Trend validation of SHIPFLOW based on the bare hull upright

Transcription

Trend validation of SHIPFLOW based
on the bare hull upright resistance
of the Delft Series
January 6, 2006
K.J. van Mierlo
0
Trend validation of SHIPFLOW based
on the bare hull upright resistance
of the Delft Series
Master thesis
Submitted in fullfillment of the requirements
of the degree of Master of Science (MSc.)
at the
Delft University of Technology,
accepted by chairman department Aerodynamics
prof. dr. ir. P.G. Bakker,
wednesday, January 25, 2006 at 14:00 hrs,
by
Koen van Mierlo
student Aerospace Engineering
born in Delft, the Netherlands.
This thesis is approved by the following supervisor:
Prof. dr. ir. P.G. Bakker
Examination Committee:
Prof. dr. ir. P.G. Bakker
Dr. ir. L.L.M. Veldhuis
Dr. ir. J.A. Keuning
Dr. C.E. Janson
Delft University of Technology
Chalmers University of Technology, Göteborg
c 2006 by K.J. van Mierlo, Delft University of Technology. All rights reserved.
Copyright For personal use only. Copies of this report can be obtained by contacting the author or downloaded from: www.hsa.lr.tudelft.nl.
ii
Contents
Abstract
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Acknowledgements
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Symbols
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Glossary
xvii
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Preliminary objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
2
2
2 Governing equations of the hydrodynamic flow around a sailing yacht
2.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Characteristic flow parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
7
3 EFD: Towing tank testing
3.1 Description of the DSYHS . . . . . . . . . . .
3.2 Towing tank test setup and testing procedure
3.3 Froude number similarity . . . . . . . . . . .
3.4 Postprocessing of towing tank measurements
3.5 Errors and uncertainty in towing tank tests .
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11
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4 CFD: non linear free surface potential flow code SHIPFLOW
4.1 Basics of potential flow . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Linearization of the free surface boundary conditions . . . . . . .
4.3 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Linear free surface potential flow . . . . . . . . . . . . . .
4.3.2 Non linear free surface potential flow . . . . . . . . . . . .
4.3.3 Special features of the solution method . . . . . . . . . .
4.4 Determination of the wave resistance . . . . . . . . . . . . . . .
4.4.1 Pressure integration . . . . . . . . . . . . . . . . . . . . .
4.4.2 Wave cut analysis . . . . . . . . . . . . . . . . . . . . . .
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iv
CONTENTS
5 Verification
5.1 Basics of verification . . . . . . . . . . . . . . .
5.2 Submerged sphere . . . . . . . . . . . . . . . .
5.2.1
Objectives . . . . . . . . . . . . . . . .
5.2.2
Description of the test case . . . . . . .
5.2.3
Results . . . . . . . . . . . . . . . . . .
5.2.4
Explanation of the results . . . . . . .
5.3 Delft 1 . . . . . . . . . . . . . . . . . . . . . .
5.3.1
Objectives . . . . . . . . . . . . . . . .
5.3.2
Description and results of the test case
5.3.3
Explanation of the results . . . . . . .
5.4 Conclusions of the verification . . . . . . . . .
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8 Conclusions and recommendations
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A SHIPFLOW input files
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6 Validation
6.1 Basics of validation . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Error and uncertainty in the DSYHS towing tank data
6.1.2 Error and uncertainty in SHIPFLOW results . . . . .
6.2 Trend validation . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Description of the cases used in the trend validation . . . . .
6.4 Comparison of the CFD and the towing tank results . . . . .
6.5 Explanation of the results of the comparison . . . . . . . . . .
6.6 Conclusions of the trend validation . . . . . . . . . . . . . . .
7 Determination of the correction equations
7.1 Basics of linear regression . . . . . . . . . .
7.2 Regression strategies . . . . . . . . . . . . .
7.3 Regression of the CFD results . . . . . . . .
7.3.1 Selection of the response . . . . . . .
7.3.2 Selection of the predictors . . . . . .
7.3.3 Regression of kpi . . . . . . . . . . .
7.3.4 Regression of kwc . . . . . . . . . . .
7.4 Test of the correction equations . . . . . . .
7.5 Conclusions about the correction equations
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B Comparison of the trim and sink of the CFD calculations with the towing
tank results
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List of Figures
3.1
Sailing yacht model for towing tank testing . . . . . . . . . . . . . . . . . . .
13
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
Meshing of the submerged sphere at Fn 0.40 . . . . . . . . . . . .
Meshing problem at the stagnation point of the sphere . . . . . .
Extrapolation of the drag area for a submerged sphere at Fn 0.35
Panel distribution on the hull of Delft 1 . . . . . . . . . . . . . .
Stretched and uniform panel distribution on the hull of Delft 1 .
Example of 2 free surface meshes for Delft 1 at Fn 0.40 . . . . .
Results for Delft 1 at Fn 0.30 . . . . . . . . . . . . . . . . . . . .
Longitudinal wavecut at y/L = 0.3 for Delft 1 at Fn 0.40 . . . .
Wavepattern of Delft 1 at Fn 0.40 . . . . . . . . . . . . . . . . .
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6.1
6.2
6.3
6.4
6.5
6.6
6.7
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
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7.1
7.2
Test of the correction equations for Delft 4 . . . . . . . . . . . . . . . . . . .
Test of the correction equations for Delft 43 . . . . . . . . . . . . . . . . . . .
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B.1
B.2
B.3
B.4
B.5
B.6
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
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96
of
of
of
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of
of
the
the
the
the
the
the
the
the
the
the
the
the
the
wave
wave
wave
wave
wave
wave
wave
trim
trim
trim
trim
trim
trim
resistance
resistance
resistance
resistance
resistance
resistance
resistance
for
for
for
for
for
for
Fn
Fn
Fn
Fn
Fn
Fn
for
for
for
for
for
for
for
0.25
0.30
0.35
0.40
0.45
0.50
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Fn
Fn
Fn
Fn
Fn
Fn
Fn
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0.25 .
0.30 .
0.35 .
0.40 .
0.45 .
0.50 .
0.55 .
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vi
LIST OF FIGURES
B.7
B.8
B.9
B.10
B.11
B.12
B.13
B.14
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
of
of
of
of
of
of
of
of
the
the
the
the
the
the
the
the
trim for Fn 0.55
sink for Fn 0.25
sink for Fn 0.30
sink for Fn 0.35
sink for Fn 0.40
sink for Fn 0.45
sink for Fn 0.50
sink for Fn 0.55
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97
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100
100
List of Tables
3.1
Range of the principal hull parameters of the Delft series . . . . . . . . . . . .
11
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
Domain size of the submerged sphere test case . . . . . . . . . . . . .
Drag area of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . .
Drag area of the sphere using the ’manual’ meshing . . . . . . . . . . .
Wave drag area of a submerged sphere at Fn 0.35 . . . . . . . . . . . .
Pressure integration results of a submerged sphere at Fn 0.35 . . . . .
Wave cut analysis results of a submerged sphere at Fn 0.35 . . . . . .
Pressure integration results of a submerged sphere at Fn 0.45 . . . . .
Drag area of Delft 1 for different number of hull panels . . . . . . . . .
Drag area of Delft 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence criteria used during the investigation . . . . . . . . . . .
Wave resistance of Delft 1 at Fn 0.30 for different convergence criteria
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30
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31
31
32
32
33
34
34
38
39
40
41
41
41
6.1
6.2
Average results and maximum deviation for Delft 1 . . . . . . . . . . . . . . .
Size of the free surface domain for the different Fn . . . . . . . . . . . . . . .
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57
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
Range of the principal hull parameters for which k pi and
Mean values and standard deviation of k pi . . . . . . . .
Regression coefficients for kpi at Fn 0.25 . . . . . . . . .
ANOVA table for the regression of k pi at Fn 0.25 . . .
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kwc are valid
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viii
7.14
7.15
7.16
7.17
7.18
7.19
7.20
7.21
7.22
7.23
7.24
7.25
7.26
7.27
7.28
7.29
7.30
7.31
7.32
7.33
7.34
7.35
LIST OF TABLES
ANOVA table for the regression of k pi at Fn 0.50
Regression coefficients for kpi at Fn 0.55 . . . . . .
ANOVA table for the regression of k pi at Fn 0.55
Mean values and standard deviation of k wc . . . .
Regression coefficients for kwc at Fn 0.25 . . . . . .
ANOVA table for the regression of k wc at Fn 0.25
Test of kpi for Delft 4 . . . . . . . . . . . . . . . .
Test of kwc for Delft 4 . . . . . . . . . . . . . . . .
Test of kpi for Delft 43 . . . . . . . . . . . . . . .
Test of kwc for Delft 43 . . . . . . . . . . . . . . .
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70
71
71
71
72
72
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73
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74
74
75
75
75
75
76
76
77
78
Abstract
In 1977 Dawson published a paper in which he describes how to linearize the non linear
boundary conditions at the free surface such that the potential flow around a ship can be
calculated. This paper started the development of non linear free surface potential flow solvers
which are used in a lot of ship research facilities nowadays. This CFD method is now mature
and it has come within reach of the average sailing yacht designer.
Most sailing yacht designers use empirical formulas based on the towing tank results of
the Delft series to determine the resistance of a sailing yacht. Before they will start using
CFD their question will be: How reliable are the CFD results? This leads to the main subject
of this report: validation of a non linear free surface potential flow solver (SHIPFLOW) based
on the bare hull upright resistance of the Delft series.
First the possible errors in the towing tank data are analyzed. Then a verification of the
non linear free surface potential flow solver is done. The predicted wave resistance values
of the CFD code are compared with the residuary resistance from the towing tank tests.
Correction equations based on the characteristic hull parameters are derived using linear
regression analysis.
The error and uncertainty in the residuary resistance can not be estimated because of
the way the towing tank data is post processed. The verification of the non linear free
surface potential flow is not possible due to the simultaneous change of attitude and resistance
when the grid is refined. Another problem of the verification is that the grid refinement is
constrained by the requirement of a single valued free surface. Calculations, with a grid which
is fine enough to show the main wave components, show that the predicted wave resistance
differs from the residuary resistance values. This is caused by viscous effects during the towing
tank tests which are not present in the potential flow method. The predicted resistance is
improved by the calculation of correction equations using a stepwise regression.
The conclusion is that a pure validation of the non linear free surface potential flow is
not possible due to uncertainties in both the towing tank results and the calculations. A
trend validation shows that the prediction of the wave resistance is poor for the low Froude
numbers but improves when the Froude number increases. Correction equations improve the
predicted resistance a lot but will never be totally accurate.
x
Abstract
Acknowledgements
This report describes my thesis work which is the last part of my study Aerospace Engineering
at Delft University of Technology, faculty Aerospace Engineering, department of aerodynamics. A large part of the work described in this report has been done at Chalmers University
of Technology in Göteborg, Sweden.
I would like to thank professor Bakker and professor Larsson for allowing me do a large
part of this thesis in Sweden. This way I have been able to combine my study with my
passion for sailing.
I would also like to thank:
Associate professor J.A. Keuning for allowing me to use the confidential data of the Delft
series towing tank tests.
Assistent professor C.E. Janson who always had an answer to my questions about the nonlinear free surface potential flow.
The innebandy team of Flowtech, SSPA and Chalmers department of naval architecture.
I really enjoyed playing innebandy.
Both the sailing teams of Fastertoo and Imagine for the nice sailing we have done.
My parents and family for their unlimited support.
xii
Acknowledgements
Symbols
ACC
AN OV A
Aw
Awc
B
C
Cb
CF
CF D
Cm
Cp
CR
CT
D
DSY HS
E
EF D
Eu
e
Fn
Fs
g
H
h
h
IT T C
k
k
L
Lcb
Lcf
Lwl
n
Americas cup class
analysis of variance
waterplane area
waterplane area coefficient
beam
cavitation number
block coefficient
frictional resistance coefficient
computational fluid dynamics
midship area coefficient
prismatic coefficient
residuary resistance coefficient
total resistance coefficient
data
Delft systematic yacht hull series
comparison error
experimental fluid dynamics
Euler number
error
Froude number
factor of safety
gravitational constant
free surface location estimate
free surface location perturbation
parameter which represents grid size
International Towing Tank Conference
correction factor
form factor
(reference) length
longitudinal centre of buoyancy
longitudinal centre of flotation
water line length
waves per unit length
xiv
Symbols
p
p
pa
pv
R
R2
RE
Re
RT
Rw
r
S
S
Sf
s
T
T
t
U
UV
UD
USN
Ureqd
U∞
V
V PP
We
order of the numerical method
pressure
atmospheric pressure
vapor pressure
radius of curvature
percentage of variance explained
Richardson extrapolation
Reynolds number
total resistance
wave resistance
refinement ratio
wetted surface area
simulated result
scale factor
standard deviation
draft
truth
time
uncertainty
uncertainty of the validation
uncertainty of the data
numerical uncertainty in the simulation
required uncertainty
freestream velocity
velocity vector
velocity prediction program
Weber number
γ
δS
δSM
δSN
η
λ
ν
ρ
Φ
φ
φ0
ϕ
∇
coefficient of surface tension
simulation error
modelling error
numerical error
residual
location of the free surface
wave length
coefficient of kinematic viscosity
density
velocity potential estimate
velocity potential
extrapolated result
velocity potential perturbation
displacement
Symbols
xv
Subscripts
0
fs
mod
n
pi
wc
x
y
z
reference value
full scale
model
derivation in normal direction
pressure integration
wave cut analysis
derivation in x direction
derivation in y direction
derivation in z direction
xvi
Symbols
Glossary
waterline: The intersection of the hull with the (undisturbed) free surface.
waterline length: The distance between the most forward and aft point on the waterline. In this report the still waterline length is often used as reference length.
length over all: The total length of the ship.
overhang: Part of the ship above the waterline that extends in front of the waterline at
the bow or aft of the waterline at the stern. If the length over all is larger than the waterline
length the ship will have overhang(s)
beam: The maximum width of the hull at the still waterline
draft: The depth of the hull measured from the undisturbed free surface.
displacement: The volume of the immersed part of the yacht.
longitudinal center of buoyancy: The longitudinal location of the center of gravity of
the displaced volume of water.
block coefficient: The ratio of the volume that the hull displaces to the volume of a block
with the same length, beam and draft as the hull.
waterplane area: The horizontal area enclosed by the still waterline.
water plane area coefficient: The ratio of waterplane area to the area of a rectangle
with the same length and beam as the hull at the waterline.
longitudinal center of floatation: The longitudinal position of the centroid of the waterplane.
section: Tranverse cut of the hull
midship area: The area below the waterline of a section at the middle of the ship
xviii
Glossary
midship area coefficient: The ratio of the midship area to a rectangle with the same
beam and depth.
prismatic coefficient: The ratio of the volume that the hull displaces to the volume of
a prism with the same length as the hull and the maximum cross section area below the
waterline of the hull.
heel angle: The angle of rotation around the longitudinal axis.
leeway angle: The angle of rotation around the vertical axis.
trim angle: The angle of rotation around the transverse axis.
sink: Vertical movement of the hull (usually due to wave formation)
Chapter 1
Introduction
The increase in computer power has brought Computational Fluid Dynamics (CFD) within
reach of the yacht designer. Yacht designers are using CFD to study the different design
options and to optimize the final design. The correct ranking of different design options is
only possible if the reliability and accuracy of the CFD code has been proven. This can be
done by a validation procedure such as described in the ITTC Quality Manual.
Validation consist of an analysis of the error between the truth and the simulated result. This
error consists of a modelling error and a numerical error. The modelling error is caused by
the fact that the truth (or reality) is represented by a mathematical model which is derived
by making certain assumptions or simplifications. The numerical error is caused by the fact
that the mathematical model can not be solved directly and needs to be discretised. The
process of determining the numerical error is called verification.
1.1
Motivation
The development of non linear free surface potential flow codes started around 1990 and
nowadays fully developed codes are used in research institutes all around the world. The
quality of the results generated by these codes is difficult to judge since all the research
institutes work on a commercial basis and usually the customers demand to keep the results
secret. This study will change that by doing a (trend)validation of a non linear free surface
potential flow solver (SHIPFLOW) using the measurements of the bare hull upright resistance
of the Delft Systematic Yacht Hull Series (DSYHS).
