Ship squat prediction using a potential flow Rankinesource method

Kevin McTaggart

Defence Research and Development Canada – Atlantic Research Centre, P.O. Box 1012,Dartmouth, Nova Scotia, B2Y 3Z7, Canada

Abstract

A numerical method has been developed predicting ship squat, with the goal of

providing guidance to ship operators for avoidance of grounding in shallow wa-

ter. The method assumes potential flow, and computations are sufficiently fast

to allow evaluation for large numbers of conditions within practical time scales.

Boundary conditions on the hull, free surface, and canal walls are satisfied using

planar Rankine sources on solid boundaries and point Rankine sources placed a

nominal distance above the calm waterline or free surface. Four different models

of the free surface boundary condition have been implemented, ranging from a

nominally flat free surface to fully nonlinear. For ships with a transom stern,

the influence of flow separation is modelled using a virtual stern, which extends

aftward from the transom to a location with zero width. The numerical method

is generally robust, except for in the vicinity of critical flow conditions. Valida-

tion with model tests has been used to develop an envelope for over-prediction

of bottom clearance, which can be considered when developing guidance for ship

operators.

Keywords: heave, pitch, Rankine, method, sinkage, squat, trim

Email address: kevin.mctaggart@drdc-rddc.gc.ca (Kevin McTaggart)

Preprint submitted to Elsevier February 8, 2017

DRDC-RDDC-2017-P105 @Her Majesty the Queen in Right of Canada (Department of National Defence), 2017@Sa Majesté la Reine en droit du Canada (Ministère de la Défense nationale), 2017

1. Introduction

A ship moving at speed in shallow water is subjected to squat, a reduction in

clearance from the water bottom due to hydrodynamic forces acting on the ship.

Briggs et al. (2010) give a comprehensive desciption of squat and its prediction.

Squat increases the risk of grounding, and is thus of significant concern to ship

operators. Ideally, a ship operator should have a good understanding of squat

for combinations of ship loading condition, water depth, and ship speed that

might be encountered.

Increasingly complex methods are available for predicting squat in both open

and confined waters. Various simple methods are available, including those de-

scribed by Dand and Ferguson (1973), Millward (1990), and Lataire et al. (2012).

Gourlay (2008) used a slender body approach for predicting squat. Yao and Zou

(2010) developed a potential flow panel method and compared it with results

from model tests for a Series 60 hull with block coefficient of 0.60. Computa-

tional fluid dynamics using Reynolds-averaged Navier-Stokes modelling (RANS

CFD) can provide high fidelity modelling of ship steady flow, including squat

(Jachowski, 2008; Toxopeus et al., 2013).

The present work was motiviated by the requirement to predict squat for

operational vessels, each operating with various loading conditions, water con-

ditions, and ship speeds. It was decided that a potential flow Rankine panel

method, similar to that used by Yao and Zou (2010), would provided a suitable

balance of accuracy and computational time. Furthermore, potential flow sea-

keeping software was already available (McTaggart, 2015) that could be used as

the foundation for developing the steady flow method. Some exploratory work

was done using RANS CFD; however, computational times were very large, typ-

ically exceeding one day for each combination of ship loading condition, water

depth, and speed.

2

2. Solution of flow surrounding ship and resulting sinkage and trim

The solution of ship squat uses an iterative process to determine the ship

sinkage and trim that results in zero net forces acting on the ship. The flow

surrounding the ship and resulting ship forces must be solved during each itera-

tion. The flow surrounding the ship is solved using a boundary element method,

for which Bertram (2012) presents an overview. The axis system has its origin

on the calm waterlane and is aligned vertically with the ship centre of gravity.

The x-axis is + forward, the y-axis is + port, and the z-axis is + upward. The

flow potential in the vicinity of the hull and free surface is modelled as:

Φ(~x) = −U x + φ(~x) (1)

where Φ is the total steady potential, ~x is a location in the fluid domain, U is

the steady ship speed, and φ is the scattered potential due to the presence of

the hull. The discretized scattered potential on the hull and free surface can be

expressed based on source strengths σ(~x) on the hull and free surface as follows:

{φ} = [E] {σ} (2)

where [E] is the influence matrix giving scattered velocity potentials from source

strengths. For a ship in deep water, the elements of the influence matrix [E]

are given by:

Eij =1

4 π

∫Aj

1

R(~xi, ~xs) dA (3)

where Aj is the area of source panel j, and R is the distance from a source at

~xs to a field point at ~xi. The source strengths must be set such that bound-

ary conditions at the hull and free surface are satisfied. The zero normal flow

boundary condition on the hull is:

∂φ

∂n= U nx on Shull (4)

The free surface Sfree is denoted by z = ζ(x, y), at which the following kinematic

boundary condition must be satisfied:

∂φ

∂z=

(−U +

∂φ

∂x

)∂ζ

∂z+

∂φ

∂y

∂ζ

∂yat z = ζ (5)

3

The pressure at a point in the fluid domain is given by:

p = ρ

[−g z +

1

2

(U2 −

(−U +

∂φ

∂x

)2

−(∂φ

∂y

)2

−(∂φ

∂z

)2)]

