

dΓ=dyy0 � y

dy (17)

Above integral is computed numerically using Gauss-quadratureintegration rule (11 gauss quadrature points are used for this purpose).From this, effective angle of attack at each nodal point is obtained usingαeff ðy0Þ ¼ αðy0Þ � αiðy0Þ . Effective angle of attack is used to calculatequasi-steady lift using Eq. (15). Circulation is computed at the center ofeach element from calculated lift L ¼ ρV∞Γ. For the first iteration,

cambered airfoil co-ordinate.

A. Halder et al. Ocean Engineering 154 (2018) 94–105

suitable values of circulation (based on elliptical lift distribution) areassumed and circulation is updated from lift calculated at each iterationuntil convergence is obtained. At each iteration, circulation is updatedusing Γnew ¼ Γold þ DðΓnew � ΓoldÞ, where D is an update parameter andsuitably chosen for proper convergence (0.05 in present simulations).

2.3. Unsteady effects

Rapid pitching of rotor blade produces strong unsteady phenomenasuch as leading-edge vortex and shade wake. To capture effect of shedwake, an unsteady model based on Theodorsen's approach has beendeveloped. Theodorsen's function has been originally developed tocalculate circulatory lift produced by sinusoidal variation of angle ofattack and pitch angle. Due to inclusion of virtual camber and itsnonlinear behavior, the oscillatory part of the lift of a cycloidal propellerblade contains not just the first harmonic, rather it shows high frequencycomponents with varying phase delays. To handle this, an automatedfunction has been developed that would perform Fourier series decom-position of oscillatory part of lift (due to shed wake) and compute cor-responding frequency and Theodorsen's function to generate unsteadylift. Fourier decomposition of quasi-steady lift coefficient can beexpressed as following (Eq. (18)).

Clquasi steady ¼ a0=2þXni¼1

an cos iΩt þ bn sin iΩt (18)

First harmonic of quasi-steady lift coefficient Clquasi steadyis Ω and cor-

responding reduced frequency is kf ¼ Ωc=2V . Theodorsen's functioncorresponding to higher harmonics are computed to calculate unsteadylift coefficient (Eq. (19)).

Clunsteady ¼ Clnc þ a0=2þXni¼1

ðan cos iΩt þ bn sin iΩtÞC�ikf � (19)

In the above equation, Clnc is non-circulatory lift co-efficient andCðikf Þ ¼ Fðikf Þ þ jGðikf Þ is Theodorsen's function corresponding toreduced frequency ki ¼ ikf , where i varies from 1 to n.

2.4. Leading-edge vortex

High amplitude pitching of rotor blade creates strong leading-edgevortex on cycloidal propeller, as shown by some of the on-going in-house experimental studies (Walther et al., 2017). Fig. 10 shows PIVmeasured flow field showing large leading-edge vortex formation.Leading-edge vortex delays flow separation on rotor blade whichsignificantly improves lift performance of rotor at high pitch anglesenabling rotor to operate even at �45� pitch amplitudes. Polhamusleading-edge-suction analogy (Polhamus, 1966; 1968) is applied to

Fig. 10. Formation of large leading-edge vortex from PIV measurements atpitch amplitude¼�45� and 90� azimuth.

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model leading-edge vortex. Additional lift (vortex lift) due toleading-edge vortex and additional drag due to leading-edge separationare modeled using this suction analogy.

Clle ¼ Clwole

�cos2 αþ ηle sin

2 α�þ Kv cos α sin2 α (20)

Cdle ¼ Clwole sin α cos αð1� ηleÞ þ Kv sin3 α (21)

In the above equation, Clle ; Cdle are lift and drag coefficient withleading-edge-suction and Clwole ; Cdwole are lift and drag coefficient withoutleading-edge-suction. Kv is non-dimensional coefficient related to addi-tional vortex lift generated by leading-edge vortex. ηle represents per-centage of leading-edge-suction. ηle ¼ 1 means no leading-edge-suctionand ηle ¼ 0 means complete leading-edge-suction.

2.5. Force computation

Once lift and drag coefficients, Cl and Cd, are computed, lift and dragforce in blade co-ordinate system is obtained using lift, L ¼ 1

2 ρV2 SCl

and drag, D ¼ 12 ρV

2SCd. Forces in blade co-ordinate are transformedinto inertial co-ordinate using proper transformation matrices. Tocalculate time-averaged propeller thrust, blade force is multiplied withsolidity, σ ¼ Nbc/(2πR) and averaged over one revolution.