The Delft series is used here because it consists of a large set of different hulls which are
tested over a relatively large speed range. This makes it the ideal set of data to use for a
(trend)validation. The results of the validation will show both the strong and weak points of
non linear free surface potential flow codes and this will give yacht designers a better insight
in the capabilities of such codes.
2
1.2
Introduction
Preliminary objectives
As can be read in section 1.1 the main objective of this thesis is:
Validation of a non linear free surface potential flow code (SHIPFLOW) based
on the upright bare hull resistance measurements of the Delft systematic yacht
hull series (DSYHS).
During this validation the different errors within the result of the simulation are investigated.
By using the assumption of potential flow a modelling error is introduced. This modelling
error should be small to achieve validation but this may not be the case. If there is a significant
modelling error, will it be possible to correct the results? This leads to the second objective
of this thesis:
Derivation of correction formulas for the modelling error which will be based on
the characteristic sailing yacht hull parameters.
The correction formulas will be based on the characteristic hull parameters such that they
can be used for other sailing yacht hulls within the range. The correction formulas will be
determined using a linear regression analysis.
1.3
Structure
This section describes the structure of the report. Chapter 2 describes the mathematical
model of the hydrodynamic part of the flow around a sailing yacht. It starts with the basic
conservation laws and treats the different boundary conditions. The last part of this chapter
is about the different characteristic flow parameters and their influence on the flow around a
sailing yacht.
Chapter 3 gives an overview of towing tank testing procedure. The influence of the scale on
the different characteristic parameters is explained. The post processing of the measurements
is described and possible errors and uncertainties in the towing tank results are given.
Chapter 4 describes the basics of the potential flow and the treatment of the non linear boundary condition. The iterative solution method and the way the wave resistance is determined
are explained.
Chapter 5 describes the verification procedure and starts with an explanation of the basics of
verification. The verification is done for two different cases: a submerged sphere and sailing
yacht hull nr 1 of the Delft series.
Chapter 6 deals with the validation procedure. It starts with an explanation of the basics
of validation. The possible errors in both the CFD and towing tank results are given. The
results of the CFD calculations and the towing tank measurements are compared and an
explanation for the difference is given.
1.3 Structure
3
Chapter 7 is about the derivation of the correction equations. First a brief description of
linear regression analysis is given. Then the linear regression is applied to the results and the
resulting correction formulas are described. In the end the correction equations are tested on
two hulls which were not included in the regression.
Chapter 8 gives the conclusions and recommendations for future work. The appendices contain the used SHIPFLOW input files and the figures which compare the trim and sink results
with the measurements.
4
Introduction
Chapter 2
Governing equations of the
hydrodynamic flow around a sailing
yacht
The equations which govern any flow are the so called Navier Stokes equations. These equations are the result of applying Newton’s second law on a fluid element. The derivation of
these equations can be found in almost all textbooks about aero- or hydrodynamics so only
the results are mentioned here. The equations govern the flow of air as well as the flow of the
water. The flow of air is neglected here because there is only a small interaction between the
air and water.
2.1
Conservation laws
As described in [1] there are three basic equations: conservation of mass, conservation of
momentum and conservation of energy. The conservation of energy is excluded from this
report because the flow around a sailing yacht is a low speed, incompressible fluid flow and
the temperature difference between body and fluid is assumed to be small.
Conservation of mass is described by the continuity equation.
Dρ
+ ρ divV = 0
Dt
(2.1)
D
∂
∂
∂
∂
=
+u
+v
+w
Dt
∂t
∂x
∂y
∂z
(2.2)
In which:
6
Governing equations of the hydrodynamic flow around a sailing yacht
and:
divV =
∂u ∂v ∂w
+
+
∂x ∂y
∂z
(2.3)
Water is an incompressible fluid and thus ρ = const. This simplifies the continuity equation
(2.1).
divV = 0
(2.4)
The conservation of momentum equation is acquired when Newton’s second law is applied to
a fluid particle. A continuous, isotropic and linear viscous fluid is assumed. The so called
Navier Stokes (N S) equation is written here using indicial notation.
"
∂p
∂
Dui
µ
= ρgi −
+
ρ
Dt
∂xi ∂xj
∂ui
∂uj
+
∂xj
∂xi
!
+ δij λdivV
#
(2.5)
Water is incompressible and the continuity equation for an incompressible fluid (2.4) can
be used to simplify the NS equation. When the viscosity of the water is assumed to be
constant the NS equation is further simplified to the Navier Stokes equation for incompressible,
constant viscosity flow:
∂
∂p
Dui
+µ
= ρgi −
ρ
Dt
∂xi
∂xj
∂uj
∂ui
+
∂xj
∂xi
!
(2.6)
The viscosity of liquids, like water, is temperature dependent but the temperature differences
in the fluid are assumed to be small so the assumption of constant viscosity is justified.
2.2
Boundary conditions
Before the boundary conditions are explained first the coordinate system is explained. The
coordinate system has the same speed as the ship but does not follow the ship movements such
as trim and sink. The origin is located at the intersection of the bow with the undisturbed
still water plane. The x-axis is horizontal and points in the downstream direction. The z-axis
is vertical and the y-axis points to starboard (to the right when looking towards the bow of
the ship).
At the upstream or inlet boundary the values of the velocity (V) and the pressure (p) must
be known.
V
= V0
(2.7)
p = p0
(2.8)
2.3 Characteristic flow parameters
7
The far field boundary conditions can be split into two parts: the boundary to the side of the
ship and the boundary behind the ship. The disturbances to the side will disappear and the
boundary conditions are the same as the upstream conditions (2.7 and 2.8). The disturbances
(waves) behind the ship will not disappear and this leads to a different boundary condition. In
practice Neumann boundary conditions (2.9 and 2.10) are used at the downstream boundary.
∂V
∂x
∂p
∂x
= 0
(2.9)
= 0
(2.10)
The particles at the hull surface have the same speed as the yacht, the so called the no-slip
condition. This leads to the following boundary condition for the velocity at the solid surface:
V =0
(2.11)
At the free surface there has to be kinematic equivalence between liquid and gas which means
that the velocity of the flow at the free surface has to be tangent to the free surface.
w(x, y, η) =
Dη
∂η
∂η
∂η
=
+u
+v
Dt
∂t
∂x
∂y
(2.12)
in which η(x, y, t) is the equation which describes the location of the free surface.
There also has to be pressure equilibrium at the free surface.
p(x, y, η) = pa − γ
1
1
+
Rx Ry
!
(2.13)
where γ is the coefficient of surface tension and R x and Ry are the radii of curvature of the
free surface. In a wave trough p < pa and in a wave crest p > pa
2.3
Characteristic flow parameters
The NS equations are valid for different kind of fluids with different properties. To make
the equations independent of the properties of the fluid (dimensionless) they are divided by
reference values.
x∗ =
x
L
8
V
V0
w
V0
tV0
L
ρ
ρ0
L∇
p + ρ0 gz − p0
ρ0 V02
µ
µ0
η
L
V∗ =
w∗ =
t∗ =
ρ∗ =
∇∗ =
p∗ =
µ∗ =
η∗ =
p∗ is the dimensionless pressure in the fluid because the gravity term cancels out against the
hydrostatic pressure (which is a part of the pressure p). When these variables are substituted
in the continuity and NS equations (2.6) the following equations are obtained:
divV∗ = 0
ρ
∗ DV
∗
Dt∗
∗ ∗
= −∇ p +
(2.14)
µ0
µ∗ ∇∗2 V ∗
ρ0 V0 L
(2.15)
These equations resemble the original equations except for one important parameter: the
Reynolds number (Re).
ρV L
VL
=
µ
ν
Re =
(2.16)
The boundary conditions are also made dimensionless. For the inflow conditions and the
conditions at infinity (2.7 and 2.8) this becomes:
V∗ =1
∗
p =0
(2.17)
(2.18)
And for the boundary condition (2.11) on the solid surface:
V∗ =0
The dimensionless kinematic boundary condition has the following form:
(2.19)
2.3 Characteristic flow parameters
9
w∗ =
Dη ∗
Dt∗
(2.20)
The boundary condition which requires pressure equilibrium at the free surface (2.13) is also
non-dimensionalised:
gL
γ
pa − p 0
+ 2 η∗ −
p∗ =
2
ρ0 V0
V0
ρ0 V02 L
1
1
+ ∗
∗
Rx Ry
!
(2.21)
Three characteristic parameters arise: the Euler number (Eu), the Froude number (Fn) and
the Weber number (We).
Eu =
Fn =
We =
pa − p 0
ρ0 V02
V
√0
gL
ρ0 V02 L
γ
(2.22)
(2.23)
(2.24)
In hydrodynamics the Euler number is usually replaced by the cavitation number (C). This
is done because the pressure level in the fluid is unimportant unless the vapor pressure is
reached.
C=
pa − p v
ρ0 V02
(2.25)
In total there are four characteristic parameters: Reynolds number, cavitation number, Froude
number, and Weber number. The Reynolds number determines the viscous behavior of the
flow which is important for the viscous and pressure resistance. The cavitation number is
only important when the pressure in the flow reaches the vapor pressure. When this happens,
it creates gas bubbles in the fluid which have a lot of influence on the flow. This only happens
in low pressure areas for example around propellers or hydrofoils and is not important for the
flow around normal sailing yachts. The Froude number is important for free surface flows and
wave formation and thus for the wave resistance. The Weber number is connected to surface
tension effects such as spray and breaking waves. The common opinion is that these effects
have a small influence on the flow and the resistance.
10
Chapter 3
EFD: Towing tank testing
Towing tank tests are frequently used to determine the resistance of ships because a decrease
in the resistance can save lots of fuel cost during the lifespan of a merchant ship. Sailing yachts
are not often tested because of the high costs and the small number of yachts produced to
share this costs. Towing tank test are used to optimize the design of racing yachts in high
profile races such as the Americas Cup and the Volvo Ocean Race. This chapter will first give
a description the Delft systematic hull series, the towing tank setup and the testing procedure.
The influence of the characteristic flow parameters will be described and the post processing
of the test results will be explained.
3.1
Description of the DSYHS
The Delft systematic sailing yacht hull series (DSYHS) consists of 50 systematically varied
sailing yacht models. The variation of the principal hull parameters is given in table 3.1.
These models have been tested at different speeds, leeway and heel angles in the Delft ship
hydromechanics laboratories towing tank. The evolution of the Delft systematic yacht hull
series can be found in references [2] to [11]
Table 3.1: Range of the principal hull parameters of the Delft series
Length - Beam ratio
Beam - Draft ratio
Length - Displacement ratio
Longitudinal centre of buoyancy
Longitudinal centre of flotation
Prismatic coefficient
Midship area coefficient
Loading factor
L
B
B
T
L
∇1/3
Lcb
Lcf
Cp
Cm
Aw
∇2/3
2.73
2.46
4.34
0.0 %
-1.8 %
0.52
0.65
3.78
to
to
to
to
to
to
to
to
5.00
19.38
8.50
-8.2 %
-9.5%
0.60
0.78
12.67
12
In [11] the total sailing yacht resistance has been divided in five components: upright resistance, change in resistance due to heel, change in resistance due to leeway, change in resistance
due to the keel and a change in resistance due to the rudder. The towing tank test results
have been used to derive empirical formulas for all these resistance terms. These empirical
formulas are polynomial expressions with Froude number dependent constants which can be
found . This report focuses on the upright resistance of the bare hull (hull without keel and
rudder).
The resistance formulas are often used in a velocity prediction program (VPP). A VPP
combines the hydrodynamic resistance formulas with an empirical aerodynamic model of the
sail forces to determine the velocity of a sailing yacht. The VPP can be used to evaluate
the performance of different sailing yacht designs. The results of this evaluation are usually
quite accurate as long as the characteristic hull parameters stay within the range of the tested
models.
3.2
Towing tank test setup and testing procedure
A towing tank test facility consists of a number of different parts. The main parts are the
tank, the carriage, the model and the measurement systems. The tank is a rectangular basin
filled with water and sometimes equipped with a wave generator. A large towing tank can
handle large models and this will decreases the influence of the scale effects.
The carriage usually is a steel construction which moves on rails along the sides of the tank.
The construction should be rigid so that it does not deform during the tests. The range and
accuracy of the velocity depend on the power and control system of the electric motors which
move the carriage.
The geometry of the model is an accurate scale model of the ship or sailing yacht. Different
construction techniques exist to build a model. Two requirements are that the model should
be rigid enough that it does not deform during the tests and it should be lightweight such that
ballast can be added to achieve the correct trim and sink. The scale of the model depends
mainly on the size of the towing tank. A large model will decrease the influence of the scale
effects but increases the blockage and interference effects.
The measurement system consists of a couple of load cells and a computer. The sensors
measure the forces in different directions and send this information to the computer. The
data collected during the test is stored and post processed to get resistance and side force
values. The water temperature during tests is measured and will be used in the post processing
to determine the viscosity.
Two different test methods exists: free sailing and semi captive. The free sailing method
resembles actual sailing because the towing forces act at the center of effort of the sails and
the yacht attains an equilibrium heel and leeway angle. This way the number of tests needed
to cover the whole range of sailing conditions is minimized. The drawback of this method is
the expensive model which needs to have the same center of gravity as the full scale yacht
3.3 Froude number similarity
13
Figure 3.1: Sailing yacht model and the attachment to the towing tank carriage (source [11])
and an active rudder system.
The Delft series are tested using the semi captive method in which the model is free to heave
and pitch but restricted in all other motions. A disadvantage of this method is that a lot
more tests are needed to cover all possible sailing conditions. The main advantage is that the
model for this test method is much simpler because the heel is fixed and thus only the shape
of the model is important. The correct trim of the model is achieved by moving the internal
ballast.
3.3
Froude number similarity
The characteristic parameters (Re, Fn, We, C) during towing tank test should be the same
as in full scale to achieve the same flow situation. This is usually not possible. Until now, no
liquid has been found which has the right properties to keep all the characteristic parameters
equal. Water is used in towing tanks and this will cause a change in some of the characteristic
parameters compared to the full scale values. The implications on the flow around the model
will be described in this section.
An important feature of the flow around a sailing yacht (or ship in general) is the wave
formation and the resistance this causes. The Froude number is the characteristic parameter
which governs this flow phenomenon and during towing tank tests the Froude number of the
model is equal to the full scale value. This means that the other characteristic parameters
during the tests are different from full scale. The latest models of the Delft systematic series
14
are 2 m long and represent a 10 m sailing yacht so the scale factor (S f ) is 1 : 5. This model
is used as an example to show what happens to the other characteristic parameters (Re, We
and C) when the Fn is kept constant between test and full scale.
F nf s = F nmod
U
U
√ f s = p mod
gSf L
gL
Umod =
q
S f Uf s
(3.1)
(3.2)
(3.3)
To keep Fn constant we have to use a towing speed which is smaller than the full scale speed.
For our example this means a towing speed of 44.72 % of full scale speed.
The other characteristic parameters are calculated with this towing speed and compared with
the full scale values.
Remod =
p
3
S f Uf s Sf L
= (Sf ) 2 Ref s
ν
(3.4)
The Re of the model is much lower than the full scale Re and because of this laminar regions
are present on the model. In full scale the flow becomes turbulent after a small distance
and this means that there is a big difference between the test and full scale situation. The
laminar flow will have a large influence on the resistance of the model and to avoid this effect
a turbulence stimulator is applied to trip the boundary layer. There are a number of different
turbulence stimulator and the most common one is a row of cylindrical studs. The turbulence
stimulator used during the Delft series tests consists of carburundum strips on hull, keel and
rudder. The carburundum has grain size 20 and is applied on the models with a density of
approximately 10 grains/cm2 . To determine the resistance of the strips a couple of tests are
carried out with single and double strips. The difference in resistance is assumed to be equal
to the resistance of one strip and this value is then used to correct all the test results.
W emod
ρ( Sf Uf s )2 Sf L
= Sf2 W ef s
=
γ
p
(3.5)
As can be seen from equation 3.5, the influence of the scale factor on We is even larger than
on Re. This difference in We has some influence on the wave breaking and spray around the
model. The amount of breaking waves and spray differs between full scale and the towing
tank tests. The Weber number represents the influence of the surface tension compared to
the dynamic pressure as can be seen in 2.24. So even if the Weber number is lower during
towing tank tests than in full scale it is assumed to be high enough that its influence on the
resistance can be neglected.