(6)

Given that the pressure must be zero at the free surface, the free surface elevation

ζ is given by:

ζ =1

2 g

[U2 −

(−U +

∂φ

∂x

)2

−(∂φ

∂y

)2

−(∂φ

∂z

)2]

(7)

The associated x and y derivatives of water elevation are:

∂ζ

∂x=

1

g

[−∂

2φ

∂x2

(∂φ

∂x− U

)− ∂2φ

∂x∂y

∂φ

∂y− ∂2φ

∂x∂z

∂φ

∂z

](8)

∂ζ

∂y=

1

g

[− ∂2φ

∂x ∂y

(∂φ

∂x− U

)− ∂2φ

∂y2∂φ

∂y− ∂2φ

∂y∂z

∂φ

∂z

](9)

For a ship in shallow water, the following boundary condition at the water

bottom must be satisifed:

∂φ

∂z= 0 on Sbottom (10)

The present work considers only cases with constant water depth, allowing the

influence of the water bottom to be solved using image sources (Garrison, 1978).

The elements of the influence matrix [E] are then given by:

Eij =1

4 π

∫Aj

1

R(~xi, ~xs) +

1

Rimage(~xi, ~xs)dA (11)

where Rimage is the distance to the image source reflected about the ocean

bottom.

Vertical walls from a canal or other stationary body can also be modelled.

In such cases, the veritcal wall must be panelled and the following boundary

condition must be satisified on each panel:

∂φ

∂y= 0 on Swall (12)

The ability of the solution to satisfy the boundary condition at each free

surface point can be evaluated using the following error term based on Bertram

(2012):

εf =1

U g

[a1

(∂φ

∂x− U

)+ a2

∂φ

∂y+ (a3 + g)

∂φ

∂z

](13)

4

A nonlinear solution with successful convergence of wave elevation gives bound-

ary condition errors that are effectively zero.

2.1. Double body solution

The simplest approach for modelling the free surface assume that it is nom-

inally flat, satisifying the following boundary condition:

∂φ

∂z= 0.0 on z = 0.0 (14)

where φ is referred to as the double body potential. Solution of the resulting

flow field can be done without panelling of the free surface by using image

sources of the hull reflected about z = 0.0 (Garrison, 1978). The double body

solution can give good results at low ship speeds, which can be useful because

free surface wavelengths are very small at such speeds, thus requiring large

numbers of panels for solutions using a panelled free surface. The double body

solution is very robust. Garrison (1978) describes evaluation of the double body

solution for both deep water and water of finite depth.

2.2. Uniform linearized free surface solution

A linearized solution based on the uniform incident flow provides a simple

approach to evaluate the free surface, as shown by Tarafder and Khalil (2006).

The scattered velocity components are assumed to be small relative to the ship

speed, and the flow kinematic terms at z = 0.0 are assumed to be representative

of the free surface. The free surface kinematic condition of Equation 5 is then

simplified to:

∂2φ

∂x2+

g

U2

∂φ

∂z= 0.0 on z = 0.0 (15)

The free surface elevation is given by:

ζ =U

g

∂φ

∂x(16)

The uniform linearized solution can give excellent results for submerged bodies

and for surface-piercing bodies that are slender at the free surface.

5

2.3. Double body linearized free surface solution

To improve upon the accuracy provided by the uniform linearized solution,

the free surface can be linearized with respect to the double body flow. This

method was first developed by Dawson (1977). The scattered velocity potential

is expressed in terms of the double-body potential φ and a potential φ that

accounts for the free surface:

φ = φ + φ (17)

It is assumed that the z derivatives of flow velocities are small near the free

surface, allowing the kinematic boundary condition of Equation 5 to be satisfied

at the mean free surface:

∂φ

∂z=

(−U +

∂φ

∂x

)∂ζ

∂x+

∂φ

∂y

∂ζ

∂yat z = 0.0 (18)

Combining Equation 18 with Equations 8 and 9 for the wave slope, the boundary

condition for the mean free surface is:

∂φ

∂z=

(−U +

∂φ

∂x

)1

g

[−∂

2φ

∂x2

(∂φ

∂x− U

)− ∂2φ

∂x∂y

∂φ

∂y− ∂2φ

∂x∂z

∂φ

∂z

]+∂φ

∂y

1

g

[− ∂2φ

∂x∂y

(∂φ

∂x− U

)− ∂2φ

∂y2∂φ

∂y− ∂2φ

∂y∂z

∂φ

∂z

]at z = 0.0 (19)

Noting that the double potential derivatives ∂φ/∂z, ∂2φ/∂x∂z, and ∂2φ/∂y∂z

are all equal to zero at z = 0.0, the double body linearized equation for the free

surface potential is:

