A STUDY ON THE MODELING OF MARINE PROPELLER TIP FLOWS US1NG BEM

J. Baltazar**, J.A.C. Falcao de Campos1 and J. Bosschers2

1: Department of Mechanical Engineering

Instituto Superior Técnico (IST) Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

e-mail: baltazar@marine.ist.utl.pt,fcampos@hidrol.ist.utl.pt web: http://www.dem.ist.utl.pt/dem/

2: Maritime Research Institute Netherlands (MARJN) PO Box 28, 6700 Wageningen, the Netherlands

e-mail: j.bosschers@marin.nl web: http://www.marin.nl

ABSTRACT

It has been recognized [1] that Boundary Element Methods (BEM) are able to predict the pres­sure distribution in the steady case with reasonable accuracy for the major part of a marine propeller blade. However, there are still some difnculties in obtaining a reliable solution close to the blade tip with the so-called conventional grids, especially if fine discretizations are ap-plied at the tip. In thiscase, the solutions appear to converge forthemajorpart of the propeller blade, but an irregular (non-smooth) behaviour of the pressure distribution near the tip is often seen. These difficulties give origin to divergence of the iterative scheme used to impose the Kutta condition of pressure equality on both sides of the propeller blade at the trailing edge or large pressure peaks at the trailing edge. Results of BEM calculations for a new panel arrangement with a "hydrodynamic tip" are pre-sented and compared with the conventional grid for two tested propellers: DTRC propeller P4119, without skew, and a modified propeller based on DTRC propeller P4842. The "hydro­dynamic tip" is defined as the location on the trailing edge where the vortex wake ends and no longer coincides with the geometrical tip. A rigid wake model and a wake alignment model with roll-up are considered. The results show that the use of "hydrodynamic tip" grids reduces the pressure oscillations edge inboard of the "hydrodynamic tip". The use of the present wake alignment model with roll-up in combination with the conventional grid introduces distur-bances in the pressure distribution near the tip due to the proximity of the tip vortex from the blade suction surface. With the "hydrodynamic tip" grids the rolled-up wake at the tip leaves the blade smoothly. In these cases, the effect of wake roll-up on the pressure distribution near the tip appears to be small.

REFERENCES

[ 1 ] K. Koyama (ed.), Comparative Calculations of Propellers by Surface Panel Method - Workshop Organized by 20t/l ITTC Propulsor Committee, Papers of the Ship Re­search Institute, Supplement No. 15, (1993).

Congreso de Métodos Numéricos en Ingenierïa 2005 Granada, 4 a 7 de julio, 2005

©SEMNI, Espaiia, 2005

A STUDY ON THE MODELING OF MARINE PROPELLER TIP FLOWS USING BEM

J. Baltazar1", J .A.C. Falcao de Campos 1 and J. Bosschers2

1: Department of Mechanical Engineering Instituto Superior Técnico (IST)

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal e-mail: baltazar@marine.ist.utl.pt,fcampos@hidrol.ist.utl.pt

web: http://www.dem.ist.utl.pt/dem/

2: Maritime Research Institute Netherlands (MARIN) PO Box 28, 6700 Wageningen, the Netherlands

e-mail: j.bosschers@marin.nl web: http://www.marin.nl

Keywords: BEM, Propeller Blade, Wake Models, Hydrodynamic Tip.

Abstract. Results of BEM calculations for a new panel arrangement with a "hydrody­namic tip" are presented and compared with the conventional gridfor two tested propellers: DTRC propeller P4119 and a modified propeller based on DTRC propeller P484S. A rigid wake model and a wake alignment model with roll-up were considered. The results show that the use of "hydrodynamic tip" grids reduces the pressure oscillations inboard of the "hydrodynamic tip". Also, the wake alignment model produces a rather smooth roll-up of the vortex sheet.

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

1 I N T R O D U C T I O N

The Boundary Element Methods (BEM) applied to marine propellers has achieved a considerable degree of maturity both for steady and unsteady calculations, as it can be seen from the results of the comparative calculations of the two workshops organized by the Propulsor Committee of the ITTC [l]-[2). However, there are still a number of difficulties in the application of BEM to the prediction of propeller blade pressure distributions, even in the simple steady case. One of the important difficulties concerns the prediction of the pressure distribution in the tip region. The correct prediction of the pressure distribution is important not only for the thrust and torque calculations by the integration of the pressure distribution on the blade surface, but also for the cavitation inception estimation.

