_Measurements of hydrodynamic forces_ surface pressure_ and wake for obliquely towed tanker model and uncertainty analysis for CFD validation

J Mar Sci Technol (2006) 11:65–75DOI 10.1007/s00773-005-0209-y

Original articles

Measurements of hydrodynamic forces, surface pressure, and wakefor obliquely towed tanker model and uncertainty analysis forCFD validation

Kenichi Kume, Jun Hasegawa, Yoshiaki Tsukada, Junichi Fujisawa, Ryohei Fukasawa,and Munehiko Hinatsu

National Maritime Research Institute, 6-38-1 Shinkawa, Mitaka, Tokyo 181-0004, Japan

The National Maritime Research Institute (NMRI)carried out several kinds of tank tests for a practicaltanker model under oblique towing conditions in whichthe drift angles were primarily 0°, 6°, and 12°. Thetested items were the hydrodynamic forces and mo-ments, surface pressures, and wakes. These items weremeasured in the 400-m-long towing tank at the NMRIwith a width and depth, respectively, of 18m and 8m.The side force distributions were estimated by integra-tion of the surface pressures measured along the girthdirection. In the following sections, procedures andresults of the experiments and the uncertainties arepresented.

2 Ship model

The principal dimensions of the model ship tested areshown in Table 1. The ship was designed by the KoreanInstitute of Ships and Ocean Engineering (KRISO).The model has no appendages and was made of paraffinwax with a wooden frame. It was trimmed by a numeri-cally controlled shaping machine and hand finished.

Studs with trapezoidal heads were nailed to thesurface of a ship hull at 10-mm intervals at S.S.9 1/2(x/LPP = −0.45, see Fig. 1) and at the midsection of thebow bulb to stimulate turbulent flow.

3 Experimental apparatus

3.1 Hydrodynamic forces

The model ship was supported at two fixed positionsunder oblique towing conditions. The system is shownin Fig. 2. The hydrodynamic forces FX (along the x-direction, see Fig. 3) and FY (along the y-direction)were measured using two load cells. The ship’s fixedcoordinates are designated (x, y, z), as shown in Fig. 3.

Abstract This article presents hydrodynamic forces and mo-ments, surface pressures, estimated side force distributions,and wakes under oblique towing conditions for a practicaltanker model (model KVLCC2M), which was designed by theKorea Research Institute of Ships and Ocean Engineering(KRISO). Ship offset data is readily available and can beobtained from the Internet. The model ship has no append-ages and no rudder. Trim and sinkage were adjusted to zeroin the static condition and the model ship was constrainedagainst any motion. Although the drift angle b was primarilyset to 0°, 6°, and 12°, other settings were used in some experi-ments. All experimental results were processed using uncer-tainty analysis. The uncertainty analyzing method follows theANSI/ASME Performance Test Code (PTC19.1-1985) andthe AIAA Standard S-071-1995. Only a few error componentswere considered here and they were empirically chosen be-cause they had a heavy weighting when used in the uncertaintycalculation. The results of these towing tank experiments willcontribute to the development of computational fluid dynam-ics (CFD) research in ship hydrodynamics.

Key words Towing tank test · Oblique towing · Uncertainty ·CFD validation

1 Introduction

Computational fluid dynamics (CFD) software is rap-idly developing and has come to be applied not only tostraightforward towing conditions but also to obliquetowing conditions. Higher accuracy is required in ex-perimental results when they are used as validation datafor CFD research. Although such experimental data areinvaluable for CFD researchers, only a few data sets arepresently available.

Address correspondence to: K. Kume ([email protected])Received: May 13, 2005 / Accepted: October 5, 2005

66 K. Kume et al.: Obliquely towed tanker model

with an electrical strain generator at the beginning ofeach day’s experiments.

3.2 Surface pressure

Surface pressures on the ship hull were measured by asemiconductor type transducer, which was connected topressure holes on the hull surface with vinyl tubes filledwith water. The pressure holes were manufactured asfollows. Firstly, a hole was drilled orthogonal to the shiphull and then a metal tube was mounted in it. The insidediameter of the pressure hole was 1.0 mm. Calibrationof the pressure transducers was carried out before eachday’s experiments by changing the head of water bymoving the transducers up and down. Degassed waterwas used to fill the vinyl tubes to avoid air bubblesblocking the tubes.