∂φ

∂z=

(−U +

∂φ

∂x

)1

g

[−∂

2φ

∂x2

(∂φ

∂x− U

)− ∂2φ

∂x∂y

∂φ

∂y

]+∂φ

∂y

1

g

[− ∂2φ

∂x∂y

(∂φ

∂x− U

)− ∂2φ

∂y2∂φ

∂y

]+

(∂φ

∂x− U

)1

g

[−(∂2φ

∂x2− ∂2φ

∂x2

) (∂φ

∂x− U

)− ∂2φ

∂x2

(∂φ

∂x− ∂φ

∂x

)−(∂2φ

∂x∂y− ∂2φ

∂x∂y

)∂φ

∂y− ∂2φ

∂x∂y

(∂φ

∂y− ∂φ

∂y

)]+∂φ

∂y

1

g

[−(∂2φ

∂x∂y− ∂2φ

∂x∂y

) (∂φ

∂x− U

)− ∂2φ

∂x∂y

(∂φ

∂x− ∂φ

∂x)

)

6

−(∂2φ

∂y2− ∂2φ

∂y2

)∂φ

∂y− ∂2φ

∂y2

(∂φ

∂y− ∂φ

∂y

)]at z = 0.0 (20)

2.4. Nonlinear free surface solution

For a given ship orientation, speed, and water depth, the nonlinear free

surface boundary conditon can be solved to very high accuracy using an iterative

approach Jensen et al. (1986); Bertram (2012). During each iteration, the free

surface elevations and velocity potentials on the free surface from the previous

iteration must be used, and these are denoted φ and ζ respectively. The solution

of equations to be solved can be written as follows: Hhull,hull Hhull,free

Hfree,hull Hfree,free

σhull

σfree

=

Phull

P free

(21)

where [H] is the influence matrix giving boundary condition values from source

strengths, and {P} is the vector of boundary condition values to be satisfied.

The influence matrix [H] and boundary condition vector {P} have been parti-

tioned into portions based on the hull and free surface. The boundary condition

on the hull is satisfied by:

Pi = U nx(~xi) for 0 ≤ i < Nhull (22)

Hij =∂Eij

∂nfor 0 ≤ i < Nhull (23)

Note that panels on the hull are numbered from 0 to Nhull − 1. The boundary

condition on the free surface is much more complex because it includes the

influence of potential derivatives up to third order. The following terms are

introduced for each free surface panel during the solution process:

a1 =

(∂φ

∂x− U

)∂2φ

∂x2+

∂φ

∂y

∂2φ

∂x∂y+

∂φ

∂z

∂2φ

∂x∂z(24)

a2 =

(∂φ

∂x− U

)∂2φ

∂x∂y+

∂φ

∂y

∂2φ

∂y2+

∂φ

∂z

∂2φ

∂y∂z(25)

a3 =

(∂φ

∂x− U

)∂2φ

∂x∂z+

∂φ

∂y

∂2φ

∂y∂z+

∂φ

∂z

∂2φ

∂z2(26)

7

b =1

(g + a3)

(∂φ

∂x− U

)2∂3φ

∂x2∂z+

(∂φ

∂y

)2∂3φ

∂y2∂z

+

(∂φ

∂z

)2∂3φ

∂z3+ g

∂2φ

∂z2

+ 2

[(∂φ

∂x− U

)∂φ

∂y

∂3φ

∂x∂y∂z

+

(∂φ

∂x− U

)∂φ

∂z

∂3φ

∂x∂z2

+∂φ

∂y

∂φ

∂z

∂3φ

∂y∂z2

+∂2φ

∂x∂za1 +

∂2φ

∂y∂za2 +

∂2φ

∂z2a3

]}(27)

The boundary condition terms to be satisfied on the previous guess of the free

surface ζ are:

Pi = 2 a1∂φ

∂x+ 2 a2

∂φ

∂y+ 2 a3

∂φ

∂z

−b

1

2

(∂φ∂x− U

)2

+

(∂φ

∂y

)2

+

(∂φ

∂z

)2

+ U2

− g ζ + U

(∂φ

∂x− U

)}for Nhull ≤ i < Nhull +Nfree (28)

The influence matrix terms giving the free surface boundary values from source

strengths are:

Hij = 2

[a1

∂Eij

∂x+ a2

∂Eij

∂y+ a3

∂Eij

∂z

+

(∂φ

∂x− U

)∂φ

∂y

∂2Eij

∂x∂y

+

(∂φ

∂x− U

)∂φ

∂z

∂2Eij

∂x∂z

+∂φ

∂y

∂φ

∂z

∂2Eij

∂y∂z

]

8

+

(∂φ

∂x− U

)2∂2Eij

∂x2+

(∂φ

∂y

)2∂2Eij

∂y2

+

(∂φ

∂z

)2∂2Eij

∂z2+ g

∂Eij

∂z

− b

[(∂φ

∂x− U

)∂Eij

∂x+

∂φ

∂y

∂Eij

∂y+

∂φ

∂z

∂Eij

∂z

]for Nhull ≤ i < Nhull +Nfree (29)

The solution proceeds by solving for the source strengths σ, then new flow

potentials φ and associated derivatives at the previous guess ζ of the free surface.

