FLOW ANALYSIS, DESIGN AND TESTING OF DUCTED PROPELLERS Gert-Jan Zondervan, Martin Hoekstra and Jan Holtrop (visitors)1

ABSTRACT

This paper addresses the design and the testing of ducted propellers and illustrates how flow analysis can be used as a support. After a review of the working principles of a ducted propeller system by referring to well-known theoretical analyses, it describes developments in computational methods. In particular a numerical code simulating the viscous flow around a ducted propeller system in open water shows promising results. Since propellers in ducts are designed as quasi-open propellers, the estimation of the effective advance speed, influenced by the velocity field induced by the duct, is an important aspect. New estimates for dividing the induced flow field into a propeller and a duct contribution are given, which lead to better correspondence with experimental information. Polynomials describing the performance of the B-4-70 propeller in nozzle 19A are also provided, in addition to an empirical relation for the virtual pitch of the Ka-series propellers. Model testing of ducted propellers is evaluated against numerical simulation tools. It is concluded that, in spite of rapid developments in computational fluid dynamics, the propulsive performance of ducted propellers can still most accurately be assessed by model tests.

1 All three authors are employed by the Maritime Research Institute Netherlands (MARIN)

NOMENCLATURE Ae/Ao expanded blade area ratio Ao - propeller disk area c - section chord length Cq - discharge coefficient CT - thrust loading coefficient D - propeller diameter E - kinetic energy g - gravitational acceleration J - advance ratio KT - thrust coefficient KQ - torque coefficient L - length of duct n - rotation rate p - pressure P - pitch of propeller PD - delivered power Pv - virtual pitch of propeller Q - torque, volume flow Rn - Reynolds number T - (total) thrust Tp - propeller thrust Tn - nozzle thrust t/l - thickness-duct length ratio u1 - induced velocity (at propeller) u2 - induced velocity (downstream) up - induced velocity by propeller un - induced velocity by nozzle V - speed of advance Vp - flow speed at propeller disk Vn,b - transpiration velocity vj - jet velocity (downstream) Z - number of blades ηi - ideal efficiency ρ - mass density τ - ratio Tp/T INTRODUCTION The advantages of propulsion systems using thrust generating ducts around propellers are well acknowledged by today’s naval architects. Ever since Kort (1934) in the 1930’s discovered the aerofoil shaped duct as a way to increase the efficiency of heavily loaded propellers, the ducted propeller concept has proven its merits and has seen a substantial number of applications in the marine industry. Propeller ducts are fitted to a wide range of vessels such as tugs, push boats, dredgers, trawlers and tankers and they are often an essential part of azimuthing thruster systems.

The design and the prediction of the hydrodynamic performance of ducted propellers are far from trivial, however. The scope and validity of available computational models used in practical

propeller design and analysis is limited, while model test results on these propulsors are complicated and subject to scale effects. Moreover, the feedback from trials with ducted propellers at full scale is often imprecise, incomplete and/or non-systematic. A designer must therefore take due care in interpreting his information and in properly choosing important propeller parameters such as the pitch.

Simple momentum balance considerations and the testing of systematic series of ducted propellers have provided the basic design information, which is used until today. But research on ducted propellers has not stopped there. Further developments in analysis tools have been pursued and more experimental data have been acquired. Although the new analysis tools are seldom used directly in the design process, they can fruitfully be applied to verify a design or to get a better understanding of the flow behaviour, which may well lead to design corrections.

This paper aims at reviewing the subject, providing an update based on some recent computational work and proposing some new views on the design and analysis of propellers and nozzles. WORKING OF PROPELLER IN A DUCT A duct around a ship propeller can be considered as an annular hydrofoil on which a lift force is created due to pressure differences around the foil. Only if a propeller is operating, the lift force has an axial component. The propeller induction causes the duct sectional profile to experience an angle of attack, which can be either positive or negative, depending on the duct shape and the loading of the propeller. Thus two categories of nozzles are distinguished. The accelerating nozzle type creates increased flow velocity (i.e. lower pressures) inside the nozzle at the propeller location; the force on the duct has a net contribution in axial direction, an additional thrust. Conversely, the decelerating nozzle reduces the axial flow (and increases the pressure) in way of the propeller at the expense of an additional drag contribution. Notice that a particular duct can happen to be a flow-accelerating nozzle in a certain propeller loading regime, and be a flow-decelerating nozzle in another regime.

The accelerating nozzle offers efficiency benefits and has therefore gained popularity. The benefit of decelerating nozzles must be sought in the possibility to suppress cavitation, a feature which is important e.g. for low-noise shrouded propellers of naval vessels. Simple momentum theory To understand the basic working principles of the ducted propeller, the simple momentum theory for a shrouded actuator disk is instructive. Consider a

ducted propeller system advancing steadily at a speed V (Figure 1). Inside the ducting, in way of the propeller disk of a diameter D, we suppose that the induced velocity is u1, while in the slipstream the induced velocity is supposed to be u2.

