Trend Validation of CFD Prediction Results

for Ship Design

(Based on Series 60)

vorgelegt von

YEON-SEUNG LEE, MSc.

aus Berlin

von der Fakultät V — Verkehrs- und Maschinensysteme

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

—Dr.-Ing.—

genehmigte Dissertation

Promotionsausschuß:

Vorsitzender: Prof. Dr.-Ing. W. Nitsche

Berichter: Prof. Dr.-Ing. Dr.h.c. H. Nowacki

Berichter: Prof. Dr.-Ing. G. Clauss

Tag der wissenschaftlichen Aussprache: 12. Mai 2003

Berlin 2003

D 83

Acknowledgement This product, my doctorate thesis, has been accomplished in Institute of Naval Architecture, Marine and Ocean Engineering of Technical University of Berlin.

Raised in a big traditional Korean family, I had suffered from a lonely life in a foreign country for 7 years. I however, gained a precious present through the hard times. First, I could be more independent in my life and my academic world as well. Moreover, It was confirmed that I had a tremendous support network, and they always would be with me.

Prof. Dr.-Ing. Kim, Soo-Young who led me to the world of Naval Architecture and gave me a chance to study in Berlin University is forever my supporter. Whenever I am in trouble or get lost, he shows the solution and leads me in right way.

I can’t say my 7 years in Berlin without mentioning him, my supervisor Prof. Dr.-Ing. Dr.h.c. Horst Nowacki. He taught me the modest and respectful attitude towards study. I appreciate and admire his passion and patience for teaching me. I should like to thank Prof. Dr.-Ing. G.. Clauss and Prof. Dr.-Ing. W. Nitsche who examined my thesis.

In addition, I must express my gratitude for Dr. Min, CEO in Hyundai Heavy Industries where I work. His unbelievable passion for R&D and unconditional affection toward researchers encouraged me to finish my thesis successfully. His advice is an excellent map for me to understand and to get along with social relationship as well. Through the long time with him, his existence becomes almost fatherly. His passionate attitude for life is what I want to keep in my mind forever.

I thank especially Wegener family of Rotary club, Klaus, Marion, Arne and grandmother Thea. They became my new ‘family’ with great love and hospitality for me. Thanks to them, I could experience the German culture deeply. The moments with the Wegener family are my most beautiful memories that make me smile whenever I recall them. How could I forget the day when Mr.Wegener taught me how to ride a bicycle all day long, in spite of his uncomfortable leg?

The next appreciation with my whole heart must be directed to my parents and family. My parents, who are very strong-minded and supportive of their children, have inspired me to go ahead whenever I wanted to give up. While I was in Berlin, my parents sent me letters often. The lovely letters were like vitamins that made me cheer up! I also send my sincere gratitude to my sisters and brother. They are my best friends.

I would like to appreciate all the people I have studied and worked with in the research institute, Dr. Stefan Harries, Frauke Baumgaertel, Justus Heimann, and Bernd-Leopold Kaether. I can’t forget Stefan Lange and Andreas Hoffmeister who were my friends and German language partners to answer my SOS. I thank Korean students in Berlin too. We sometimes shared Korean foods and consoled our homesickness together.

My favorite place while studying in Berlin was the riverside nearby the laboratory. I used to drown my grief and anxiety, lying on the grass of the riverside by myself. I remember clearly how much I liked the magically beautiful blue sky, even though the sky made me cry sometimes. Now, I am looking up at the blue sky again, recalling the old times in Berlin. I hope to meet a young woman, by the name of Yeon-Seung Lee, who did her best to be an independent person. I miss her so much.

i

Trend Validation of CFD Prediction Results for Ship Design Based on Series 60 Abstract This study deals with the issue of the difference existing between predictions in ship hull form wave resistance derived from experiments (EFD data) and those based on the results of state of the art CFD systems. It concentrates on the question whether any observed errors between measured results and numerical predictions are primarily systematic functions of the hull form parameters or contain random errors to any significant degree. Thereby the study intends to make a contribution to the issue of the reliability of CFD results on wave-making resistance for ship design purposes relative to experimental data (EFD results).

This problem was addressed here by means of a specific sample of hull shapes taken from a systematic test series, Series 60, and compared with numerical predictions from the numerical solver SHIPFLOW. The study was confined to the limited subset of Series 60 (CB from 0.6 to 0.7, lcb from -1.5% to 0.5% of LBP, L/B from 6.5 to 7.5 and B/T from 2.5 to 3.5) in order to investigate the trends in a physically coherent domain. The Series 60 results were obtained from the documented experimental data for full scale ships of standard length of 400 feet and were reinterpolated for a model size of L=20 feet by reverse application of the ATTC method in Series 60 to obtain the experimental values of CR and CW using the Froude and Hughes methods respectively. These data were inherently fair and free of random errors because the interpolation and presentation in the Series 60 documents had already removed any existing noise. This does not imply that the experimental data are error free, but any remaining error trends are systematic. The results obtained by computational fluid dynamics are thoroughly verified for their consistency. This was done by systematic sensitivity studies on the body panel number, free surface panel resolution, truncation domain and iterative convergence until the computational parameters were all set at such a level that further refinements did not produce relevant changes in excess of very small tolerances. These data are called and used as verified CFD data. The numerical and computational errors were investigated systematically over the whole range of the Series 60 subset by evaluating 58 distinct hull forms covering the domain of the free hull form parameters of Series 60 as densely as possible in practice. This yields a prediction of CWCFD for each individual hull form at various FN. The differences between experimental and computational results denoted by ER und EW can thus be obtained.

The principal study concerns the dependence of two error function ER und EW on the four variables in the hull form definition, as in Series 60, and separately also on ship speed (FN). This analysis which was newly conceived in this work is called trend validation. It was performed by the methodology of regression analysis being applied to the whole sample data volume of the error function vs. the free design variables at any given FN. The evaluation of the data by stepwise, multivariate regression analysis showed very clear, significant trends in the whole sample. The errors therefore show a clear and simple dependence on the design variables. The errors are generally not of a negligible magnitude because they comprise the cumulative effects of many modeling and procedural approximations. But they are systematic enough to allow direct conclusions for design purposes. The observation of a low noise level or small random error influences in the regression analysis of systematic series CFD results, validated against a systematic experimental series, is encouraging regarding the reliability of current state of the art CFD systems in ship design.

ii

Trendvalidierung für Ergebnisse von CFD-Prognosen für den Schiffsentwurf, basierend auf Series 60

Zusammenfassung Diese Arbeit thematisiert den existierenden Unterschied zwischen zwei Kategorien von Vorhersagen bezüglich des Wellenwiderstands von Schiffsformen: zwischen den auf der einen Seite aus experimentellen (EFD) Daten abgeleiteteten und den auf der anderen Seite auf den Resultaten aktueller CFD-Systeme basierenden Prognosen. Die Untersuchung konzentriert sich auf die Frage, ob die durch den Vergleich von Messdaten mit numerischen Prognosen festgestellten Fehler vorrangig systematische Abhängigkeiten von den Formparametern des Schiffsrumpfs aufweisen oder Zufallsfehler in signifikanter Größenordnung enthalten. So liegt die Intention der Arbeit darin, eingegrenzt auf die Thematik des Wellenwiderstands als Problem des Schiffsentwurfs, einen Beitrag zum Problemfeld der Zuverlässigkeit von CFD-Resultaten im Vergleich zu experimentell gewonnenen (EFD) Daten zu leisten.

Die Herangehensweise an die Problematik stützt sich auf eine spezifische Auswahl von Rumpfformen, die einer systematischen Testserie, der Series 60, entnommen und mit numerischen Prognosen des numerischen Solvers SHIPFLOW verglichen wurde.

Die Arbeit beschränkt sich auf einen bestimmten Bereich der Series 60 (CB von 0.6 bis 0.7, lcb von -1.5% bis 0.5% von LBP, L/B von 6.5 bis 7.5 und B/T von 2.5 bis 3.5), um die Trends in einem physikalisch zusammenhängenden Bereich untersuchen zu können.

Die Resultate der Series 60 wurden den dokumentierten Experimentaldaten für Schiffe in Originalgröße mit einer Standardlänge von 400 Fuß entnommen. Anschliessend wurden diese Resultate für eine Modellgröße von L=20 Fuß reinterpoliert, indem die ATTC-Methode der Series 60 umgekehrt angewendet wurde, so dass die Experimentalwerte von CR und CW mittels der Methoden von Froude und Hughes ermittelt wurden. Diese Daten waren geglättet und frei von Zufallsfehlern, da in den Dokumenten der Series 60 bereits jegliches Rauschen durch Interpolation und Präsentation eliminiert worden war. Dies impliziert nicht, dass die experimentellen Daten fehlerfrei sind, aber alle verbleibenden Fehlertrends sind systematisch. Die durch CFD erhaltenen Ergebnisse wurden gründlich hinsichtlich ihrer Konsistenz verifiziert. Dies erfolgte mittels systematischer Sensitivitäts-Untersuchungen bezüglich der Anzahl der Panele, der Auflösung der freien Wasseroberfläche, des Abbruchbereichs und der iterativen Konvergenz, bis die Berechnungsparameter eine Größenordnung erreicht hatten, dass weitere Verfeinerungen keine relevanten Änderungen jenseits sehr kleiner Toleranzen bewirkten. Diese Daten werden als verifizierte CFD-Daten bezeichnet bzw. verwendet. Die numerischen und rechnerischen Fehler wurden systematisch für den gesamten Bereich der zugrunde gelegten Teilmenge von Series 60 untersucht, indem 58 unterschiedliche Rumpfformen evaluiert wurden, die den Bereich der freien Rumpfformparameter der Series 60 in der Praxis so dicht wie möglich abdecken. Dies ergibt für jede individuelle Rumpfform eine Prognose von CWCFD für unterschiedliche FN. So erhält man die Differenzen zwischen den experimentellen und den rechnerischen Resultaten (bezeichnet durch ER und EW). Der Hauptteil der Arbeit befasst sich mit der Abhängigkeit der zwei Fehlerfunktionen ER und EW von den vier Variablen der Rumpfformdefinition, wie in der Series 60, und zusätzlich noch von der Schiffsgeschwindigkeit

iii

(FN). Dieser Analysetyp wurde für die vorliegende Arbeit neu konzipiert und wird als Trendvalidierung bezeichnet. Die Durchführung der Analyse erfolgte unter Anwendung der Methode der Regressionsanalyse auf das gesamte Datensample für die Fehlerfunktionen in Abhängigkeit von den freien Entwurfsvariablen für jede gegebene FN.

Die Evaluation der Daten mittels schrittweiser, multivariater Regressionsanalyse ergab sehr klare, signifikante Trends im gesamten Sample. Die Fehler weisen deshalb eine klare und einfache Abhängigkeit von den Entwurfsvariablen auf. Die Größenordnung der Fehler ist im allgemeinen nicht vernachlässigbar, da sie die kumulierten Effekte vieler modellbezogener und prozeduraler Approximationen beinhalten. Sie sind jedoch hinreichend systematisch, um direkte Folgerungen für Entwurfszwecke zu erlauben. Ein niedriger Rauschpegel beziehungsweise ein geringer Einfluss durch Zufallsfehler in der Regressionsanalyse der systematischen Serie von CFD-Ergebnissen konnte beobachtet werden, was gegen eine systematische experimentelle Serie validiert wurde. Dies wirkt ermutigend bezüglich der Zuverlässigkeit aktueller, den neuesten Stand der Technik repräsentierender CFD-Systeme für den Schiffsentwurf.

Contents

List of Tables 4

List of Figures 6

List of Symbols 9

1 Introduction 13

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 State of the art in validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Methodology 17

2.1 Overview and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Problem analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Validation state variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Trend validation concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Error types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Methods and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 SHIPFLOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.2 Series 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.3 Trend validation domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1

3 Verification 51

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Verification methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.1 Numerical parameters and verification range . . . . . . . . . . . . . . . . . . . . . 52

3.2.2 Criteria for error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Panel convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Effect of the body panel distribution on the CWCFD prediction . . . . . . . . . . . 58

3.3.2 Design of the panel sensitivity test for free surface panels . . . . . . . . . . . . . . 62

3.3.3 Panel convergence tendencies and error estimation (case 1) . . . . . . . . . . . . . 65

3.3.4 Panel error dependency on the change of flow phenomena (Case 2) . . . . . . . . . 71

3.3.5 Panel error dependency according to the change of the hull form variation (case 3

and 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4 Computational domain convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4.1 Design of a sensitivity test of the computational domain . . . . . . . . . . . . . . . 82

3.4.2 Computational domain dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4.3 Estimation of the computational domain convergence error . . . . . . . . . . . . . 84

3.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5 Iterative Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.1 Tendency of iterative convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.2 Sensitivity of iterative convergence to numerical parameters . . . . . . . . . . . . . 90

3.5.3 Iterative tendency to the change of the ship form . . . . . . . . . . . . . . . . . . . 92

3.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6 Numerical error and contribution of each error type . . . . . . . . . . . . . . . . . . . . . 95

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4 Trend Validation 98

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2 Experimental data for trend validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2.1 Series 60 test results[101][102][103] . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2.2 Interpolation of Series 60 test results in the validation domain . . . . . . . . . . . . 100

4.2.3 Qualitative error investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3 Discussion of form parameter effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2

4.3.2 Univariate form parameter effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3.3 Bivariate form parameter effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.3.4 Trivariate form parameter effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.3.5 Significant results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.4 Systematic error trends analysis by regression analysis . . . . . . . . . . . . . . . . . . . . 136

4.4.1 Approach to regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.4.2 Regression analysis and error functions[95][96] . . . . . . . . . . . . . . . . . . . . . 138

4.4.3 Graphical presentation and discussion of results . . . . . . . . . . . . . . . . . . . . 143

4.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5 Summary and Conclusions 148

Bibliography 151

3

List of Tables

2.1 An overview of characteristics of each error type . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Overview of sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Principal characteristics of Series 60 parent models . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Ranges of form parameters of Series 60 parents . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Variation of lcb as a function of CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 The bounds and values of each form parameter . . . . . . . . . . . . . . . . . . . . . . . . 45

2.7 The list of ship variations from raw data and as supplemented by interpolation . . . . . . 47

2.8 The list of ship variations from raw data and as supplemented by interpolation . . . . . . 48

3.1 Ranges of sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 CPXI(×104) according to the change of the panel size on the body . . . . . . . . . . . . . 60

3.3 Variation of the panel distribution by means of the stretching function . . . . . . . . . . . 60

3.4 CWCFD for different body panelizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Total number of panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Predictions of CWCFD and their extrapolation for the Ship No.1 at FN=0.25 . . . . . . . 68

3.7 Differences of predictions related to the panel variations . . . . . . . . . . . . . . . . . . . 74

3.8 Range of the computational domain convergence in percent of CWCFD for XY1 . . . . . . 84

3.9 Design of the range of the iterative convergence test . . . . . . . . . . . . . . . . . . . . . 89

3.10 Number of iterations / Estimated error percentage for given tolerance criteria at each FN 91

4.1 The variation of draft and displacement with respect to the t1 . . . . . . . . . . . . . . . . 109

4.2 Residuary resistance and viscous pressure drag (CPV = k × CFITTC) of ship No.1 . . . . 112

4.3 Form factor k as a function of ti, ∆ki=|Max(ki)-Min(ki)|, ti= 0, 0.25, 0.5, 0.75, 1.0 . . . . 118

4.4 Comparison of CR, CW and CWCFD for 0 ≤ ti ≤ 1, i = 1, ..., 4 . . . . . . . . . . . . . . . 119

4.5 Variational range of resistance coefficients over the variational range of each form para-

meter as a percentage of CR[0000]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4

4.6 Sensitivities of CW and CWCFD as a univariate functions of form parameters . . . . . . . 121

4.7 Error trends ER and EW as a univariate functions of form parameters . . . . . . . . . . . 122

4.8 Sensitivities of ER and EW as a univariate functions of form parameters . . . . . . . . . . 123

4.9 Surface of ER [ B/T L/B t3 t4] at Fn=0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.10 Surface of EW [ B/T L/B t3 t4 ] at Fn=0.25 . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.11 Surface of ER [ t1 t2 LCB CB] at Fn=0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.12 Surface of EW [ t1 t2 LCB CB] at Fn=0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.13 Surface of ER [B/T L/B t3 0] at Fn=0.316 . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.14 Surface of EW [B/T L/B t3 0] at Fn=0.316 . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.15 Surface of ER [B/T t2 lcb 0] at Fn=0.316 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.16 Surface of EW [B/T t2 lcb 0] at Fn=0.316 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.17 Surface of ER[t1 L/B lcb 0] at Fn=0.316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.18 Surface of EW [t1 L/B lcb 0] at Fn=0.316 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.19 The range of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.20 CR and CR CORRECTED as a function of ti starting from [0 0 0 0] . . . . . . . . . . . . . 145

4.21 CR and CR CORRECTED as a function of ti starting from [1 1 1 1] . . . . . . . . . . . . . 146

5

List of Figures

2-1 Conceptual stages of simulation modeling and related errors with verification and validation 21

2-2 Error types contributing to the total error ET . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-3 Types of numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2-4 Verification process for the wave resistance problem . . . . . . . . . . . . . . . . . . . . . . 32

2-5 Lines of the Series 60 parent ship of CB = 0.65 (Model No.4211W) . . . . . . . . . . . . . 40

2-6 A visualization of the Series 60 parent ship of CB = 0.65 (Model No.4211W) . . . . . . . 41

2-7 L/B and B/T variation for a parent ship of CB = 0.6 . . . . . . . . . . . . . . . . . . . . 42

2-8 Validation range of CB and L/B as a part of Series 60 variational range . . . . . . . . . . 44

2-9 Validation range of CB and lcb as a part of Series 60 variational range . . . . . . . . . . . 44

2-10 Relation of range of ti and range of each form parameter . . . . . . . . . . . . . . . . . . . 45

2-11 Four-dimensional hyperspace of free variables . . . . . . . . . . . . . . . . . . . . . . . . . 49

3-1 Panel variations and their sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3-2 Cases of panel configuration sensitivity analysis in the verification range . . . . . . . . . . 64

3-3 Grid convergence tendency as a function of the number of panels (Fn = 0.25, Ship No. 1) 65

3-4 Convergence rate of CWCFD as a function of the number of panels . . . . . . . . . . . . . 67

3-5 Exact solutions extrapolated on the panel variations at FN = 0.25 . . . . . . . . . . . . . 69

3-6 Variation of various error estimates based on panels (hi,AR1) . . . . . . . . . . . . . . . . 70

3-7 Actual and estimated fractional error based on various AR as a function of the number

of panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3-8 Panel convergence tendencies at different Froude numbers . . . . . . . . . . . . . . . . . . 72

3-9 Convergence rates as a function of the number of panels at each FN with relation to the

panel types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3-10 Estimated error for each panel type as a percentage of CWCFD (h1,AR1) as a function of

the change of FN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3-11 Panel convergence tendencies of different hull forms at each Froude number . . . . . . . . 77

6

3-12 CWCFD for different panel numbers and extrapolated solution by Richardson method of

the ship CB = 0.6, B/T = 3.5 (No. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3-13 Estimated panel convergence errors of the hull form at different Froude numbers . . . . . 78

3-14 Estimated panel convergence errors of the hull form at FN = 0.225 and 0.275 . . . . . . . 79

3-15 Comparison of the panel convergence error with the CWCFD corresponding to the hull

form variation at FN = 0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3-16 Variation of the computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3-17 Computational domain convergence of CWCFD at different Froude numbers . . . . . . . . 85

3-18 Computational domain dependencies at FN = 0.25 of different ship shapes . . . . . . . . . 86

3-19 Computational region dependency of S60 expressed by the estimated error at each FN . . 87

3-20 Computational domain convergence error by means of two different scales of region XY3

and XY1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3-21 Normalized residuals in boundary conditions for the initial run at different Froude Numbers 90

3-22 Oscillating iterative history of CWCFD by using the panel type (h1, AR2) at FN = 0.25 . 91

3-23 Comparison of iterative convergence errors as a function of numerical parameters. . . . . . 93

3-24 Iterative convergence error related to the change of ship shapes at each FN . . . . . . . . 94

3-25 Degree of contribution of each error type to the total numerical error at FN = 0.25 for S60 96

3-26 The estimated errors of individual error types and the total numerical error of S60 at

FN = 0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3-27 Comparison of total error and numerical error. . . . . . . . . . . . . . . . . . . . . . . . . 97

4-1 Sensitivity of CR to 1% error of individual error types . . . . . . . . . . . . . . . . . . . . 102

4-2 Sensitivity of CW to 1% error of individual error type . . . . . . . . . . . . . . . . . . . . 102

4-3 Experimental data error ED in Cw and CR due to 1% changes in each input variable type 103

4-4 CR and CW based on different friction lines . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4-5 Differences of CW and CR based on each friction line of ITTC and ATTC . . . . . . . . . 104

4-6 Geometric range for single form parameter effect investigation . . . . . . . . . . . . . . . . 108

4-7 Comparison of CR and CWCFD with respect to the t1 variation as a function of FN . . . 110

4-8 ER for each FN as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4-9 Sensitivity of CR as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4-10 Sensitivity of CWCFD as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4-11 Comparison of CW and CWCFD with respect to the t1 variation as a function of FN . . . 114

4-12 EW for each FN as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4-13 Sensitivity of CWCFD as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4-14 Sensitivity of ER as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7

4-15 Sensitivity of EW as a function of t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4-16 Projection of groups of ships on the level of B/T (t1) . . . . . . . . . . . . . . . . . . . . . 133

4-17 Comparison of [t1 L/B lcb CB ] and [t1 0 0 0] at FN = 0.25 . . . . . . . . . . . . . . . . . . 134

4-18 Comparison of [B/T t2 lcb CB ] and [0 t2 0 0] at FN = 0.25 . . . . . . . . . . . . . . . . . 134

4-19 Comparison of [B/T L/B t3 CB] and [0 0 t3 0] at FN = 0.25 . . . . . . . . . . . . . . . . . 135

4-20 Comparison of [B/T L/B lcb t4] and [0 0 0 t4] at FN = 0.25 . . . . . . . . . . . . . . . . . 135

8

List of Symbols

ADD Straight addition equation

A Actual fractional error

ARi Aspect ratio of a panel

B/T Beam-draft ratio

BBC Body boundary condition

CA Model-ship correlation allowance

CB Block coefficient

CF Frictional resistance coefficient

CFATTC Frictional resistance coefficient based on ATTC friction line

CFITTC Frictional resistance coefficient based on ITTC1957 friction line

CM Maximum section coefficient

CP Prismatic coefficient

CPV Viscous pressure resistance coefficient (k × CFITTC)CPXI x-directional integration of the pressure distributed on a deeply

submerged body at zero speed

CR Residual resistance coefficient

CRATTC Residual resistance coefficient based on the CFATTC

CRITTC Residual resistance coefficient based on the CFITTC1957

CTM Total resistance coefficient of the model ship

CTS Total resistance coefficient of the full scale ship

CFM Frictional resistance of the model ship

CFS Frictional resistance of the full scale ship

CR CORRECTED Corrected residual resistance coefficient

CROSS Cross correlation equation

CT Total resistance coefficient

CV Volumetric coefficient

9

CW Wave resistance coefficient

CWP Waterplane area coefficient

CW CORRECTED Corrected wave resistance coefficient

CWCFD NO.i CWCFD prediction computed by the panel type No.i

CWCFD Wave resistance coefficient predicted by the CFD simulation

CWEXACT Exact solution of discretized numerical model describing wave resistance

problem

CWTRUE Analytical solution of mathematical model describing the wave resistance

problem

DFSBC Dynamic free surface boundary condition

ED Experimental data error

EDE Evaluation error

EDM Measurement error

EFD Experimental fluid dynamics

EGEOM Geometric data error

E Estimated fractional error

E 2(r = 1.2) Estimated fractional error by the Richardson 2nd order extrapolation

with the panel refinement ratio r = 1.2

A 4(r = 1.44) Actual fractional error by the Richardson 4th order extrapolation with

the panel refinement ratio r = 1.44

E rel Relative difference between the predictions

ER Difference between CR and CWCFD

ES Simulation error

ESN Numerical error

ESN ADD Numerical error combined by straight addition

ESN CONVERGE Numerical error with respect to the iterative convergence parameters

ESN CROSS Numerical error combined by cross correlation equation

ESN PANEL Numerical error with respect to the panel parameters

ESN REGION Numerical error with respect to the computational domain parameters

ESN RSS Numerical error combined by the root-sum-squares

ESP Physical model error

ET Total error, Difference between experimental data and simulation data

EW Difference between CW and CWCFD

10

FFE Fractional factorial experiment

FN Froude number

FSBC Free surface boundary condition

g Gravitational acceleration

GCI Grid convergence index

hi Longitudinal panel size, grid spacing

k Form factor

KFSBC Kinematic free surface boundary condition

L Ship length in general

L/B Length-beam ratio

LBP,LPP Length between perpendiculars

lcb Longitudinal center of buoyancy

LE Length of the entrance part

LR Length of the run part

LX Length of the parallel part

PDE Partial differential equation

q Disturbed velocity

RSS Root sum of squares

S Wetted-surface area

SAC Sectional area curve

TC Tolerance criterion for the iterative convergence test

ti Nondimensional free variable

U Undisturbed velocity

Region Xi Computational domain in longitudinal direction

Region XYi Computational domain accounting for the Kelvin angle

Region Yi Computational domain in transverse direction

∆CPXI ij Difference between the total pressure coefficients computed by the

panel type No.i and No.j

∆CWCFD ij Difference between the CFD predictions computed by the panel type

No.i and No.j

Aij Area of the panel

Nij Total number of panel type (hi, ARj)

(hi, ARj) Individual panel type with longitudinal size hi and its aspect ratio ARj

h = i All panel types of which longitudinal size is hi

11

CRij Panel convergence rate between panel type (hi, ARj) and (hi+1, ARj)

AR = j All panel types of which aspect ratio is ARj

to the change of the number of panels

∆CWCFD (hi,ARj) Total panel convergence range between the maximum and the minimum

predictions by all the panel variations

Rij Computational domain refinement ratio of region XYi/XYj

S60 Series 60 mother ship of CB = 0.6

∆ki Variation range of k when ti is varied from 0 to 1

∆CRi Variation range of CR when ti is varied from 0 to 1

12

Chapter 1

Introduction

1.1 Motivation

Ship designers and users of CFD (Computational Fluid Dynamics) simulations, those who are affected by

the decisions based on these simulations, are justly concerned with whether the simulations are reliable,

i. e., whether the level of credibility of these simulations is acceptable for the intended purposes. This

concern has been the principal problem since the inception of using CFD simulations for ship design.

Although much progress has been made in the development of CFD methods for flow prediction on

ship hulls, both with respect to the physical models and to the numerical processes, limits of accuracy

still exist. Moreover, the magnitude of these errors depends on many influences, especially on hull shape

characteristics, ship speed (Froude number), and the conditions of tests and CFD calculations. In many

cases, these errors are not of negligible magnitude.

Many studies to assess the reliability of CFD predictions have been performed based on verification1

and validation2. However, these were either limited to a certain scope or mainly conducted to develop and

improve the accuracy of simulation models. As the known accuracies of partial simulation models cannot

represent the accuracy of overall simulation code, it is difficult to assess the accuracy of CFD prediction

for each intended purpose only by the accuracy of the code in general. The characteristics of error types

are different and the degree of influence of each error type on the CFD prediction varies depending

on the design configurations. These reasons not only cause inadequate prediction of performance and

resistance but also prevent an objective comparison of CFD predictions between ships being designed.

This results in risky decisions about optimal ship shape from a hydrodynamic point of view when using

1Verification is the process of testing the accuracy of the numerical implementation of the mathematical model.2Validation is the process of testing the ability to accurately model the physical phenomena.

13

CFD simulation in the preliminary ship design stage.

For ship design purposes, it is necessarily required to systematically assess the reliability of CFD

prediction in a wide range of applications. A systematic validation of a large sample of ships is undertaken

here in order to shed some light on systematic error tendencies as a function of the principal shape

characteristics of ships. This study is performed by one state of the art CFD solver SHIPFLOW in

comparison to Series 60. The results will serve to ascertain to what extent the error trends are systematic

enough to rely on CFD predictions for hull form development at the design stage despite such errors.

1.2 Objectives

This study deals with the issue of the existing difference between predictions in ship hull form wave

resistance derived from experiments (EFD data) and based on the results of state of the art CFD

systems. It concentrates on the question whether any observed errors between measured results and

numerical predictions are primarily systematic functions of the hull form parameters or contain random

errors to any significant degree. Thereby the study intends to make a contribution to the issue of the

reliability of CFD results for wave-making resistance for ship design purposes relative to experimental

data (EFD results). This study is performed based on the following subgoals:

1. A systematic trend validation methodology should be developed for wave resistance prediction.

This methodology suggests how to design the validation domain and how to deal with the error

trends in the validation domain.

2. The numerical accuracy of the CFD prediction should be investigated in the validation domain

before the trend validation is conducted. Input parameters which lead to the best use of the code

SHIPFLOW and lead the CFD predictions to a certain level of accuracy should be chosen.

3. The deficits of the simulation models describing the wave resistance problem should be analyzed

and the error trends should be established as a function of the hull forms at different ship speeds.

The characteristics of the error trends over the validation domain will be analyzed by statistical

methods. The statistical methods distinguish the systematic error trends related to the hull form

parameters from the total error trends. This process results in reliable best fit models which may

correct the CFD prediction error, i. e., differences between real and model flow phenomena.

14

1.3 State of the art in validation

Computational fluid dynamics (CFD) has been applied to increasingly complex physics and geometries

in part due to algorithmic and computer hardware development. The significance of the CFD simula-

tions depends solely on their reliability. CFD simulation users expect that the CFD simulation offers

a reliable response for the intended purpose. As a result, there is a need to systematically assess and

report the degree of accuracy of CFD predictions. In 1991, Mehta [67][68] proposed a guide to credible

computational fluid dynamics simulations. He emphasized that the departure of the simulation from

reality is an uncertainty in simulation and proposed to identify the individual uncertainty sources and to

conduct a sensitivity-uncertainty analysis. Although there has been little agreement on establishing the

credibility of CFD simulation during the past decade, discussion and methodology for estimating errors

and uncertainties in CFD simulation have reached a certain level of maturity [20][83][99][37][38][39].

This progress responds to the need for achieving consensus on concepts, definitions, and useful method-

ology and procedures, as CFD is applied for the design purpose in a wide range of applications. The

reliability of CFD simulation is established by conducting verification and validation of simulation mod-

els. Verification assesses whether the problem is solved correctly and validation assesses whether the

right physical model is solved, i. e., verification is a process of testing a simulation model to establish

its ability to numerically solve the physical model (mathematical model) and validation is a process of

testing the ability to accurately model the physical phenomena. These processes are generally judged

by the uncertainty analysis assuming uncertainties are inherent in computational fluid dynamics (CFD)

[66][67]. The verification and validation are achieved by identifying the intended use of the simulations

and the uncertainty sources (error types) in the simulations and by conducting the sensitivity-uncertainty

analysis [68].