A grid oriented along constant radii, the so-called conventional grid, has been previously employed with the BEM. This grid arrangement results in high aspect ratio of the panels at the propeller tip. Pyo [3], among others. showed that using the conventional grid arrangement, it is difficult to obtain convergence of the iteration required to enforce a pressure Kutta condition at the trading edge in the tip region.

Grid convergence studies [4] were carried out for marine propellers using conventional and orthogonal grids. At the corresponding design conditions the solutions obtained with the conventional grids appear to converge for the major part of the propeller blade, but an irregular (non-smooth) behavior of the pressure distribution near the tip was evident. The use of orthogonal grids showed some promising results in dealing with the occurrence of pressure peaks at the trailing edge after the application of the pressure Kutta condition. However, the predicted pressure distributions close to the tip remained rather erratic and much influenced by the smoothness of the grid generated.

BEM computations [5] were carried out for an elliptical wing with conventional and orthogonal grids using a rigid wake model. Both with the conventional and orthogonal grids the pressure distributions near the tip of the wing showed unrealistic pressure peaks at the trailing edge when the iterative pressure Kutta condition was applied. From these studies it became more clear that the lack of wake alignment with the flow near the tip should be responsible for the occurrence of the unrealistic pressure peaks near the trailing edge.

To avoid the irregular behaviour of the pressure distribution at the tip, the flow adapted grid arrangement was introduced by Kinnas et al. [6], which is orthogonal at the leading edge and aligned with the resulting mean flow at the trailing edge. Pyo [3] showed that this grid arrangement improved the convergence of the iterative pressure Kutta condition and the pressure distribution at the tip of the propeller blade. Furthermore, it was observed for elliptical wings [7] that the mean velocity normal to the trailing edge near the tip may be directed towards the inside of the lifting surface. Since this is not consistent with the condition of vortex shedding at the edge, the concept of the "hydrodynamic tip" is introduced. The "hydrodynamic tip" is defined as the location on the trailing edge where

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J. Baltazar, J .A.C. Falcao de Campos and J. Bosschers

the vortex wake ends and no longer coincides with the geometrical tip. Results of BEM calculations for a new panel arrangement with a "hydrodynamic tip"

are presented and compared with the conventional grid arrangement for two tested pro­pellers: DTRC propeller P4119, without skew, and a modi Bed skewed propeller based on DTRC propeller P4842. A rigid wake model and a wake alignment model with roll-up were considered.

The paper is organized as follows: Section 2 briefly presents the mathematical formula-tion for the steady potential flow. Section 3 describes the major features of the numerical method including the algorithm for the vortex wake alignment. Results of BEM calcula­tions for the two propellers are presented in section 4. In section 5 the main conclusions are drawn.

2 MATHEMATICAL F O R M U L A T I O N

Consider a screw propeller of radius R rotating with constant angular velocity Q around its axis and advancing with constant axial speed U in an incompressible ideal fluid at rest in a domain extending to infinity in all directions. The propeller is made of K blades symmetrically distributed around an axisymmetric hub. Figure 1 shows the coordinate system used to describe the propeller geometry and the fluid flow around the propeller.

Figure 1: Propeller coordinate system.

We introducé a cartesian coordinate system (x, y, z) fixed to the rotating propeller blades, with the positive x - axis direction opposite to the propeller axial motion, the y - axis coincident with the propeller reference line, passing through the reference point of the root section of the blade k — 1, and the z - axis completing the right-hand system. We use a cylindrical coordinate system (x,r,6) related to the cartesian system by the transformation

y = rcos0 , z = rsinO. (1)

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J. Battazar, .J.A.C. Falcao de Campos and J. Bosschers

In the coordinaie system rotating with the propeller blades the flow is steady and is assumed to be irrotational so that we may introducé a potential function $ related to the flow velocity by V = V $ . We write for the potential $ = 0TO + <p, where (j>oo is the potential of the undisturbed flow, Uco = Ui + Vtrëe = V ^ » , and 4> is the perturbation potential due to the presence of the body. The perturbation potential satisfies the Laplace equation V2<fi = 0. The boundary of domain consists of the blade surfaces SB and the hub surface Sn. The kinematic boundary condition (dfi/dn) = —ÜQQ • n is satisfied on the boundary of domain, where d/dn denotes differentiation along the normal and n is the unit vector normal to the surface directed outward from the body. At infinity the flow disturbance due to the body vanishes V<£ —> 0.