As explained in section 2.3 the effect of the cavitation number is only important when the
pressure in the flow reaches the vapor pressure. This is not the case for full scale and is also
3.4 Postprocessing of towing tank measurements
15
unimportant during towing tank tests because the velocity is lower than in full scale (see 3.3).
So the difference in cavitation number has no significant influence on the flow around the
model.
3.4
Postprocessing of towing tank measurements
The resistance values of the towing tank test are post processed to convert them into full
scale resistance values. This is done according to Froude’s method which states that the wave
resistance can be extrapolated directly between model and full scale if the Froude number is
equal. The postprocessing consists of a number of different steps which will be described in
this section. The first step is to correct the test results for the resistance of the turbulence
stimulator and blockage effects. The resistance of the turbulence stimulator is determined by
a double width test as described in section 3.3. The blockage effects depend on the towing
tank size and are based on measurements with the same model in different towing tanks and
on the experience of the test facility. After this the values are made dimensionless by dividing
the resistance with the dynamic pressure and the reference area (wetted surface area at zero
speed).
CTmod =
RTmod
1
2
2 ρVmod Smod
(3.6)
The next step is to subtract the frictional resistance from the total resistance to get the
residuary resistance. Two different methods exist to calculate the frictional resistance: ITTC
1957 and ITTC 1978. The ITTC 1957 formula calculates the frictional resistance coefficient
by using a flat plate friction formula which includes a form factor of 12%. The form factor
will be explained at the end of this section.
CF =
(10
0.075
log Re − 2.0)2
(3.7)
The frictional resistance coefficient of the model is calculated using 3.7. The Reynolds number
used in the post processing of the DSYHS is based on a reference length of 70% of the
waterline. The residuary resistance coefficient of the model is computed by subtracting the
frictional resistance coefficient from the total resistance coefficient.
CRmod = CTmod − CFmod
(3.8)
The Froude number of the model test and the full scale are the same and this means that the
residuary resistance coefficient is equal for both cases.
CRf s = CRmod
(3.9)
16
The frictional resistance of the full scale ship is calculated using the ITTC 1957 formula
(3.7) again. The residuary and frictional resistance coefficients are added to get the full scale
resistance coefficient.
(3.10)
C Tf s = C F f s + C R f s
This total resistance coefficient is used to calculate the total resistance of the full scale ship.
1
RTf s = CTf s ρVf2s Sf s
2
(3.11)
The ITTC 1978 method is based on the ITTC 1957 method but takes the so called form drag
into account. Form drag is the viscous pressure resistance caused by the displacement effect
of the boundary layer. The form drag is taken into account using a form factor k. The form
factor can be determined using Prohaska’s method. Prohaska’s method assumes that for low
Froude numbers (0.05 ≤ F n ≤ 0.20) the ratio of form drag to viscous drag is constant. The
CT
residuary resistance is assumed to be a linear function of F n 4 . When C
is plotted against
F
F n4
CF
the points should lie on a straight line which is described by equation 3.12. The form
factor can be determined by extrapolating the line to
F n4
CF
= 0.
F n4
CT
=m
+ (1 + k)
CF
CF
(3.12)
The ITTC 1978 method calculates the residuary resistance by subtracting the viscous and
form drag from the total resistance. For the viscous drag the same formula is used as in the
ITTC 1957 method. (see equation 3.7)
CRmod = CTmod − (1 + k)CFmod
(3.13)
According to Froude’s assumption the residuary resistance coefficient is the same for model
and full scale. (see equation 3.9) The total resistance coefficient is calculated by adding the
viscous and form drag. The form factor is assumed to be the same for model and full scale.
CTf s = CRf s + (1 + k)CFf s
(3.14)
The total drag can be calculated by formula 3.11
3.5
Errors and uncertainty in towing tank tests
Although towing tank testing is the most realistic experiment to represent the full scale physics
it should be noted that the results will contain an error. This error can be split into two parts:
3.5 Errors and uncertainty in towing tank tests
17
a measurement error and a post processing error. Possible errors which can propagate into the
measurements are for example errors in: the geometry of the model, calibration of the load
cells, displacement of the model, alignment of the model (initial trim and leeway), velocity of
the carriage, temperature of the water and waiting time between tests.
The error in the final results is not only caused by an error in the measurements but also by
errors in the post processing. The ITTC 1957 and ITTC 1978 methods are standard methods
in the post processing of towing tank results. The ITTC 1957 frictional coefficient formula
is based on a best fit to experimental results. This formula will approximate the frictional
resistance but is not exact.
Another error in the post processing of the results is caused by the fact that the reference
area (the wetted surface area at zero speed) is used to determine the frictional resistance.
The wetted surface area during the test differs from this reference area because of the wave
formation. The difference will be small for low speeds but may increase for the higher speeds.
A full uncertainty analysis of the whole towing tank set up and the postprocessing, as described in [12], would be the best way to determine the influence of the different errors on the
wave resistance. Unfortunately this procedure will take a lot of time and effort which makes
it too expensive for most towing tank facilities.
A more practical way of checking the errors is to repeat some tests and compare the measurements. In [13] Fasardi describes how to recognize and how to avoid possible errors in
the measurements. In [14] an assessment of the accuracy and repeatability of towing tank
tests for ACC yacht development is made. Three different towing tank facilities (INSEAN,
QinetiQ and SSPA) describe their testing procedures and give a explanation of the sources
of the variation in the measured forces. The conclusion is that a long term accuracy and
repeatability of drag measurements in the order of ±1% can be achieved. This is based on
faired or averaged results incorporating a number of individual test runs.
18
Chapter 4
CFD: non linear free surface
potential flow code SHIPFLOW
This chapter will give an overview of the potential flow and the solution method used in the
non linear free surface case. The description is general, details about SHIPFLOW can be
found in the user manual [15]. First the basics of potential flow and the linearisation of the
free surface boundary conditions are explained. Then the non linear solution method and its
special features will be treated. The last part of this chapter deals with the two different ways
to determine the wave resistance within the non linear free surface potential flow.
4.1
Basics of potential flow
The principal assumptions in potential flow are: inviscid, irrotational, incompressible, and
steady flow. These assumptions are valid for the flow around a ship because the Reynolds
number is relatively high and the effect of viscosity will be limited to a thin layer close to the
hull and the wake. The large scale flow features such as the wave pattern are not affected
much and this justifies the use of the potential flow assumption to model this large scale flow
features. The potential flow assumptions are used to simplify the Navier Stokes equations
(2.6) which leads to the following equation:
1
∇V2 = ρg − ∇p
2
(4.1)
The continuity equation stays the same and is repeated here for easy reading.
divV = 0
(4.2)
The velocity vector can be written as the gradient of a scalar. This scalar is called the velocity
potential.
20
CFD: non linear free surface potential flow code SHIPFLOW
V = ∇φ
(4.3)
This is substituted in the Bernoulli and continuity equations:
1
p + ρgz + (∇φ · ∇φ) = const.
2
(4.4)
∇2 φ = 0
(4.5)
The continuity equation (2.4) transforms into the Laplace equation (4.5) which is linear and
homogeneous. This allows the superposition of different solutions. The pressure and velocity
are decoupled which makes it possible to solve the Laplace equation first and compute the
pressure later.
The no-slip boundary condition (2.11) at the body changes to the tangential flow boundary
condition. The simplifications introduced by the potential flow assumptions have decreased
the degrees of freedom which make it impossible to maintain the no-slip condition. The
tangential flow condition means that the fluid cannot flow through the body.
φn = 0
(4.6)
The velocity potential is substituted in the kinematic boundary condition at the free surface
(2.12) and the time dependent terms are dropped because of the steady flow assumption. The
location of the free surface is described by the single valued function η(x, y) which makes it
impossible to calculate overturning (breaking) waves and spray. These effects are considered
to have a small influence on the global wave pattern and thus is this single valued free surface
approach allowed. Two boundary conditions exist at the free surface. The velocity vector at
the free surface is tangential to the free surface.
φx ηx + φ y ηy − φ z = 0
at z = η(x, y)
(4.7)
And the pressure in the water at the free surface has to be equal to the atmospheric pressure.
1
pa + ρgz + (∇φ · ∇φ) = const.
2
at z = η
(4.8)
which can be rewritten as
gη +
i
1h
2
(φx )2 + (φy )2 + (φz )2 − U∞
=0
2
(4.9)
4.2 Linearization of the free surface boundary conditions
21
By reducing the Navier Stokes equations to the potential flow equations a lot of information
is lost. This leads to the situation that the solution of the potential flow equations is no
longer unique and more than one solution exist. This can give non physical solutions such
as waves upstream of the bow. An extra condition is added to avoid non physical solutions:
the radiation condition. The radiation condition states that free surface waves generated by
a ship can not travel in the upstream direction.
4.2
Linearization of the free surface boundary conditions
The free surface conditions are non linear and this makes it difficult to solve them. To
overcome this problem the boundary conditions are linearized. This is done by dividing the
potential (φ) in two parts: an estimated flow or base flow (Φ) and a perturbation (ϕ). A
good estimate will result in a small perturbation which justifies the linearization.
∇φ = ∇Φ + ∇ϕ
(4.10)
η = H +h
(4.11)
This can then be substituted into the free surface boundary conditions 4.7 and 4.9.
Φx ηx + Φ y ηy + ϕ x Hx + ϕ y Hy − Φ z − ϕ z = 0
η=
1 2
U∞ − Φ2x − Φ2y − Φ2z − 2Φx ϕx − 2Φy ϕy − 2Φz ϕz
2g
(4.12)
(4.13)
These equations have to be satisfied at the free surface. The location of this free surface is
unknown and therefore these equations need to be transferred to the estimated surface (z =
h).
∇φ(z=η) ≈ ∇Φ(z=H) + ∇ϕ(z=H) + h
∂∇Φ
∂z
(4.14)
Dawson proposed to neglect the transfer term h ∂∇Φ
∂z and the higher order terms which leads
to the combined linear free surface boundary condition on the known surface z = H:
1
∂ 2
Φx + Φ2y + Φ2z + 2Φx ϕx + 2Φy ϕy + 2Φz ϕz
Φx
2g ∂x
1
∂ 2
− Φy
Φx + Φ2y + Φ2z + 2Φx ϕx + 2Φy ϕy + 2Φz ϕz
2g ∂y
+ϕx Hx + ϕy Hy − Φz − ϕz = 0
−
(4.15)
22
4.3
Solution method
There are a lot of different ways to solve the Laplace equation for the velocity potential. In
[16] Raven compares the advantages and disadvantages of the possible solution strategies and
his conclusion is that a panel method using Rankine sources on the hull and free surface will
probably be the most efficient. A Rankine source is a point source which potential can be
σ
such that it satisfies the Laplace equation ∇ 2 Φ = 0. The Laplace
described by Φ = − 4πr
equation is homogeneous which makes it possible to add different solutions to create a new
solution, the superposition principle. Since a Rankine source satisfies the Laplace equation, a
combination of different Rankine sources can be used to represent a body in a potential flow.
A more detailed description of the basics of panel methods can be found in [17].
Lifting surfaces can easily be included in a panel method and are needed to model appendages
such as keels and rudders. This report does not consider the appendages but a detailed
explanation about lifting surfaces in free surface flows can be found in [18].
4.3.1
Linear free surface potential flow
The linear case starts with the calculation of the estimate or base flow. For this base flow
the slow ship approximation is used which means that no free surface waves are present (the
free surface is flat). It is possible to calculate this base flow by meshing both the hull and
free surface but it is more efficient to make use of symmetry by mirroring the under water
part of the hull in the water plane. This eliminates the need to mesh the free surface which
saves computing time. The flow around this so called ”double body” is calculated and the
slow ship approximation at the free surface is immediately satisfied.
To determine the perturbation both hull and free surface are meshed. The result of the double
body flow calculation is used as an estimate and then the perturbation can be calculated using
the equations from section 4.2. The resulting perturbation is added to the base flow which
gives the linear free surface potential flow solution.
A problem of the linear method is that it does not take into account the shape of the hull
above the still waterline. This is an important drawback because most sailing yachts have
overhangs which will have a lot of influence on the flow around the hull.
4.3.2
Non linear free surface potential flow
The non linear solution method is an extension of the linear case. After the linear solution
has been calculated this result is used as a new estimate. The hull and free surface panels are
moved and the perturbation is calculated again. These steps are repeated until a converged
solution is achieved. According to [16] convergence is achieved when the change in wave
height for two consecutive iterations is below a set tolerance.
The first advantage of the non linear solution method is that it gives a solution of the system
4.4 Determination of the wave resistance
23
of equations and is no longer an approximation (as in the linear case). The second advantage
of the non linear solution is that when the panels are moved they are adjusted to fit the
new intersection between the hull and free surface. This way the shape of the hull above the
waterline is taken into account.
4.3.3
Special features of the solution method
The radiation condition (see section 4.1) is satisfied by using an upwind approximation for
the second derivatives of the potential in the longitudinal direction at the free surface. This
upwind discretisation eliminates the formation of upstream waves. A central scheme is used
for the second derivatives of the potential in the transverse direction.
The stability and convergence of the solution method is improved by using raised panels on the
free surface. This means that the panels are raised above the free surface but the collocation
points are on the actual free surface. Solutions using this raised panels showed point to
point oscillations in the calculated source strength. An effective way to avoid this is to use
a forward shift in the collocation point location. The phase and amplitude of the calculated
waves show a dependency on the distance that the panels are raised and the forward shift
of the collocation point. The two dimensional case of the non linear free surface potential
flow has been studied in detail in [16] and also in [18] and the conclusion is that each upwind
scheme has its own optimal raised distance. In general a forward shift of 25 to 30% of the
panel length and a raised distance of more than 1 panel length lead to accurate results in the
two dimensional case. For the three dimensional case the raised distance has some influence
on the condition of the system of equations. This leads to a decrease in the raised distance
when the Froude number increases: approximately 70% of the panel length for Fn 0.25 to
approximately 30% of the panel length for Fn 0.55.
The non linear free surface potential flow can be calculated using a fixed location of the hull
or a hull which is free to trim and sink. In case the hull is free to trim and sink an extra set of
equations is added to the solver. The weight distribution of the hull has to be in equilibrium
with the hydrodynamic forces. After each iteration the trim and sink are adjusted to maintain
the equilibrium. This gives two extra convergence criteria: the change of the trim angle and
the sink should be within a given tolerance.
4.4
Determination of the wave resistance
There are two ways to determine the wave resistance of the ship: pressure integration and wave
cut analysis. This section will describe the two methods starting with pressure integration.
24
4.4.1
Pressure integration
The pressure integration method determines the wave resistance by integrating the pressure on
the hull panels. The pressure on the hull consists of the hydrostatic and the hydrodynamic
pressure. For the linear solution the hydrostatic pressure sums to zero and this makes it
possible to integrate only the dynamic pressure to get the wave resistance. For the non
linear solutions the hydrostatic pressure does not cancel and thus both pressures need to
be integrated. The magnitude of the hydrostatic pressure is often larger than that of the
hydrodynamic pressure and this can cause some problems concerning the accuracy of the
pressure integration method. The solution to this problem is to use a sufficient number of
panels on the hull surface.
4.4.2
Wave cut analysis
The wave cut analysis technique determines the wave resistance by analyzing the wave pattern.
Longitudinal or transverse wave cuts can be used but the transverse method is preferred
because it puts less demands on the size of the free surface. The method determines the wave
elevation in a number of transverse wave cuts behind the ship. The first requirement with
respect to the location of the wave cuts is that the wave cuts need to be in a region where
the wave pattern is relatively smooth. This means that the first wave cut can not be too
close to the stern of the ship. In SHIPFLOW a minimum distance of 40% of the ship length
is used. The second requirement is that the wave cuts cover at least one wavelength and the
distribution of the wave cuts can not be equidistant.
The wave cut method approximates the wave elevation in each wave cut by the sum of a series
of elemental waves. The wave resistance is determined with the result of this approximation.
A detailed description of the method can be found in [19]. The advantage of the wave cut
analysis is that it is less dependent on the number of panels on the hull. This will make
the wave cut method more robust than the pressure integration method for hulls with a
complicated geometry (high curvature areas).