2.6. Inflow model

Previously in the analysis of cycloidal rotors and vertical axis windturbines, Single Streamtube inflow or Double Multiple Streamtube (D-MS) models have been mostly used (Benedict et al., 2011b). For SingleStreamtube model, the inflow is considered uniform along the azimuthand inflow at top and bottom halves of the blade is assumed to be same.Therefore, no blade interaction or wake effect is considered after the flowpasses through the upper-half. This assumption is not physical. On theother hand, for D-MS model, there is azimuthal variation in inflowmagnitude, however, inflow direction is assumed to be radial to bladepath in the upper half in the hover state, which is also not physicallyrealistic.

In the present hydrodynamic model (schematic shown in Fig. 11), theproposed inflow model (modified D-MS model) relaxes these assump-tions. In this model, it is assumed that various streamlines interact withthe rotor blade twice, upstream and downstream, with different inflow

Fig. 11. Schematic of inflow model.

A. Halder et al. Ocean Engineering 154 (2018) 94–105

velocity magnitude and direction. Two adjacent streamlines form astreamtube. Unlike D-MS model, this modified model calculates theinflow direction based on resultant force direction on the blade at thatparticular azimuthal location (Eq. (22)). The underlying reasoningbehind this approach is that at a local azimuthal location, whatever hy-drodynamic force the propeller blade experiences, it exerts same force tothe nearby fluid in the opposite direction. When the rotor blade startsrotating, it would accelerate nearby fluid opposite to the direction ofresultant force blade is experiencing (Fig. 12).

γu ¼ π � ϕ� tan�1

�DL

�(22)

In Eq. (22), γu is inflow angle at upstream locations, ϕ is the anglebetween local tangent and resultant flow velocity with respect to pro-peller blade (Fig. 12). Once the inflow direction is calculated based onabove approach, actuator surface theory and mass and momentum con-servation laws were applied to determine magnitude of inflow velocity(Eqs. (14) and (16)) at upstream (Vu) and downstream (Vd). The wakevelocity of completely expanded flow after interacting with upstreamblade is denoted as Vu wake and wake velocity after interacting withdownstream blade is denoted as Vd wake . Using actuator surface theory,inflow velocities can be expressed as Eq. (23) and Eq. (24).

VuðsÞ ¼ 12ðV∞ þ Vu wakeðsÞÞ (23)

VdðsÞ ¼ 12ðVwðsÞ þ Vu wakeðsÞÞ (24)

In the above equations, s denotes position in propeller disc arc andthus it is basically a function of azimuth. Force exerted by fluid on thepropeller blade at upstream (Fu) and downstream (Fd) can be expressed asEq. (25) and Eq. (26), respectively.

FuðsÞ ¼ _mðV∞ � Vu wakeðsÞÞ (25)

FdðsÞ ¼ _mðVu wakeðsÞ � VwðsÞÞ (26)

where, _m is the mass flow rate. Using continuity theory, above equationscan be rewritten as Eq. (27) and Eq. (28).

Fu ¼ ρVuA sin γuðV∞ � Vu wakeÞ (27)

Fd ¼ ρVdA sin γdðVu wake � VwÞ (28)

By comparing Eq. (23) and Eq. (27), upstream inflow velocity (Vu)

Fig. 12. Schematic of inflow model.

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and upstream wake velocity (Vu wake) can be obtained (Eq. (29) and Eq.(30)).

VuðsÞ ¼ V∞

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

∞

4� fuðsÞ2ρ sin γuðsÞ

s(29)

Vu wakeðsÞ ¼ V∞ � fuðsÞρVuðsÞsin γuðsÞ

(30)

In the above equations, f is force per unit area along the direction offlow. In the similar procedure, downstream inflow velocity (Vd) anddownstream wake velocity (Vd wake) can be obtained (Eq. (31) and Eq.(32)) by comparing Eq. (24) and Eq. (28).