3.3 Wake

The wake was measured by the use of an 8-hole spheri-cal pitot tube. It had an additional 3 holes comparedwith the conventional 5-hole pitot tube to accommodatea large angle of flow. The present pitot tube could beused over the ranges −30° to +60° for vertical flows and−30° to +30° degrees for horizontal flows. The pressuremeasuring system used was the same as that employedfor the surface pressure measurements.

Calibration of the 8-hole pitot tube was carried out inadvance. The calibration was done at the towing tank ata velocity of 1.0m/s. A newly designed calibration appa-ratus was used to set inflow angles of the pitot tubeprecisely. The apparatus had a mechanism to adjust theinflow angles by rolling the pitot tube around the axisand turning the supporting point of the pitot tube. Thecalibration range was the same as mentioned above, i.e.,from −30° to +60° in the vertical plane and from −30° to+30° in the horizontal plane. The measurement pointswere arranged in a lattice-like configuration at intervalsof 5° to 10° to give the calibration plane of the pitottube.

Table 1. Principal dimensions of the model tested

Item Symbol Unit Value

Length between perpendiculars LPP m 4.9700Breadth (molded) B m 0.9008Draft (molded) d m 0.3231Wetted surface area without appendages SW m2 6.5597Displacement without appendages ∇ m3 1.1712Center of buoyancy from midship (forward, +) lcb %LPP 3.50Blockage coefficient CB — 0.8098

-0.5(F.P.)

5.0 0(A.P.)x/LPP

Fig. 1. Location of studs for turbulence stimulation (dashedlines)

Fig. 2. Oblique towing system (top view)

Fig. 3. Coordinate systems for hydrodynamic forces andmoment

The capacity of the each load cell was 294 N for both FXand FY. The moment around the z-axis was calculatedby multiplication of force, FY, and the arm length,which is the distance between the midship and the loadcell position. Calibration of FX and FY was undertaken

67K. Kume et al.: Obliquely towed tanker model

4 Experimental procedures and conditions

4.1 Hydrodynamic forces

The model ship used was supported at two points to fixthe ship’s motions as shown in Fig. 2. The front supportpoint was movable in the transverse direction to rotatethe model around the support pillar that was connectedto a towing carriage. The drift angles were calculated bymeasuring the displacement of one point on the modelship in the Y-direction. Because this oblique towingsystem may cause internal forces as a result of adverseconditions in the two supports, the system was set upcarefully to minimize these forces.

Hydrodynamic forces were measured in the longitu-dinal direction (x) and the transverse direction (y). Driftangles were set at every one degree between −3° to +6°and at every three degrees for angles between +6° to+18°. In addition, six to eight repeat measurements wereperformed at angles of 0°, 9°, and 18° for the uncertainty

analysis. The Froude number ( F U gLn pp= ) and

Reynolds number (Rn = ULpp/v) in the tests were 0.1424and 3.945 × 106, respectively.

4.2 Surface pressure

The drift angles were 0°, 6°, and 12° and six semi-conductor pressure transducers were employed. Thecapacity of the transducers was 0.03MPa. The modelship had over 400 measurement points on its hull.The points on the hull were equally divided into threegroups, located fore, mid, and aft, and the surfacepressures in each group were measured using aScanivalve (Liberty Lake, WA, USA) that had six setsof input-switching devices of 24 steps each. Using thismeasurement system, 144 sets of pressure data wereobtained per run. The measurement duration at eachpoint was approximately 5s.

The measurements were performed eight timesat each point for the uncertainty analysis. This meantthat the precision errors were calculated for all measur-ing points. The precision errors were defined as theproduct of the standard deviation of the data and thestudent value (t), which was affected by the samplenumber.

The Froude number was 0.1424 in the tests, and theReynolds numbers were 4.280 × 106 (fore), 4.440 × 106

(mid) and 4.743 × 106 (aft). The differences were theresult of differences in water temperature.

4.3 Wake

Measurements were taken at drift angles of 0°, 6°, and12°. The ranges of the measurement planes were deter-

mined for each drift angle so as to cover large vortexregions, as shown in Fig. 4. The measurement planeswere not in the y–z plane but in the plane vertical to thetowing direction instead. Those planes cut throughx/LPP = 0.48 at the centerline of a ship.

