Non-linear hydrodynamic hull forces derived from segmented model tests

Marine Simulation and Ship Manoeuvrability, Chislett(ed.)© 1996 Batkema, Rotterdam. ISBN 90 5410 8312

Non-linear hydrodynamic hull forces derived from segmented model tests

J.P Hooft & F.H.H.A.Quadvlieg

Maritime Research Institute Netherlands. Wageningen, Netherlands

ABSTRACT: I n many situations the manoeuvering performance of ships is largely effected by the non-linear contributions of the hydrodynamic reaction forces on the hull. The origin of these non-linear components can be ascertained from experiments with segmented models.

In this paper a review is given of the analysis of the segmented model tests which have been reported in literature so far.

From the results derived from this analysis it will be shown that the non-linear hull force and moment components can be described in a robust manner by means of a longitudinal distribution of the so-called cross-flow drag coefficient. Thus, a good description of the ship's manoeuvrability is acquired for the entire speed range of a vessel, including the transition from the manoeuvres at service speed with relatively small drift angles to the manoeuvres at reduced speed (harbour manoeuvering) with large drift angles.

The paper also discusses the linear hydrodynamic coefficients as a result of the segmented model tests. The values of these coefficients sometimes deviate significantly from the general accepted coefficients presented i n literature which were derived from measurements of the overall hull forces and moments on the complete model.

In the paper also the validity of manoeuvres predicted with the resultant mathematical model w i l l be discussed.

1 INTRODUCTION

For the prediction of the ship's manoeuvrability in the design stage use is made o f computer simulation programs such as described e.g. by Inoue Ref [1], Kijima Ref. [2], Hooft Ref. [3] and many others see e.g. Barr R e f [4].

In such simulation programs the lateral force and the yawing moment N f j on the bare hull are mostly described by a linear and a non-linear component:

Y H = ^min +YH„„nHlin = '^m "^Hnl ( 1 1 ) N H = NH,i„ + NH„„„^,i„ s N H , + N H „ I (1.2)

Often, the non-linear components are described by higher order contributions of which the coefficients are estimated from empirical relations with the overall dimensions of the ship; see e.g. Ref. [4]. However in this way it w i l l not be possible to predict the non-linear contributions o f the lateral hull forces satisfactorily accurate. Generally three reasons are given for this inaccuracy of the hull force estimation:

1. The non-linear hydrodynamic coefficients are rather sensitive to the local form parameters of the ship. This means that much more information is required about the non-linear coefficients as a function of the large number of parameters. 2. The non-linear hydrodynamic coefficients have

been determined experimentally only f o r a limited number of ships because many experiments have only been performed at a limited range of small drift angles and small turning rates. This means that only a limited level of confidence has been achieved in the description o f the coefficients as a function o f the ship's main dimensions.

3. Often only the hydrodynamic coefficients have been published without the actual model test results. Some of the authors use quadratic linear coefficients while others apply tertiary non-linear coefficients. In this way the validity o f the coefficients is Umited since a specific non-linear coefficient presented in one publication cannot be compared with the corresponding coefficient for a different hull form in another publication. It was thought that a better prediction of the non-linear component of the lateral force could be

achieved i f it would be possible to describe the distribution of the non-linear force component over the length of the ship instead of describing the non-linear component of the global lateral force on the total hull. Such a non-linear component of the local lateral force is described by the local cross flow drag coefficient; see for example Oltman [5] and Hooft [6]. In this way the non-linear component of the global lateral force is described by the integration of the local non-linear lateral force contributions:

F P

Ynnon-lin = "O-^P J C D ( X ) T ( X ) V ( X ) | V(X) Idx (1.3) AP

FP

NH„on-Un = "O-Sp ƒ X C D ( X ) T ( X ) V ( X ) | V ( X ) |dx (1.4) AP

in which CpCx) is the local cross flow drag coefficient, T(x) is the local draught and v(x) is the local lateral velocity:

v„ = v(x) = V + x * r (1.5) In earlier studies it was suggested to describe the

cross flow drag coefficient Cp as a constant over the length of the ship. In this way the variation of the local lateral velocity w i l l result in a lateral non-linear hull force and yawing moment component as a function of the combination of v and r. Thus, also the longitudinal variation of the draught could be taken into account. However, the application of this assumption still resulted in the prediction of unrealistic manoeuvering performances.