To allow for the existence of circulation around the propeller blades, vortex sheets are shed from the trailing edge of the blades constituting the propeller wake. We assume that the vortex sheet is shed from the trailing edge between a point on the blade root and a point called "hydrodynamic tip". The boundary conditions on the vortex wake surfaces Sw in steady flow are the zero normal velocity component to the surface and the continuity of the pressure across the surface. In order to specify uniquely the circulation around the blades it is necessary to impose the Kutta condition at the trailing edge. In its original form, the Kutta condition states that^the velocity must remain finite

V<£ < co (2)

at the blade sharp trailing edge. Applying the second Green's identity, assuming for the interior region to SB and S-H,

4> = 0, we obtain the integral representation of the perturbation potential at a point p on the body surface,

2*0= ƒ [Gfedg-^jgJdS-ƒ/*(,)§£* (3)

where G(p,q) ~ — 1/R(p:q), R(p,q) is the distance between the field point p and the point q on the boundary S = SB U <S U SW. With the d<$>/dnq on the surfaces S& and SH known from the Neumann boundary condition on the body surface,

— = n • Vè = —n • Uoo on SB and SH- (4) on

Eq.(3) is a Fredholm integral equation of the second kind in the dipole distribution fi(q) = —<j>{q) on the surfaces SB and SH- The Kutta condition (2) yields the additional relationship between the dipole strength A<£(<?) in the wake and the surface dipole strength at the blade trailing edge.

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J. Baltazar, J.A.C- Falcao de Campos and J. Bosschers

3 N U M E R I C A L M E T H O D

3.1 Surface discretization

For the numerical solution of the integral equation (3) we discretize the blades surfaces <Ss, the hub surface Sn, and the blade wake surfaces S\y in bi-linear quadrilateral elements which are defined by four points on the body surface. At grid singularities, triangular elements are used with one node coincident with the singular point.

The conventional grid has been used traditionally for BEM calculations on propeller blades. It consists of families of grid lines running along the contour of blade sections on cylindrical surfaces of constant radius. This approach allows the definition of blade strips roughly aligned with the flow along the blade sections for most part of the blade. Near the tip, where three-dimensional effects on the flow are rather strong, the blade strips are no longer aligned with the flow. For wide blades with strong reduction of chord at the tip, eventually with zero tip chord, the conventional grids give origin to highly skewed panels near the leading and trailing edges with angles between the two families of grid lines approaching zero.

To avoid the disadvantages of the conventional grids near the tip, a new panel arrange­ment with a "hydrodynamic tip" was introduced. In this new panel arrangement, the grid singularity (which coincides with the "hydrodynamic tip") is displaced along the trail­ing edge and an orthogonal grid is generated on the blade surface. This grid generation technique was devised by Baltazar and Eca [8].

The blade wake surface is discretized in the spanwise direction, extending downstream from the trailing edge the corresponding strips on the propeller blade. A gradual variation of the panel dimensions from the trailing edge to the ultimate station far downstream was adopted.

3.2 B E M

The integral equation (3) is solved by the collocation method with the element center point as collocation point. We assume a constant strength of the dipole and source distributions on each element. The influence coefficients are determined analytically using the formulations of Morino and Kuo [9]. The expressions are exact for the dipole influence coëfficiënt. For the source influence coëfficiënt the expressions are only exact for a plane element. In some cases, for the highly distorted panels near the tip, numerical integration is to be preferred, since the implementation of the analytical integration routine is not sufliciently robust to handle every possible combination of field point and panel geometry. In the present work, the wake influence coefficients are computed numerically. In this way, we adopted a simpler numerical calculation of the influence coefficients, evaluated in a local panel coordinate system (£:?/). An adaptive integration algorithm, which provides a subdivision of the reference element, is used to achieve a specified numerical tolerance (10~5). The resulting matrix equation is solved by a LU factorization method.

