Theoretical Calculation of Propulsive Performances of

(Read at the Autumn Meeting of The Society of Naval Architects of Japan, Nov. 1989)17

Theoretical Calculation of Propulsive Performances

of Stator-Propeller in Uniform Flow

by Vortex Lattice Method

by Mitsuhisa Ikehata*, Member Subrata Chanda**

Summary

By fitting stator vanes behind a screw propeller energy loss of rotational flow in the slip stream of

the propeller can be reduced and hence the propeller efficiency can be improved. Then the stator

generates thrust to propel a ship. Therefore the screw propeller with the stator behind it acts as one

propulsor. We call it stator-propeller in the present paper. So far, experimental methods were used for designing stator vanes and for obtaining propulsive

performances of the stator-propeller. For experiments we need scale models of the stator-propeller and the ship, and the experimental work is time-consuming and expensive.

The aims of the present research are to develop a method for theoretical calculation of propulsive

performances of the stator-propeller and to develop a method for design of the stator by theoretical calculation in case that the geometry of the installed screw propeller is known.

The vortex lattice method has been employed for the above-mentioned purposes. Kerwin's roll-up model has been applied as the vortex lattice model of the screw propeller. A stator has been represented by one spanwise line vortex along with rectangular vortex rings trailing in downstream. The outside duct has been replaced by a circular vortex ring in the duct and the trailing vortex lattice cylinder in

downstream. The developed methods have been applied to two example models of the stator-propeller and the

calculated performances have been compared with the results of open water tests. They have shown

good agreement. The results of performance calculation show an appreciable improvement in propeller efficiency in open water and additionally the load of the screw propeller has reduced due to the effect

of the stator. The twisting angle of the stator vane designed by the developed method for theoretical design as for

the same two examples has been also in good agreement with that of actual models designed by the conventional method based on experimental data of rotational angle of flow velocity in the race of the

screw propeller. This fact concludes the completion of the theoretical design method of the stator-

propeller.

1. Introduction

The well-known contra-rotating propeller can reduce

the rotational energy loss in its race so much that the

propulsive efficiency can be improved remarkably. But

because of greater mechanical complications of its

shaft it has not been popular till recently. Nevertheless

there have been made many studies on the contra-

rotating propeller by some researchers1)•`5). The simplest

way of reducing the rotational energy loss due to the

same principle is to employ stator vanes or fins either

upstream6)•`8) or downstream9),10) of the screw propeller.

The present paper is on the theoretical study of

improvement of propulsive performance by fitting

stator vanes behind a screw propeller. As shown in Fig.

1, several stator vanes are set between the inside hub

and the outside duct. Hereafter this whole propulsive

device, i. e. the stator along with the screw propeller

will be referred to as "Stator-Propeller". Experiment

results showed appreciable improvement in propulsive

performances using the stator-propeller9),11). There have been, so far, no theoretical method avail-

able to calculate propulsive performances of the stator-

* Yokohama National University, Departement of Naval Architecture and Ocean Engineering.

** Graduate Student of Engineering, Yokohama National University at the time of research.

Now living in India

Fig. 1 Configuration of stator-propeller

18 Journal of The Society of Naval Architects of Japan, Vol. 166

propeller and also to design the stator suitable to the

given screw propeller. So the aims of the present research are to develop a theoretical method for calcu-

lating the propulsive performances of stator-propellers

and to develop a theoretical method for designing

stators.

The vortex lattice method has been employed for

theoretical calculation. Kerwin's model' has been em-

ployed for the vortex model for the screw propeller and new models have been proposed for the stator and the

outside duct. The computer program for calculations of

propulsive performances of the stator-propeller has been applied to two examples. The calculated results

has been found to be in good agreement with those of

model tests in the experimental tank.

It has been proved also by such theoretical calcula-

tion that by fitting the stator behind the screw propeller,

not only the efficiency of the propulsor is improved but

also an appreciable reduction in load on the blade of the

screw propeller is brought about. This implies that the

stator-propeller can be used for improving not only

propulsive efficiency but also cavitation characteristics. The method for theoretical design of the stator has

been applied to two examples of the model stator-

propeller. The calculated propulsive performances of

such designed stator-propeller has shown appreciable

improvement as expected.