Also f r o m a theoretical point of view one finds that the Cp distribution w i l l not be constant over the ship's length; see e.g. Ref. [7]. This was confirmed by the experimental results with segmented models; see [8] to [11]. A n analysis o f these results of available experiments with segmented models is presented in Ref. [12] i n which a description is given about the distribution of the cross flow drag coefficient Ci3(x) over the length of the ship. I n this description use was made of the following test results with segmented models as presented in the references [8] to [11]:

In Ref. [8] and [9] the results were presented of the tests with a model of a "Tanker" with Lpp/T=17.68 and CB=0.809 at even keel and with a model of a "Mariner cargo ship" with Lpf/T=21.46 and CB=0.595 at a bow up trim angle of 0.434°.

In Ref. [10] the results were presented of the tests with a model o f a "Container ship" with Lpj/r=20.55 and CB=0.562 at a bow up trim angle o f 0.325° and with a model of a "Tanker" with Lpp/T=15.31 and CB=0.825 at even keel.

In Ref. [11] the results were presented of the drift tests with a model of a "Series 60" hull form

with CB=0.70 at three draughts (Lpp/T= 14.2, 17.5 and 22.81) and two additional trim angles 3.4° bow up and 3.4° bow down.

2 DERTVA'nON OF THE CROSS FLOW DRAG COEFFICIENT FROM SEGMENTED M O D E L TESTS

While considering the ship's manoeuvering performance three velocity components can be discerned at each moment of time:

•• u; the ahead velocity component in the centre of gravity G

v; the lateral velocity component in G

•• r; the rate of turning around the vertical axis in G For the description of the total lateral hull force use is made of a subdivision in a linear component Y j j i and a non-linear contribution Y j q ^ ; see equation (1.1). The linear part of the lateral force is associated with l i f t phenomena on a slender body and is most effectively described by two coefficients Yjj^y and

^ H u r

-Y H I = -Y H U V • u v c o s( P ) + Y H „ * u r (2.1)

in which the drift angle p is defined by:

P

=atanW (2.2)

W

From the segmented model tests not only the global lateral force on the total model was derived but also the local lateral forces on each of the various segments. For example, i n Figure 1 some results are shown of the local lateral force measurements during the tests under a drift angleJn this figure the results are presented of the lateral forces on the first and the sixth segment behind the bow which had been measured by Beukelman Ref. [11]. It is noticed that in the forward part of the ship the linear contribution of the local lateral force is expressed by a negative coefficient while in the aft part of the ship the linear contribution of the local lateral force is expressed by a positive coefficient.

From the segmented model test results one determines the non-linear component of the lateral force Yn^j per segment by subtracting the linear contribution from the measured lateral force Y n per segment. According to equation (1.1) one determines for each segment:

Yn„,(v„) =Yn(measured) - Yn,(v,r) (2.3)

in which v„ is local lateral velocity as described by equation (1.5).

fitst segment behind ihc bow; Fn = 0.i5 O measurement • 1 4 » A 1 7 . » • 1420 a 17-50 O 2 1 K O S D

sixlh segment behind the bow; Fn=:0.15 O measurement

lir.ear cootributioci

Fig. 1. Results of drift tests with a segmented model of the "Series 60", Lpp/T = 17.5, Cg = 0.70, at zero trim, from [11]

In the non-dimensional form equation (2.3) reads:

Yn;;((p,Y) =Yn"(measured) - Y n, " ( P, Y ) (2.4) in which the local lateral force is written in a non dimensional way by using the local lateral area of the segment S^:

Yn Yn

0 . 5 p U ^ S „

(2.5)

with Up, being the resultant velocity of the model

According to equation (2.1), the linear component o f the lateral force Yn on the n-th segment is described by:

Yn,(v,r) =Yn„^ * u v c o s( P ) + Yn„^ * u r (2.6) which in the non-dimensional form reads:

Y n, " ( p, Y ) =C|3„ * s i n( p ) * c o s 2( p ) + C * Y c o s( p ) (2.7) From a combination of equations (2.4) and (2.7) one finds the non-linear component of the local lateral force on each segment for each drift angle and rate of rotation. From this the local cross f l o w drag coefficient is derived:

• Mai

A 1 7 »

Fig. 2. Cross flow drag coefficients with segmented model of the "Series 60", C d = 0.70, at zero trim, from [11]

CDn(P.Y) = C D ( X ; P , Y )

- Y n l ^ ( P , Y )

( s i n( P ) + x' Y ) l s i n( P ) + x' Y l (2.8) In Figure 2 some examples are shown of the distribution of the cross f l o w drag coefficient over the ship length as derived f r o m the experiments at a drift angle with a segmented model.