The ranges of the measurement fields were Y/LPP =−0.04 to 0.04, Z/LPP = −0.07 to −0.01 at b = 0°; Y/LPP =−0.04 to 0.12, Z/LPP = −0.07 to −0.01 at b = 6°; and Y/LPP

= −0.04 to 0.17, Z/LPP = −0.07 to 0.00 at b = 12°, where(X, Y, Z) is the space-fixed coordinate system, as shownin Fig. 3. Then the coordinate X can be written as:

X x= ⋅cos β (1)

where b is the drift angle and x = 0.48LPP.The Froude number was 0.1424 and the Reynolds

numbers were 4.011 × 106 (b = 0°), 3.967 × 106 (b = 6°),and 4.000 × 106 (b = 12°).

5 Experimental results and uncertainty analysis

The uncertainty was calculated using the followingequations:

U B P

B B B

Pt SD

M

RSS = +

= ⋅( ) + ⋅( ) + ⋅ ⋅ ⋅

= ⋅

2 2

1 1

2

2 2

2θ θ

(2)

where URSS is a 95% confidence uncertainty, Bi is themagnitude of the bias limit, qi is the sensitivity of eachbias limit, t is the student value, SD is the standarddeviation of measured values, and M is the number ofsample data for calculation of SD.

β

β

β Fig. 4. Ranges of measured velocity field in WAKE 1 plane

68 K. Kume et al.: Obliquely towed tanker model

5.1 Hydrodynamic forces

The hydrodynamic forces analyzed in the present studywere the x- and y-direction forces CX and CY and themoment around the z-axis CN and N/Y. Capitalizing theletter C indicates that it is a dimensionless variable.The hydrodynamic forces and moment were defined asfollows:

C FX V L d

C FY V L d

C MZ V L d

N Y MZ FY L

X W PP

Y W PP

N W PP

PP

=

=

=

= ( )

121212

2

2

2 2

ρ

ρ

ρ

(3)

where FX and FY are the hydrodynamic forces [N], MZis the moment [Nm], r is the density of water [Ns2/m4],VW is the towing speed relative to the water [m/s], LPP isthe length between the perpendiculars [m], and d is thedraft [m].

The experimental results are shown in Figs. 5–8. Be-cause the data at 0°, 9°, and 18° in Fig. 6 were obtained

by averaging eight sets of data, this data deviates slightlyfrom the curve. It is assumed that the differences in thedata are caused by a bias error as a result of the experi-ments being conducted on different days.

The error bars in the figures indicate the 95% confi-dence uncertainty levels. The components contributingto the total bias limit were considered to be the standarderror of the estimate (SEE) of the load cells (B1) and theSEE of the calibration line of the current meter (B2).The bias limits of r, LPP, and d were neglected herebecause we knew empirically they are relatively smallcompared to the other error components. The standarddeviation was considered as the principal component ofthe precision index (P1). The sensitivities at every biaslimit are given by:

θ ∂∂

θ ∂∂

θ ∂∂

θ ∂∂

θ ∂∂

θ ∂∂

θ∂

∂θ

∂∂

1 2

1 2

1 2

1 2 0

= =

= =

= =

=( )

=( )

=

CFX

CV

CFY

CV

CFY

CV

N Y

FY

N Y

V

X X

W

Y Y

W

N N

W

W

(4)

-0.020

-0.019

-0.018

-0.017

-0.016

-0.015

-0.014

-0.013

-0.012

-6 -3 0 3 6 9 12 15 18 21(deg)b

CX

Fig. 5. Hydrodynamic force coefficient, CX

b

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

-6 -3 0 3 6 9 12 15 18 21(deg)

CY

Fig. 6. Hydrodynamic force coefficient, CY

b

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

-6 -3 0 3 6 9 12 15 18 21

(deg)

CN

Fig. 7. Hydrodynamic moment coefficient, CN

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-6 -3 0 3 6 9 12 15 18 21

(deg)

N/Y

b

Fig. 8. Length between midship and center of pressure, N/Y

69K. Kume et al.: Obliquely towed tanker model

The averages, bias limits, precision indices, sensitivities,and uncertainties for CX, CY, CN, and N/Y are summa-rized in Tables 2–5, respectively. The percentages foruncertainty, URSS, at drift angle 0° in Tables 3–5 take

large values because these are affected by the smallaverage values. That is not to say that the absolutevalues of the URSS in these conditions are especiallylarger than the others.

