Trend validation of SHIPFLOW based on the bare hull upright resistance of the Delft Series

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Trend validation of SHIP FLOW based on the bare hull upright

resistance of the Delft Series

K . J . van.Mierlo

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List of Figures

3.1 Sailing yacht model and the attachment to the carriage 11 5.1 Meshing of the submerged sphere at Fn 0.40, 20 panels per unit length . . . . 26

5.2 Meshing problem at the stagnation point of the sphere (correct manual meshing

on the right) 28 5.3 Extrapolation of the drag area for a submerged spere at Fn 0.35 29

5.4 Extrapolation of the drag area for a submerged sphere at Fn 0.40 30 5.5 Extrapolation of the drag area for a submerged sphere at Fn 0.45 31

5.6 Mesh of Delft 1 30 by 5 panels and 90 by 15 panels 35 5.7 Mesh of Delft 1, 60 by 10 panels stretched (left) and uniform (right) 36

5.8 2 free surface meshes for Delft 1 at Fn 0.40: fine (top) and coarse (bottom) . 39

5.9 Results for Delft 1 at Fn 0.30 40 5.10 Results for Delft 1 at Fn 0.35 41 5.11 Results for Delft 1 at Fn 0.40 42 5.12 Results for Delft 1 at Fn 0.45 43 5.13 Results for Delft 1 at Fn 0.50 44 5.14 Results for Delft 1 at Fn 0.55 45 5.15 Longitudinal wavecut at y = 0.3 for Delft 1 at Fn 0.40 (green = coarse grid,

red — fine grid) 46 5.16 Wavepattern Delft 1 at Fn 0.40 (bottom = coarse grid, top = fine grid) . . . 47

6.1 Comparison of the wave resistance for Fn 0.25 53 6.2 Comparison of the wave resistance for Fn 0.30 54 6.3 Comparison of the wave resistance for Fn 0.35 54 6.4 Comparison of the wave resistance for Fn 0.40 55 6.5 Comparison of the wave resistance for Fn 0.45 55 6.6 Comparison of the wave resistance for Fn 0.50 56 6.7 Comparison of the wave resistance for Fn 0.55 56

B.1 Comparison of the t r i m for Fn 0.25 84 B.2 Comparison of the trim for Fn 0.30 84 B.3 Comparison of the t r i m for Fn 0.35 85 B.4 Comparison of the t r i m for Fn 0.40 85 B.5 Comparison of the trim for Fn 0.45 86 B.6 Comparison of the trim for Fn 0.50 86 B.7 Comparison of the t r i m for Fn 0.55 87

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L I S T O F F I G U R E S iv

B.8 Comparison of the sink for Fn 0.25 87 B.9 Comparison of the sink for Fn 0.30 88 B.10 Comparison of the sink for Fn 0.35 88 B.11 Comparison of tlie sink for Fn 0.40 89 B.12 Comparison of the sink for Fn 0.45 89 B.13 Comparison of the sink for Fn 0.50 90 B.14 Comparison of the sink for Fn 0.55 90

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List of Tables

3.1 Range of the principal hull parameters 1Q 5.1 Size of the submerged sphere test case (diameter of the sphere is used as

ref-erence length) 25 5.2 Initial drag area of a sphere 27

5.3 Final drag area of a sphere using the correct settings 27 5.4 Wave drag area of a submerged sphere at F n 0.35 27 5.5 Wave drag area of a submerged sphere at Fn 0.40 28 5.6 Wave drag area of a submerged sphere at Fn 0.45 28 5.7 Extrapolation of the pressme integration resuhs of a submerged sphere at F n

0-35 29

5.8 Extrapolation of the wave cut analysis results of a submerged sphere at Fn 0.35 29 5.9 Extrapolation of the wave cut analysis results of a submerged sphere Fn 0.40 30 5.10 Extrapolation of the pressure integration results of a submerged sphere at Fn

0-45 31

5.11 Extrapolation of the wave cut analysis results of a submerged sphere at Fn 0.45 3 1

5.12 Drag area of Delft 1 for different nr of hull panels 35 5.13 Drag area of Delft 1 for different stretching of the huU panels 36

5.14 Convergence criteria used during the investigation 37 5.15 Wave resistance of Delft 1 at Fn 0.30 for different convergence criteria . . . . 37