3 THEORETICAL BACKGROUND 3.1 General

It should be realized that up till now no analytical methods have been developed completely to determine either the linear drift coefficient Cg and turning coefficient C^ or the non-linear contribution Yn„|. For the theoretical derivation of the linear coefficients one is for instance referred to the study by Hearn and Clarke Ref [13] and the one by Gadd Ref. [14]. The non-linear contribution has been studied for example by Price and Tan R e f [15].

In the earlier studies of the cross f l o w drag it was only assumed that the local cross flow drag coefficient Cj)(x)=Co„ varied over the ship length; see e.g. Ref [5] and [16]. However no changes were considered as a function of the ship's motion components P and Y¬

In the following further attention w i l l be paid to the variation of Cp^ with the change of the ship's motion: ^ D n ^ D n( P' Y ) indicated in equation (2.8).

3.2 Cylinder in a cross flow

One first considers a long cylinder in a homogeneous stationary cross flow; see for instance Ref. [17]. Usually one only considers the average value of the force from the measurement in such a flow. I f any, less attention is devoted to the oscillatory part of the force; see e.g. Sarpkaya Ref. [18].

In case of an infinitely long cylinder one generally describes the lateral force Y by its dimensionless value Y ' which corresponds to the so-called drag coefficient Cp:

- Y - Y ' = .

O . S p S v | v |

(3.1)

in which S is the lateral area o f the cylinder corresponding to the product of the height and the length of the cylinder.

It has been found that in an unlimited space the drag coefficient will vary with the flow conditions indicated by the Reynolds number Rn:

C p = CpCRnp) with: R n r V *D

(3.2)

with D|, being the hydrodynamic diameter of the cylinder. This Reynolds number dependency of Cp as expressed in equation (3.2), has been confirmed by experimental results presented by various authors; see e.g. Ref [17]. It means that due attention must be paid in determining the f u l l scale values of Cp from model tests results because scale effects are to be expected as a consequence of the influence of the Reynolds number on Cp.

It is further known that the drag coefficient of an infinitely long beam in a cross flow depends on the dimensions and form of the cross section of the cylinder defined by the width/height ratio and the rounding radius of the cross .section; see Hoerner Ref [17]. In terms of the ship dimensions these parameters correspond to the ship's beam/draft ratio and the radius of the bilge keel.

In case that the cylinder is placed in the free water surface or close to it then it has been found that the drag coefficient Cp is also influenced by the Froude number Fn relative to the height or the draught T of the cylinder:

Fn^. = \ / g * T

(3.3)

In case that the length of the cylinder is restricted with respect to its height or beam (such as in the case of a ship) then also a longitudinal force is generated on the cylinder in a homogeneous cross flow if both ends of the cylinder are not symmetrical relative to the midship section.

* C„r»«iMr L f p / T = . 2 l l 5 S t = < l U . r O T i n i e r L p p / T = I 5 J l no trim

O O O

* * •

Fig, 3. Longitudinal distribution of the drag coefficient Cpgg in a cross flow P = 90°; from [10]

In Figure 3 the results of segmented model tests in a beam flow are given which were presented by Matsumoto Ref [10]. I n this figure the local drag force coefficient Cp<,o^p(ipi=90°) was derived from:

Cp9o(x) - Y n( | p | =90°) 0 . 5 p S„ v i v |

(3.4)

From the results in Figure 3 it is .seen that at the midship section the drag coefficient C|3^5=Cp9Q(x=0) of the full bodied ship (CB=0.82) is much larger than the one for the container ship.