Table 4. Averages and uncertainties for the moment force CN

Drift q1B1

angle (SEE of q2B2 (SEE of Total Total precision Uncertaintyb Average load cells) current meter) bias limit B index P (= P1) URSS

0° −6.449 × 10−5 1.728 × 10−5 7.069 × 10−7 1.729 × 10−5 1.013 × 10−4 1.028 × 10−4

(99.8% of B) (0.2% of B) (2.8% of URSS) (97.2% of URSS) (159% of CN)9° 1.940 × 10−2 1.728 × 10−5 −5.963 × 10−4 5.966 × 10−4 5.856 × 10−4 8.360 × 10−4

(0.1% of B) (99.9% of B) (50.9% of URSS) (49.1% of URSS) (4.3% of CN)18° 3.494 × 10−2 1.728 × 10−5 −1.075 × 10−3 1.075 × 10−3 7.178 × 10−4 1.292 × 10−3

(0.0% of B) (100.0% of B) (69.2% of URSS) (30.8% of URSS) (3.7% of CN)

Table 5. Averages and uncertainties for the point of application of the moment force N/Y

Drift q1B1

angle (SEE of q2B2 (SEE of Total Total precision Uncertaintyb Average load cells) current meter) bias limit B index P (= P1) URSS

0° 2.713 × 10−1 1.756 × 10−1 0.000 1.756 × 10−1 1.502 1.5124(100.0% of B) (0.0% of B) (1.3% of URSS) (98.7% of URSS) (558% of N/Y)

9° 4.260 × 10−1 6.016 × 10−4 0.000 6.016 × 10−4 1.050 × 10−2 1.052 × 10−2

(100.0% of B) (0.0% of B) (0.3% of URSS) (99.7% of URSS) (2.5% of N/Y)18° 2.820 × 10−1 1.319 × 10−4 0.000 1.319 × 10−4 4.636 × 10−3 4.638 × 10−3

(100.0% of B) (0.0% of B) (0.1% of URSS) (99.9% of URSS) (1.6% of N/Y)

Table 3. Averages and uncertainties for the hydrodynamic force CY

Drift q 1B1

angle (SEE of q2B2 (SEE of Total bias Total precision Uncertaintyb Average load cells) current meter) limit B index P (= P1) URSS

0° −4.169 × 10−5 1.559 × 10−4 2.581 × 10−6 1.560 × 10−4 2.259 × 10−4 2.745 × 10−4

(100.0% of B) (0.0% of B) (32.3% of URSS) (67.7% of URSS) (658% of CY)9° 4.553 × 10−2 1.559 × 10−4 −1.368 × 10−3 1.377 × 10−3 4.619 × 10−4 1.452 × 10−3

(1.3% of B) (98.7% of B) (89.9% of URSS) (10.1% of URSS) (3.2% of CY)18° 1.239 × 10−1 1.559 × 10−4 −3.708 × 10−3 3.711 × 10−3 9.685 × 10−4 3.835 × 10−3

(0.2% of B) (99.8% of B) (93.6% of URSS) (6.4% of URSS) (3.1% of CY)

Table 2. Averages and uncertainties for the hydrodynamic force CX

Drift q1B1

angle (SEE of q2B2 (SEE of Total bias Total precision Uncertaintyb Average load cells) current meter) limit B index P (= P1) URSS

0° −1.756 × 10−2 1.842 × 10−4 −5.285 × 10−4 5.597 × 10−4 1.477 × 10−4 5.788 × 10−4

(10.8% of B) (89.2% of B) (93.5% of URSS) (6.5% of URSS) (3.3% of CX)9° −1.729 × 10−2 1.842 × 10−4 −5.288 × 10−4 5.600 × 10−4 2.689 × 10−4 6.212 × 10−4

(10.8% of B) (89.2% of B) (81.3% of URSS) (18.7% of URSS) (3.6% of CX)18° −1.370 × 10−2 1.842 × 10−4 −4.213 × 10−4 4.598 × 10−4 2.209 × 10−4 5.101 × 10−4

(16.1% of B) (83.9% of B) (81.2% of URSS) (18.8% of URSS) (2.9% of CX)

q, sensitivity of the bias limit; B, magnitude of the bias limit; SEE, standard error of the estimate

70 K. Kume et al.: Obliquely towed tanker model

5.2 Surface pressure

The experimental results showing the surface pressuresat drift angles of 0°, 6°, and 12° are shown in Figs. 9–11.The pressure distributions are represented by a dimen-sionless number called CP. CP is defined as:

CP

UP =

12

2ρ (5)

where P is the pressure on the ship’s hull [N/m2], r is thedensity of water [Ns2/m4], and U is the towing speedrelative to the ground [m/s].

