Towards a model for propeller-ice interaction

Date 2012 ,

Author Tom van Terwisga, Adam Peddle and Jie Dang

/jC

TUDelft

Ship Hydromechanics and Structures Laboratory Mekelweg 2, 2628 CD Delft

Deift University of Technology

T o w a r d s a model for propeller-ice i n t e r a c t i o n b y

Tom van T e r w i s g a , Adam Pedlle and Jie dang

Report No. 1 8 4 7 - P 2012 Proceedings of tlie ASME 2 0 1 2 3 1 ^ ' I n t e r n a t i o n a l Conference

on O c e a n , Offshore and Arctic E n g i n e e r i n g , O M A E 2 0 1 2 , July 1¬ 6, 2 0 1 2 , Rio de J a n e i r o , B r a z i l , Paper O M A E 2 0 1 2 - 8 3 0 8 8

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OMAE2012-83088

TOWARDS A MODEL FOR PROPELLER-ICE INTERACTION

Adam Peddle Dr. Ir. Jie Dang Prof. Dr Ir Tom van Terwisga

MARIN MARIN MARIN RD Department Ships Department RD Department

RO. Box 28, 6700 AAWageningen RO. Box 28, 6700 AA Wagenlngen RO. Box 28, 6700 AA Wageningen The Netherlands The Nethedands The Nethedands

ABSTRACT

A reduced level o f Ai'ctic sea ice in recent years has resulted in an increase in commercial interest i n shipping through previously inaccessible waters, such as the Northwest Passage. This interest has been significantly reinforced by the fact that vast amounts o f natural resources are expected to exist in the Arctic regions. As such, the operation o f vessels in ice-covered waters is currently a research topic o f great relevance.

Computational methods such as those based on potential flow have become invaluable in the evaluation and optimization o f marine propellers, though as o f this writing there exists a deficit o f proven software tools to evaluate the operation o f propellers i n icy waters. I n order to properly develop a model and a tool, a physical understanding o f the involved processes is vital.

A Cooperative Research Ships (CRS) project has been initiated with the ultimate aim o f developing a software tool for the evaluation o f propellers operating in ice which w i l l operate within the CRS PROCAL environment. The research described in this paper is the first step in the development o f such a tool.

This paper is intended to serve primarily as a review o f the existing literature on mathematical models o f propeller-ice interaction as well as their numerical solutions. Such a study serves as a vital first step in the development o f a complete method for propeller ice interaction. In addition to providing an ovei-view ot^ the current level of understanding in the field, this paper intends also to identify which areas are well understood and which require further investigation. It w i l l highlight the strengths and shortcomings o f current models with the intention of advancing towards the final model.

INTRODUCTION

O f prime consideration in the design o f vessels operating in ice-covered waters is the effect o f ice on the propeller. The designer must develop a propeller design which is both strong enough to resist the significantly increased loads placed upon it by interaction with ice as well as capable o f operating efficiently i n conditions which vary significantly from open-water.

Propellers which are intended to operate in ice differ primarily fi-om propellers operating in open-water i n that they feature thicker blades as well as significant geometric adaptations such as reinforced blade roots and thickened edges and tips designed to ensure sufficient structural integrity at higher loading. While such strengthening is necessary for safe operation, it often comes at a cost i n terms o f efficiency. However, with a sufficiently descriptive mathematical model, computational methods such as panel-based Boundary Element Methods ( B E M ) may be employed to optimize the geometry o f a propeller for both efficiency and reliability in ice.

Several mathematical models have been proposed and validated experimentally for propeller-ice interaction over approximately fifty years o f research, primarily in Canada, Finland and the former USSR. Attempts have been made at both a complete model o f interaction processes and at modelling o f only certain specific phenomena. The complexity o f ice mechanics and the large number o f numerical methods for simulation leads to the existence o f several mathematical models with significant disparity i n robustness.

The aim o f this paper is to provide an outline o f the most prevalent mathematical models employed for the simulation o f ice interaction with screw propellers. Furthermore, any shortcomings in the current academic understanding o f the physical phenomena w i l l be highlighted.

LOADING SCENARIOS

In order to be able to model interaction phenomena, it is first necessary to define specific loading conditions as well as specific types o f interaction. Loading conditions may be considered as belonging to one o f four possible quadrants [6].