It is assumed that for the midship section the dras coefficient Cp^^ due to the ship's lateral motion ( I p i =90°) will correspond to the values determined for an infinitely long cylinder. In that case the value of C p ^ 5 w i l l depend on:

1. the Reynolds number (see equation (3.2)) which means that the model test results might be effected by scale effects and

2. the .ship's form at the midship section expressed by the midship beam-draft ratio BH" and the midship section coefficient C^, which is an indication of the midship bilge rounding. However this approach does not lead to a successful description of Cp^^ as a function of the ship's parameters. It is assumed that this negative result is caused by three dimensional effects due to the limitation of the ship's length relative to the ship's width or draught. I t seems that better relations can be achieved when Cp^^^ at f p 1=90° is related to the block coefficient Cg though being an overall characteristic of the ship. It is most reasonable to assume that also the trim of the ship w i l l affect the midship drag coefficient C p „ , a t | p | = 9 0 .

From the results in Figure 3 it is also observed that for the f u l l bodied ship (CB=0.82) the drag coefficient Cp decreases towards the bow and the stern of the ship while for the slender body ship (CB=0.56) the drag coefficient Cp increases towards the ends of the ship.

It is assumed that the change of Cp towards the bow is caused by the fact that a more rounded bow form (full hull form) will lead to a reduction of the drag coefficient at the bow instead of an increase with a sharper bow form (slender hull form).

For the description of the fullness of the fore body various parameters can be imagined to be indicative to the change of Cp towards the bow; • The dimensions of the cross section somewhere

near the bow (e.g. 15% behind the forward perpendicular) expressed by the local beam/draft ratio or the local cross sectional area coefficient. • The water line angle at the bow or the angle of

the curve of cross sectional area at the bow. • The block coefficient of the fore ship or the

location of the centre of buoyancy of the fore ship.

It is assumed that the change of Cp from the raid.ship section towards the stern is caused by the fact that a fuller stern form (full hull form) will lead to a reduction of the drag coefficient at the stern instead of an increase for a sharper stern form (slender hull form).

For the description of the fullness of the aft body various parameters can be imagined to be indicative to the change of Cp towards the stern:

The dimensions of the cross .section somewhere near the stern (e.g. 15% before the aft perpendicular) for instance expressed by the local beam/draft ratio or the local cross sectional area coefficient

• The shape of the hull at the stem which can have either V-form sections or more S-form sections.

• The block coefficient of the aft body or the location of the centre of buoyancy of the aft ship.

3.3 Slender bodies in a homogeneous flow at small

angles of attack

The next aspect to be considered is the lateral force on a slender body at a small angle of attack in a homogeneous flow without boundaries. As mentioned in section 2 the lateral force in this case of a .slender body at a small angle of attack consists of both a linear (lift) contribution and a non-linear (drag) contribution.

In Figure 4 the measured cross flow drag coefficients have been plotted which were presented by Tinling and Allen Ref. [7]. They showed that the drag force on a slender body depends on:

• the strength of the local vortex shedding from the body.

• the separation point at which the vortex sheet starts. This point moves forward with increasing angles of attack p . O Kn = 3 - I 0 " P^ZW' r 0 O 0 O c 1 0 c O f. 5 4 .1 2 1 U

Discanoe along axis. P(^-^<VR

Fig. 4. Distribution of the cross flow drag coefficient Cp from measurements by Tinling and Allen [7]

From these effects one finds that the value of the cross flow drag coefficient Cp will vary as a function of:

• the location on the hull defined by the distance ^ from the nose of the body: Cp = Co(^) with ^=Lpp/2-x.

the angle of attack P of the flow to the slender

body: Cp = C p ( 5 ; P )

It should be noted that the increment o f the value of Cp at short distances behind the nose (t/R < abt 4; R being the radius of the cylinder) depends largely on inertial phenomena o f the fluid. It was therefore suggested that the value of Cp on this part of the cylinder is independent of the Reynolds number and does not contain any scale effects.

However the value of Cp at large distances from the nose (Z/R « ) corresponds to the value of Cp of the cylinder concerned with infinitely length in a homogeneous cross flow; see equation (3.1). According to equation (3.2) it w i l l be obvious that C D ( ^ R ~ ) will depend on the Reynolds number and will therefore be effected by scale effects when determined from model test results.