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

3.3 Wake mode l s

Two wake models were considered in the present work: a rigid wake model and a wake alignment model with roll-up.

In the rigid wake formulation, the Kutta condition (2) is replaced by the condition of equal pressure on both sides of the blade at the trailing edge. Morino used a numerical implementation of the Kutta condition at the collocation points of the panels adjacent to the trailing edge on the two sides of the blade in the form:

(A<t>)mSw = <t>+-f', (5)

where <f>+ and 4>~ denote the potential on the back and face of the propeller blade. It was verified [10] that this condition does not ensure the equality of the pressure on both sides of the blade at the trailing edge in the three-dimensional case. Due to the non-linear character of the pressure, the application of the condition for the pressure equality on the collocation points on the panels adjacent to the trailing edge creates a non-linear system of equations. In the present work, its soliition is obtained by the method of Newton-Raphson.

A wake alignment model which predicts the roll-up of the vortex sheet behind the propeller blades is being under development. The method aligns the vortex lines with the local fluid velocity by displacing the nodes of the wake surf ace grid with the mean velocity. A discretized form of the streamline equation is applied to find the new location of the wake surface which is aligned with local mean velocities:

Vw Vw Vw \VW\ ~f~ = -ir- = ir- = const = -ir1 (6) Ax Ay Az As

where V^ denotes the cartesian components of the mean velocity, and Ax, Ay, Az are the differences between the coordinates of two adjacent panel corners along the streamwise direction. The method is iterative. If As is defined as the distance between two adjacent panel corners i and i + 1 at a given wake strip:

A* = y(x<:»-x!"»)2 + (y<;»1-,,'»»)2

+(^)-,,'"))2 (?) at the iteration level (n), then the new coordinates at the [n + l)th iteration level are determined as follows:

j / , ( : r " = w n ) + ^ - A S (8) \VW\

4:tn = * , ! " > + ^ A ,

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

where V^ViZ is calculated by taking the average of the mean velocity between two adjacent panel corners:

(9)

The calculation of velocities, Eq.(9), have been inspired by the model of Moulijn and Kuiper [11]. The mean velocities at the panel corner points are obtained from linear interpolation from the values calculated at the mid points along each spanwise strip, except the trailing edge strip (or first strip) where the velocities are calculated at the corner points. The induced velocities on the wake surface can be evaluated by directly differentiating Eq.(3). A scheme is applied to accelerate the roll-up of the wake, which consists in updating all the nodes downstream of the current node along a streamwise grid line simultaneously with the displacement of the current node.

3.4 Calculation of velocities and pressure

From the potential solution on the surface the covariant surface velocity components are calculated by means of a second order differentiation scheme of the potential relative to the are lengths on the body surface grid. From Bernoulli's equation, the pressure coëfficiënt Cp can be determined from Cp = (p - Poo)/(l/2pUl) = 1 - (\V\fUoo)2.

4 RESULTS

Since we me interested in the tip region and in the results of the vortex wake alignment model in that region, the iteration of the blade vortex sheet with the propeller hub has not vet been considered in this work. Accordingly, all calculations were carried out without the reference of the propeller hub, allowing the wake roll-up to occur at the blade root. In the unlimited inviscid incompressible flow, the propeller operating conditions are characterized by a single non-dimensional parameter: the advance coëfficiënt J = U/nD, where n — Q/27T is the rate of revolution and D = 2R the propeller diameter.

4.1 Propeller D T R C P4119

The DTRC propeller P4119 is a three-bladed propeller without skew or rake. The blade root section is closed with four rows of panels matching the panel distribution on the first panel strip on the blade. Since the propeller has zero chord at the tip, a strip of triangular panels is presented in all grids.

Calculations were carried out at the design condition J = 0.833 for the conventional grid and the new panel arrangement with the "hydrodynamic tip" located at r/R — 0.97 using the rigid wake and the wake alignment models. Although a sniall displacement of the "hydrodynamic tip" from the geometrie tip would be sufficiënt to avoid mean velocities normal to the trailing edge pointing into the blade, the "hydrodynamic tip" location was selected to be comparable to the one used by Pyo in his work with the flow adapted grid

Vw = i rx,y,z 2

K x,y,z

|(") , yw |(n)

7

{•

3. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

Figure 2: Blade back panel arrangement for the propeller P4119: conventional grid (left) and new panel arrangement with "hydrodynamic tip" (right).