Va

Quadrant U Quadrant I

n

Quadrant I V

Figure 1: Ice/Propeller Interaction Quadrants of a FP Propeller (Koskinen et al. 1996)

In Figure Irigure 1, different operating conditions for a propeller are defined by the advance speed and rotational speed o f the propeller. For a fixed-pitch (FP) propeller, quadrant I is the most common case, where the leading edge o f the blade comes into contact with the ice and the loads are o f the milling type. Quadrant I I I is the backing case and is very similar to quadrant I , with the notable difference being that the trailing edge contacts the ice instead o f the leading edge.

In quadrants I I and IV, the face or back o f the blade directly impacts the ice piece, and not the leading or trailing edges. These conditions are o f paiticular concern due to the fact that they lead to vvoiTyingly large propeller loadings. Both conditions, however, do not occur frequently and are associated with manoeuvring conditions where the ship speed and propeller thrust act in opposite directions. Neither quadrant has been studied or modelled in depth.

Wang [16] considered a propeller operating i n ice milling conditions, whereby the propeller cuts thi-ough a piece o f ice with the leading edge o f the blade. He proposed the distinction between three types o f propeller loading: milling loads, separable hydrodynamic loads and inseparable hydrodynamic loads. The separable hydrodynamic loads are the loads which would arise from the open-water operating conditions o f the propeller. Theoretically, these loads could be directly subtracted from the total load in ice to determine the net result o f the ice-related forces. However, he notes that this is not exactly the case as blade l i f t may not completely develop while the blade is in the ice or immediately after exiting the ice. Therefore, the separable hydrodynamic loads are approximate values.

The combination o f milling loads and inseparable hydrodynamic loads are referred to as the ice-related loads. M i l l i n g loads arise from the direct mechanical interaction o f the propeller and the ice. Inseparable hydrodynamic loads, on the

other hand, arise from ice-induced blockage o f flow, proximity effects and cavitation.

Terms relating to impact have been used in several interchangeable fashions with respect to propeller-ice interaction. For the purposes o f this paper, an ice impact refers to a dynamic interaction between an ice piece and the blade face/back. Ice interactions with the leading or trailing edges o f the blade may be considered to be milling interaction, as they are fiindamentally similar at the local level regardless o f the dynamics o f the ice piece.

SIMPLE MILLING

The milling condifion has often been studied for an idealized case whereby the propeller blade interacts with a piece o f ice advancing at ship speed, and where the dynamic response o f the ice piece is neglected. I n these models it is effectively just the material interaction at the blade-ice interface which is modelled. As such, the defining feature o f a milling-type interaction is the leading/trailing-edge interaction with the ice, as opposed to the face, back or tip o f the blade.

The first published model o f milling was the Jagodkin model [ 5 ] , which was intended to calculate the propeller torque in an ideal milling situation. The model is based on an idealization o f a propeller blade and calculates the blade torque as a sum o f the shear and crushing torques, which are expressed analytically as opposed to numerically. It assumes uniaxial values for crushing and shear strength and neglects friction.

A model was developed for determining the required strength o f a propeller blade by Ignatjev [4]. The blade-bending moment was calculated in the weakest direction on the root section as the sum o f the ice load, hydrodynamic torque and hydrodynamic thrust loading. The propeller blade was modelled as a simplified cantilevered beam, and the actual failure mode was not studied for the ice itself, which limits the effectiveness o f the model. It is relevant, however, as the first effort to consider the loads on the blades themselves as opposed to the shaft torque.

The first model to consider discretization o f the propeller was that proposed by Kotras et al. [ 7 ] . Rather than panelizing the propeller blades, they are instead considered as finite cylindrical strips located at constant blade sections. The pitch angle, thickness, chord length and camber are constant for each section, and as such they vary radially.

The Kotras model introduces the idea o f shadowing, whereby a portion o f each blade is operating in the groove cleared by the previous blade, as opposed to intact ice. This implies that the interaction proceeds in such a fashion as to cause the ice load to jump to the following blade once the contact area on the leading blade reaches a cetlain limiting value. Such a formulation permits the working quadrants o f the propeller to be included in the Kotras model in principle.