The new free surface ζ is then evaluated using the previous guess of the free

surface ζ, and the previous and newly solved potentials φ(ζ) and φ(ζ) at the

old free surface:

ζ = ζ +ζNζD

(30)

ζN = −

(∂φ

∂x− U

) (∂φ

∂x− U

)− ∂φ

∂y

∂φ

∂y− ∂φ

∂z

∂φ

∂z

+1

2

(∂φ∂x− U

)2

+

(∂φ

∂y

)2

+

(∂φ

∂z

)2

+ U2

− g ζ (31)

ζD = g +

(∂φ

∂x− U

)∂2φ

∂x∂z+

∂φ

∂y

∂2φ

∂y∂z+

∂φ

∂z

∂2φ

∂z2(32)

Once the location of the new free surface ζ has been solved, the flow potentials

and their derivatives can be evaluated on the new free surface. During each

iteration, the quality of the solution can be evaluated by examining the free

surface boundary condition errors from Equation 13.

To obtain successful convergence of the nonlinear free surface solution, the

most important factor is the quality of the initial guess. The double body

linearized solution provides a suitably accurate initial guess for most cases.

2.5. Modelling of sources

Figure 1 shows an example computational source mesh for a Series 60 hull

with block coefficient of 0.60. The shown view is a close-up in the vicinity of

9

the ship, and includes only a portion of the free surface. The hull is modelled

by planar panels, which also serve as source panels. The free surface is also

modelled by planar panels; however, a point source with vertical offset above

the free surface is used for each free surface panel, avoiding singularities during

evaluation of 1/R. For linearized free surface computations, the point sources

can be located at a constant elevation above the calm free surface. For nonlinear

free surface computations, good results are obtained when the point source

associated with each free surface panel is located at a specified distance above

the centroid elevation of that panel.

Although using raised sources for the free surface provides advantages, it

introduces the complexity of requiring additional panelling between the hull

and the raised sources for the free surface. Figure 1 shows the usage of a

virtual hull wall (green) between the actual hull (yellow) and raised free surface

sources (blue). The virtual hull wall has vertical panels extending from the ship

waterline to the raised free surface sources. Note that the raised free surface

sources do not include coverage above the waterplane of the ship.

2.6. Evaluation of potential derivatives and radiation condition

The evaluation of potential derivatives and the radation at the fore boundary

of the domain are closely related. Solution of the free surface flow requires

evaluation of flow potential derivatives up to third-order. Dawson (1977) and

Raven (1996) used a forward difference method to evaluate derivatives of flow

potentials. A forward difference method can easily statisfy the radiation of the

flow velocity being undisturbed at the fore boundary. Futhermore, the usage

of a finite difference method such as forward difference reduces the order of

derivatives that must be evaluated for the term 1/R and its associated integrals

over source panels.

The present work evaluates potential derivatives directly, including second

and third-order derivatives. The radiation boundary condition at the fore

boundary of the domain is satisfied by shifting sources associated with the fore-

most part of the domain to immediately aft of the domain, as described by

10

Bertram (2012). In comparison with finite difference methods, this method has

the advantage of requiring many fewer panels for resolution of the flow field.

2.7. Modelling of transom sterns

The current potential flow solution does not include modelling of flow sep-

aration. For ships with transom sterns, a potential flow solution can give un-

realistically high velocity gradients in the vicinity of the transom. The present

method can use a virtual stern on a ship to approximate the influence of flow

separation. Figure 2 shows a DTMB 5512 destroyer model (Irvine et al., 2013)

with a transom stern and with a virtual narrow stern. The narrow stern extends

aft 1/10 of the length of the ship and is a vertical line at its aftmost location.

The virtual transom stern is not included when integrating vertical forces acting

on the ship. It is postulated that the influence of a virtual transom stern will

have only a very minor influence on squat predictions, which are dominated by

vertical forces.

2.8. Solution of sinkage and trim

The sinkage and trim are solved in an iterative manner using the following

equation: ∆η3

∆η5

=

C33 C35

C53 C55

F3 − 4 g

F5

(33)

where η3 and η5 are heave and pitch displacements, Cij is hydrostatic stiffness,

η3 and η5 are heave and pitch displacements, and 4 is ship mass displacement.

The flow solution is re-evaluated for each iteration of sinkage and trim. Forces

on the hull are integrated using the actual wetted surface for each iteration.

Relaxation methods are applied to prevent changes in heave or pitch from being

excessively large between iterations.

When solving sinkage and trim using linearized free surface methods, the

calm free surface is repanelled for each iteration because the ship waterplane

will vary with sinkage and trim. For solution of sinkage and trim using the

11

nonlinear free surface method, the sinkage and trim are typically solved initially

using the double body linearized method. This initial solution of sinkage and

trim increases the likelihood of successful convergence for the nonlinear free

surface.

3. Software implementation

The required software for solution of the free surface and resulting sinkage

and trim was obtained by extending the ShipMo3D ship motion library (McTag-

gart, 2010, 2015). ShipMo3D is written in C# (Michaelis and Lippert, 2016),

which facilitates efficient software development and includes a numerical capa-

bility for complex numbers. Linear systems of equations are solved utlizing the

Intel MKL library, (Wang et al., 2014).