Figure 1: Actuator disk representation of ducted

propeller

The velocity of the fluid in the propeller disk, Vp, is the sum of the entrance velocity, V, and the induced velocity u1. The momentum added to the flow through the system per unit of time must be equal to the exerted force. The momentum balance is applied to the whole propulsion system of which the total thrust, T, in steady forward motion is the sum of the nozzle thrust, Tn and the propeller thrust, Tp. This results into:

p n 2T=T +T =ρQ u , (1) where ρ is the mass density of the water and Q is the volume flow through the duct:

2P 0 1 o 1Q=V A =(V+u )A =(V + u )¼πD . (2)

Next, we apply Bernoulli’s law both upstream and downstream of the actuator disk under the assumption that the pressure sufficiently far upstream and downstream of the actuator disk is equal to p∞. By putting:

2p o p a bT /A =T /(¼πD )=p -p , (3)

we find:

22 2 pρu (V+½ u )¼πD = T (4)

Apparently, the induced velocity in the slipstream far downstream, u2, is a function of only the propeller thrust Tp and independent of the nozzle thrust Tn.

For sake of convenience we introduce the symbol τ to represent the fraction of the propeller thrust to the total thrust: τ=Tp/T. By using the thrust-loading coefficient CT with

P N T-TotT 2 2 2 2 2

T +T K8 T 8C = = =π½ ρ V ¼ π D πρV D J

, (5)

we find for the induced velocities:

2T

u = 1+τC -1

V (6)

and

( )1T

u 1 = 1+τ C 1-2τV 2τ

+ . (7)

In contrast to u2, the induced velocity at the propeller disk, u1, is a function of both the propeller thrust and the nozzle thrust. According to the definition of τ the nozzle is fully inactive for τ=1. With Tn=0, the expressions for the induced velocities lead to: u2 = 2 u1, and are fully in line with the classical actuator-disk theory of open propellers. The mean axial velocity at the propeller disk, Vp = V + u1, becomes:

PT

V 1= ( 1+τC +1)V 2τ

. (8)

Moreover, from the given relations it follows that:

J 2

P 1

V+V 2V+uτ= =

2V 2(V+u ). (9)

For some applications it is convenient to express the equations for the mean axial velocity, Vp, and the induced velocity, u1, in terms of J and the thrust coefficient of the propeller:

2PT-prop

V 1 8= (J+ J + K )nD 2τ π

(10)

and

21T-prop

u 1 8= (J+ J + K )-JnD 2τ π

. (11)

In the case of zero entrance speed, i.e. J=0, the first expression reduces to:

PT-prop

V 1 8= KnD 2τ π

(12)

or

P 2

2 TV =τ ρπD

. (13)

The equations (6) and (7) for the induced velocities imply a change in shape of the streamtube that passes through the edge of the propeller disk with varying τ. Comparing an open propeller (τ=1) with a flow-accelerating ducted propeller (τ<1) with the same diameter and loading, the latter’s streamtube has greater cross-sectional area at both the upstream and the downstream end. In other words, the streamtube contraction is greater at the upstream side, but less at the downstream side, than for an open propeller. Efficiency aspects The kinetic energy added per unit of time to the slipstream is equal to the total power supplied to the propulsion system, PD, minus the viscous losses in the ducting. The latter are not present in non-viscous potential flow. The power supplied is equal to 2πQn for rotating propellers (note that Q denotes here the torque, instead of the volume flow rate elsewhere in this paper). The power supplied is supposed to be transferred to pressure and momentum added to the flow in axial direction. This assumption implies an impeller efficiency (or a pump efficiency in the case of a water jet) of 100 per cent because no energy is being lost in the transfer from power to pressure and to axial momentum of the fluid. The kinetic energy added to the slipstream per unit of time, E, is equal to:

2 2 2 2j 2

2 22 2 2

E = ½ρQ (v -V )=½ρQ [(u +V) -V ]

= ½ρQ(u +2Vu ) = ½ρQu +TV (14)

where TV is the "thrust power" and the term ½ρQu2

2 is commonly referred to as the "loss in the slipstream". Substituting the expression of the flow rate, Q, gives:

2 21 2 2E=½ρ(V + u ) ¼πD (u +2 V u ) . (15)

The ideal efficiency of a ducted propeller system is defined as the work done by the system divided by the power supplied, or:

i 212 2

222

TV TVη = =E TV+ ρQu

2 2= =2 + u V2 + ρ Q u (TV)

(16)

Substitution of the expression for u2/V in equation (6) gives:

iT

2η = 1 + 1+ τ C

. (17)

So, the highest efficiency is potentially attained if the nozzle carries a relatively large part of the total thrust, i.e. if τ has a low value. In Figure 2 the ideal efficiency is given for a range of thrust loadings and thrust ratios.

Figure 2: Ideal efficiency versus thrust loading

coefficient

Again, for τ equal to 1, a condition in which the

nozzle does not impart any axial force, the expressions become identical to those for an actuator disk representing a single open propeller.

Although the simple momentum theory for a shrouded actuator disk reveals some important trends, it cannot be applied without due care. For example, the efficiency according to equation (17) can in a real flow never be attained, due to effects not accounted for. The major ones are viscous losses, rotational losses, losses originating from the finite number of blades and losses related to non-optimum radial loading distribution. In this paper the section on propeller design gives another example of the global nature of the theory summarised above: it does not explicitly tell what the contribution of the duct is to the induced velocity at the propeller plane.

Other theoretical/numerical models Several attempts have been made to improve and enhance the numerical models for describing the hydrodynamic performance of ducted propellers. The problem here is not so much the formulation of the equations governing the flow, but rather to find a numerical model which combines sufficient realism with an acceptable effort required to solve it. We summarize here what has been done at MARIN.