Each blade secfion is modelled in a simplified fashion, as shown i n Figure 2 below.

(Xfl.yo)

Figure 2: Simplified Blade Section Geometry (Kotras et al. 1985)

The angle o f advance, (3, may be either positive or negative, with a positive angle o f advance indicating that the blade edge is cutting the ice, and a negative angle o f advance indicating that the blade face or back is crushing the ice.

As the blade advances through the ice, layers o f ice are removed. The thickness o f these layers is given for any blade section as:

S . = f s i n p (1)

Where r is the radius at the particular blade section being studied and Z is the number o f blades. Furthermore, the height of the shadows w i l l be:

In order to determine the contact forces, the ice contact length must be found for each blade section. I f the edge o f the contact lies between coordinates 1 and 2 as in Figure 3, its Cartesian coordinates are found via simple geometry to be:

^1 -(yz - y i ) - t a " « + ( x i - x j - ( x p +y, - t a n a ) ( y 2 - y i ) - t a n a + X i - X 2

y = ( y 2 - y r ) - - ^ + y r

(3)

(4)

Where a is the angle o f attack o f the blade section and the other parameters may be found in Figure 2. Contact loads arise through crushing, bending and shearing o f the ice. However, the Kotras model describes only the crushing mode, and derives the total force fl-om the total volumetric work done on the blade sections. The work done by facet i o f the strip j is:

W i j = Oc • Vij (5) Where o^ is the compressive strength o f the ice and is the crushed volume o f facet i o f strip j . I f a facet moves through a distance Asj, the cleared volume is:

where:

2Tir

c - s i n ( ( p - p ) - — ' S i n P ; (3 > 0 c - s i n ( ( p - p ) + — ' S i n P ; p < 0 hs = the shadow height

c = the section length

(p = the pitch angle o f the section

(2)

The interaction between a given blade section and ice as modelled by [7] is shown in Figure 3.

V „ = A s , - b , ( L . c o s ? p (6)

Where bj is the width o f strip j , 1^- is the length o f facet i o f strip j and is the inclination angle between the facet and the advance direction. Under the assumption that the facet w i l l do equivalent work via normal and frictional forces, as well as o f simple Coulomb friction, the normal force on the facet can be found per:

™" sin^,+^:•cos^, (7)

1 ^^tf"^^ T-^5^<'*-*•:•••: •'•::'^.•S••^••

'^(xi.yi)

a

Figure 3: Cutting of Ice Block With and Without Edge (Kotras et al. 1985)

The Kotras model was a step forward for propeller-ice interaction models as it introduced the idea o f discretizing the propeller. Viilually all milling models to follow would in some way include this idea. However, it is limited in its simple representation o f blade geometry and its failure to consider modes o f ice failure other than crushing. While understandable in light o f the limited computational power available at the time, these limitations prevent contemporary use o f the Kotras model for propeller optimization.

M i l l i n g interaction was also studied by Belyashov and Shpakov [IJ.This model assumes the failure mechanism to be the shear failure o f the chip elements to the preceding groove. Based on cutting tests o f freshwater ice, the mechanism was taken to involve crushed ice being extruded in the contact zone between the propeller blade and the ice. As this contact zone

grows, the force increases in magnitude until an ice chip separates, corresponding to a drop in the force.

As with [7], Belyashov and Shpakov discretized the propeller as two-dimensional profile sections. The load on each section was then integrated over the entire blade. Only the first quadrant was studied and Mohr-Coulomb failure was assumed.

The Belyashov and Shpakov model determines the geometry o f the interaction in detail based on the relative speed o f the ship to the ice, as well as the geometry o f the blade and its cutting angle. Theoretically, it is one o f the most robust models, but is not reproduced in detail here in the interest o f space. It has since been superseded by other models, but provides a valid basis for interaction geometiy.

Yet another method for calculating the milling loads in the first quadrant based on division o f the blade into circular sections was developed by Sasajima and Mustamaki [11]. This model was based on model tests, and assumes that the contact force on the blade is due to the shear failure o f the ice. This shear failure is further divided into three types o f failure: shear o f ice to the cut groove on the face side, shear failure o f ice towards the bottom o f the ice piece on the blade face, and shear failure o f ice towards the bottom o f the ice piece on the blade back.