Verification includes the investigation of numerical error and uncertainty characteristics, stochastic or

deterministic, and may apply the generalized Richardson extrapolation [91][92] for input parameters and

the concept of correction factors based on analytical benchmark. Much effort is contributed to estimate

the numerical accuracies and to verify these error estimation methods [13] [14] [15] [16] [45] [93] [106].

Roache [93] discussed the limitation of the Richardson extrapolation and provided the use of a Grid

Convergence Index (GCI) for the uniform reporting of grid refinement studies. Jameson and Martinelli

[45] discussed also the difficulty of the Richardson extrapolation based on integer grid refinement and

suggested to use alternate implementations of submodules of the complete program.

Validation is performed by comparing the CFD simulation with the benchmark experimental data

and estimating the modeling error. This approach takes into account the uncertainties in both the

simulation and the experimental data in assessing the level of validation [2] [99] [28] [106]. The validation

process is distinguished by the intended purpose of CFD simulation: single code validation, multiple

15

code validation and trend validation of predictions. A single code validation is undertaken in most cases

of validation. Marvin [65] emphasizes the importance of benchmark experimental data and suggests

a benchmark methodology of experimental fluid dynamics for physical model validation. Stern [99]

provided a case of validation for a cargo and container ship where issues with regard to the practical

application of the methodology and procedures and the interpretation of verification and validation

results are discussed. Multiple code validation is undertaken based on several types of ship and diverse

codes [62] [63] [39]. Stern and Longo [63] procure considerable model-scale data based on surface-ship

resistance and propulsion of modern hull forms. This data base encompasses a wide range of ship types,

cargo/container ship, combatant, tanker, and diverse flow states. Although the validation is not directly

undertaken, CFD predictions are presented by the Gothenburg 2000 CFD workshop and the 22nd

International Towing Tank Conference (ITTC) [39]. Trend validation of CFD prediction was carried

out by Harries and Schulze [36]. CFD predictions are compared to the D-Series of 13 monohull family

test results. The ships are varied by three independent form parameters, beam to draft ratio (B/T ),

volumetric coefficient (CV ), and prismatic coefficient (CP ). This study was a pioneer trend validation

for ship design purposes which was based on systematically varied series hull forms.

As has been pointed out, much effort to establish the reliability of CFD simulation and to improve the

accuracy of CFD predictions has been invested. However, validation can be achieved at the required level

only when the experimental data is accurate enough. Therefore, the validation should be accompanied

by the development of experimental fluid dynamics (EFD). Moreover, a large test data base for diverse

flow states and ship types is needed.

16

Chapter 2

Methodology

2.1 Overview and objectives

In recent years, the accuracy and reliability of ship resistance prediction by CFD calculations have been

much improved [10], [54]-[59], both with respect to the physical models and to the numerical processes.

For the wave resistance, it is now state of the art that the nonlinear free surface condition for finite

wave amplitudes is taken into account at the surface panel control points [54]-[59]. Trim and sinkage

of the model can be approximated and accounted for. For viscous flow modeling, similar physical

improvements, e.g. for turbulence models, are also making gradual progress. Numerical methods in

CFD solvers have also gained more accuracy. Despite all of these improvements during the last decades

there remain errors between the results of experiments and CFD predictions, both in wave-making

and viscous flows. The magnitude of these errors depends on many influences, especially on hull shape

characteristics, ship speed (Froude number), and the conditions of tests and CFD calculations. In many

cases these errors are not of negligible magnitude. This is why the verification and validation remain

essential. A systematic validation of a large sample of ships is undertaken here in order to shed some

light on systematic error tendencies as a function of the principal shape characteristics of ships. This

study is performed by one state of the art CFD solver SHIPFLOW in comparison to Series 60. The

results will serve to ascertain to what extent the error trends are systematic enough to rely on CFD

predictions for hull form development at the design stage despite such errors.

This study will investigate whether the following claims are true:

• CFD simulation deviations from experimental results are not negligible.

• Error trends as a function of the form parameters are systematic, although there exists some

small noise. This can be true provided that the flow in the range of hull form variation is

17

generally stable. This means that a small change in monotonic ship variation will normally

lead to small change in flow. In this range nature does not make jumps:

”NATURA NON FACIT SALTUS”.

Hull forms from the Series 60 are chosen because this series covers a wide range of systematic hull form

changes. The hull forms in this series show family resemblance and the shapes do not differ much with

respect to their local shape character. If the experimental and computational results show monotonic or

simple dependencies on the shape parameters, then both results can agree at least in trend unless large

discrete errors occur. This suggests that the flow character changes monotonically and smoothly over

the whole range of variation, too.

This does not mean that certain other local hull form changes are not able to cause sudden, abrupt

changes in flow, e.g., flow instabilities like wave breaking and separation. But the validation promises

to hold at least in the investigated Series 60 range because in this range there is no evidence of small

changes causing great effects (no unstable behavior). Because of this, the CFD results can be analyzed

for systematic error trends. Average corrections can also be suggested for the difference between the

CFD results and the systematic experiments.

The problem analysis including validation state variables and concept and error types will be discussed

in section 2.2. Verification and validation processes will be discussed in section 2.3 and then the validation

method and tool will be discussed in section 2.4.

2.2 Problem analysis

2.2.1 Validation state variables

Wave resistance is very sensitive to the hull shape and the flow state, while viscous resistance is mainly

dependent on the wetted surface area of the ship and flow property. Even a small change in the local

shape or hull form characteristics will generate a new wave system by interference of individual dispersive

waves. Therefore, the wave-making problem around a ship has been an important subject of interest

for ship designers who intend to improve and optimize the hull form from the hydrodynamic point of

view. In this thesis, we will concern ourselves with the wave resistance prediction. As state variables,

both global measures like the wave resistance coefficient, the residuary resistance coefficient and local

flow phenomena like the wave profile and the pressure distribution can be used.

A physical validation using local flow variables like the wave profile and the pressure distribution

can help to acquire a more detailed understanding of the physical phenomena of the local flow. Also,

local variables are more suitable for validating the characteristics of individual models in the process

18

of developing hull forms. However the resulting physical phenomena are very complicated so that the

isolated validation of each individual phenomenon would be very difficult to be undertaken due to

complex interactions. Besides, this requires a high level of experimental accuracy for the consistency

of comparisons between experiment and computation. It is too expensive to stage experiments for all

cases. As a consequence, such experiments are seldom carried through for a large number of systematic

series ships.

In contrast, since variables such as wave resistance coefficient or residuary resistance coefficient

represent the flow characteristics of interest in a single value, error trends of prediction can be obtained

as a function of flow states and hull form parameters. These validation state variables enable to judge

relative merits of each hull form variant. For the purpose of ship design, design oriented resistance

and performance coefficients which offer a criterion to decide on a best hull form in a modification or

optimization process must be validated from a hydrodynamic point of view.

In this thesis, the wave resistance coefficient (CW ) and the residuary resistance coefficient (CR) are

chosen as validation state variables. In fact, these quantities predicted by EFD (CW , CR) and CFD

simulations (only wave resistance coefficient CWCFD) have conceptually different physical meaning,

because CWCFD is computed by integrating the longitudinal component of the pressure force over the

entire wetted part of the hull below the calculated wave profile under some physical assumptions for

specific flow situations, while CR and CW are extrapolated by the two hypotheses of Froude and Hughes

if the CT and form factor k are known from model tests. The conceptual difference between CR, CW and

CWCFD1 must be correctly understood and clarified in detail. These matters are discussed in chapter

4.2. The differences between CFD simulation and experimental data are discussed in a wide range of

shape variations. This may help to extract the relationship between hull form parameters at each flow

state and any systematic errors of the CFD simulation model.

2.2.2 Trend validation concept

The goal of CFD simulation is to predict the true properties of real physical phenomena as closely as

possible. However, the simulation models are built on mathematical equations for the physical models

and are based on some restrictions and simplifications of complex physical phenomena. In addition, the

physical models are transformed to numerical models which are discretized approximations. Generally,

CFD simulation models have many modeling assumptions and such assumptions result in specific types

of error. The CFD prediction is affected by such diverse types of errors simultaneously. For the design

purpose, it is very important to define a validity range of the simulation and to estimate the magnitude

1CWCFD is defined as the wave resistance coefficient calculated by a CFD simulation.

19

of its errors in this range. Therefore, the ability of CFD simulation to predict the correct trends of

results as a function of free shape variables should be tested in a wide range of parameters: Trend

validation. Trend validation is testing whether simulation results are in sufficient agreement with reality

with respect to the hull form dependent trends. The trend validation is accomplished by comparing the

simulation results with experimental data. This difference is defined as ”total error”, ET . As shown in

Fig. 2-1, in our case of dealing with the wave resistance problem, the total error is denoted by ER or

EW which are defined as

ER = CR − CWCFD (2.1)

EW = CW − CWCFD (2.2)

where CR and CW are the Series 60 experimental data[104] of the residuary and the wave resistance

coefficients. CR and CW are defined as follows :

CR =RR

1/2 ρ S v2, CW =

RW1/2 ρ S v2

(2.3)

where, RR and RW are the residuary resistance and the wave resistance respectively, ρ is the mass

density of water, S is the wetted surface area, v is the speed of ship. CR and CW are derived from the

published CT of full scale ships for 20ft models. The process about how to extrapolate CR and CW from

CT will be fully discussed in chapter 4.2. CWCFD is the wave resistance coefficient calculated by CFD

simulation.

These total errors concerned with the wave resistance problem, ER and EW , contain two categories

of error types: experimental data error and CFD simulation error. To begin with, it is necessary to

classify the details of the error types and to analyze the degree of influence of each error type on the

results. There are many kinds of error types in both EFD (Experimental Fluid Dynamics) and CFD

simulation. A systematical list of error types which contribute to the total error will be discussed in

detail in the next subchapter. These error types have different characteristics. In addition, they are

interdependent or are propagated to the results of CFD or EFD to different degrees. The effects are

based on the sensitivity and error propagation analysis which provides the quantitative estimation of

each error and the degree of its propagation to the simulation results.

From the design point of view, Series 60 test results can be taken as a reference in CFD validation

because they are interpolated and graphically faired in the variational range and hence the experimental

data of Series 60 are free of random error. Their accuracy and trends are generally accepted as accurate

enough for design purposes.

The CFD simulation errors consist of two categories of error types, physical model error and nu-

20

Figure 2-1: Conceptual stages of simulation modeling and related errors with verification and validation

21

merical error: It is neither possible to precisely model the real world in mathematical form nor to get

the exact numerical solution CWCFD for the complex nonlinear physical model in an analytical form.

Therefore, the trend validation can successfully be carried out only when the accuracy of the math-

ematical models and of the numerical algorithms is sufficiently ensured. The process of verifying the

computational and numerical accuracy is called verification2[68]. This process determines the numerical

accuracies of the numerical model. This verification aims not only at estimating the numerical errors

but also at finding the best numerical inputs to reduce the numerical errors. It may provide the best

use of the code bounded by the limits of the computational capacity. In practice, since there is no best

set of parameters which guarantees a zero numerical error, a residual numerical error always exists after

the verification has been carried out. Verification will be described in detail in chapter 3.

Since the same results are obtained for each single hull form if the same input variables are used for

a simulation run, a single prediction by simulation has deterministic character. However, in the domain

of trend validation, the remaining numerical errors in each prediction of simulation may be distributed

randomly even after the verification process has been carried through. This concept makes it possible

to model best fits expressing the systematic error trends in the validation domain for general use and to

measure the remaining random noise. This approach filters out the random numerical errors and shows

the systematic tendencies of the simulation. Conceptual modeling, numerical verification, and trend

validation processes are shown in context in Fig.2-1.

2.2.3 Error types

Overview

Since trend validation is established by comparison of the CFD prediction to EFD (Experimental Fluid

Dynamics) data in the validation domain, there are many kinds of error types involved in the total error.

In order to reduce the numerical errors and to finally establish the systematic error trends, error types

and their characteristics should be carefully investigated. In fact, it is difficult to classify a complete list

of error types in both CFD and EFD. Moreover, the error characteristics and the degree of propagation

of each error source to the total error are not known. A list of error types and their characteristics in

the wave resistance problem will be discussed. General error types in CFD prediction and EFD data

are shown in Fig. 2-2.

2Verification is defined as a process to test whether the results of the numerical analysis of a physical model are in agree-ment with the assumptions of the model (or as expected from the model), i.e., testing for numerical errors/implementationerrors via sensitivity analysis.

22

Figure 2-2: Error types contributing to the total error ET

Errors in experimental data (EFD data) from model tests

Errors in EFD consist of measurement errors due to the limited capability of tests and measurements

in capturing all physical phenomena, and evaluation errors due to the evaluation procedure based on

Froude’s or Hughes’ hypotheses for CR and CW . We restrict ourselves to the model scale test for a towing

tank. Since each towing tank has unique facilities and its own data acquisition standards, general aspects

concerning validation state variables will be discussed in overview.

ED 1 Measurement errors (raw data errors)

Generally, the following error types are considered as affecting the measurement error; they are

referred to in detail in the ITTC proceedings [38] [39].

(a) Speed measurement

The model velocity is measured relatively to the ground or to the water. The velocity is

averaged over the data acquisition time.

(b) Resistance measurement

Generally load cells are used today for resistance measurement. The resistance is also averaged

over the data acquisition time.

(c) Water temperature, kinematic viscosity

This is measured differently depending on the time to be tested and on the location to be

measured. Temperature is typically taken as the mean over the measured distance at half of

the mean draft of the model or at one or several fixed depths at fixed locations in the tank.

Kinematic viscosity is calculated by means of the measured temperature.

23

(d) Water quality (absence of contamination)

(e) Finite tank size (blockage effects)

ED 2 Evaluation errors for CR and CW

Froude’s and Hughes’ hypotheses separate certain physical phenomena and are based on data

reduction schemes such as friction lines. In this evaluation process, the following phenomena are

ignored or several error types are added.

ED 2.1 Surface tension

Surface tension is present, but is not accounted for.

ED 2.2 Viscous effects approximated

(a) Froude method(CR)

• Separability assumption error: wave-viscous interaction is neglected.• Errors in using a single friction line: The ITTC line or any other friction line is at

best valid for an ”average hull form”. A single line cannot account for friction of an

individual hull form.

• Model-ship correlation allowance CA is not applied here. Thus, an indirect correctionfrom tank experience is not applicable. Absence of CA correction means no friction

line correction takes place.

• Effects of pressure and velocity variation on the hull surface upon friction are onlyapproximated by ”form factors” implicit in the friction line.

(b) Hughes method(CW )

• Separability assumption and friction line error still exist.• Hughes-Prohaska evaluation error (extrapolation from finite FN to FN = 0)

• The error in the assumption k = const over FN does not account for variable wave-

viscous interaction.

ED 2.3 Nonlinear wave-making effects not accounted for or not properly scaled

(a) Wave breaking, flow instabilities are neglected in the evaluation, hence arbitrarily asso-

ciated with CW alone.

(b) Wave-viscous interaction is not included in CF , hence arbitrarily included in CR or CW ,

though not only dependent on FN .

ED 2.4 Data fairing errors

(a) Limited size of discrete data points sample in the Series 60 results

(b) Averaging effects

24

Errors in computational data(CFD data), CR and CW , based on SHIPFLOW

CFD simulation which consists of physical and numerical models has two categories of error, namely

the physical model error and the numerical error. Physical model errors are caused by simplified Partial

Differential Equations (PDEs) describing the flow, the boundary conditions for PDEs and by improper

modeling of certain aspects of the flow. These models are based on restrictions and simplification of real

phenomena. Numerical errors are due to discretization error and an approximate solution of the discrete

equation (convergence error) and computational implementation error.

ESP Physical model errors

Physical model errors result from a simplification of physical phenomena (idealization of fluid

dynamic phenomena) and improper modeling by partial differential equations (PDEs) describing

the flow and its boundary conditions. An isolation of phenomena is made by assuming that there is

either absolutely no influence or a perfectly known influence on these phenomena by other natural

phenomena. However, there are inevitable consequences of this separation and idealization so that

it is impossible to address all relevant phenomena simultaneously, although their interactions which

may exist are not known. The reason can be either a lack of knowledge necessary for modelling or a

lack of computational capacity. In the design process, the overall effect of the unknown interaction

of phenomena related to the body such as the hull form may cause noticeable changes in their

prediction and they are not likely to be simply superimposable. The wave resistance problem is

described under some physical simplifications for idealized flow situations. The flow situation is

idealized as follows: still water around a hull, steady forward motion in the water and infinitely

deep and wide water. As the complexity of the flow increases, more information is required in the

physical modeling and more error types are introduced. Despite such idealizations, the resulting

flow phenomena are complicated by a number of different phenomena. In the present treatment

of wave making by using SHIPFLOW, some complicated phenomena are neglected by assuming

incompressible, inviscid, irrotational and hence potential flow. This idealized wave resistance

problem is described by the Laplace equation as the equation of state. The Laplace equation is

developed from the conservation law of mass based in a fluid continuum and does not take into

account the following effects:

ESP 1 Simplification of physical phenomena

ESP 1.1 Surface tension is neglected.

ESP 1.2 Viscous effects are neglected.

ESP 1.3 Wave-viscous interaction is neglected.

25

Regarding the modeling of the boundary conditions the following further simplifications are intro-

duced in SHIPFLOW:

ESP 1.4 The domain is limited to a finite control volume(truncation error).

ESP 1.5 Strongly nonlinear effects are neglected: Wave breaking, even steep waves.

The boundary conditions are required for the solution of the governing equation. These condi-

tions limit the flow analysis to the situation of interest. Therefore, it is important to model the

appropriate boundary conditions for expressing the intended physical phenomena of interest.

ESP 2 Modeling errors in the boundary conditions

ESP 2.1 Body Boundary Condition (BBC)

This conveys the fact that normal velocity components are zero on the wetted part of the

body. This is usually called the body boundary condition and expressed by the normal

derivative of the potential on the body surface. Simplifications on a body shape and boundary

discontinuities may cause a physical modeling error.

ESP 2.2 Free Surface Boundary Condition (FSBC)

The particle velocity normal to the fluid must be equal to normal velocity of the free surface.

This is the kinematic free surface boundary condition(KFSBC). However, it is not sufficient

because the free surface is an unknown boundary. A further boundary condition is the dynamic

free surface boundary condition (DFSBC) where the pressure on the free surface must be

atmospheric. This is formulated by the Bernoulli equation, assuming the surface tension

on the free surface is negligible. This model includes the dynamic pressure and hydrostatic

pressure simultaneously. A combined FSBC can be derived simultaneously in a convenient

way by substituting the kinematic condition into the dynamic condition. This FSBC is an

exact, but nonlinear equation. Moreover, this condition should be imposed on the unknown

free surface. Since these characteristics cause numerical difficulties, some simplifications in the

structure and parameters of the mathematical equation such as linearization are sometimes

made. These simplifications are based on both physical and numerical points of view. Errors

in the linearized FSBC increase when the nonlinear effects of the situation are large, for

example for a faster ship or a full block ship.

ESP 2.3 Radiation condition

A body moving steadily has no waves (fluid motion) far ahead. But the body has outgoing

waves at infinity in all other directions in a real fluid. This situation will be taken care of

26

by imposing a radiation condition, which ensures asymptotic decay of the wave system far

behind the model.

ESN Numerical errors

There are two sources of numerical errors, discretization and evaluation errors. Discretization errors

are inherent in the distinction of the simulated model and the conceptual model. This distinction

is caused generally by the need for an efficient and robust computational algorithm. For example,

a dimensional reduction of the situation of interest, an asymptotic global solution such as steady

state, periodic solution cause the errors. Also, an improper implementation of the initial and

boundary conditions changes the conceptual model. In practice, the simulation model cannot be

the same as the conceptual model because of the discretization. This process is one cause of the

lack of accuracy in simulation results. Computational accuracy in CFD results is determined by

the type of discretization, convergence scheme and criteria, and the implementation. The detailed

computational error types are discussed in the following paragraphs.

ESN 1 Discretization errors[10][54][55][56]

ESN 1.1 Finite panel size on body surface, hull shape resolution error

ESN 1.2 Finite panel size and number on free surface, resolution of wave shape error

ESN 1.3 Domain bounded (tank wall error, truncation error aft)

ESN 1.4 Pressure integration error

ESN 1.5 Free surface condition is met only at discrete points (panel control points)

ESN 2 Convergence errors[10][54][55][56]

ESN 2.1 Grid convergence error

ESN 2.2 Domain convergence error

ESN 2.3 Residual error at cut off tolerance

ESN 2.4 Convergence to false level (e.g. by numerical dissipation)

ESN 3 Trim and sinkage error

CFD trim and sinkage are either neglected or approximated by an ideal fluid pressure integral. In

both cases they do not agree with the experiment in a real fluid.

ESN 4 Round off errors in the implementation (computational error)

27

Errors in geometric data used in this thesis

EGEOM 1 Error in the hull form variation based on the Lackenby transformation

In this thesis, Series 60 hull forms were reconstructed for this thesis in the chosen validation domain.

A single ship of Series 60 CB = 0.65(4211W ) was chosen as a parent ship, i. e., one of the Series

60 parents. All other 57 ships which belong to the validation sample were derived by means of the

Lackenby transformation. In order to vary the hull form as similarly as possible to the Series 60,

additional design parameters such as CP , CM , lcf , LE/LBP , LR/LBP were treated as being the

same as in the Series 60 as well as the 4 main parameters of L/B, B/T , lcb, CB. Thus while in

this thesis all hull shapes in the sample were derived from a single parent, is possible that some of

the corresponding shapes in Series 60 stem from some other parent. Therefore, differences in hull

shape may exist between the original Series 60 and the Series 60 varied for this CFD work.

EGEOM 2 Surface modeling and fairing error

In Series 60, the hull surface geometries were derived at least indirectly from graphical representa-

tion in lines plans which took into account the SAC and the given section shapes of the parent. In

this thesis, however, the surface definition was obtained by computational interpolation of given

section shapes using the MULTISURF software system. Thus the resulting surface interpolant

may deviate slightly from the actual Series 60 shapes.

EGEOM 3 SHIPFLOW discretization error

SHIPFLOW contains its own preprocessor that remodels the given offset data for shape discretiza-

tion. Therefore, some internal shape resolution errors are added to the geometric model of the

input data.

Conclusion

Each of these error types has its own characteristics. They can be predominantly random or systematic.

These categories should be dealt with in different ways according to their characteristics. An overview

of predominantly assumed error characteristics of each error type is shown in Table 2.1

Random errors are much reduced by using the faired experimental data (in Series 60) and by fairing

the CFD data for CW (by regression). They may not be negligible, but they are not known here, neither

for Series 60 nor for CFD by SHIPFLOW for any specific ship. But they are not relevant to the intended

error trend analysis (analysis of systematic error trends), which is based on average values.

Systematic error trends of EW = CW − CWCFD or ER = CR − CWCFD encompass influences

combined from all types of systematic error terms in Table 2.1 plus maybe some residues of random

28

Systematic error Random errorExperimental error(ED) ED 2.1-ED 2.3 ED 1, ED 2.4

Physical model error(ESP )ESP 1ESP 2

ESP 1.5

Numerical error(ESN)ESN 1.3ESN 2.4ESN 3

ESN 1ESN 2.1−ESN 2.3ESN 4

Geometric data error(EGEOM) EGEOM 1 EGEOM 2, EGEOM 3

Table 2.1: An overview of characteristics of each error type

errors if they contain non-zero averages. The magnitude of this ”residual noise” will be estimated as a

by-product of the regression analysis.

The errors EW and ER, obtained by subtracting faired(verified) CFD data from faired experimen-

tal data, are for simplicity’s sake called ”total systematic errors”. They will be investigated in their

dependence on hull form parameters ti to determine whether they follow simple, monotonic or nearly

monotonic trends.

2.3 General approach

2.3.1 Overview

Since trend validation is undertaken by a direct comparison of the CFD prediction with experimental

data, the total error includes many error types, classified in section 2.2.3, at the same time. Therefore, it

is meaningless if the trend validation is assessed without a thorough understanding of the error types and

their degree of influence. For example, a faulty model may appear to be correct if positive and negative

computational errors cancel each other or if the computational error corrects the physical model deficits

quantitatively. Sensitivity and error propagation analyses are introduced in order to at least quantify

the sensitivity of each error type upon the validation state variable and to establish the degree of errors

in the chosen validation domain. These analyses are used as background both for the verification process

and the validation process. Statistical analysis assists to model the best fit of the systematic error trends

over the validation domain. These approaches serve to improve the reliability of the CFD prediction by

assessing CFD simulation errors and by modeling the best fit and reconstructing corrected predictions.

1. Sensitivity analysis

Sensitivity analysis is generally defined as a procedure for determining the reaction of output

quantities to an input parameter [68]. Sensitivity analysis is used to get to know how a model will

behave if one or more of the model inputs are varied. This will here provide an identification of the

29

important contributors to the validation state variables of fluid dynamic systems. If the simulated

quantities (output) are identified the input values are chosen according to the intended purpose.

For example, the input values can be many types of elements and parameters of the CFD model

such as discretization models, panel size, iteration criteria for the solution, or turbulence model,

governing equations, free surface boundary conditions for validation, etc. The input variables can

be chosen either from the error types themselves or from the simulation input which exerts influence

on the error types. The ideal objective is to seek input parameters which lead to simulations that

are essentially independent of the computational process and to simulations that essentially capture

the real world for the intended application. Sensitivity analysis measures the sensitivities and the

degree of contribution of each input parameter. The influence coefficient of each parameter is

computed by perturbing the input parameters and obtaining the response in the output quantities

while neglecting the influence of other inputs.

2. Error propagation analysis

Error propagation analysis is defined as a scheme to estimate the effect of the errors involved in

all stages of a process on the final response, namely on the simulated results. There are several

possibilities to estimate the total error when each individual error is known. If there exists a data

reduction equation and it can be used as the propagation equation which expresses the relationships

between inputs and output, the total error can be estimated by differentiating the equation with

respect to each individual error type. The most important errors for trend validation are those

which remain and propagate through the simulation process and systematically affect the reliability

of the CFD prediction. We will restrict ourselves to systematic errors only.

3. Regression equation

A regression equation is an equation expressing the dependence of a random variable Y on a scalar

quantity X or vector of quantities X, the independent variables, which are not random. There are

two types of models in the classical regression approach: linear and nonlinear models. A linear

model is one in which the parameters appear linearly even though the relations between dependent

and independent variables are nonlinear while nonlinear models may also contain combinations

(products) of parameters. Linearity and nonlinearity always refer to parameters, not to variables.

A linear multiple regression is used in this study.

2.3.2 Verification

Verification assesses whether the computational model represents the analytical model correctly and

determines the level of the computational accuracy of the numerical model by estimating the errors owing

30

Figure 2-3: Types of numerical errors

to numerical error types. Generally, the numerical model is developed by using two procedures. One is

the discretization process which replaces the mathematical model by algebraic equations. The other is

the process to numerically solve such algebraic equations. The two procedures result in discretization

error and solution error including convergence error, implementation error and round-off error in the

implementation. These error types were classified in section 2.2.3 in detail.

These numerical error types are combined with each other and are propagated to the CFD prediction,

each with a different degree of influence. For the CFD simulation for the wave resistance problem, results,

CWCFD, are affected by the numerical parameters(Xi) which are directly related to the numerical error

types as follows:

CWCFD = f(X1,X2, ...,Xn) (2.4)

The error can be defined by comparing the prediction to the analytical solution as follows:

ESN = CWCFD − CWTRUE (2.5)

where ESN is the numerical error of the simulation and CWTRUE is an analytical solution as shown in

Fig. 2-3. In practice, since an analytical solution for complicated phenomena does not exist, CWTRUE

can at best be estimated for each numerical parameter. The estimation of the analytical solution is re-

placed by a basic assumption that the error magnitude converges to zero when each of the numerical

input parameters such as the panel size or the iterative criteria are sufficiently small. The criteria such

as Richardson extrapolation, GCI(Grid Convergence Index) and iterative convergence error criterion are

fully described for individual error estimation in chapter 3.2.2. These processes are based on the sensi-

31

Figure 2-4: Verification process for the wave resistance problem

32

tivity analysis. Since a single code SHIPFLOW is used, a numerical model is already given such as the

order of discretization, the discretization scheme or solution method, etc. Based on the given numerical

model which is influenced by the numerical parameters such as panel parameters, computational domain

parameters and iterative convergence parameters as shown in Fig. 2-4, in this thesis, the sensitivities of

the numerical model are identified by perturbing each numerical parameter systematically as follows:

δCWCFD = f(δX1, δX2, ......, δXn) (2.6)

Moreover, since the explicit form of a data reduction equation such as (2.4) is not known in advance,

one needs to perform an error propagation analysis in order to estimate the total numerical error resulting

from each error type due to the individual numerical parameters. If an individual error is estimated for

a numerical parameter, for example, X1, the numerical error of X1 is estimated as follows:

ESN X1= f(X1, C2,..., Cn)− CWTRUE 1 (2.7)

where ESN X1 is an individual numerical error due to the numerical parameter X1, f(X1, C2,..., Cn) is

a solution calculated by a chosen variable X1 by fixing other variables as constant, CWTRUE 1 is an

estimated analytical solution concerned with X1.

If the sensitivity analysis and individual error estimations are undertaken for all numerical para-

meters, the total numerical error will be determined by the root-sum-square (RSS) of the individual

errors:

ESN RSS =n

i=1

E2SN Xi

1/2

(2.8)

and if the interactions between the input variables are to be accounted for, the cross correlation terms

are added to (2.8). This is denoted as CROSS and defined as:

ESN CROSS =

⎛⎝ n

i=1

n

j=i

ESN XiESN Xj

⎞⎠1/2

(2.9)

The straight addition of the individual numerical errors is denoted by ADD and defined as:

ESN ADD =n

i=1

ESN Xi

This verification process should be performed for at least some applications and flow situations at

their extreme values in order to establish the numerical credibility over the whole range of validation.

This process results not only in estimating the degree of numerical error in the simulation prediction,

33

but also in finding the numerical input parameters such as grid refinement and computational domain

and iteration convergent criteria which lead to the best use of the code.

2.3.3 Validation

Validation is the process of testing the ability to accurately predict the physical phenomena and deal

with the accuracy of the solution as it relates to the real world [39] [68]. The validation is accomplished

by comparison of the CFD results with the experimental data. The validation process is distinguished

by the intended purpose of CFD simulation: Single ship validation and trend validation. The single

ship validation process aims at establishing the degree of the physical model error and improving the

reliability of the physical model. This is performed by separating the physical model error from the

total error by filtering out the experimental error (ED) and the numerical error (ESN ) from the total

error (ET ). The single ship validation can take place only when the accuracy and limitations of the

experimental data are known and only when the accuracy of the numerical algorithms of the code is

determined.

The trend validation is testing whether simulation results are in sufficient agreement with reality

with respect to the hull form dependent trends for the design purpose. This process is undertaken as

a function of the free shape variables in a wide range of the hull form design. This process aims at

establishing the systematic error trends as a function of the form parameters in a validation domain. In

this thesis, the verified prediction of simulation and the faired model test results are used for the trend

validation. The difference is called ”systematic error trends” and is denoted by ET . This trend validation

process is undertaken by a sensitivity analysis based on the form parameter variation. The simulated

quantities are the systematic error trends for the purpose of ship design. The ability to predict the

physical phenomena will be tested for various shapes with distinct values of B/T, L/B, lcb, CB at each

speed of FN . These form parameters are varied combinatorially. Univariate, bivariate, and multivariate

form parameter effects on the error trends will be discussed and visualized. The validation domain is

planned to check their interaction by choosing the nonlinear and multivariate validation domain.