2. Numerical lifting surface theory for stator-

propeller

2. 1 Vortex lattice model for stator-propeller

The principle of the stator is illustrated in Fig. 2. The

rotational velocity of the slipstream of the screw pro-

peller makes the angle of attack of the resultant inflow velocity to the stator blade section. Due to Kutta-

Joukowski theorem the lift is generated on the stator blade. The direction of the lift is inclined forward, so

that the axial component of it becomes thrust to propel

the stator blade forward. On designing the twist angle

of the stator, axial and rotational components of veloc-

ity of the slipstream at each section of the stator blade

with various radius should be known. So far they have

been measured in the downstream of a screw propller model. But the screw propeller model is expensive and the measuring experiments are time-consuming. Hence a theoretical method to calcuate such velocity compo-nents of the slipstream of the screw propeller should be developed by any means.

The vortex lattice method has been chosen to this aim because of its feasibility to take into consideration the combination of a screw propeller, the stator and the duct of stator. The thickness of propeller blades, stator blades and the duct of stator has been considered to be small. Hence modeling of the screw propeller blade, the stator blade and the duct of stator has been made as follows. ( 1 ) According to the concept of Kerwin's roll-up model, a propeller blade have been replaced by the lattice of vortices and sources on its mean camber surface, along with a distribution of vortices shed in the transition wake and roll-up in the ultimate wake. The sources distribution has been assumed to be independent of time and determinable by application of the thin wing theory. Fig. 3 shows the vortex lattice model of the screw propeller. ( 2 ) A blade of the stator is re-

placed by one spanwise line vortex at the position of a quarter chord from the leading edge on the mean cam-ber surface, which is followed by the lattice of trailing and shed vortices in a radial plane in downstream wake. The displacement effect of thickness of the stator blade is represented by three lines of sources on the mean camber surface. Fig. 4 shows the vortex lattice model of the stator. ( 3 ) The duct of the stator is replaced by a octagon bound vortex ring at the position of a quarter chord from the leading edge on the mean camber sur-face which is followed by the cylindrical lattice of trailing and shed vortices in downstream. Fig. 5 shows the vortex lattice model of the duct.

The number of vortex elements is as follows. On a propeller blade : 7 (spanwise) x 8 (chordwise), On a stator blade : 7 (spanwise) x 1 (chordwise) ,

On a duct of stator : 8 (peripheral) x 1 (chordwise). 2. 2 Determination of circulation of vortices and

streangth of sources By applying the principle of conservation of circula-

tion at each lattice point and Kerwin's formula we can

Fig. 2 Resultant inflow velocity and forces on a section

of stator blade

Fig. 3 Vortex lattice model of propeller

Theoretical Calculation of Propulsive Performances of Stator-Propeller in Uniform Flow by Vortex Lattice Method 19

express the circulation strengths of chordwise, trailing

and shed vortices by the corresponding spanwise vor-

tex. Hence the circulation strengths of spanwise vor-

tices are remained as unknown variables. The circula-

tion strengths of the spanwise vortices are determined

by satisfying the kinematic boundary condition at each

control point. The control points are assigned at the

geometric center of each vortex lattice element. The kinematic boundary condition is given by the

equation :

( 1)

where V is the resultant velocity vector at the i-th

control point and n, is the outward normal vector to the

mean camber surface at the i-th control point. The

resultant velocity V, at the i-th control point is com-

posed of the summation of all induced velocities of the vortex lattices of the propeller, the stator and the duct

of stator, and undisturbed inflow velocity, namely,

( 2 )

where V,I(i is the velocity at the i-th control point

induced by the vortices related to propeller, V,C,, is the

velocity at the i-th control point induced by the sources

of propeller, is the velocity at the i-th control point

induced by the vortices related to stator, V is the


of stator, V, (). is the velocity at the i-th control point

induced by the vortices related to duct, VI()) is the


of duct and V,, is the undisturbed inflow velocity.