5.16 Wave resistance of Delft 1 at Fn 0.40 for different convergence criteria . . . . 38 5.17 Wave resistance of Delft 1 at Fn 0.40 for different convergence criteria . . . . 38

6.1 Average results and maximum deviation for Delft 1 51 6.2 Size of the free surface domain for the different Fn 53 7.1 Range of the principal hull parameters used in the regression of kpi and k^^jc • 61

7.2 Me&n values and standard deviation of kpi for the different Froude numbers . 61 7.3 Regression coefficients and standard error of the coefficients for kpi at Fn 0.25 62

7.4 ANOVA table for the regression of kpi at Fn 0.25 62 7.5 Regression coefficients and standard error of the coefficients for kpi at Fn 0.30 62

7.6 ANOVA table for the regression of k^i at Fn 0.30 62 7.7 Regression coefficients and standard error of the coefficients for kpi at Fn 0.35 63

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L I S T O F T A B L E S 1

7.11 Regression coefücients and standard error of tire coefficients for kpi at Fn 0.45 64

7.16 ANOVA table for the regression of kpi at Fn 0.55 65 7.17 Mean values and standard deviatioir of k^c for the different Froude numbers 65

7.18 Regression coefficients and standard error of the coefficients for k^c at Fn 0.25 66

7.19 ANOVA table for the regression of ku,c at Fn 0.25 66 7.20 Regression coefficients and standard error of the coefficients for /c^c at Fn 0.30 66

7.21 ANOVA table for the regression of ku,c at Fn 0.30 66 7.22 Regression coefficients and standard error of the coefficients for k^c at Fn 0.35 67

7.23 ANOVA table for the regression of ku,c at Fn 0.35 67 7.24 Regression coefficients and standard error of the coefficients for kyjc at Fn 0.40 67

7.25 ANOVA table for the regression of fc^c at Fn 0.40 67 7.26 Regression coefficients and standard error of the coefficients for kyjc at Fn 0.45 68

7.27 ANOVA table for the regression of ky,c at Fn 0.45 68 7.28 Regression coefficients and standard error of the coefficients for k^jc at Fn 0.50 68

7.29 ANOVA table for the regression of /c„c at Fn 0.50 69 7.30 Regression coefficients and standard error of the coefficients for k^^c at Fn 0.55 69

7.31 ANOVA table for the regression of /c^^ at Fn 0.55 69

7.32 Test of kpi for Delft 4 70 7.33 Test of kyjc for Delft 4 70 7.34 Test of kpi for Delft 43 70 7.35 Test of k^c for Delft 43 71

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Chapter 1

I n t r o d u c t i o n

1.1 Motivation

The increase in computer power has brought Computational Fluid Dynamics (CFD) within reach of the yacht designer. Yacht designers are using CFD to study the different design options and to optimize the final design. The correct ranking of different design options is only possible if the reliability and accuracy of the CFD code has been proven. This can be done by a validation procedure such as described in the ITTG Quality Manual.

Validation consist of an analysis of the error between the truth and the simulated result. This error consists of a modelling error and a numerical error. The modelling error is caused by the fact that the truth or reality is represented by a mathematical model which is derived by making certain assumptions or simplifications. The numerical error is caused by the fact that the mathematical model can not be solved directly and needs to be discretised. The process of determining the numerical error is called verification.

In this study the measurements of the bare hull upright resistance of the Delft Systematic Yacht Hull Series (DSYHS) will be used to validate a non linear free surface potential flow solver (SHIPFLOW). The advantage of using the DSYHS results for vahdation is that the validation will cover a part of the design space instead of just a single point. The non linear free surface potential flow solver is used because this is a mature method which can be run on a modern desktop computer and is within reach of the yacht designer.

1.2 Preliminary objectives

As can be read in the section 1.1 the main objective of this thesis is:

Validation of a non hnear free surface potential flow code (SHIPFLOW) based on the upright bare hull resistance measurements of the Delft systematic yacht hull series (DSYHS).

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1.3 Structure ₃

During this validation the different errors within the result of the simulation are investigated. By using the assumption of potential flow a modelling error is introduced. This modelling error should be small to achieve validation but this may not be the case. If there is a significant modeUing error is it possible to correct the results? This leads to the second objective of this thesis:

Derivation of correction formulas for the modelling error which will be based on the charac-teristic sailing yacht hull parameters.

The correction formulas will be based on the characteristic hull parameters such that they can be used for other saihng yacht hulls within the range. The correction formulas will be determined using a linear regression analysis.