The above considerations about the influence of the Reynolds number may lead to the following effect. In general, it is found that the Cp value of an infinitely long cylinder i n a cross flow w i l l decrease with increasing values of the Reynolds number. With this observation and the above consideration it might be found that at f u l l scale the lateral resistance at the aft body of the cylinder at a drift angle is smaller than at model scale. This leads to the conclusion that

due to this effect of the Reynolds number the ship will be "less course stable" (more correct is the statement: "more course unstable") than experienced during the model tests.

The above considerations will not only apply to ogive cylindrical bodies but also to slender bodies of any cross sectional shape and slenderness. It might be expected however that the distribution and the absolute values o f the cross flow drag coefficients

w i l l differ for different hull forms.

The tendencies of the Cp variations from the measurements in Ref. [7] correspond to the general tendencies shown in Figure 2 derived from segmented ship models. When comparing the segmented model test results for a wide variety of hull forms then i t was observed that the Cp distribution over the ship's length could be described systematically for all hull forms as a function of the distance t, behind the forward perpendicular FP relative to the ship's length ^ ' = ^ p p instead of the findings in Ref. [7] where the distance ^ behind the nose was described by ^ ' = ^ R . From a hydrodynamic point of view this latter description seems more appropriate.

From the experiments with segmented ship models (see Refs. [8] to [10]) it was found that the local cross flow drag coefficient did not only depend on the drift angle P but also on the turning rate r (expressed in the non-dimensional form by 7) being the other motion component in the horizontal plane: C p ^ p ( ^ ; P , Y ) ; see also equation (2.8).

3.4 Slender bodies on the free water surface at small

angles of attack

I f the speed of the ship is small then the same phenomena due to its motions at the water surface will be observed as those described above which arise during the motions of a slender body in an unbounded fluid. This result is explained by the fact that at low velocities the water surface w i l l not be disturbed so that the same phenomena will be generated as i f the water surface is fixed. With the image theory it then can be shown that such a condition corresponds to that o f an unbounded space.

At higher forward speeds it w i l l be observed that the water surface around the ship is largely changed. Due to this change of water level the following phenomena are observed which were derived f r o m the segmented model test results; see for example Figure 2:

1. Some part of the cross flow drag distribution will correspond to the values for the deeply submerged condition while,

2. the variation of the pressure along the ship (especially at the bow and the stern) w i l l cause additional contributions to the cross flow drag coefficient ACp(x;Fn); see Ref. [12].

From these considerations it follows that when sailing at the water surface then the cross flow drag coefficient Cp is influenced by the Froude number in addition to the effects mentioned above. The distribution o f the total cross flow drag coefficient can thus be described by the following schematic representation:

C D „ ( P . Y ) =CDnO * C D „ f + C p „ , (3.5)

with Cp„o being the cross flow drag coefficient for Fn=0 at which the free water surface is undisturbed. The contribution Cp^f describes the additional drag coefficient at the bow as a function o f the Froude number Fn:

Conf =CD„f(Fn) for ^ < ^ f ( 3 . 6 )

while Cp„3 describes the additional drag coefficient at the stern as a function of the Froude number:

for 4 > ^,(3.7)

It is assumed that the contributions of C p ^ and ^Dna largely influenced by the wave distribution around the ship (see e.g. Ref. [10]) by which they are independent of scale effects. However the contribution of Cp^^Q contains scale effects at the aft body which correspond to the scale effects mentioned in section 3.3. This w i l l lead to the same conclusion as found in section 3.3: the influence of the Reynolds number on the Cp w i l l cause the ship to be "more course unstable" than experienced during the model tests.

4 APPLICATION OF CROSS FLOW D R A G COEFHCIENTS

In this section a discussion w i l l be given about the consequences of the description of the non-linear lateral force component by means of the distribution of the cross flow drag coefficient over the ship's length. For this purpose one considers the measured global lateral hull force and yawing moment on the complete ship in comparison to the calculated force and moment according to equations (1.1) to (1.4) taking into account the above descriptions.

In the present considerations the linear component of the lateral hull force and yawing moment N{j have been derived from a summation of the local linear lateral force components derived from the segmented model tests; see for example the results in Figure 1.

For the linear lateral force coefficient due to drifting one thus finds:

Lpp 1

while for the yawing moment coefficient due to drifting one finds:

NHP M

' ^ y n ? .

LppT (4.2)

For the rotation coefficients the same type of summations were performed.