Falcão de Campos (1983) developed an inviscid flow model, modelling the duct by a vorticity distribution on the duct surface in its true shape and representing the propeller by an actuator disk. To this model he added the effect of viscosity on the flow around the duct by employing an integral method for the calculation of the boundary layer. The results of this method were compared with Laser-Doppler velocimetry measurements on a type 37 nozzle with generally good results. His research efforts have not only provided valuable insight into the performance of ducted propellers but also a useful data set for the 37 duct.

Next to lifting line and lifting surface methods, surface panel methods (or also called: Boundary Element Methods) have been set up for propellers. These comprise also an inviscid flow model, and lend themselves well to include the duct. They allow the interaction between propeller and duct to be evaluated, because the propeller is represented in true shape instead of as an actuator disk. Bosschers has recently carried out some exploratory research with this tool, analysing the propeller-duct combination Ka 4-55 in a 19A duct. In his studies he paid special attention to the modelling of the flow through the gap between the propeller blade tip and the nozzle. In panel methods the flow through the gap can be modelled with additional source panels with a prescribed transpiration velocity in the gap, using a discharge coefficient as originally defined by van Houten and discussed by Kerwin et al. (1987) and Moon et al. (2002). The discharge coefficient Cq is defined as the ratio between the actual transpiration velocity Vn and a transpiration velocity Vb based on the pressure difference between pressure and suction side of the propeller as obtained from Bernoulli:

nq

b

VC =

V (18)

where the velocity component Vb from the Bernoulli equation is given by:

b2∆pVρ

= . (19)

Typical values for the discharge coefficient compiled by van Houten range from 0.76 to 0.92. Bosschers used a much greater variation and found a strong effect on the radial loading distribution of the propeller (see Figure 3 for an example). A careful tuning with experimental data is therefore needed.

r/R

Fx0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3 no gapCq= 1Cq= 4Cq= 8Cq=12

J= 0.4

Figure 3: Influence of gap discharge coefficient on

the radial blade loading distribution for J=0.4

A problem with inviscid flow models for ducted

propellers is the choice of the stagnation point (i.e. imposition of the Kutta condition) on the rear of the duct. In a viscous flow model that problem is not encountered, which makes it attractive for analysing ducts with blunt trailing edges. Today, the solution of the Reynolds-averaged Navier-Stokes (RANS) equations is feasible for many practical applications, and has also been applied to ducted propellers [e.g. Abdel-Maksoud & Heinke, 2003]. But the huge computational effort involved is an impediment for the application in a design process.

The RANS equations, mentioned above, are also the basis for a new tool developed at MARIN for the viscous-flow analysis of ducted propellers in open water (Hoekstra, 2006). However, considering that the viscous flow around the duct will not be drastically affected by the details of the propeller geometry, the propeller is represented by an actuator disk of finite thickness (distribution of volume forces), while the duct is retained in its true shape. This keeps the computational effort low and makes the simulation model particularly suited for duct design.

With this tool several flow simulations have been made for the 19A and 37 ducts. Particularly for the 19A duct, we did elaborate numerical investigations and were able to reproduce most experimental

information, among which the τ-CT relation (Figure 4), and the effects of slipstream swirl and hub diameter on Tn (Hoekstra, 2006).

The benefit of the analysis tool, next to providing global performance figures, is that insight is obtained in the flow behaviour. Thus we were able to confirm that flow separation occurs on the outer face of the 19A duct when the propeller is lightly loaded. We further found that the duct experiences a so-called shock-free angle of attack when CTprop is about 4. For higher propeller loading the front stagnation point moves to the outer face of the duct. For the high angles of attack occurring beyond CTprop = 10, the foil would stall if not the propeller suction prevented it. Indeed, our results indicate that massive flow separation from the leading edge on the inner face of the 19A duct does not occur under any operating condition. What might happen under extremely high propeller loading (e.g. bollard pull) is that a small separation bubble appears on the nose, because in the pressure distribution on the duct surface a local pressure minimum develops with increasing propeller loading, implying an adverse pressure gradient over a small distance along the duct surface. But even if this would cause the flow to separate, reattachment would occur immediately afterwards, because of the favourable pressure gradient further downstream. So a tiny separation bubble could appear, but it would be an almost invisible correction of the duct shape and can not be expected to influence the duct thrust to any significant extent.

CTP

tau

100 101 1020.4

0.5

0.6

0.7

0.8

0.9

1

1.1

PolynomialsComputed

Figure 4: Comparison of measured and computed

duct thrust as a function of the propeller thrust

What does happen, though, is flow detachment

from the inner duct face just ahead of the propeller

position. This gives generally rise to a small separation bubble, but depending on the radial loading distribution of the propeller, the tip clearance and the Reynolds number (see below) it may grow in size and have an appreciable influence on the duct performance. An example will be shown in the sequel.

More applications of this flow simulation tool for ducted propellers are underway and results will be published in due time. DUCT DESIGN ASPECTS Classical nozzle shapes At the former NSMB (now MARIN) extensive systematic experiments on ducted propellers have been carried out. This work has promoted the practical application of these systems, Oosterveld (1973). The investigated nozzles include accelerating as well as decelerating types (Figure 5). In various articles by Van Manen et al. (1953, 1957, 1962), Oosterveld et al. (1968, 1970) and Kuiper (1992) the experimental investigations have been reported (Nozzle Nos. 2 to 37).