1.3 Structure

Chapter 2 describes the mathematical model of the hydrodynamic part of the flow around a sailing yacht. I t starts with the basic conservation laws and it treats the different boundary conditions. The last part of the chapter is about the different characteristic flow parameters and their influence on the flow around a sailing yacht.

Chapter 3 gives an overview of towing tank testing procedure. The influence of the scale effect on the different characteristic parameters is explained. The post processing of the measurements is described and possible errors and uncertainty in the towing tank results are given.

Chapter 4 describes the basics of the potential flow and the treatment of the non linear bound-ary condition. The iterative solution method and the way the wave resistance is determined are treated.

Chapter 5 treats the verification procedure and starts with a description of the basics of verification. The veriflcation is done for two different cases: a submerged sphere and sailing yacht hull nr 1 of the Delft series.

Chapter 6 deals with the validation procedure. I t starts with an explanation of the basics of validation. The possible errors in both the CFD and towing tanlc results are given. The results of the CFD calculations and the towing tank measurements are compared and an explanation for the difference is given.

Chapter 7 is about the derivation of the correction formulas. First a brief description of linear regression analysis is given. Than the linear regression is applied to the results and the resulting correction formulas are given. I n the end the correction formulas are tested on two hulls which were not included in the regression.

Chapter 8 gives the conclusions and recommendations for future work. The appendices con-tain the used SHIPFLOW input files and figures of the trim and sink results.

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Chapter 2

Governing equations of the

hydrodynamic part of the flow

around a saihng yacht

The equations which govern any flow are the so called Navier Stokes equations. These equa-tions are the result of applying Newton's second law on a fluid element. The derivation of these equations can be found in almost all textbooks about aero- or hydrodynamics so only the results are mentioned here. The equations govern the flow of air as well as the flow of the water. The flow of air is neglected here because there is only a small interaction between the air and water. This is a reasonable assumption because there is almost no difference i n velocity during the towing tank test and the large difference i n the properties of air and water.

2.1 Conservation laws

As described in [1] there are three basic equations: conservation of mass, conservation of momentum and conservation of energj^. The conservation of energy is excluded from this report because the flow around a sailing yacht is a low speed, incompressible fluid flow and the temperature difference between body and fluid is assumed to be small.

Conservation of mass is described by the continuity equation.

| ^ + p d i v V = 0 (2.1)

Water is an incompressible fluid and thus p = const. This simplifies the continuity equation (2.1).

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2.2 Boundary conditions ₅

The conservation of momentum equation is acquired when Newton's second law is applied to a fluid particle. A continuous, isotropic and hnear viscous fluid is assumed. The so called Navier Stolces (NS) equation is written here in vector form using indicial notation.

DY

PS „ d (2.3)

Water is incompressible and the continuity equation for an incompressible fluid (2.2) can be used to simplify the NS equation. When the viscosity of the water is assumed to be constant the NS equation is fm'ther simplified to the Navier Stokes equation for incompressible, constant viscosity flow:

DV

P ^ - Pg - V p + pV^Y (2.4)

The viscosity of liquids, like water, is temperature dependent but the temperature differences in the fluid are assumed to be small so the assumption of constant viscosity is justifled.

2.2 Boundary conditions

At the upstream or inlet boundary the values of the velocity (V) and the pressure (p) must be known.

V = VQ (2.5)

P = Po (2.6)

At infinity the disturbances disappear- and the boundary conditions are the same as the upstream conditions. (2.5 and 2.6) During computations the domain is not infinitely large and this requires different boundary conditions.

The particles at the hull surface have the same speed as the yacht, the so called the no-slip condition. This leads to the following boundary condition for the velocity at the solid surface:

V = 0 (2.7)

At the free surface there has to be kinematic equivalence between liquid and gas which means that the velocity of the flow at the free surface has to be tangent to the free smface.

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2.3 Characteristic flow parameters 6

ri{x,y,t) is the equation which describes the location of the free surface.