Chapter 5
Verification
Verification is done to determine the numerical error and uncertainty within the computed
solution and is a necessary step in the validation process. This chapter will start with a
introduction to the basics of verification and the available verification methods. Then two
different test cases will be described. The first test case is the free surface flow around a
submerged sphere and the second test case is the free surface flow around sailing yacht model
nr 1 of the Delft series. This chapter will end with the conclusions of both the test cases.
5.1
Basics of verification
According to the ITTC quality manual [20] the simulation error (δ S ) is the difference between
the truth (T) and the simulated result (S). This simulation error consists of a modelling error
(δSM ) and a numerical error (δSN )
δS = S − T = δSM + δSN
(5.1)
Verification is a process to estimate the numerical error and the uncertainty in that error
estimate. There are different ways to estimate the numerical error and the basics will be
described here.
The best way to determine the numerical error in the simulation is to compare the result
with the analytical solution. Unfortunately the three dimensional non linear free surface
potential flow is too complicated to solve analytically and thus are there no analytical solutions
available. One of the characteristics of the potential flow is the so called d’Alembert’s paradox
which states that the drag of an arbitrary body in a 2D potential flow is zero. This is only
valid for flows without the influence of the free surface because the free surface waves extract
energy from the body. (see [16])
Since no analytical solution exists for most aero- or hydrodynamic flow cases a different
26
Verification
strategy needs to be used to determine the numerical error and uncertainty. A method which
is often used is Richardson extrapolation (RE). RE is based on the assumption that the
numerical error (e) can be represented by the following Taylor series expansion.
e φi = φ i − φ 0 =
n
X
αp hpi
(5.2)
p=1
in which φi is the numerical solution, φ0 is the exact solution, αj are constants, hi is a
parameter which represents the grid size and p are exponents related to the order of accuracy
of the method. The assumption used in RE is that the results are within the asymptotical
range, which means that the first term in equation 5.2 is dominant over the higher order
terms. This assumption is used to simplify equation 5.2 into the following equation.
φi − φ0 = αhpi
(5.3)
There are three unknowns in 5.3 and thus three solutions on different grids are required to
determine φ0 , α and p. An extra requirement for using this method is that the convergence
is monotonic. For a constant refinement ratio r = hh21 = hh32 the numerical error in the solution
on the finest grid can be estimated with the next formulas.
δREh1 =
p=
εh21
rp − 1
ln(εh32 /εh21 )
ln(r)
(5.4)
(5.5)
In which εh32 = φ3 − φ2 is the change in solution between coarse (h 3 ) and medium grid (h2 )
and εh21 = φ2 − φ1 is the change in solution between medium (h 2 ) and fine grid (h1 ). A
constant refinement ratio is not required but makes RE easier to apply.
The uncertainty in this error estimate is determined by multiplying the error estimate by a
factor of safety (FS ). The recommended value for the factor of safety is 1.25 for careful grid
studies (when p is close to the theoretical value) and 3 for all other cases.
Uh1 = FS |δREh1 |
(5.6)
A disadvantage of the generalized Richardson extrapolation based on three solutions is that
it can not recognize oscillatory convergence which can lead to wrong conclusions about the
accuracy of the solution.
In [21] Hoekstra and Eça derive a method which is similar to RE. This method is based on
the same basic assumptions as RE but the error estimate uses a larger number of numerical
solutions. The parameters φ0 , α and p are found by minimizing the following function.
5.2 Submerged sphere
27
v
u n
uX
S(φ0 , α, p) = t (φi − (φ0 + αhpi )2
(5.7)
i=1
This clearly is a more robust method to determine the numerical error but it is also more
expensive because it requires 4 or more solutions. For the estimation of the uncertainty a
safety factor approach is used similar to equation 5.6. In [21] conservative ways to bound the
upper value of the safety factor are given.
A problem with both described verification procedures is that they assume that the error can
be expressed by a Taylor series expansion of the grid size. This assumption is probably true
for local flow quantities in simple cases such as two dimensional potential or Euler flows on a
cartesian grid. It is questionable if this assumption holds for global quantities in complex flow
situations. A number of workshops on this topic have been organized in the last couple of
years in which different CFD codes are used to calculate the flow in two and three dimensional
test cases. Often the observed numerical order of accuracy differs from the theoretical value
and the discussions about this topic are ongoing (as can be seen in [22]).
5.2
Submerged sphere
The verification procedure was started with a sailing yacht hull of the DSYHS but this immediately caused a problem. When the hull is left free to trim and sink a change in wave
resistance between different cases is not only caused by a refinement of the grid but also by a
change in trim and sink. An option to eliminate the influence of the trim and sink is to calculate the flow around a fixed sailing yacht hull. The disadvantage of this option is that there is
no control over the displacement which may not stay constant when the grid is refined. Since
the displacement has a lot of influence on the wave resistance a similar problem occurs as for
the free sailing yacht case: a change in the wave resistance will be caused by a combination
of grid refinement and a difference in displacement. The solution to the previous mentioned
problems is to use a fixed submerged body for the verification procedure. The submerged
body used in this test case is a sphere because points on the surface of this geometry are
described by a simple analytical expression.
5.2.1
Objectives
The objective of this test case is to do a verification of the wave resistance of a submerged
sphere calculated by a non linear free surface potential flow solver (SHIPFLOW)
5.2.2
Description of the test case
The test case is split in two parts: the potential flow around a sphere without and with free
surface. For the case without free surface the drag should be zero because of d’Alemberts
28
Verification
paradox. Different distributions of panels on the sphere are used to asses the influence of
grid refinement on the numerical error. The number of panels in longitudinal and transverse
direction is equal and goes from 16 to 56. The panels used for this case are so called first
order panels: flat panels with a constant source strength. Since there is no free surface the
resistance is determined by pressure integration.
For the case with free surface three different Froude numbers are tested: 0.35, 0.4 and 0.45.
The reference length for the Froude number is the diameter of the sphere. The distance
between the undisturbed free surface and the center of the sphere is different for each Froude
number. For Fn 0.35 and 0.40 this distance is equal to 1 diameter while for Fn 0.45 this
distance is equal to 1.25 diameter. The reason to increase the distance for the higher Froude
number is to avoid breaking waves. When wave breaking occurs the free surface is no longer
a single valued surface and this will stop the computation.
The origin of the coordinate system is located in the most forward point of the sphere. For
Froude number 0.35 and 0.40 the inflow boundary is located one diameter upstream, the
outflow boundary 3 diameters downstream and the width of the domain is 2 diameters. For
the Froude number 0.45 the inflow boundary is located 2 diameters upstream, the other
dimensions are equal to the Fn 0.35 and 0.40 case. The size of the domain (see table 5.1) is
relatively large compared to the standard case to make sure that the calculated results are
not affected by the boundaries.
The panel distribution on the submerged sphere is the same as for the case without free
surface. The number of panels on the free surface in longitudinal direction (on a reference
length of 1 diameter) is equal to the number of longitudinal panels on the body. The width
of the panels on the free surface is two times the length which results in an aspect ratio of
0.5. This is done to reduce the number of free surface panels and thus the computational
time and costs. Both the panels on the sphere and the free surface are first order panels (flat
panels with a constant source distribution). An example of the free surface mesh is given in
figure 5.1.
Table 5.1: Domain size of the submerged sphere test case (diameter of the sphere is used as reference
length)
Froude number
depth of the sphere center
upstream boundary
downstream boundary
width of the free surface
0.35
1
-1
3
2
0.40
1
-1
3
2
0.45
1.25
-2
3
2
A free surface computation is converged / stopped when the maximum wave change is below
the convergence tolerance for wave height change. The default value is 5.0 · 10 −5 but during
this investigation the value was changed to 1.0 · 10 −8 to be sure that the results are free of
any iterative convergence error.
Figure 5.1: Meshing of the submerged sphere at Fn 0.40 with 20 panels per unit length
29
30
Verification
5.2.3
Results
The increase in the number of panels leads to a change in the resistance coefficient and a
change in the reference area. The comparison of the different panel distributions is done by
the so called drag area which is the resistance coefficient multiplied by the reference area.
Values of the drag area for the sphere in a flow without free surface can be found in table 5.2.
Table 5.2: Drag area of the sphere
nr of panels
Cd · S · 105
162
2.62
202
1.53
242
3.68
282
5.37
322
6.02
362
5.45
402
4.02
442
-6.11
482
-7.03
522
-7.84
562
-8.54
The resistance of the sphere shows no clear trend with an increasing number of panels. This
is unexpected because normally the numerical error should decrease with a decreasing panel
size. A further investigation shows that the reasons for the unexpected result are caused by
the mesh generator and the iterative matrix solver.
A closer inspection of the mesh shows that the corner points of the panels are not exactly
on the surface of the sphere. This is caused by the mesh generator which uses splines to
approximate the sphere. The difference is very small but it may affect the result of the
calculations. To eliminate this problem the mesh generator is run in ’manual’ mode in which
the panel corner points are read from an input file (see figure 5.2).
Another reason for the unexpected trend of the drag area is the iterative matrix solver which
stops when certain convergence criteria are met. By changing these convergence criteria or
by using a Gaussian elimination procedure to solve the system of equations, the drag area of
the sphere reduces to values which are of machine accuracy as can be seen in table 5.3.
Table 5.3: Drag area of the sphere using the ’manual’ meshing and more strict convergence criteria
nr of panels
Cd · S · 1016
162
0.71
202
-3.30
242
-0.62
282
-3.45
322
-3.11
362
1.89
402
-6.55
442
7.57
482
2.52
522
-1.30
562
-1.33
During the free surface calculations of the submerged sphere the manual input option and
correct convergence criteria for the matrix solver are used. The results for Fn 0.35, 0.40 and
0.45 can be found in tables 5.4, 5.5 and 5.6
Table 5.4: Wave drag area of a submerged sphere at Fn 0.35
panels
Cwpi · S · 103
Cwwc · S · 103
16
1.2359
0.69117
20
1.1491
0.74197
24
1.0918
0.77275
28
1.0431
0.79028
32
1.0004
0.80193
36
0.96241
0.81071
40
0.92837
0.81836
44
0.89708
0.82529
The least squares method is used to determine the numerical order and the extrapolated drag
area of the submerged sphere. First the least squares method is applied using the results from
31
Figure 5.2: Meshing problem at the stagnation point of the sphere with automatic meshing on the
left and correct ’manual’ meshing on the right
panels
Cwpi · S · 103
Cwwc · S · 103
16
5.6248
3.9864
20
5.6287
4.0810
24
5.5969
4.1149
28
5.5475
4.1276
32
5.4947
4.1337
36
5.4429
4.1374
40
5.3961
4.1416
44
5.3521
4.1459
panels
Cwpi · S · 103
Cwwc · S · 103
16
1.0556
1.0706
20
1.0149
1.0980
24
0.99121
1.1148
28
0.97675
1.1268
32
0.96762
1.1364
36
0.96145
1.1443
40
0.95759
1.1518
32
Verification
all the available grids. Then the result from the coarsest grid is removed and the least squares
method is applied again, which is repeated until only the 6 finest grids are left. The method
needs 4 different grids to work but the reliability increases when more results are used.
Table 5.7: Extrapolation of the pressure integration results of a submerged sphere at Fn 0.35
nr of grids
8
7
6
φ0 · 103
-0.8912
-
α · 103
1.791
-
p
0.169
-
δ · 103
1.788
-
Table 5.8: Extrapolation of the wave cut analysis results of a submerged sphere at Fn 0.35
nr of grids
8
7
6
φ0 · 104
8.5321
8.5574
8.7992
α · 105
-2.9048
-3.1725
-5.5131
p
1.686
1.613
1.092
δ · 105
2.7919
3.0445
5.4627
−4
13
x 10
DA pi
DA wc
12
Drag area
11
10
9
8
7
6
0
0.5
1
1.5
hi/h1
2
2.5
3
Figure 5.3: Extrapolation of the drag area for a submerged sphere at Fn 0.35 using the 6 finest grids
For Fn 0.35 a negative value is found for the extrapolated drag area using all 8 available
grids. When the 7 or 6 finest grids are used no solution is found in the range 0.01 ≤ p ≤ 5
33
which means that the wave resistance determined by pressure integration diverges. The wave
resistance determined by wavecut analysis converges but the extrapolated value depends on
the number of grids included in the extrapolation procedure. The difference between the
result on the finest grid and the extrapolated value is between 4 and 7%.
For Fn 0.40 no extrapolated value is found in the range 0.01 ≤ p ≤ 5 which means that
the pressure integration results diverge. The wave resistance determined by wave cut analysis
converges and the results of the least squares method can be found in table 5.9. The difference
between the result on the finest grid and the extrapolated value is less than 1%
nr of grids
8
7
6
φ0 · 103
4.1456
4.1474
4.1581
α · 105
-0.28128
-0.39102
-1.3218
p
3.988
3.581
1.936
δ · 105
0.03368
0.1470
1.2206
−3
5.8
x 10
DA pi
DA wc
5.6
5.4
5.2
Drag area
5
4.8
4.6
4.4
4.2
4
3.8
0
0.5
1
1.5
hi/h1
2
2.5
3
For Fn 0.45 both the wave resistance determined by pressure integration and the wave resistance determined by wave cut analysis converge. The pressure integration results converge to
a value which differs less than 3% from the result on the finest grid. The wave cut analysis
results show convergence but the extrapolated value depends on the number of grids included
34
Verification
in the extrapolation procedure. The difference between the extrapolated value and the result
on the finest grid is between the 5 and 8%. Although both methods to determine the wave
resistance converge, there is a significant difference between the extrapolated values.
Table 5.10: Extrapolation of the pressure integration results of a submerged sphere at Fn 0.45
nr of grids
7
6
φ0 · 103
0.93644
0.94014
α · 105
2.0555
1.7203
p
1.920
2.121
δ · 105
2.1147
1.7445
nr of grids
7
6
φ0 · 103
1.2125
1.2558
α · 104
-0.61608
-1.0431
p
0.907
0.595
δ · 104
0.60707
1.0395
−3
1.3
x 10
DA pi
DA wc
1.25
1.2
Drag area
1.15
1.1
1.05
1
0.95
0.9
0
0.5
1
1.5
2
2.5
hi/h1
5.2.4
35
Explanation of the results
Although the drag area of the sphere without free surface reduces to machine accuracy this
does not mean that the numerical error is zero. Because of the symmetric mesh the error
cancels when the pressure is integrated. This makes it difficult to determine the quality of
the panel distribution on the sphere. The difference between the mesh generated panels and
the manual input is so small that it can be neglected in practical cases.
The grid refinement shows that the calculations for the Froude numbers 0.35 and 0.40 diverge
when the wave resistance is determined by pressure integration. The convergence is around
second order for Froude number 0.45. There are a couple of different reasons for this behavior.
The wavelength (λ) of the waves generated by a ship or submerged body depends on the
Froude number and can be calculated by the following formula:
λ = 2πF n2 L
(5.8)
This formula (5.8) can be used to determine the number of waves per unit length (n).
n=
1
2πF n2
(5.9)
The number of waves per unit length decreases when the Froude number increases as can be
seen from equation 5.9. The finest grid for Froude numbers 0.35 and 0.40 have 44 panels per
unit of length which results in respectively 33.87 and 44.23 panels per wavelength. The finest
grid for Fn 0.45 uses 40 panels per unit of length which results in 50.89 panels per wavelength.
So although the grid for this case is coarser than the ones used for the lower Froude numbers,
it has a better resolution of the flow features (waves). This higher resolution may be one of
the reasons for the better convergence of the drag of the submerged sphere at Fn 0.45.
A requirement of both Richardson extrapolation and the least squares method is that the grid
needs to be systematically refined. This requirement is not totally satisfied here. The number
of panels is systematically refined but this is not the case for the location of the collocation
points. The corner points of the first order panels are exactly on the surface of the sphere but
the collocation points are a small distance inside the sphere (since the panels are flat). This
distance becomes smaller when the number of panels increases. The result of this is that a
refinement of the panels results in slightly different shape of the submerged body which may
affect the resistance values and the extrapolation.
Another reason which may explain the better convergence of the drag of the submerged sphere
at Fn 0.45 is the location of the inflow boundary. A ship does not generate waves upstream
of the bow which makes it possible to put the inflow boundary at 0.5 times the waterline
length upstream of the bow. The disturbance caused by the sphere is so large that it does
generate a Bernoulli wave which starts upstream of the body. To minimize the influence of
this effect the inflow boundary is put further upstream than for the conventional case. For the
submerged sphere at Fn 0.35 and 0.40 the inflow boundary is located one diameter upstream
36
Verification
of the body while for the submerged sphere at Fn 0.45 the inflow boundary is located two
diameters upstream of the body. A further increase in the upstream distance for Fn 0.35
and 0.40 may improve the convergence. Another solution is to use a more slender submerged
body with a smaller volume such that a strong Bernoulli wave is avoided.