When the local hft coefficient per unit length is denoted by Yp^':

then one finds according to Jones' theory Ref. [19] that Yp^' is determined by the local variation of non-dimensional lateral added mass of water 9myyo'(x')/9^' (at zero frequency) alongside the ship per unit length with

my/(0.5pLppT): „ / -myy^'((0 =0) = -dm. _yyO 9^' (4.4) This means that theoretically on a drifting ship the linear component of the lateral force coefficient Yug' on a segment between ^ f ' and (with ^=Lpp/2-x being the distance behind the FP) follows

Ynp' = J y J ^ d ^ ' = - ƒ myyo^'(O) =0)d^' =

" y y o + m y.vo (4.5)

In Figure 5 the results from equation (4.5) for a tanker are shown in compari.son to the experimental results by Matsumoto Ref [10]. In Ref [12] the consequences of Jones' theory have been examined in more detail. I n that study also the equations for the lateral force distribution as a response to the ship's mming rate have been described; see Figure 6.

From the results in Figures 5 and 6 it is concluded that in general the largest deviations between theory and experiments occur in the aft body. Hoerner R e f [17] and Fedyayevski Ref [20] present the following explanation for the fact that the experimental results in the aft body do not correspond to the theoretical description in an ideal fluid. Due to the viscosity of the fluid one finds that when the curvature at the aft body becomes too large then the flow w i l l not be able to follow the stream lines which occur in an ideal fluid. Therefore the flow will "separate" and the linear lateral force at the stern will be smaller than might be expected from the theory for an ideal fluid.

T3

Fig. 5. Comparison between the measured lateral force coefficient Yn"n per segment from drift tests and the results from Jones' theory. Measurements by Matsumoto [10]

Fig. 6. Comparison between the measured lateral force coefficient Yn".^per segment from rotation tests and the results from Jones' theory. Measurements by Matsumoto [10]

As a consequence of the "reduced" linear lateral force component at the stern one finds a nett linear lateral force which is opposite to the ship's lateral velocity. Also a linear yawing moment is mea.sured that is smaller than the theoretically derived value (the so-called Munk moment). This means that the course stability of the ship is less dramatic than one would expect from the theory.

Based on his explanation, Fedyayevski further assumed that for the ship (full scale values) one finds a nett linear lateral force due to the ship's lateral velocity which is relatively smaller than measured at model scale because the linear lateral force component at the stem will deviate less from the theoretical values than found on model scale. This expectation is based on the fact that the curvature of the ship is smaller than that of the model such that

the flow behind the ship will follow the stream lines in a ideal fluid much more closely than at model scale, leading to a better agreement between theory and experiments. According to potential theory the nett linear lateral force is equal to zero corresponding to the paradox of d'Alembert. Based on the above assumption also the linear yawing moment on the ship w i l l be relatively larger (closer to the Munk moment) than on model scale.

The effect of the hull curvature on the linear lateral force component and the linear yawing moment component has to be denoted also as scale effects on the model test results. Both the smaller linear lateral force and larger linear yawing moment on a drifting ship relative to the model scale values, w i l l cause the ship to be "more course unstable" than was expected from model tests.

Having described the linear contribution of the global lateral force on the complete ship, then the following results are obtained.

In Figure 7 the total forces are shown which were measured by Beukelman on a "Series 60" model with a block coefficient of CB=0.70; see Ref [11], In this figure the contributions of the linear and of the non-linear components are shown which were obtained as follows:

Linear components:

Y H , ' ( P . 7 )

N H | ' ( P , Y ) = N p ' *sin(p) *cos^(P) -t-N,^' * Y c o s ( P ) (4.7) with Yp'=-0.2235 and Np'=-0.1I01 for the hull form in Figure 7. Non-linear components: FP Y H H I ' = - ƒ C D W v(.v)' I v(x)' Idx' (4.8) AP V T j FP , NHnl' =

- ƒ

' ' ' C D ( > ' ) P ^

liiwai ciimpnnciii iif btcr-il fertc

liikml fiircc Jue li> - - ' ' ^ ^ ^ LiDss fliiw drag ciiefficni

10

p ,

N H ' " '