There also has to be equality of normal momentum flux at the free surface. Because of the requirement of tangential flow at the free surface in equation 2.8, the velocity has dropped from the normal momentum equation. The normal momentum equation states that the pressure and surface tension in the liquid is equal to the pressure in the gas.

f 1 1 \

p{x, y, 77) = patm - 7 + (2-9)

where 7 is the coefhcient of surface tension and and Ry are the radii of curvature of the free surface. I n a wave trough p < Patm and in a wave crest p > Patm

2.3 Characteristic flow parameters

The NS equations are valid for different kind of fluids with different properties. To make the equations independent of the properties of the fluid (dimensionless) they are divided by reference values. V* w = t* p = L V_ Vo w Vo fVo L P_ Po V * = LV p + pogz - Po JL Po

p* is the dimensionless dynamic pressure in the fluid because the gravity cancels out against

the hydrostatic pressure. When these variables are substituted in the continuity and NS equations (2.4) the following equations are obtained:

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2.3 Characteristic flow parameters

Tliese equations resemble the original equations except for one important parameter: the Reynolds number (Re).

pVL VL

Re = = 2.12 p V

The boundary conditions are also made dimensionless. For the inflow conditions and the conditions at infinity (2.5 and 2.6) this becomes:

V* = 1 (2.13)

/ = 0 (2.14)

And for the boundary condition (2.7) on the sohd surface:

F * = 0 (2.15) The dimensionless kinematic boundary condition has the following form:

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The normal momentum equation (2.9) is also non-dimensionalised:

PoV,' + 1 / ^ ^ PoV^LiRl^Rt

Three characteristic parameters arise: the Euler number (Eu), the Froude number (Fn) and the Weber number (We).

Eu = P-*-^ P° (2.18)

PoV^o

We = (2.20)

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2.3 Characteristic flow parameters ₈

I n hydrodynamics the Euler number is usually replaced by the cavitation number (C). This is done because the pressure level in the fluid is unimportant unless the vapor pressure is reached.

In total there are four characteristic parameters: Reynolds number, cavitation number, Froude number, and Weber number. The Reynolds number determines the viscous behavior of the flow which is important for the viscous and pressm-e resistance. The cavitation number is only important when the pressm-e in the flow reaches the vapor pressure. When this happens, it creates gas bubbles in the fluid which have a lot of influence on the flow. Tliis only happens in low pressure areas for example around propellers or hydrofoils and is not important for the flow around normal sailing yachts. The Froude number is important for free surface flows and wave formation and thus for the wave resistance. The Weber number is connected to surface tension effects such as spray and breaking waves. The common opinion is that these effects have a small influence on the flow and the resistance.

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Chapter 3

EFD: Towing t a n k testing

Towing tank tests are frequently used to determine the resistance of ships. A decrease i n the resistance can save lots of fuel cost during the live span of a merchant ship and towing tank tests are often used to select the optimum design. Sailing yachts are not often tested because of the high costs and the small number of yachts produced to share this costs. Towing tank test are used to optimize the design of racing yachts i n high profile races such as the Americas Cup and the Volvo Ocean Race. This chapter will first give a description the Delft systematic hull series, the towing tank setup and the testing procedure. The influence of the characteristic flow parameters will be described and the post processing of the test results will be explained.

3.1 Description of the DSYHS

The Delft systematic sailing yacht hull series consists of 50 systematicaUy varied sailing yacht models. The variation of the principal hull parameters is given in table 3.1. These models have been tested at different speeds, leeway and heel angles in the Delft ship hydromechanics laboratories towing tank. The evolution of the Delft systematic yacht hull series can be found in references [2] to [11]

The total saihng yacht resistance has been divided in five main components: upright resis-tance, heeled resisresis-tance, resistance due to leeway, resistance of the keel and resistance of the rudder. The towing tank test results have been used to derive empirical formulas for all these resistance terms. These emperical formulas are polynomial expressions with Froude number dependent constants which can be found in [11]. This report focuses on the upright resistance of the bare hull (hull without keel and rudder).

The resistance formulas are often used in a velocity prediction program (VPP). A VPP combines the hydrodynamic resistance formulas with an empirical aerodynamic model of the sail forces to determine the velocity of a sailing yacht. The VPP can be used to evaluate the performance of different sailing yacht designs. The results of this evaluation are usually

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3.2 Towing tank test setup and testing procedure 10

Table 3.1: Range of the principal hull parameters Length - Beam Ratio Lwt

Bwl 2.73

to 5.00 Beam - Draft Ratio Bwl

Tc

2.46 to 19.38 Length - Displacement Ratio _{V r l / 3}Lwl 4.34 to 8.50 Longitudinal Centre of Buoyancy LCB 0.0 % to -8.2 %

Longitudinal Ceirtre of Flotation LCF -1.8 % to -9.5% Prismatic Coefhcient Cp 0.52 to 0.60 Midship Area Coefficient Cm 0.65 to 0.78 Loading Factor _{V r 2 / 3}Aw 3.78 to 12.67

quite accurate as long as the characteristic hull parameters stay within the range of the tested models.