The last explanation of the poor convergence of the wave resistance determined by pressure
integration may be found in the way the the influence coefficients are calculated. The calculation of the influence coefficient between two panels depends on their relative distance. A
panel within close range is treated exact (flat panel with constant source strength), while a
mid range panel is approximated by four point sources in the panel corner points and a far
field panel is treated as a single point source. This approximation is done to reduce the computing time and costs. Refinement of the panels on the submerged body and the free surface
will lead to a rapid increase in the number of far field panels. The error in this approximation
is small for a single panel but it may have a lot more influence when it is summed over a
large number of panels (which happens when the pressure is integrated). This may explain a
decrease in the accuracy of the results when the grid is refined, but it does not explain why
the results are so poor for the submerged sphere at Fn 0.35 and 0.40 while the results for the
submerged sphere at Fn 0.45, on a similar grid, are satisfactory.
The wave resistance determined by wave cut analysis converges for all the three cases but the
calculated numerical order of this method shows some strange values. For Froude numbers
0.35 and 0.45 the numerical order is lower than the theoretical value while for Fn 0.40 the
numerical order is higher than the theoretical value.
The high value of the numerical order for Fn 0.40 may be caused by the coarse grids which
are outside the asymptotical range. The numerical order becomes closer to the theoretical
value when the two coarse grids are removed from the least squares extrapolation method.
The low numerical order of the wave cut analysis method for Fn 0.35 and 0.45 is probably
caused by an inaccuracy in one or more of the steps taken to determine the wave resistance.
The basics of the wave cut analysis are explained in section 4.4 and in reference [19] and
consist of the following steps: a spline through the control points, a redistribution of points
on the spline, a discrete fourier transform, a least squares approximation and a summation.
The numerical order of these steps is unknown and may deteriorate the numerical accuracy
of the calculations.
5.3
Delft 1
Verification of the wave resistance for a sailing yacht hull which is free to trim and sink is not
possible. As explained in section 5.2, the change in resistance, when the grid is refined, is a
combination of a numerical error and a change in attitude (trim and sink). Another reason
which makes verification impossible is the requirement that the solutions need to be in the
asymptotic range. One of the assumptions used in the derivation of the non linear free surface
potential flow method is that the free surface can be described by a single valued function.
This assumption makes it impossible for the method to find a solution when wave breaking
5.3 Delft 1
37
is present. The single valued free surface assumption limits the grid refinement because there
are always regions close to the hull where the waves are breaking. This leads to the following
problem for the verification: a solution can be found on a coarse grid, but this solution is
probably not in the asymptotic range, while for a fine grid in the asymptotic range no solution
can be found due to wave breaking.
Experience with the method has shown that a minimum of 20 to 25 panels per wavelength are
necessary to get a solution which gives a good representation of the large wave scales. The
number of waves per unit of length depends on the Froude number and can be calculated by
the following formula.
n=
1
2πF n2
(5.10)
This test case will investigate the influence of the most important parameters on the results
within the practical range of 20 to 30 panels per wavelength.
5.3.1
Objectives
The objective of this test case is to investigate the influence of the different parameters on the
wave resistance. The parameters which will be investigated are: the number of hull panels,
the convergence criteria and the number of free surface panels.
5.3.2
Description and results of the test case
Hull nr 1 of the DSYHS, which is the parent hull form of the first part of the series, will be
used for the investigation. The investigation will start with the calculation of the potential
flow around the hull without free surface, the double model. Then the free surface is included
and the influence of convergence criteria and free surface panels are investigated. To keep
things clear the results are given immediately after the description of the tests.
Panel distribution on the hull
A way to judge the quality of the panel distribution on the hull is to calculate the resistance
without a free surface, the double body resistance. This double body resistance should be zero
since it is symmetric and does not generate lift but the calculated resistance value will not
be zero because of the discretization error. The increase in panels should give a decrease in
the discretization error and thus a decrease in the resistance. Second order panels, parabolic
panels with a linear source distribution, are used to represent the hull. The advantage of the
second order panels is that less panels are needed, compared to first order panels, to give an
accurate representation of the hull. 16 different panel distributions are tested: all possible
combinations of 30, 60, 90 and 120 panels in longitudinal direction and 5, 10, 15 and 20 panels
in transverse direction. The results can be found in table 5.12.
38
Verification
Figure 5.6: Example of two different panel distributions on the hull of Delft 1, 30 by 5 panels (top)
and 90 by 15 panels (bottom)
Table 5.12: Drag area of Delft 1 for different number of hull panels
Cd · LS2 · 106
nr of longitudinal panels
30
60
90
120
nr of transverse panels
5
10
15
20
4.253
6.698 7.970 8.679
0.5426
2.314 3.191 3.585
-0.5394 1.188 1.775 2.077
-1.077 0.7495 1.172 1.328
5.3 Delft 1
39
The mesh of 60 by 10 panels is used to investigate the influence of stretching the mesh in
longitudinal direction. Since the highest pressure is found in the bow and stern regions it is
expected that refining the panels in these areas will improve the quality of the grid. Stretching
is done by specifying the size of the first and last panel as percentage of the length. All 16
combinations of 0.4, 0.8, 1.2 and 1.6 percent are calculated and compared.
Figure 5.7: Example of stretched and uniform panel distribution on the hull of Delft 1: stretched on
the left and uniform on the right
Table 5.13: Drag area of Delft 1 for different longitudinal stretching of the hull panels
Cd · LS2 · 106
last panel in % of Lwl
0.4
0.8
1.2
1.6
first
0.4
2.909
2.674
2.820
2.972
panel in % of
0.8
1.2
2.735 2.524
2.616 2.302
2.514 2.328
2.540 3.092
Lwl
1.6
2.568
2.126
2.121
2.979
The drag area values of the sailing yacht hull are low but not as close to zero as for the
submerged sphere. The mesh is not symmetric and generated by the mesh generator which
may cause a small error. When the 4 results on the diagonal in table 5.12 are compared it
40
Verification
shows that the drag area decreases for systematic refinement of the mesh, but the decrease is
relatively slow.
The results of the stretched grid do not show a clear trend. All the values are in the same
range and it is not possible to determine the best grid based on these results. This is caused
by the fact that the total number of panels is constant and a refinement of the panels in bow
and stern will lead to coarse panels in the midship region. (see figure 5.7) So the improve in
the bow and stern is cancelled by a decrease in accuracy in the midship region.
Convergence criteria
In this section the influence of the convergence criteria on the non linear potential flow solution
will be investigated. This is done first because if there is a convergence error it will show up
in all other calculations. A number of free surface calculations are done in which the hull is
free to trim and sink just as during the towing tank tests.
There are three different convergence criteria in the non linear calculations: change of wave
height, change of trim and change of sink. To investigate the influence of the convergence
criteria three calculations are done: one with the default setting, one with a strict setting
which is an order smaller than the default values and one with a medium setting ( see table
5.14). This is done for the individual criteria and for a combination of all criteria. The
calculations are done on the default free surface domain of 0.5 ship length upstream, 1 ship
length downstream of the stern and the width of the free surface is 0.7 ship length. The hull
is discretized using 60 by 10 second order panels. Longitudinal stretching of the hull panels is
applied with a first and last panel length of 0.8% of the water line length. The free surface has
a panel distribution of 25 panels per wavelength in the longitudinal direction. The number
of panels in the transverse direction is chosen such that the aspect ration of the free surface
panels is around 1. The (full scale) wave resistance determined by pressure integration is used
to compare the results. The investigation is done for the Froude numbers 0.30, 0.40 and 0.50.
Table 5.14: Convergence criteria used during the investigation
convergence tolerance
waveheight
L
trim angle
sink
L
default
5 · 10−5
1 · 10−2
1 · 10−5
medium
1 · 10−5
5 · 10−3
5 · 10−6
strict
5 · 10−6
1 · 10−3
1 · 10−6
The results in tables 5.15, 5.16 and 5.17 show that the influence of more strict convergence
criteria is less than 0.1% of the calculated wave resistance. The conclusion of the investigation
is that the convergence error, using the default settings, is negligible.
5.3 Delft 1
41
Table 5.15: Wave resistance of Delft 1 at Fn 0.30 for different convergence criteria
wave resistance (N)
convergence setting default medium
wave height
258.64
258.50
trim angle
258.64
258.64
sink
258.64
258.57
combination
258.64
258.50
strict
258.45
258.64
258.57
258.45
wave resistance (N)
wave height
1736.3
1737.1
trim angle
1736.3
1736.3
sink
1736.3
1737.1
combination
1736.3
1737.0
strict
1737.1
1736.7
1737.1
1737.1
wave resistance (N)
wave height
5282.8
5284.6
trim angle
5282.8
5282.8
sink
5282.8
5284.6
combination
5282.8
5284.8
strict
5284.5
5284.8
5284.6
5284.7
42
Verification
Panel distribution on the free surface
The influence of the free surface panels in combination with the hull panels is investigated.
Four different panel distributions on the hull are used which are combined with three or four
different panel distributions on the free surface. The number of free surface panels is in the
range of 20 to 30 panels per wavelength. The number of transverse panels on the free surface
is chosen such that the aspect ratio stays reasonable. Six different speeds are tested from
Fn 0.30 to Fn 0.55. The width and downstream boundary of the free surface domain are
extended such that both pressure integration and wave cut analysis can be used to determine
the wave resistance. An example of the coarse and fine free surface mesh for Fn 0.40 can be
seen in figure 5.8. The results of the calculations are shown in figures 5.9 to 5.14.
Figure 5.8: Example of 2 free surface meshes for Delft 1 at Fn 0.40: fine at the top and coarse at
the bottom
5.3 Delft 1
43
144
Rw wavecut analysis (N)
Rw pressure integration (N)
320
300
280
260
240
22
24
26
28
nr of panels per wavelength
142
140
138
134
22
30
0.15
30 x 5
60 x 10
90 x 15
120 x 20
136
24
26
28
30
24
26
28
30
−0.0406
−0.0408
−0.041
sink (m)
trim (deg)
0.145
0.14
−0.0412
−0.0414
0.135
−0.0416
0.13
22
24
26
28
30
−0.0418
22
Figure 5.9: Results for Delft 1 at Fn 0.30
44
Verification
660
640
620
600
580
20
25
30
350
345
340
335
0.28
−0.0555
0.275
−0.056
0.27
−0.0565
0.265
0.26
0.255
20
30 x 5
60 x 10
90 x 15
120 x 20
330
325
20
35
sink (m)
trim (deg)
355
680
25
30
35
25
30
35
−0.057
−0.0575
25
30
35
−0.058
20
5.3 Delft 1
45
1330
1900
1850
1800
1750
1700
1650
20
25
30
1290
25
30
35
25
30
35
−0.074
sink (m)
trim (deg)
1300
−0.073
30 x 5
60 x 10
90 x 15
120 x 20
0.72
0.7
0.68
20
1310
1280
20
35
0.76
0.74
1320
−0.075
−0.076
−0.077
25
30
35
−0.078
20
46
Verification
3700
3650
3600
3550
3500
3450
3400
22
trim (deg)
2800
24
26
28
30
2780
2760
2740
30 x 5
60 x 10
90 x 15
120 x 20
2720
2700
22
32
1.82
−0.092
1.8
−0.093
24
26
28
30
32
24
26
28
30
32
1.78
sink (m)
3750
1.76
−0.094
−0.095
1.74
−0.096
1.72
1.7
22
24
26
28
30
32
−0.097
22
5.3 Delft 1
47
3600
5600
5400
5200
5000
4800
20
25
30
3520
3500
25
30
35
25
30
35
−0.101
sink (m)
trim (deg)
3540
−0.1
30 x 5
60 x 10
90 x 15
120 x 20
3
2.9
2.8
2.7
20
3560
3480
20
35
3.2
3.1
3580
−0.102
−0.103
25
30
35
20
48
Verification
4100
6600
6500
6400
6300
6200
6100
26
28
30
32
34
3950
30 x 5
60 x 10
90 x 15
120 x 20
3900
28
30
32
34
36
28
30
32
34
36
−0.1
3.95
−0.102
3.9
sink (m)
trim (deg)
4000
3850
26
36
4
3.85
−0.104
−0.106
3.8
3.75
26
4050
28
30
32
34
36
−0.108
26
5.3 Delft 1
5.3.3
49
Explanation of the results
Influence of the hull panel distribution
The panel distribution on the hull has a lot of influence on the wave resistance determined
by pressure integration. The drag area of the double body does not show a clear trend (see
table 5.13) but the calculations with free surface do show that an increase in the number of
panels on the hull gives a more accurate wave resistance value. The wave resistance, trim
and sink are all calculated by pressure integration and the results show that for an increase
in the number of panels the results get closer together. The difference in the values of the
wave resistance, trim and sink of the two finest panel distributions on the hull are so small
that a further increase in the number of panels on the hull will not give a significant change
in the results.
The panel distribution on the hull has almost no influence on the wave resistance determined
by wave cut analysis. Apparently even the coarse grid gives a good enough representation
of the hull to get the same wave pattern behind the hull. The value of the wave resistance
determined by wave cut analysis is a lot lower than the pressure integration value.
Influence of the free surface panel distribution
The free surface panel distribution has a significant influence on all the results (resistance,
trim and sink). This makes it difficult to explain the results because wave resistance, trim
and sink all change at the same time.
The first observation is that wave resistance determined by pressure integration and by wave
cut analysis show a different behavior when the number of free surface panels is increased.
This is caused by the fact that the strongest non linear effects (wave interactions) happen
close to the hull in the bow region and the length scale of these effects is a lot smaller than
the length scale of the global wave pattern.
Because the strong non linear effects happen close to the hull they will have a lot of influence
on the wave resistance determined by pressure integration. The influence of the strong local
non linear effects disappears further away from the ship when they interact with the global
wave pattern and will have a small influence on the wave resistance determined by wavecut
analysis.
A refinement of the free surface panels will have more influence on the local non linear effects
close to the hull than on the global wave pattern behind the hull as can be seen in figures
5.15 and 5.16. This leads to different trends in the wave resistance values for the pressure
integration and the wave cut analysis method.
The second observation is that the trend of the resistance, trim and sink is independent of
the number of panels on the hull. Although coarse hull panels give different values compared
with fine hull panels the trend is the same. Apparently the quality of the coarse grid is good
50
Verification
Figure 5.15: Longitudinal wavecut at y/L = 0.3 for Delft 1 at Fn 0.40, the green line is the result
on the coarse grid and the red line is the result on the fine grid
enough to show the resistance trend but not the exact value.
5.4
Conclusions of the verification
The quality of the panel distribution on the body should not be judged by using only the
double model resistance. The symmetrical mesh on the submerged sphere combined with
the anti symmetrical source strength gives a resistance value close to zero independent of the
number of panels used to represent the body. The double model resistance of the sailing yacht
hull does not give a clear indication of the quality of the mesh: all values are of the same
order.
The numerical order of the non linear free surface potential flow around a submerged sphere
differs from the theoretical value. The error in the wave resistance values determined by
pressure integration and wave cut analysis does not follow the assumed Taylor series expansion which is caused by the fact that the error in the wave resistance values is not a pure
discretization error but a combination of a discretization error and other small errors. In
practice it is difficult to fulfill all the requirements of the extrapolation procedures.
The default convergence criteria for a non linear free surface potential flow calculation around
a hull free to trim and sink are strict enough to reduce the iterative convergence error such
that it becomes negligible.
Verification of the resistance calculated by the non linear free surface potential flow around a
5.4 Conclusions of the verification
51
Figure 5.16: Wavepattern of Delft 1 at Fn 0.40 coarse grid at the bottom and fine grid at the top
52
Verification
hull free to trim and sink is not possible. The wave resistance, trim and sink all depend on the
panel distribution on the hull and free surface. In general it can be said that an increase in
the number of panels on the hull will give more accurate results. An increase in the number of
panels on the free surface will give a better representation of the small wave components but
if the panels are refined too much iterative convergence problems occur (the solution blows
up). These iterative convergence problems are caused by wave breaking which can not be
captured by the non linear free surface potential flow due to the assumption that the free
surface can be described by a single valued function.
