MARCO CONCEPTUAL

The thrust and torque coefficients for deeply submerged propellers subject to an in-line inflow will here be termed KT J and KQJ. These coefficients are

experimentally determined by so-called open water tests, usually performed in a cavitation tunnel or a towing tank. For a specific propeller geometry, KT J

and KQJ are often given as functions of the advance number Ja:

Ja=

Va

nD, (2.16)

where Va is the propeller advance (inflow) velocity. This relationship is com-

monly referred to as an open-water propeller characteristics. In general, the coefficients are written as KT J = KT J(Va, n) and KQJ = KQJ(Va, n). In the

following, the arguments Va and n will mostly be omitted. The corresponding

open-water efficiency η₀is defined as the ratio of produced to consumed power for the propeller:

η₀= VaTa 2πnQa = VaKT J 2πnKQJD = JaKT J 2πKQJ . (2.17)

Systematic tests with similar propellers are typically compiled in a propeller series. The perhaps most well-known series is the Wageningen B-series from MARIN in the Netherlands, see van Lammeren et al. (1969), Oosterveld and van Oossanen (1975), and the references therein. In the latter reference, KT J

and KQJ are given from:

KT J = f1 µ Ja, P D, Ae A0 , Z ¶ , (2.18) KQJ = f2 µ Ja, P D, Ae A0 , Z, Rn, t c ¶ , (2.19)

where P/D is the pitch ratio, Ae/A0 is the expanded blade-area ratio, Z is the

number of blades, Rn is the Reynolds number, t is the maximum thickness of

Z P/D Ae/A0 KT 0 KQ0 KT 0r KQ0r

4 1.0 0.70 0.445 0.0666 0.347 0.0628

Table 2.2: Wageningen B4-70 example propeller data: Diameter D, blade num- ber Z, pitch ratio P/D at radius 0.7D/2, expanded blade area ratio Ae/A0,

nominal thrust coefficient KT 0, nominal torque coefficient KQ0, reverse nomi-

nal thrust coefficient KT 0r, and reverse nominal torque coefficient KQ0r.

−0.60 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 J a [−] K_TJ 10K_QJ η₀

Figure 2.1: Open-water characteristics for the Wageningen B4-70 propeller.

shows an example propeller characteristics for the Wageningen B4-70 propeller with main parameters given in Table 2.2. The Wageningen B4-70 propeller will be used in examples and simulation studies throughout this thesis. The open-water characteristics in Figure 2.1 is extended into the fourth quadrant, i.e. negative Va. If the propeller operating direction is reversed, this can be

modelled by a similar open-water characteristics for negative shaft speeds. For normal operation, KT J and KQJ are both positive. In off-design conditions,

however, this may not be the case. If Ja is increased slightly beyond the value

for which KT J is zero (i.e. a high Va compared to n), the thrust becomes

negative while the torque is still positive. For a further increase in Ja, both the

thrust and torque become negative. The propeller is then said to be windmilling, and the propeller is in practice producing power instead of absorbing it.

Remark 2.2 The open-water characteristics is specific to a certain propeller geometry. However, the general shape, as shown in Figure 2.1, will remain the same, regardless of the propeller. This can be explained by the physics of the propeller; the lift and drag of the propeller blades will give decreasing thrust

2.1 Propeller characteristics 15

and torque for increasing Ja, and the unsteady operating conditions for small

negative Ja will lead to a drop in the thrust and torque.

Model representations

For simulation purposes, various representations of the open-water characteris- tics may be used. By expressing KT J and KQJ as second order polynomials in

Ja, Taand Qacan be written explicitly as functions of Vaand n:

KT J = KT 0+ αT 1Ja+ αT 2Ja|Ja| , (2.20)

KQJ = KQ0+ αQ1Ja+ αQ2Ja|Ja| , (2.21)

⇓

Ta = Tnnn |n| + Tnv|n| Va+ TvvVa|Va| , (2.22)

Qa = Qnnn |n| + Qnv|n| Va+ QvvVa|Va| , (2.23)

where αT 1, αT 2, aQ1, and αQ2 are constants, and:

Tnn= ρD4KT 0, Qnn= ρD5KQ0,

Tnv= ρD3αT 1, Qnv = ρD4αQ1,

Tvv = ρD2αT 2, Qvv = ρD3αQ2.

(2.24)

With αT 2= αQ2= 0, this reduces to the linear approximation commonly used

in the control literature (Blanke, 1981; Fossen, 2002):

KT J = KT 0+ αT 1Ja, (2.25)

KQJ = KQ0+ αQ1Ja, (2.26)

⇓

Ta = Tnnn |n| + Tnv|n| Va, (2.27)

Qa = Qnnn |n| + Qnv|n| Va. (2.28)

It can be argued that the quadratic polynomial is a physically more reasonable representation that the linear one (Kim and Chung, 2006). (2.22) and (2.23) clearly show the dependence of the thrust and torque on the advance velocity. Figure 2.1 indicates that the linear and quadratic polynomial representations in reality only are applicable in the first quadrant: they clearly cannot capture the drop in KT J and KQJ for small negative Ja. In addition, as will be discussed

in Section 2.1.6, the open-water characteristics of a ducted propeller is usually significantly different, and less linear, than for an open propeller. Hence, these approximations must be used with care both for simulations and controller- observer design, and verified against the open-water characteristics of the actual propeller.