3.2 Towing tank test setup and testing procedure

A towing tank test facility consists of a number of different parts. The main parts are the tank, the carriage, the model and the measmement systems. The tank is a rectangular basin filled with water and sometimes equipped with a wave generator. A lai'ge towing tank can handle large models and this decreases the influence of the scale effects.

The carriage usually is a steel construction which moves on rails at the side of the tank. The construction should be rigid so that it does not deform dming the tests. The range and accuracy of the velocity depend on the power and control system of the electric motors. The geometry of the model is an accurate scale model of the ship or sailing yacht. Different construction techniques exist to build a model. Two requirements are that the model should be rigid enough that i t does not deform during the tests and it should be lightweight such that ballast can be added to achieve the correct t r i m and sink. The scale of the model depends mainly on the size of the tank. A large model will decrease the influence of the scale effects but increases the blockage and interference effects.

The measurement system consists of a couple of load cells and a computer. The sensors measure the forces in different directions and send this information to the computer. The data collected dming the test is stored and post processed to get resistance and side force values. The water temperature during tests is measured and will be used in the post processing to determine the viscosity.

Two different test methods exists: free sailing and semi captive. The free sailing method resembles actual sailing because the towing forces act at the center of effort of the sails and the yacht attains an equilibrium heel and leeway angle. This way the number of tests needed to cover the whole range of sailing conditions is minimized. The downside of this method is the expensive model which needs to have the same center of gravity as the full scale yacht and an active rudder system.

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3.3 Froude number similarity ₁₁

The Delft series are tested using the semi captive method i n which the model is free to heave and pitch but restricted in aU other motions. A disadvantage of this method is that a lot more tests are needed to cover all possible sailing conditions. The main advantage is that the model for this test method is much simpler because the heel is fixed and thus only the shape of the model is important. The correct trim of the model is achieved by moving the internal baUast.

3.3 Proude number similarity

The characteristic parameters (Re, Fn, We, C) during towing tank test should be the same as in full scale to achieve the same flow situation. This is usually not possible. Until now, no liquid has been found which has the right properties to keep all the chai-acteristic parameters equal. Water is used in towing tanks and this will cause a change i n some of the characteristic parameters compared to the full scale values. The implications on the flow around the model wiU be described in this section.

An important feature of the flow around a sailing yacht (or ship i n general) is the wave formation and the resistance this causes. The Fi-oude number is the characteristic parameter which governs this flow phenomenon and during towing tank tests the Fi-oude number of the model is equal to the full scale value. This means that the other characteristic parameters during the tests are different from f u l l scale. The latest models of the Delft systematic series are 2m long and represent a 10m sailing yacht so the scale factor (Sf) is 1 : 5. This model is used as an example to show what happens to the other characteristic parameters (Re, We

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3.3 Pr oude number similarity ₁₂

and C) when the Fn is kept constant between test and full scale.

Frifs Umod Umod VgSfL V s f U f s (3.1) (3.2) (3.3)

To keep Fn constant we have to use a towing speed which is smaller than the full scale speed. For our example this means a towing speed of 44, 72 % of full scale speed.

The other characteristic parameters are calculated with this towing speed and compared with the full scale values.

R e m o d = ^ ^ ^ ^ ^ f ^ = S f l R e f , (3.4)

The Re of the model is much lower than the full scale Re and because of this laminar regions are present on the model. I n full scale the flow becomes turbulent after a small distance and this means that there is a big difference between the test and full scale situation. The laminar flow will have a large influence on the resistance of the model and to avoid this effect a turbulence stimulator is applied to trip the boundary layer. There are a number of different tmbulence stimulator and the most common one is a row of cylindrical studs. The tmbulence stimulator used during the Delft series tests consists of carburundum strips on hull, keel and rudder. The carburundum has grain size 20 and is applied on the models with a density of approximately 10 grains/cm^. To determine the resistance of the strips a couple of tests are carried out with single and double strips. The difference in resistance is assumed to be equal to the resistance of one strip and this value is then used to correct all the test results.