∆EW or ∆ER = f(∆L/B, ∆B/T, ∆lcb, ∆CB) (2.10)

Table 2.2 shows each simulated quantity and input variable for the sensitivity test of experimental

data and simulation models for the wave resistance problem. The systematic error trends between the

experimental data and the simulation results are used. In this case, the direct sensitivity analysis of the

error characteristics as a function of design configurations such as ship shape variation and ship speed

change becomes possible.

34

Experimental data Systematic error trendsSimulated− CR,CW ERQuantities EWInput− Evaluation functionVariables CT , CF , k B/T, L/B, lcb, CB

B/T,L/B, lcb, CB FNFN

Table 2.2: Overview of sensitivity analysis

The trend validation will at last be performed by the methodology of regression analysis being applied

to the whole sample data volume of the error function vs. the free design variables at any given FN .

The evaluation of the data will be undertaken by stepwise, linear, multivariate regression analysis [24]

[100].

2.4 Methods and tools

The objective of the trend validation is to test the ability of the CFD simulation to predict the trends

of variables such as design configurations and, if possible, to approximate the systematic error trends for

general use. For our subject, SHIPFLOW is used as a simulation tool and Series 60 as reference ships.

The domain of trend validation is chosen as a part of the range of Series 60. In the following sections,

a general review of SHIPFLOW and Series 60, and a scheme to select the validation domain will be

presented.

2.4.1 SHIPFLOW

In this thesis, CFD simulation is based on the commercial code SHIPFLOW[54]-[59]. SHIPFLOW is

a code developed for the special need for the use by naval architects in ship design. SHIPFLOW offers

many kinds of subjects corresponding to the ”numerical towing tank”, although it is based on some

restrictions and simplifying assumptions. Physical and numerical model implemented by SHIPFLOW

will be briefly reviewed for the wave resistance prediction[58]. Sinkage and trim are not included in the

CFD calculations of this thesis.

Physical model

1. Governing equation

The flow is assumed steady, incompressible, inviscid, irrotational. U and q represent the undis-

turbed and disturbed velocities respectively. Assumption of irrotational flow permits assuming the

35

existence of velocity potential such as

q =∇φ (2.11)

Then the Laplace equation represents the equation of state (conservation of mass).

∇2φ(x,y, z) = 0 (2.12)

2. Boundary conditions

(a) Free Surface Boundary Condition (FSBC)

i. Kinematic Free Surface Boundary Condition(KFSBC): This is a mathematical expression

of the fact that the surface is the boundary between two immiscible fluids. This means

that the particle velocity normal to free surface must be equal to normal velocity of the

free surface. If a potential φ exists, then

φxηx+φyηy−φz= 0 on z = η(x,y) (2.13)

where, z = η(x, y) refers to the elevation of free surface. φx =∂φ∂x , ηx =

∂η∂x and the same

for φy, φz, ηy.

ii. Dynamic Free Surface Boundary Condition(DFSBC): The pressure on the free surface

is atmospherically constant. Air-pressure and surface tension on the free surface are

negligible. In the presence of a free surface, gravity must be taken into account throughout

the calculations because the height fluctuation due to the surface deflection changes the

hydrostatic pressure3, although the total pressure on the surface remains constant as

atmospheric. This means that there is an interaction between gravitational forces acting

on the fluid and the dynamics of the fluid motion. This is expressed by Bernoulli’s

equation which includes the dynamic pressure and hydrostatic pressure simultaneously

with the acceleration of gravity g and ship speed U.

gη + 1/2(∇φ)2= 1/2U

2 on z = η(x,y) (2.14)

iii. Combined Free Surface Boundary Condition (FSBC): This nonlinear free surface bound-

ary condition is derived by substituting equation (2.14) into the equation (2.13) or directly

3This is equal to the static weight per unit area of the fluid column at the given depth.

36

by applying the kinematic boundary condition to the total pressure on the free surface.

gφz+1/2∇φ ·∇[(∇φ)2] = 0 on z = η(x,y) (2.15)

(b) Body Boundary Condition(BBC): This requires that no fluid particle penetrate the body

boundary.

φn= 0 on S0, z < η(x,y) (2.16)

where S0 is ship hull surface below the free surface z = η(x, y) and n is the unit normal vector

outward from the hull

(c) Bottom Boundary Condition for infinite depth water. This means the fluid is at rest at

the deep bottom.

φz= 0 as z→ −∞ (2.17)

(d) Radiation Condition: Upstream disturbance by a moving ship vanishes at infinity.

limr→∞ ∇φ =U as r = x2 + y2 + z2 (2.18)

Numerical model and implementation

1. Implementation

The wave resistance problem which is posed in the physical model will be solved in SHIPFLOW

by a boundary integral technique where Rankine sources are distributed on the hull and on a part

of the free surface. SHIPFLOW offers a nonlinear free surface boundary condition option

by solving exact FSBCs satisfied at points on the exact location of the wavy free surface. This is

realized by an iterative solution method. As initial approximation, the double model solution,ΦDM

is used on the undisturbed free surface η = 0 [22]. At each iteration, the disturbed solution is

substituted into undisturbed solution, Φ = φ,and the changed wavy free surface is calculated and

substituted for the free surface from the previous step of iteration. This iteration will be continued

until the differences are within the convergence criteria.

2. Discretization

SHIPFLOW offers first and high order discretization both on the hull and a part of the free surface.

The first order method assumes that constant strength sources are distributed on flat panels. The

higher order method assumes that linearly varying strengths of sources are distributed on bi-

parabolic panels. Panels are quadrilateral and can be optionally distributed on the raised free

37

surface. Boundary conditions are only satisfied at one point (collocation point) on each panel.

SHIPFLOW offers a choice of the number of panels on the hull and free surface, and the domain

of the computational free surface to be discretized.

3. Solution

The system of equations for the panel source strengths is solved either directly by Gaussian elimina-

tion, or by an iterative technique. CWCFD is computed by integrating the longitudinal component

of the pressure force over the entire wetted part of the hull under the calculated wave profile.

SHIPFLOW offers two ways of pressure integration on the body by assuming constant pressure

on each panel acting in the negative normal direction or by assuming linearly varying pressure on

each curved panel surface.

Numerical options for wave resistance prediction

SHIPFLOW offers three modules of XMESH, XPAN and XPOST for wave resistance prediction. XMESH

is a panel generator for the potential flow module XPAN. XMESH can be executed as separate program

to check the panelization of the body and free surface before the potential flow computation is executed.

The XMESH module is also executed during the potential flow computation when nonlinear iterations

are performed and the geometry is updated in each iteration. XPAN is a module for computing the

potential flow around three dimensional bodies based on a surface singularity panel method. This module

includes flows with or without free surface, ship flows with or without transom stern, ship flows with

or without sinkage and trim, multiple ship speeds and the influence of the propeller. This module also

offers several flexible options such as first or higher order panel method, linear or nonlinear free surface

boundary condition, double or single model approximation, symmetry feature, specifying the initial

position of the ship, computing the velocity and pressure at specified body points. This code yields

results in wave resistance, wave pattern, potential streamlines, pressure contours, velocity vectors, and

sinkage and trim.

2.4.2 Series 60

Overview

The David Taylor Model Basin of the United States Navy have carried out a long term methodical

research for a single screw ship design from 1948 to 1962. The results are published in [101][102][103][104]

Series 60 is the latest systematic series of large scope which was intended as a benchmark for single

screw container ship design. This Series consists of five parent ships which are distinguished by basic

form parameter CB , and 57 systematically varied ships by changing three other form parameters L/B,

38

B/T 4 and lcb5. This series of ships covers a wide field of design proportions of single screw merchant

ships.

Series 60 Parent Ships

There are five parent ships of CB = 0.6, 0.65, 0.7, 0.75, 0.8 in Series 60. Many methodical series were

derived from a single parent form by proportional geometrical change. However, this variation over a

large range inevitably led to unfavorable forms. Moreover, a series of models which are interpolated

between the high quality parent shapes from the hydrodynamic point of view does not necessarily show

good resistance qualities6 . Therefore, three Series 60 parent ships of CB = 0.6, 0.7, 0.8 were derived

to achieve better resistance and performance qualities than Series 57 through more detailed benchmark

series model test, although the geometry became more flexible. Two more parents of CB = 0.65, 0.75

were derived by generating the area and waterline contours through the spots derived from the three first

parents. Therefore, Series 60 is not a single parent systematic series. It has five parent shapes of different

characteristics of hull form geometry. Overall, Series 60 parents have the following characteristics, as

shown in Table 2.3.

CM , CP and bilge radius are almost linearly dependent on the CB. There is no deadrise. The

position of lcb in the parents is not entirely systematically derived. lcb was chosen which showed the

best resistance performance in Series 57. Load waterline half-angle of entrance were chosen depending

on the CB. Sectional area and waterline coefficients contours are interpolated by contour lines. Entrance

position, the length of parallel middle body for Series 60 parents depend on CB. The body plans are

also plotted by the contour lines of waterline-half-breadth coefficients. Ratios of LE/LBP , CPE/CPR

are given as a function of CB and the position of lcb. Bow & stern contours are the same as in Series 57.

Table 2.3[104] shows the principal characteristics of 5 parents ships in model scale. Fig. 2-5 and Fig.

2-6 shows one of the parent ships No.4211W.

Free parameters and parameter ranges

Four free parameters, CB, lcb, L/B, B/T were chosen defining the Series ships. Block coefficient CB is

a basic parameter, although CP is often considered more meaningful parameter from the hydrodynamic

point of view. Series 60 forms are not very much different in midship area coefficient CX . The resistance

qualities can be related, therefore, either to CB or CP or both. lcb is defined as percent of LBP from

midships. lcb is directly related to the resistance quality of the ship. The position of lcb is chosen

4Todd used draft H instead of T in several publications.5 lcb is defined as percent of LBP from amidships, positive to the F.P. and negative to the A.P.6Mr. Todd described this results in the way : ”Apparently in ship models, as in human beings, the selection of good

parents doesn’t necessarily lead to better - or even as good- offspring.”.

39

Model Number 4210W 4211W 4212W 4213W 4214W-B4LBP (ft) 20 20 20 20 20L/B 7.50 7.25 7.00 6.75 6.50B/T 2.50 2.50 2.50 2.50 2.50LE/LBP 0.5 0.472 0.410 0.350 0.290LX/LBP 0 0.035 0.119 0.210 0.300LR/LBP 0.5 0.493 0.471 0.440 0.410CB 0.60 0.65 0.70 0.75 0.80CX 0.977 0.982 0.986 0.990 0.994CP 0.614 0.661 0.710 0.758 0.805CWP 0.706 0.746 0.785 0.827 0.87112αE, deg 7.0 9.1 14.5 22.5 43.0lcb % LBP from amidships 1.5A 0.5A 0.5F 1.5F 2.5F

Table 2.3: Principal characteristics of Series 60 parent models

Figure 2-5: Lines of the Series 60 parent ship of CB = 0.65 (Model No.4211W)

40

Figure 2-6: A visualization of the Series 60 parent ship of CB = 0.65 (Model No.4211W)

CB 0.6 0.65 0.7 0.75 0.8lcb -2.5-0.5 -2.5-1.5 -2.0-2.5 0.5-3.5 0.75-3.5L/B 6.5-8.5 6.25-8.25 6.0-8.0 5.75-7.75 5.5-7.5B/T 2.5-3.5 2.5-3.5 2.5-3.5 2.5-3.5 2.5-3.5

Table 2.4: Ranges of form parameters of Series 60 parents

independently from the L/B and B/T . L/B and B/T were chosen as free parameters which are suitable

to take into account finer, fuller or relatively longer and narrower ship types. The ranges of the four

form parameters are chosen to cover practically existing designs. The range of the Series 60 family is

shown in Table 2.4

Series 60 variation

Each parent ship was varied by the combination of each of three points of L/B and B/T in the range

defined in Table 2.4. The range of L/B is changed as a function of CB while that of B/T is the same for

all parents. L/B points are always taken by dividing its range equally into two subranges. Therefore,

for any one parent ship a total of nine models are derived. This variation is not always based on the

parent ship. For the case of CB = 0.65, 0.7, new parents for the combinational variation of L/B and

B/T are chosen which have 1.5A and 0.5A of lcb, respectively. The model numbers are No. 4218 and

4221. In this variation, the displacement depends on the value of L/B and B/T and they are related by

the expression of∆L100

3 =CB

LB

2 BT

× 28570 (2.19)

The dimensions are in feet and displacement in tons, salt water. A variation by the combination of L/B

and B/T for a parent ship of CB = 0.6 is shown in Fig. 2-7.

Since there were fewer correlations between CB and lcb based on the methodical model test results,

a linear variation of position of lcb with fullness was adopted here. This line is based on a mean of the

available test data. The lcb variation is shown in Table 2.5.

41

7 4

6 9 8

3

2 1 5

6.5 7.5 8.5

2.5

3.0

3.5

L/B

B/T

CB = 0.6lcb =-1.5

Figure 2-7: L/B and B/T variation for a parent ship of CB = 0.6

CB lcb (%)0.6 -2.5, -1.5, -0.5, 0.50.65 -2.5, -1.5, -0.5, 0.5, 1.50.7 -2.0, -0.5, 0.5, 1.5, 2.50.75 0.5, 1.5, 2.5, 3.50.8 0.75, 1.5, 2.5, 3.5

Table 2.5: Variation of lcb as a function of CB

Series 60 has a total of 62 ships in its family. It is consist of 5 parents, 40 ships varied by L/B and

B/T combination and 12 additional ships with lcb variation.

2.4.3 Trend validation domain

Overview

It would be ideal if validation could be undertaken for all applications of interest. However, the central

basis of the trend validation process is bench marking, whereby a limited number of CFD predictions

is made and compared to experimental data. Even if such a process is performed successfully, if the

validation domain is too limited in scope, it may be not appropriate for minimizing the risk of providing

a faulty prediction. Especially, the wave resistance coefficient is not necessarily a monotonically varying

function in relation to the ship speed and shape. The distribution of the wave resistance coefficients

on the change of FN shows humps and hollows which are caused by longitudinal and transverse wave

interferences. In addition, they are very sensitive to the change of hull form characteristics at each ship

speed. Therefore, a suitable domain for trend validation should be selected by considering both efficiency

and accuracy.

42

Form parameters to be used for selecting the validation domain.

The four form parameters L/B7, B/T 8, lcb9, CB10 will be used as validation parameters similarly to

Series 60. In this variation the LBP is normalized as 1. Therefore, the displacement of the ship is a

dependent variable in relation to the change of the three validation form parameters L/B, B/T , CB.

The change of L/B results in the change of the displacement and the width of the ship. By decreasing

L/B both the displacement and the breadth of the ship are increased. Wider ship shapes will be derived.

The change of B/T also results in the change of the displacement and the draft by fixing the breadth.

When increasing B/T the displacement and the draft T of the ship will be decreased. A flatter ship

form will be derived. L/B and B/T are the form parameters which scale the ship shape without any

nondimensional form change. This variation is realized by affine transformation.

In contrast to the L/B, B/T variations lcb and CB are form parameters which change the hull form

geometrically. The position of lcb is moved to the fore and after ship without changing its displacement.

However, the change of CB produces very complicated hull form changes. In this study, some global and

local form parameters such as CM 11 , CWP12 , CP

13 , LE14 , LR

15 , LP16 etc. are varied at the same time

deliberately when changing CB . These additional form parameters help to derive hull forms being as

similar as possible to Series 60.

7Length-beam ratioL : Length between perpendicularsB : Beam amidships at the designed waterline8Beam-draft ratioT : Draft to the designed waterline9Longitudinal center of buoyancy in percent of LBP from amidships, positive forward and negative aft.10Block coefficient : The ratio of the volume of displacement to the volume of the rectangular block with a length

appropriate to the type of the ship and a beam and draft equal to that at the maximum section area

CB =∇LBT

11Midship section coefficient : The ratio of the midship section area at draft T (AM ) to the area of a rectangle withwidth B and draft T

CM =AM

BT

12Waterplane area coefficient : The ratio of the waterplane area at the designed waterline (AW ) to the area of thecircumscribed rectangle

CWP =AW

LWLB

13Prismatic coefficient : The ratio of the volume of displacement to the volume of a cylinder with a length L and a crosssection equal in area to that of the maximum section at the designed waterline (AX)

CP =∇LAX

14Length of entrance15Length of the run: the length of the portion of the bottom aft from the widest section to the trailing edges of the stern16Length of the parallel middle body

43

The range of each form parameter

A subdomain of the whole range of Series 60 is chosen while the number of variational cases was made

more numerous. This is shown in Fig. 2-8 and Fig. 2-9 (shaded).

Figure 2-8: Validation range of CB and L/B as a part of Series 60 variational range

Figure 2-9: Validation range of CB and lcb as a part of Series 60 variational range

Each form parameter has 5 combinatorial values respectively. This is indicated in Table 2.6. The

bold numbers in the combinatorial values designate the reference parent ship No. 1 in the validation

domain.

In particular, CB and L/B are form parameters which mainly affect the change of the physical

44

Range of form parameter Combinatorial values0.6 < CB < 0.7 0.6, 0.625, 0.65, 0.675, 0.7−1.5 < lcb < 0.5 -1.5, -1.0, -0.5, 0.0, 0.56.5 < L/B < 7.5 6.5, 6.75, 7.0, 7.25, 7.52.5 < B/T < 3.5 2.5, 2.75, 3.0, 3.25, 3.5

Table 2.6: The bounds and values of each form parameter

1.00 0.25 0.5 0.75

3.52.5 2.75 3.253

6.57.5 7.25 6.757.0

0.5-1.5 -1.0 0.0-0.5

0.70.6 0.625 0.6750.65

ti

B/T (t1) range

L/B (t2) range

lcb (t3) range

CB (t4) range

Figure 2-10: Relation of range of ti and range of each form parameter

phenomena while B/T has relatively weak influence. In practice here, the CB and L/B ranges are

determined by half of the total range of the Series 60 variation, i.e., the validation range of CB contains

half of the total range in the direction of increasing displacement from the starting point of the ship.

The L/B range is also half of the total range of the Series 60 variation in the direction of increasing

displacement and width. The lcb has a large range containing the Series 60 variation in all possible

CB ranges. The B/T range is identical to Series 60. The increase of B/T results in a decreasing

displacement.

The range of each form parameter i is normalized by means of the free variable ti. This notation assists

in describing a four dimensional variational space in the parameters ti, i = 1, ..., 4. Namely, the values of

each form parameter B/T , L/B, lcb and CB are represented by free variables t1 to t4 respectively. Each

ship is uniquely associated with a set of discrete ti values from the raster in Fig. 2-10. This notation is

shown in Fig. 2-10.

Selection of the geometric validation domain

In general, validation requires intensive computer simulations. For example, the number of the total

combinations of each variational case given in Table 2.6 will be 625 = 54 terms for one speed. If all

variations would be validated at six FN respectively, 3750 CFD calculations were required. A systematic

45

approach based on a sequential experimentation scheme is applied [17] [73]. This aims at conducting a

minimum ordered set of validations to analyze the error tendency of the CFD prediction. This involves

performing single (univariate) validation cases over the entire validation domain in order to identify

the characteristic error tendency and then performing combined cases to build on the interacting error

tendency in this range of validation. This makes it possible to select a validation domain requiring only

a relatively limited number of runs[11] [73]. Among various possible fractional factorial experiments

(FFEs), a two level FFE with center points is recommended for screening the experiment design because

of its efficiency and its insensitivity to response nonlinearity. Three- and higher-level FFEs usually

yield little improvement, but show results with great design and experimental complexity. However, in

the validation, the corners and the edges consisting of the bounds of the validation domain are very

important in order to know whether the CFD simulation covers the furthest term from the validation

starting point. Therefore, we use three points of t = 0, 0.5 and 1 in each normalized range, the validation

domain is determined by the three level factorial design with 4 factors (the 34 design [73]). Among them

hull forms showing more than 2 middle points (t = 0.5) were excluded. Only for the main axes of form

parameters, on which one form parameter is varied by fixing the other form parameters as zero, two

points of t = 0.25 and 0.75 are added. Under such considerations the validation domain is designed by

the following process:

1. The ranges of four form parameters are divided into two ranges with three points of ti =0, 0.5, 1

respectively. All possible factorial combinations of validation cases yield a total of 81 terms:

34 = 81(terms) (2.20)

2. We avoid those combinational terms which have more than two center points of each form parame-

ter, i.e., we accept only the validation terms which are corners and center points along the edges.

This means that the center points which are on the two dimensions (24) or in the hexahedron in

three (8) and four dimensions (1) are removed from factorial combination of the validation domain.

34 1− 4C22

3

21

3

2

+4 C32

3

1

3

3

+4 C41

3

4

= 48(terms) (2.21)

3. The principal axes stretched from the validation starting point are the t1(B/T ), t2(L/B), t3(lcb),

t4(CB) axes and have additionally 2 more points, t = 0.25 and 0.75. They are divided into 4 ranges

with 5 points on each axis.

2(steps)× 4(axis) = 8(terms) (2.22)

46

No. B/T L/B lcb CBof ship Dim. t1 Dim. t2 Dim. t3 Dim. t4

1 2.5 0 7.5 0 -1.5 0 0.6 02 2.75 0.253 3 0.54 3.25 0.755 3.5 16 2.5 0 7.25 0.257 2.5 0 7 0.58 3.5 19 2.5 0 6.75 0.7510 2.5 0 6.5 111 3 0.512 3.5 113 2.5 0 7.5 0 -1 0.2514 2.5 0 7.5 0 -0.5 0.515 3.5 116 2.5 0 6.5 117 3.5 118 2.5 0 7.5 0 0 0.7519 2.5 0 7.5 0 0.5 120 3 0.521 3.5 122 2.5 0 7 0.523 3.5 124 2.5 0 6.5 125 3 0.526 3.5 1

Table 2.7: The list of ship variations from raw data and as supplemented by interpolation

4. We add two more terms which own the same form parameters as the parent ship of Series 60 with

CB = 0.65 and 0.7 (2 terms).

5. The total number of validation terms in their domain is therefore 58 terms:

(2.21) + (2.22) + 2(terms) = 58(terms)

The overview of the detailed table and figure are shown in Table 2.7, 2.8 and Fig.2-11. Here, it

will be examined how to determine the level and terms for a two and three dimensional range of

validation in Fig.2-11.

Validation range of ship speed

The validation range of ship speed is chosen to range from FN = 0.2 to 0.316 in steps of 0.025, i.

e., validation will be done for six Froude numbers FN = 0.2, 0.225, 0.25, 0.275, 0.3, 0.316. This is a

47

27 2.5 0 7.5 0 -1.5 0 0.625 0.2528 2.5 0 7.25 0.25 -1.54 -0.02 0.65 0.529 2.5 0 -1.5 0 0.65 0.530 3.5 131 2.5 0 6.5 132 3.5 133 2.5 0 7.5 0 0.5 134 3.5 135 2.5 0 6.5 136 3.5 137 2.5 0 7.5 0 -1.5 0 0.675 0.7538 2.5 0 7.5 0 -1.5 0 0.7 139 3 0.540 3.5 141 2.5 0 7 0.542 3.5 143 2.5 0 6.5 144 3 0.545 3.5 146 2.5 0 7 0.5 -0.55 0.475 0.7 147 2.5 0 7.5 0 -0.5 0.548 3.5 149 2.5 0 6.5 150 3.5 151 2.5 0 7.5 0 0.5 152 3 0.553 3.5 154 2.5 0 7 0.555 3.5 156 2.5 0 6.5 157 3 0.558 3.5 1

Table 2.8: The list of ship variations from raw data and as supplemented by interpolation

48

t3(lcb)

t2(L/B)

t1(B/T)

t4(CB)

t4=0(CB=0.6)

123

4

5

15

2120

19 22 24

16

10

11

128

6 7 917

26 2523

1314

18

2733

35

31

3230

29

37

40 42 45

44

43

49

5651 5452

53

4839

55

47

4138

5758

50

3436

t3(lcb)

t2(L/B)

t1(B/T) t3(lcb)

t2(L/B)

t1(B/T)

t4=0.5(CB=0.65)

t4=1(CB=0.7)

Figure 2-11: Four-dimensional hyperspace of free variables

49

choice taking into account our computational capacity and time limits. For the reason of the limited

experimental data, the validation range of the flow state will be reduced according to the CB of the ship.

The validation range of CB = 0.6 will be validated over the total flow range. However, in the case of

CB = 0.65 the validation will be carried out only in the flow range from FN = 0.2 to FN = 0.275 and

for CB = 0.675 and 0.7 only up to FN = 0.25.

50

Chapter 3

Verification

3.1 Overview

Verification assesses whether the computational model represents the analytical model correctly and

determines the level of the computational accuracy of the numerical model by estimating the errors

owing to numerical error types. At the same time, this process enables to choose those initial variables

for the CFD code which will generally lead to a numerical error small enough to be neglected. If it

is impossible to determine such inputs due to limits in computational effort and capacities, alternative

inputs optimal to the situation will be chosen and the degree of the systematic errors will be estimated

by error propagation analysis. This process will be performed here by sensitivity analysis within the

validation domain by using SHIPFLOW. The degree of individual numerical error is estimated by using

the known criteria [83][91][92][93][94][109]. This process makes it possible to consider the remaining

numerical errors in the CFD prediction as random errors over a wide application region. The goal of

the verification in this task is not only to estimate the numerical errors in the CFD predictions, but

also to determine the best initial variables for SHIPFLOW within a required numerical accuracy range

which lead to the best use of the code.

The following processes are performed in this chapter:

1. Choose numerical input parameters which are expected to exert a noticeable influence on the

numerical error.

2. Determine a specific class of applications and a specified range of flow conditions to be verified (a

verification range).

3. Find criteria for the error estimation of the individual error types due to input parameters.

51

4. Plan the sensitivity analysis including the bounds and levels of the perturbation of the input

variables.

5. Estimate the convergence solutions to be compared with the CFD predictions.

6. Estimate the individual uncertainties and influences of the inputs.

7. Determine the initial values of SHIPFLOW.

8. Estimate the total numerical errors over the verification range.

3.2 Verification methodology

3.2.1 Numerical parameters and verification range

Three convergence analyses will be carried out : a panel convergence test, a computation domain

convergence test for discretization error estimation, and an accuracy test for equation solving error

estimation.

The panel convergence test is carried through by changing the grid size, its aspect ratio, and stretching

the panels in their distribution both on the body and on the free surface. The computation domain of

the free surface is changed in the longitudinal (X) and the transverse direction (Y), and the XY region

accounting for the Kelvin angles respectively. The accuracy test for nonlinear equation iteratively solving

error estimation, which will be called as ” iterative convergence” is related to the termination criteria

and their convergence tendencies. In addition, since the wave resistance coefficient is calculated by

integrating the pressure on the body, the post-processing error is very sensitive to the panel on the

wetted hull surface. The geometrical fairness of the ship, the panel generation method, and the mesh

fineness play an important role for the accuracy of the predictions.

The influence of each numerical parameter on the solution will be different for each design application.

This leads to carrying through the verification in a range at least containing the extreme cases of

the sample. The influence of ship speeds and shapes on the numerical accuracy will be verified for 6

ship speeds and 5 ship variations in the hull form. Table 3.1 shows the numerical parameters for the

convergency analysis and the verification range.

3.2.2 Criteria for error estimation

The quality of the numerical model is related to the degree of accuracy in the model. The difference

between the simulated ’reality’ and the conceptual ’reality’ is caused by the necessity of computational

simplifications and approximations. The lack of accuracy is caused mainly by the discretization which

52

Ship Speeds Hull form variations Numerical parameters

Fn⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0.20.2250.250.2750.30.316

ShipsParent ship(No.1), CB(No.38),L/B(No.5), B/T (No.10), lcb(No.19) Panel⎧⎨⎩ Panel size

Aspect ratioStretching

Computational Domain⎧⎨⎩ Longitudinal region(X)Transverse region(Y )Kelvin angle region(XY )

Iterative Convergency⎧⎪⎪⎨⎪⎪⎩Iterative convergence criteriaPanel sizeAspect ratioComputational region

Table 3.1: Ranges of sensitivity analysis

includes imprecise shape definition, truncation error, and discretization error, approximate solution

procedures and post processing of simulated results. Although the numerical model is adequate in

consistency and stability, no guarantee can be given for the accuracy of solutions under all conditions.

Moreover, since the consistency and stability conditions provide necessary and sufficient conditions only

for linear PDEs, they are not sufficient for the convergence of a numerical solution to the continuum

solution for nonlinear problems. The accuracy depends on how well the continuum information can

be transferred through the discrete points and the degree of accuracy is estimated by approximating

the exact solution of the PDEs or errors based on the solutions of the simulation. There are three

different convergence tendencies satisfying stability criteria. The convergence tendency of solutions can

be classified by such categories.

x 20151050

1.5

1

0.5

0

0.5

-1

[a] Pure oscillation

x 20151050

1.5

1

0.5

0

0.5

-1

[b] Oscillatingly convergent

x 20151050

2

1

0

-1

-2

[c] Monotonically convergent

53

1. Pure oscillation (εn+1/εn = −1) :

The mean value of the oscillating predictions is calculated as

Φmean = 1/2(Φmax +Φmin) (3.1)

Its uncertainty and error are

UΦmean= 1/2(Φmax − Φmin) (3.2)

Error = Φmean − UΦmean (3.3)

2. Oscillatingly convergent (−1 < εn+1/εn < 0) :

The uncertainty is equal to half of the upper and lower limits of the solution envelope. The

uncertainties are calculated in a similar way as before. However, a cautious determination of the

solution envelope is important when the convergence rate is fast relative to the change of the input

variables.

3. Monotonically convergent (0 ≤ εn+1/εn < 1) :

Since the error is the difference between the convergent exact solution and the calculated solution

as follows:

Error = Φexact − Φn (3.4)

it is very important in this case to decide how to get the exact solution. For the discretization error

estimation the Richardson extrapolation [91][92] will be used. This method can only be applied to

the monotone convergence cases. For an iterative convergence error estimation an approximation

method is utilized using the largest eigenvalue of the matrix [93].

In the following sections methods for estimating the exact solutions and their uncertainty will be

discussed.

Richardson Extrapolation (1910, 1927) [91][92]

The Richardson extrapolation is an h2 extrapolation based on an asymptotic estimation of the grid

convergence accuracy of a particular discretized solution. The discrete solutions f are assumed to be

represented in the grid spacing h:

f = fexact + g1h+ g2h2 + g3h

3+... (3.5)

54

where g1, g2 are independent continuum functions. For two grids of different size the equations are as

follows:

f1 = fexact + g1h1 + g2h21 + g3h

31 +

... (3.6)

f2 = fexact + g1h2 + g2h22 + g3h

32+

... (3.7)

where h1 is the fine grid spacing and h2 is the coarse grid spacing.