Is it then possible to formulate a better simulation model in terms of KT J

and KQJ? If a higher-order polynomial in Ja is chosen for KT J and KQJ, the

resulting representation is singular for n = 0. Another option is to tabulate KT J and KQJ as functions of Ja, and use interpolation and equations (2.3,

2.4) to calculate Ta and Qa. However, since n → 0 from (2.16) implies that

Ja→ ±∞, a zero-crossing of n is not covered by the open-water characteristics

unless special precautions are taken. In addition, since Ta and Qa are given

as quadratic functions of n, Ta = Qa = 0 for n = 0, regardless of Va (and

hence Ja). For simulation purposes, the singularity for n = 0 is therefore an

inherent weakness in this model representation. The reason is probably that the open-water characteristics originally was developed for propellers on vessels in transit, i.e. for the first quadrant of operation, where only nonzero n were considered.

In order to capture the correct quasi-static behavior for all 4 quadrants, and also give physically reasonable results for time-varying inflow and shaft speed, it appears necessary to use another parametrization than the open-water characteristics.

2.1.2

4-quadrant model

A more accurate propeller characteristics model was apparently first defined by Miniovich (1960), and later used by amongst others van Lammeren et al. (1969) for some of the Wageningen B-series propellers. It is based on the angle of attack β of the propeller blade at radius 0.7R:

β = arctan( Va

0.7πnD) = arctan( Va

0.7ωR), (2.29) where R = D/2 is the propeller radius, and ω = 2πn is the propeller angular velocity. The four quadrants of operation are now defined as:

1st_{: 0}◦_{≤ β ≤ 90}◦_{, V} a≥ 0, n ≥ 0, 2nd_{: 90}◦_{< β ≤ 180}◦_{, V} a≥ 0, n < 0, 3rd: ₋₁₈₀◦_{< β ≤ −90}◦, Va< 0, n < 0, 4th: ₋₉₀◦_{< β ≤ 0}◦, Va< 0, n ≥ 0. (2.30)

This model hence covers also the windmilling regime. The non-dimensional thrust and torque coefficients CT and CQ are defined as:

CT = Ta 1 2ρ(Va 2_{+ (0.7ωR)}2₎π 4D2 = ₁ Ta 2πR2ρV0.72 , (2.31) CQ = Qa 1 2ρ(Va 2 + (0.7ωR)2₎π 4D3 = Qa πR3_ρV 0.72 , (2.32)

where the undisturbed incident velocity to the propeller blade at radius 0.7R is defined as:

V0.7=

q

Va2+ (0.7ωR)2. (2.33)

For a specific propeller, CT is in van Lammeren et al. (1969) modelled by a 20th

2.1 Propeller characteristics 17 −3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1 1.5 β [rad] C T 10C_Q

Figure 2.2: CT and CQ for the 4-quadrant representation of the Wageningen

B4-70 propeller.

modelled with coefficients AQ(k) and BQ(k):

CT(β) = 20 X k=0 (AT(k) cos βk + BT(k) sin βk), (2.34) CQ(β) = 20 X k=0 (AQ(k) cos βk + BQ(k) sin βk). (2.35)

Note that the Fourier series coefficients in (2.34, 2.35) have zero-based indexing. This formulation has many advantages over the open-water characteristics, since it is based on a physically more sound foundation: it is valid for any shaft speed and inflow, and covers all four quadrants of operation. However, it requires significantly more model knowledge than the open-water characteristics, and the necessary data may not be available.

CT(β) and CQ(β) may be parameterized in other ways than the Fourier

series representations in (2.34, 2.35), e.g. by tabulating CT and CQ as functions

of β and using interpolation in this table. Figure 2.2 shows CT(β) and CQ(β)

for the Wageningen B4-70 propeller with main parameters given in Table 2.2. Numerical values for the Fourier coefficients can be found in van Lammeren et al. (1969), and are also reproduced in Appendix A.

Since both the 1-quadrant model in (2.6, 2.7) and the 4-quadrant model in (2.31, 2.32) express Taand Qaas functions of n and Va, it is possible to establish

by:

β = arctan( Va

0.7πnD) = arctan( Ja

0.7π). (2.36) By equating (2.6) and (2.31), the relationship between KT and CT is found to

be: KT = CT π 8(J 2 a+ 0.72π2), (2.37)

and similarly betwenn KQ and CQ from (2.7) and (2.32):

KQ= CQ

π 8(J

2

a+ 0.72π2). (2.38)