W e ^ = ^ i ^ M l ^ ^ S f W e , . (3.5)

As can be seen from equation 3.5, the influence of the scale factor on We is even larger than on Re. This difference in We has some influence on the wave breaking and spray around the model. The amount of breaking waves and spray differs between full scale and the towing tank tests. There is no correction for this effect and its influence on the resistance is assumed to be so small that it can be neglected.

As explained in section 2.3 the effect of the cavitation number is only important when the pressure in the flow reaches the vapor pressure. This is not the case during towing tank test or in full scale and thus the difference in cavitation number has no signiflcant influence on the flow around the model.

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3.4 Postprocessing of towing tank measurements ₁₃

3.4 Postprocessing of towing tank measurements

The resistance values of the towing tank test are post processed to convert them into full scale resistance values. This is done according to Fronde's method which states that the wave resistance can be extrapolated directly between model and f u l l scale if the Proude number is equal. The postprocessing consists of a number of different steps which will be described in this section. The first step is to correct the test results for the resistance of the turbulence stimulator and blockage effects. The resistance of the turbulence stimulator is determined by a double width test as described in section 3.3. The blockage effects depend on the towing tanlf size and are based on measurements with the same model in different towing tanks and on the experience of the test facility. After this the values are made dimensionless by dividing the resistance with the dynamic pressure and the reference area (wetted surface area at zero speed).

(3.6)

The next step is to subtract the frictional resistance from the total resistance to get the wave making resistance. Two different methods exist to calculate the frictional resistance: I T T C 1957 and I T T C 1978. The I T T C 1957 formula calculates the frictional resistance coefficient by using a flat plate friction formula.

0,075 , ^

= ( - l o g i ? e - 2 . 0 ) ^ ^'-'^ The frictional resistance coefficient of the model is calculated using 3.7. The residuary

resis-tance coefficient of the model is computed by subtracting the frictional resisresis-tance coefflcient of the total resistance coefficient.

'^Rmad = C'r^od - (^F^,a (3.8)

The Froude number of the model test and the f u l l scale are the same and this means that the residuary resistance coefficient is equal for both cases.

CR,,, = CR^^, (3.9) The frictional resistance of the fuU scale ship is calculated using the I T T C 1957 formula

(3.7) again. The residuary and frictional resistance coefficieirts are added to get the full scale resistance coefficient.

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3.5 Errors and uncertainty in towing tank tests ₁₄

This total resistance coefficient is used to calculate the total resistance of the full scale ship.

RT,.=CT,\pVlSf, (3.11)

The I T T C 1978 method is based on the I T T C 1957 method but takes the so called form drag into account. Form drag is the viscous pressure resistance caused by the displacement effect of the boundary layer. The form drag is taken into account using a form factor k. The form factor can be determined using Prohaska's method. Prohaska's method assumes that for low Fn (0.05 < Fn < 0.20) the ratio of form drag to viscous drag is constant. The residuary resistance is assumed to be a linear function of Fn'^. When is plotted against ^ the points should lie on a straight line which is described by equation 3.12. The form factor can be determined by extrapolating the hue to ^ = 0.

C > = ™ c 7 + ^^ + ^^ (3.12) The I T T C 1978 method calculates the wave resistance by subtracting the viscous and form

drag from the total resistance. For the viscous drag the same formula is used as in the I T T C 1957 method, (see equation 3.7)

= CTrr^oi " (1 + k)CF^,^ (3.13)

According to R-oude's assumption the wave resistance coefficient is the same for model and fuU scale, (see equation 3.9) The total resistance coefficient is calculated by adding the viscous and form drag. The form factor is assumed to be the same for model and full scale.

CTS = CRS + (1 + k)CFs (3.14)

The total drag can be calculated by formula 3.11

3.5 Errors and uncertainty i n towing tank tests

Although towing tank testing is the most realistic experiment to represent the full scale physics it should be noted that the results will contain an error. This error can be split into two parts: a measurement error and a post processing error. Possible errors which can propagate into the measurements are for example errors in: the geometry of the model, calibration of the load cells, displacement of the model, ahgnment of the model (initial t r i m and leeway), velocity of the carriage, temperature of the water and waiting time between tests.