Figure 5: Overview of MARIN nozzle series

The practical offspring of these investigations is a

well-known propeller series (Ka-series) and two widely applied nozzles (No. 19A and No. 37). The 19A nozzle was derived from a small series of nozzles (No. 18 to 20). This nozzle series was based on the NACA 250 section type, having a 250-meanline in combination with a NACA 0015 thickness distribution. The nozzle series has a varying camber ratio (18: f/c=0.09, 19: f/c=0.07 and 20: f/c=0.05) and a nose-tail line at an angle of 10.2 degrees relative to the propeller shaft. The length-diameter ratio c/D of these nozzles is 0.5. As a result, the diffuser angle of these nozzles varied between 3.5 to 6.5 degrees.

From the shapes of nozzles 18 and 19 the more practical shapes of standard nozzles 18A, 19A and 19B were derived to meet a few practical requirements:

• a relatively thick tail of the section for structural purposes, stiffness and improved astern performance,

• a cylindrical part in way of the propeller, • a straight line at the exterior side of the

nozzle to reduce building costs and • a straight line in the diffuser part of some

nozzles at the inside (18A, 19B). At a later stage the square trailing edge shape of the 19A and 19B nozzles was modified to allow a circular ring and flow-separation edge being fitted for reasons of preventing steering problems by asymmetric flow separation. Such problems had been reported occasionally in the years 1960-1975. In all cases, the performance of the nozzles with and without a separation edge was considered identical.

The diffuser angle of the 19A nozzle gradually increases to a value a little above 5.5 degrees at the aft end. Notice that this series of nozzles was initially developed to exceed the maximum angle of the diffuser of 3.5 degrees, a diffuser angle which was considered to be the very maximum at that time. The shape of the sectional profile of the nozzles 21 (c/D=0.7), 22 (c/D=0.8), 23 (c/D=0.9) and 24 (c/D=1.0) is the same as that of the 19A nozzle.

A similar adaptation was made to the geometry of the standard propeller commonly used in nozzles. Since the Ka-type of propeller with the very thin tip edges was found to be too vulnerable in practice, a new planform and thickness distribution was designed and communicated to the Dutch propeller manufacturers in August 1962. This became the Kc-type of standard propeller. Its performance was considered equivalent to that of the Ka-type propeller (see also Yossifov et al., 1989a+b)

Nozzle 18 has been further simplified by the use of a straight, conical diffuser. The diffuser angle of the 18A nozzle has become 2 degrees, 51 minutes. Lack of interest for this type of nozzle has caused that the rear rounding and the separation ring were never applied to this variant of nozzle 18.

In agreement with the simplification from the 18 to the 18A nozzle with a straight diffuser, a 19B type nozzle has been developed with the same diffuser shape as that of the 18A nozzle, again with a diffuser angle of 2 degrees, 51 minutes. In all cases, both in the 19A and in the 19B nozzle, the separation ring is applied as a standard.

The general-purpose nozzle 19A, simplified as it is, is suited for a wide range of loadings. It is well known, from both experiments and numerical flow simulations, that in extremely lightly loaded conditions flow separation occurs at the leading on the external

face of the duct. But where ducts are typically applied to heavily loaded propellers, this is of little practical relevance.

Figure 6: Type 37 nozzle

The astern performance of the 19A nozzle is unfortunately rather poor. To improve the backing performance, a nozzle shape has been developed at MARIN with a more symmetrical section shape. This nozzle is known as standard nozzle number 37. The 37-type nozzle has only slightly lower efficiency in ahead conditions, but has superior qualities for running astern. In addition, in approaching bollard condition-ahead, nozzle 37 is a little better than nozzle 19A. The geometry of nozzle 37 is shown in Figure 6. New duct shapes

The last few years have seen attempts to design, develop and implement new shapes of ducts. This has led to dedicated designs for specific loadings and applications. Examples are the nozzles developed for Aquamaster (now part of Rolls-Royce), the Nautican nozzle, licensed by Wärtsilä Propulsion and Ulstein (part of Rolls-Royce), the Van der Giessen - Optima nozzle, the Hodi Superior Nozzle of Van Voorden.

In some cases excessive improvements are claimed which are hard to verify as they usually depend on single full-scale measurements in which systematic effects are difficult to quantify. It is striking though that it is usually admitted that model testing does not show a clearly superior performance of these new duct shapes over the classical ones. This would mean that the greater part of any substantial gain must be attributed to differences in scale effects. It is relevant therefore to consider possible scale effects in more detail. To that end MARIN has applied the viscous-flow analysis tool for ducted propellers, in open water, already introduced above (Hoekstra, 2006). Scale effects on flow around ducts For studying scale effects on the flow around a duct, a viscous flow model is indispensable. Having developed such a model with that purpose, we have

made calculations for the 19A duct with varying Reynolds number, i.e. scale effect studies. These have confirmed that the effectiveness of the duct increases with increasing Reynolds number. Figure 7 shows the computed changes in τ as a function of the Reynolds number for three different propeller loading coefficients; also the choice of the turbulence model has been varied. These results indicate that changes are monotonous and become drastic only when entering the domain of Reynolds numbers below 5*104 (where the Reynolds number is based on the chord length of the duct and the advance speed of the system).