Even if verification of the non linear potential flow solver is not possible it will still be a useful
tool for (yacht) designers if it predicts the correct resistance trend when different hulls are
compared. This will be investigated in the next chapter. If the predicted resistance trends
differ a lot from the measured values than it will be necessary to use a more advanced flow
model such as Reynolds averaged Navier Stokes (RANS). RANS codes already exist but are
too computationally intensive to be used on a regular basis which would be needed for a trend
validation.
Chapter 6
Validation
6.1
Basics of validation
Validation is the process of determining the modelling error of the result of a simulation. As
stated in [20] the error between the truth and the result of a simulation consists of two parts:
a numerical error(δSN ) and a modelling error (δSM ). The numerical error is assessed by the
verification process and the modelling error will be assessed using a validation process.
Validation is done by comparing data (D) with the results from a simulation (S) which leads to
the comparison error (E). The uncertainty of the validation (U V ) is the sum of the uncertainty
in the data (UD ) and the numerical uncertainty in the simulation(U SN ).
E = D − S = δD − (δSM + δSN )
(6.1)
2
2
+ USN
UV2 = UD
(6.2)
Somebody who uses a simulation will have requirements about the uncertainty (U reqd ) which,
when combined with |E| and UV , leads to six possible situations:
|E|
|E|
Ureqd
UV
UV
Ureqd
< UV
< Ureqd
< |E|
< |E|
< Ureqd
< UV
< Ureqd
< UV
< UV
< Uregd
< |E|
< |E|
For the first three situations |E| < U V which means that validation is achieved. This does not
mean that the simulation is useful because the uncertainty may be too high. Somebody who
54
Validation
uses a simulation is mainly interested in situations one, two and four in which |E| < U reqd
because then the simulation error is smaller than required.
In case |E| > UV the validation is unsuccessful because the simulation error is larger than
the uncertainty level of the validation. This simply means that the model is inappropriate for
the simulation. Although the model is not suitable for the simulation it can be used to get a
better understanding of the modelling error. Especially for the situation in which U V << |E|
because then the simulation error is directly related to the modelling error and the simulation
error can be used to correct / improve the model.
E ≈ −δSM
6.1.1
(6.3)
Error and uncertainty in the DSYHS towing tank data
The experimental data of the Delft towing tank, which is used for the validation, is mainly
based on single tests. Some tests were repeated to check the accuracy and repeatability.
According to the test facility the accuracy of the total resistance measurements is within 2%
The total resistance values of the Delft towing tank tests are postprocessed according to the
ITTC 1957 method to get the residuary resistance values (see section 3.4). The Reynolds
number used in equation 3.7 to calculate the frictional resistance coefficient is based on a
reference length of 0.7 · Lwl instead of Lwl which is normally used. This frictional resistance
coefficient formula is based on a best fit to flat plate resistance measurements. The uncertainty
in the coefficient can be estimated using the scatter in the original data. According to Stern
[12] the uncertainty can be estimated at 7% of C F in the range 105 < Re < 1010 .
The percentages seem to be reasonable low but this changes when the actual values are
computed. The average ratio of total resistance to residuary resistance on model scale goes
from around 5.5 for the lowest Fn (0.25) to around 1.4 for the highest Fn (0.55). This means
that for the low Froude numbers even a small error in the total or frictional resistance will
lead to a lot of uncertainty in the residuary resistance values.
The frictional resistance formula 3.7 is based on flat plate resistance measurements which
already include some form drag. The wetted surface of the flat plate is constant and independent of the speed. During the postprocessing of towing tank results the frictional resistance
is determined using this formula and the reference area (wetted surface area at zero speed).
The actual wetted surface area will differ from the reference area and this leads to an error in
the frictional resistance. The error will probably be small at low speeds and increase with the
Fn. Just as for the CF value, a small error in the wetted area will cause a significant change
in the residuary resistance. There is no information available which can be used to estimate
this error and the uncertainty in the error. This makes it impossible to give a qualitative
value of the uncertainty in the residuary resistance.
6.2 Trend validation
6.1.2
55
Error and uncertainty in SHIPFLOW results
The results of the verification show that it is not possible to determine the numerical error of
the method for a sailing yacht which is free to trim and sink. The figures in chapter 5 show
that there is a dependency of the results on the free surface panel distribution. The average
resistance of the different free surface panel distributions (using the finest hull panels) and
the maximum deviation of this average are calculated and given in table 6.1. This does not
tell anything about the numerical accuracy or numerical uncertainty but it does show that
the difference between different free surface grids in the practical range is relatively small.
Table 6.1: Average results (from the verification procedure) and maximum deviation for Delft 1
Fn
0.30
0.35
0.40
0.45
0.50
0.55
6.2
Rwpi (N)
245.2 ± 0.37%
597.6 ± 0.30%
1711 ± 2.86%
3507 ± 2.20%
5142 ± 4.75%
6288 ± 1.42%
Rwwc (N)
138.6 ± 2.52%
338.4 ± 3.45%
1302 ± 1.65%
2746 ± 1.55%
3533 ± 1.37%
3980 ± 2.46%
trim (◦ )
0.136 ± 2.30%
0.261 ± 1.98%
0.710 ± 6.08%
1.752 ± 2.02%
2.925 ± 7.32%
3.873 ± 2.17%
sink (m)
−0.0409 ± 0.08%
−0.0566 ± 0.59%
−0.0754 ± 1.66%
−0.0940 ± 0.89%
−0.1009 ± 0.23%
−0.1018 ± 2.07%
Trend validation
Validation of the resistance values is not possible because there is not enough information
to determine the numerical uncertainty of the calculations and the uncertainty in the data
but it is possible to see if SHIPFLOW gives the same trend as the towing tank tests. If the
predicted trend is the same as for the towing tank tests than the program will be useful in
the decision making process.
The trend validation is done for the non linear free surface potential flow and is based on the
upright bare hull towing tank results of the Delft series. The calculations are done with the
hull free to trim and sink just as during the towing tank tests. This will probably lead to a
difference in trim and sink between the measurements and the simulation but this difference
is regarded as a part of the modelling error in the simulation.
The numerical error in the simulation can not be determined as shown in the previous section.
A change in the numerical error between different hulls may give a false trend which will lead
to the wrong conclusions. The assumption used in this report is that the influence of the
numerical error will be of the same order when the panel distribution for the different cases is
equal. This assumption can not be proven but some explanations which show it is a reasonable
assumption are given here.
The panel distribution on the hull is important to give an accurate representation of the hull.
In chapter 5 it is shown that a relatively coarse panel distribution on the hull is good enough
to get an accurate wave cut resistance value. The number of panels on the hull is mainly
56
Validation
determined by the need for accurate pressure integration resistance values. By using the same
number of panels and the same distribution (stretching) the same accuracy will be achieved
for different hulls.
The free surface panel distribution has a strong influence on the results. The size of the free
surface panels is directly connected to the wavelength that can be resolved on the mesh. On
a fine grid more wave details can be resolved which should give a more accurate result. A
problem with the free surface is that no solution is found when the panel size becomes too
small. This is caused by the basic assumptions which are used to derive the non linear free
surface potential flow. The free surface panel distribution needs to be fine enough to capture
the most important waves but not too fine such that iterative convergence problems are
avoided. By using the same panel distribution on the free surface, the same wave components
can be represented which means that the resolution of the solutions will be equal. Since the
resolution is equal for the different cases it will be possible to compare the different results.
6.3
Description of the cases used in the trend validation
This section describes the cases which are used in the trend validation. The number of panels
on the hull is 60 by 10 and the size of the first and last panel is 0.8% of the water line length.
Figures 5.9 to 5.14 show that there still is a difference between the pressure integration results
using this panel distribution and the finer ones but this is not very important since the results
are used for trend validation. Results from finer panel distributions may be more accurate
but the computational time will increases a lot and this makes the panel distribution with
600 panels the most suitable.
The number of free surface panels in the longitudinal direction is determined as 25 panels
per wavelength. This is enough to get a good resolution of the main wave components as
shown by Kim [23] and avoids iterative convergence problems. The number of panels in the
transverse direction is chosen such that the aspect ratio of the panels is reasonable (between
0.5 and 2).
The upstream boundary is 0.5 times the waterline length in front of the bow and this is
sufficient since there are no waves in front of the ship. The other dimensions of the free
surface are determined by the requirements of the wave cut analysis. The wave cut analysis
uses 8 wave cuts which are distributed over 1 fundamental wavelength. The first wave cut
is located 40% of the water line length behind the stern. To avoid boundary effects the free
surface is extended 40% of the water line length behind the last wave cut. The total length
behind the stern is thus 0.8 times the waterline length plus one fundamental wavelength. The
fundamental wavelength depends on the Fn and can be calculated with equation 5.8 The free
surface need to be wide enough that the waves leave the free surface through the outflow
boundary. The Kelvin wave angle, the angle between the free surface wave pattern and the
center line of the ship, is independent of the speed and equal to 19.5 degrees. To make sure
that the waves will leave the outflow boundary the width of the free surface is chosen such that
the angle between centerline and a line from bow to widest point on the outflow boundary is
23 degrees. The size of free surface domain for the different Fn can be found in table 6.2.
6.4 Comparison of the CFD and the towing tank results
57
Table 6.2: Size of the free surface domain for the different Fn, the still water line length is used as
reference
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
upstream
0.5
0.5
0.5
0.5
0.5
0.5
0.5
midship
1
1
1
1
1
1
1
downstream
1.19
1.37
1.57
1.81
2.07
2.37
2.70
width
0.93
1.00
1.09
1.19
1.30
1.42
1.56
Default values for the convergence criteria have been used because they are strict enough to
avoid convergence errors. The actual input files used for the computations can be found in
appendix A
6.4
Comparison of the CFD and the towing tank results
The results of the CFD are compared with the towing tank measurements. The full scale
wave or residuary resistance values are used for the comparison. To see if the CFD predicts
the same wave resistance trend as the towing tank test the results are presented in figures
6.1 to 6.7. To make interpretation of the results easier the hulls are ordered such that the
measured wave resistance increases from left to right. The trim and sink are also compared
and the figures can be found in appendix B
200
180
Rtow
Rwpi
Rwwc
160
140
Rw (N)
120
100
80
60
40
20
0
Figure 6.1: Comparison of the wave resistance for Fn 0.25, the hulls are ordered such that the towing
tank resistance is increasing from left to right
58
Validation
600
Rtow
Rwpi
Rwwc
500
Rw (N)
400
300
200
100
0
1200
Rtow
Rwpi
Rwwc
1000
Rw (N)
800
600
400
200
0
6.4 Comparison of the CFD and the towing tank results
59
3000
Rtow
Rwpi
Rwwc
2500
Rw (N)
2000
1500
1000
500
0
7000
Rtow
Rwpi
Rwwc
6000
5000
Rw (N)
4000
3000
2000
1000
0
60
Validation
14000
Rtow
Rwpi
Rwwc
12000
10000
Rw (N)
8000
6000
4000
2000
0
18000
16000
Rtow
Rwpi
Rwwc
14000
Rw (N)
12000
10000
8000
6000
4000
2000
0
6.5 Explanation of the results of the comparison
61
The calculated wave resistance does not always give the same trend as the towing tank results.
Especially for the low Froude numbers (0.25 and 0.30) there is a lot of difference between the
towing tank results and the CFD calculations. For the higher Froude numbers the trends are
better predicted but not exactly the same. For the highest Froude numbers (0.50 and 0.55)
the difference between the towing tank resistance and the CFD values becomes larger.
6.5
Explanation of the results of the comparison
The results of comparison seem to improve with increasing Froude number. This can be
partly explained by the fact that the low speed towing tank results are more sensitive for
measurement errors than the high speeds results. The largest part of the difference is the
modelling error caused by the basic assumptions of the potential flow solution. The towing
tank results are calculated by subtracting the frictional resistance, computed according to the
ITTC 57 formula, from the measured values. This is a standard method to postprocess the
measurements but this method does not take the viscous pressure resistance into account.
The viscous pressure resistance will be different for each hull and this leads to a difference
in the trend predicted by CFD compared to the towing tank measurements. The viscous
pressure resistance is usually a couple of percent of the frictional resistance and its influence
will decrease with increasing speed because at higher speeds the ratio of frictional resistance
to wave resistance decreases.
A second reason for the difference in the predicted trend is that the influence of the viscous
effects on the trim is not accounted for. The influence of the frictional resistance on the trim
is assumed to be small because it acts close to the center of gravity and thus generates only a
small trimming moment. The Reynolds number during the towing tank tests is lower than at
full scale and this will lead to an increased influence of the frictional resistance on the trim.
Also when the boundary layer leaves the hull at the stern it forms a wake behind the ship.
The viscous effects in the wake will dampen the waves close to the stern which will change
the trim. Both these effects are not present in the non linear free surface potential flow. The
difference in trim leads to a slightly different shape of the under water part of the hull. The
difference in shape will lead to a small difference in wave resistance but the relative difference
may be considerable especially for the low Froude numbers.
The fact that for the high speeds the calculated resistance is lower than the measured resistance is partly caused by wave breaking and spray during the towing tank test. These effects
increase the resistance but can not be modelled in the non linear free surface potential flow.
6.6
Conclusions of the trend validation
The non linear free surface potential flow has problems to predict the same trend as the
towing tank results for the low Froude numbers. For the higher Froude numbers the prediction
improves but still shows differences compared to the towing tank results. Part of the difference
can be explained by the fact that the residuary resistance determined in the towing tank is
62
Validation
very sensitive to measurement errors at low Froude numbers but most of the difference is
caused by the modelling error in the non linear free surface potential flow. The towing tank
results contain both wave resistance and viscous pressure resistance. The viscous pressure
resistance and the influence of the viscous effects on the trim are not accounted for in the non
linear potential flow. The influence of these effects on the residuary resistance at low speeds
is significant but decreases at the higher speeds.
Chapter 7
Determination of the correction
equations
As can be seen in section 6.4 the non linear free surface potential flow code has problems to
predict the same resistance trend as measured in the towing tank. In this chapter correction
equations will be derived to improve the resistance prediction by the non linear free surface
potential flow code. These correction equations will be based on the general hull characteristics such that they can be used for sailing yacht hulls similar to those in the Delft series. The
determination of the SHIPFLOW correction formulas is done using linear regression analysis
which has also been used by Harries [25] and Lee [26]. First the basics of linear regression
are described and after this the most common regression strategies are explained. Then the
actual linear regression of the results is done to get the correction equations. These equations
are tested on two different hulls which are not included in the regression analysis. The last
part of this chapter gives the conclusions of the results of the regression analysis.
7.1
Basics of linear regression
Linear regression is a data analysis technique which is useful to get an idea about the relation
between the variables xi (predictors) and the result y (response). Linear regression (see
equation 7.1) uses a linear regression equation (with respect to the parameters) to describe
the relation between the predictors and the response. The parameters (b i ) in the equation
are determined by minimizing the error or residual () using a least squares method.
ŷ = b0 + b1 x1 + b2 x2 + b3 x3 + (7.1)
The simplest approximation of the response would be to use only a constant which results in
the mean of the response. (see equation 7.2)
64
Determination of the correction equations
ȳ =
n
1X
yi
n i=1
(7.2)
The standard deviation about the mean gives an idea about the variance in the data.
v
u
u
s=t
n
1 X
(yi − y)2
n − 1 i=1
(7.3)
The goal of the regression analysis is to find the regression equation which gives the best description of the data. The quality of the regression formula can be judged using the percentage
of variance explained (R 2 ) value.
Pn
i=1
ŷi − yi
R2 = 1 − P n
i=1
yi − ȳ
2
2
(7.4)
The R2 value compares the variance about the regression with the variance about the mean.
The simplest regression (the mean) leads to R 2 = 0 while the perfect regression will give
R2 = 1. A problem with using R 2 to judge the regression is that it always increases when
the number of predictors used in the regression is increased. This may lead to complicated
regression equations with too many predictors. A better way to judge the quality of the
regression is to use the adjusted R 2 value. The calculation of the adjusted R 2 is similar
to R2 only the variance is divided by the number of degrees of freedom used in the model
compared to the total number of degrees of freedom. This gives a decrease in the R 2 value if
a complicated model is used. A lot more information about regression analysis can be found
in [24].