The error in the final results is not only caused by an error in the measurements but also by errors in the post processing. The I T T C 1957 and I T T C 1978 methods are standard methods

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3.5 Errors and uncertainty in towing tank tests 15

in the post processing of towing tank results. The I T T C 1957 frictional coefficient formula is based on a large number of towing tanlc experiments using a flat plate. The formula for the frictional resistance coefficient was determined by performing a best fit to the results of these tests.

Another error in the post processing of the results is caused by the fact that the reference area (the wetted surface area at zero speed) is used to determine the frictional resistance. The wetted surface area during the test differs from this reference area because of the wave formation. The difference will be small for fow speeds but may increase for the higher speeds. A fuU uncertainty analysis of the whole towing tank set up and the postprocessing, as de-scribed in [12], would be the best way to determine the influence of the different errors on the wave resistance. Unfortunately this procedure will take a lot of time and effort which makes it too expensive for most towing tank facilities.

A more practical way of checking the errors is to repeat some tests and compare the mea-surements. In [13] Fasardi describes how to recognize and how to avoid possible errors in the measurements. In [14] an assessment of the accmacy and repeatabihty of towing tank tests for ACC yacht development is made. Three different towing tank facilities (INSEAN, QinetiQ and SSPA) describe their testing procedm-es and give a explanation of the sources of the variation i n the measured forces. The conclusion is that- a long term accm-acy and repeatability of drag measmements in the order of ± 1 % can be achieved. This is based on faired or averaged results incorporating a number of individual test runs.

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Chapter 4

CFD: non linear free surface

potential flow code S H I P F L O W

This chapter will give an overview of the potential flow and the solution method used in the non linear free surface case. The description is general, details about SHIPFLOW can be found in the user manual [15]. The basics of potential ffow and the linearisation of the free surface boundary conditions are explained. After this the non linear solution method and its special features will be treated. The last part of this chapter deals with the two different ways to determine the wave resistance.

4.1 Basics of potential flow

The principal assumptions in potential flow are: inviscid, irrotational, incompressible, and steady flow. These assumptions are valid for the flow around a ship because the Reynolds number is relatively high and the effect of viscosity will be limited to a thin layer close to the hull and the wake. The lai-ge scale flow features such as the wave pattern are not affected much and this justifies the use of the potential flow assumption to model this large scale flow featm-es. The potential flow assumptions are used to simplify the Navier Stokes equations (2.4) which leads to the following equation:

^ V V 2 = p g - V p (4.1) The continuity equation stays the same and is repeated here for easy reading.

divV = 0 (4.2) The velocity vector can be written as the gradient of a scalar. This scalar is called the velocity

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4.1 Basics of potential flow 17

V = V(/) (4.3)

This is substituted in the bernouUi and continuity equations:

P + P52 + ^ W - V < ^ ) = C (4.4)

V^4> = 0 (4.5)

The continuity equation (2.2) transforms into the Laplace equation (4.5) which is linear and homogeneous. This allows the superposition of different solutions. The pressme and velocity are decoupled which makes it possible to solve the Laplace equation first and compute the pressure later.

The no-shp boundary condition (2.7) at the body changes to the tangential flow boundary condition. The simplifications introduced by the potential flow assumptions have decreased the degrees of freedom which make it impossible to maintain the no-slip condition. The tangential flow condition means that the fluid cannot flow through the body.

^ = 0 (4.6)

on

The velocity potential is substituted i n the kinematic boundary condition at the free smface (2.8) and the time dependent terms are dropped because of the steady flow assumption. The location of the free surface is described by the single valued function r]{x,y) which makes i t impossible to calculate overturning (breaking) waves and spray. These effects are considered to have a small influence on the global wave pattern and thus is this single valued free surface approach allowed. Two boundary conditions exist at the free smface. The velocity vector at the free surface is tangential to the free surface.

(/-xl^ + < ^ y | ^ = 0 at z = vix,y) (4.7)

And the pressme in the water at the free surface has to be equal to the atmospheric pressure.

Patm + pgz + ^ (V<?!» • W(t>) = const, at z = rj (4.8)

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4.2 Linearization of the free surface boundary conditions ₁₈

An extra condition, which is needed to make sure that the results are physical, is the radiation condition. This condition states that there are no waves in front of the bow of the ship.

4.2 Linearization of the free surface boundary conditions

The free surface conditions are non linear and this makes it difficult to solve them. To overcome this problem the boundary conditions are hnearized. This is done by dividing the potential in two parts: an estimated flow and a pertm-bation. A good estimate wiU resufl in a small pertmbation which justifles the linearization.