Rn

tau

103 104 105 106 1070.5

0.55

0.6

0.65

0.7

0.75

0.8 Duct 19A CTP= 6 k-omega modelDuct 19A CTP=10 Menter modelDuct 19A CTP=40 k-omega model

Figure 7: Scale effect on duct thrust

It is interesting now to see what changes in the

flow cause the breakdown of the duct thrust. We have plotted in Figure 8 the pressure distributions and streamline patterns for Rn=104 and Rn=105, while CTP=6, which show that the primary change in the flow is the growth of the separation region near the propeller position. This changes the effective shape of the duct section profile, which affects the pressure distribution, causing a reduced duct thrust.

We have found no evidence at all that for Reynolds numbers in the range from 105 to 107 appreciable changes occur in the flow field, which could significantly affect the duct performance. Although small changes occur (as we mentioned, leading to a small improvement in the duct thrust with increasing Rn) there is not the least indication of flow features that might explain model test results for the 19A duct to be misleading due to scale effects. The computations show considerable portions of laminar flow in the boundary layer on the duct which get smaller with increasing Reynolds number. But that in itself is not a reason for strong scale effects. Only if the laminarity of the flow leads to flow separation

which would not occur or have a different appearance if the flow had been turbulent significant effects can be expected. This would mean that model tests can well be used to evaluate the performance of ducted propellers, provided they are carried out at sufficiently high Reynolds number. Admittedly, they still suffer then from scale effects, but these are small and predictable.

x/L-0.6 -0.4 -0.2 0 0.2 0.4 0.6

x/L-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Figure 8: Streamline patterns and pressure

distribution around 19A duct at Rn=104 (top) and Rn=105 (bottom)

PROPELLER DESIGN ASPECTS Most of the ducted propellers are designed as open propellers. An important parameter in this choice is the speed of advance of this “quasi-ducted propeller”. The propeller is designed in a flow field in which the action of the nozzle has to be accounted for. Eventually, the designer deduces the pitch to be applied on the basis of the correctness of the nozzle-induced flow field. Additional empirical factors to correct for the conditions in which the ducted propeller operates are not discussed here. They concern e.g. corrections for the presence of vertical struts, gearbox housings of a steerable thrusters or a rudder close to the rear end of the nozzle.

Fortunately, in many cases the choice of the pitch of the blades can be substantiated by a parallel analysis using the characteristics of systematic series of ducted propellers determined in model experiments. The extensive series of Ka-propellers in various types of

nozzles can be used to that purpose. Below, we will describe how the induced velocity of the duct at the propeller position can be estimated and how the results can be corroborated with experimental data. Induced velocities The induced velocity in the plane of the propeller disk, u1, can be considered as the sum of the ‘nozzle-induced velocity’, un, and the ‘propeller-induced velocity’, up:

p 1 p nV = V+u = V + u + u . (20) The reason to discriminate between the propeller-induced velocity, up, and the velocity induced by the nozzle, un is found in the use of propeller design tools for non-ducted propellers. In this manner a “quasi-ducted propeller” can be designed as a single, open propeller without further considering the nozzle, except for adaptations regarding the pressure changes, which are to be referred to later. The equivalent speed of advance of the open propeller is the sum of the entrance velocity, V, and the velocity induced by the nozzle, un. Naturally, the “quasi-ducted propeller” should produce the same velocity through the system, Vp.

Following the simple momentum theory presented earlier, Oosterveld (1970) and Van Manen and Oosterveld (1966) expressed these induced velocities in terms of the total thrust coefficient CT and the thrust ratio τ as:

PT

u 1= ( 1+τC -1)V 2

(21)

and

nT

u 1-τ= ( 1+τC +1)V 2τ

. (22)

A reformulation in terms of J and KT yields:

p 2T-Prop

u 1 8 = - J J KnD 2 π

⎛ ⎞+ +⎜ ⎟⎜ ⎟

⎝ ⎠ (23)

and

2nT-Prop

u 1-τ 8 = J + K + JnD 2τ π

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

. (24)

Thus the induced velocities for J=0 are given by:

T-Propp

8K τ 2Tu = ½ n D =π D πρτ

(25)

and:

T-Propn

8K1-τ 1-τ 2Tu = ½ n D =τ π D πρτ

. (26)

It has been found, however, that the expressions given above lead to a serious under-estimation of the nozzle-induced velocities for high thrust loadings, both in accelerating nozzles and in cylindrical tunnels. Alternative method The subdivision of the two components of the induced velocity is certainly not unique and an alternative approach is shown here by which the nozzle and propeller-induced velocities can be determined.

The basis for the design of the equivalent non-ducted propeller is still the entrance velocity which should be the sum of the original entrance speed, V, and the nozzle-induced velocity, un. Deviations of the induced velocity far upstream or downstream from other sources can be accepted as long as the volume flow rate through the propeller disk is the same in the equivalent single, non-ducted propeller configuration as for the ducted propeller system. This requirement leads to a somewhat different formulation of the nozzle- and propeller-induced velocities and, hence, to a different formulation of the equivalent entrance speed to be used in the design of the propeller as an open propeller.