Pn
i=1
adjusted R2 = 1 − P
n
i=1
7.2
ŷi − yi
2
yi − ȳ
/d.o.f.model
2
(7.5)
/d.o.f.total
Regression strategies
There are different regression strategies and the basic four are described here. The simplest
strategy is to do all regressions which are possible with a given set of predictors. This
method uses all possible regressions starting with one predictor and increasing the number
of predictors until all predictors are included. This leads to a large number of regression
equations which all need to be judged to determine the most suitable one. Since this method
determines all possible regressions it will always find the best regression possible with the
7.3 Regression of the CFD results
65
given set of predictors. The major drawback of the method is its inefficiency because all
regressions have to be calculated and analyzed.
Regression using a forward selection procedure. This method starts with a constant and then
searches for the predictor which describes most of the variance. This predictor is included
in the regression equation and the search is repeated for the next predictor. The procedure
stops when the user defined maximum number of predictors or the desired explanation of the
variance is reached. This method is efficient but a disadvantage is that after predictors are
included they will always stay in the regression equation even if they do not contribute much
anymore.
Regression using a backward elimination procedure. This method starts with a regression
equation using all possible predictors. After the coefficients are determined the method checks
the influence of the different predictors and then removes the predictor with the least influence
from the regression equation. This is repeated until the user defined explanation of the
variance is reached. A disadvantage of the method is that it starts with all predictors and
this makes it less efficient than the forward selection method.
Stepwise regression with both forward selection and backward elimination. This procedure
uses the strong points of both previous methods: it first does a forward selection step and
then a backward elimination step. This sequence is repeated until all the predictors in the
equation have the desired influence level (α). The advantage of the method over forward
selection is that it removes predictors which have lost their influence and this keeps the
regression equation as compact as possible.
7.3
Regression of the CFD results
The regression is done using the stepwise regression with forward selection and backward
elimination because this is the most efficient regression method. The influence level (α) for
entering or removing predictors from the regression equation have been tested and α values
of 0.15 and 0.20 have been used. This leads to regression equations which are usually the
same but when there is a difference the regression equation with the largest adjusted R 2 is
used. These α values give regression equations which explain most of the variation but are
still strict enough to avoid complicated models and over fitting. The important decision in
regression analysis is the selection of the response and the predictors which will be treated
next.
7.3.1
Selection of the response
Different options exist for the response but here is chosen for the correction factor k (see
equations 7.6 and 7.7). The advantage of using this correction factor as response is that it is
dimensionless and can be easily applied by future users to correct SHIPFLOW results.
66
7.3.2
kpi =
Rtow
Rwpi
(7.6)
kwc =
Rtow
Rwwc
(7.7)
Selection of the predictors
The main problem in regression analysis is to select the predictors which will be used. In
the ideal situation the variation of parameters within an experiment is designed such that
the results can be used for regression analysis. This is unfortunately not the case for the
DSYHS. The first 21 hulls are varied systematically but the newer hulls are based on different
parent models to keep up with the trends in yacht design. Also not all of the 50 models
have been tested in the bare hull upright condition. This leads to a set of hulls which is not
ideal for regression analysis. The predictors used in this regression analysis are the general
T
∇1/3
∇2/3
characteristics of the sailing yacht hull: B
L , B , Lcb, Lcf, L , Cp, Cb, S and Awc. Not only
the linear combination but also all second order combinations are included to make sure that
higher order variations in the response can be captured by the regression equation.
An inspection of the calculated kpi and kwc values shows that the correction values k for
B
hulls with a large B
T ratio are far from the mean value. The variation in the T ratio of the
hull is not equally spaced since not all 50 hulls from the DSYHS are tested in the bare hull
upright condition. The couple of hulls with a B
T ratio larger than 10 will have a more than
proportional influence on the result of the regression and are thus removed from the data.
This decreases the range in which the correction formulas are valid but will have little effect in
practice since these hulls are not very common in actual sailing yacht design. The correction
equations are valid for the range of principal hull parameters as given in table 7.1.
Table 7.1: Range of the principal hull parameters for which kpi and kwc are valid
Length - Beam ratio
Beam - Draft ratio
Length - Displacement ratio
Longitudinal centre of buoyancy
Longitudinal centre of flotation
Prismatic Coefficient
Loading factor
7.3.3
L
B
B
T
L
∇1/3
Lcb
Lcf
Cp
Aw
∇2/3
2.73
2.46
4.34
0.0 %
-1.8 %
0.52
3.78
to
to
to
to
to
to
to
5.00
6.94
7.52
-7.9 %
-9.1%
0.60
7.58
Regression of kpi
Before the results of the regression are shown first the mean and standard deviation of k pi are
shown for the different Froude numbers. As can be seen in table 7.2 the standard deviation
67
around the mean is relatively large for the low Froude numbers and decreases for the higher
Froude numbers. The standard deviation around the mean is too high to use the mean value
as correction factor. The regression equations will approximate the correction factor and this
reduces the variation. The results of the linear regression are presented in so called ’analysis
of variance’ (ANOVA) tables. The coefficients of the regression equation and the standard
error in the coefficient are given as well. These give a good indication of the quality and
reliability of the regression formulas.
Table 7.2: Mean values and standard deviation of kpi for the different Froude numbers
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
mean value
0.930
0.931
0.934
1.004
1.130
1.329
1.485
standard deviation (s)
0.371
0.243
0.185
0.120
0.107
0.104
0.126
The regression equation for kpi at Fn 0.25
kpi = 3.84 − 101 · Cb ·
∇2/3
∇2/3
+ 52.8 · Lcb ·
− 4.08 · Lcb · Lcf
Sc
S
s = 0.195, R2 = 75.4%, adjusted R2 = 72.4%
Table 7.3: Regression coefficients and standard error of the coefficients for k pi at Fn 0.25
predictor
constant
2/3
Cb · ∇S
2/3
Lcb · ∇S
Lcb · Lcf
coefficient
3.8440
−100.91
52.796
−4.078
standard error
0.7450
11.83
6.966
1.872
Table 7.4: ANOVA table for the regression of kpi at Fn 0.25
source
regression
residual
total
degrees of freedom
3
25
28
sum of squares
2.9004
0.9479
3.8483
mean square
0.9668
0.0379
F
25.50
(7.8)
68
kpi = 4.08 − 59.7 · Cb ·
∇2/3
T ∇1/3
+ 12.5 · ·
S
B
L
(7.9)
s = 0.130, R2 = 73.1%, adjusted R2 = 71.1%
predictor
constant
2/3
Cb · ∇S
T ∇1/3
B · L
coefficient
4.0843
−59.744
12.454
standard error
0.3711
7.293
2.436
source
regression
residual
total
degrees of freedom
2
27
29
sum of squares
1.2463
0.4596
1.7059
mean square
0.6231
0.0170
F
36.61
kpi = 2.82 − 19.1 · Cb ·
∇1/3
∇2/3
− 6.24 · Lcf ·
S
L
(7.10)
s = 0.084, R2 = 81.0%, adjusted R2 = 79.6%
predictor
constant
2/3
Cb · ∇S
1/3
Lcf · ∇L
coefficient
2.8162
−19.126
−6.243
standard error
0.1776
4.786
2.451
source
regression
residual
total
degrees of freedom
2
27
29
sum of squares
0.8069
0.1897
0.9966
mean square
0.4034
0.0070
F
57.41
69
∇1/3
∇2/3
kpi = 2.414 − 9.59 · Lcf ·
− 7.60 · Cp ·
+ 8.40 ·
L
S
∇1/3
L
!2
(7.11)
s = 0.047, R2 = 86.3%, adjusted R2 = 84.7%
predictor
constant
1/3
Lcf · ∇L
2/3
Cp · ∇S
1/3
( ∇L )2
coefficient
2.4136
−9.591
−7.602
8.401
standard error
0.1927
2.499
2.177
4.276
source
regression
residual
total
degrees of freedom
3
26
29
sum of squares
0.3627
0.0575
0.4202
mean square
0.1209
0.0022
F
54.67
kpi = 2.71 − 14.0 · Cp ·
T
∇2/3
− 1.53 · Lcb · Lcf + 2.19 ·
S
L
(7.12)
s = 0.043, R2 = 85.7%, adjusted R2 = 84.1%
predictor
constant
2/3
Cp · ∇S
Lcb · Lcf
T
L
coefficient
2.7108
−13.978
−1.5262
2.194
standard error
0.2050
2.069
0.4622
1.100
70
source
regression
residual
total
degrees of freedom
3
26
29
sum of squares
0.2858
0.0476
0.3334
mean square
0.0953
0.0018
F
52.01
kpi = 3.98 − 13.0 · Cb ·
∇2/3
B
B
− 4.12 · Lcb + 5.54 · · Lcf − 3.36 · · Cp
S
L
L
(7.13)
s = 0.062, R2 = 70.4%, adjusted R2 = 64.8%
predictor
constant
2/3
Cb · ∇S
Lcb
B
L · Lcf
B
L · Cp
coefficient
3.9754
−12.955
−4.1163
5.536
−3.359
standard error
0.5043
2.259
0.9653
2.076
1.665
source
regression
residual
total
degrees of freedom
4
21
25
sum of squares
0.1902
0.0799
0.2701
mean square
0.0475
0.0038
F
12.49
kpi = 2.67 + 15.6 ·
B
L
2
− 11.0 ·
s = 0.048, R2 = 87.3%, adjusted R2 = 85.3%
∇1/3
B
· Lcb − 10.9 · · Lcf
L
L
(7.14)
71
predictor
constant
2
(B
L)
Lcb · Lcf
T
L
coefficient
2.6708
15.589
−11.027
10.940
standard error
0.2068
1.918
2.010
2.191
source
regression
residual
total
7.3.4
degrees of freedom
3
19
22
sum of squares
0.3053
0.0444
0.3497
mean square
0.1018
0.0023
F
43.56
Regression of kwc
Before the results of the regression are shown first the mean and standard deviation of k wc are
shown for the different Froude numbers (see table 7.17). The wave resistance values calculated
with wave cut analysis are in general lower than the pressure integration values which results
in a higher mean value of the correction factor k wc . The variation in kwc is also higher than
the variation of the correction factor for the pressure integration.
Table 7.17: Mean values and standard deviation of kwc for the different Froude numbers
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
mean value
1.972
1.667
1.595
1.374
1.457
1.881
2.413
standard deviation
1.014
0.551
0.400
0.249
0.189
0.252
0.389
The regression equation for kwc at Fn 0.25
kwc = 11.5 − 72.0 ·
∇1/3
· Cb + 89.7 ·
L
s = 0.317, R2 = 91.3%, adjusted R2 = 90.2%
B
L
2
− 107 ·
B
· Cb
L
(7.15)
72
Table 7.18: Regression coefficients and standard error of the coefficients for k wc at Fn 0.25
predictor
constant
∇1/3
L · Cb
2
(B
L)
B
L · Cb
coefficient
11.4692
−72.04
89.662
−107.004
standard error
0.7672
11.45
5.774
9.522
Table 7.19: ANOVA table for the regression of kwc at Fn 0.25
source
regression
residual
total
degrees of freedom
3
25
28
sum of squares
26.2671
2.5064
28.7735
mean square
8.7557
0.1003
F
87.33
kwc = 23.3 − 147 ·
∇2/3
T
∇1/3 ∇2/3
· Lcf + 33.9 · − 18.9 · Awc + 67.0 ·
·
S
L
L
S
(7.16)
s = 0.227, R2 = 85.3%, adjusted R2 = 83.0%
predictor
constant
∇2/3
S · Lcf
T
L
Awc
2/3
· ∇S
∇1/3
L
coefficient
23.271
−146.91
33.91
−18.932
67.03
standard error
2.337
13.84
10.36
2.824
36.01
source
regression
residual
total
degrees of freedom
4
25
29
sum of squares
7.5063
1.2905
8.7968
mean square
1.8766
0.0516
F
36.35
73
kwc = 11.0 − 60.5 ·
∇2/3
∇1/3 ∇2/3
· Lcb − 13.6 · Lcf · Awc + 30.6 ·
·
S
L
S
(7.17)
s = 0.147, R2 = 87.9%, adjusted R2 = 86.5%
predictor
constant
∇2/3
S · Lcb
Lcf · Awc
∇1/3 ∇2/3
L · S
coefficient
10.9578
−60.453
−13.577
30.64
standard error
0.8866
8.598
2.228
12.63
source
regression
residual
total
degrees of freedom
3
26
29
sum of squares
4.0798
0.5611
4.6409
mean square
1.3599
0.0216
F
63.01
kwc = 3.59 − 67.8 ·
∇1/3
B ∇2/3
∇1/3
· Lcb + 29.8 · ·
+ 31.5 ·
· Lcf
L
L
S
L
(7.18)
s = 0.111, R2 = 82.1%, adjusted R2 = 80.1%
predictor
constant
∇1/3
L · Lcb
B ∇2/3
L · S
∇1/3
L · Lcf
coefficient
3.5915
−67.76
29.805
31.48
standard error
0.2836
11.29
5.809
10.88
74
source
regression
residual
total
degrees of freedom
3
26
29
sum of squares
1.4734
0.3202
1.7936
mean square
0.4911
0.0123
F
39.88
kwc = −25.0+83.8·
∇2/3
B
∇2/3
+2.10· ·Lcf −168·Cp·
+24.3·Cp+45.0·Awc−80.6·Lcb·Awc+44.7·Lcb 2
S
L
S
(7.19)
s = 0.059, R2 = 92.5%, adjusted R2 = 90.1%
predictor
constant
B
L
∇2/3
S
· Lcf
2/3
Cp · ∇S
Cp
Awc
Lcb · Awc
Lcb2
coefficient
−24.986
83.82
2.1028
−167.60
24.337
45.03
−80.65
44.66
standard error
7.152
30.83
0.6727
56.00
9.639
13.52
26.49
17.16
source
regression
residual
total
degrees of freedom
7
22
29
sum of squares
0.9614
0.0778
1.0393
mean square
0.1374
0.0035
F
38.82
kwc = 4.82 − 9.50 · Lcb · Lcf + 66.6 ·
s = 0.102, R2 = 86.2%, adjusted R2 = 83.6%
B
L
2
− 133 ·
T
B ∇1/3
·
+ 24.4 ·
L
L
L
(7.20)
75
predictor
constant
Lcb · Lcf
2
(B
L)
1/3
B ∇
L · L
T
L
coefficient
4.8200
−9.499
66.64
−133.28
24.419
standard error
0.4453
1.380
10.33
22.42
5.228
source
regression
residual
total
degrees of freedom
4
21
25
sum of squares
1.3651
0.2186
1.5837
mean square
0.3413
0.0104
F
32.79
kwc
T
= 5.96 − 9.55 · Lcf · Cb + 6.83 · · Lcf + 25.5 ·
B
B
L
2
− 44.3 · Lcf ·
∇1/3
L
(7.21)
s = 0.113, R2 = 93.1%, adjusted R2 = 91.6%
predictor
constant
coefficient
5.9590
−9.546
6.829
25.478
−44.29
standard error
0.5789
3.256
3.712
5.892
13.51
source
regression
residual
total
degrees of freedom
4
18
22
sum of squares
3.1066
0.2303
3.3369
mean square
0.7766
0.0128
F
60.71
76
7.4
Test of the correction equations
The regression equations are tested by comparing the predicted correction factors with the
actual correction factors. Two hulls are used to test the regression: an older one and a
modern one which are randomly selected and not included in the regression analysis. The
two hulls are nr 4 and nr 43 of the Delft series. Tables 7.32, 7.34, 7.33 and 7.35 give the
actual correction factor and the values calculated according to the regression equations. The
resistance prediction improvement of the correction equations can be seen in figures 7.1 and
7.2 which show the residuary resistance from the towing tank, the calculated wave resistance
using both wave cut analysis and pressure integration and the predicted resistance using the
correction equations.