A l l three modes o f shearing are based on the cutting o f prismatic chips fi-om the ice, each with a thickness

approximately equal to that o f the section. The geometry o f the chips is shown in Figure 4.

Figure 4: Cutting of Ice in Mode 1, after Sasajima & Mustamaki 1984

From here, it is assumed that the failure angle, is constant and independent o f the local blade geometry. The internal energy was determined based on the motion o f one slice due to shearing stresses and compared to the external energy, which arose from the nominal contact pressure. Based on conservation o f energy, the total force on a circular strip, i , was found to be:

Fi = TsbiSN f A n r + ^ , ) (8) Vcos Yi sin i|i cos i|; 2 r i S i n i | ; /

Where Yt is the local inclination o f the leading edge, b; is the thickness o f the blade section, I'l is the radius o f the section,

Ts is the shear strength o f the ice and S n is the advance o f the blade to the normal direction o f the cut ice surface. The force which is found is technically the force component normal to the direction o f projecdon o f the contact width.

The second mode occurs when the propeller is advancing at a sufficient rate so as not to permit ice to be broken to the previous groove. This geometry is depicted i n Figure 5.

Figure 5: Cutting of Ice in Mode 2, after Sasajima & Mustamaki, 1984

Applying the same conservation o f energy method to this altered geometry yields an elemental force component of:

/ Ci 1 \

^' - ^^^"P' [ T ; ; ^ + sin m cos 4 / j

Where cpi is the projection o f the section length o f the plane normal to the section speed (relative to the ice speed) and

Ci is the local chord length. The third failure mode is described

by this equation as well, with the notable difference being that the contact load acts upon the suction side o f the blade instead o f the pressure side.

As the force calculated by this model is not the total force, but rather the normal component o f the force, the total force is determined based on the vector geometry and with consideration o f the frictional force. From there, the total thrust, ice torque, blade bending moment and blade spindle torque are determined based on integration o f the loads on the blade sections.

A more advanced model o f milling based on Finite Element Methods (FEM) was subsequently developed by Chernuka et al. [2]. Loads were calculated for two-dimensional blade sections using a failure criterion originally developed for concrete. The model considers loads on both the pressure and suction sides as well as friction, but only in the first operating quadrant. From here, F E M calculations determine the pressure distribution assuming complete contact between the ice and the blade.

The Chernuka model includes several simplifications i n order to facilitate numerical computation. It is assumed that the propeller blade is rigid and the advance speed is constant.

Furthermore, the relationship between the contact pressure and the ice parameters is talcen to be the uniaxial crushing strength o f the ice. Finally, the interaction behavior was studied in two-dimensional space. This has the resuh o f reducing the propeller to a series o f cylindrical strips, as with the majority o f milling models.

In order to distinguish between crushing and shearing modes o f ice failure, the perpendicular distance between successive cut grooves is compared to the projection o f the blade section i n the direction o f motion. For a ratio greater than two, crushing is assumed to occur, as shown in Figure 6 below.

Figure 6: Distinction Between Shearing and Crushing Modes (Chernuka et al. 1989, after Jussila and Soininen 1991)

The model applies a pressure distribution on the contact length o f the strip when the ice fails by crushing and a point load acting on the leading edge o f the blade when shearing is the failure mode. The pressure distributions are assumed to be fiinctions o f the contact length, section curvature and the perpendicular distance from the leading edge to the surface o f the path made by the previous blade. These values are normalized to the uniaxial crushing strength o f ice.

The form o f the pressure distributions was determined with the non-linear concrete model in the A D I N A F E M software. Due to computational problems encountered as the result o f the strain-softening o f ice, the analysis is static and does not account for deterioration o f ice mass.

The distributions o f contact pressures and forces used in the model are shown below in Figure 7.

Figure 7: Distributions of Contact Pressures and Forces, Chernuka et al. 1989)

Soininen [14] has developed a model based on the crushing and extrusion o f ice. It considers only milling, i.e. it neglects hydrodynamic and dynamic effects. It assumes two-dimensional behavior on the blade section as well as a grain orientation o f ice corresponding to the propeller radius.