Since the strengths of sources will be given by after-

mentioned relations and the undisturbed inflow velocity

are known, VC), V, VI", and Va are known at the

outset and V,1,1, V,s,, and 17,1(), can be expressed in terms

of the circulation strengths of spanwise vortices of

propeller, stator and duct respectively. Hence the circu-lation strengths of spanwise vortices of propeller, stator

and duct are determined as the solutions of equation

( 1 ) . Substituting eq. ( 2 ) into eq. ( 1 ) and modifying it

with respect to the circulation strengths of spanwise

vortices,

( 3 )

where

( 4 )

K, Ks: no. of propeller blades and stator blades re.

spectively

M, Ms, MI): no. of spanwise vortex elements of propel.

ler blade, stater blade and duct respectively

N : no. of chordwise vortex elements of propeller

blade

N,, Nis, no. of elements of trailing vortex of pro.

peller blade, stator blade and duct respectively u, : normal component of the velocity at the i-th

control point induced by unit ring vortex on

propeller blade, stator blade and ductz6∫, normal component of the velocity at the i-th

control point induced by unit ring vortex on

propeller wake, stator wake and duct wake11κ: circulation strength of spanwise vortex at time

Lit

7m: total circulation around the m-th all spanwise

vortices at time Ot, that is

(5)

The superscripts P, S, D stand for propeller, stator and duct respectively.

The control points are taken on all blades of the

propeller, on all blades of the stator and on the duct of stator. Thus a set of simultaneous algebraic linear equations with respect to circulation strengths of spanwise vortices are deduced from eq. (3) and solved at the time tilt. At the time (t+1)Jt, another set of

Fig. 4 Vortex lattice model of stator

Fig. 5 Vortex lattice model of duct

20 Journal of The Society of Naval Architects of Japan, Vol. 166

simultaneous equations are deduced and solved because

the propeller blades rotate relative to the blades of

stator.

The distribution of sources on the mean camber

surface can represent the thickness effect of propeller

blades, stator vanes and duct. The density of source

distribution is proportional to the derivative of thckness

of the blade according to thin wing approximation.

Since we replace the continuous source distribution by

some discrete line sources, the strength of the discrete

line source representing the chordwise element between

sn and sn+i is given by the formula :

( 6 )

where s is the chordwise coordinate measured from the

leading edge,

(7)

V: axial component of incident velocity at the

position of line source (the advance velocity for the propeller blade)

14 : circumferential component of incident velocity at the position of line source (271-nr for the

propeller blade, where n =no. of revolutions per second, r is the radial distance)

t(s) : blade thickness at position sThe correction factor, 0.83, has been multiplied for the strength of source calculated by eq. ( 6 ) and no correc-tion has been made for sinks.

The number of discrete line sources and sinks along a certain chord is 8 for a propeller blade, 3 for a stator blade and 3 for the duct. The locations of such discrete line sources and sinks coincide with those of spanwise vortex lattices for the propeller blade and are 1/6, 1/2, 5/6 of chord for the stator blade and for the duct.

The calculation starts when the key blade of the

propeller is at the same position of the top blade of the stator, and proceeds increasing the angular position of the key blade of the propeller by JO 7rnZlt until the angular position of the key blade coincides the angular

position of the next blade of the stator. 2. 3 Hydrodynamic forces of stator-propeller Once the circulation strengths of all vortex elements

have been determined, the hydrodynamic forces of the

propeller, the stator and the duct can be calculated by using the same formulas as the previous paper") at the time of tilt.

The total hydmodynamic force of the stator-proleller

is the vector summation of the following forces : 1) Lagally forces acting on sources and sinks,

2) Kutta-Joukowski forces acting on spanwise and chordwise vortices on blades and duct,

3) Unsteady forces due to the time derivative of sum of spanwise vortices between the leading edge and the concerning point along the chord,

4) Viscous drag along the chord of blade element. Finally the thrust and the torque of the propller and the thrust of the stator and the duct are calculated from the

above-mentioned hydrodynamic force of the stator-

propeller. Thus the propulsive performances of the stator-propeller can be calculated.

3. Propulsive performances of stator-propeller

in uniform flow

The two models of stator-propeller has been chosen

as examples for applying the developed theoretical

calculation method. One is the propulsor for a 6m long

tanker model, which is named P 175. The other is the

propulsor for a 5m long bulker model, which is named

P 137. The principal particulars of P 175 and P 137 are

Table 1 Principal particulars of stator-propeller P175

Table 2 Principal particulars of stator-propeller P137


given in Table 1 and Table 2 respectively. The thrust and the torque in the uniform flow have been calculated

in order to obtain the open water characteristics, KT

(thrust coefficient) , KQ (torque coefficient) , 7) (propeller

efficiency) , versus J (advance coefficient) . The thrust of

KT is the total thrust of the propeller and the stator

with the duct. The distribution of the twist angle of

stator blades is shown in Fig. 6 for P175 and in Fig. 7 for

P 137. The maximum twist angle is about 12 degrees for

P 175 and about 9 degrees for P137 at the root of blades.

