SK Validation Study – 2D Coefficients and Forces

The most fundamental computation performed by SeaKeeping is the solution of the two-

dimensional radiation problem for each section. The solution leads to the two-

dimensional hydrodynamic added mass and damping coefficients, and the two-

dimensional complex diffraction forcing amplitude for the section. To confirm the

efficacy of the code, the results for simple geometries were compared to model test data

published by Vugts (1968).

Three different geometries were analyzed: a box section, a triangular section, and a

cylindrical section, as shown by Figure 1. For the box, three beam-to-draft (B/T) ratios

were also analyzed. In all cases, the beam factor (See SEAKEEPING command

documentation regarding /BF) was taken at the default value of 8. The resulting non-

dimensional coefficients, and the total heave, sway, and roll forcing amplitudes are given

in the following section. In most cases, fair to good agreement between the experimental

data and the computational result is obtained. Appreciable differences, such as the case in

roll for the triangular section, are attributable to neglecting viscosity, flow-separation, and

other significant nonlinearities evident in the real-world flow. Further deviation is

observed in the cylindrical section in roll and sway-into-roll. Here, theoretical results are

practically zero, as viscous effects account for most of the experimental observations.

These assumptions are typical of two-dimensional potential flow based solvers, such

as the one used by SeaKeeping, and the deviations are expected.

Figure 1 Vugts 2D Section Geometry

In the case of roll for the box section at B/T=4 and B/T=8, the experimental data is

corrected for the difference between the origin “O” and the section’s vertical center-of-

gravity “CG” as outlined by Vugts (1968). In these cases, a note has been included in the

plot legend. The computational results were also obtained about the origin, or CG, to

maintain consistency with Vugts’ conventions. Results about the origin when B/T=4 and

B/T=8 were obtained by setting the VCG equal to the draft. In all other cases, the VCG

was taken at 0.2 m as indicated.

In all cases, a heading of 90 degrees (beam seas) was used to match the experimental

conditions imposed by Vugts (1968). The total forcing amplitude in sway, heave, and roll

are in good agreement with the experimental results. For completeness, the roll moment

is also computed about the origin for the box section for beam-to-draft ratios other than 2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a2

2/ρA

ω(B/2g)1/2

SK, B/T=2

SK, B/T=4

SK, B/T=8

Vugts, B/T=2

Vugts, B/T=4

Vugts, B/T=8

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b2

2/ρA

)(B

/2g

)1

/2

ω(B/2g)1/2

Figure 2 2D Sway Added Mass and Damping (a22/b22) – Box Section

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a3

3/ρA

ω(B/2g)1/2

SK, B/T=2

SK, B/T=4

SK, B/T=8

Vugts, B/T=2, ζa=0.01

Vugts, B/T=4, ζa=0.01

Vugts, B/T=8, ζa=0.01

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b3

3/ρA

)(B

/2g

)1

/2

ω(B/2g)1/2

Figure 3 2D Heave Added Mass and Damping (a33/b33) – Box Section

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a2

4/ρA

B

ω(B/2g)1/2

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b2

4/ρA

B)(

B/2

g)

1/2

ω(B/2g)1/2

SK, B/T=2, CG=O

SK, Vugts, CG=O

Figure 4 2D Sway-into-Roll Added Mass and Damping Coefficient (a24/b24) – Box

Section, B/T=2

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a2

4/ρA

B

ω(B/2g)1/2

SK, B/T=4, CG

SK, B/T=4, O

Vugts, O-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b2

4/ρA

B)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 5 2D Sway-into-Roll Added Mass and Damping Coefficient (a24/b24) – Box

Section, B/T=4

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a2

4/ρA

B

ω(B/2g)1/2

SK, B/T=8, CG

SK, B/T=8, O

Vugts, O

-0.1

-0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b2

4/ρA

B)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 6 2D Sway-into-Roll Added Mass and Damping Coefficient (a24/b24) – Box

Section, B/T=8

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a4

4/ρA

B2

ω(B/2g)1/2

SK, B/T=2, CG=O

Vugts, CG=O, φa=0.05

Vugts, CG=O, φa=0.10

Vugts, CG=O, φa=0.20

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b4

4/ρA

B2)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 7 2D Roll Added Mass Moment of Inertia and Damping Coefficient (a44/b44) –

Box Section, B/T=2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a4

4/ρA

B2

ω(B/2g)1/2

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b4

4/ρA

B2)(

B/2

g)

1/2

ω(B/2g)1/2

SK, B/T=4, CG

SK, B/T=4, O

Vugts, CG, φa=0.05

Vugts, CG, φa=0.20

Figure 8 2D Roll Added Mass Moment of Inertia and Damping Coefficient (a44/b44) –

Box Section, B/T=4

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a4

4/ρA

B2

ω(B/2g)1/2

SK, B/T=8, CG

SK, B/T=8, O

Vugts, CG, φa=0.05

Vugts, CG, φa=0.20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b4

4/ρA

B2)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 9 2D Roll Added Mass Moment of Inertia and Damping Coefficient (a44/b44) –

Box Section, B/T=8

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fy/ρ

gA

kζ

a

ω(B/2g)1/2

SK, B/T=2

Vugts

Figure 10 2D Total Sway Forcing (Fy) – Box Section B/T=2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fz/ρ

gB

ζa

ω(B/2g)1/2

SK, B/T=2

Vugts

Figure 11 2D Total Heave Forcing (Fz) – Box Section B/T=2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(Mx/ρ

gk

ζa)(

12

/B3)