VdðsÞ ¼ Vu wakeðsÞ2

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVu wakeðsÞ

4

2

� fuðsÞ2ρ sin γuðsÞ

s(31)

Vd wakeðsÞ ¼ Vu wakeðsÞ � fdðsÞρVdðsÞsin γdðsÞ

(32)

Once the inflow is updated using above Equations (29)–(32), hydro-dynamic forces are calculated based on new inflow and circulation. Thesesteps are repeated until both circulation and inflow are converged.

3. Model validation

Due to the dearth of experimental data in the present literature ontime-history of hydrodynamic forces on cycloidal propeller blade, in-house experiments were carried out and the present hydrodynamicmodel is validated with the results obtained from these experiments.

3.1. Experimental setup

For model validation, a test setup (Fig. 13) is developed where asingle bladed cycloidal propeller is tested inside a water tank at a Rey-nolds number of around 18,000. Forces and moments are measured usinga miniature 6-component force balance at the blade root. A 12-channelslip ring is used to transfer the signals from the force balance in therotating frame to the data acquisition equipment in the stationary frame.The blade pitching is controlled using an analog servo, which allows us toprescribe specific pitching kinematics for a test. The experiments are firstconducted in air which provide the inertial forces; then the same ex-periments are repeated in water. The pure hydrodynamic forces are ob-tained by subtracting the inertial forces from the force measurementsmade in water.

Fig. 13. Single-bladed cycloidal propeller test rig in water tank.

Fig. 14. Radial force coefficient as a function of azimuth (Experimentvs. Analysis).

A. Halder et al. Ocean Engineering 154 (2018) 94–105

3.2. Results

Dynamic pitching experiments are conducted with a single bladedcycloidal propeller at 40 rpm. Propeller blade has chord of 2 inches andspan of 12 inches and pitched at quarter chord location. Radius ofcycloidal propeller is 3.43 inches. Figs. 14 and 15, respectively, show thecomparison between measured forces and predictions from the presenthydrodynamic model. The figures show the variation of radial andtangential force coefficients as a function of blade azimuthal locationover a range of pitch amplitudes (�20� to�45�). The results show overallreasonable correlation between the present hydrodynamic model pre-diction and test data. It is also interesting to see both from the experimentand analysis that the blades are producing higher radial and tangentialforces at the lower half (ψ ¼ 180�–360�) than the upper half(ψ ¼ 0�–180�). The reason for this will be explained in the discussionsection.

4. Discussion

Once the hydrodynamic model is validated with in-house experi-mental data, it is used to investigate physics behind thrust production of acycloidal propeller. In the following subsections details of various phe-nomena are discussed in depth.

4.1. Cause and effect of dynamic virtual camber

As shown by Eq. (16), virtual camber manifests as additional lift onthe blade. Fig. 16a shows variation of additional lift coefficient (Clo ) dueto virtual camber along azimuthal location for a cycloidal propeller bladerotating at 40 rpm with 35� pitch amplitude and Fig. 16b and c showscorresponding prescribed pitch and pitch rate (measured values),respectively, as blade goes through various azimuthal locations. It showseffects of curvilinear flow, pitch, pitch-rate and inflow on virtual camberand therefore, on additional lift. From Fig. 16a it can be observed thatvirtual camber effect due to only curvilinear geometry (magenta line) isstatic in nature and it always causes negative virtual camber leading tonegative Clo . While pitch, pitch rate and inflow creates time-dependencyof virtual camber effect making it a dynamic virtual camber. Fig. 16areveals that pitch and especially pitch rate creates a very dominant andcharacteristic virtual camber effect unlike inflow distribution which ismore random in nature.

Fig. 16a shows that blade pitch decreases negative virtual camber andopposes the effects of curvilinear geometry. The effect of blade pitch ismore prominent near 90� and 270� azimuth since pitch angle reaches atits peak at those locations (Fig. 16b). It is also observed that pitch ratecreates positive virtual camber effect near 0� azimuth, which almostnullifies the effects of curvilinear geometry; while at 180�, it createsnegative virtual camber which together with curvilinear effect produceseven larger negative lift. For this reason, as seen from Fig. 14, the netradial force coefficient is near zero at 0� azimuth, while it is much belowzero at 180� azimuth, although pitch angle is near 0� at both azimuthlocations (Fig. 16b). Pitch rate effect on virtual camber is dominant at0� and 180� azimuth because pitch rate reaches its peak near these twolocations (Fig. 16c). Fig. 17 shows graphically how pitch rate is creatingopposite virtual camber effects at different azimuth locations.