The distribution curve of the twist angle is like a

hyperbola similarly for P175 and P137.

The calculated open water characteristics of the two

stator-propeller models are shown in Fig. 8 for P175 and

in Fig. 9 for P137. It is found in Figs. 8 and 9 that the

thrust of the stator-propeller (i. e. with stator) is

greater than that of the propeller only (i. e. without stator) , the torque of the stator-propeller is less than

that of the propeller only, so that the efficiency of the

stator-propeller is higher than that of the propeller only.

The most gain of calcuated efficiency by the stator is

7.2% at J=0.2 for P175, and is 5.7% at J =0.2 for P137.

But the difference between with and without stator

decreases as the advance coefficient increases. This

means the effect of the stator is greater as the load of

the propeller is higher at the lower advance coefficient.

This is reasonable because the rotational velocity in

the slipstream of the propeller is the greater, the higher

is the load of the propeller.

The experimental open water characteristics are

shown in Fig. 10 for P175 and in Fig. 11 for P137. The

results in Figs. 10 and 11 show similar trends as those of

the calculated open water characteristics in Figs. 8 and

9.

The comparison between calculated and experimen-

tal open water characteristics of the stator-propeller

Fig. 6 Angle of twist of stator blade (P175)

Fig. 7 Angle twist of stator blade (P137)

Fig. 8 Calculated open water characteristics (P175)

Fig. 9 Calculated open water characteristics (P137)

Fig. 10 Experimental open water characteristics

(P175)

22 Journal of The Society of Naval Architects of laDan. Vol. 166

(P175) is shown in Fig. 12. They are in good agreement

with each other. This proves the accuracy of the devel-

oped calculation method.

Fig. 13 shows the changes of thrust coefficient, torque

coefficient and open water efficiency of propeller in

percentage of the coefficients for propeller only when

the stator is fitted. I)KT, DIN and DETA are all

positive. This means that thrust and torque of the

propeller are reducing due to the influence of the stator.

So it can be concluded that the load on the propeller

itself is reduced by the contribution of the stator. This

fact is favorable from the point of view of cavitation

performance. Fig. 14 shows the calculated contribution of the stator (with duct) in the total thrust of the

stator-propeller P175. It is noticed that the contribution

of the stator decreases with increasing J because of low

angle of attack of incident flow into the stator blade at

high values of advance coefficient.

Fig. 15 shows the calculated variations of K, , KQ and

77 as propeller blades rotate. It is observed in Fig. 15 that

the unsteadiness of the open water characteristics of the

stator-propeller is negligible small.

The circulation around the m-th section of the propel-

ler blade, is given by eq. ( 5 ). And the radial distri-

butions of the circulation of propeller blades are shown

in Fig. 16 in the non-dimensional form :

( 8 )Fig. 11 Experimental open water characteristics

(P137)

Fig. 12 Comparison between calculated and experi-

mental open water characteristics of stator-

propeller (P175)

Fig. 13 Decrease in contribution of propeller when

stator is fitted (P175)

Fig. 14 Contribution of stator in total thrust (P175)

Fig. 15 Variation of open water characteristics of

stator-propeller for rotation of propeller

(P175)


,where D is propeller diameter and VA is the advance

speed of the propeller. There are differences between

propeller blades due to the different angular positions against fixed stator blades, but they are not so much as

to be considered. The circulations of propeller only (i. e.

without stator) are nearly mean of all blades of the

stator-propeller (i. e. with stator) . The circulations

around stator blades are shown in Fig. 17 in the non-

dimensional form :

( 9 )

, where Ds is diameter of stator. The differences between blades are not so much as to be taken into account. The circulations of stator blades distribute

greater as r/R gets smaller. This distribution can be nearly regarded as the distribution of the thrust of the stator blade.

4. Theoretical design of stator

We shall consider only hydrodynamic design of

stator. It is at first assumed that the principal particu-

lars of the propeller are given and the working condi-

tion of the propeller, that is the design point of advance

coefficient, is settled. The geometrical parameters of the

stator are presumed as follows :

1) The diameter of the stator wheel and the outside

duct is the same as that of the propeller.