In Soininen's model, tensile cracks form in the solid ice ahead o f the propeller blade. The ice within the formed spall is then ciTJshed and extruded through a channel between the solid ice and the propeller blade. New cracks w i l l continually be formed at the location o f maximum pressure. Figure 8 shows this process.

Figure 8: Ice Cmshing on Back Side (after Wang 2007, Soininen 1998)

The instantaneous maximum pressure is based on the stress required to induce macrocracking. Soininen used a macroscopic slip surface model based on that o f Kujala [8,9] to explain the response o f ice to the indentation o f the blade, such that the crack path is determined by the stress state, with the total failure load being the integral o f the loads required to induce failure among the individual elements. The failure used was the Mohi'-Coulomb criterion, which generally predicts that spalling

of solid ice should create curved surfaces. These cui-ved surfaces were simplified as wedges.

Failure loads were illustrated using two simplified examples of pressure distribution/solid wedge geometry: symmetrically-loaded and an asymmetrically-loaded wedges. Based on these models and an analysis o f ice strength, the critical peak pressure was proposed to be:

Piïsnire

(10)

With O c being the uniaxial compressive strength o f solid ice. A pressure distribution was then developed which takes into account the pressures in the crushed ice as it is extruded between the blade and the solid ice, past the critical point. A granular model was applied at the leading edge and a viscous model was applied to the compacted crushed ice extrusion towards the trailing edge.

The load induced by the flow fi-om the narrowest point o f the channel to the trailing edge is given by the following viscous model: 1 2 HV "h. h i hi + ax h2 2(hi + ax)^ 2 h ' a Vhi + ax (11)

With the boundary condition p = p^ at x = L and h2 = hi + aL. The channel coordinates and velocities are defined as in Figure 9.

y V

L Figure 9: Channel Coordinate Definition (Soininen 1998)

Towards the leading edge the granular model is used, with the load required to induce spalling becoming dominant. The effective load is the time average o f the instantaneous pressure distributions, which vary both temporally and, to a smaller extent, spatially, as new spalls are formed in response to the motion o f the blade.

Based on this theoiy as well as empirical evidence, Soininen developed an idealized pressure distribution along the blade section as shown in Figure 10.

Face side Leading

edge

Figure 10: Definition Points of Pressure Distribution Shape (Soininen 1998)

The pressure distribution is therefore defined by the maximum value o f average pressure, PMA, the location o f this maximum along the blade, X , the pressure, po, at the first point o f calculation. A , and the pressure value, PL, at L2 + L j = A D . A constant pressure value is assumed along segment ED as an approximation o f the entire parallel-walled channel section.

The values o f the definition points may be calculated as follows. Firstly, the leading edge pressure, PLE, was found to be:

p^^=4.5 + 2 6 w [MPa] (12) The maximum value o f the pressure distribution was defined as:

PMA =1 0 a ° (13)

Assuming that PMA/PWA takes a value o f 0.9, the position o f the pressure maximum bears a strong dependence on the ratio o f PLE/PWA> as below A C •7AD = ( 1.18 - 0.018 X f o . 5 5 - 0 . 4 1 — ' \ P w a / (14)

Po/pwA is strongly dependent on PLE/PWA, and slightly dependent on ij) - (3. I f ^ - (3 is in the range o f 15-20°:

^ V p w A = 0-54 + 0.33 PLE PWA

(15)

Finally, a second-order curve is used to yield the distribution o f effective load at the back side between the definition points as follows:

P =

f - . - ( f ^ ) ( X - x ) ^ cnAC

(16)

The Soininen model is one o f the most complete models o f milling-type interaction. By applying different material models

at different points along the blade surface it provides a more robust model for the ice, which often evades mathematical simplification.

INSEPARABLE HYDRODYNAMIC LOADS

Inseparable hydrodynamic loads arise from alterations i n the flow around the propeller due to the presence o f ice, i.e. blockage effects, proximity effects or altered cavitation characterisdcs of the propeller [16]. Compared to milling loads, these are significantly less well-understood from a mathematical perspective.

The blockage effect occurs when ice upstream o f the propeller disc restricts the flow o f water to the propeller. Thus, the propeller acts as though it is operating at an artificially low advance velocity, with the result being higher thrust and torque loads. The loading may i n fact be even higher than loading under bollard pull conditions due to difficulties with drawing water from upstream. Furthermore, it has been noted that the presence o f the ice block enforces a turbulent flow regime into the propeller [10].