The majority of the software was written in C#, which runs in the .NET

runtime. Some portions of the software were written in native C, with the

goal of improving computational performance. Recent experience suggests that

software written in C# is now giving very similar performance to native C code.

Parallel processing was achieved with only modest development efforts re-

quired. The solution of linear systems with the Intel MKL library includes

parallel processing. Solution influence terms involving 1/R and its derivatives

are solved in parallel.

4. Computational experience

Computational experience with the solution of squat for a variety of ships

and operating conditions has provided guidance for application of the developed

method. Froude numbers based on ship length and water depth are defined

respectively by:

FL =

√U

g L(34)

Fh =

√U

g h(35)

12

Most of the present work has been for ship length Froude numbers up to 0.4.

Comments below regarding numbers of panels apply to half the total domain

because lateral symmetry applies.

4.1. Ship in open deep water

A ship in open deep water is among the simpler cases, and is further sim-

plified if the ship is fully submerged below the water surface. The free surface

panelling can extend in the x direction from −2.5L to 1.0L, and in the y direc-

tion to 1.2L. The ship can usually be represented sufficiently accurately with

1000 panels on the wet portion.

Selection of panel size is of critical importance for achieving an appropriate

balance between accuracy and computation time. Furthermore, satisfaction of

the radiation condition has been found to be dependent on panel size, likely due

to the approach of moving sources from the foremost location of the domain

to aft of the domain. In practice, the current method works well when using

a constant panel x length in an inner region surrounding the ship, and then

increasing plane x length gradually to the outer boundaries of the domain. An

inner region of −2.0L ≤ x ≤ 0.7L has proven to work well. It has been

found that a suitable maximum x panel length in the outer region is 2.0 times

the x panel length in the inner region. Very good results are obtained when the

free surface sources are located 1.0 times the maximum x panel length above

the calm water level for linear computations, and above the computed wave

elevation for nonlinear computations. An inner x panel length of 0.026L gives

very good results for Froude numbers of 0.13 and greater, corresponding to

having 8 or more panels in the x direction per wavelength. At lower Froude

numbers, the free surface in deep water is effectively flat and the double body

solution can be used, requiring no panelling of the free surface. It is suggested

that the y panel length be approximately equal to the x panel length; however, it

has been found that it is sometimes necessary to increase the panel length in the

y direction, likely due to numerical instabilities associated with wave breaking

immediately adjacent to the hull. For a surface-piercing ship, panelling must

13

include the virtual hull wall between the free surface and its raised sources. It

has been found that having 4 panels in the vertical direction works well.

Various checks can be performed to verify that a numerical solution is valid.

The free surface should be flat at the foremost portion of the fluid domain. In

deep water, the scattered flow velocity components should have magnitudes less

than 0.01 U . The condition number for the influence matrix to solve source

strengths should not be excessively large. Also, the non-dimensional boundary

condition errors on the free surface from Equation 13 should be small. Lin-

earized solutions should typically provide RMS εf errors less than 0.01 based

on sampling off all free surface panels.

When solving the nonlinear free surface, the implemented software has demon-

strated very good convergence for cases in open deep water. During successive

iterations, both the horizontal and vertical dimensions of the free surface are

repanelled, with the change of horizontal dimensions dependent on how much

the ship hull walls vary from being vertical. A relaxation factor of 0.6 is typically

applied to successive iterations of the free surface.

4.2. Ship In restricted water

The above comments regarding deep water provide useful guidance for ships

in restricted water; however, succesful solution of cases in restricted water is

more challenging. Wave elevations are often higher in shallow water. For

subcritical flow (water depth Froude number Fh < 1.0) waves will propogate

upstream from the ship, introducing challenges for modelling flow at the fore

boundary of the free surface. It has been found that solutions are very difficult

to obtain in the transcritical region 0.8 < Fh < 1.2, likely due to very steep

flows.

Flow solutions in restricted water are more sensitive to panelling that solu-

tions for open water. For linear solutions, often more than one panel size must

be tried for the free surface. The presence of the ship can significantly affect flow

at the fore boundary of the domain; thus, free surface elevation slopes at the fore

boundary of the domain should be checked to verify that they are acceptably

14

small.

Convergence for nonlinear free surface solutions is more difficult in restricted

water than for open deep water, likely due to increased wave slopes. Convergence

has been achieved for cases with subcritical depth Froude numbers.

4.3. Initial solution for nonlinear free surface iteration

The nonlinear free surface method requires an initial guess for the free sur-

face. The double body linearized solution can be used. To improve the like-

lihood of a convergent nonlinear solution, the sinkage and trim for the double

body linearized solution can be determined before proceeding with the nonlinear

solution. If multiple ship speeds are being evaluated, a very efficient nonlinear

solution for a given ship speed can be obtained by using the nonlinear solution

for the previous ship speed as the initial solution. This approach has proven to

work very well for increments in ship length Froude number of 0.01.