In Eq. 5, the towing speed relative to the ground,U, is used instead of VW, because the authors don’tconsider that VW gives a good estimation of thecharacteristic velocity, especially for surface pressuremeasurements under oblique towing condition. Inspatially nonuniform residual flow, the characteristicvelocities for each pressure hole would be slightly dif-ferent by location, even if the pressure data were mea-

0.1

0.2

-0.1

-0.18

0

x/Lpp

z/Lp

p

0.30 0.35 0.40 0.45 0.50 0.55

-0.06

-0.04

-0.02

0

Fig. 9. Surface pressure distribution for b = 0° (contour lineinterval ∆CP = 0.01)

-0.05-0.1-0.15-0.2

-0.25

-0.3-0.35-0.4-0.45

-0.3

-0.25

-0.05-0.1

-0.1

5

-0.2-0.25

-0.3

-0.3

-0.45 -0.4 -0.35

-0.3

-0.2

5 -0.2-0.15 -0.1

-0.05

-0.05

-0.10-0.15-0.20-0.25

-0.25

x/Lpp-0.5 -0.4 -0.3 -0.2 -0.1 0

-0.05

-0.1

-0.2

-0.1 0 0.1

0.25

-0.0 5

-0.1

-0.15 -0.2

-0.15

-0.2

-0.1 0

0.1

0.2

0-0.1 0.1

0.20.25

-0.05-0

.1 -0.2 -0.1 0

0.1

0.25

x/Lpp0 0.1 0.2 0.3 0.4 0.5

Fig. 10. a Distribution of surface pressure coefficient for b = 6°(fore) (contour line interval ∆CP = 0.05). b Distributionof surface pressure coefficient for b = 6° (aft) (contour lineinterval ∆CP = 0.05)

a

b

-0.05-0.1-0.2

-0.3-0.4-0.5

-0.25

-0.1

-0.2-0.3

-0.5

-0.6

-0.5

-0.4 -0.3-0.2 -0.1

0-0.05-0.1

0

-0.15-0.15

x/Lpp-0.5 -0.4 -0.3 -0.2 -0.1 0

-0.05-0.1

-0.15 -0.1 0 0.1 0.2

-0.2-0.3

-0.1

-0.2

-0.1

-0.25-0.25

0

0.1

0.2

0.250.2

0.10

-0.1

-0.15

-0.2

-0.3

-0.2-0.1-0

.05

-0.2

-0.25

-0.1 0 0.1

-0.2

0.25

x/Lpp0 0.1 0.2 0.3 0.4 0.5

a

b

Fig. 11. a Distribution of surface pressure coefficient for b =12° (fore) (contour line interval ∆CP = 0.05). b Distribution ofsurface pressure coefficient for b = 12° (aft) (contour lineinterval ∆CP = 0.05)

71K. Kume et al.: Obliquely towed tanker modelC

P

y/LPP

-0.05 0 0.05-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Fig. 12. Uncertainty of surface pressure coefficient for b = 12°at S.S.1

sured at the same time. The effect of the residual flow isimplicitly included in the error of surface pressure, P, asa consequence. The pressure measurements were re-peated eight times at each pressure hole under the sameconditions. The average values were used to draw thecontours.

As the drift angle became larger, lower pressure areaswere observed in the region of the stem bottom on portside and stern bottom on starboard side. The uncertain-ties of the measured data are shown in Fig. 12. Althoughthis figure shows only the data for a drift angle of 12° atS.S.1, the uncertainty levels were almost the same overthe hull surface and for every other drift angle.

The components of the total bias limit were the SEEof the calibration line of the pressure transducers (B1),an estimated maximum error in the carriage speed (B2),and an estimated maximum difference of drift angles(B3). The bias limit of r was neglected. The standarddeviation of measured data was considered as a princi-pal component of the precision index (P1). The sensitivi-ties of each bias limit were given by:

θ ∂∂

θ ∂∂

θ ∂∂ β1 2 3= = =C

PCU

CP P P (6)

Averages, bias limits, precision indices, sensitivities,and uncertainties for CP at y/LPP = 0 at S.S.1 are summa-rized in Table 6. From the table, we see that the uncer-tainty of CP mainly consists of B1. Similar trends wereobserved for all measuring points and drift angles.