The fundamental idea of the extrapolation is to solve for the unknown continuous functions gi at the

grid points. The leading order error terms in the assumed error expansion are eliminated by combining

each discrete solution f on a different grid spacing h.

1. First order extrapolation

For a second-order method, g1is zero. g2 is calculated from (3.6) and (3.7) respectively:

g2 =f1 − fexact

h21=f2 − fexact

h22(3.8)

fexact =h21f2 − h22f1h21 − h22

+H.O.T 1 (3.9)

Let the grid refinement ratio be r = h2/h1.

fexact = f1 +f1 − f2r2 − 1 +O(h

4) (3.10)

Generally, the grid refinement ratio should be an integer and commonly be used with a grid

doubling or halving. Also the estimated exact solution is fourth-order accurate if f1,and f2 are

second-order accurate. However the order of accuracy is dependent on the discretization scheme.

2. Second order extrapolation with 3 grids

It is done by an iterative extrapolation by using the estimated corrections as follows. The second

order extrapolation results in 6 orders of accuracy.

Panel size (h) Solutions O(h2) 1st order ext. O(h4) 2nd order ext. O(h6)

h1(fine) f1

h2(medium) f2 f1 +f1 − f2r2 − 1 = f

1st1

h3(coarse) f3 f2 +f2 − f3r2 − 1 = f

1st2 f1st1 +

f1st1 − f1st2

r4 − 1 = f2nd1

1Higher-order terms

55

3. Higher order extrapolation (by Roache)[93][94]

fexact ∼= f1 + f1 − f2rP − 1 (3.11)

where p is the order of the method and r is the grid refinement ratio. If central differences are

used, then the extrapolation is (p+2) order accurate. For a coarse grid solution the extrapolation

is

fexact ∼= f2 + (f1 − f2)rP

rP − 1 (3.12)

The estimated exact solution can be used to estimate the error of the fine grid solution. An Estimated

fractional error E1 for the fine grid solution f1is expressed as:

E1[fine grid] = ε/(rP − 1) (3.13)

E2[coarse grid] = εrp/(rP − 1) (3.14)

where, ε = (f2 − f1)/f1.The Actual fractional error A1 of the fine grid solution is expressed as usual:

A1 = (f1 − fexact)/fexact (3.15)

A1 = E1 +O(hP+l, E21) (3.16)

The estimated fractional error E1 is an ordered error estimator, i.e., an ordered approximation of the

actual fractional error of the fine grid solution. The general error defined as ε which is the quantity

commonly reported in the grid refinement studies is not conclusive evidence. However, there are some

disadvantages related to the Richardson extrapolation.

• The grid refinement ratio r must be an integer.• The truncation error convergence must be monotone in the mesh spacing h.• It is not conservative.

• High order extrapolation is not valid for a coarse grid.• The extrapolation magnifies round off errors and incomplete iteration errors.

Grid Convergence Index (GCI by Roache) [93]

This method suggests the uniformity of grid convergence studies and uncertainty estimation in the

reporting of results. This is based on the assumption that consistency and stability are satisfied and

56

Taylor expansion is valid asymptotically. This is an objective asymptotic approach to the quantification

of the uncertainty of grid convergence by incorporating a safety factor into the Richardson-based error

estimates. This can approximately relate the results from any grid refinement test to the expected results

from a grid doubling using a second order method. Roache has recommended a conservative value of 3

as a safety factor which has the advantage of relating any grid convergence study having any r and p to

one with a grid doubling and a second-order method (r = 2, p = 2)

GCI[fine grid] = 3|ε|/(rP − 1) (3.17)

for coarse grid,

GCI[coarse grid] = 3|ε|rP /(rP − 1)= rPGCI[fine grid] (3.18)

where ε = fi+1 − fi. From the solutions with three different grid spacings, the order of solutions can be

observed as follows:

P k =ln(ε23/ε12)

ln(rk)

In multiple dimensional theory separate convergence studies are possible. In the case of (rx = ry = rz)

the total GCI can be obtained by simple addition as follows:

GCI = GCIx +GCIy +GCIz (3.19)

A mixed order method, non Cartesian grid such as a streamline panel or a stretching panel can be

also considered by a separated GCI method. Decreasing (Increasing) ε, GCI indicates grid convergence

(divergence) with uncertainty estimates based on ε for the finest grids.

Iterative convergence error estimation

Based on the evaluation of the iteration records for CWCFD, the level of iterative convergence is deter-

mined by the number of orders of magnitude reduction and the magnitude in the residuals. Generally,

a difference ranging between |Φn − Φn−1| < ε is used as the iteration cutoff criterion. However the

difference may be small when the error is not small. Also, the residual can be used as the termination

criterion. The iteration is terminated when the residual norm has been reduced to a fraction of its origi-

nal size. The residual and the error usually do not fall in the same way at the beginning of the iteration

process, the criterion is also needed because if the matrix is poorly conditioned, the error may be large

57

even when the residual is small. Therefore, the iterative convergence error should be estimated even

when the solution Φn is satisfying the iterative criteria given as the residual norm. The error estimation

is carried through as follows:

1. Pure oscillatory:

Error =Mean(Φ)− 12(Max(Φ)−Min(Φ)) (3.20)

2. Mixed convergent/oscillatory:

Error = Φn − 12(Upp(Φ)− Low(Φ)) (3.21)

where, Upp(Φ) is upper bound and Low(Φ) is lower bound of the solution Φ.

3. Monotonically convergent:

Error = Φ− Φn ≈ ε

λ1 − 1 (3.22)

where ε is the difference between two solutions (Φn − Φn−1) and λ is the largest eigenvalue which

is approximated as(ε12ε23

)[93][94].

3.3 Panel convergence analysis

3.3.1 Effect of the body panel distribution on the CWCFD prediction

The body panel structure exerts much influence on the wave resistance prediction. Sometimes, an

unfavorable panel distribution error even induces a negative wave resistance on the free surface due to

the use of the double body model for the first iteration of the nonlinear wave resistance calculation. In

addition, the body panel structure is directly related to the change of the wetted surface area. This

affects the resistance coefficient although the same physical situation is considered. Therefore, the body

panel sensitivity analysis must be tested for the given ship shape variations. The hull shapes variable four

design parameters, namely CB , lcb, L/B, and B/T,are normalized by unity. According to the changes

of CB and lcb the ships may have quite different forms. Full shape hulls showing a big local change of

curvature should have densely distributed panels. The parts of the stern and the bow require a greater

density of the panel. However, if a thin and simple formed ship is covered with too many panels, the

errors may even increase due to the round off error. Therefore, the optimal number and structure of the

panels should be determined before the wave resistance calculations are done.

58

Criterion for checking the body panel error

There exist analytical benchmark tests for choosing the body panels. Based on potential theory, the

integral of the local distributed pressure on a deeply submerged body should be zero at zero speed

because neither rotational flow nor energy loss exist near the body. The criterion determining a suitable

panel structure on the body is the total pressure coefficient (CPXI) which is calculated by x directional

integration of the pressure distributed on a deeply submerged body at zero speed. Indeed, a panel

distribution with a zero total pressure coefficient (CPXI) on the body represents the ideal case. Panel

arrangements having a minimal total pressure coefficient within 0 < CPXI < 5e− 06 are chosen for 58varied ships. It is proven that the panel error within the chosen criterion 0 < CPXI < 5e − 06 affectsCWCFD of the ship No.1 within 1 % at FN = 0.2 through the following tests. The ship No.1 is the

thinnest ship among the 58 varied ships and CWCFD at FN = 0.2 is the smallest one regarding the speed

range.

Choice of panel size and arrangement on the body

The sensitivity of the error according to the change of the hull form by CB and lcb will initially be tested.

The determination of the panel sensitivity according to the change of the scaling factors L/B and B/T

will then proceed. The shapes of the panels are quadrilateral. The panel changing elements include the

length and the aspect ratio of the panel. Two distinct discretization regions of the body (in the stern and

the bow region) are tested, while quadrilateral panel shapes which are long in the longitudinal direction

of the ship are used. The first setting of the panel size variation for Series 60 CB = 0.6 is as follows:

-Transverse panel size: ∆y/L = 0.1×Draft-Longitudinal panel size: ∆x/L = 5×∆y/L-Number of transverse panels: Ny/L = Lg/∆y/L, Lg = (B/2)2 + T 2

The size of ∆x/L and ∆y/L is varied systematically. The size of ∆y/L is changed by a refinement

ratio of 1.2. After the first setting, two smaller and two larger sizes are tested. Since the size of ∆x/L is

set depending on the size of ∆y/L, namely 5×∆y/L, the variation is done in steps of integer multiplesof the transverse panel size ∆y/L. The search process to find a suitable panel size begins with the size

variation in ∆x/L. From the first setting in the size ∆x/L = 5 ×∆y/L, CPXI is calculated in order tofind the size with a minimum CPXI . At first this is stopped at the size 4×∆y/L. From this chosen pointthe search process to find a panel size ∆y/L is run. CPXI increases when the panel size ∆y/L increases.

For a decreasing panel size ∆y/L/1.2, CPXI decreases but with too small panel sizes ∆y/L/1.22 it may

increase slightly. We will choose the panel size 4 × ∆y/L, ∆y/L/1.2 as the actual panel size in thisrefinement stage. From the point of variation, a finer panel refinement will be carried through. Since the

sensitivity of CPXI is larger according to the change of the panel size in the longitudinal direction, the

59

Panel size in y directionCPXI(×104) ∆y/L/1.2

2 ∆y/L/1.2 ∆y/L ∆y/L × 1.2 ∆y/L × 1.22(0.0037) (0.0044) (0.0053) (0.0064) (0.0077)

Panel size in x3×∆y/L 0.2107

3.75×∆y/L4×∆y/L

4.25×∆y/L0.1911

0 .19510.18510 .2031

0 .19450.2016 0.2406

5×∆y/L 0.24486×∆y/L 0.27777×∆y/L 0.3486

Table 3.2: CPXI(×104) according to the change of the panel size on the body

No. of stretching Bow Stern CPXI0 No No 0.1851e− 041 0.015 0.015 0.5593e− 042 0.01 0.007 0.2697e− 043 0.01 0.006 −0.1498e− 054 0.01 0.0064 0.9473e− 07

Table 3.3: Variation of the panel distribution by means of the stretching function

panel size should be more refined like 3.75×∆y/L and 4.25×∆y/L. This is tested in the neighborhood

of the chosen point and shown in italics. CPXI increases for all refined panel sizes. The chosen point

may be very close to the real minimum. However, the magnitude of CPXI still exceeds the criterion for

the panel choice.

A stretched panel distribution can help to find a better panel choice because the form of the ship is

changed very quickly in the bow and the stern regions. In order to cover the region of rapid curvature

change more smoothly and at the same time to reduce the round off error in the flat region, the stretching

can be useful. In this thesis, a tangential stretching will be used in these two regions, bow and stern.

As shown in Table 3.3, the stretched panel can cover the ship surface better and reduce CPXI up to

0.9473e− 07. This process is carried through iteratively for all 58 ships.

Effect of the body panel error on the accuracy of CWCFD

In order to investigate the influence of the body panel error on CWCFD the wave resistance is computed

for three body panel types which show positive, negative, and minimum CPXI .They are all of the same

panel size (4×∆y/L,∆y/L/1.2) but they have different stretching types of No.0, 3, and 4 shown in Table3.3. If the relationship between the body panel error and the CWCFD prediction is known, it is possible

to recognize the degree of the panel error propagating into the CWCFD prediction. Since the direct error

estimation in CWCFD is difficult we consider the CWCFD calculated by using the panel of No.4 in Table

60

3.3 as a reference prediction. It is denoted as CWCFD No.4 . The other predictions are compared with

CWCFD No.4 and the relative differences in predicting CWCFD will be related to the body panel error.

This makes it possible to investigate the degree of propagation of the body panel error to the CWCFD

prediction.

Table 3.4 shows the differences in the CWCFD predictions computed by different body panel types,

No. 0, 3, and 4. The difference between the CWCFD predictions is denoted by ∆CWCFD 04 and

∆CWCFD 34 respectively and defined as

∆CWCFD 04 = CWCFD No.0 − CWCFD No.4

∆CWCFD 34 = CWCFD No.3 − CWCFD No.4

The difference in CPXI is denoted by ∆CPXI 04 and ∆CPXI 34 respectively, defined in the same

way.

These differences describe the degree of the influence caused by the body panel on CWCFD. In

fact, when the nonlinear wave resistance is calculated iteratively, the size of the body panels and their

distributions are changed in each iteration. The body panels used in the last iteration are somewhat

different from the first panels. However, the differences in CWCFD are caused by the initially given

number of panels and their arrangement.

As shown in Table 3.4, it is evident that a positive CPXI causes an increase in CWCFD and a negative

CPXI causes a decrease in CWCFD. CWCFD No.0 is slightly overpredicted and its relative magnitudes

decrease for increasing FN . Their percentage rate to the best prediction CWCFD No.4, as shown by the

parameter ∆CWCFD 04/CWCFD No.4(%)2 , is within 5 % over FN . ∆CWCFD 34/CWCFD No.4(%) are

within 1 % over FN . The sensitivity of the CWCFD predictions to the change in CPXI is investigated by

the parameters ∆CWCFD 04/∆CPXI 04 and ∆CWCFD 34/∆CPXI 34. The ratio shows that the degree of

the influence of CPXI on CWCFD is dependent on the magnitude of CPXI but not directly proportional

to the magnitude of CPXI . The ratio of differences in CWCFD,|∆CWCFD 04||∆CWCFD 34| shows that the ratio of

differences in CWCFD remains unaffected by the change of FN .

Conclusion

This investigation shows that the number and the arrangement of the panels on the body has much

influence on the accuracy of the CWCFD predictions. The panelization error on the body can be estimated

by the global pressure coefficient in the longitudinal direction (CPXI) of the submerged body at zero

speed. The magnitude of CPXI influences directly the magnitude of CWCFD and it remains unchanged

2∆CWCFD 04/CWCFD No.4(%) =(CWCFD No.0−CWCFD No.4)

CWCFD No.4× 100

61

FN0.2 0.225 0.25 0.275 0.3 0.316

∆CWCFD 04(×e− 05) 1.2 1.4 1.4 1.5 1.7 1.7∆CWCFD 34(×e− 05) -0.26 -0.35 -0.32 -0.37 -0.43 -0.46∆CWCFD 04/CWCFD No.4(%) 5 5 4 1 1 1∆CWCFD 34/CWCFD No.4(%) -1 -1 -1 0 0 0∆CWCFD 04/∆CPXI 04 0.6 0.7 0.8 0.8 0.9 0.9∆CWCFD 34/∆CPXI 34 1.8 2.2 2.0 2.3 2.7 2.9|∆CWCFD 04||∆CWCFD 34| 4 4 4 4 4 4

Table 3.4: CWCFD for different body panelizations

with the change of the ship speed FN . This means that the degree of the panelization error in CWCFD

decreases for increasing FN proportionately to the increasing value of CWCFD.

The characteristics of the body panel error - the degree of the propagation and the sensitivity to the

change of FN - can be used to find a suitable body panel configuration within the required accuracy. If

the accuracy of the body panelization is required within 1 % of CWCFD for a Series 60 CB = 0.6 ship,

CPXI should be smaller than 5e-06 in the region 0.2 ≤ FN ≤ 0.25 and smaller than 2e-05 in the region0.25 < FN ≤ 0.316. The criterion for choosing the panel size is chosen as 5e-06. The number and thearrangement of the panels are chosen independently for each of the 58 ships. The same panels will be

used for all speeds.

3.3.2 Design of the panel sensitivity test for free surface panels

Panel variation

The input parameters for the panel sensitivity test are the panel sizes hi, their aspect ratio AR and

the stretching coefficient. Five different panel sizes hi are varied by the panel refinement ratio r which

remains constant as r = hi+1/hi = 1.2, h1 representing the finest panel size and h5 the coarsest one.

This panel size variation is iteratively carried through for three different shapes of panels showing an

aspect ratio of ∆xi : ∆yj = 1 : 1, 1 : 1.5, and 1 : 2 . Each size of a panel (in both directions) can be

expressed in terms of the coarsest or finest panel size as follows:

∆xi = hi = ri−1 · h1 or r−(i−1) · h5 (3.23)

∆yij = hi ·ARj (3.24)

where i = 1 to 5, j = 1, 1.5, 2

62

Figure 3-1: Panel variations and their sizes

Panel sizeh1 h2 h3 h4 h5

Aspect ratio 0.0174 0.0208 0.0250 0.0300 0.03601 6608 4931 3689 2830 23431.5 4762 3752 2816 2190 18742 3768 2942 2331 1950 1673

Table 3.5: Total number of panels

The area of the panel is simply calculated as follows:

Aij = ∆xi ·∆yj = ARj · h2i = ARj · (ri−1 · h1)2 (3.25)

Fig. 3-1 shows each panel variation on the plane consisting of the axes of the panel size (h) and the

aspect ratio(AR). Each individual point of a panel variation is expressed as (hi,ARj), where i = 1 to 5

and j = 1, 1.5, 2. The CWCFD predictions calculated with those panels are expressed as CWCFD (hi,ARj).

Sometimes, a groupwise notation for the panel variation will be used by using symbolic forms such as

AR = j, or h = i, where AR = j means the five panel size variations hi from i = 1 to 5 on the axis

of AR being equal to j. Likewise, h = i means the three aspect ratio variations ARj from j = 1 to

3 on the axis of h being equal to i. The wave resistance and its sensitivity to the panel size will be

observed on the h − AR plane. Table 3.5 shows the total number of panels covering the free surface

63

FN

Ship

Panel

s60 L/B B/T LCB CB

0.25

s60 L/B B/T LCB CBS60(No.1)

0.25

S60(No. 1)

Case 1 Case 2 Case 3 Case 4

0.2 0.225 0.25 0.275 0.3 0.316 0.2 0.225 0.25 0.275 0.3 0.316

Figure 3-2: Cases of panel configuration sensitivity analysis in the verification range

by each panel type. In this case, the computational region for the panel convergence test is composed

by 0.5LPP upstream from FP, LPP downstream from AP in the longitudinal direction, and 0.7LPP

in a half symmetry transverse direction. The response of the wave resistance coefficients to the panel

sensitivity test can be expressed not only for the panel size and its aspect ratio but also for the total

number of panels and the number of panels constituting a wave length. This analysis will be tested in

the verification range including five ships and six FN .

Cases of panel convergence analysis

The panel sensitivity test will begin with reference ship No.1 at the speed of FN = 0.25 (case 1). The

reference ship No.1 is denoted by S60 in the following. This is shown in Fig. 3-2. The accuracies of

the panels for six different FN will be tested for the same ship (case 2). Then, the panel convergence

analysis will be systematically carried out for five ship shapes at a ship speed of FN = 0.25 (case 3).

The tendencies of the panel convergence and its error will be investigated related to five ship shapes and

six ship speeds (case 4). For all cases, each panel convergence analysis is based on the panel variation

including the 15 panel types depicted in Fig. 3-1 and the panel convergence tendencies are investigated

as a function of the panel size and shapes. These tests aim at determining a suitable panel structure

arrangement for each simulation case and at establishing the degree of the error of the used panel.

64

3.3.3 Panel convergence tendencies and error estimation (case 1)

CWCFD predictions according to the change of the panelization for the ship No.1 at FN = 0.25

Fig. 3-3 shows the wave resistance coefficients CWCFD calculated by the 15 different panel variations

at FN = 0.25. These are the predictions for the Series 60 parent ship of CB = 0.6, the reference ship

No.1. The predictions are shown on the axes of the total number of panels and the number of panels per

wave length3. They show a convergence tendency towards an increasing number of panels. This diagram

shows that increasing the panel number in the longitudinal direction causes a more rapid convergence

of the predictions than that in the transverse direction at FN = 0.25, while the predictions with the

panels having an aspect ratio of 2 (AR = 2)4 show a somewhat oscillating convergence tendency. If the

number of panels has to be limited, the use of a denser panel distribution in the longitudinal direction, i.

e., when the aspect ratio is larger than one, is more efficient and yields a more rapid convergence. The

total difference of the solutions calculated by the 15 panel variations is 10 ∗ E − 05. This magnitude isabout 25 % of the solution CWCFD (h1,AR1) based on the finest panel. About 23 % is due to the change

of the panel size h on the axis of AR = 1 and about 2 % is due to the AR variation at h = 1. This

clearly shows the influences of the aspect ratio and the panel size on the solution.

3.8E-04

4.3E-04

4.8E-04

5.3E-04

2319161311

The number of panels per wave length( Wave length: 0.39 m )

CW

CF

D

AR=1

AR=1.5

AR=2

3.75E-04

4.25E-04

4.75E-04

5.25E-04

1000 3000 5000 7000

The total number of panels

CW

CFD

Figure 3-3: Grid convergence tendency as a function of the number of panels (Fn = 0.25, Ship No. 1)

3This expression of the wave resistance coefficient based on the panel number per wave length is useful in comparisonof the predictions at various speeds.

4This expression AR = 2 includes the five panel sizes on the axis of h having an aspect ratio of 2, i. e., the five panelsof 5

i=1(hi,2) are refered to shortly as AR = 2.

65

Panel convergence rate

The convergence rate of the CWCFD predictions is different in each case according to the used panel

sizes and types. In order to investigate the degree of the effect of the panels on the prediction, the panel

convergence rate to the change of the number of panels depending on both hi and ARj is investigated.

Fig 3-4 shows the convergence rate on the average number of two panel types. The convergence rate as

a function of the panel number is defined as

CRij =CWCFD (hi,ARj) − CWCFD (hi+1,ARj)

(Ni,j −Ni+1,j) (3.26)

where CRij is a convergence rate between the panel (hi,ARj) and (hi+1,ARj), Ni,j is the number

of panels (hi, ARj), and (Ni,j − Ni+1,j) is the difference between the number of panels (hi, ARj) and(hi+1, ARj). Each convergence rate is expressed on the average number of panels calculated as (Ni,j +

Ni+1,j)/2. Here, we assume that the discrete values calculated above (3.26) approach the real convergence

rate of CWCFD with the increase of the number of the panels.

From Fig. 3-4, we can see that the domain of the convergence rate as a function of hi on AR = j

is larger than that as a function of ARj on h = i. When the number of panels is higher than 2500,

the convergence rate due to the h variation is larger than that due to the AR variation, even when the

finest panel changes from (h2,AR1) to (h1,AR1) Looking at the convergence rate due to the change of

AR, h = 1, 2, 3, the convergence rate of the panel change from AR = 1.5 to AR = 1 is almost zero.

This means no further grid refinements are required and the aspect ratio of 1 is not deemed necessary.

When the panel number gets near 4500, the convergence rates due to the change of the panel size h

at AR = 1 and AR = 1.5 are similar and still larger than the convergence rate due to the change of

AR. If a panel has to be chosen based on the convergence rate, the panel (h1,AR2) is preferred in

comparison with (h2 , AR1.5), (h3 , AR1) near 3300 and the panel (h1,AR2) is preferred in comparison

with (h2 , AR1) near 4500. Around the panel number 3300, the convergence rate decreases when the

aspect ratio decreases. Generally, longitudinally denser panels yield a rapid convergence if the number

of panels is limited, since the solution is much more sensitive to the increase of the panel numbers in the

longitudinal direction than to that in the transverse direction.

The absolute magnitude of the convergence rate cannot always be the criterion for the choice of a

panelization. The convergence rate can be affected by the tested number of panels and the convergence

tendencies. When the predictions show oscillating tendencies with respect to the change of the panel

number, the convergence rate has to be considered together with the exact solution. In comparison to

the panel convergence rate around the panel number 3300, the convergence rate of the prediction with

AR = 2 is very small when the panel size is refined from h2 to h1. However, the second derivative

66

of the panel convergence rate is very high in the above refinement since the predictions are somewhat

oscillating. This will be discussed in the next section.

-2.0E-07

-1.5E-07

-1.0E-07

-5.0E-08

0.0E+00

5.0E-08

1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

The number of panels

CR

I j

AR=1 AR=1.5 AR=2

h=1 h=2 h=3

h=4 h=5

Figure 3-4: Convergence rate of CWCFD as a function of the number of panels

Comparison of error estimates

The exact solutions can be approximated by means of the Richardson extrapolation as discussed in

3.2.2. These values offer a fundamental benchmark for the error estimation. However, the exact value

and its error are estimated differently according to the order and refinement ratio of the extrapolation

and depend strongly on the tendency of the predictions used for extrapolation. There are studies about

extrapolation and its reliability in [15] and [91].

Three different extrapolations are performed for the panel size h variation: 2nd and 4th order ex-

trapolations with r = 1.2, and 2nd order extrapolation with r = 1.44. For the aspect ratio AR variation,

the 2nd order extrapolation with r = 1.44 is carried out5. These different extrapolation methods result

in different exact solutions as shown in Table 3.6.

For the predictions on the axis of h = 4, the Richardson extrapolation is not applicable as they

are oscillating. In addition, the predictions on AR = 2 are oscillatingly convergent. Therefore, the 4th

5The aspect ratio is not varied by the panel refinement 1.44. This requires three predictions on AR = 1, 1.44 and 2.07which offer an exact refinement of r = 1.44. The predictions are interpolated for AR = 1.44 , 2.07, and an extrapolationof the exact solution is carried through. The extrapolated value for both types of AR is almost the same. We use thedistance of AR, 1, 1.5, and 2, continuously.

67

CWCFD (∗E − 04) Panel size Higher order extrapolationPanel size h5 h4 h3 h2 h1 2nd.1.2 4th.1.2 2nd.1.446

Ext.Sol.(2nd1.44) 4.86 3.28 4.22 4.07 3.96 3.79*

AR1 4.88 4.50 4.22 4.07 3.96 3.76 3.68 3.80AR1.5 4.91 4.52 4.22 4.07 3.96 3.74 3.65 3.80AR2 4.96 4.50 4.30 4.01 4.01 4.41 5.41 3.79

Table 3.6: Predictions of CWCFD and their extrapolation for the Ship No.1 at FN=0.25

order extrapolation in which all predictions on AR = 2 are used result in noticeably different values in

comparison with the other exact values extrapolated on AR = 1 and AR = 1.5. However, for the case

of the extrapolation having r = 1.44 on AR = 2 in the table, the three predictions of h1, h3, h5 are all

monotonically convergent. Their extrapolated values from AR = 1, 1.5 and 2 are similar. The 4th order

extrapolation (r = 1.2) of AR = 2 also yields a too high value. This is due to the divergent tendency

of the predictions in the region of (h2,AR2), (h3,AR2) and (h4,AR2). The Richardson extrapolation

is only applicable to the monotonically convergent predictions. In the region where the convergence

rate is too high, the extrapolation tends to exaggerate the true value and finer panel sensitivity tests

are required. Moreover, if the panel refinement ratio is too small7 the extrapolated true value is likely

to express a local convergence tendency. 2nd order extrapolation having r = 1.44 proves suitable to

estimate the exact value. The value having * in the Table 3.6 is the exact value reextrapolated by using

the Ext.Sol.(2nd1.44) which are extrapolated by the solutions according to the AR variations. This is

similar to other values of 2nd order extrapolation with r = 1.44. The true wave resistance coefficient,

which is independent of the panel refinement, lies near 3.79E − 04 and 3.80E − 04. Fig. 3-5 shows eachextrapolated solution by means of predictions according to each axis of the panel variation8. This is

done by second order extrapolation using the panel refinement ratio 1.44.

These extrapolated solutions can be used as benchmarks for estimating the errors of the predictions.

Various error measures for the predictions are shown in Fig. 3-6 as a function of the panel refinement

ratio. Not only the usual error (E rel.) which is defined as the relative difference between the predictions

calculated by two different panel sizes, but also the estimated fractional error E expressed in (3.13), and

the actual fractional error A in (3.15), and GCI are compared in the figure. A 4 (r = 1.44) means

that the actual fractional error by means of the exact solution is estimated by 4th order extrapolation

with the panel refinement ratio 1.44. Similarly, E 2 (r = 1.2) means the estimated fractional error by

the Richardson 2nd order extrapolation with the panel refinement ratio r = 1.2. The relative difference

of the predictions defined as E rel in Fig. 3-6 is lower in comparison to the estimated errors. GCI 2

(r = 1.44) amounts to 3 times of the magnitude of E 2 (r = 1.44). Given that E 2 (r = 1.44) and

7 In the Richardson extrapolation, an integer refinement ratio is recommended.8The values are printed bold face in Table 3.6.

68

3.0E-04

3.5E-04

4.0E-04

4.5E-04

5.0E-04

5.5E-04

1500 2500 3500 4500 5500 6500 7500 8500 9500 10500

Total number panels

CW

CFD

h=5

h=4

h=3

h=2

h=1

AR=1

AR=1.5

AR=2

Figure 3-5: Exact solutions extrapolated on the panel variations at FN = 0.25

A 2 (r = 1.44) are based on the exact solutions, they are slightly conservative and their magnitudes are

similar. The actual and estimated errors by 2nd order extrapolation with r = 1.44 are almost inside a

5% error region when the panel refinement factor equals 1. Fig. 3-7 shows the actual and estimated

errors in CWCFD predicted by means of each panel type as a function of the number of panels. Overall,

the magnitude of the error decreases monotonically and the errors of the panel with AR = 2 are the

least among the panel types having the same number. The three panels which yield predictions near and

within 5 % error, (h1,AR1), (h1,AR1.5), and (h1,AR2), are chosen. Among them the panel (h1,AR2) is

the most favored candidate since it provides accurate and fast solutions.

Conclusions

The choice of a panel type for the free surface panelization results in a noticeably different accuracy in

the CFD calculation to predict the wave resistance. This investigation shows a panel type dependency

test for the CWCFD prediction of Series 60 CB = 0.6 at FN = 0.25.

1. The CWCFD prediction is very sensitive to a change of the panel size. When the number of the

panels for the free surface is varied from 1600 to 6600 with corresponding panel size and aspect

ratio, the convergence range of CWCFD is nearly 25 % of the CWCFD predicted by the finest panel

(h1,AR2). However, when a number of panels higher than 3000 is used, the convergence range of

CWCFD decreases by at most 5 %.

69

Variation of various uncertainty estimates of AR=1

0

5

10

15

20

25

30

35

0,80 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40

Panel refinement factor

Erro

r(%

)

GCI_2(r=1.44)

A_4(r=1.2)

A_2(r=1,2)

A_2(r=1,44)

E_2(r=1.2)

E_2(r=1.44)

E_rel.

Figure 3-6: Variation of various error estimates based on panels (hi,AR1)

0

5

10

15

20

25

30

35

1000 2000 3000 4000 5000 6000 7000

The number of panels

Erro

r(%

)

A_AR=2

A_AR=1.5

A_AR=1

E_AR=2

E_AR=1.5

E_AR=1

Figure 3-7: Actual and estimated fractional error based on various AR as a function of the number ofpanels

70

2. In the panel dependency test, when the range of the panel variation is too small or when the

panels are distributed too coarsely, the extrapolated exact value is estimated as faulty. This value

may only correspond to a local panel convergence. Indeed, when the convergence tendency of the

prediction is oscillating or oscillatingly convergent the panel size must be chosen very carefully.