Figure 3: Helicoidal wake grid (left) for the propeller P4119 with a detail over the blade tip (right).

arrangement [3]. Figure 2 shows the blade back panel arrangement with 60 chordwise and 30 spanwise panels for the conventional grid (left) and the new panel arrangement with "hydrodynamic tip" (right). Cosine spacing in spanwise and chordwise directions has been used in the conventional grid. For the new panel arrangement, a cosine distribution was also used for the boundaries. Near the propeller tip, the panels present a high aspect

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

Figure 4: Rolled-up wake grid for the propeller P4119 obtained using the conventional grid (top) and the new panel arrangement with "hydrodynamic tip" (down).

ratio for. the conventional grid, due to the existence of a grid singularity at the tip. The maximum deviation from orthogonality for the conventional grid is 85.2 degrees and 33.7 degrees for the new panel arrangement.

Figure 3 presents the helicoidal grid with 30 streamwise and 30 chordwise panels for the conventional grid. generated using the same pitch of the propeller blade for the helicoidal lines on the wake surfaces. Figure 4 shows the aligned wake grid obtained with the wake ahgnment model for the conventional grid and the new panel arrangement with

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

0.0 •-

-> 1 < 1 ' 1 ' r

- * r/R=0.95 -0 r/RsO.99 -e r/Rs0.995

0.2 0-4 0.6 s/l-

0.8 1.0 0.2 s/c

0.6

0.0

"> r i f

r/R=0.95 r/R=0.99 r/R=0.995

0.2 04 ,/r °6 0.8 1.0

1.6

1.2

0.0

n ! , r

- h r/R=0.95 - ö r/R=0.99 -© r/R=0.995

j I i L 0.2 0.4 0.6

s/c

Figure 5: Pressure distribution on the propeller P4119 for the conventional grid using the rigid wake model (top-left) and the wake alignment model (top-right), and for the new panel arrangement with "hydrodynamic tip" using the rigid wake model (down-left) and the wake alignment model (down-right).

"hydrodynamic tip". The iterative pressure Kutta condition converged in all cases to | A C P I T £ < 10~3. It can be seen that the wake alignment model produces a rather smooth roll-up of the vortex sheet.

Figure 5 presents the pressure distribution close to the tip at sections r/R = 0.95, 0.99 and 0.995 for the conventional grid (top) and for the new panel arrangement with "hydrodynamic tip" (down), using both wake models. From the pressure distribution

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

z d

Figure 6: Streamlines from the pressure side around the tip region with the new panel arrangement for the propeller P4119.

obtained for the conventionai grid with the rigid wake model, some oscillations and a small pressure peak at the trailing edge are observed for section r/R = 0.995. Using the wake alignment model, the oscillations signincantly increased, due to the presence of the rolled-up wake sheet on the suction side of the blade near the tip, as we can see in figure 4 (top-right). Notice that the section r/R — 0.995 is extremely close to the tip. It may be socn. that the oscillations at section r/R = 0.99 are already considerably less. For the new panel arrangement with "hydrodynamic tip", similar pressure distributions were obtained using both wake models. Pressure peaks at the trailing edge are seen at sections r/R. = 0.99 and 0.995. Analysing the potential flow, figure 6, we observe that some streamlines from the pressure side pass to the suction side flowing around the thin and sharp "leading edge" between the "hydrodynamic tip" and the geometrie tip, which is located at the geometrie trailing edge. Due to the shape of this "leading edge", a pressure peak is produced since, from the theoretical point of view the pressure is singular at the sharp edge. This singularity is unavoidable for any location of the "hydrodynamic tip" on the trailing edge non-coincident with the geometrie tip as, by definition, we do not allow vortex shedding outboard of the "hydrodynamic tip".