This phenomenon is clearly observed in Fig. 18. Fig. 18 shows theactual chord-line of cycloidal propeller blade and virtual chord-line dueto virtual camber effect along different azimuth locations. It can beobserved again at 0� azimuth, virtual camber is minimum producingalmost negligible negative lift while at 180� azimuth it has huge negativevirtual camber producing large negative lift.

Moreover, viewing from inertial reference frame, it can be observedfrom Fig. 18 that there is significant negative virtual camber when theblade operates in the upper half, which causes flow to separate very early.While, in the entire lower half, it creates a positive virtual camber, whichdelays flow separation and blade can attain much higher lift at these

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locations. Moreover, the virtual incidence decreases the effective angle ofattack in the upper half and increases the angle of attack in the lower half.

Fig. 15. Tangential force coefficient as a function of azimuth (Experimentvs. Analysis).

Fig. 16. Effect of Virtual Camber due to various phenomena.

Fig. 17. Effect of Pitch rate on dynamic virtual camber at two extremeazimuthal locations (0� and 180�).

A. Halder et al. Ocean Engineering 154 (2018) 94–105

These are the reasons why the magnitude of maximum radial andtangential force coefficients in the downstream half (ψ ¼ 180�–360�) is

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significantly higher than that at upstream (ψ ¼ 0�–180�) (Figs. 14 and15).

4.2. Reason for time-averaged side forces

Another non-intuitive physical phenomenon observed during theoperation of cycloidal propeller is asymmetric or non-negative net sideforce even with a symmetric prescribed pitch without any phase offset(Ty force shown in Fig. 2). From the kinematics shown in Fig. 2,

Fig. 18. Virtual chord-line due to virtual camber effect along azimuth.

Fig. 20. Comparison instantaneous vertical and side forces along azimuth(Pitch Amplitude: 45�).

A. Halder et al. Ocean Engineering 154 (2018) 94–105

intuitively, one may expect all the side forces due to drag and lift tocancel each other giving only a net vertical force; however, in reality,experiments have shown the presence of a dominating side force (Figs. 19and 20). Fig. 19 shows experimental comparison of time-averaged forcesover a range of pitch amplitude. The results show existence of a non-zerotime-averaged side force (Ty), even though significantly smaller inmagnitude than the vertical force (Tz).

To investigate this phenomenon, instantaneous vertical and sideforces are plotted and compared in Fig. 20. Figs. 19 and 20, show overallreasonable correlation between results obtained from experiments andanalysis. Instantaneous vertical and side forces are obtained usingfollowing expressions.

Fz ¼ FR sin ψ � FT cos ψ (33)

FY ¼ �FR cos ψ � FT sin ψ (34)

In the above equations, FR is the radial force, FT is tangential force, FZis vertical force and FY is side force at azimuth location, ψ . This phe-nomenon can be explained by dynamic virtual camber effect due to pitchrate and flow curvature. It can be observed from Eq. (34) that side force

Fig. 19. Comparison time-averaged forces at different pitch amplitude.

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comes from two sources, radial and tangential forces. Radial force ismostly dominated by lift while tangential force is mostly dominated bydrag. The relation between lift - drag and radial - tangential force aregiven by equations (35) and (36), where α is angle of attack.

FR ¼ L cos αþ D sin α (35)

FT ¼ L sin α� D cos α (36)

Therefore, it should be possible to trace back the source of side forceto lift and drag on the blade. The validated hydrodynamic model is usedto investigate underlying reason behind non-zero time-averaged sideforce. In Fig. 21, side force due to lift and drag are plotted separately todistinguish their effects.