4.4. Visualization of ship and free surface

Visualization of the ship and free surface are important for verification and

interpretation of numerical predictions. Figure 3 shows a Series 60 hull with

block coefficient of 0.60 travelling through the channel used by Jiang (1998)

for model tests and computations. The color axis ranges from blue to red. The

minimum and maximum values of the color axis are usually set to have the same

absolute values, which then gives green for zero elevation. Panels on the free

surface are plotted as nonplanar surfaces. The local elevations for each panel

are determined using the elevation and its first and second x and y derivatives at

the centroid; thus, no smoothing is performed using data from adjacent panels.

Smooth variation of free surface elevations is a positive indicator of acceptable

results. Small variation of wave elevation at the foremost part of the domain

indicates appropriate modelling of the radiation condition.

5. Validation of numerical predictions

The numerical predictions were validated using experimental data published

in the open literature for 3 different ships.

15

5.1. DTMB 5515 destroyer in deep water

Validation was performed using experimental data from Irvine et al. (2013)

for model DTMB 5515, a geosim of the widely used DTMB 5415 model of a

naval destroyer. Figure 4 shows experimental measurements and numerical pre-

dictions from the present method. The sinkage s is given at midships and is

positive down. The trim angle τ is positive for trim by stern. The double body

method gives good results for ship length Froude numbers of 0.20 and less.

The double body linearized and nonlinear methods give very similar results,

which are very close to experimental values. The uniform linearized method ex-

hibits noticeable deviation from the more accurate methods for Froude numbers

greater than 0.25.

5.2. Series 60 hull with 0.60 block coefficient in deep and restricted water

Validation was performed for a Series 60 hull with block coefficient of 0.60

in both deep and restricted water. The computations were performed at full

scale, with a ship length of 121.92 m, beam of 16.255 m, and draft of 6.501 m.

The wetted hull was modelled using approximately 1500 panels. The ship has a

narrow stern; thus, approximate modelling of a transom stern was not required.

Figure 5 shows comparisons of predictions in deep water with experimental

results from Takeshi et al. (1987). The results for sinkage are very good, with

accuracy generally improving as the sophistication of the numerical method

increases. The trim results show greater variation, which can be partly explained

by the small magnitude of the trim relative to the sinkage. The significant

differences between the double body linearized and nonlinear results at higher

Froude numbers indicate that free surface nonlinearities have a significant effect

on trim. A possible explanation for the trim sensitivity is the narrowness of the

stern, giving lower pitch hydrostatic stiffness.

Computations in shallow water were performed for the experimental condi-

tions reported by Jiang (1998). The water depth was 2.0T (13.002 m) and the

channel width was 2.09L (254.8 m). Figure 6 gives sinkage and trim for the

Series 60 hull in shallow water. The uniform linearized, double body linearized,

16

and nonlinear methods give excellent results for ship length Froude numbers up

to 0.25, corresponding to a water depth Froude number of 0.80. The uniform

linearized and double body linearized methods were unable to provide solutions

in the vicinity of critical flow. The uniform linearized and double body linearized

methods give good predictions at high Froude numbers. The nonlinear free sur-

face was only able to achieve convergence for Froude numbers up to 0.255, with

excellent agreement of predicted sinkage and trim with experiments.

5.3. KVLCC2 very large crude carrier in restricted water

Computations were performed for the KVLCC2 very large crude carrier for

experimental conditions described by Lataire et al. (2012), with a channel width

of 5 times the ship breadth and channel depths of 1.1, 1.35, and 1.5 relative to the

draft of the ship. Figures 7 to 9 show predicted and experimental values. The

numerical predictions consistently underpredict the sinkage, but give very good

agreement for trim. The quality of numerical predictions generally improves

with the level of fidelity in modelling the free surface.

5.4. Error in predicted bottom clearance

If the current method is going to be utilized to provide guidance to ship

operators, then it is essential to consider errors in predicted bottom clearance.

Figure 10 gives over-prediction of bottom clearance non-dimensionalized by ship

draft when the double body linearized method is used to model the free surface.

It was assumed that the bottom of the ship was at the ship baseline when

evaluating bottom clearance at the bow and stern. The greater over-prediction

occurs in the transcritical flow region, where bottom clearance using the present

method is evaluated by interpolation of available predictions at subcritical and

supercritical ship speeds.

6. Discussion

Comparisons with model tests indicate that the modelling of flow separation

effects from a transom stern by using a narrowing virtual stern has minimal

17

influence on squat predictions. This result is not surprising given that squat is

dominated by vertical forces.

The double body method (nominally flat free surface) provides robust and

efficient predictions of squat at lower ship length Froude numbers, with valida-

tion results suggesting a suitable maximum value of FL ≈ 0.1. This result is

very useful from a practical perspective because it avoids the requirement for

using large numbers of free surface panels caused by small wavelengths at low

ship speeds.

For detailed modelling of the free surface (uniform flow linearized, double

body linerized, and nonlinear methods), the current approach using direct eval-

uation of higher order flow potential derivatives allows for much larger free

surface panels than approaches that use finite difference methods. It is recom-

mended that 6 or more panels per wavelength be used for the current approach.