3. The degree of the error depends on the error estimator. The degree of the error of each CWCFD

prediction with 15 panel type variations is estimated and compared by means of various error

estimators. The actual and the estimated fractional errors show similar degrees of errors for

different panel type variational cases when the number of panels is near 4000.

4. Regarding the panel type variation under the same number of panels, careful determination of the

panel type yields a faster convergence and more accuracy. For the free surface panelization, the

denser panels in the longitudinal directions are more efficient. The panel type of (h1,AR2) seems

to be suitable for the CWCFD prediction for the ship No.1 at FN = 0.25 because its accuracy is

within 5 % and this panel yields the fastest solution among other panel candidates resulting in a

solution with a similar accuracy.

In the process of the panel dependency test, the ability of the panel to predict the wave resistance

coefficients should be determined by comparing the convergence tendency of the prediction and the

convergence rate as well as their estimated fractional error at the same time because any of these criteria

may give partly local information.

3.3.4 Panel error dependency on the change of flow phenomena (Case 2)

CWCFD predictions of the ship No.1 related to the change of FN

The tendencies of panel convergence related to the change of FN have been investigated in the region

of the verification. For six different ship speeds, the panel dependency tests of the CWCFD predictions

are undertaken for 15 panel variations and the convergence tendencies are shown at each speed in Fig.

3-8. The individual convergence tendencies of the predictions due to the panel variation are noticeably

dependent on the ship speeds. At each speed, the predictions are differently convergent, i.e., increasingly,

decreasingly, or slightly oscillatingly, as a function of the number of panels. The convergence ranges of

the predictions are also very different at each speed even though the same panel variations are used for

all FN . The convergence ranges are large at FN = 0.25 and 0.275 where the flow states vary largely with

the change of ship speed. Conversely, they are low at other speeds.

Table 3.7 shows the convergence ranges of the predictions to the panel variations and their rates to

the prediction. In the table, ∆CWCFD (hi,ARj) means the total convergence range between the maximum

71

Fn=0.2

2.00E-04

2.50E-04

3.00E-04

3.50E-04

4.00E-04

0 2000 4000 6000 8000

The number of panels

CW

CF

D Fn=0.225

2.00E-04

2.50E-04

3.00E-04

3.50E-04

4.00E-04

0 2000 4000 6000 8000

The number of panels

CW

CF

D

Fn=0.25

3.50E-04

4.00E-04

4.50E-04

5.00E-04

5.50E-04

0 2000 4000 6000 8000

The number of panels

CW

CF

D

Fn=0.275

1.13E-03

1.18E-03

1.23E-03

1.28E-03

1.33E-03

0 2000 4000 6000 8000

The number of panels

CW

CF

D

Fn=0.3

1.55E-03

1.60E-03

1.65E-03

1.70E-03

1.75E-03

0 2000 4000 6000 8000

The number of panels

CW

CF

D

Fn=0.316

1.55E-03

1.60E-03

1.65E-03

1.70E-03

1.75E-03

0 2000 4000 6000 8000

The number of panels

CW

CF

D

Figure 3-8: Panel convergence tendencies at different Froude numbers

72

AR=1

-2.0E-07

-1.5E-07

-1.0E-07

-5.0E-08

0.0E+00

5.0E-08

1.0E-07

1.5E-07

1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

The number of the panels

CR

i 1

0.2 0.225

0.25 0.275

0.3 0.316

Fn

AR=1.5

-2.0E-07

-1.5E-07

-1.0E-07

-5.0E-08

0.0E+00

5.0E-08

1.0E-07

1.5E-07

1500 2000 2500 3000 3500 4000 4500

The number of the panels

CR

i 1.

5

0.2 0.225

0.25 0.275

0.3 0.316

Fn

AR=2

-2.0E-07

-1.5E-07

-1.0E-07

-5.0E-08

0.0E+00

5.0E-08

1.0E-07

1.5E-07

1500 2000 2500 3000 3500

The number of the panels

CR

i 2

0.2 0.225

0.25 0.275

0.3 0.316

Fn

Figure 3-9: Convergence rates as a function of the number of panels at each FN with relation to thepanel types

73

FNPanel convergence 0.2 0.225 0.25 0.275 0.3 0.316(1) ∆CWCFD (hi,ARj) (∗E − 05) 3.3 1.2 10.0 17.6 6.0 6.9(2) ∆CWCFD (hi,ARj)/CWCFD (h1,AR1)(%) 13 4 25 15 4 4(3) ∆CWCFD (hi,AR1)/CWCFD (h1,AR1)(%) 13 4 23 14 2 4(4) ∆CWCFD (h1,ARj)/CWCFD (h1,AR1)(%) 1 1 2 1 0 1

Table 3.7: Differences of predictions related to the panel variations

and the minimum predictions by 15 panel variations at each speed. ∆CWCFD (hi,AR1) is the convergence

range among five predictions according to the change of the panel size in the longitudinal direction h

on the axis of AR = 1. Similarly, ∆CWCFD (h1,ARj) is the convergence range between 3 predictions

according to the change of the aspect ratio on the axis of h = 1 on the panel variation plane.

In (2), (3), (4), each range is divided by CWCFD predicted by the finest panel type. This table shows

that the total convergence range ∆CWCFD (h1,AR1) at FN = 0.275 is almost 15 times larger than the

range at the speed of FN = 0.225. These large differences of the convergence ranges suggest the necessity

of panel sensitivity tests at different speeds.

Convergence rates

Fig. 3-9 shows convergence rates of each panel type on the average number of two panel types hi and

hi+1 at each ship speed. The convergence rate with the average panel numbers of less than 3000 is

noticeable. We can limit our region of choosing the number of panels to higher than 3000 because the

high convergence rate of the prediction means that it is not good enough to predict an accurate solution

and finer panels or better distributions are needed. This tendency is stronger in the region of speeds

of FN = 0.25 and 0.275. If the degree of the estimated fractional error is taken into account at the

same time, the solutions by the panels (h1, ARj) are clearly more accurate in comparison to the panels

(h3, ARj) as shown in Fig. 3-10. Where the number of panels is around 3800, the accuracies become

rapidly higher and the degree of accuracy converges with the increasing number of panels. At the ship

speed of FN = 0.275, the panel error is too high to accept its predictions. The prediction at this speed

undoubtedly needs much finer panels. In this case of the convergence rate being too high, the results of

the extrapolation tend to be exaggerated and therefore the estimated fractional error is likely to be too

high or too low. Except for the case of FN = 0.275, the solution corresponding to the panel (h1,AR2)will

be used for each FN as well as three calculations using the panel variations (hi,AR2),where i = 1, 3, 5

for estimating their accuracies. Additionally, the panel dependencies can be investigated not only by

means of the number of panels but also by means of other parameters which are correlated to the panel

size such as the number of panels per wave length and the area of the panels. By a panel dependency

test with different panel configuration parameters based on the correlation with the wave resistance

74

coefficients, at FN = 0.2, 0.225, 0.3, and 0.316, the area of the panels has a much higher correlation

while at FN = 0.25 and 0.275 the correlation of the longitudinal panel size hi and the number of the

panels per wave length are more suitable. The number of the panels per wave length is suitable to

compare the panel dependencies for different flow states at the same time.

-6

-3

0

3

6

9

12

2000 3000 4000 5000 6000 7000

Erro

r (%

)

0.2 0.2250.25 0.30.316

(h3, AR2)

(h3, AR1.5)

(h1, AR2) (h3, AR1)

(h1, AR1.5) (h1, AR1)

30

36

42

48

54

60

66

2000 3000 4000 5000 6000 7000

Erro

r (%

)

0.275

The number of panels

Figure 3-10: Estimated error for each panel type as a percentage of CWCFD (h1,AR1) as a function ofthe change of FN

Conclusion

1. The convergence ranges of the CWCFD predictions due to the panel variation are different at

each FN . At FN = 0.25 and 0.275, the convergence rates are very high and a considerably finer

panelization is required.

2. The degree of the panel dependency error is not directly proportional to the change of the ship

speed, although in the low speed region the degree of the error tends to be small and in the high

speed region it increases. A panel convergence analysis should be performed for each FN .

3. Except for the speed FN = 0.275, the degree of the estimated panel convergence error is within

5% of each prediction. At FN = 0.275, the degree of the error is over 30% of CWCFD.

75

3.3.5 Panel error dependency according to the change of the hull form vari-

ation (case 3 and 4)

The panel error dependency on the CWCFD predictions of the five different ship shapes is tested ac-

cording to the change of FN . The panel types of (hi,AR2), i = 1, 3, 5 are used for the panel conver-

gence tests and for the error estimation. Fig. 3-11 shows the convergence tendencies of each hull

form on the number of the panels at the speeds. In these cases, three types of panels having the sizes

(h1,AR2), (h3,AR2), (h5,AR2) are tested. The predictions and their convergence rates decrease when the

number of panels increase. The panel convergence tendencies of each ship remain relatively parallel to

the change of the number of panels. There are no changes in the order of magnitude of the predic-

tion with relation to the increase of the panel number. Here, one point must be considered before the

validation is carried through. One must compare the effects of the hull form variations on the panel

convergence error and on the physical phenomena. The relative change in the numerical error cannot be

accepted to go beyond the physically caused change due to the form variation. E. g., the ship designer

using CFD simulation may choose a hull form of which resistance or performance are better based on

the simulation prediction. However, if the errors vary more than the CFD prediction for different hull

forms, the relative comparison of the design candidates is impossible.

Fig. 3-12 shows a comparison of the CWCFD predictions for different panel types with the extrapo-

lated prediction. The ship varied by B/T variation (No.5) is used and CWCFD is shown as a function of

FN . CWCFD for coarse panels (h3, 2) is overpredicted over FN . Especially, a large difference between the

CFD prediction and the extrapolated value is shown at FN = 0.275 as discussed for S60. For the finest

panel type (h1, 2), the difference remains still large, although CWCFD converges to the extrapolated

value depending on the panel fineness. The CWCFD predictions for the coarse panels are more smooth

than the extrapolated value over the FN region.

Fig. 3-13 and 3-14 shows the estimated fractional error of five hull forms at each tested FN . In this

case, at the speed of FN = 0.225 and 0.275, the estimated fractional error of the two hull forms varied by

the form parameters L/B and CB is high. The panel convergence error is high in the speed range where

the variation of the prediction to the speed (∆CWCFD/∆FN ) is great. This is because the convergence

rate of the prediction according to the panel variation increases. The degree of the error depends more

on the change of FN . However, few systematic trends related to the hull form variations are shown. At

the speeds of FN = 0.225 and FN = 0.275, the predicted wave resistance coefficient contains too large

numerical errors.

Fig. 3-15 shows a qualitative and quantitative comparison of the predictions and the degree of the

error at FN = 0.25 when the hull forms are varied. If we consider the ship No.1 as the reference ship, the

parameters CWCFD/CWCFD(S60) and E/E(S60) express the qualitative rates of the predictions and

76

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

1500 2000 2500 3000 3500 4000

The number of panels

CW

CF

D

S60 B/T L/B

lcb CB

Fn=0.2

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1500 2000 2500 3000 3500 4000

The number of panels

CW

CF

D

S60 B/T L/B

lcb CB

Fn=0.225

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1500 2000 2500 3000 3500 4000

The number of panels

CW

CF

D

S60 B/T L/B

lcb CB

Fn=0.25

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

1500 2000 2500 3000 3500 4000

The number of panels

CW

CF

DS60 B/T L/B

lcb CB

Fn=0.275

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

1500 2000 2500 3000 3500 4000

The number of panels

CW

CF

D

S60 B/T L/B

lcb CB

Fn=0.3

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

1500 2000 2500 3000 3500 4000

The number of panels

CW

CF

D

S60 B/T L/B

lcb CB

Fn=0.316

Figure 3-11: Panel convergence tendencies of different hull forms at each Froude number

77

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

0.200 0.225 0.250 0.275 0.300 0.316Fn

CW

CF

D

Extrapolation

CWCFD_(h1, 2)

CWCFD_(h3, 2)

CWCFD_(h5, 2)

Figure 3-12: CWCFD for different panel numbers and extrapolated solution by Richardson method ofthe ship CB = 0.6, B/T = 3.5 (No. 5)

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

S60 B/T L/B lcb CB

Estim

ated

Pan

el C

onve

rgen

ce E

rror

0.2

0.225

0.25

0.3

0.316

Figure 3-13: Estimated panel convergence errors of the hull form at different Froude numbers

78

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

S60 B/T L/B lcb CB

Estim

ated

Pan

el C

onve

rgen

ce E

rror

0.225

0.275

Fn

Figure 3-14: Estimated panel convergence errors of the hull form at FN = 0.225 and 0.275

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

S60(No.1) B/T(No.5) L/B(No.10) LCB(No.19) CB(No.38)

Rat

e

CWCFD/CWCFD(S60) E/E(S60)

E/CWCFD dE/dCWCFD

Figure 3-15: Comparison of the panel convergence error with the CWCFD corresponding to the hull formvariation at FN = 0.25

79

the estimated errors to those of the reference ship. When CB increases from 0.6 to 0.7 the predictions

increase to more than twice of CWCFD(S60) . By means of the parameter E/E(S60), a qualitative

comparison of the estimated error of each varied ship is possible. As shown in Fig. 3-15, the estimated

fractional error of the panel is not necessarily proportional to the magnitude of the prediction. The

degree of the error of the ship of CB = 0.7 is noticeably small. The parameter E/CWCFD describes the

rate of the panel convergence error to the prediction. The degree of the error stays generally within 10 %

of the predictions. The positive (negative) symbol of the error means that the solution is overpredicted

(underpredicted). In the case of FN = 0.25, the estimated errors are all positive, i. e., the solutions are

overpredicted. ∆E/∆CWCFD is the rate of the estimated error range to the prediction range when the

hull form is changed from S60 to each type. The parameter for e. g. the B/T variations is calculated

as follows:∆E

∆CWCFD=

E(S60)−E(B/T )CWCFD(S60)− CWCFD(B/T )

(3.27)

A positive ∆E/∆CWCFD means that when CWCFD increases (decreases) the degree of the panel error

increases (decreases) also. In Fig. 3-15, ∆E/∆CWCFD is negative for the ship shapes varied by lcb and

CB. In this case, since the predictions of CWCFD are higher than those of S60, the error decreases. If the

difference in the error is larger than that in the predictions, i. e., |∆E/∆CWCFD| ≥ 1, the comparison ofthe resistance or the performance based on the CFD prediction is very difficult because the degree of the

error exceeds the effects of the physical phenomena. Therefore, it is very important to check whether

∆E/∆CWCFD is not much different for the candidate ships. If ∆E/∆CWCFD is within ±0.5 whenE/CWCFD is smaller than 0.1, a rough comparison of each case is possible. These four parameters give

an understanding of the possibility of a comparison of the resistance and the performance coefficients

based on the CFD simulation. The panel convergence error is in most cases small enough. In this case,

the CFD predictions can be compared for several ship shapes.

The panel convergence error is mainly affected by the change of the ship speed. The magnitude

of the error increases when a ship having a large displacement and waterplane area is calculated for

a ship speed at which the wave system is rapidly changed. Therefore, the ship L/B(No.10) having a

larger waterplane area in comparison to the displacement shows the largest panel error at the speed

FN = 0.275. The panelization should be carefully chosen in any given design configuration where the

physical phenomena are much changed due to such parameters.

3.3.6 Conclusion

1. It is known that body panel errors are propagated into CWCFD with almost the same scale so

that a positive CPXI results in an overprediction of CWCFD and a negative CPXI results in an

80

underprediction. This makes it possible to choose an acceptable body panelization within a re-

quired accuracy by using CPXI .Through the body panel sensitivity test varying the panel size and

distribution, each panel on the body for the 58 ships is found within 1 % accuracy.

2. The CWCFD prediction is very sensitive to a change of the panel size. When the number of the

panels is varied from 1600 to 6600, the prediction range of CWCFD is nearly 25 % of CWCFD by the

finest panel (h1,AR1). However, when more than 3000 panels are used, the prediction of CWCFD

converges up to 5 % of the solution by the finest panel. Careful determination of the shape of the

panel yields faster convergence with the same accuracy. On the free surface, denser panels in the

longitudinal direction are more efficient.

3. At FN = 0.25 and 0.275, the predictions are very sensitive to the change of the panel size and

especially finer panels are required. By contrast, at the speeds FN = 0.225 and 0.3, the predictions

are not very sensitive to the change of the panel size and converge with a relatively small number

of panels. The degree of the panel convergence error is not directly proportional to the change of

the ship speed, although in the low speed region the degree of the error tends to be small and in

the high speed region it increases. A panel convergence analysis should be carried through for each

FN . Except for the speed FN = 0.275, the degree of the estimated panel convergence error stays

within 5% of each prediction. The panel type of (h1,AR2) seems to be suitable for the CWCFD

prediction for the ship No.1.

4. The panel convergence error is depending on the ship speed while any major systematic trends

related to the hull form variations are not shown. However, the degree of the panel convergence

error increases when a ship having a large displacement and waterplane area is calculated with a

ship speed at which the wave system is rapidly changed. The panel should be carefully chosen in

each given design configuration where the physical phenomena are considerably changed due to

such parameters. For each hull form, the same panel type as for the ship No.1 will be used which

results in 10 % of the prediction by the finest panel. For the case of the ships such as L/B(No.10)

and CB(No.38) at difficult FN , finer panel types are used to get reliable CFD predictions within

at least 10 % of (h1,AR1).

3.4 Computational domain convergence

The computational domain is a user defined finite region of the analytical region in a simulation for

computational purposes. This is bounded by ignoring small physical phenomena in infinity. The limited

computational domain sometimes leads to unstable behavior and affects the accuracy of the CFD

81

predictions. With regard to the wave resistance problem, the choice of a suitable computational domain

seems to be important to predict reliable wave resistance coefficients. Too small a computational domain

can not represent the physical phenomena and can also cause numerical reflections where the boundary

of the computational domain meets with waves. This results in difficulties concerning the numerical

convergence and accuracy. Conversely, too wide a computational domain results in a lot of computational

time and costs although the accuracies in the predictions are not changed much. The most important

factor in arriving at a decision for a limited computational domain is therefore finding a minimal domain

which attempts to reflect the reality as closely as possible. In this section, the quantitative dependency

of the CWCFD prediction on the computational domain will be investigated based on sensitivity analysis.

3.4.1 Design of a sensitivity test of the computational domain

The computational domain of the free surface is defined as a quadrilateral plane expressed by two points

in its diagonal, a starting point (−0.5, 0) LPP upstream and one of each variational point downstream inFig. 3-16. The point (−0.5, 0) LPP upstream remains unchanged for all domain variations. The initial

domain of the sensitivity analysis is composed by the two points of (−0.5, 0) LPP and (2, 0.7) LPP . I. e.,the longitudinal dimension is half of the length in the upstream and twice the length in the downstream

direction. The transverse coordinates are considered to be depending on the longitudinal length and are

determined by the Kelvin angle, e. g., Y = X · tan19◦28 . Here, X and Y are lengths originating from

the stem of the ship, (0, 0). In this case, the computational domain consists of the four points (−0.5, 0),(2, 0), (−0.5, 0.7), and (2, 0.7) LPP . This computational domain is considered as a reference domain.A variation of the computational domain results in three types of region groups: region X (longitu-

dinal region), region Y (transverse region) and region XY (Kelvin region). Region X varies from the

reference domain by adding and subtracting a distance of 0.025 LPP twice respectively. In this case, the

transverse length is fixed as the length of the reference domain. Therefore, the region X is composed

of five equally distanced points on the axis of Y = 0.7. The region Y is defined similarly to the region

X. Each coordinate in Y is computed by X · tan19◦28 respectively. Region Y is composed of the 5

points on the axis of X = 2. The transverse length of the region XY is dependent on the function

Y = X · tan19◦28 based on the X variation of the region X. Therefore, the region XY is composed of

the five points on the diagonal line of the function Y = X · tan19◦28 . The largest region of X = 2.5 will

be denoted as X1 and the smallest region of X = 1.5 as X5. For the regions of Y and XY , the same

notation is used such as Yi, XYi, i = 1, ..., 5. X3, Y3 and XY3 regions are the same.

82

Figure 3-16: Variation of the computational domain

3.4.2 Computational domain dependency

An analysis of the computational domain dependency is carried through by calculating the wave resis-

tance coefficients for each computational domain given in Fig. 3-16. This analysis is repeated in the

verification range given in Table 3.1. The degree of CWCFD dependency on the computational domain

will be investigated as a function of the ship shapes and the flow states. This is similar to the process

involved in the panel convergence analysis in Fig. 3-2.

Computational domain dependency of the Ship No. 1 at each FN

The computational domain dependency of CWCFD predictions is investigated in relation to each domain

variation, i. e., X, Y and XY regions. The CWCFD predictions related to each region are shown in Fig.

3-17 for different FN as a function of the area of each computational domain. The areas resulting from

the variations in regionX and Y are quite similar, while the area due to the variation of regionXY covers

a wider area range. The prediction by the reference computational domain, where all predictions meet,

lies at area 1.75. Generally, an increase of the transverse region Y results in a little change in CWCFD,

while an increase of the longitudinal region X causes large oscillations in the predictions. The CWCFD

predictions show large oscillations in small computational domains and have convergent tendencies when

the computational domain is greater than the reference domain. The predictions are more sensitive to

the change of the region X. This may be a result of the loss of the high wave energy downstream or of

the numerical reflections due to the limited computational domain. Observing the predictions based on

the variations in the region X and Y carefully, their averages are partially similar to CWCFD based on

the variation in the region XY in the area 1.5 ≤ area ≤ 2.25. This is depicted by the line representative

83

FN 0.2 0.225 0.25 0.275 0.3 0.316S60 1 4 2 4 2 1B/T 1 3 2 4 2 0L/B 1 4 1 4 1 2lcb 1 4 3 4 2 1CB 2 7 1 3 1 0

Table 3.8: Range of the computational domain convergence in percent of CWCFD for XY1

of the ’average’ of Fig. 3-17. Predictions by the region XY of area ≥ 1.75, the line ’XY region’,and

the line ’average’ are likely to be similar. The magnitudes of the convergent domain are within 2% for

FN = 0.2, 0.3, 0.316, given the percentage of CWCFD predicted by the region of the area 1.75. They are

between 5-7 % for other speeds, where the change of CWCFD is faster; given these speeds the oscillation

in CWCFD due to the change of X regions is noticeable.

Computational domain dependency of five different ship shapes at each FN

The computational domain dependencies are taken into account for five ship shapes chosen from the

verification range. Each prediction of CWCFD at FN = 0.25 is shown in Fig. 3-18. The first, third,

and fifth variations in the region XY are chosen for the computational domain dependency test for

five varying ship shapes. As shown in Fig. 3-18, the computational domain dependencies have minor

influence on each prediction. The effect of a computational domain dependency does not exceed the

differences in effects due to hull form variations. The magnitudes of the convergent domains remain

unchanged to the hull form variations at each FN . Each convergence domain given the percentage of

CWCFD calculated by the computational domain XY3 is compared in Table 3.8. The computational

domain dependencies from the change of the ship speeds are larger than those due to the ship shape

variations except for the case of the CB variation.

3.4.3 Estimation of the computational domain convergence error

The Richardson extrapolation is applied to the estimation of computational domain errors. The wave

resistance coefficients are extrapolated by considering the direct rate of the computational domains as a

refinement ratio since the refinement ratio is given as a ratio between two sizes of different computational

domains, e. g., such as r12 = XY1XY2

, where XY1 is a larger computational domain than XY2. For

predictions by using each computational domain shown in Fig. 3-16, various types of extrapolations are

possible related to the order and the bounds of the extrapolation. We have carried through first and

second order extrapolations with the computational domain variations. Since the refinement ratio of the

given computational domains is slightly different, each prediction is interpolated for a uniformly divided

84

Fn=0.25

3,95E-04

4,00E-04

4,05E-04

4,10E-04

4,15E-04

4,20E-04

1 1,25 1,5 1,75 2 2,25 2,5 2,75

Fn=0.2

2,54E-04

2,56E-04

2,58E-04

2,60E-04

2,62E-04

1 1,25 1,5 1,75 2 2,25 2,5 2,75

The area of computational region

C W C F D X region

Y region

xy region

Average

Fn=0.225

2,80E-04

2,85E-04

2,90E-04

2,95E-04

3,00E-04

3,05E-04

3,10E-04

1 1,25 1,5 1,75 2 2,25 2,5 2,75

Fn=0.275

1,10E-03

1,12E-03

1,14E-03

1,16E-03

1,18E-03

1,20E-03

1 1,25 1,5 1,75 2 2,25 2,5 2,75

Fn=0.3

1,66E-03

1,67E-03

1,68E-03

1,69E-03

1,70E-03

1,71E-03

1,72E-03

1 1,25 1,5 1,75 2 2,25 2,5 2,75

Fn=0.316

1,63E-03

1,64E-03

1,65E-03

1,66E-03

1,67E-03

1 1,25 1,5 1,75 2 2,25 2,5 2,75

Figure 3-17: Computational domain convergence of CWCFD at different Froude numbers

85

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-03

1.1E-03

XY1XY3XY5

Compuational domain (XY region)

CW

CFD

S60 B/T L/B lcb CB

Figure 3-18: Computational domain dependencies at FN = 0.25 of different ship shapes

computational domain. Here, the errors estimated by the general 2nd order Richardson extrapolation

suggested by Roache [91][92][93][94] are shown and discussed. The predictions based on the XY region

variation with the refinement ratio r = 1.2 are used.

We show the estimated error for two cases. One is to show the degree of the computational domain

dependency of the ship Series 60 CB = 0.6 (symbolized as S60) as a function of the computational

domain for each FN . The other is to show the degree of the computational domain dependency as a

function of the ship shapes for each FN . The first case is shown in Fig. 3-19. The degree of the error is

within ±5% for all test cases and it decreases noticeably in the computational domain XY1 within ±1%.A further expansion of the computational domain does not seem to be of much help for the S60 ship

to reduce the degree of the computational domain error when allowing for the increase of the number

of panels required by the expansion of the computational domain. These characteristics are similar for

all tested ship speeds. Such computational domain dependency tests are carried through for five varied

ship shapes defined in the verification range. The computational domain dependency is also similar as

in Fig. 3-19 for all ship shapes tested. Among them, Fig. 3-20 shows the estimated errors of each ship

in the computational domain with the region XY3 and XY1 respectively. The estimated error is less

sensitive to the change of the ship shape. Its absolute magnitude is distributed around ±1% for XY3

and ±0.5% for XY1. For the case of the ship varied by L/B and CB at FN = 0.25, the error increases

slightly more. The computational domain XY3 seems to be a suitable choice independent of the ship

86

speed and the shape variation. The computational domain is only weakly dependent on the accuracy of

the wave resistance prediction if the downstream region is larger than the ship length, and the transverse

region is larger than the region marked by the Kelvin angle with the longitudinal region.

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

XY1XY3XY5

Computational region XY

Est.

erro

r (%

)

0.2 0.225

0.25 0.275

0.3 0.316

Fn

Figure 3-19: Computational region dependency of S60 expressed by the estimated error at each FN

3.4.4 Conclusion

A computational domain dependency analysis is undertaken in the verification range by varying the

computational domain systematically.

1. A variation in the computational region X results in an oscillating tendency of the CWCFD pre-

diction. The state of the truncated wave system due to the bounded downstream region affects

the CWCFD prediction.

2. A variation in the region Y results in a relatively small effect on the CWCFD prediction in compar-

ison to the region X. The CWCFD predictions show a monotonous tendency related to the change

of the scale of the region.

3. Predictions related to the region XY are generally similar to the average value of those related to

the region X and the region Y in the same area.

87

Computational region XY 3

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

S60 B/T L/B lcb CB

Ship shape variation

Est.

erro

r(%

)

0.2 0.225

0.25 0.275

0.3 0.316

Computational region XY 1

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

S60 B/T L/B lcb CB

Ship shape variation

Est.

erro

r(%

)

0.2 0.225

0.25 0.275

0.3 0.316

Figure 3-20: Computational domain convergence error by means of two different scales of region XY3and XY1

88

Numerical parameters Parameter range Initial configurationTolerance criteria 1e− 04, 5e− 05, 1e− 05 5e− 05Number of panels (hi, ARj), i = 1, ..., 5, j = 1, 1.5, 2 (h1, AR2)Computational domain XYi region, i = 1, 3, 5 XY3

Table 3.9: Design of the range of the iterative convergence test

4. The computational domain dependency is somewhat larger in respect to the change of the ship

speed than to the ship shapes in the verification range. The degree of the estimated error becomes

small enough to be neglected for both cases when the computational domain increases.

5. The computational domain dependency on the CWCFD prediction is small when the domain is

larger than the reference computational domain of which the longitudinal length is twice as long

as the length of the ship downstream and half of the length in the upstream direction, and the

transverse length is determined by the Kelvin angle Y = X ·tan19◦28 . This computational domainresults in a small error around ±1%. No further enlargement of the computational domain is neededin the verification range.

3.5 Iterative Convergence

The iterative convergence criterion applied in SHIPFLOW is the change of the maximum wave elevation

between iterations. For difficult convergence cases, the maximum iteration number and the relaxation

parameters can be used also for stopping or accelerating the iteration. The level of the iterative conver-

gence for a run is investigated based on the residuals of the normalized total and the dynamic boundary

conditions as well as the number of iterations. If the solution satisfies the given iterative criteria the

iterative convergence error in the predictions of CWCFD can be estimated as described in 3.2.2.

The level of the iterative convergence is investigated as a function of the numerical parameters.

Numerical parameters such as the panel size and the aspect ratio, the computational domain as well

as tolerance criteria will affect the level of the iterative convergence and result in a different degree of

errors in CWCFD. This iterative convergence test will be repeated in the verification range including

ship speed and shape variations. We will compare the level of influence of the numerical parameters in

the verification range on the iterative convergence. The initial configurations for a simulation run and

their sensitivity tests are given in Table 3.9

3.5.1 Tendency of iterative convergence

Iterative convergence tendencies from the initial configuration are shown by means of the normalized

boundary condition residuals in Fig. 3-21. The speed of the convergence is quite fast. In the 5th

89

iteration the magnitude of the normalized BC residuals arrives at around 1e − 04. The residuals areless than 10 % of the magnitude of CWCFD at each FN except for the speed FN = 0.25 where the

magnitude is almost 20 % of CWCFD due to its oscillating convergence tendency. Due to the same given

tolerance criteria 5e−05 for all FN , the number of iterations at high FN where the wave phenomena areremarkably higher increases slightly. Each iteration is finished within five to seven iterations, namely

five iterations for FN = 0.2 and 0.225, six iterations for FN = 0.275 and seven iterations for FN = 0.3

and 0.316. At the speed FN = 0.25, the solution is oscillating, therefore it cannot be stopped before

the 15th iteration. This is shown in Fig. 3-22. The iterative convergence errors in the wave resistance

coefficient represented as a percentage about the simulated CWCFD are -0.2, -0.5, 0.9, 0.3, 0.0, 0.1 %

respectively. All iterative convergence errors are within 1 %.