- i n

5 l u 1 5 2Ü

linear ctimpimenl uf y j w mumcal

-20 J i n u l y j w momeot •30 yaw miiiiieiii duu tu ' ^ ' ^

Fig. 7. Decomposition of global lateral force and yaw moment on the bare hull. According to the results in [11]

in which the distribution of the cross flow drag coefficient Cj)(x) is given in Figure 2. In equations (4.8) and (4.9) the non-dimensional local lateral speed v(x)' follows from:

(4.10)

In the past, the total lateral force Y j j ' and yawing moment N , ^ ' were generally described by the global coefficients Y p ' , Yppp', Np' and Npip,' according to:

Y H ' ( P ) = Y p ' * s i n ( P ) + Y p p p ' * s i n ^ p ) (4.11) N H ' ( P ) = N p ' * s i n ( P ) + N p | p | ' * s i n ( P ) * | s i n ( P ) | ^

or something equivalent. From a regression of the results in Figure 7 the following values were derived for the coefficients in equations (4.11) and (4.12): = Y B ' * s i n ( P ) *cos^(P) *Y • *Ycos(p) v(x)' = ^ = s i n ( P ) + x' * Y

^ (4.6)

Lateral force

Linear component: Yp' = -0.2235 Yppp' = +0.2235

Non-linear component: Yp' = -0.0705 Y p p p ' = -1.3731

Total lateral force Yp' = -0.2940 Y p p p ' = -1.1496

From the results in this table it is seen that the components on each of the segments: for equation linear coefficient for the global lateral force Y p ' = - (4.6) it was stated that Yg'=-0.2235 for this hull 0.2940 does not correspond to the earlier given value form of the "Series 60" wifh a block coefficient " that was derived from a summation of the CB=0.70; see Ref [11].

Yawing moment

Linear component: Np' = -0.1161 Npip,' = +0.0541

Non-linear component: Np' = +0.0044 N p p ' =+0.0189

Total lateral force Np' = -0.1117 Npipi' = +0.0730

It is remarkable that the linear coefficient for the global yaw moment Np'=-0.1117 in this Table corresponds so well to the earlier given value that was derived from a summation of each of the coefficients of the yaw moments due to the local lateral forces on each of the segments: for equation (4.7) it was stated that Np'=-0.110l for this hull form of the "Series 60" with a block coefficient of CB=0.70; see Ref. [ l l j .

From the above results it is concluded that from the segmented model tests better descriptions are obtained for both the linear as well as the non-linear contributions in the lateral force and yawing moment on the bare hull in reaction to the ship's motion components u, v and r;

* The distribution o f the local linear hydrodynamic manoeuvering coefficients derived from the segmented model tests provide more clear insight in the hydrodynamic phenomena around the manoeuvering ship. As a consequence of this insight it w i l l be possible to find a correlation between the experimental findings and the results from theoretical approaches such that better relations can be achieved of the hydrodynamic linear coefficients as a function of the ship's hull form parameters.

* From the segmented model test results also the concept of the cross flow drag coefficients could be developed by means o f which a more systematic description could be derived for the distribution of the local non-linear hydrodynamic lateral force component over the length of the ship.

5 SIMULATING THE SHIP'S M A N O E U V R A B I L I T Y

Far more than other manoeuvering models, the cross flow drag model yields reliable hull forces for a wide range of drift angles and a wide range of Froude numbers. This allows not only the applicability of the so-called hull forces to the calculation of standard ship manoeuvres, but gives the opportunity to perform calculations on harbour manoeuvering, on drifters (ships under influence o f wind, currents and waves) or even for studies on dynamic positioning. It suffices to come up with sufficiently accurate description of propellers, thrusters, rudders and environmental forces.

In view of the recent emphasis given to IMO

requirements with regard to manoeuvres, the application of the model with respect to the so-called IMO manoeuvres is of utmost importance. However, not only the comparison of calculations with model tests should be analyzed, but more important is the comparison between the measurements during the trials and the predictions made by the computer program. Extensive validation with full scale measurements is carried out. Many ships, varying in length between 70 and 320 in, built at Dutch and foreign yards, were simulated. Calculated and measured results are compared in terms of the I M O criteria. The SURSIM package allows simulation of all the standard manoeuvres such as the turning circle test, the zig-zag test, the meander manoeuvre and the overshoot test. Some results with respect to the tactical diameter are presented in Figure 8.