It is notable that cavitation may occur more easily i n blocked flow due to the fact that heightened local flow velocities w i l l result in a reduction o f pressure [16].

A two-dimensional model o f blocked flow was developed by Shih and Zheng [12] based on the direct boundary element method. They found that the maximum non-contact force in blocked flow is 5.3 times that in open water. This was followed up with the development o f a thi-ee-dimensional B E M model [13]. This model predicted that the maximum thrust and torque o f the propeller reaches a value o f

1.6 times the normal value, with local blade section loadings o f as much as five times the open-water amount.

The thi-ee-dimensional B E M model was based on the potential flow around a propeller with an ice block in the close (i.e. proximity) condition. The viscosity and compressibility o f the fluid were neglected. Furthermore, the fluid was assumed to be irrotational, which implies the existence o f a velocity potential.

The interaction is modelled in keeping with conventional boundary element formulations for propeller operation, however the boundary surface is assumed to include three surfaces, i.e. the propeller surface, the wake surface and the ice block surface. The perturbation potential, P(x,y,z), at any point in the fluid domain was then formulated as follows, using Green's Theorem:

4.Ect,(P) = / / ^ < } ^ ( Q ) A ( ^ ) d S

rr dm)

And where F(P,Q) is the distance fi-om the field point, P(x,y,z) to the boundary point Q ( x ' , y ' , z ' ) and 5/on is the derivative normal to the boundai-y surface, S. The boundary conditions were defined on the blade surface as:

£ = - ( V , + n x r ) - n (18)

And on the ice-block surface as:

9c() _ r 0 before reaching the propeller an " (—VA • n after reaching the propeller It was further assumed that the wake surface was infinitely thin and that no flow or pressure jump occurs across the surface, though potential jump was permissible. A n equal-pressure Kutta condition was applied at the trailing edge o f the propeller along with a Newton iterative procedure.

A lifting-surface method for analysis o f propeller performance in blocked flow was developed by Yamaguchi [18] which considered the effect o f blocked flow on the propeller to be based on the combination o f flow separation and a proximity effect which resulted i n increased flow velocity between the propeller blade and the ice block.. The method predicted an increase of thrust and torque loading on the propeller by a factor o f two at maximum, more in keeping with the three-dimensional B E M than the two-three-dimensional model. However, comparison between predicted and numerical results showed a discrepancy thought to be the result o f the proximity effect being neglected.

In practice (for example, [16]), for the performance o f numerical simulations the axial flow velocity downstream o f the ice block was simply assumed to be 0.01 times the fi-ee-stream velocity.

ICE DYNAMICS

In all o f the models outlined i n the section on simple milling above, it was assumed that the ice piece is constrained, i.e. that it may not translate or rotate. It was fiirther assumed that the piece does not erode in response to the propeller. Relatively little research has been performed to model direct impact cases including ice piece dynamics, with the simplified milling case being the subject o f much further study. However, it is highly unlikely that the propeller would advance into constrained ice and so the response o f the ice piece to the action of the propeller is o f great interest. The idealized milling models explained above sei-ve as a basis for the physical phenomena taking place at the propeller-ice interface, but a complete model o f the interaction must be necessity include a method o f describing the kinematic and dynamic response o f the ice piece.

Wind [17] provides the first model to distinguish between milling o f ice and direct impact. The mass fiow o f ice blocks was taken as:

rh, = p , - B ' h i ' V s (19) Where: nii is the mass f l o w o f the ice

Pi is the density o f the ice B is the ship's beam h, is the mean ice thicloiess Vs is the ship speed

The mass f l o w was assumed to follow a lognormal distribution, with the mass o f the largest fragment being:

(20)

The impact loading on the propeller blades was based on the linear momentum o f the ice piece:

At (21)

Where: F i is the ice force

At is the duration o f the ice impact, i.e. the time in which the ice block w i l l be linearly accelerated to ship speed

V i is the initial speed of the ice block The blade stresses can then be found to be:

Fi • lj • cos (|)

(22)

Where: li = the moment arm o f the impact on the blade section

Zr = blade sectional modulus at radius r It is o f note that this model does not account for cutting o f the ice block, or any motion other than linear acceleration ahead to ship speed. As such, it provides an interesting basis from which to proceed, but is not particularly useful on its own.