5.3 Interpolation accuracy of surface pressure

Since the pressure data was measured at coarse inter-vals, it needed to be interpolated to draw smooth con-tours over the hull surface. The interpolation methodused was “Kriging,” which was carried out by thegraphic software Tecplot (Bellevue, WA, USA).

Generally speaking, the interpolation process cancause errors and therefore its accuracy must be vali-dated by another method. The validation method was asfollows. First, a certain arbitrary function expressed interms of the coordinates was defined and the true valuesover the whole hull surface were calculated. The equa-tion adopted was as follows:

f x y z x z, ,( ) = ⋅sin sin3 3 (7)

Then, using the true values of the pressure measure-ment points, the values at the other points on the hullsurface were interpolated by the Kriging method. Com-parison of the interpolated values with the true valuesrevealed the accuracy of the interpolation method. Acomparison is made in Fig. 13.

The interpolated values at the fore and aft of themodel ship coincide well with the true values becauseenough measurement points were taken in these re-gions; however, at the mid region, interpolation datatends to be slightly inaccurate because of the insuffi-ciency of measurement points.

Table 6. Averages and uncertainties for the surface pressure CP at y/LPP = 0 of S.S.1

Drift q1B1 (SEE of q2B2

angle pressure (Carriage q3B3 Total bias Total precision Uncertaintyb Average transducers) speed) (Drift angle) limit B index P (= P1) URSS

0° −5.794 × 10−2 1.662 × 10−2 4.008 × 10−3 −2.812 × 10−4 1.710 × 10−2 3.038 × 10−3 1.736 × 10−2

(94.5% of B) (5.5% of B) (0.0% of B) (96.9% of URSS) (3.1% of URSS) (30.0% of CP)6° −7.481 × 10−2 1.662 × 10−2 4.008 × 10−3 −7.760 × 10−4 1.711 × 10−2 3.071 × 10−3 1.739 × 10−2

(94.3% of B) (5.5% of B) (0.2% of B) (96.9% of URSS) (3.1% of URSS) (23.2% of CP)12° −1.511 × 10−1 1.662 × 10−2 4.008 × 10−3 −1.271 × 10−3 1.714 × 10−2 6.662 × 10−3 1.839 × 10−2

(94.0% of B) (5.5% of B) (0.5% of B) (86.9% of URSS) (13.1% of URSS) (12.2% of CP)

Fig. 13. Contours of interpolated and true values. Upper,interpolated values; lower, true values; dots are the locationsof pressure holes

72 K. Kume et al.: Obliquely towed tanker model

5.4 Lateral force distribution

The lateral force distributions were estimated by inte-gration of the surface pressures measured along thegirth direction and are shown in Figs. 14 and 15. Theerror bars in the figures indicate the uncertainty levels,which were calculated using the precision error basedon the student value and the standard deviations of thedata at locations where it was measured eight times.

5.5 Wake

The experimental results of wake at drift angles of 0°,6°, and 12° are shown in Figs. 16–18. The axial velocity,u, and the tangential velocities, v and w, are dimension-less values divided by the ship speed. Generally, onlyone measurement was performed at each point.

Both Figs. 17a and 18a reveal a large vortex on theright-hand side for the conditions of b = 6° and 12°. Itwas generated at the fore part of the ship. There wasa region above the propeller boss where the wakecouldn’t be analyzed, particularly at b = 12°. This wasbecause the lateral flow angle exceeded the calibration

range of the pitot tube. It may be possible to estimatethe wake values at these points by the use of adjacentdata, but the contour lines in this region were not indi-cated because it is difficult to extrapolate wake valuesexactly and incorrect information can be generated.