Compiruon of full Bcale IrUlt and calculated reaulta

Fig. 8. Calculated tactical diameter in comparison to sea trial results

Due attention should be paid in predicting the manoeuvrability of full block short ships (high block coefficient and low L/B ratio). For some of these ships large discrepancies may be found between measured and predicted results. It is assumed that this is caused by the fact that f u l l short ships have an unstable behaviour, and that only slight differences in the exciting forces (rudder, wind) can cause large effects in view of resulting manoeuvering behaviour.

The application to areas other than the I M O manoeuvres is illustrated in Figure 9. In this figure, a comparison is made on the motions of a tanker solely under influence of a bow thruster. Two methods are compared:

Transverse distance

Heading

Fig. 9 . Simulation results of behaviour of a tanker responding to the bow thruster

a specially for this purpose tuned model o f a tanker, based on model test results and

the cross f l o w drag model (SURSIM). The graphs present the heading and the transverse displacement of the tanker under influence of the bow thruster. The starting situation is the equilibrium position at zero speed. Results show a very good agreement between the two mathematical models, even for this very low speed.

6 CONCLUSIONS

In this paper a description has been presented of the non-linear component of the lateral hull force by means of the distribution of the cross flow drag coefficient over the length of the ship.

A detailed explanation is given about the physical aspects with respect to the cross flow drag coefficient. The origin of the hydrodynamic phenomenon is discussed and also the influence of the ship's motions on the distribution over the length of the ship.

Also the consequences are shown on the

development of a empirical method by means of which the relations are established between the hydrodynamic coefficients and the hull form parameters.

Also a few results are shown which were obtained by applying a simulation program using the concept of the cross flow drag for determining the bare hull forces. When comparing these results with the results from sea trials then it is concluded that the lateral hull force is acceptably well described by means of the cross flow drag distribution over the ship's length. NOMENCLATURE AP B C B "-090 C D ( X ) '(3 Y ' Fn FP G g L Lpp SL r T T(x) u v(x) aft perpendicular

ship's breadth at the midship section block coefficient; V/(Lpp.B.T) drag coefficient; Y„|/('/2.p.SL.v.lvl) cross flow drag coefficient varying over the ship's length for the condition of 90'= drift angle; Cj^g^M = CD(X;IPI

= 9 0 ° )

local drag coefficient; Yn„,/[V2.p.S„.v(x).lv(x)l]

C D ( X )

coefficient describing the linear component o f the lateral force as a function of

p =

3Y,'/ap

coefficient describing the linear component of the lateral force as a function of 7 = dY{/dj

Froude number; U/(g.Lpp)°-^ forward perpendicular center of gravity

acceleration due to gravity length between perpendiculars lateral area o f the ship (mostly LppT is taken)

lateral area of the n-th segment ship's rate of taming; yaw rate ship's draft at the midship section local draft at the distance x f r o m the midship section

ab.solute velocity; (u^ + v^)"-^ longitudinal velocity component; U,.cos(P)

lateral velocity component of the midship; U^.sin(P)

v(x)

local lateral velocity component at the location x; (v + x.r)

distance from the midship section in the longitudinal direction; positive when forward of the midship section : distance of center of gravity f r o m the

direction; positive when forward of the midship section

Y = lateral force component Y ' = non-dimensional lateral force

component; Y/['/2.p.SL.v.lvl] Yn = lateral force on the n-th segment of

the ship

Y n " = non-dimensional local lateral force component; Yn/['/2.p.Sj,.v(x).lv(x)l] P = drift angle = atan(v/u)

Y = non dimensional rate of turning; r.Lpp/Ur

p = density of water

q = distance along the ship behind the FP; Y = volume of displaced water

Furthermore:

1. Primes denote that forces respectively moments have been made non-dimensional by dividing by (0.5pL T) respectively by ( 0 . 5 p L ^ T ) . 2. Double primes denote that a local force been

made non-dimensional by dividing by (O.SpSjj). 3. Subscripts of physical variables denote rate of change of some quantity respect to the subscripted variable, thus: Y^ = 8(Y)/a(v). 4. Subscripts of items denote the relation of a

quantity to the subscripted item, thus: Yj^ = transverse force Y on the bare hull.

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