Modelling of the forward boundary radiation condition is satisfactorily achieved

by shifting foremost sources to immediately aft of the domain.

The uniform flow linearized and double body linearized methods were able

to model shallow water squat in subcritical and supercritical flow regions, but

were unable to provide solutions at depth Froude numbers approaching 1.0.

The nonlinear method only achieved solutions in subcritical flow. Previous

work by Jiang (1998) assuming a slender ship suggests that the current work

should perhaps be extended using a time domain approach for solution at depth

Froude numbers near 1.0.

Among the free surface models presented here, the double body linearized

method is recommended for most squat predictions for forward speed Froude

numbers above 0.1. It provides a suitable balance of fidelity and robustness.

7. Conclusions

A potential flow method has been developed for predicting ship squat. The

four different approaches for modelling the free surface are double body, uni-

form flow linearized, double body linearized, and nonlinear, given in order of

18

increasing fidelity. Direct evaluation of higher order derivatives of flow poten-

tial allows the free surface to be modelling using a moderate number of panels.

Flow separation from a transom stern can be modelled using a narrowing virtual

transom. The method gives robust and efficient prediction of ship squat, with

the exception of near critical flow conditions (Fh ≈ 1.0).

References

Bertram, V., 2012. Practical Ship Hydrodynamics, 2nd Edition. Butterworth-

Heinemann, Oxford.

Briggs, M., Vantorre, M., Uliczka, K., Debaillon, P., 2010. Handbook of Coastal

and Ocean Engineering. World Scientific Publishing Company, Ch. Chapter

26 – Prediction of Squat for Underkeel Clearance, pp. 723–774.

Dand, I., Ferguson, A., 1973. The squat of ships in shallow water. Transactions,

Royal Institution of Naval Architects 115, 237–255.

Dawson, C., 1977. A practical computer method for solving ship-wave prob-

lems. In: Second International Conference on Numerical Ship Hydrodynam-

ics. Berkeley, California.

Garrison, C., 1978. Hydrodynamic loading of large offshore structures. In:

Zienkiewicz, O., Lewis, R., Stagg, K. (Eds.), Numerical Methods in Offshore

Engineering. John Wiley & Sons, Chichester, England, pp. 87–139.

Gourlay, T., 2008. Slender-body methods for predicting ship squat. Ocean En-

gineering 35 (2), 191–200.

Irvine, M., Longo, J., Stern, F., December 2013. Forward speed calm water

roll decay for surface combatant 5415: Global and local flow measurements.

Journal of Ship Research 57 (4), 202–219.

Jachowski, J., 2008. Assessment of ship squat in shallow water using CFD.

Archives of Civil and Mechanical Engineering VIII (1), 27–36.

19

Jensen, G., Soding, H., Mi, Z.-X., 1986. Rankine source methods for numerical

solutions of the steady wave resistance problem. In: Sixteenth Symposium on

Naval Hydrodynamics. Berkely, California, pp. 575–581.

Jiang, T., 1998. Investigation of waves generated by ships in shallow water. In:

Twenty-Second Symposium on Naval Hydrodynamics. Washington.

Lataire, E., Vantorre, M., Delefortrie, G., 2012. A prediction method for squat

in restricted and unrestricted rectangular fairways. Ocean Engineering 55,

71–80.

McTaggart, K., 2010. Verification and validation of ShipMo3D ship motion pre-

dictions in the time and frequency domains. In: International Towing Tank

Conference Workshop on Seakeeping: Verification and Validation for Non-

linear Seakeeping Analysis. Seoul, Korea.

McTaggart, K., 2015. Ship radiation and diffraction forces at moderate forward

speed. In: World Maritime Technology Conference. Providence, Rhode Island.

Michaelis, M., Lippert, E., 2016. Essential C# 6.0. Addison-Wesley Professional,

New York.

Millward, A., 1990. A preliminary design method for the prediction of squat in

shallow water. Marine Technology 27 (1), 10–19.

Raven, H., 1996. A solution method for the ship wave resistance problem. Ph.D.

thesis, Technical University of Delft.

Takeshi, H., Hino, T., Hinatsu, M., Tsukada, Y., Fujisawa, J., 1987. ITTC

cooperative experiments on a series 60 model at ship research institute – flow

measurements and resistance tests. Tech. rep., Ship Research Institute, Japan.

Tarafder, M. S., Khalil, G. M., 2006. Calculation of ship sinkage and trim in

deep water using a potential based panel method. International Journal of

Applied Mechanics and Engineering 11 (2), 401–414.

20

Toxopeus, S., Simonsen, C., Guilmineau, E., Visonneau, M., Xing, T., Stern,

F., 2013. Investigation of water depth and basin wall effects on KVLCC2

in manoeuvring motion using viscous-flow calculations. Journal of Marine

Science and Technology 18 (4), 471–496.

Wang, E., Zhang, Q., Shen, B., Zhang, G., Lu, X., Wu, Q., Wang, Y., 2014.