Veitch [15] studied the interaction between a rigid propeller with constant rotational and translational velocities and a spherical ice piece. Two-dimensional sections are used to analyze the contact geometry. A major limitation lies in the fact that changes in the mass o f the ice piece due to cutting are neglected.

Veitch used a set o f inertial axes which were fixed with respect to the undisturbed free surface. He further used a set o f axes centred on the ice piece and a third set centred on the propeller as in Figure 11. A t several stages in the calculations, transformations are required between the various coordinate systems. These transformations follow the principles o f tensor analysis and do not require elaboration herein.

I C E BODY AXES

PKOPELLER BODY AXES

Figure 11; Coordinate Systems Used by Veitch (1995)

The motion o f the ice piece is defined by a system o f six differential equations, which are derived directly f r o m the kinematics o f a three-dimensional submerged body. For a detailed derivation, refer to [15].

- C m 0 0 0 0 0 - f i n r X j \ 0 m-C„ 0 0 0 0 V Y T 0 0 m - C „ , 0 0 0 w Z T 0 0 0 I 0 0 p K T 0 0 0 0 1 0 q M T 0 0 0 0 0 I ^ r . N T -(23) Such that. ( C „ - m ) ( w q - v r ) - ( W - B ) s i n e - C „ - p „ D ^ u | v | + X , 8 ( C „ - m ) ( u r - w p ) + ( W - B ) s i n * c o s e - C„ J p „ D ^ v | v | + 8 ( C „ - m ) ( v p - u q ) + ( W - B ) c o s * c o s 6 - Co j p „ D ^ v | v | + Z , T i p „ D ' 3 2 i i p „ D 3 2 ~ i i p „ D -3 2 I C f p l t i i l s i n ^ I Cpqltolsin'' Jo I C p r l c i j l s i n ^ Jo eds ede ede (24)

Where: Cm is the added mass o f the body m = the mass of the ice piece W = the weight of the ice piece B = the buoyant force on the ice piece O = the rotation about the y-axis 0 = the rotation about the x-axis C f is the coefficient o f friction o f the ice C d is the drag coefficient of the ice piece Also, the ice contact components are given by Xc, Yc, Zc, Kc, Mc and Nc. The acceleration o f the ice piece is thus found by a solufion to the system o f equations. Denoting the velocity components as U i , U2,...U6 and the position components as Zoi, Zo2,---Zo6. the above can be expressed as below, i n light o f the fact that the contact force is dependent on the position o f the ice

piece relative to the propeller and that the weight and buoyancy terms contain orientation angles.

d U , ( t ) dt

= f ( t , U ^ , U , , . . . U , , Z „ . , Z „ , , . . . Z ^ y . i = 1,2...6 (25)

From this formulation, the velocity components in the ice-body system may be found. Position and orientation o f the ice piece may be found by integration. It is noteworthy, however, that there is a direct relationship between the Isinematics o f the ice piece and the dynamics o f the entire system. As such, Veitch proposed a method for the determination o f the interaction forces between the propeller blade and the ice piece. This model, were it to be applied independently o f the above kinematic model, could be thought o f as an alternate milling model. However, i t w i l l be continued in this section for the sake of continuity.

The ice cutting geometry used by Veitch is shown i n Figure 12, and is based on a two-dimensional model o f orthogonal cutting [3]

ICE B O M

lAMCBT TO SnctlOS S118 AT POINT « SBfRESraiED

IS A mm cimiK mi

lASCRU TO PRESSU81 SIDE AT POINT Of SEPBE.'iSSIED AS A PLANAR CÜTtlNC PACE

NOMAL TO BLADE SECTION VELOCITY YEaOB

With the total length extending to the contact limit point, 1, at Xp = Xc. Through the use o f model tests a conceptual model was developed. B E M models were used to model the indentation contact between the indentor and the ice body.

First, the early stages o f contact and penetration were modeled, secondly, the more advanced stages o f interaction where chips had formed on the pressure side o f the tool were considered. The third stage which was analyzed was that o f contact between a semicircular indentor and a closely conforming indent on an ice body.