2) The number of stator blades is taken as 8 from

experimental experience.

3) The chord of the stator blade is taken as one

tenth of the propeller diameter from experimental

experience.

4) For the blade section of the stator NACA 4412 is

chosen because it generates positive lift even at a

negative angle of attack of around degrees.

5) The position of stator is such that there are

adequate clearances between propeller and rudder.

Thus the problem of theoretical design of stator leads

to determine the radial distribution of the twist angle of

stator blades. The principle of determination of twist

angle of stator blades has already been developed in

Appendix of reference 9) by one of the authors. So far

the experimental method has been employed in order to

know axial and rotational velocities induced by the

propeller. The measurement of these axial and rotational velocities by 5-holes Pitot tube has been

carried out at the position of the stator in the slipstream

of the propeller. Furthermore we must know the incre-

ment of the rotational velocity in the far rear of the

propeller. So far it has been obtained by empirical way such as measurement or prediction from data. In theo-

retical design of stator such velocities are calculated by

the vortex lattice method. The procedures to determine

the twist angle of stator is as follows.

( 1 ) The axial and rotational components of the

velocities at points on the spanwise vortex elements of

the stator blade are calculated considering the propeller

only. Let the components be u and v respectively.

( 2 ) The rotational component of the induced veloc-

ity of the propeller at infinite distance from the propel-

ler (supposing 10 complete revolutions of ultimate

wake of the propeller to be such infinity) . Let this

rotational induced velocity at infinity be v-. The ratio

of the increment of the rotational induced velocity of

the propeller to that at the position of the stator is

calculated by

(10)

( 3 ) The optimum stator induces the opposite rotational velocity to cancel (1 + E)v at infinity. There-fore the self induced rotational velocity of the stator at its own position is (1 + E)v/2. Thus the resultant inflow velocity into the section of the stator blade is determinable.

( 4 ) Since the resultant inflow velocity (vector) has been determined, the twist angle of the section of the stator blade can be calculated by the iterative proce-dure according to the principle described in Appendix of reference 9) . The computer program for theoretical designing the distribution of the twist angle of stator blades has been

Fig. 16 Distribution of circulation on propeller blades

(P175)

Fig. 17 Distribution of circulation on stator blades

(P175)

24 Journal of The Society of Naval Architects of Japan. Vol. 166

developed and applied to the same two examples, P175

and P137. Figs. 18 and 19 show the comparison between

the twist angle designed by the present method and the

experimental methods. It can be mentioned from Figs.

18 & 19 that the result of theoretical calculation is in

very good agreement with that of experiment. Figs. 20

and 21 show the comparison of open water characteris-

tics of stator-propeller with experimentally given and

theoretically designed twist angle. It is of course seen in

Figs. 20 and 21 that the agreement between design and

given is quite good. So conclusively the accuracy of the

present design method by theoretical calcuation has been found very satisfactory even for practical use.

5. Final remarks and acknowledgments

In the present paper the authors has developed a

method for theoretical caciculation of propulsive per-

formances of the stator-propeller and a method for theoretical design of the stator behind a given propeller employing the vortex lattice models for the propeller and the stator with the duct. The results of numerical examples has shown good agreement with experimental results. The gain of the efficiency of the stator-propeller is proved to be about 5% to 7% by theoretical calcula-tion of the present method.

The authors appreciate discussions and encourage-ments given by Professor Emeritus H. Maruo and Associate Professor K. Suzuki. They are grateful for their assistances of Mr. Z. Takusagawa and Mr. I. Okada in the laboratory of Marine Hydrodynamics of Yokohama National University. They used HITAC M-280D computer system of Information and Process-ing Center of Yokohama National University and HITAC M-680H/M-682H computer system of Tokyo University Computer Center with great thanks.

References

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Fig. 18 Comparison of angle of twist of stator blade

between calculation and experiment (P175)

Fig. 19 Comparison of angle of twist of stator blade

between calculation and experiment (P137)

Fig. 20 Comparison of open water characteristics of

stator-propeller between designed and given

angle of twist (P175)

Fig. 21 Comparison of open water characteristics of

stator-propeller between designed and given

angle of twist (P137)


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Documents

Theoretical Calculation of Propulsive Performances of