In addition, uncertainty analysis was performed forthe velocities on the line of z/LPP = −0.0348. Theyare given in Figs. 19–21. The error bars in the figuresindicate the uncertainty levels. The components thatcontribute to the total bias limit are the estimated maxi-mum difference of the setting angles of the pitot tube incalibration (B1), the SEE of the calibration line of thepressure transducers (B2), and the SEE of the currentmeter (B3) as an error of the residual flow. The standarddeviation calculated from data that were measuredeight times was the principal component for calculatingthe precision index (P1). The sensitivities at every biaslimit are given by:

∆YP

[=F

Y P/(

0.5 ρ

U2 L

PPd)

/dx]

x/LPP

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Fig. 14. Estimated lateral force distribution for b = 6°

∆YP

[ =F

Y P/(

0.5 ρ

U2 L

PPd)

/dx]

x/LPP

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Fig. 15. Estimated lateral force distribution for b = 12°

0.9

0.8

0.70.6 0.4

0.3

0.3 0.30.2

0.4

0.9

0.8

0.70.60.5

0.4

0.50.4

0.3

0.2

0.5

-0.04 -0.02 0 0.02 0.04-0.08

-0.06

-0.04

-0.02

y/LPP

z/L

PP

-0.04 -0.02 0 0.02 0.04-0.08

-0.06

-0.04

-0.02

0.2

L

z/L

PP

y/LPP

Fig. 16. a Velocity field contours in WAKE 1 plane for b = 0°.b Cross-flow vectors in WAKE 1 plane for b = 0°

a

b

73K. Kume et al.: Obliquely towed tanker model

0.9

0.8

0.7

0.60.5

0.9

0.8

0.7

0.8

0.4 0.5

0.4

0.4

0.70.6

0.5

0.9

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

-0.06

-0.04

-0.02

0

y/LPP

z/L

PP

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

-0.06

-0.04

-0.02

0

0.2

y/LPP

z/L

PP

Fig. 17. a Velocity field contours in WAKE 1 plane for b = 6°.b Cross-flow vectors in WAKE 1 plane for b = 6°

0.9

0.8 0.7

0.6

0.50.4

0.9

0.8

0.7

0.9

0.8

0.7

0.60.5

0.7

0.6

0.5

0.6

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.08

-0.06

-0.04

-0.02

0

y/LPP

z/L

PP

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.08

-0.06

-0.04

-0.02

0

0.2

y/LPP

z/L

PP

Fig. 18. a Velocity field contours in WAKE 1 plane for b =12°. b Cross-flow vectors in WAKE 1 plane for b = 12°

y/LPP

u/U

, v/U

, w/U

u

v

w

/U

/U

/U

-0.04 -0.02 0 0.02 0.04-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 19. Axial and tangential velocities in WAKE 1 plane atz/LPP = −0.0348, b = 0°

y/LPP

u/U

, v/U

, w/U

u

w

v

/U

/U

/U

-0.04 -0.02 0 0.02 0.04-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 20. Axial and tangential velocities in WAKE 1 plane atz/LPP = −0.0348, b = 6°

y/LPP

u/U

, v/U

, w/U

u

v

w/U

/U

/U

-0.04 -0.02 0 0.02 0.04-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 21. Axial and tangential velocities in WAKE 1 plane atz/LPP = −0.0348, b = 12°

θ∂

∂ α φθ

∂

∂θ

∂∂

θ∂

∂ α φθ

∂

∂θ

∂∂

θ∂

∂ α φ

1 2 3

1 2 3

1

=( )( ) =

( )( ) =

( )

=( )( ) =

( )( ) =

( )

=( )( )

u U

f

u U

f P P P P P

u U

U

v U

f

v U

f P P P P P

v U

U

w U

f

T B C S P

T B C S P

, , , , ,

, , , , ,

,θθ

∂

∂θ

∂∂2 3=

( )( ) =

( )w U

f P P P P P

w U

UT B C S P, , , ,

(8)

where f(a, f) is the function used to calculatethe influence on the wake velocities caused by anymisalignment of the pitot tube during calibration andf(PT,PB, PC, PS, PP) is the function used to calculate theinfluence on the wake velocities caused by bias limitsof the total pressures measured by the five holes ofthe eight-hole pitot tube. Parameters a and f are thedifferences in the setting angles of the pitot tube duringcalibration.