In this alternative method, we are aware to have fully separated the propeller from the ducted propeller system. By using the formulation of the induced velocity of an open propeller, which is represented by an actuator disk, we determine the propeller contribution to the induced velocity. When viewed from the perspective of a single, non-ducted propeller, now fully isolated from the duct, the equivalent speed of advance is V+un. Hence, the velocity to be used in the load coefficient of the single propeller, τCT , should now become V+un, instead of V as in Oosterveld’s method. Moreover, the velocity induced by the propeller, up, is now to be made non-dimensional by division by V+un, instead of by V as in Oosterveld’s method. These adaptations lead to the following subdivision of u1 into propeller-induced and nozzle-induced velocities up and un, respectively:

p 2T

n n

u 1 V = 1+ ( ) τ C - 1V+u 2 V+u

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(27)

or

p 2n nT

u u u1 = (1 ) τ C (1 )V 2 V V

⎛ ⎞+ + − +⎜ ⎟⎜ ⎟

⎝ ⎠ (28)

and un = u1 - up = Vp – V - up Notice that the same definition and nomenclature for the thrust loading coefficient has been used as before. By arithmetic manipulation we can express these implicit relationships by explicit ones:

( )pT

u τ = 1+ τ C - 1V 2

(29)

and

( )2 2nT

u 1 = (1-τ ) 1+τ C +(1-τ)V 2τ

. (30)

When τ=1 the results of the momentum theory for an open propeller are recovered. On the other hand, significant differences with equations (21) and (22) occur where in accelerating nozzles in zero-speed condition values of τ as low as 0.5 are being found.

Again, the equations of the second method can be expressed in terms of the advance coefficient J and the thrust coefficient of the propeller, leading to the following formulation:

p 2

T-Prop

u τ 8 = - J + J + KnD 2 π

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(31)

and

2 2 2nT-Prop

u 1 8 = (1-τ ) J + K +J(1-τ)nD 2τ π

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

. (32)

Notice that these relations satisfy the requirement that in the propeller plane the sum of the propeller- and nozzle-induced velocity agrees with the induced velocity of the whole system:

p p pn

2T-Prop

u v vu+ = -J = -J

nD nD nD nD1 8= J+ J + K -J2τ π⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(33)

In bollard conditions, J=0, these last equations reduce to:

2

pτ 2Tu = D πρτ

(34)

and 2 2

T-Propn

8 K1-τ 1-τ 2Tu = nD =2τ π D πρτ

(35)

T-Prop1 p n

8K1u = u + u = ½ n D τ π

1 2T=D πρτ

(36)

In bollard condition, the propeller-induced velocities of the two methods differ by a factor τ. In order to check the validity of the above formulations experimental data are needed of ducted propeller experiments of which the characteristics of the single propeller in open-water conditions are known as well. Some model test results in open-water on B-series propellers in a few accelerating nozzles are known, while the characteristics of the single B-series propellers in uniform flow are defined by the polynomial representation published since 1972 (Kuiper, 1992). The characteristics of the B-4-70 propellers tested in nozzle No.19A have been compared to those of the B-4-70 propellers in open water.

Some results are plotted in relation to the experimental data in Figure 9. The nozzle-induced velocities as calculated by the formulae of the alternative method are evidently better in agreement with the experimental data, particularly in heavily loaded conditions, including the bollard condition where the τ-values are lowest. The same exercise has been repeated with a few B-series propellers tested in MARIN nozzle No. 7. These concerned the B-4-55 propellers in nozzle No. 7, see e.g. Oosterveld (1970). Also here the agreement with the experimental data is better for the alternative method for calculating the nozzle-induced velocities.

Figure 9: Nozzle-induced velocity

It is noted that the use of the alternative method to calculate the nozzle-induced velocities has resulted in a closer agreement between the anticipated and actual rotation rate of tunnel thrusters as well.

Of course, the magnitude of the nozzle-induced velocity varies spatially. This indicates that the simple concept of the actuator disk model discussed here, giving an average induced velocity only, is too coarse for a complete assessment of the flow field inside the duct. Pressure changes When a ducted propeller is designed as an open propeller, also the pressure depends on the method by which the nozzle-induced velocities are calculated. The pressure drop ∆p in way of the propeller disk follows from Bernoulli's law:

n n ∆p = ρu (V + ½ u ) (37) This reduction of the local pressure can be treated as a reduction of the static pressure when the design problem is split up as indicated. Additional pressure reductions due to the thickness of the nozzle should be added. Systematic series Systematic model test results of Ka series propellers in several types of nozzles help to assess the efficiency and the appropriate value of the pitch ratio to be applied. For results of systematic model experiments on series of propellers in nozzles, tested at MARIN, reference is made to Kuiper (1992).

At MARIN it is customary to compare propellers on the basis of the virtual pitch, a parameter, which, by its definition, is the average pitch of the zero-lift direction. The zero-lift direction of the sectional profiles is determined by Glauert’s method, incorporating the effects of the viscosity on the zero-lift angle using Burrill’s method.

It is to be noted that effects of the tip contour can be substantial in heavily loaded ducted propeller configurations. An assessment of pitch corrections due to a rounded narrow tip is therefore needed. These pitch corrections can be derived from the comparison of B-series and Ka-series propellers in a 19A nozzle. The difference in pitch due to the width of the tip can be of the order of several per cents.

Similarly, effects of a rounded planform on the propulsive efficiency might be assessed using the characteristics of the B 4-70 series tested in nozzle 19A in comparison to tests on the Ka 4-70 propellers tested in the same nozzle. These results are

complimentary to more recent tests at SVA, reported by Heinke and Philipp (1995). The trends are the same, but the magnitudes differ somewhat.