4.2 Propeller DTRC P4842M

The DTRC propeller P4842 is a five-bladed propeller with high skew. The modified DTRC propeller P4842, designated P4842M, was obtained from the original P4842 propeller geometry by increasing the chord length in the tip region while keeping the same mid-chord skew distribution. The propeller has finite chord at the tip so the conventionai grid

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J. Baliazar, J.A.C. Falcao de Canipos and J. Bosschers

does not require the use of triangular panels. The blade root section is closed with four rows of panels matching the panel distribution on the first panel strip on the blade.

Figure 7: Blade back panel arrangement for propeller P4842M: conventional grid (left) and new panel arrangement with "hydrodynamic tip : ' (right).

Calculations were carried out at the design condition J ~ 0.905 for the conventional grid and the new panel arrangement with the "hydrodynamic tip" located at r/R — 0.973. using the rigid wake and the wake alignment models. The location of the "hydrodynamic tip" was selected after preliminary calculations in order to allow a smooth roll-up to occur. Figure 7 shows the blade back panel arrangement with 50 chordwise and 25 spanwise panels for the conventional grid (left) and the new panel arrangement with "hydrodynamic tip" (right). Cosine spacing in spanwise and chordwise directions has been used in the conventional grid. For the new panel arrangement, a cosine distribution was also used for the boundaries. The maximum deviation from orthogonality for the conventional grid is 84.2 degrees and 55.8 degrces for the new panel arrangement. Figure 8 presents the rolled-up wake grid obtained with the wake alignment model for the conventional grid (top) and the new panel arrangement with "hydrodynamic tip" (down).

The Kutta condition did not converge for the conventional grid with both wake models. Since the iterative Kutta condition is applied at the collocation points of the panels adjacent to the trailing edge on both sides of the blade, the pressure distribution is presented at figure 9 for the last strip of collocation points (J = 25), obtained using the linear Kutta condition. This last strip corresponds to section r/R = 0.998, which is extremely close to the tip. Irregular oscillations and very large pressure peaks were obtained at the last strip which determines the lack of convergence of the iterative Kutta condition. This is due to the highly skevved panels near the propeller tip, causing loss of

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

Figure 8: Rolled-up wake grid for the propeller P4842M obtained using the conventional grid (top) and the new panel arrangement with "hydrodynamic tip"'(down).

accuracy in the calculation of the infiuence coefficients. In addition, the wake alignment model starting from the linear Kutta condition solution. produced distorted panels and an irregular rolled-up geometry near the tip vortex, as we can see in figure 8 (top-left).

For the new panel arrangement with "hydrodynamic tip", the iterative pressure Kutta condition converged to | A C P | T E < 10~3 using the rigid wake model, and to \ACP\TE < 8 x 10"2 using the wake alignment model. In order to avoid the problems associated with the convergence of the Kutta condition in conventional grids, it was necessary to

13

J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

0.12 0.12

0.08 -

0.04 -

0.00,

Figure 9: Pressure distribution on the propeller P4842M for the conventional grid using the rigid wake model (left) and the wake alignment model (right). Linear Kut ta condition.

1.6

- r/R=0.95 - r/R=0.97 - r/R=0.99

- r/R=0.95 - r/R=0.97 - r/R=0.99

- r/R=0.95 - r/R=0.97 - r/R=0.99

1.2

- r/R=0.95 - r/R=0.97 - r/R=0.99

0.8

V" 0.4

. j I i L 0.2 0.4 0.6

s/c 0.8 1.0

-04.

- r/R=0.95 - r/R=0.97 - r/R=0.99

- r/R=0.95 - r/R=0.97 - r/R=0.99 1.2

- r/R=0.95 - r/R=0.97 - r/R=0.99

0.8

o-

0.4

0.0 0.4 0.6 S/C

Figure 10: Pressure distribution on the propeller P4842M for the new panel arrangement with "hydro­dynamic tip" using the rigid wake model (left) and the wake alignment model (right).

displace the "hydrodynamic tip" far from the geometrie tip. In fact, in the prehminary calculations with the "hydrodynamic tip" located between the current position and the geometrie tip. difheulties werc met in the convergence of the iterative Kutta condition. A more regular rolled-up wake geometry is obtained with the wake alignment model. Figure

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J. Baltazar, J.A.C. Falcao de Campos and J. Bosschers

10 presents the pressure distribution at sections r/R = 0.95, 0.97 and 0.99. Similar pressure distributions were obtained with the rigid wake and the wake alignment models. At section r/R = 0.95 the pressure distribution presents a region of negative load near the trailing edge. In addition, at sections r/R = 0.97 and 0.99 there are large pressure peaks at the trailing edge. This is due to the fact that the flow passes from the pressure side to the suction side and crosses the thinner and sharp "leading edge" located in the geometrie trailing edge of the propeller.