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

1th-ite 2th-ite 3th-ite 4th-ite 5te-ite 6th-ite 7th-ite

The number of iterations

Nor

mal

ized

BC

Res

idua

ls

0.2

0.225

0.25

0.275

0.3

0.316

FN

Figure 3-21: Normalized residuals in boundary conditions for the initial run at different Froude Numbers

3.5.2 Sensitivity of iterative convergence to numerical parameters

For the initial configurations, the iterative convergence error was small enough to be ignored. The

sensitivity of the iterative convergence to the change of the numerical parameters based on Table 3.9 will

be discussed in the following. The degree of error is shown in terms of the percentage of the estimated

error of CWCFD.

90

3.90E-04

4.00E-04

4.10E-04

4.20E-04

4.30E-04

4.40E-04

4.50E-04

0 5 10 15 20 25 30 35

The number of iterations

CW

CF

D

4,01E-04Lower Limits

Upper Limits4,09E-04

Figure 3-22: Oscillating iterative history of CWCFD by using the panel type (h1, AR2) at FN = 0.25

TC \ FN 0.2 0.225 0.25 0.275 0.3 0.3161e− 04 4 / 1.93 4 / -1.26 15 / -0.11 5 / -0.53 6 / 0.11 6 / 0.225e− 05 5 / -0.72 5 / -0.64 15 / -0.09 6 / 0.28 7 / -0.03 7 / -0.051e− 05 6 / -0.16 7 / 0.08 15 / -0.09 8 / 0.03 9 / 0.02 9 / 0.00

Table 3.10: Number of iterations / Estimated error percentage for given tolerance criteria at each FN

Accuracy of the CFD prediction to the change of iterative tolerance criteria

The tolerance criterion (TC) given here is the change of the maximum wave elevation between iterations.

Three magnitudes of the criterion are tested, respectively. As a result, the relative magnitudes of the

criterion to the prediction of CWCFD at FN = 0.2 are 0.38, 0.19, 0.04 for the tolerance criteria 1e− 04,5e− 05 and 1e− 05 respectively. At FN = 0.316, they are 0.06, 0.03 and 0.01 respectively. The iterativeconvergence error as a function of the convergence tolerance is shown in Fig. 3-23. The degrees of error

are within ±1% when the tolerance criterion is 5e− 05. If the tolerance criterion is equal to 1e− 05, theerrors belong within the range of ±0.1%, but the number of iterations increases. This is shown in Table3.10.

The magnitude of the estimated errors is greater than ±1% when the applied tolerance criterion

is larger than 20 % of CWCFD at each FN . If the applied tolerance criterion is smaller than 15 % of

CWCFD then the error lies within 0.5 % of CWCFD. We will use the tolerance criterion 5e− 05 for theother calculation cases when both the CPU time and the accuracies are considered.

91

Sensitivity of the iterative convergence to the change of the panelization and the compu-

tational domain

The iterative convergence error was within 1 % of CWCFD for the initial configuration. Table 3.9 shows

a design for the sensitivity test of the iterative convergence in relation to the change of the panel size, the

aspect ratio and the computational domain. The results show that the degree of the errors is relatively

small, they generally lie within 1% of CWCFD, except for the errors at FN = 0.25 and 0.275. At those

speeds where a rapid change of the wave resistance coefficient takes place, the solutions converge with

a large iterative convergence error or oscillate. Unlikely, the solutions are less sensitive to the change of

the computational domain and the aspect ratio of the panel while the change of the panel size exerts a

rather noticeable influence on the number of iterations and their convergence accuracies.

The computational domain variation has little influence on the iterative convergence of the wave

resistance coefficient. In fact, the estimated errors in CWCFD are those which remain after the iteration

satisfies the given tolerance criteria. Therefore, the errors can take into account the randomly distributed

errors which bear little relationship with the parameters. Convergence characteristics such as oscillating

tendencies or low convergencies can be handled by changing the numerical parameters. In order to

achieve a better convergence, the panel size should be not too small and the panel shape should be as

rectangular as possible. Also, a wider computational domain prevents oscillation.

3.5.3 Iterative tendency to the change of the ship form

The tendency of the iterative convergence is investigated as a function of the ship shapes, S60 (No. 1),

B/T (No. 5), L/B (No. 10), lcb (No. 19), CB (No. 38) defined in the verification range. Fig. 3-24 shows

the effect of the ship shape variation on the iterative convergence. The numerical parameters which were

chosen before by each sensitivity analysis are used directly to get CWCFD for each ship. However, for

all cases the same tolerance criterion of 5e− 05 is used. The magnitudes are within ±1% and they are

nearly invariant in response to the ship shape variations except for FN = 0.25. At the speed FN = 0.25,

the iterative error increases up to 2 % for the ship shape varied by the form parameter B/T . Generally,

full ships with wider but shallow hull forms in which the increase of the wetted surface of the ship is

larger in comparison with the increase of the displacement show a more difficult convergence9.

9As a reference the rates of the wetted surface to the displacement of each ship are 40, 48, 35, 40, 38 for S60, B/T ,L/B, lcb, CB respectively.

92

-1.0

-0.5

0.0

0.5

1.0

1.5

XY1XY3XY5

Computational domain (XY region)

Itera

tive

erro

r (%

)0.2

0.225

0.25

0.275

0.3

0.316

FN

-1.0

-0.5

0.0

0.5

1.0

1.5

AR1AR1.5AR2

Aspect ratio

Itera

tive

erro

r (%

)

0.2

0.225

0.25

0.275

0.3

0.316

FN

-10.00

-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

h1h2h3h4h5

Panel size

Itera

tive

erro

r (%

)

0.2

0.225

0.25

0.275

0.3

0.316

FN

Figure 3-23: Comparison of iterative convergence errors as a function of numerical parameters.

93

-1.0

0.0

1.0

2.0

3.0

S60 B/T L/B LCB CB

Ship shape variation

Erro

r/CW

CFD

(%)

0.2

0.225

0.25

0.275

0.3

0.316

Fn

Figure 3-24: Iterative convergence error related to the change of ship shapes at each FN

3.5.4 Conclusion

The level of the iterative convergence is investigated as a function of the numerical parameters such as

the panel size and the aspect ratio, the computational domain as well as tolerance criteria. This iterative

convergence test is repeated in the verification range including ship speed and shape variations. The

results show that the degree of the errors is relatively small, they generally lie within 1% of CWCFD,

except for the errors at FN = 0.25 and 0.275. At those speeds where a rapid change of the wave resistance

coefficient takes place, the solutions converge with a large iterative convergence error or oscillate. In

contrast, the solutions are less sensitive to the change of the computational domain and the aspect ratio

of the panel while the change of the panel size exerts a rather noticeable influence on the number of

iterations and their convergence accuracies. For the different ship shapes the iterative convergence errors

lie within ±1% and they are nearly invariant in response to the ship shape variations. At the speed

FN = 0.25, the iterative convergence error increases up to 2 % for the ship shape varied by the form

parameter B/T . Generally, full ships with wider but shallow hull forms in which the increase of the

wetted surface of the ship is larger in comparison with the increase of the displacement show a more

difficult convergence.

94

3.6 Numerical error and contribution of each error type

The estimated errors by panel convergence error, computational domain convergence error and iterative

convergence error are propagated to the CFD predictions. Although the explicit form of a data reduction

equation is not known in advance, one needs to perform an error propagation analysis in order to

estimate the total numerical error resulting from each estimated error by convergence analysis. As already

discussed in section 2.3.2 the total numerical error can be determined by the RSS (root summation of

the square), CROSS (cross correlation equation) and ADD (addition of the individual numerical errors)

of each estimated error:

ESN RSS =n

i=1

E2SN Xi

1/2

(3.28)

ESN CROSS =

⎛⎝ n

i=1

n

j=i

ESN XiESN Xj

⎞⎠1/2

(3.29)

ESN ADD =n

i=1

ESN Xi (3.30)

Fig. 3-25 shows the estimated individual errors and the degree of the total numerical errors by the

three data reduction equations of S60 at FN = 0.25. The three data reduction equations give a similar

degree of the numerical error. However, these equations should be used generally according to the error

characteristics. In this case, three error types are interdependent. The CROSS data reduction equation

is used for the numerical error estimation in this thesis. Fig. 3-26 compares the estimated individual

errors and the degree of the total numerical errors by the CROSS data reduction equations of S60 at

FN = 0.25. Each error type produces a different degree of error. The panel convergence error is still

large in comparison with the other error types. However, they lie all within ±2% of the CFD prediction

CWCFD. At FN = 0.275 the numerical error is too large. This is shown in Fig. 3-27. The propagated

errors are much smaller in comparison to the total errors ER and EW . At FN = 0.275, the numerical

error is beyond the degree of the total error. However, the numerical errors at the other FN are small

enough to be ignored. They are neither systematically distributed nor large.

3.7 Conclusion

Verification is performed in order to estimate the numerical errors in the CFD predictions and to choose

the best input parameters for SHIPFLOW. The verification is carried out by panel convergence analysis,

computational domain convergence analysis, and iterative convergence analysis in a verification range

encompassing different ship speeds and hull forms. Due to the fact that there are no absolute analytical

95

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

GridConvergence

Error

IterativeConvergence

error

ComputationalDomain Error

Total NumericalError

(ESN_cross)

Total NumericalError

(ESN_RSS)

Total NumericalError

(ESN_ADD)

Erro

rs

Figure 3-25: Degree of contribution of each error type to the total numerical error at FN = 0.25 for S60

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

Grid ConvergenceError

IterativeConvergence error

ComputationalDomain Error

Total Numerical Error(ESN_cross)

Erro

r/CW

CF

D(%) 0.2

0.225

0.25

0.3

0.316

Figure 3-26: The estimated errors of individual error types and the total numerical error of S60 atFN = 0.25

96

-4.0E-04

0.0E+00

4.0E-04

8.0E-04

1.2E-03

1.6E-03

2.0E-03

0.2 0.225 0.25 0.275 0.3 0.316

Fn

CW

CFD

and

err

ors

EW ER

CWCFD E_SN

Figure 3-27: Comparison of total error and numerical error.

values, extrapolated values are compared. The numerical errors due to the panel and computational

domain convergence errors are estimated by means of several criteria based on Richardson extrapolation.

The iterative convergence error is approximated based on the iterative tendency of the prediction. The

method verifies the numerical consistency. The results have to be consistent and accurate. A few open

issues still exist and require later attention. By and large, the numerical results obtained by SHIPFLOW

are sufficiently consistent when all system parameters are tuned suitably. Then it becomes possible to do

a meaningful comparison between the experimental results and the CFD results yielded by SHIPFLOW

providing a foundation for the validation of significant trends in Series 60.

97

Chapter 4

Trend Validation

4.1 Overview

While the validation of a CFD simulation to predict the wave-making resistance has been extensively

studied in relation to a single hull form at any given ship speed, the analysis of the systematic error

trends concerning a systematic hull form variation has seldom been performed. In this chapter, we will

investigate and discuss error trends as a function of hull shape parameters for a systematically varied

ship shape at different ship speeds, namely in a 4-dimensional hyperspace validation domain. The error

trends as a function of geometrical parameters and ship speed are investigated by sensitivity analysis

[9][13][65][67][68] in the validation range, i. e., as a function of hull form parameters and FN . Univariate

and multivariate form parameter effects on the error trends are visualized and discussed by means of

response surface methodology1 [11][24][28][73]. The errors due to physical model deficits and due to

numerical inaccuracies are composed of two characteristic error types, namely systematic error (bias)

and random error (noise). This makes it possible to formulate a best fit model describing the systematic

error trends as a function of geometrical parameters at each FN by means of a regression equation. A

regression analysis is undertaken to investigate the correlation of error trends with form parameters and

to model a best fit. Best fit models will be selected to test the ability of CFD prediction to be reliable

for the purpose of ship design despite the errors in the physical model and in the numerical analysis.

A trend validation based on 58 Series 60 family hull forms which are varied by four form parameters,

namely L/B, B/T , lcb, CB, is carried out in this study. Before the trend validation which is undertaken

by comparison of the CFD prediction and the experimental data begins, a brief overview of Series 60 test

1Although I have used a certain aspect of this methodology of thinking, my own approach was more specifically designedfor the current validation problem.

98

results and an investigation of their relative accuracies will be presented in section 4.2. CFD prediction

results were discussed in the previous chapter 3.

4.2 Experimental data for trend validation

4.2.1 Series 60 test results[101][102][103]

A methodical experiment with 62 Series 60 models was carried out in a wide range of ship speeds.

These experiments were carried through with turbulence stimulation2. Series 60 experimental results

are presented as circular c against circular k, and CTS against V/√LW for 400 ft LBP for the standard

sea water temperature of 15◦C (57◦F ). These total resistance coefficients were extrapolated from the

model test results CTM of a 20 ft model based on Froude’s hypothesis using the ATTC1947 line and the

model-ship correlation allowance CA = 0.0004 as follows:

CR ATTC = CTM − CFM ATTC (4.1)

CTS = CR ATTC + CFS ATTC + CA (4.2)

The validation state variables, CW and CR required in this thesis are not directly presented in Series

60. A backward extrapolation to reconstruct the total wave resistance coefficient of the model scale must

be performed. This is carried out in the following steps:

1. The model test results CTM can be reconstructed from the equations (4.1) and (4.2) reversing the

Series 60 extrapolation

CR ATTC = CTS − CFS ATTC − CA (4.3)

CTM = CR ATTC + CFM ATTC (4.4)

2. We will here use the ITTC1957 friction line for separating CR from the total resistance coefficient[104]

CR ITTC = CTM − CFM ITTC (4.5)

3. The Hughes method based on the Prohaska evaluation [85] is used to separate the wave resistance

2Studs are used for the Series 60 parents and LCB Series and trip wires are used for the final series of the L/B andB/T ratios. Model test results with trip wires gave slightly higher resistance than those with studs for the high CB .

99

coefficient CW from the total resistance coefficient CTM given by equation (4.4).

CW HUGHES = CTM − (1 + k)CFM ITTC (4.6)

where the form factor k is extrapolated by the Prohaska method referring to the ITTC1957 friction

line.

4.2.2 Interpolation of Series 60 test results in the validation domain

The trend validation domain consists of 58 different ships at six FN . This validation domain contains a

denser population than the original Series 60. Therefore, the test results of Series 60 must be interpolated

in the validation range. In order to get the test results at the FN required in the validation domain, an

interpolation for FN is also carried out. At first, the interpolations are carried out for the total resistance

coefficient CTM . The validation state variables are then calculated from equations (4.5) and (4.6). The

residuary resistance coefficients CR are directly calculated from the interpolated CTM . However, in

order to get the wave resistance coefficient, the form factor k must also be interpolated in the validation

domain. The required test data are interpolated by a cubic-spline interpolation method if the number

of the given data is more than 3, by a polynomial interpolation method if the number of the given data

is 3 or less. FN interpolation is done for each single ship respectively. At each FN , CTM as a function of

the form parameters B/T, L/B, lcb3 and CB is interpolated in the validation domain by the 4 following

steps, case by case:

Step 1. Interpolation on the B/T axis

The total resistance coefficients CTM at B/T = 2.75 and 3.25 is interpolated from three given

CTM at B/T = 2.5, 3.0, 3.5. This interpolation will be successively carried out for CB = 0.6, 0.65,

and 0.7 in the dependently varying L/B range on each CB .

Step 2. Interpolation on the L/B axis

L/B is linearly varied according to the change of CB. Therefore, for a given CB and B/T , three

or more than three points can be interpolated on the L/B axis.

Step 3. Interpolation on the CB axis

The total resistance coefficient CTM at CB = 0.625 and 0.675 is interpolated in the range 0.6 ≤CB ≤ 0.7. Three data for CB = 0.6,0.65, and 0.7 are given. This interpolation is successively

undertaken for all required L/B and B/T combinations.

3 lcb is dependent on the change of CB . Therefore, both form parameters can be treated as interdependent variables inthe interpolation.

100

Step 4. Interpolation on the lcb axis

lcb is varied totally independently from the L/B and B/T variations. The test results for CTM of

four or five lcb series are given for 5 parent ships respectively. CTM according to the change of lcb

is interpolated at each CB.

The same interpolation procedure is carried out for the form factor interpolation

Although this interpolation process adds errors to the experimental data due to the lack of more raw

data, the Series 60 test results remain fair in the validation domain.

4.2.3 Qualitative error investigation

In the validation process, the accuracy of the experimental data is a decisive element in determining the

level of validation. Therefore, it is necessary to get an understanding of the consistency and accuracy of

each error type in the experimental data. However, there is no report on the accuracy of measurement

of Series 60 tests. Regarding the state variables CR and CW which are derived by means of the Froude

or Hughes hypotheses, error types due to the frictional resistance coefficients and the form factor k

are added to the experimental data. As already discussed in section 4.2.1, the ITTC line and the

Prohaska extrapolation are used in this thesis. The degree of each individual error propagated to the

experimental data CR and CW is not known. This procedure introduces further errors. However, a

qualitative assessment of the sensitivity in the end results for CR and CW to an assumed normalized

error magnitude of 1% in the input variables CT , CF , and k can be performed to develop a feeling for

error propagation.

Fig.4-1 shows the sensitivities of the residuary resistance coefficient CR to 1% changes in the input

variables CT and CF with the test results of a parent ship of Series 60 having CB = 0.6. Fig. 4-1 shows

that ±1 % error in CT causes a larger than ±5 % error in CR at FN = 0.2. Similarly, 1 % error in CF

causes almost ±5 % error in CR. These sensitivities are reduced as FN increases. This absolute error in

CR is of course dependent on the magnitude of the individual errors in CF and CT , which is not known.

Fig. 4-2 shows the sensitivity of the wave resistance coefficient CW to the input variables, CT and CF

and the form factor k. Sensitivities of CW to each variable are higher than those of CR. They reach up

to 18% at low speed and 5 % in the high speed region according to Fig. 4-2. The effect of the form factor

k on CW is relatively small in comparison with CT and CF . It is within -3 %. These high sensitivities

are well anticipated due to the small magnitude of the wave resistance coefficient in comparison with

CT and CF .

Fig. 4-3 shows the total experimental error propagated by each input variable type to the validation

criteria CR and CW . The total experimental error is calculated by an error propagation method, Root

101

-6

-4

-2

0

2

4

6

8

0.2 0.225 0.25 0.275 0.3 0.316

Fn

Sens

itivi

ty(%

)Sensitivity of CR to CT

Sensitivity of CR to CF

Figure 4-1: Sensitivity of CR to 1% error of individual error types

-20

-15

-10

-5

0

5

10

15

20

0.2 0.225 0.25 0.275 0.3 0.316

Fn

Sens

itivi

ty(%

)

Sensitivity of Cw to CTSensitivity of Cw to CFSensitivity of Cw to k

Figure 4-2: Sensitivity of CW to 1% error of individual error type

102

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.2 0.225 0.25 0.275 0.3 0.316FN

ED (%

)

Error in CW

Error in CR

Figure 4-3: Experimental data error ED in Cw and CR due to 1% changes in each input variable type

Sum of Square (RSS). In the low speed region, the error of CW is much higher than that of CR, while in

the high speed region between FN=0.275 and FN=0.316 the differences become smaller. Experimental

errors in CR are near 7% in the low FN region and near 3% in the high FN region. Experimental errors

in CW are in the range of 14% - 25% in the low FN region and in the range of 3% - 5.5% in the high

FN region. This shows that the validation of the wave resistance coefficient in the low FN region will

be very difficult. In the low speed region, either very accurate measurement or a different state variable

like CR are required in order to carry through a reliable validation.

As a reference, the differences of CR and CW derived by each friction line4 of ITTC and ATTC

respectively are compared in Fig. 4-4 and Fig. 4-5. Each CR ATTC and CW ATTC can be described by

CR ITTC and CW ITTC derived by the ITTC friction line as follows:

CR ATTC = CR ITTC + (CFM ITTC − CFM ATTC) (4.8)

4ATTC line:

0.242

CFATTC= log10(Rn ×CFATTC )

Hughes line:

CFo =0.066

(log10Rn − 2.03)2ITTC line:

CF ITTC =0.075

(log10Rn − 2)2

103

CW ATTC = CW ITTC + ( 1 + k) · (CFM ITTC − CFM ATTC) (4.9)

The friction lines give a similar tendency while the differences decrease for increasing FN . This is

caused by the different inclination of the two friction lines as a function of the ship speed. The ATTC

line results in larger CR and CW in comparison with those yielded by the ITTC line. The hull forms

dealt with in this thesis are of 20ft length and their error trends will be investigated at each different FN

case by case. Therefore, the choice between the two friction lines does not affect the systematic error

trends which are qualitatively investigated as a function of the form parameters.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0.2 0.225 0.25 0.275 0.3

FN

CR, C

W

CR ATTCCR ITTCCW ATTCCW ITTC

Figure 4-4: CR and CW based on different friction lines

DIfferences in magnitude

0.E+00

2.E-05

4.E-05

6.E-05

8.E-05

1.E-04

0.2 0.225 0.25 0.275 0.3FN

difference in CR

difference in CW

Differences in percent

0

10

20

30

40

50

0.2 0.225 0.25 0.275 0.3 0.316FN

CR

CW

Figure 4-5: Differences of CW and CR based on each friction line of ITTC and ATTC

104

4.2.4 Summary

The Series 60 test results are used as experimental data for the trend validation. The thesis validation

domain, which is a subdomain of Series 60, contains a denser population than the original Series 60.

The test results of Series 60 are interpolated in the validation range. The validation state variables, CW

and CR, required in this thesis are not directly presented in Series 60. Therefore, they are reconstructed

by reverse Froude’s and Hughes’ extrapolation by using the ITTC1957 friction line and the form factor

which is extrapolated by the Prohaska method based on the ITTC1957 friction line. In order to carry out

a reliable trend validation, measurements of high accuracy are required, though random errors (noise)

are removed in the validation process.

1. In the low speed region, the accuracy in the experimental data determines the reliability of the

validation. Especially, the magnitude of CW is so small that it is influenced up to 25 % by a 1%

error of input variable magnitude. In the low speed region, experimental data with high consistency

and accuracy, at least within 1 % error range, are necessarily required. CR will offer more reliable

values than CW . We hope, by using CW as the state variable for the trend validation, that the

systematic trend of CW as a function of the form parameter is stronger than the perturbation of

CW due to the propagated error by each error type.

2. In the high speed region, the experimental data are in a sufficiently accurate range for the trend

validation.

3. Since the frictional resistance law is the same for all 58 ships at each FN , the frictional resistance

error remains without influence on the trend validation as a function of the form parameters. This

error type does not affect the relative comparison of error trends as a function of ship shapes.

4. The choice of friction line also does not affect the relative comparison of error trends as a function

of ship shape variables. Only the magnitude of error of CT and of the form factor k influence the

accuracy of CW .

4.3 Discussion of form parameter effects

4.3.1 Approach

The sensitivities of the predictions and the error trends will be investigated as a function of the form

parameters in this study. In order to discuss the sensitivities of the predictions and the error trends

on the 4-dimensional validation domain systematically, the response surface methodology is introduced

[17][73]. This methodology is generally applied to situations where several input variables are potentially

105

influencing some performance measure or quality characteristics of a process. Based on this methodology,

the validation domain can be divided into several domain groups which yields univariate effects and

multivariate effects (bivariate, trivariate, quadrivariate) of form parameters. By this kind of treatment

of each validation case, there are 16 combinatorial possibilities to form groups of investigation which are

defined by 4 hull form parameters (24). The normalized form parameters serving as the free variables in

the regression were denoted by ti, i =1,...,4 as introduced in section 2.4.3 and are related as follows:

t1 to B/T , t2 to L/B, t3 to lcb, t4 to CB .

• Univariate form parameter effects (4):

t1, t2, t3, t4

• Multivariate form parameter effects (11):

— Bivariate (6):

t1 vs t2, t1 vs t3, t1 vs t4, t2 vs t3, t2 vs t4, t3 vs t4

— Trivariate (4):

t1 vs t2 vs t3, t1 vs t2 vs t4, t1 vs t3 vs t4, t2 vs t3 vs t4

— Quadrivariate (1):

t1 vs t2 vs t3 vs t4

To analyze and discuss the univariate form parameter effects on the predictions and the error trends,

each individual form parameter is varied systematically while fixing all other form parameters. Regarding

the graphical visualization domain shown in Fig. 2-11, this effect can be investigated e. g. by showing the

error trends on each edge of the validation hyperspace domain. Multivariate effects can be investigated

when more than 2 form parameters are changed simultaneously. Multivariate effects consist of three sets

of combination: Bivariate combination when two form parameters are varied while the other two form

parameters remain fixed, trivariate combination and quadrivariate combination correspondingly.

In order to illustrate each variational case and a group of variational cases briefly, we define a

four column matrix symbol expressed by three types of notation: nondimensionalized number (0,...,

1), nondimensionalized form parameter (ti, i = 1, ..., 4) and the form parameter (B/T , L/B, lcb, CB).

The first column of the matrix is used for t1, the second column for t2 and similarly for t3 and t4. A

nondimensional number in a matrix corresponds to any point in the 4-dimensional validation hyperspace

domain. If no form parameter is varied, i. e., ti = 0, i = 1, ..., 4, this corresponds to the ship No.1 and is

106

denoted by 0 0 0 0 . Conversely, if all ti = 1, i = 1, ..., 4, this is denoted by 1 1 1 1 , i.

e., the last ship of the validation domain, Ship No. 58. The form parameters B/T , L/B, lcb, CB in the

matrix are defined as denoting a group of ti variations. B/T in the matrix means a group containing

5 points determined by the t1 variation, i. e., t1 = 0, 0.225, 0.25, 0.75, 1.0. This is, e. g., denoted

by B/T 0 0 0 corresponding to the edge departing from 0 0 0 0 in the hyperspace

validation domain. B/T L/B 0 0 means a group containing all validation cases determined

by the combinatorial variation of t1 and t2 and corresponds to the plane which is created by t3 = 0

and t4 = 0. B/T L/B lcb CB will contain all 58 ships and corresponds to the 4-dimensional

hyperspace validation domain.

In the matrix the symbol ti is used for the variable of interest. Therefore, each univariate form para-

meter variation ti which is departing from 0 0 0 0 is denoted by t1 0 0 0 , 0 t2 0 0 ,

0 0 t3 0 , 0 0 0 t4 respectively. When ti is varied from 0 to 1, each univariate matrix

corresponds to a point on any axis of interest in the 4-dimensional validation hyperspace domain. For

a bivariate form parameter variation t1 vs t2 departing from 0 0 0 0 , the matrix is denoted by

t1 t2 0 0 and corresponds to a point on the plane which is made by t3 = 0 and t4 = 0. This holds

also for the trivariate and the quadrivariate form parameter variation, e. g., such as t1 t2 t3 0

and t1 t2 t3 t4 . These matrices correspond to any point determined by each ti in a 3-dimensional

space and in a 4-dimensional hyperspace respectively. As a last explanation for the matrix notation,

a mixed notation such as B/T L/B t3 t4 will be used for a bivariate form parameter effect

investigation. Since the form parameter in the matrix is a parameter denoting a group and ti is a vari-

able of interest, this matrix means a bivariate form parameter variation of t3 and t4 by means of each

B/T and L/B plane. In fact, this is a 4-dimensional analysis because a two dimensional investigation is

undertaken by means of the variation of the plane.

We will discuss the predictions and the error trends related to the univariate and multivariate effects

for each subset of interest.

4.3.2 Univariate form parameter effects

This process aims at analyzing the sensitivity of the predictions and the error trends relative to the

individual form parameter variation. This is carried out by changing stepwise only one form parameter

of interest in five steps of ti = 0.0, 0.25, 0.5, 0.75, 1.0, while fixing the other form parameters. Here,

we will investigate univariate form parameter effects shown on the edges of the validation domain which

departs from the Ship No. 1, i.e., 0 0 0 0 . The four form parameter groups of t1 0 0 0 ,

0 t2 0 0 , 0 0 t3 0 , 0 0 0 t4 will be considered. Different ships correspond to

107

[ t1 0 0 0 ] [ 0 t2 0 0 ] [ 0 0 t3 0 ]

[ 0 0 0 t4 ]

t1(B/T)

t3(lcb)

t4(CB)

t2(L/B)t2(L/B) t2(L/B) t2(L/B)

t1(B/T)t1(B/T) t1(B/T)

t3(lcb) t3(lcb) t3(lcb)

t4(CB)t4(CB)t4(CB)

Figure 4-6: Geometric range for single form parameter effect investigation

varying each ti = 0.0, 0.25, 0.5, 0.75 , 1.0 for each form parameter. The row matrix symbols correspond

to the 4 edges in the 4 dimensional hyperspace validation domain as shown in Fig. 4-6.

This analysis consists of four steps. We will start by comparing the experimental data CR and CW

with the CFD predictions. As a next step, the error trends ER and EW are discussed as a function of

the form parameters. The topic of the third and fourth steps is the trends in CR, CW , CWCFD and the

trends in ER, EW respectively.

Univariate form parameter effect of t1

Fig. 4-7 shows a comparison of CR with CWCFD with respect to the t1(B/T ) variation of t1 = 0.0, 0.25,

0.5, 0.75 , 1.0, i. e., t1 0 0 0 . When t1 increases from 0 to 1, B/T increases from 2.5 to 3.5.

In Series 60 variation L/B, B/T and ∆/(L/100)3 are not independent but are related by the following

expression with dimensions in feet and displacement in tons, salt water :

∆L100

3 =CB

LB

2 BT

× 28570 (4.10)

This leads to decreasing draft and total displacement for the same length and block coefficient because

the breadth B is fixed as that of the ship No.1. Table 4.1 shows the variation of draft and displacement

for the model ship of 20 ft(6.096 m) with respect to the t1.

The behavior of CR and CWCFD is quite similar for both the t1 and FN variations. In the low speed

region (0.2 ≤ FN ≤ 0.25), CR increases slowly whereas CWCFD remains more or less unchanged. ER

gradually increases. At FN=0.275, CR and CWCFD lie close together. ER decreases. What is also

108

t1 0.0 0.25 0.5 0.75 1.0B/T 2.5 2.75 3.0 3.25 3.5T (m) 0.325 0.296 0.271 0.250 0.232∆(ton) 0.976 0.887 0.813 0.751 0.697

Table 4.1: The variation of draft and displacement with respect to the t1

noticeable is the fact that in the high speed region (0.3 ≤ FN ≤ 0.316), CR increases while CWCFD

decreases. ER increases. This is shown in Fig. 4-8 as a function of t1. The degree of ER is about 40% of

CR in the low speed region 0.2 ≤ FN ≤ 0.25 and 20-30% of CR in the high speed region. The range of CRfor the t1 variation is wider in the low speed region (0.2 ≤ FN ≤ 0.25) compared to the range of CWCFD.

But at FN = 0.275, this gets inverted: the range of CR is narrower than the range of CWCFD, while

the ranges are almost the same at FN = 0.3 and 0.316. The range of CR of the ships t1 0 0 0

is 19% of CR[0000] at FN = 0.275, the range of CWCFD is over 25%. For the t1 variation, ER shows

a large sensitivity at FN = 0.275. This is shown in Fig. 4-14. At the speed, there were numerical

convergence problem and CWCFD were somewhat largely predicted. This results in a small magnitude

of ER. Moreover, the sensitivity of CFD prediction as a function of t1 is much stronger than those of

experimental data. These two reasons lead to a large sensitivity of ER at FN = 0.275.