High-Performance Computing on the Intel R©Xeon PhiTM. Springer Interna-

tional Publishing, Switzerland.

Yao, J., Zou, Z., 2010. Calculation of ship squat in restricted waterways by using

a 3D panel method. In: Ninth International Conference on Hydrodynamics.

Shanghai, China.

21

Figure 1: Panelling of Series 60 CB = 0.60 hull (yellow), source panels above the calm freesurface (blue), and hull virtual wall (green) between free surface and its source panels.

Figure 2: DTMB 5512 destroyer hull with transom stern (top) and with virtual stern (bottom).

Figure 3: Series 60 hull with 0.60 block coefficient in shallow channel, ship length Froudenumber 0.24.

Figure 4: DTMB 5512 sinkage and trim in deep water.

Figure 5: Series 60 hull with block coefficient 0.60 sinkage and trim in deep water.

Figure 6: Series 60 hull with block coefficient 0.60 sinkage and trim in shallow channel.

Figure 7: KVLCC2 sinkage and trim in shallow channel, h/T = 1.1.

Figure 8: KVLCC2 sinkage and trim in shallow channel, h/T = 1.35.

Figure 9: KVLCC2 sinkage and trim in shallow channel, h/T = 1.5.

Figure 10: Nondimensional bottom clearance over-prediction error.

22

Each of the following pages has a single figure prepared for publi-

cation, as required by the publisher.

23

24

25

26

0.00 0.10 0.20 0.30 0.40 0.500.00

0.02

0.04

0.06

0.08

0.10

0.12

Ship length Froude number U/√g L

Sinkages/Tm

id

Experiments

Nonlinear

Double body linearized

Uniform linearized

Double body

0.00 0.10 0.20 0.30 0.40 0.50−0.10

−0.05

0.00

0.05

0.10

0.00

Ship length Froude number U/√g L

TrimτL/(2Tm

id)

27

0.20 0.25 0.30 0.350.00

0.02

0.04

0.06

0.08

0.10

Ship length Froude number U/√g L

Sinkages/Tm

id

Experiments

Nonlinear

Double body linearized

Uniform linearized

Double body

0.20 0.25 0.30 0.35−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.00

Ship length Froude number U/√g L

TrimτL/(2Tm

id)

28

0.10 0.20 0.30 0.40−0.20

0.00

0.20

0.40

0.00

0.4 0.6 0.8 1 1.2

Ship length Froude number U/√g L

Sink ages/Tm

id

Depth Froude number U/√g h

Experiments

Nonlinear

Double body linearized

Uniform linearized

Double body

0.10 0.20 0.30 0.40−0.40

−0.20

0.00

0.20

0.40

0.60

0.80

0.00

0.4 0.6 0.8 1 1.2

Ship length Froude number U/√g L

TrimτL/(2Tm

id)

Depth Froude number U/√g h

29

0.05 0.06 0.07 0.08 0.09 0.10 0.110.00

0.02

0.04

0.06

Ship length Froude number U/√g L

Sinkage−ξ 3(mid)/Tm

idExperiments

Double body linearized

Uniform linearized

Double body

0.2 0.3 0.4

Depth Froude number U/√g h

0.05 0.06 0.07 0.08 0.09 0.10 0.11−0.04

−0.03

−0.02

−0.01

0.000.00

Ship length Froude number U/√g L

Trim−η5L/(2Tm

id)

Experiments

Double body linearized

Uniform linearized

Double body

0.2 0.3 0.4

Depth Froude number U/√g h

30

0.05 0.10 0.150.00

0.05

0.10

0.15

Ship length Froude number U/√g L

Sinkage−ξ 3(mid)/Tm

idExperiments

Double body linearized

Uniform linearized

Double body

0.2 0.3 0.4 0.5

Depth Froude number U/√g h

0.05 0.10 0.15−0.06

−0.04

−0.02

0.000.00

Ship length Froude number U/√g L

Trim−η5L/(2Tm

id)

Experiments

Double body linearized

Uniform linearized

Double body

0.2 0.3 0.4 0.5

Depth Froude number U/√g h

31

0.05 0.10 0.150.00

0.02

0.04

0.06

0.08

0.10

0.12

Ship length Froude number U/√g L

Sink age−ξ 3(mid)/Tm

idExperiments

Double body linearized

Uniform linearized

Double body

0.2 0.3 0.4

Depth Froude number U/√g h

0.05 0.10 0.15−0.05

−0.04

−0.03

−0.02

−0.01

0.000.00

Ship length Froude number U/√g L

Trim−η5L/(2Tm

id)

Experiments

Double body linearized

Uniform linearized

Double body

0.2 0.3 0.4

Depth Froude number U/√g h

32

0.0 0.2 0.4 0.6 0.8 1.0 1.2−0.30

−0.20

−0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.00

Water depth Froude number U/√g h

Clea ran

ceover-predictionε c

Series 60 Cb = 0.60

KVLCC2 h/T = 1.1

KVLCC2 h/T = 1.35

KVLCC2 h/T = 1.5

Envelope

33