5 CONCLUSIONS

Results of BEM calculations for a new panel arrangement with a "hydrodynamic tip" are presented and compared with the conventional grid for two propellers: DTRC propeller P4119, without skew, and a modified skewed propeller based on DTRC propeller P4842. A rigid wake model and a wake alignment model with roll-up were considered.

The results show that the use of "hydrodynamic tip" grids reduces the pressure os-cillations edge inboard of the "hydrodynamic tip". At the edge, outboard of the "hy­drodynamic tip", a pressure peak occurs associated to the flow around the sharp edge from the pressure to the suction side of the blade. The use of the present wake alignment model with roll-up in combination with the conventional grid introduces disturbances in the pressure distribution near the tip due to the proximity of the tip vortex from the blade suction surface. With the "hydrodynamic tip" grids the rolled-up wake at the tip leaves the blade smoothly. In these cases, the effect of wake roll-up on the pressure dis­tribution near the tip appears to be small. Most likely, for less extreme locations of the "hydrodynamic tip" the effect of roll-up will become more significant. These points will be the subject of future investigations.

6 ACKNOWLEDGMENTS

The first author acknowledges the financial support granted by Fundacao para a Ciência e a Tecnologia, Ph.D. grant SFRH/BD/14334/2003.

REFERENCES

[1] K. Koyama (ed.), Comparative Calculations of Propellers by Surface Panel Method -Workshop Organized by 20"* ITTC Propulsor Committee, Papers of the Ship Research Institute, Supplement No. 15, (1993).

[2] B. Gindroz, T. Hoshino and J.V. Pylkkanen (eds.), 22"d ITTC Propulsion Committee Propeller RANS/Panel Method Workshop, Grenoble (1998).

[3] S. Pyo, Numerical Modeling of Propeller Tip Flows with Wake Sheet Roll-Up in Three Dimensions, Ph.D. Thesis, Department of Ocean Engineering, Massachusetts Institute of Technology, (1995).

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J. Baltazar, J.A.C. Falcao de Campos and .1. Bosschers

[4] J.A.C. Falcao de Campos and P. Ferreira de Sousa, Grid Convergence Studies for Pro­pellers DTRC P4119 and P4842M with the Panel Code PROPAN, IST-MARETEC RT-D087-5, (2001).

[5] J. Baltazar and J.A.C. Falcao de Campos, Grid Convergence Studies for Elliptical Wings with the Panel Code PROPAN, IST-MARETEC RT-D087-7, (2001).

[6] S.A. Kinnas, S. Pyo, C.-Y. Hsin and J.E. Kerwin, Numerical Modelling of Propeller Tip Flows, In Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics, (1993).

[7] J. Baltazar and J.A.C. Falcao de Campos, Potential Flow Calculations on Elliptical Wings with Hydrodynamic Tip Grids, IST-MARETEC TR-D087-10, (2004).

[S] J. Baltazar e L. Eca, Geracao de malhas estruturadas eni superffcie, Mêtodos Cora-putacionais em Engenharia, Lisboa (2004). (in Portuguese).

[9] L. Morino and C.-C. Kuo, Subsonic potential aerodynamics for complex configura-tions: a general theory, AÏAA Journal, Vol. 12, No. 2, pp. 191-197, (1974).

[10] J.E. Kerwin, S.A. Kinnas, J.-T. Lee and W.-Z. Shih, A surface panel method for the hydrodynamic analysis of ducted propellers, SNAME Transactions, Vol. 95, pp. 93-122, (1987).

[11] J.C. Moulijn and G. Kuiper, The influence of the wake model on induced veloci-ties in the propeller plane, Proceedings of the International Conference on Propeller Cavitation, PROPCAV'95, pp. 213-217, (1995).

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