Contact pressure distributions were therefore developed as a function o f cutting angles and ice thicknesses as follows:

Pmax 1 -/s \ 1 • - ( l - s i n m ( Y - k ) ) ; Y c < y < 7 2 \ ' p / ^ (27) l - ( ^ ) • - ( l - s i n m ( Y - k ) ) ; ^ Y ^ 0

Where the subscripts p and s denote the pressure and suctions sides o f the blade, respectively, and m and k are defined as: -; Y c < Y < 7 2 (28) Y 2 - Y i

72

With contact limits defined as functions o f the ratio hp/R, where R is the leading edge radius o f the section:

DETAIL OP SEalOS NEAR LEADINC E K E SEOIIliC CONTACT PATHS AND PATH LENCTIIS

Figure 12: Cutting Geometry (after Veitch 1995)

The path lengths along the surface are measured from the kinematic leading edge such that the distance f r o m the K L E to any point, P, satisfies:

(26) Yi = Yc - 2 Y C P / / R TT Y 2 = T 7 ^ / R + 7 4 . p / 2 h, ' / R + V S / (30) (31)

Yc refers to the angle - 7 r / 3 , as it is assumed that the pressure decreases to zero between cutting angles o f - TC /2 and - TT /3. The pressure is calculated as above for each incremental area, AAij centered on the point Pij. The pressures can then integrated in order to yield the forces and moments on the propeller due to contact:

I t l n Fp,c = - ^ J ^ P i j A A i j - i ï i j (32) i = i j = i in n Mp,c = ^ [ r p , i j ( p i j A A i j • Hi,)] (33) i = i j = i

Where Tp y is the position vector o f point Pij in the propeiler body axis and njj is the normal vector at the centroid o f the elemental area. It is o f note that the calculations are performed in the propeller frame, with two coordinate transformations being required in order to find the force components in the ice frame. From these coordinate transformations, the forces and moments on the ice piece may be found, which w i l l in turn yield the contact force and moment components Xc, Yc, Zc, Kc, Mc, and Nc required to complete the equations o f ice modon.

CONCLUSIONS AND RECOMMENDATIONS

The problem o f propeller-ice interaction can be considered as an extension o f the normal problems faced when designing a propeller. Considerations on the basis o f hydrodynamics and cavitation must still be made, but are made more complex by effects such as blockage and proximity induced by the presence o f ice pieces. Also, while it is an important component o f the propeller designer's task to ensure sufficient structural integrity o f the propeller blade in ice-free waters, it is made significantly more challenging due to the high loads caused by ice pieces as they move and deform in response to the action o f the propeller.

The main fi-amework used here has been one which considers the loads on the propeller as the sum o f the separable hydrodynamic loads, the inseparable hydrodynamic loads and the milling effects, as per [16]. It is o f note that this is simply the model which has been deemed the easiest with which to work and to better understand propeller-ice interaction, but it is by no means the only framework which has been applied in the past.

M i l l i n g refers to the action o f the edges o f the propeller blade (generally the leading edge) cutting thi-ough the ice. Although the majority o f milling models consider only a static piece o f ice (i.e. neglecfing translation, rotation and deformation o f the ice piece), milling is not in fact limited to only these situations. Models developed for static ice can veiy often be extended to the more realistic case o f dynamic interaction, as with the inclusion o f the Soininen model o f milling in the Koskinen et al. model o f overall interaction ([14] and [6], respectively).

There exist several different models o f milling. This is a direct effect of the difficulties involved with modelling the material properties o f ice. Hybrid models which utilize different material models for different mechanical situations seem to provide the most reasonable description o f milling-type loading.

Unfortunately, there is a dearth o f research into the effects o f direct impact on the face and back o f the blade. This is recognized as a field which requires further study. The modelling o f the strongly-coupled fiuid-structure interaction problem o f dynamic response o f the ice also demands fiirther study.

Ultimately, a model o f propeller-ice interaction should be able to consider the loading on the propeller due to hydrodynamics, milling of the ice, and the motion o f the ice in response to the propeller.

ACKNOWLEDGMENTS

The authors wish to acknowledge the support o f the CRS ProPolar working group.

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