Averages, bias limits, precision indices, sensitivities,and uncertainties for u/U, v/U, and w/U are presented in

a

b

a

b

74 K. Kume et al.: Obliquely towed tanker model

Table 8. Averages and uncertainties for tangential velocity v/U at y/LPP = 0 and z/LPP = −0.0348

Drift q1B1 q2B2 (SEE of q3B3 Totalangle (Misalignment pressure (Residual Total bias precision Uncertaintyb Average in calibration) transducers) flow) limit B index P (= P1) URSS

0° −2.276 × 10−2 1.734 × 10−3 3.724 × 10−2 0.000 3.728 × 10−2 2.198 × 10−2 4.328 × 10−2

(0.2% of B) (99.8% of B) (0.0% of B) (74.2% of URSS) (25.8% of URSS) (190% of v/U)6° 7.677 × 10−2 1.734 × 10−3 3.983 × 10−2 0.000 3.987 × 10−2 9.305 × 10−3 4.094 × 10−2

(0.2% of B) (99.8% of B) (0.0% of B) (94.8% of URSS) (5.2% of URSS) (53.3% of v/U)12° 7.342 × 10−2 1.734 × 10−3 1.808 × 10−2 0.000 1.817 × 10−2 3.510 × 10−3 1.850 × 10−2

(0.9% of B) (99.1% of B) (0.0% of B) (96.4% of URSS) (3.6% of URSS) (25.2% of v/U)

Table 9. Averages and uncertainties for tangential velocity w/U at y/LPP = 0 and z/LPP = −0.0348

Drift q1B1 q2B2 (SEE of q3B3 Totalangle (Misalignment pressure (Residual Total bias precision Uncertaintyb Average in calibration) transducers) flow) limit B index P (= P1) URSS

0° −2.764 × 10−1 2.002 × 10−3 4.201 × 10−2 0.000 4.206 × 10−2 1.075 × 10−2 4.341 × 10−2

(0.2% of B) (99.8% of B) (0.0% of B) (93.9% of URSS) (6.1% of URSS) (15.7% of w/U)6° −1.441 × 10−1 2.002 × 10−3 4.046 × 10−2 0.000 4.051 × 10−2 9.866 × 10−3 4.169 × 10−2

(0.2% of B) (99.8% of B) (0.0% of B) (94.4% of URSS) (5.6% of URSS) (28.9% of w/U)12° −4.260 × 10−2 2.002 × 10−3 2.022 × 10−2 0.000 2.032 × 10−2 3.477 × 10−3 2.062 × 10−2

(1.0% of B) (99.0% of B) (0.0% of B) (97.2% of URSS) (2.8% of URSS) (48.4% of w/U)

Table 7. Averages and uncertainties for axial velocity u/U at y/LPP = 0 and z/LPP = −0.0348

Drift q1B1 q2B2(SEE of q3B3 Totalangle (Misalignment pressure (Residual Total bias precision Uncertaintyb Average in calibration) transducers) flow) limit B index P (= P1) URSS

0° 2.931 × 10−1 2.017 × 10−6 4.523 × 10−2 1.507 × 10−2 4.768 × 10−2 7.871 × 10−3 4.832 × 10−2

(0.0% of B) (90.0% of B) (10.0% of B) (97.3% of URSS) (2.7% of URSS) (16.5% of u/U)6° 3.779 × 10−1 2.017 × 10−6 1.489 × 10−1 1.507 × 10−2 1.496 × 10−1 8.123 × 10−3 1.498 × 10−1

(0.0% of B) (99.0% of B) (1.0% of B) (99.7% of URSS) (0.3% of URSS) (39.7% of u/U)12° 6.452 × 10−1 2.017 × 10−6 4.679 × 10−2 1.507 × 10−2 4.916 × 10−2 4.659 × 10−3 4.938 × 10−2

(0.0% of B) (90.6% of B) (9.4% of B) (99.1% of URSS) (0.9% of URSS) (7.7% of u/U)

Tables 7–9. These tables suggest that the error compo-nent B2 is dominant at any drift angle.

6 Conclusion

Experimental results and uncertainties of hydrody-namic forces and moments, surface pressures, estimatedlateral force distributions, and wake are presented. Theseries of tank experiments has been successfully com-pleted, but the uncertainties in some cases still neededto be reduced. Further ways of reducing the error mustbe considered. These data are valuable for all CFDresearchers and we believe that they will help acceleratethe research and development of numerical shiphydrodynamics.

Acknowledgments. The authors wish to thank Mr. T.Kanai at the Shipbuilding Research Center of Japanand Dr. Y. Ukon at NMRI for their assistance withpreparing offset data for the model ship.

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