Since the growing share of controllable pitch propellers in nozzles and the application of skew to ducted propellers there appears to be an increased interest in using the characteristics of the B4-70 propellers in nozzle No. 19A. Because this systematic series has not yet been published on earlier occasion, a polynomial representation of these characteristics is included in Appendix I.

The following empirical formula has been derived for the virtual-pitch ratio of the Ka-series propellers:

0.92226 0.328054vP P Z P= + 0.018426 ( ) ( )D D Ae/Ao D

⋅ (38)

Here P is the nominal pitch, D the propeller diameter, Z the number of blades and Ae/Ao is the expanded blade-area ratio.

In general, the virtual pitch of the propeller required to match the rotation rate of the propeller to the available engine power, depends on the blade area, the planform, the tip-duct clearance and the radial load distribution. Apparently, all these factors affect the interaction between the propeller and the nozzle-induced velocity field. MODEL TESTS ON DUCTED PROPELLERS In spite of rapid developments in computational fluid dynamics, the propulsive performance of ducted propeller systems can still most accurately be assessed by model experiments. Indeed, the widely used characteristics of the extensive systematic series of ducted propellers, has a purely experimental background. There are, however, some factors which make testing ducted propellers on model scale more critical than similar experiments with open propellers. First, many ducted propeller configurations are integrated into the hull by headboxes, tunnels, skirts and brackets. Hence, the traditional treatment of the propulsion problem by a hull resistance test, a propulsor open-water test and a propulsion experiments is sometimes somewhat artificial as the subdivision into propulsive parts and hull appendages is not always obvious. Secondly, the non-rotating parts of the propulsion system, as e.g. the nozzle, experiences a low Reynolds number flow where scale effects are bound to be greater than in the experiments on open propeller configurations.

In model tests on the propulsive performance of ducted propeller systems two categories can be distinguished:

• Comparative testing in open-water, i.e. uniform, axial flow conditions, to verify the performance of various designs of nozzles and propellers. These tests should accurately discriminate between variants and their results should be compared with “standard solutions” as e.g. the Ka-series propellers in the 19A nozzle.

• Tests on the actual configuration by which the quality of the propulsive performance including the interaction with the hull and the appendages as the rudder is investigated. These tests allow a reliable extrapolation to full scale.

With reference to the quality of the experiments, MARIN has learned to obey some basic rules for preserving accuracy: • Comparative testing is essential. This will often

involve the re-testing of a reference ducted propeller configuration

• In general, a sufficiently high Reynolds number is to be preserved in the model experiments. It is realised that in open water experiments higher Reynolds numbers can be obtained than in propulsion experiments, but since MARIN considers the latter as the most realistic tests for the prediction of the “real-life” propulsive performance, care is taken that the propulsion units on model scale are of sufficient size for the conditions in the propulsion test. Testing complex propulsors of small dimensions, how attractive it may seem from a cost point of view, has often led to disappointingly poor correlations with full scale trial results.

Further, it is standard practice to trip the flow to turbulence by means of leading edge roughness. Turbulence tripping may help to bring flow conditions in the boundary layer at model scale closer to those prevailing at full scale. Unfortunately, the turbulence tripping by the use of leading-edge roughness seems to lower the propulsive efficiency a little. Therefore we intend in future testing to use the RANS simulations mentioned in this paper to explore the need for tripping.

In various committees of the ITTC the experiments on complex propulsors have been the subject of discussion; see e.g. the report of the Specialist Committee on Unconventional Propulsors (1999). Important recommendations are to test only the complete units, preferably at higher Reynolds numbers than those which would prevail if Froude scaling would be adhered to and to carry out load-variation tests in propulsion experiments.

The extrapolation of tests on complex propulsors might easily lead to erroneous conclusions if methods are being applied which are only suitable for open

propeller configurations. Similarly, wrong conclusions are drawn if tests are analysed in which the propeller loading conditions significantly deviate from those in reality and load-variation experiments, as recommended by the ITTC to be carried out for complex propulsors, have been omitted. Tests on substantially overloaded propellers will then show too favourable results for the ducted propeller arrangements, owing to the natural benefit they receive from a high propeller load. It will be no surprise to see that tests on small models or at too low Reynolds numbers provide wrong results. As on steady wings and rotating propeller blades the increase of the Reynolds number leads to an increase of the lift and a reduction of the drag coefficients.

Instead, care is to be taken that the separate scale effect corrections for the various components of the ship are properly applied. This implies that the wake scale effect correction, the treatment of the drag of the hull appendages and the scale effect on the friction of the propeller blades are properly accounted for. Applying scale effects corrections on the nozzle thrust by using simple rules based on flat-plate friction formulations, as indicated by Holtrop (2001), gives no more than a first-order correction under the important assumption that the flow regime is similar on model and full scale. In that paper it is shown how the component scale effects are being applied without turning to the somewhat artificial subdivision into the three basic powering model experiments as in the ITTC-1978 method. The use of the classical propulsion model factors, thrust deduction, wake fraction and relative rotative efficiency fails in many cases owing to the character of the influence of the loading. Moreover, the assumption, implicitly made in the ITTC-1978 method that the traditional propulsion factors are loading independent, is often unrealistic.

