Wake field scaling

5 The wake field

5.6 Wake field scaling

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74 Marine propellers and propulsion

Figure 5.8 (a) The radial variation of the wake coefficient for models having twin screws (D/L=0.03) and (b) the radial variation of the wake coefficient for twin-screw ships (D/L=0.03) (Reproduced with permission from

Reference 6)

Figure 5.9 Composition of the wake field

Figure 5.10 (T–I) approach to effective wake field estimation

Choi and Kinnas (References 33 and 34) developed an unsteady effective wake prediction methodology by coupling an unsteady lifting surface cavitating pro- peller procedure with a three-dimensional Euler code.

In this arrangement the propeller effect is represented by unsteady body forces in the Euler solver such that the unsteady effective wake both spatially and tempo- rally can be estimated. Using this method it was found that the predicted total velocity distribution in front of the propeller was in good agreement with measured data. Leeet al.in Reference 35 studied rudder sheet cavitation with some success when comparing theoreti- cal predictions with experimental observation. In this procedure a vortex lattice method was coupled to a three-dimensional Euler solver and boundary element method; the latter being used to calculate the cavitating flow around the rudder.

Considerable progress is being made in the estima- tion of effective wake using advanced computational analyses and this trend looks set to continue for the foreseeable future.

The wake field 75

Figure 5.11 Comparison of model and full-scale wake fields – meteor trials (1967)

numbers between the ship and model, a disparity in Reynolds number exists which leads to a relative dif- ference in the boundary layer thickness between the model and the full-scale ship; the model having the rel- atively thicker boundary layer. Consequently, for the purposes of propeller design it is necessary to scale, or contract as it is frequently termed, the wake mea- sured on the model so that is becomes representative of that on the full-size vessel. Figure 5.11 illustrates the changes that can typically occur between the wake fields measured at model and full scale and with and without a propeller. The results shown in Figure 5.11 relate to trials conducted on the research vesselMeteor in 1967 and show respectively pitot measurements made with a 1/14th scale model; the full-scale vessel being towed without a propeller and measurements, again at full scale, made in the presence of the working propeller.

In order to contract nominal wake fields in order to estimate full-scale characteristics two principal meth- ods have been proposed in the literature and are in com- paratively wide use. The first method is due to Sasajima et al. (Reference 13) and is applicable to single-screw ships. In this method it is assumed that the displacement wake is purely potential in origin and as such is inde- pendent of scale effects, and the frictional wake varies linearly with the skin friction coefficient. Consequently, the total wake at a point is considered to comprise the sum of the frictional and potential components. The

total contraction of the wake field is given by c=Cfs+Cfs

C_fm

whereC_fsandC_fmare the ship and model ITTC 1957 friction coefficients expressed by

Cf = 0.075 (log₁₀Rn−2)²

andCfsis the ship correlation allowance.

The contraction in Sasajima’s method is applied with respect to the centre plane in the absence of any potential wake data, this being the normal case. However, for the general case the contraction procedure is shown in Figure 5.12 in which the ship frictional wake (w_fs) is given by

wfs=wfm

(1−wps) (1−wpm)

The method was originally intended for full-form ships having block coefficients in the order of 0.8 andL/B values of around 5.7. Numerous attempts by a num- ber of researchers have been made to generalize and improve the method. The basic idea behind Sasajima’s method is to some extent based on the flat plate wake idealization; however, to account for the full range of ship forms encountered in practice, that is those with bulbous sterns, flat afterbodies above the propeller and

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76 Marine propellers and propulsion

Figure 5.12 Basic of Sasajima wake scaling method

so on, a more complete three-dimensional contraction process needs to be adopted. Hoekstra (Reference 14) developed such a procedure in the mid-1970s in which he introduced, in addition to the centre-plane contrac- tion, a concentric contraction and a contraction to a horizontal plane above the propeller.

In this procedure the overall contraction factor (c) is the same as that used in the Sasajima approach. How- ever, this total contraction is split into three component parts:

c=ic+jc+kc (i+j+ |k| =1)

whereiis the concentric contraction, jis the centre- plane contraction andkis the contraction to a horizontal surface above the propeller.

In Hoekstra’s method the component contractions are determined from the harmonic content of the wake field;

as such the method makes use of the first six Fourier coefficients of the circumferential wake field at each radius. The contraction factors are determined from the following relationships:

i= Fi

|Fi| + |Fj| + |F_k| j= F_j

|Fi| + |Fj| + |F_k|

and k= Fk

|F_i| + |F_j| + |F_k|

in which F_i=

_2R

rhub

S_i(r) dr,F_j= _2R

rhub

S_j(r) dr and F_k=

_2R

r_hub

S_k(r) dr with

Si=1−A0

+

A2+A4+A6−¹₂S_k if S_k≥Sj

−S_j+(A₂+A₄+A₆) if S_k <S_j Sj= −[A2+A4+A6

+ |max(A₂cos 2φ+A₄cos 4φ +A6cosφ|] (φ=0,π, 2π) S_k=2(A1+A3+A5)

whereA_n (n=0, 1,. . ., 6) are the Fourier coefficients and at the hubS_iis taken as unity withS_j=S_k=0.

The method as proposed by Hoekstra also makes an estimation of the scale effect on the wake peak vel- ocity in the centre plane and for the scale effect on any bilge vortices that may be present. The method has been shown to give reasonable agreement in a limited number

The wake field 77

Figure 5.13 Relationship between model and ship wake field

of cases of full scale to model correlation. However, there have been very few sets of trail results available upon which to base any firm conclusions of this or any other wake field scaling procedure.

Figure 5.13 essentially draws the discussions of effective wake and wake scaling together. In most design or analysis situations the engineer is in posses- sion of the model nominal wake field and wishes to derive the ship or full-scale effective wake field char- acteristics. There are essentially two routes to achieve this. The most common is to scale the derived nominal wake field from model to full scale and then to derive the effective wake field at ship scale from the derived nominal full-scale wake.