Table 7.32: Test of kpi for Delft 4
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
actual kpi
1.198
1.030
0.977
1.004
1.117
1.343
1.509
regression kpi
1.115
1.019
0.990
1.010
1.105
1.310
1.490
Table 7.33: Test of kwc for Delft 4
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
actual kwc
2.514
1.786
1.688
1.290
1.424
1.917
2.480
regression kwc
2.415
1.728
1.683
1.358
1.362
1.911
2.456
7.4 Test of the correction equations
77
8000
Rtow
Rpi
Rpic
7000
Rwc
Rwcc
6000
Resistance (N)
5000
4000
3000
2000
1000
0
0.25
0.3
0.35
0.4
0.45
Fn
0.5
0.55
Figure 7.1: Test of the correction equations for Delft 4
Table 7.34: Test of kpi for Delft 43
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
actual kpi
0.758
0.886
0.954
1.031
1.209
1.494
1.602
regression kpi
0.829
0.881
0.960
1.021
1.190
1.444
1.599
0.6
0.65
78
Table 7.35: Test of kwc for Delft 43
Fn
0.25
0.30
0.35
0.40
0.45
0.50
0.55
actual kwc
1.520
1.785
1.945
1.653
1.662
2.084
2.490
regression kwc
2.224
1.887
1.844
1.541
1.541
2.338
2.703
9000
Rtow
Rpi
Rpic
8000
Rwc
Rwcc
7000
Resistance (N)
6000
5000
4000
3000
2000
1000
0
0.25
0.3
0.35
0.4
0.45
Fn
0.5
0.55
0.6
0.65
Figure 7.2: Test of the correction equations for Delft 43
7.5
Conclusions about the correction equations
The regression for the correction factor of the pressure integration fits better than the one for
the correction factor of the wave cut analysis. As can be seen by comparing table 7.2 and table
7.17 the standard deviation in the wave cut results is a lot larger than the standard deviation
in the pressure integration results. The correction formulas determined by regression can
reduce this but not eliminate it entirely. This causes that the correction formulas for the
wave cut results still have a larger standard deviation compared to the correction formulas
for the pressure integration results.
7.5 Conclusions about the correction equations
79
The quality of the regression equations clearly improves when the Froude number increases.
As shown in the previous chapter the trend of the calculated wave resistance becomes better
with increasing Froude number. This leads to smaller variation in the correction factors
which can be better approximated by the regression equation. The standard deviation in
the correction factors after regression shows that the correction formulas for the low Froude
numbers (0.25, 0.30 and 0.35) are not very accurate. Apparently it is not possible to get
an accurate description of the modelling error at low speeds with a formula based on the
characteristic hull parameters. It should be noted however that the wave resistance at low
speeds is only a small part of the total resistance. The correction factors at the higher speeds
can be described quite accurately by formulas based on the main hull characteristics.
An important note about the results of the regression is that the correction equations are
only valid when the following requirements are satisfied. The main characteristics of the
hull has to lie within the range given in table 7.1. The panel distribution on hull and free
surface should be the same as the one used in this report (see appendix A) with the hull free
to trim and sink. The main restriction of the correction equations is that they do not give
the absolute wave resistance value but a residuary resistance value which approximates the
DSYHS results. The residuary resistance of the DSYHS is determined using the ITTC 1957
friction formula (with a Reynolds number based on 70% of the water line length). This ITTC
1957 friction formula does not take the viscous pressure resistance (form factor) into account.
80
Chapter 8
Conclusions and recommendations
8.1
Conclusions
Validation of the non linear free surface potential flow code SHIPFLOW is not possible because
the error and uncertainty in both the Delft towing tank results and the CFD calculations can
not be estimated accurate enough. The first reason is that the total resistance measurements
in the towing tank are quite accurate but the ratio of total resistance to residuary resistance
will lead to an increase of the error in the residuary resistance at low Froude numbers. The
influence of the post processing on the error and the uncertainty in the error estimate can not
be determined. The second reason is that the numerical accuracy of the non linear potential
flow can not be determined because not only the wave resistance but also the trim and sink
change when the free surface panels are refined. Also the maximum refinement of the free
surface panels is limited by the assumption that the free surface can be described by a single
valued function. These two reasons make it impossible to use extrapolation techniques to
determine the numerical error in the CFD results.
The submerged sphere test case shows that the wave resistance determined by pressure integration does not always converge. The wave resistance determined by wave cut analysis
converges but the order of the method differs from the theoretical value. This is caused by
the fact that the error in the resistance does not follow the assumed Taylor series expansion.
The error is not a pure discretization error but consists of different small errors and together
these do not behave according to the assumption. In theory Richardson extrapolation should
work but in practice it is difficult to satisfy all the requirements.
The trend validation shows that the resistance trend predicted by the non linear potential
flow is reasonable for the higher Froude numbers but not very accurate for the low Froude
numbers. This is mainly caused by the fact that the residuary resistance from the towing tank
measurements still contains some viscous pressure resistance while the non linear free surface
potential flow calculates the wave resistance without any viscous effects. Other reasons are
differences in trim and sink and the fact that at low Froude numbers the residuary resistance
from the towing tank is relatively sensitive to measurement errors. The level of the calculated
82
Conclusions and recommendations
resistance at higher Froude numbers is lower than the level of the measured values because
the non linear free surface potential flow can not model spray and breaking waves. The wave
resistance determined by pressure integration predicts the resistance trend better than the
wave resistance determined by the wave cut analysis. The pressure integration method works
here because the shape of a sailing yacht hull is very smooth. The wave resistance determined
by wave cut analysis is more robust and less dependent on the panel distribution on the hull
which is why it will work better for more complicated hull shapes.
The correction factors can be approximated by functions of the characteristic hull parameters.
These correction equations are relatively accurate for the higher Froude numbers but less
accurate for the low Froude numbers. The correction equations can reduce the difference
between the CFD calculations and towing tank measurement but will not explain it entirely.
One should be aware of the restrictions when these equations are used to correct the results
of the non linear free surface potential flow calculation.
8.2
Recommendations
The first recommendation is related to the mesh generator. In the submerged sphere test case
it is shown that the mesh generator has a small problem to create the correct mesh close to
the stagnation point of the sphere. This problem does not occur for the sailing yacht hull but
it is something which should be investigated further. Another problem of the mesh generator
is related to the meshing of heeled hulls. The mesh generator uses the centerline of the hull
to divide the mesh into two parts and this works fine as long as then the centerline stays
under water when heeled. The centerline of wide light displacement hulls leaves the water
when heeled and this causes a meshing problem. This problem can be solved by dividing the
mesh differently and this approach works fine for the linear free surface calculations but does
not work yet for the non linear calculations.
Another recommendation is to continue the cooperation of Chalmers with Delft university
and do a trend validation for the rest of the Delft series. Especially the prediction of the
lift and induced drag of hulls with keel and rudder combined with the non linear free surface
effects will be interesting for sailing yacht designers.
The last recommendation is to repeat this validation when more advanced flow solvers become
available. The two main problems encountered during the validation are the fact that the
non linear free surface potential solver is not able to capture all the important physics and
that it is not possible to separate the viscous resistance components from the measured total
resistance in the towing tank. When a more advanced flow solver such as RANS is used to
calculate the flow at model scale it will capture more of the physics (viscous effects, wave
breaking and spray) and will eliminate the uncertainty caused by the post processing of the
towing tank results.
References
[1] White, F.M., ”Viscous fluid flow”, McGraw-Hill international editions, 1991.
[2] Gerritsma, J., Moeyes, G. and Onnink, R., ”Test results of a systematic yacht hull
series.”, 5th HISWA Symposium on Yacht Architecture, 1977.
[3] Gerritsma, J., Keuning, J.A. and Versluis, A., ”Geometry, resistance and stability of the
Delft systematic yacht hull series”, Delft University of Technology, 1981.
[4] Gerritsma, J. and Onnink, R., ”The experimental results of the Delft systematic yacht
hull series.”, Delft University of Technology, 1983.
[5] Gerritsma, J., ”Calculation of sailing yacht resistance”,Delft University of Technology,
1983.
[6] Gerritsma, J. and Keuning, J.A., ”Experimental analysis of five keel-hull combinations”,
Delft University of Technology, 1984.
[7] Gerritsma, J. and Keuning, J.A., ”Further experiments with keel − hull combinations”,
Delft University of Technology, 1985.
[8] Gerritsma, J., Keuning, J.A. and Versluis, A., ”Upright resistance of sailing yacht hull
forms”, Delft University of Technology, 1989.
[9] Gerritsma, J., Keuning, J.A. and Onnink, R., ”Sailing yacht performance in calm water and in waves”, 12th International Symposium on Yacht Design and Construction,
HISWA, 1992.
[10] Keuning, J.A. and Sonnenberg, U.B., ”Developments in the velocity prediction based on
the Delft systematic yacht hull series”, Delft University of Technology, 1998.
[11] Keuning, J.A. and Sonnenberg, U.B., ”Approximation of the calm water resistance on
a sailing yacht based on the Delft systematic yacht hull series”, Delft University of
Technology, 1999.
[12] Longo, J. and Stern, F., ”Uncertainty assessment for towing tank tests with example for
surface combatant DTMB model 5415”, Journal of Ship Research, Vol. 49, No. 1, March
2005, pp. 55 − 68.
84
REFERENCES
[13] Fasardi, C., ”Tank testing and data analysis techniques for the assessment of sailboat
hydrodynamic characteristics”, High Performance Yacht Conference, Auckland, 2002.
[14] Brown, M., Campbell, I. and Robinson, J., ”The accuracy and repeatability of tank testing, from experience of ACC yacht development”, High Performance Yacht Conference,
Auckland, 2002.
[15] Flowtech International AB, ”Shipflow user manual”, Gothenburg, Sweden, 1999.
[16] Raven, H.C., ”A solution method for the nonlinear ship wave resistance problem”, PhD
Thesis, MARIN/Delft University of Technology, Netherlands, 1996.
[17] Katz, J. and Plotkin, A., ”Low speed aerodynamics”, Cambridge University Press, 2001.
[18] Janson, C.E., ”Potential flow panel methods for the calculation of free surface flows with
lift”, Chalmers University of Technology, 1997
[19] Janson, C.E. and Spinney, D., ”A comparison of four wave cut analysis methods for wave
resistance prediction”, Ship Technology Research Vol. 51, 2004.
[20] ITTC, ”CFD General, Uncertainty Analysis in CFD, Verification and Validation,
Methodology and Procedures”, ITTC Quality Manual 7.5 − 03 − 01 − 01, 2002.
[21] Eça, L. and Hoeksta M., ”An evaluation of verification procedures for CFD applications”,
24th Symposium on naval hydrodynamics, Fukuoka, Japan, 2002.
[22] Eça, L. and Hoeksta M., ”Workshop on CFD Uncertainty Analysis”, Lisbon, Spain, 2004.
[23] Kim, K., Janson, C.E. and Larsson, L., ”Nonlinear calculations of the free surface potential flow around ships with special hull configurations”, 2nd International Conference
on Hydrodynamics, Hong Kong, 1996.
[24] Draper, N. R. and Smith, H., ”Applied regression analysis”, John Wiley and Sons, 1998
[25] Harries, S. and Schulze, D., ”Numerical investigation of a systematic model series for
the design of fast monohulls”, 4th international conference on fast sea transportation,
Sydney, 1997
[26] Lee, Yeon-Seung, ”Trend Validation of CFD Prediction Results for Ship Design (Based
on Series 60)”, PhD Thesis, Berlin University of Technology, Germany, 2003.
Appendix A
SHIPFLOW input files
The SHIPFLOW input file for Fn 0.25
xflow
titl(
prog(
hull(
offs(
titl="Sailing yacht, Delft No 23" )
xmesh, xpan )
yacht, fsflow)
file = "off_delft23.txt", lpp = 10.00, ysign = -1,
xori = 0.541, zori = 0.704)
vshi( fn = [0.250], rn = [0.0] )
end
xmesh
body( grno = 1, offsetgroup = "hull", expa = 2, station = 61, point = 11,
str2 = 5, df2 = 0.008, dl2 = 0.008 )
free( grno = 2, nbd2 = 1, ibd2 = [1], y4side = -0.93, xdow = 2.19,
point = 28, str1 = 1, df1 = 0.04, stau = 25, stam = 65, stad = 77,
stru = 1, dlu = 0.0157 )
\\
\\ the use of a transom group depends on the hull shape
\\
tran( grno = 3, nbd1 = 1, ibd1 = [1], point = 3, stad = 77 )
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
86
xflow
titl(
prog(
hull(
offs(
xmesh, xpan )
yacht, fsflow)
xori = 0.541, zori = 0.704)
vshi( fn = [0.30], rn = [0.0] )
end
xmesh
str2 = 5, df2 = 0.008, dl2 = 0.008 )
stru = 1, dlu = 0.0226 )
\\
\\
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
87
xflow
titl(
prog(
hull(
offs(
xmesh, xpan )
yacht, fsflow)
xori = 0.541, zori = 0.704, ztem = 0.05, itte = 5)
vshi( fn = [0.35], rn = [0.0] )
end
xmesh
body( grno = 1, offsetgroup = "hull", expa = 2,
station = 61, point = 11, str2 = 5, df2 = 0.008, dl2 = 0.008 )
stru = 1, dlu = 0.0308 )
\\
\\
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
88
xflow
titl( titl="Sailing yacht, Delft No 23" )
prog( xmesh, xpan )
hull( yacht, fsflow)
offs( file = "off_delft23.txt", lpp = 10.00,
ysign = -1,
vshi( fn = [0.40], rn = [0.0] )
end
xmesh
str2 = 5, df2 = 0.008, dl2 = 0.008 )
stru = 1, dlu = 0.0402 )
\\
\\
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
89
xflow
titl(
prog(
hull(
offs(
xmesh, xpan )
yacht, fsflow)
vshi( fn = [0.45], rn = [0.0] )
end
xmesh
str2 = 5, df2 = 0.008, dl2 = 0.008 )
point = 24, str1 = 1, df1 = 0.04, stau = 11, stam = 21, stad = 42)
\\
\\
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
90
xflow
titl(
prog(
hull(
offs(
xmesh, xpan )
yacht, fsflow)
vshi( fn = [0.50], rn = [0.0] )
end
xmesh
str2 = 5, df2 = 0.008, dl2 = 0.008 )
\\
\\
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
91
xflow
titl(
prog(
hull(
offs(
xmesh, xpan )
yacht, fsflow)
vshi( fn = [0.55], rn = [0.0] )
end
xmesh
str2 = 5, df2 = 0.008, dl2 = 0.008 )
\\
\\
end
xpan
cont( nonl, free)
iter( maxit = 20)
twcut( on)
end
92
Appendix B
Comparison of the trim and sink of
the CFD calculations with the
towing tank results
Comparison of the trim and sink of the CFD calculations with the towing tank
94
results
0.5
tow
cfd
0.4
0.3
trim (deg)
0.2
0.1
0
−0.1
−0.2
−0.3
Figure B.1: Comparison of the trim for Fn 0.25
0.6
tow
cfd
0.5
0.4
0.3
trim (deg)
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
95
0.8
0.6
trim (deg)
0.4
0.2
0
−0.2
−0.4
tow
cfd
−0.6
1.4
tow
cfd
1.2
1
0.8
trim (deg)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
96
results
3.5
3
2.5
trim (deg)
2
1.5
1
0.5
0
tow
cfd
−0.5
5
4.5
4
3.5
trim (deg)
3
2.5
2
1.5
1
0.5
tow
cfd
0
97
6
5
trim (deg)
4
3
2
tow
cfd
1
0
−0.005
tow
cfd
−0.01
−0.015
sink (m)
−0.02
−0.025
−0.03
−0.035
−0.04
Figure B.8: Comparison of the sink for Fn 0.25
98
results
0
tow
cfd
−0.01
−0.02
sink (m)
−0.03
−0.04
−0.05
−0.06
−0.07
−0.01
tow
cfd
−0.02
−0.03
sink (m)
−0.04
−0.05
−0.06
−0.07
−0.08
−0.09
−0.1
99
0
tow
cfd
−0.02
−0.04
sink (m)
−0.06
−0.08
−0.1
−0.12
−0.14
−0.02
tow
cfd
−0.04
−0.06
sink (m)
−0.08
−0.1
−0.12
−0.14
−0.16
−0.18
−0.2
100
results
0
tow
cfd
−0.05
sink (m)
−0.1
−0.15
−0.2
−0.25
−0.02
tow
cfd
−0.04
−0.06
−0.08
sink (m)
−0.1
−0.12
−0.14
−0.16
−0.18
−0.2
−0.22

Trend validation of SHIPFLOW based on the bare hull upright

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