Regarding the variation of t1, both CR and CWCFD show decreasing tendencies for increasing t1. A

systematic tendency between the CR decrease and the CWCFD decrease can be noted. These decreasing

ratios of CR and CWCFD to those of the ship No.1 are shown for each FN in Fig. 4-9 and Fig. 4-10

respectively. Since the lines in both figures show decreasing tendencies for most cases, it is evident that

CR and CWCFD decrease for the t1 variation. However, in Fig. 4-9 in the case of t1 = 1.0 at FN = 0.2,

0.225, the sensitivities are slightly increasing. At these speeds, CR[1000](for B/T = 3.5) is larger than

CR[0.75000](for B/T = 3.25). The sensitivities of CR are somewhat parabolic in the low speed region

0.2 ≤ FN ≤ 0.25 and monotonous in the high speed region 0.275 ≤ FN ≤ 0.316 while the sensitivities ofCWCFD are strictly monotonous in all speed region. The sensitivity of CWCFD in the high speed region

is higher than those of CR. In this speed region, CWCFD is more sensitive to the t1 variation than CR.

Similar tendencies are shown in a comparison of CW with CWCFD in Fig. 4-11. There are systematic

differences between CR and CW . These differences of the two coefficients CR and CW are explained by

the viscous pressure drag coefficient, defined as k × CF , since CW = CR − kCF . Table 4.2 shows arelatively high viscous pressure drag for Series 60 No.1 ship. Conceptually, the magnitude of the viscous

pressure drag should be independent of the friction line and the form factor extrapolation method.

However, the use of ITTC1957 friction line containing correlation for the ship form additionally and

Prohaska method may cause several uncertainty sources to extrapolate the form factor.

Regarding the difference between CR and CW as a function of FN for a ship, it decreases propor-

109

FN

CR,C

WC

FD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CR [ t1 0 0 0 ]

Figure 4-7: Comparison of CR and CWCFD with respect to the t1 variation as a function of FN

t1

E R

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-8: ER for each FN as a function of t1

110

t1

Sens

itivi

tyof

CR

0 0.25 0.5 0.75 10.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-9: Sensitivity of CR as a function of t1

t1

Sen

sitiv

ityof

CW

CFD

0 0.25 0.5 0.75 10.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-10: Sensitivity of CWCFD as a function of t1

111

FN 0.200 0.225 0.250 0.275 0.300 0.316CR(∗E3) 0.6717 0.7303 0.8137 1.4846 2.2871 2.3641

CW (∗E3) 0.2126 0.2806 0.3721 1.0502 1.8593 1.9400CWCFD

(∗E3) 0.3967 0.4104 0.4716 1.2347 1.6886 1.6226CFITTC(∗E3) 3.1106 3.0468 2.9913 2.9425 2.8989 2.8734CPV (∗E3) 0.4591 0.4497 0.4415 0.4343 0.4279 0.4241

Table 4.2: Residuary resistance and viscous pressure drag (CPV = k × CFITTC) of ship No.1

tionally to the CF line when FN increases because k remains constant. The absolute magnitudes of the

total model resistance coefficient of the ship No.1 are compared in Table 4.2 at each FN . The decreasing

rate of k × CF as a function of FN is small enough to be assumed as constant in comparison with an

increasing rate of CR[0000]. However, in the speed range of 0.2 ≤ FN ≤ 0.275, the magnitude of k × CF

is larger than ER, CWCFD are overpredicted (EW is negative) in this speed range and underpredicted

(EW is positive) in the other higher speed ranges. Fig. 4-12 shows these trends of EW as a function of

t1. EW is more than 100% of CW at FN = 0.2 and about 50 % at FN = 0.225. From the speed FN =

0.25, EW decreases from 20% to 7% of CW for increasing FN . However, at FN = 0.316, EW increases

up to 20-30% of CW .

Regarding the difference of CR and CW as a function of t1, it is proportional to the change of the

form factor k. The form factor k for each ship according to the change of ti is shown in Table 4.3. ∆ki

is the variation range of k when ti is varied from 0 to 1 and defined as

∆ki = |Max(ki)−Min(ki)| , ti = 0, 0.25, 0.5, 0.75, 1.0 (4.11)

where ki contains the 5 form factors corresponding to ti = 0.0, ..., 1.0. ∆ki/k[0000](%) is the percentage

ratio of the ∆ki to the form factor k of ship No.1, i. e., k[0000] = 0.1476. k has a concave parabolic

trend when t1 is varied from 0 to 1. In the range of t1, the form factor k is changed by 10.5% of that

of the ship 0 0 0 0 , i. e., ∆k1/k[0000](%) = 10.5. If we investigate the form factor effects in the

sensitivities of CR and CW shown in Fig. 4-9 and 4-13 respectively, the decreasing tendencies remain

in the sensitivity of CW . The degree of sensitivity of CW becomes larger in comparison with that of

CR in the low speed region. This is certainly because CW [0000] is very small in the low speed region in

comparison with CR[0000]. Also it can be noticed that the form factor has considerable influence on the

magnitude of CW in the low speed region. The sensitivity of CW shows a decreasing tendency for all t1

variations while the sensitivity of CR increases slightly for the case of t1 = 1.0 at FN = 0.2 and 0.225.

A relatively large viscous pressure drag may occur around a ship having wider breadth compared to the

depth such as B/T = 3.5 in the speed region of FN = 0.2 and 0.225. If we investigate the form factor

effects in the error trends ER and EW , ER is influenced much by the form parameter variation. Its

112

parabolic trends as a function of t1 variation are similar to those of form factor while EW shows rather

monotonous trends.

Regarding the trends of the experimental data as a function of FN , in the low speed region, the

experimental data are changing only slowly. However, beyond FN = 0.25, the data begin to increase

rapidly. This tendency is changed when FN becomes larger than 0.3. Up to FN = 0.316 the increasing

ratio decreases. CFD predictions in the low speed region are less sensitive to the change of FN in

comparison with the experimental data. They are almost unchanged. However, beyond FN = 0.25,

the CFD predictions also begin to increase rapidly. The increasing ratio is higher than that of the

experimental data. Therefore, the CFD prediction will be close to the experimental data. However,

if FN is larger than 0.3, the CFD prediction decreases while the experimental data increase. In this

speed region, the tendency of both data is opposite. The error increases for increasing FN . The degree

of error becomes very small only for FN = 0.275 because the CFD prediction approach is close to the

experimental data.

Regarding the experimental data as a benchmark, CWCFD is overpredicted in the low speed region.

This could be caused by the following reasons. Firstly, CWCFD by definition has no viscous effects or

wave-viscous interaction. Secondly, trim and sinkage effects are not considered in the CWCFD calculation

although these effects are small in the low speed region. Thirdly, the form factor may be too great. Hence

the experimental data CW become too small. The main reason of the underprediction of CWCFD in high

speed region could be the neglected trim and sinkage effects in the calculation. These effects increase

for increasing FN . The trim and sinkage effects become noticeable and when FN is larger than 0.3, this

effect is large enough to change the qualitative characteristics of CWCFD. Although there still exists

some viscous effect on the wave system in the high speed region, its relative magnitude is not significant

enough to change the qualitative tendency of the error.

The form factor k is assumed to remain constant for all speed regions. However, it is not clear that

the physical phenomenon defined as the viscous pressure drag is directly proportional to k×CF . For eachship, there is a speed region where extreme viscous phenomena begin to occur. However, the change rate

of k×CF is only dependent on the inclination of the CF line when the ship speed is changed. Therefore,k may be a function of the ship speed as long as the Hughes method is used for the CW extrapolation.

Fig. 4-14 and Fig. 4-15 show the sensitivities of each error ER and EW as a function of the form

parameter variation for t1. The trends of the sensitivity of ER are similar except those at FN=0.275.

As discussed before, the ER and EW are small because CR or CW and CWCFD become closest at FN

= 0.275, while sensitivity of CWCFD for t1 is higher than that of CR or CW . The sensitivity of ER is

similar to the form factor as a function of t1. The trends of EW cover a wider range of sensitivity. For

FN < 0.275, EW is negative, for FN > 0.275, EW is positive. The sensitivities of EW at FN = 0.275

113

FN

CW

,CW

CFD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CW [ t1 0 0 0 ]

Figure 4-11: Comparison of CW and CWCFD with respect to the t1 variation as a function of FN

t1

E W

0 0.25 0.5 0.75 1-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-12: EW for each FN as a function of t1

114

t1

Sens

itivi

tyof

CW

0 0.25 0.5 0.75 10.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-13: Sensitivity of CWCFD as a function of t1

t1

Sen

sitiv

ityof

ER

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-14: Sensitivity of ER as a function of t1

115

t1

Sen

sitiv

ityof

EW

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

Figure 4-15: Sensitivity of EW as a function of t1

are increasing greatly. The reason is the same as in the case for ER at FN = 0.275. As discussed in

Chapter 3, the numerical error at FN = 0.275 is relatively high. Except for the sensitivity of ER and

EW at FN = 0.275, the ER and EW trends are changed within 25 % when t1 is varied throughout the

validation domain. The trends of ER and their sensitivities are systematic.

As a next step, in order to investigate the change of the error trends with respect to the different

form parameters, all the results for ti, i = 1, 2, 3, 4 are summarized in Tables 4.4 - 4.8.

Comparison of the experimental data with the CFD predictions

Table 4.4 shows comparisons of CR with CWCFD and CW with CWCFD. The speed range in the

validation domain is 0.2 ≤ FN ≤ 0.316 for ti, i = 1, 2, 3 and 0.2 ≤ FN ≤ 0.275 for t4.Regarding the variation of FN , the sensitivities of the resistance coefficients are higher for increasing

FN than those for the variations of the form parameters in the validation domain. Additionally, the

sensitivities of the experimental data are apparently a little higher than the sensitivities of the CFD

predictions. For all cases in the speed region of 0.2 ≤ FN ≤ 0.25, CR and CW increase slightly while

CWCFD appears relatively unchanged. For ti, i = 1, 2, 3, in the FN region of 0.3 ≤ FN ≤ 0.316, CR andCW increase continuously while CWCFD decrease. The errors ER and EW increase when FN increases.

Regarding the variation of the form parameter ti = 0, ..., 1, the value ranges of the resistance coef-

ficients are quite different for ti, i =1,2,3,4. The value ranges of the resistance coefficients are shown

116

at each speed as a percentage of CR of CR[0000]in Table 4.5, where CR[0000]

is the residual resistance

coefficient of the ship No.1, i. e., 0 0 0 0 . As defined for the variation range of the form factor

k, ∆CRi is the variation range of CR when ti is varied from 0 to 1. The ratio of the variation range to

CR[0000]in percent is denoted by γRi. These are defined as

∆CRi = |Max(CRi)−Min(CRi)| , ti = 0.0, 0.25, 0.5, 0.75, 1.0 (4.12)

γRi = ∆CRi/CR[0000]× 100 (4.13)

where CRi contains the 5 CR values corresponding to the ships at ti = 0.0, ..., 1.0. For the case of

ti, i = 2, 3, 4, the symbols are analogous. ∆CWi, ∆CWCFDi are used for the value ranges of the

wave resistance coefficients given by the experiment and the CFD simulation for any form parameter

ti respectively. The ratios γWi, γWCFDi are defined likewise. Additionally, the viscous pressure drag

coefficient, CPV 5= k × CF is introduced. Its notation is the same.When we look at Tables 4.4 and 4.5 in combination, in the low speed region of 0.2 ≤ FN < 0.25,

the residual resistance CR is more sensitive to the change of t2(L/B) and t4(CB) in comparison with

t1(B/T ) and t3(lcb). This holds for CWCFD. Therefore, the ratio of the variation range of γRi, γWCFDi

is larger for t2 and t4 while the ratio is very small for t3. γWi decreases proportionally with the degree

of the influence of γPV i on γRi. Since the effect of the viscous pressure drag is relatively large in

this speed region of FN , γWi is changed much, especially for t2 and t3. γR2 decreases while γR3

increases. Therefore, in this region, the accuracy of the form factor k influences the comparison of CW

and CWCFD considerably. Since the magnitude of wave resistance is relative small in the low speed

region, small change in form factor influence much the accuracy in the wave resistance CW . In addition,

when the effects of hull form variation on the wave resistance are as small as those of lcb variations,

small errors in form factor of each hull form lead to a faulty comparison not only quantitatively but also

qualitatively. In the high speed region of 0.275 ≤ FN ≤ 0.316, γRi becomes larger although it decreasesslightly when FN increases.

Concerning the individual case, γW1is small in the low speed region and large in the high speed

region. This means CWCFD is less sensitive in the low speed region and more sensitive in the high speed

region in comparison to CR and CW . This can be shown by the ratios γW1/γR1 and γWCFD1/γR1. For

the case of the t2 variation, γW2 becomes very small. Therefore, the sensitivity of EW will increase in

the low speed region. In the high speed region, γW2 and γWCFD2 are similar. For the case of the t3

variation, since both CR and CWCFD are less sensitive to the change of the form variation t3, γR3 is

5The actual pressure viscous drag could be larger than k×CFITTC because the friction line CFITTC contain additionalcorrelation factor 0.125CF HUGHES . The form factor k was extrapolated here based on the CFITTC friction line.

117

1.27 and γWCFD3 is 1.52. However, since γW3 becomes very much larger due to the effect of γPV 3,

the sensitivity of EW increases. In the low speed region, γWCFD3 is more similar to γR3. In the high

speed region, these tendencies remain almost unchanged. The variation in t4 causes a wider range of the

form factors and both the residual and the wave resistance. All the value ranges increase sensitively for

increasing FN . The ratio γW4/γR4 increases monotonously while the ratio γWCFD4/γR4 has its peak

point at FN = 0.25.

Sensitivity of CW and CWCFD

The following figures in Table 4.6 show the sensitivity of CW and CWCFD. Since the difference between

CR and CW depends only on the form factor k, this is shown in Table 4.3, only CW and CWCFD

are compared here. These sensitivities are calculated by dividing each resistance coefficients CW and

CWCFD by the resistance coefficients of the ship No.1 at each FN respectively.

These figures of sensitivities show two important characteristics to be considered. One is the relative

change rate of CW and CWCFD to the change of form parameter variation, ti = 0, ..., 1.

If the sensitivities are lower than 1, this means CW and CWCFD decrease due to the variation of

form parameter ti and if they are larger than 1, CW and CWCFD increases. If the sensitivity remains 1

over the ti variation, this means CW and CWCFD are fully independent of the ti variation.

The other is the relative relationship of each sensitivity due to the form parameter variation with

that due to FN variation. If each sensitivity line at each FN has a similar tendency, this means the

sensitivities of CW and CWCFD due to the change of form parameters are proportional to the change

rate of resistance due to FN variation. For example, if the 6 lines of sensitivity having different FN meet

together on a line, this means the change rate of CW and CWCFD due to the form parameter variation

at each FN is the same as the change rate of the CW and CWCFD due to the FN .

Regarding all figures in Table 4.6 based on general characteristics of sensitivity lines, CW and

CWCFD decrease for increasing t1 while they are for most cases increasing with increasing t2, t3, t4.

kit1(B/T ) t2(L/B) t3(lcb) t4(CB)

0.00 0.1476 0.1476 0.1476 0.14760.25 0.1371 0.1523 0.1434 0.1410ti =0.50 0.1321 0.1581 0.1397 0.14290.75 0.1325 0.1651 0.1348 0.15331.00 0.1383 0.1732 0.1310 0.1722∆ki 0.0155 0.0257 0.0166 0.0312∆ki/k[0000](%) 10.50 17.38 11.24 21.11

Table 4.3: Form factor k as a function of ti, ∆ki=|Max(ki)-Min(ki)|, ti= 0, 0.25, 0.5, 0.75, 1.0

118

CR and CWCFD CW and CWCFD

FN

CR,C

WC

FD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CR [ t1 0 0 0 ]

FN

CW

,CW

CFD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CW [ t1 0 0 0 ]

FN

CR,C

WC

FD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CR [ 0 t2 0 0 ]

FN

CW

,CW

CFD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CW [ 0 t2 0 0 ]

FN

CR,C

WC

FD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CR [ 0 0 t3 0 ]

FN

CW

,CW

CFD

0.2 0.225 0.25 0.275 0.30.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

0.316

CWCFD

CW [ 0 0 t3 0 ]

FN

CR,C

WC

FD

0.2 0.225 0.25 0.2750.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

CWCFD

CR [ 0 0 0 t4 ]

FN

CW

,CW

CFD

0.2 0.225 0.25 0.2750.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-030.00.250.50.751.0

0.00.250.50.751.0

CWCFD

CW [ 0 0 0 t4 ]

Table 4.4: Comparison of CR, CW and CWCFD for 0 ≤ ti ≤ 1, i = 1, ..., 4

119

FN 0.200 0.225 0.250 0.275 0.300 0.316CR[0000]

(∗E3) 0.67 0.73 0.81 1.48 2.29 2.36CFITTC (∗E3) 3.11 3.05 2.99 2.94 2.90 2.87

γR1 17.08 11.75 10.16 18.86 21.66 18.52γW1 10.50 6.42 6.71 17.00 20.48 17.39

t1(B/T ) γWCFD1 8.13 5.52 8.26 25.40 22.99 18.97γV P1 7.18 6.46 5.70 3.07 1.96 1.88γW1/γR1 0.61 0.55 0.66 0.90 0.95 0.94γWCFD1/γR1 0.48 0.47 0.81 1.35 1.06 1.02

γR2 13.34 17.16 16.12 22.55 21.36 20.52γW2 1.80 6.46 6.71 17.48 18.11 17.40

t2(L/B) γWCFD2 11.02 13.33 11.59 17.46 18.07 16.88γV P2 11.88 10.70 9.43 5.08 3.25 3.12γW2/γR2 0.14 0.38 0.42 0.78 0.85 0.85γWCFD2/γR2 0.83 0.78 0.72 0.77 0.85 0.82

γR3 1.27 2.75 12.90 12.34 9.17 9.28γW3 8.22 8.91 19.02 15.65 11.27 11.31

t3(lcb) γWCFD3 1.52 2.72 7.01 4.48 4.81 5.43γV P3 7.62 6.82 5.85 3.16 2.06 1.95γW3/γR3 6.49 3.24 1.47 1.27 1.23 1.22γWCFD3/γR3 1.20 0.99 0.54 0.36 0.52 0.58

γR4 33.62 44.19 65.60 116.44γW4 22.23 33.92 56.56 104.57

t4(CB) γWCFD4 13.53 42.15 66.44 89.62γV P4 14.00 12.41 10.88 5.09γW4/γR4 0.66 0.77 0.86 0.90γWCFD4/γR4 0.40 0.95 1.01 0.77

Table 4.5: Variational range of resistance coefficients over the variational range of each form parameteras a percentage of CR[0000]

120

Sensitivity of CW Sensitivity of CWCFD

t1

Sen

sitiv

ityof

CW

0 0.25 0.5 0.75 10.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

t1

Sens

itivi

tyof

CW

CFD

0 0.25 0.5 0.75 10.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

t2

Sen

sitiv

ityof

CW

0 0.25 0.5 0.75 10.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 t2 0 0]

t2

Sens

itivi

tyof

CW

CFD

0 0.25 0.5 0.75 10.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 t2 0 0]

t3

Sen

sitiv

ityof

CW

0 0.25 0.5 0.75 10.80

0.90

1.00

1.10

1.20

1.30

1.40

FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 0 t3 0]

t3

Sens

itivi

tyof

CW

CFD

0 0.25 0.5 0.75 10.80

0.90

1.00

1.10

1.20

1.30

1.40 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 0 t3 0]

t4

Sen

sitiv

ityof

CW

0 0.25 0.5 0.75 10.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

FN = 0.2FN = 0.225FN = 0.25FN = 0.275

[0 0 0 t4]

t4

Sens

itivi

tyof

CW

CFD

0 0.25 0.5 0.75 10.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60 FN = 0.2FN = 0.225FN = 0.25FN = 0.275

[0 0 0 t4]

Table 4.6: Sensitivities of CW and CWCFD as a univariate functions of form parameters

121

ER EW

t1

E R

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

t1

E W

0 0.25 0.5 0.75 1-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

t2

E R

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 t2 0 0]

t2

E W

0 0.25 0.5 0.75 1-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 t2 0 0]

t3

E R

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 0 t3 0]

t3

E W

0 0.25 0.5 0.75 1-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 0 t3 0]

t4

E R

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03 FN = 0.2FN = 0.225FN = 0.25FN = 0.275

[0 0 0 t4]

t4

E W

0 0.25 0.5 0.75 1-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04 FN = 0.2FN = 0.225FN = 0.25FN = 0.275

[0 0 0 t4]

Table 4.7: Error trends ER and EW as a univariate functions of form parameters

122

Sensitivity of ER Sensitivity of EW

t1

Sen

sitiv

ityof

E R

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

t1

Sens

itivi

tyof

E W

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00 FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[t1 0 0 0]

t2

Sen

sitiv

ityof

E R

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 t2 0 0]

t2

Sens

itivi

tyof

E W

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 t2 0 0]

t3

Sen

sitiv

ityof

E R

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 0 t3 0]

t3

Sens

itivi

tyof

E W

0 0.25 0.5 0.75 1-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

FN = 0.2FN = 0.225FN = 0.25FN = 0.275FN = 0.3FN = 0.316

[0 0 t3 0]

t4

Sen

sitiv

ityof

E R

0 0.25 0.5 0.75 10.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

FN = 0.2FN = 0.225FN = 0.25FN = 0.275

[0 0 0 t4]

t4

Sens

itivi

tyof

E W

0 0.25 0.5 0.75 1-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

FN = 0.2FN = 0.225FN = 0.25FN = 0.275

[0 0 0 t4]

Table 4.8: Sensitivities of ER and EW as a univariate functions of form parameters

123

Only in the case of t3 variation, the sensitivity of CW is opposite to that of CWCFD in the low speed

region. As discussed in Table 4.4 and 4.5, the variation range of CR and CWCFD as a function of t3 is

very small in the low speed region while the form factor variation range is larger than those of prediction

CR. This yields a spread variation range in CW as a function of t3. In this speed range, the errors in

experimental data due to the data evaluation as well as form factor extrapolation, although they are

not so large, could play a significant role because the change of wave resistance due to form variation

is too small. Trend validation will be successful when they are undertaken in the validation domain

which gives a significant difference and sensitivity in the wave resistance coefficient. The trends of the

small local change of ship shape which results in less sensitivity in the validation state variables could

be within the errors due to the experimental data and CFD predictions. For the variation of t1 and t2,

CWCFD shows strictly monotonous tendency. The degree of sensitivity of t4 is larger in comparison to

the other form parameters for both CW and CWCFD. Generally for CWCFD, the sensitivities at high

speed are larger than those in the low speed region. The range of sensitivity of CW as a function of FN

is relatively wider in the low speed region than that of CWCFD. However, in the high speed region, the

characteristics are similar for both CW and CWCFD. The difference of sensitivity of CW and CWCFD

directly influence the error EW tendencies. EW may be more systematic in the high speed region as the

hull forms are changed.

Error trends and sensitivities as a function of form parameters

The error trends between the experimental data CR, CW and CWCFD and their sensitivities are shown

as a function of the form parameter variation in Table 4.7 and Table 4.8. In Table 4.7, ER and EW

show systematic trends as a function of ti = 0.0, ..., 1.0. As shown in Table 4.4, it is evident that both

types of error ER and EW increase rapidly in the high speed region 0.275 < FN ≤ 0.316. ER and EWin Table 4.8 show significant sensitivities to the change of FN . In the low speed region, ER and EW

increase when FN increases except at FN = 0.275. At FN = 0.275, ER and EW of the ship No.1 are

the smallest. However, their increasing rate is high when ti is varied from 0.0 to 1.0. Similar tendencies

appear for all four cases ti, i = 1, 2, 3, 4, although there is some difference in the degree of sensitivity.

4.3.3 Bivariate form parameter effects

Bivariate form parameter effects on the error trends will be investigated in the 4-dimensional hyperspace

validation domain. In order to investigate the change of the bivariate form parameter effects as a

function of the other two form parameter variations as well as the bivariate form parameter effects,

the 4-dimensional hyperspace validation domain will be divided into several 2-dimensional domains.

The 2-dimensional error trends are visualized together in the following figures. All validation points

124

in the hyperspace domain can be visualized in a table. A brief notation for this case was defined in

section 4.3.1. The bivariate error trends ER and EW are investigated and visualized at FN = 0.25

and 0.316. The effects of the bivariate form parameters B/T vs L/B and lcb vs CB on ER and EW

are discussed in relation to the change of the two parameters t3 vs t4 and t1 vs t2 respectively. i. e.,

B/T L/B t3 t4 and t1 t2 lcb CB .

Bivariate form parameter effects of B/T vs L/B and lcb vs CB at FN = 0.25

The first figure of Table 4.9, i. e., the surface of the error trends ER on B/T L/B 0 0

looks similar to a developable surface generated by each univariate error trend curve departing from

0 0 0 0 . The bivariate effects of B/T vs L/B on the error trends remain monotonic. Regarding

the variational tendency as a function of t3 and t4, the surfaces are little changed as a function of t3

while the forms of the surfaces are much changed as a function of t4. Since the surfaces with any of

t3 or t4 being equal to 0.5 are generated only by four points on its edge, they remain always a plane.

Regarding the overall variational tendency of the B/T and L/B surfaces, the t4 effect on the error trends

ER and EW is large in comparison with the t3 effect. The CB variation gives a dominant effect on the

bivariate form parameter effects. Similar tendencies are shown for the EW surface in Table 4.10. The

EW surface is more sensitively changed to the t4 variation than in the case of the ER surface. Especially

in the case of t4 = 1, B/T L/B t3 1 , the surface shape is much changed by EW on the middle

point, 0.5 t2 1 1 and shows the effect of CB dominantly. The characteristics of the univariate

form parameter effects remain similar in the bivariate form parameter effects.

Tables 4.11 and 4.12 show the trend of ER and EW on t1 t2 lcb CB . These figures show the

same results as those on B/T L/B t3 t4 but from a different aspect of visualization. The error

trends surfaces on any plane of lcb vs CB are investigated and the variational tendency of the surface

variation are considered as a function of t1 and t2. As discussed before, the error trend surfaces ER

and EW on 0 0 lcb CB show an apparently dominant CB effect. Regarding the change of the

surfaces as a function of t1 and t2, all figures appear almost identical. The bivariate effects of B/T and

L/B on ER and EW are very smooth.

Bivariate form parameter effects of B/T vs L/B, B/T vs lcb and L/B vs lcb at FN = 0.316

Since the speed range in the validation is limited for the CB variation up to FN = 0.25 for CB = 0.7, the

form parameter CB is excluded here. The bivariate form parameter effects are shown in the following

tables. The three kinds of error trend surfaces on the plane by B/T vs L/B, B/T vs lcb and L/B vs lcb

are shown as a function of the remaining ti. With regard to Table 4.13 and 4.14, the small bivariate

effects on the error trends are shown at FN = 0.316. In the first figure which shows the error trend

125

t4t3 0 0.5 1

0

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

0.5

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

t2 (L/B)

t1 (B/T)

ER

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

Table 4.9: Surface of ER [ B/T L/B t3 t4] at Fn=0.25

126

t4t3 0.0 0.5 1.0

0.0

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

0.5

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

t2 (L/B)

t1 (B/T)

EW

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

1.0

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t1 (B/T)

t2 (L/B)

Ew (C

W-C

WC

FD)

Table 4.10: Surface of EW [ B/T L/B t3 t4 ] at Fn=0.25

127

t2t1 0 0.5 1

0

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

0.5

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

t4 (CB)

t3 (LCB)

ER

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

2

3

4

5

6

7

8

x 10-4

t3(LCB)

t4(CB)

ER (C

R-C

WC

FD)

Table 4.11: Surface of ER [ t1 t2 LCB CB] at Fn=0.25

128

t2t1 0 0.5 1

0

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

0.5

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

t4 (CB)

t3 (LCB)

EW

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

-2

-1

0

1

2

x 10-4

t3(LCB)

t4(CB)

EW (C

W-C

WC

FD)

Table 4.12: Surface of EW [ t1 t2 LCB CB] at Fn=0.25

129

t3 = 0 t3 = 0.5 t3 = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t1 (B/T)

t2 (L/B)

ER (C

R-C

WC

FD)

Table 4.13: Surface of ER [B/T L/B t3 0] at Fn=0.316

t3 = 0 t3 = 0.5 t3 = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t1 (B/T)

t2 (L/B)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t1 (B/T)

t2 (L/B)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t1 (B/T)

t2 (L/B)

EW (C

W-C

WC

FD)

Table 4.14: Surface of EW [B/T L/B t3 0] at Fn=0.316

t2 = 0 t2 = 0.5 t2 = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t1 (B/T)

t3 (LCB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t1 (B/T)

t3 (LCB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t1 (B/T)

t3 (LCB)

ER (C

R-C

WC

FD)

Table 4.15: Surface of ER [B/T t2 lcb 0] at Fn=0.316

130

t2 = 0 t2 = 0.5 t2 = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t1 (B/T)

t3 (LCB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t1 (B/T)

t3 (LCB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t1 (B/T)

t3 (LCB)

EW (C

W-C

WC

FD)

Table 4.16: Surface of EW [B/T t2 lcb 0] at Fn=0.316

t1 = 0 t1 = 0.5 t1 = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t2 (L/B)

t3 (LCB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t2 (L/B)

t3 (LCB)

ER (C

R-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

6

6.5

7

7.5

8

8.5

9

9.5

10

x 10-4

t2 (L/B)

t3 (LCB)

ER (C

R-C

WC

FD)

Table 4.17: Surface of ER[t1 L/B lcb 0] at Fn=0.316

t1 = 0 t1 = 0.5 t1 = 1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t2 (L/B)

t3 (LCB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t2 (L/B)

t3 (LCB)

EW (C

W-C

WC

FD)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

3

3.5

4

4.5

5

5.5

x 10-4

t2 (L/B)

t3 (LCB)

EW (C

W-C

WC

FD)

Table 4.18: Surface of EW [t1 L/B lcb 0] at Fn=0.316

131

surface on B/T L/B 0 0 , there is a certain difference between the two curves on the edge of

B/T 0 0 0 and B/T 1 0 0 while the other two curves on the edge of 0 L/B 0 0

and 1 L/B 0 0 remain almost unchanged. This means that the L/B variation exerts such a

strong influence on the error trends that the tendency of the B/T effect is changed. Regarding the

variational tendency of each surface as a function of t3, the surface shapes remain almost unchanged,

although there is some difference in their magnitudes. The lcb effects on the error trends are small

in comparison with the other form parameter effects. These characteristics are shown in the following

tables. They also show the same results but are visualized from a different aspect. In all other figures,

the dominant effect of lcb on the error trends is shown. At FN = 0.316, the bivariate effects are more

variable in comparison with the surface at FN = 0.25.