Fig. 21a shows the contribution of instantaneous blade lift to sideforce. Note that, in this case, the X-axis is plotted differently to separatethe left and right halves. It can be observed from Fig. 21a that if no virtualcamber is considered (blue line), then side force due to lift is perfectlysymmetric between left and right halves. Side force from left half ofcycloidal propeller cycle (ψ ¼ 90�–270�) cancels side force of right half(ψ ¼ 270�–450�/90�), as seen from Fig. 2. The red line in Fig. 21a showsthat once the virtual camber due to curvilinear flow geometry and pitchangle is incorporated, the side force is still symmetric between the leftand right halves producing almost zero net side force. This is becausevirtual camber due to flow curvature introduces asymmetry between tophalf (upstream) and bottom half (downstream) of the cycle; the symmetrybetween right and left halves remains intact. But once the virtual camberdue to pitch rate is introduced (black line in Fig. 21a), it causes signifi-cant asymmetry in side force between left and right halves causing non-zero net side force. This is because the propeller blade goes through pitchup motion during right half of cycle (ψ ¼ 270�–450�/90�) while it goesthrough pitch down motion during left side of cycle (ψ ¼ 90�–270�). Asexplained in the previous section, the pitch up motion causes positivevirtual camber effect while pitch down motion creates negative cambereffect. Due to this two opposite virtual camber effects in left and righthalves, the side force gets unbalanced. This phenomenon becomesevident from Fig. 18 where it is observed that cycloidal propeller motion

Fig. 21. Effect of virtual camber on asymmetry of side force (Pitch Ampli-tude: 45�).

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creates mild negative virtual camber on right side while it creates largenegative camber on left side.

From Fig. 2, one may expect the drag at the top half (upstream) to becanceled by the drag at the bottom half (downstream). Fig. 21b(contribution of instantaneous blade drag to side force) shows similarresults when no virtual camber is included (blue line is antisymmetricbetween upper and lower halves). However, once the virtual camber dueto flow curvature is included, the red line shows asymmetry in side forcedue to drag. It shows decrease in magnitude of side force in the upstreamhalf while increase in magnitude of side force in the downstream half.This is because flow curvature creates almost equal negative cambereverywhere. In the upstream half, geometric pitch is positive and thus,negative virtual camber decreases effective angle of attack which in turncauses decrease in magnitude of drag and corresponding side force.While in downstream half, the geometric pitch is negative and negativevirtual camber creates larger negative effective angle of attack (observedfrom rotating blade frame) which causes larger magnitude of drag andcorresponding side force. This phenomenon causes imbalance in sideforce due to drag.

5. Summary and conclusions

The primary goal of this study is to develop a lower order unsteadyhydrodynamic model of cycloidal propeller that can predict instanta-neous blade forces and propeller performance with sufficient accuracythat it could be utilized for design optimization of next generation ofUUVs. Towards this, several underlying physical phenomena behind theoperation of cycloidal propeller are thoroughly investigated. Dynamicvirtual camber, effects of shed and near wakes, leading-edge vortices andcomplex inflow characteristics are rigorously modeled. The developedhydrodynamic model is validated with measured time-history of fluid-dynamic forces obtained from in-house experimental data. The keyconclusions from the study are given below:

1. Chord-wise variation of incident velocity angle on cycloidal propellerblade is manifested as virtual camber/incidence effect. Virtual

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camber and incidence depend on curvilinear flow geometry, pitchangle, pitch-rate and inflow distribution. Considering all these effects,a generalized methodology is developed to model virtual camber.

2. Curvilinear geometry causes a static negative virtual camber at allazimuth locations while pitch, pitch rate and inflow distributioncause cyclic variation of virtual camber with blade azimuthal locationmaking it a dynamic virtual camber. A positive blade pitch-rate (nose-up pitch) creates positive virtual camber, while a negative pitch-rate(nose-down pitch) creates negative virtual camber.

3. Virtual camber caused by pitch rate creates asymmetry in side forcedue to blade lift between the right and the left halves. However, thevirtual camber resulting from flow curvature creates asymmetry inside force between upper and lower halves due to blade drag. Thesetwo types of asymmetries create net time averaged side force on acycloidal propeller even with zero phase offset.

4. Due to curvilinear flow, the cycloidal propeller blade experiencesreverse or negative virtual camber in the upper half (upstream half) ofits circular trajectory along with decrease in effective angle of attackdue to negative virtual incidence. However, the blade experiencespositive virtual camber (observed from inertial frame) and increase ineffective angle of attack (due to positive virtual incidence) in theentire lower half (downstream half). For this reason, blades producelarger hydrodynamic forces in the lower or downstream half asobserved from both experiments and analysis. Moreover, unsteadyphenomena such as dynamic stall keep the flow attached to thecycloidal propeller blade up to very high pitch angles, which results invery high sectional force coefficients.

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