Adhering to the measures described above is a pre-requisite to properly assess the performance of new nozzle designs by model experiments. But if done so, experience has learned that differences, small as they usually are, can accurately be assessed. Indications have not come forward, neither from the experiments on model scale, nor from correlations with results of full scale speed trials and bollard pull measurements, that serious errors of conclusion are being made by model experiments and their analysis. To be true, where the differences searched for are usually quite small, the risk of making errors of conclusion is still present due to uncertainties related to the complexity of the measurements and unexpected scale effects on e.g. the nozzle forces. But as a rule careful model tests on ducted propeller systems can be considered as sufficiently representative for the full scale situation.

CONCLUSIONS The primary conclusions of what we have brought forward are summarized as follows: • For nozzle design and analysis purposes the

recently developed RANS method gives good indications of the flow around nozzles and the resulting force components.

• The design of ducted propellers, treating the

propeller as quasi-ducted, benefits from the new formulation of the nozzle-induced velocities, based on the simple actuator disk model.

• Reviewing past and recent developments in

experimental techniques and considering results of computational fluid dynamics, we consider model tests as offering currently the best prospects for the reliable assessment of the propulsive performance of ducted propeller systems.

• It is expected that results of RANS calculations

will become more and more important in the interpretation of the results of model experiments and in the scaling of test results.

REFERENCES Abdel-Maksoud, M. and Heinke, H.-J.: “Scale effects on ducted propellers”, 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan (2003). Falcão de Campos, J.A.C.: “On the Calculation of Ducted Propeller Performance in Axisymmetric Flows, MARIN Publication No. 696, 1983. Heinke, H-J. and Philipp, O.: “Development of a Skew-Blade Shape for a Ducted Controllable Pitch Propeller System”, PropCav 95, Newcastle upon Tyne, 1995. Hoekstra, M.: “A RANS-based analysis tool for ducted propeller systems in open water”, to be published in International Shipbuilding Progress, September 2006. Holtrop, J.: “Extrapolation of Propulsion Tests for Ships with Appendages and Complex Propulsors”, Marine Technology, Vol. 38, July 2001. Kerwin, J.E., Kinnas, S.A., Lee, J-T, Shih, W-Z: "A Surface Panel Method for the Hydrodynamic Analysis of Ducted Propellers", SNAME Transactions Vol. 95, 1987.

Kort, L.: “Der neue Düsenschrauben-Antrieb”, Werft-Rederei-Hafen, Jahrgang 15, Heft 4, February 15, 1934. Kuiper, G.: “The Wageningen Propeller Series”, MARIN Publication 92-001, May 1992. Van Manen, J.D. et al.: “Open water test series with propellers in nozzles”, Jahrbuch S.T.G., 1953. Van Manen, J.D. et al.: “Recent research on propellers in nozzles”, Int. Shipb. Progress, Vol. 4, no. 36, 1957. Van Manen, J. D.: “Effect of Radial Load Distribution on the Performance of Shrouded Propellers”, NSMB/MARIN Publication No. 209, or International Shipbuilding Progress, Vol. 9, No. 93, May 1962. Van Manen, J.D. and Oosterveld, M.W.C.: “Analysis of Ducted-Propeller Design”, SNAME Annual Meeting, November 1966. Moon, I-S., Kim, C-S.: “Blade Tip Gap Flow Model for Performance Analysis of Waterjet Propulsors”, IABEM 2002. Oosterveld, M.W.C.: “Model tests with decelerating nozzles”, ASME Symp. on Pumping Machinery for Marine Propulsion, May 1968. Oosterveld, M.W.C.: “Wake adapted Ducted Propellers”, Doctors Thesis, MARIN Publication 345, 1970. Oosterveld, M.W.C.: “Ducted Propeller Characteristics”, RINA Symposium on Ducted Propellers, May 30-June 1, London, 1973. Report of the Specialist Committee on Unconventional Propulsors, Proceedings 22nd International Towing Tank Conference, 1999. Yossifov, K. and Belchev, V.: “Systematic Tests of the Wageningen Kc-Ducted Propellers”, SMSSH-86, 15th Jubilee Session, 1989, Varna, Bulgaria. Yossifov, K., Stavena, A. and Belchev, V.: “Equations for Hydrodynamic and Optimum Efficiency Characteristics of the Wageningen Kc-Ducted propeller Series”, Prads-89, 23-28 October 1989, Varna, Bulgaria.

Appendix I: Characteristics of the B-4-70 Propellers in Nozzle 19A in Polynomial Form KQ:

C1=0.068376-0.310777*P+0.550444*P2- 0.366186*P3+0.0987228*P4

C2=0.00611561*P5-0.0123253*P4 C3=-0.0384682*P+0.0294353*P2 C4=-0.0378791 C5=0.0217869-0.0102791*P

KTT: C1=0.529237*P-0.0972606+0.0290876*P5 C2=5.16347*P2-2.22734*P- 5.3479*P3+2.52105*P4-0.501966*P5 C3=0.449636*P3-0.073982-0.541458*P2 C4=0.0 C5=-0.063168*P

KTN: C1=-0.0952786+0.354927*P-

0.0430466*P2+0.0175069*P5 C2=-0.143536-0.3707*P+0.00191818*P5 C3=0.318959 C4=-0.164491+0.00758634*P4 C5=0.0