ω(B/2g)1/2

SK, B/T=2

Vugts

Figure 12 2D Total Roll Moment (Mx) – Box Section B/T=2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fy/ρ

gA

kζ

a

ω(B/2g)1/2

SK, B/T=4

Vugts

Figure 13 2D Total Sway Forcing (Fy) – Box Section B/T=4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fz/ρ

gB

ζa

ω(B/2g)1/2

SK, B/T=4

Vugts

Figure 14 2D Total Heave Forcing (Fz) – Box Section B/T=4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(Mx/ρ

gk

ζa)(

12

/B3)

ω(B/2g)1/2

SK, B/T=4, CG

SK, B/T=4, O

Vugts

Figure 15 2D Total Roll Moment (Mx) – Box Section B/T=4

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fy/ρ

gA

kζ

a

ω(B/2g)1/2

SK, B/T=8

Vugts

Figure 16 2D Total Sway Forcing (Fy) – Box Section B/T=8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fz/ρ

gB

ζa

ω(B/2g)1/2

SK, B/T=8

Vugts

Figure 17 2D Total Heave Forcing (Fz) – Box Section B/T=8

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(Mx/ρ

gk

ζa)(

12

/B3)

ω(B/2g)1/2

SK, B/T=8, CG

SK, B/T=8, O

Vugts

Figure 18 2D Total Roll Moment (Mx) – Box Section B/T=8

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a2

2/ρA

ω(B/2g)1/2

SK

Vugts

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b2

2/ρA

)(B

/2g

)1

/2

ω(B/2g)1/2

Figure 19 2D Sway Added Mass and Damping Coefficient (a22/b22) – Triangle Section

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a3

3/ρA

ω(B/2g)1/2

SK

Vugts, ζ a=0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b3

3/ρA

)(B

/2g)

1/2

ω(B/2g)1/2

Figure 20 2D Heave Added Mass and Damping Coefficient (a33/b33) – Triangle Section

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a2

4/ρA

B

ω(B/2g)1/2

SK, CG=O

Vugts, CG=O

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b2

4/ρA

B)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 21 2D Sway-into-Roll Added Mass and Damping Coefficient (a24/b24) – Triangle

Section

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a4

4/ρA

B2

ω(B/2g)1/2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(b4

4/ρA

B2)(

B/2

g)

1/2

ω(B/2g)1/2

SK, CG=O

Vugts, CG=O, φa=0.05

Figure 22 2D Roll Added Mass Moment of Inertia and Damping Coefficient (a44/b44) –

Triangle Section

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fy/ρ

gA

kζ

aω(B/2g)

1/2

SK

Vugts

Figure 23 2D Total Sway Forcing (Fy) – Triangle Section

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fz/ρ

gB

ζa

ω(B/2g)1/2

SK

Vugts

Figure 24 2D Total Heave Forcing (Fz) – Triangle Section

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(Mx/ρ

gk

ζa)(

12

/B3)

ω(B/2g)1/2

SK

Vugts

Figure 25 2D Total Roll Moment (Mx) – Triangle Section

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a2

2/ρA

ω(B/2g)1/2

SK

Vugts

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b2

2/ρA

)(B

/2g)

1/2

ω(B/2g)1/2

Figure 26 2D Sway Added Mass and Damping Coefficient (a22/b22) – Cylinder Section

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a3

3/ρA

ω(B/2g)1/2

SK

Vugts, ζ a=0.01

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b3

3/ρA

)(B

/2g

)1

/2

ω(B/2g)1/2

Figure 27 2D Heave Added Mass and Damping Coefficient (a33/b33) – Cylinder Section

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a2

4/ρA

B

ω(B/2g)1/2

SK, CG=O

Vugts, CG=O

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b2

4/ρA

B)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 28 2D Sway-into-Roll Added Mass Coefficient and Damping Coefficient

(a24/b24) – Cylinder Section

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a4

4/ρA

B2

ω(B/2g)1/2

SK, CG=O

Vugts, CG=O, φa=0.05

-0.015

-0.01

-0.005

0

0.005

0.01

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b4

4/ρA

B2)(

B/2

g)

1/2

ω(B/2g)1/2

Figure 29 2D Roll Added Mass Moment of Inertia and Damping Coefficient (a44/b44) –

Cylinder Section

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fy/ρ

gA

kζ

a

ω(B/2g)1/2

SK

Vugts

Figure 30 2D Total Sway Forcing (Fy) – Cylinder Section

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fz/ρ

gB

ζa

ω(B/2g)1/2

SK

Vugts

Figure 31 2D Total Heave Forcing (Fz) – Cylinder Section

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(Mx/ρ

gk

ζa)(

12

/B3)

ω(B/2g)1/2

SK

Vugts

Figure 32 2D Total Roll Moment (Mx) – Cylinder Section

References

VUGTS, J. H. (1968). The hydrodynamic coefficients for swaying, heaving and rolling

cylinders in a free surface = De hydrodynamische coefficienten voor het

verzetten, dompen en slingeren van cilinders in een vrije vloeistof oppervlakte.

Delft, Netherlands Ship Research Centre TNO, Shipbuilding Dept. Available at:

http://mararchief.tudelft.nl/file/409/ [Last Accessed: 15 Jan. 2018].