4.3.4 Trivariate form parameter effects

The individual trivariate form parameter effects on EW and ER should be shown in a 4-dimensional

hyperspace directly. We will investigate and visualize the error trends EW of the 4-dimensional hyper-

space on the trivariate validation domain. This is achieved by projecting all the values of EW on the

trivariate validation domain to each point of the remaining form parameter. A geometric illustration

of this approach is shown for B/T in Fig.4-16. This is done firstly by fixing t1 at the level of 0, 0.25,

0.5, 0.75, 1 and by dividing the sample into groups having the same level of t1. This can be symbolized

shortly as t1 L/B lcb CB and the characteristics of the data group will be investigated as a

function of t1. The matrix of (4.14) shows the number of ships corresponding to each t1. In this context,

56 ships are considered simultaneously.

0 L/B lcb CB : 26 ships (4.14)

0.25 L/B lcb CB : 1 ship

0.5 L/B lcb CB : 8 ships

0.75 L/B lcb CB : 1 ship

1 L/B lcb CB : 20 ships

The error trends of EW are shown for all four form parameters at FN = 0.25 in Fig. 4-17, Fig. 4-18,

Fig. 4-19 and Fig. 4-20. All points at each ti = 0.0, 0.25, 0.5, 0.75, 1.0 are the projection of the EW

hyperspace to a value of the form parameter. The mean is an average value of all EW projected at each

ti value. Regarding the distribution of individual EW points, they are widely distributed on the B/T

132

1 2 3 4 5B/T

22 24

16

10

31

7

3525

20

11

4439

3

5257

L/B40 42 45

32

12

17

2621 2334

53

4830

55

15

85

3658

50

37

38

47

51

CB

lcb

12729

3319

4341

6 949

5654

1314

18

[ 0 L/B lcb C B ] [ 1 L/B lcb C B ][ 0.5 L/B lcb C B ]

Figure 4-16: Projection of groups of ships on the level of B/T (t1)

and the L/B axis. On the lcb axis, the variation range of data points increases as a function of lcb. This

distribution shows also the dominant effect of the CB variation on the EW trends while the B/T and

L/B effects are rather small. Comparing the mean effects and the univariate effects on each axis, on

the B/T and L/B axes, the mean values are more parabolic while the univariate effects are monotonic.

This shows also the dominant influence of the CB effect on EW . On the lcb axis, both are similarly

distributed. These tendencies are similar to the univariate effects on EW .

4.3.5 Significant results

The following characteristics in the results of CR and CW CFD predictions and in their error trends are

significant.

1. CFD prediction of CR and CW is less sensitive to the hull form variation in the low speed region

while it is more sensitive in the high speed region than the experimental data.

2. In the low speed region, the trend validation of CWCFD is not accurate because the experimental

data CW is much influenced by the form factor. Therefore, a small error in the form factor

extrapolation can cause a significant error in the test evaluation. In the region of 0.25 ≤ FN ≤0.275, the numerical CFD simulation has a convergence problem. The prediction CWCFD contains

a large numerical error which influences the systematic error trends.

133

-3.00E-04

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

0 0.5 1

t1 (B/T)

E W[t1 L/B LCB CB ]

Mean[t1 L/B LCB CB]

[ t1 0 0 0 ]

Figure 4-17: Comparison of [t1 L/B lcb CB] and [t1 0 0 0] at FN = 0.25

-3.00E-04

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

0 0.5 1

t2 (L/B)

E W

[ B/T t2 LCB CB ]

Mean [B/T t2 LCB CB]

[ 0 t2 0 0 ]

Figure 4-18: Comparison of [B/T t2 lcb CB] and [0 t2 0 0] at FN = 0.25

134

-3.00E-04

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

0 0.5 1

t3 (LCB)

EW[ B/T L/B t3 CB ]

Mean [B/T L/B t3 CB]

[ 0 0 t3 0 ]

Figure 4-19: Comparison of [B/T L/B t3 CB] and [0 0 t3 0] at FN = 0.25

-3.00E-04

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

0 0.5 1

t4 (CB)

E W

[ B/T L/B LCB t4 ]

Mean [ B/T L/B LCB t4 ]

[ 0 0 0 t4 ]

Figure 4-20: Comparison of [B/T L/B lcb t4] and [0 0 0 t4] at FN = 0.25

135

3. It is quite evident that ER and EW react much more sensitively to a change of the ship speed

than to a change of the geometric parameters in the validation domain. The error trends of

EW and ER are very different depending on the kind and the change of the form parameters.

However, their characteristics are simple and smooth and their univariate form parameter effects

are dominant. The bivariate form parameter effects show a similar tendency as the univariate

form parameter effects. ER and EW show similar bivariate tendencies although the ER surface

changes more smoothly than the EW surface. At low speeds, the error surfaces are more monotonic

in comparison with the bivariate effects in the high speed region. The tendency of ER and EW

as a function of the bivariate form parameters is governed by the combined effects of two form

parameters. The trivariate form parameter effects show similar tendencies as the univariate form

parameter effects.

4.4 Systematic error trends analysis by regression analysis

4.4.1 Approach to regression analysis

A general multivariate linear regression equation form is:

Yi = β0 + β1Xi1 + β2Xi2 + ...+ βpXip + εi (4.15)

where Xip are taken as independent variables, Yi as response variables or dependent variables and βp

as regression parameters. In this study, a more general type of linear model in variables X will be

introduced as following :

Yi = β0 + β1Zi1 + β2Zi2 + ...+ βpZip + εi (4.16)

Zip are general transformed functions of Xip. In this study, the integer power transformation, the recip-

rocal transformation, the logarithmic transformation and the square root transformation were explored

and selectively used. The transformations could also involve several Xip variables simultaneously. The

purpose of applying transformations of this type is to be able to use a regression model of a simple

form for the transformed variables, rather than a more complicated one for the original variables. This

transformation of independent variables enables to model a nonlinear multivariate best fit by carrying

out a linear regression analysis

136

Criteria for the estimation of regression parameters

When the type of model is chosen, the regression parameters are then adjusted to minimize a criterion

that measures the agreement between the data and the model with a particular choice of regression

parameters. The least squares minimization based on maximum likelihood estimations is used to estimate

the regression parameters.

Best Fit selection procedures

Draper[24] discusses several methods for finding ”best” regression equations. Among these methods,

two procedures are judged as yielding the most acceptable and exact results while at the same time not

pushing the computational complexity to its limits: backward stepwise elimination and forward stepwise

regression. Their working, features, and advantages are summarized here.

1. Backward stepwise elimination

Backward elimination starts with a regression equation containing all candidate terms of variables.

In a loop, the partial F-test for each variable term is computed and compared with a chosen

threshold significance level. If the lowest F-test value is smaller than the threshold value, the

corresponding variable is removed from the equation, thus reducing the regression equation to

the most significant variables. Backward elimination has proven to be very efficient in terms

of computer run time. Draper mentions cases where ill-conditioned matrices (close to singular)

can lead to corrupted results because of rounding errors but it seems as if modern high-capacity

computer systems will alleviate most of these problems.

2. Forward stepwise regression

In contrast to backward elimination, forward stepwise regression (which is actually a refinement

of the forward selection procedure) begins with a very small regression equation to which vari-

able terms are added subsequently. Not only are individual variables tested against a preselected

significance level but also at each stage, variables are reevaluated concerning combination effects

among each other. In effect, a very finely tuned regression equation is achieved. Draper favors this

method but nevertheless points out that the task of selecting good initial variables can be quite

decisive.

The best fit model is selected by means of criteria examining the quality of model such as ANOVA

(ANalysis Of VAriance), Multiple correlation coefficient (R), Standard error of regression parameters.

This work was carried through with the SPSS[32][77][78] software. The procedures described above

were evaluated and compared. In my application, backward elimination was taken because its run time

137

behavior was better.

4.4.2 Regression analysis and error functions[95][96]

The physical model errors in CFD simulation were defined as the difference between the faired Series 60

experimental data CR, CW and the verified CFD prediction CWCFD in the validation domain. The in-

terpolated experimental data suffer from experimental error, fairing error by Series 60, and interpolation

error. All of these errors have systematic characteristics. However, although the systematic numerical

errors in CWCFD were reduced during the verification process, CWCFD still contains some random noise.

This means the dependent variables ER and EW contain random error due to numerical model errors

for each ship in the validation domain although systematic error trends are dominant.

Regression equation

Best fit models of two dependent variables ER and EW are denoted by ER fit and EW fit. A linear

multiple regression is used. A function form of independent variables is expressed in the equation (4.17)

ER fit = β0 + β1f1(B/T,L/B, lcb, CB) + β2f2(B/T,L/B, lcb, CB) + (4.17)

...+ βpfP (B/T,L/B, lcb, CB) + εi

Since EW contains one more error source due to the form factor k, we use ER as a dependent variable

for the regression model only. EW fit will be obtained by

EW fit = ER fit − kCFITTC (4.18)

Range of regression

These best fit models should cover the whole validation range. They are modelled in the validation

range consisting of 58 different ships which are varied by the combination of four form parameters and

6 increasing Froude numbers, FN = 0.2, 0.225, 0.25, 0.275, 0.3, 0.316. In this process, the best fit models

are constructed for each FN individually. There are two reasons for that. Physically, an approximation of

the wave resistance coefficients is likely to ignore the change of real physical phenomena such as hollows

and humps as a function of ship speed, especially in the low speed region, unless very small steps in FN

are taken. From the statistical point of view, the error tendencies related to the change of geometry and

FN are completely different, both in their magnitude and characteristics. The simultaneous regression

of form parameters and speed would spoil the homogeneity of the sample for regression purposes. This

would result in poor correlation and finally a poor model.

138

FN 0.2 0.225 0.25 0.275 0.3 0.316No. of data 58 58 58 37 26 26Range of CB 0.6-0.7 0.6-0.7 0.6-0.7 0.6-0.675 0.6-0.65 0.6-0.65

Table 4.19: The range of regression

The Series 60 model test results are limited for the parent ship having CB, = 0.65 under FN = 0.275

and for the ship having CB = 0.7 up to FN = 0.25. Therefore, a part of the ships are used for modeling

the best fit for FN = 0.275, 0.30, 0.316. The Table 4.19 shows the given regression range by means of

the number of data. No. of data, for example, 58, means the total number of given data is 58 ships

and the range is from ship No. 1 to ship No.58. The hull form characteristics as a function of the ship

number is shown in Table 2.7-2.8.

Independent variables and data transformation

Four form parameters,B/T, L/B, lcb, CB show up as independent variables in the regression equation.

In order to express the nonlinearity of the error trends, some additional variable terms which are trans-

formed from the four form parameters are introduced to the regression. These additional variable terms

are constructed by:

Quadratic and bivariate transformation: (B/T )2, (L/B)2, lcb2, C2B , B/T ∗L/B,B/T ∗ lcb,B/T ∗CB,L/B ∗ lcb, L/B ∗ CB, lcb ∗ CB

Logarithmic transformation: lnB/T, lnL/B, ln lcb, lnCB

Reciprocal transformation: (B/T )−1, (L/B)−1, lcb−1, C−1BSquare root transformation: (B/T )1/2, (L/B)1/2, lcb1/2, C1/2B

Best fit models

1. The backward elimination method is used to select the best correlated variables in the model. The

basic steps in the procedure are as follows[24]:

Step 1. A regression equation containing all 26 variables is computed.

ER fit = β0 + β1B/T + β2L/B + ...+ β26C1/2B

Step 2 The partial F-test value is calculated for every variable.

Step 3. The lowest partial F-test value, FL is compared with a preselected significance level Fo.

If FL < Fo, remove the variable which is just considered. Recompute the regression equation

in the remaining variables : go to step 2.

139

If FL > Fo, adopt the regression equation as calculated..

For the best fit model at FN = 0.2, the overall F value F =1.359E − 071.161E − 09 = 117. This value exceeds

Fo which is defined by taking an α risk of 0.1 in the F-distribution. This model also has a value of

multiple R of 0.9656. This model is accepted. We have used, in this study, α = 0.1 for all selection

procedures as criterion to reject variables.

In the following the six best fit models for each FN are shown.

Best Fit Model for FN= 0.2

- Multiple R = 0.9656

No. of Variables6 Sum of Squares Mean Square F

Regression 6 8.157E-07 1.359E-07 117.0

Residual 51 5.923E-08 1.161E-09

Total 57 8.749E-07Variables in the equation

Variable Coefficient Standard Error in Coefficient

(Constant) -1.927E-02 4.060E-03

(B/T )2 -5.107E-06 1.666E-06

(L/B)2 6.149E-05 1.005E-05

C2B -6.393E-03 3.205E-03

L/B ∗ CB -1.508E-03 2.181E-04

lcb ∗CB 4.174E-02 8.282E-03

lcb−1 6.337E-03 1.371E-03

Best Fit Model for FN= 0.225

- Multiple R = 0.9197

No. of Variables Sum of Squares Mean Square F

Regression 7 3.518E-07 5.025E-08 39.18

Residual 50 6.412E-08 1.282E-09

Total 57 4.159E-07Variables in the equation

6This is called generally as degree of freedom (df) in statistics.

140

Variable Coefficient Standard Error in Coefficient

(Constant) 1.472E-02 2.643E-03

(L/B)2 1.153E-04 2.632E-05

C2B -1.105E-02 2.673E-03

B/T ∗ lcb -4.693E-05 2.080E-05

L/B ∗ lcb -1.849E-03 6.890E-04

L/B ∗ CB -1.243E-03 2.321E-04

lcb ∗CB 1.883E-02 7.574E-03

C−1B -5.971E-03 1.124E-03

Best Fit Model for FN= 0.25

- Multiple R = 0.8902

No. of Variables Sum of Squares Mean Square F

Regression 8 4.290E-07 5.363E-08 23.37

Residual 49 1.124E-07 2.294E-09

Total 57 5.414E-07Variables in the equation

Variable Coefficient Standard Error in Coefficient

(Constant) 2.446E-02 2.967E-03

(L/B)2 1.482E-04 4.552E-05

lcb2 1.973E-02 8.936E-03

C2B -1.138E-02 2.509E-03

B/T ∗ L/B 5.675E-05 3.102E-05

B/T ∗ lcb -7.771E-04 4.325E-04

L/B ∗ lcb -3.164E-03 1.249E-03

L/B ∗ CB -1.152E-03 3.131E-04

C−1B -9.770E-03 1.264E-03

Best Fit Model for FN= 0.275

- Multiple R = 0.9936

141

No. of Variables Sum of Squares Mean Square F

Regression 7 4.173E-07 5.962E-08 319.5

Residual 29 5.411E-09 1.865E-10

Total 36 4.228E-07Variables in the equation

Variable Coefficient Standard Error in Coefficient

(Constant) 8.718E-02 5.433E-03

(B/T )2 1.323E-05 8.461E-07

(L/B)2 6.429E-05 1.673E-05

C2B 6.088E-02 4.418E-03

L/B ∗ lcb -3.136E-03 4.592E-04

L/B ∗ CB 1.085E-03 2.113E-04

lcb ∗CB -1.603E-01 1.062E-02

lcb−1 -2.872E-02 1.825E-03

Best Fit Model for FN= 0.3

- Multiple R = 0.9902

No. of Variables Sum of Squares Mean Square F

Regression 8 5.482E-08 6.852E-09 107.1

Residual 17 1.086E-09 6.392E-11

Total 25 5.591E-08Variables in the equation

Variable Coefficient Standard Error in Coefficient

(Constant) 6.971E-02 1.220E-02

(B/T )2 7.344E-05 1.648E-05

(L/B)2 7.848E-05 1.223E-05

lcb2 -5.972E-02 1.581E-02

B/T ∗ L/B 2.596E-05 8.289E-06

B/T ∗ lcb -1.187E-03 2.171E-04

L/B ∗ lcb -3.323E-03 4.255E-04

(L/B)−1 -2.303E-02 5.403E-03

lcb−1 -2.122E-02 3.973E-03

142

Best Fit Model for FN= 0.316

- Multiple R = 0.9553

No. of Variables Sum of Squares Mean Square F

Regression 7 1.094E-07 1.563E-08 26.82

Residual 18 1.049E-08 5.829E-10

Total 25 1.199E-07Variables in the equation

Variable Coefficient Standard Error in Coefficient

(Constant) 1.727E-01 3.545E-02

(B/T )2 5.545E-05 2.641E-05

(L/B)2 9.449E-05 3.681E-05

lcb2 -1.927E-01 4.599E-02

B/T ∗ L/B -4.226E-05 2.240E-05

L/B ∗ lcb -3.720E-03 1.282E-03

(L/B)−1 -2.899E-02 1.522E-02

lcb−1 -5.537E-02 1.162E-02

4.4.3 Graphical presentation and discussion of results

In section 4.4.2, systematic error trends are analyzed by regression analysis. The best fit models ex-

pressing the error trends ER fit and EW fit are determined at different Froude numbers. The best fits

have different residual sums of squares at each FN . However, their magnitudes are small enough to

be neglected. In this section, the corrected residuary and wave resistance coefficients, CRCORRECTED

and CWCORRECTED will be investigated by comparing them with experimental data. Table 4.20 and

Table 4.21 shows comparisons of CR and CW with CRCORRECTED and CWCORRECTED as a function

of the univariate form parameter variation. The corrected resistance coefficients CRCORRECTED and

CWCORRECTED are derived by the relation of CRCORRECTED = CWCFD+ER fit and CWCORRECTED =

CWCFD+EW fit. Table 4.20 shows the comparison of the resistance coefficients on each univariate form

parameter variation departing from the first variational ship 0 0 0 0 in the validation domain.

Table 4.21 shows the same comparison but the form parameter range is departing from the last ship

1 1 1 1 . Corrected CFD simulation data, CRCORRECTED and CWCORRECTED appear really

close to the experimental data. But small differences exist between the corrected simulation data and

the experimental data in the low speed region. When FN is larger than 0.25 both data are situated on

143

the same trend line. The two data of the wave resistance coefficient show a slightly larger difference than

those of the residuary resistance coefficient. This is conceptually possible because the best fit models

selected for ER and EW fit are derived from ER fit by subtracting k×CF . The degree of the modelingerror (residual sum of squares) of ER fit exerts direct influence on the accuracy of EW fit. Therefore,

the difference between CW and CWCORRECTED looks like being increased as much as the difference in

the magnitude of EW and ER. With regard to the validity of the corrected simulation results in terms

of their relative difference to the experimental data, the corrected CR is contained for most cases within

a 5% error range in the low speed region 0.2 ≤ FN ≤ 0.25 and within a 1% error range in the high speedregion 0.275 ≤ FN ≤ 0.316.

4.4.4 Results

Error functions are selected at each FN by means of a backward stepwise multivariate linear regression

analysis. The correlation of error trends and form parameters is high and is dominantly governed by

the univariate form parameters. Multivariate form parameter effects are for most cases similar to the

univariate form parameter effects. Also, the error trends are not changed in a complicated way in

response to the form parameter variation. They show a systematic and smooth tendency as a function

of the form parameters. These characteristics of the error trends lead to a success of the regression

analysis. The error trends can be modeled by means of a regression equation as a function of the form

parameters. In the best fit selection process, it is demonstrated that the simple regression equation is

good enough to express the error trends.

4.5 Conclusion

A brief overview on Series 60 test results and an investigation of their relative accuracies are presented

in this chapter. The accuracy of the experimental data determines the reliability of the validation. This

is especially important for the low speed region because the magnitudes of the validation state variables

are very small. This validation can only be successful when the sensitivity of CW and CR as a function

of the ship shape and ship speed is greater than the error range of CW and CR due to the experimental

error types.

Error trends are investigated and discussed as a function of ship shapes and ship speeds in a 4-

dimensional hyperspace validation domain. Univariate and multivariate form parameter effects on the

error trends are visualized and discussed. CFD predictions show similar tendencies as experimental

data for both hull form variation and ship speed variation. These similar tendencies of the two data sets

cause smooth systematic error trends. The error trends of EW and ER are very different depending on

144

CR and CRCORRECTED CW and CWCORRECTED

t

CR,C

Rco

rrec

ted

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

CR, [t1,0,0,0]CRCORRECTED, [t1,0,0,0]CR [0,t2,0,0]CRCORRECTED [0,t2,0,0]CR [0,0,t3,0]CRCORRECTED [0,0,t3,0]CR [0,0,0,t4]CRCORRECTED [0,0,0,t4]

FN = 0.20

t

CW

,CW

corre

cted

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

CW [t1,0,0,0]CWCORRECTED [t1,0,0,0]CW [0,t2,0,0]CWCORRECTED [0,t2,0,0]CW [0,0,t3,0]CWCORRECTED [0,0,t3,0]CW [0,0,0,t4]CWCORRECTED [0,0,0,t4]

FN = 0.20

t

CR,C

Rco

rrec

ted

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

CR [t1,0,0,0]CRCORRECTED [t1,0,0,0]CR [0,t2,0,0]CRCORRECTED [0,t2,0,0]CR [0,0,t3,0]CRCORRECTED [0,0,t3,0]CR [0,0,0,t4]CRCORRECTED [0,0,0,t4]

FN = 0.25

t

CW

,CW

corre

cted

0 0.25 0.5 0.75 1-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

CW [t1,0,0,0]CWCORRECTED [t1,0,0,0]CW [0,t2,0,0]CWCORRECTED [0,t2,0,0]CW [0,0,t3,0]CWCORRECTED [0,0,t3,0]CW [0,0,0,t4]CWCORRECTED [0,0,0,t4]

FN = 0.25

t

CR,C

Rco

rrec

ted

0 0.25 0.5 0.75 16.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

1.6E-03

1.8E-03

2.0E-03

CR [t1,0,0,0]CRCORRECTED [t1,0,0,0]CR [0,t2,0,0]CRCORRECTED [0,t2,0,0]CR [0,0,t3,0]CRCORRECTED [0,0,t3,0]

FN = 0.275

t

CW

,CW

corre

cted

0 0.25 0.5 0.75 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

CW [t1,0,0,0]CWCORRECTED [t1,0,0,0]CW [0,t2,0,0]CWCORRECTED [0,t2,0,0]CW [0,0,t3,0]CWCORRECTED [0,0,t3,0]

FN = 0.275

t

CR,C

Rco

rrec

ted

0 0.25 0.5 0.75 11.4E-03

1.6E-03

1.8E-03

2.0E-03

2.2E-03

2.4E-03

2.6E-03

2.8E-03

CR [t1,0,0,0]CRCORRECTED [t1,0,0,0]CR [0,t2,0,0]CRCORRECTED [0,t2,0,0]CR [0,0,t3,0]CRCORRECTED [0,0,t3,0]

FN = 0.30

t

CW

,CW

corre

cted

0 0.25 0.5 0.75 11.0E-03

1.2E-03

1.4E-03

1.6E-03

1.8E-03

2.0E-03

2.2E-03

2.4E-03

CW [t1,0,0,0]CWCORRECTED [t1,0,0,0]CW [0,t2,0,0]CWCORRECTED [0,t2,0,0]CW [0,0,t3,0]CWCORRECTED [0,0,t3,0]

FN = 0.30

t

CR,C

Rco

rrec

ted

0 0.25 0.5 0.75 11.6E-03

1.8E-03

2.0E-03

2.2E-03

2.4E-03

2.6E-03

2.8E-03

3.0E-03

CR [t1,0,0,0]CRCORRECTED [t1,0,0,0]CR [0,t2,0,0]CRCORRECTED [0,t2,0,0]CR [0,0,t3,0]CRCORRECTED [0,0,t3,0]

FN = 0.316

t

CW

,CW

corre

cted

0 0.25 0.5 0.75 11.0E-03

1.2E-03

1.4E-03

1.6E-03

1.8E-03

2.0E-03

2.2E-03

2.4E-03

CW [t1,0,0,0]CWCORRECTED [t1,0,0,0]CW [0,t2,0,0]CWCORRECTED [0,t2,0,0]CW [0,0,t3,0]CWCORRECTED [0,0,t3,0]

FN = 0.316

Table 4.20: CR and CR CORRECTED as a function of ti starting from [0 0 0 0]145

CR and CRCORRECTED CW and CWCORRECTED

t

CR,C

Rco

rrec

ted

0 0.5 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

CR [t1,1,1,1]CRCORRECTED [t1,1,1,1]CR [1,t2,1,1]CRCORRECTED [1,t2,1,1]CR [1,1,t3,1]CRCORRECTED [1,1,t3,1]CR [1,1,1,t4]CRCORRECTED [1,1,1,t4]

FN = 0.20

t

CW

,CW

corre

cted

0 0.5 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03CW [t1,1,1,1]CWCORRECTED [t1,1,1,1]CW [1,t2,1,1]CWCORRECTED [1,t2,1,1]CW [1,1,t3,1]CWCORRECTED [1,1,t3,1]CW [1,1,1,t4]CWCORRECTED [1,1,1,t4]

FN = 0.20

t

CR,C

Rco

rrec

ted

0 0.5 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

1.6E-03

CR [t1,1,1,1]CRCORRECTED [t1,1,1,1]CR [1,t2,1,1]CRCORRECTED [1,t2,1,1]CR [1,1,t3,1]CRCORRECTED [1,1,t3,1]CR [1,1,1,t4]CRCORRECTED [1,1,1,t4]

FN = 0.225

t

CW

,CW

corre

cted

0 0.5 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03CW [t1,1,1,1]CWCORRECTED [t1,1,1,1]CW [1,t2,1,1]CWCORRECTED [1,t2,1,1]CW [1,1,t3,1]CWCORRECTED [1,1,t3,1]CW [1,1,1,t4]CWCORRECTED [1,1,1,t4]

FN = 0.225

t

CR,C

Rco

rrec

ted

0 0.5 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

1.6E-03

1.8E-03

2.0E-03

CR [t1,1,1,1]CRCORRECTED [t1,1,1,1]CR [1,t2,1,1]CRCORRECTED [1,t2,1,1]CR [1,1,t3,1]CRCORRECTED [1,1,t3,1]CR [1,1,1,t4]CRCORRECTED [1,1,1,t4]

FN = 0.25

t

CW

,CW

corre

cted

0 0.5 10.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

CW [t1,1,1,1]CWCORRECTED [t1,1,1,1]CW [1,t2,1,1]CWCORRECTED [1,t2,1,1]CW [1,1,t3,1]CWCORRECTED [1,1,t3,1]CW [1,1,1,t4]CWCORRECTED [1,1,1,t4]

FN = 0.25

Table 4.21: CR and CR CORRECTED as a function of ti starting from [1 1 1 1]

146

the kind and the change of the form parameters. However, their characteristics are simple and smooth

and their univariate form parameter effects are dominant. These characteristics of the error trends lead

to a success of the regression analysis. The evaluation of the data by stepwise, multivariate regression

analysis showed very clear, significant trends in the whole sample. The errors therefore show a clear

and simple dependence on the design variables. The errors are generally not of a negligible magnitude

because they comprise the cumulative effects of many modeling and procedural approximations. But

they are systematic enough to allow direct conclusions for design purposes. Furthermore the regression

coefficients obtained from the analysis of this error function sample, which are a measure of the statistical

standard deviation and hence the noise present in the error data, are generally of a remarkably high

value indicating a low noise. This in retrospect also justifies the choice of the regression equation.

The error trends based on the verified simulation predictions and the faired experimental data of

Series 60 family ships are systematic and smooth. These error trends are mainly governed by form

parameter variation and, therefore, can be expressed as a function of the form parameters with high

accuracy.

147

Chapter 5

Summary and Conclusions

This study deals with the issue of the existing difference between predictions in ship hull form wave

resistance derived from experiments (EFD data) and based on the results of state of the art CFD

systems. It concentrates on the question whether any observed errors between measured results and

numerical predictions are primarily systematic functions of the hull form parameters or contain random

errors to any significant degree. Thereby the study intends to make a contribution to the issue of the

reliability of CFD results on wave-making resistance for ship design purposes relative to experimental

data (EFD results).

This problem was addressed here by means of a specific sample of hull shapes taken from a systematic

test series, Series 60, and compared with numerical predictions from the numerical solver SHIPFLOW.

The study was confined to a limited subset of Series 60 (CB from 0.6 to 0.7, lcb from -1.5% to 0.5% of

LBP , L/B from 6.5 to 7.5 and B/T from 2.5 to 3.5) in order to investigate the trends in a physically

coherent domain.

The Series 60 results were obtained from the documented experimental data for full scale ships of

standard length of 400 feet and were reinterpolated for a model size of L = 20 feet by reverse application

of the ATTC method in Series 60 to obtain the experimental values of CR and CW using the Froude

and Hughes methods respectively. These data were inherently fair and free of random errors because the

interpolation and presentation in the Series 60 documents had already removed any existing noise. This

does not imply that the experimental data are error free, but any remaining error trends are systematic.

The results obtained by computational fluid dynamics are subject to physical modeling, numerical

implementation, and computational accuracy errors. These error types were discussed and qualitatively

analyzed in detail. They were described as primarily systematic or random error causes and were

classified accordingly. Each numerical case study was thoroughly verified for their consistency. This

148

was done by systematic sensitivity studies on the body panel number, free surface panel resolution,

truncation domain and iterative convergence until the computational parameters were all set at such

a level that further refinements did not produce relevant changes greater than very small tolerances.

These data are called and used as verified CFD data.

The numerical and computational errors were investigated systematically over the whole range of the

Series 60 subset by evaluating 58 distinct hull forms covering the domain of the free hull form parameters

of Series 60 as densely as possible in practice. This yields a prediction of CWCFD for each individual

hull form at various FN . The differences between experimental and computational results denoted by

ER and EW can thus be obtained.

The principal study concerns the dependence of two error function ER and EW on the four variables

in the hull form definition, as in Series 60, and separately also on ship speed (FN ). This analysis which

was newly conceived in this work is called trend validation. It was performed by the methodology of

regression analysis being applied to the whole sample data volume of the error function vs. the free

design variables at any given FN .

The evaluation of the data by stepwise, multivariate regression analysis showed very clear, significant

trends in the whole sample. The errors therefore show a clear and simple dependence on the design

variables. The errors are generally not of a negligible magnitude because they comprise the cumulative

effects of many modeling and procedural approximations. But they are systematic enough to allow direct

conclusions for design purposes. Furthermore the regression coefficients obtained from the analysis of

this error function sample, which are a measure of the statistical standard deviation and hence the

noise present in the error data, are generally of a remarkably high value indicating a low noise. This in

retrospect also justifies the choice of the regression equation.

The individual error trends in this trend validation study were discussed in Section 4.4. The general

character of these trends allows the following conclusions, at least for the investigated sample:

1. Although the errors in ER and EW between the experimental and the computational results are

not of a negligible magnitude, they are still of such a smooth and systematic nature that in general

the observations made on the relative merits of one hull form variation from experiments and from

CFD calculation are not in conflict or in contradiction.

2. The physical procedural causes for the differences between experiment and CFD for each individual

hull form could not be identified in detail by the current methodology because an individual error

case study was not performed in this approach so that a separation by error types was not feasible.

However, the development of improved CFD prediction methods may benefit from the current

results by placing the top priorities on those error influences which appear to be dominant in this

149

sample. In such individual studies it is also possible to reduce the error magnitudes by removing

certain procedural simplifications such as the neglect of trim and sinkage in CFD. But this type

of individual ship validation was not intended here. Further studies dealing with the error analysis

of individual hull shapes thus remain necessary and desirable.

3. The observation of a low noise level or small random error influences in the regression analysis of

systematic series CFD results, validated against a systematic experimental series, is encouraging

regarding the reliability of current state of the art CFD systems in ship design.

150

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