Vcf) = V^è + Vip (4.10) V ^ H + h (4.11)

Tliis can then be substituted into the free surface boundary conditions 4.7 and 4.9.

^xVx + ^yVy + ^xH:o + VyHy - - ip^ = 0 (4.12)

^ = ^ [ U ^ - K - % - ~ ' ^ i - 2^x^x - 2%^y - 2 $ , ^ , j (4.13)

These equations have to be satisfied at the free surface. The location of this free surface is unknown and therefore these equations need to be transferred to the estimated smface (z = h).

V<^(^=^) « V$(^=^^) + V^(^=^) + (4.14)

Dawson proposed to neglect the transfer term and the higher order terms which leads to the combined linear free surface boundary condition on the known surface z = H:

^h^'^hl + + + 2*-^^- + '^^v'Pv + 2^-'P.) (4.15)

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4.3 Solution method 19

4.3 Solution method

There are a lot of different ways to solve the Laplace equation for the velocity potential. I n [16] the advantages and disadvantages of the possible solution strategies are compared and the conclusion is that a panel method using rankine sources on the hull and free surface will probably be the most efficient. A description of the basics of panel methods and Rankine sources can be found in [17].

Lifting surfaces can easily be included in a panel method and are needed to model appendages such as keels and rudders. This report does not consider the appendages but a detailed explanation about hfting surfaces in free surface flows can be found in [18].

4.3.1 Linear free surface p o t e n t i a l flow

For the linear case the velocity distribution of the slow ship approximation or double body flow is used as estimate. The slow ship approximation can be calculated by mirroring the under water part of the body in the undisturbed free smface plane (the so caUed "double body") and calculate the flow mound this double body as ff it is totally smrounded by the fluid. Then the perturbation can be calculated which wiU be added to the estimated basis flow to give the linear free surface potential flow solution.

The problems with the linear method are that the height of the bow wave is usually under-estimated and the resuffs in the aft region are not very accurate. These effects can lead to negative wave resistance values. Experienced users are able to use the linear method to opti-mize bow shapes by looking at the calculated wave patterns. Another problem of the linear method is that it does not take into account the shape of the hull above the still waterline. This is an important drawback because most sailing yachts have overhangs which will have a lot of influence on the flow around the hull.

4.3.2 N o n linear free surface p o t e n t i a l flow

The non linear solution method is an extension of the linear case. After the linear solution has been calculated this result is used as a new estimate. The hull and free sm-face panels are moved and the pertmbation is calculated again. These steps are repeated until a converged solution is achieved. According to [16] convergence is achieved when the change in wave height for two consecutive iterations is below a set tolerance.

4.3.3 Special features of t h e solution m e t h o d

The radiation condition is satisfled by using an upwind approximation for the second deriva-tives of the potential in the longitudinal direction. This upwind discretisation ehminates the formation of upstream waves. A central scheme is used for the second derivatives of the

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4.4 Determination of the wave resistance ₂₀

potential in the transverse direction.

The stability and convergence of the solution method is improved by using raised panels on the free surface. This means that the panels are raised above the free surface but the collocation points are on the actual free surface. Solutions using this raised panels showed point to point oscillations in the calculated source strength. A n effective way to avoid this is to use a forward shift in the collocation point location. The phase and amplitude of the calculated waves show a dependency on the distance that the panels are raised and the forward shift of the collocation point. The two dimensional case of the non linear free surface potential flow has been studied in detail in [16] and also in [18] and the conclusion is that each upwind scheme has its own optimal raised distance. I n general a forward shift of 25 to 30% of the panel length and a raised distance of more than 1 panel length lead to accurate results in the two dimensional case. For the three dimensional case the raised distance has some influence on the condition of the system of equations. This leads to a decrease in the raised distance when the Froude number increases: ±70% of the panel length for Fn 0.25 to ± 3 0 % of the panel length for Fn 0.55.

The non linear free surface potential flow can be calculated using a fbced location of the huU or a hull which is free to trim and sink. In case the hull is free to t r i m and sink an extra set of equations is added to the solver. The weight distribution of the hull has to be in equihbrium with the hydrodynamic forces. After each iteration the trim and sink are adjusted to maintain the equilibrium. This gives two extra convergence criteria: the change of the t r i m angle and the sink should be within a certain hmit.

4.4 Determination of the wave resistance

There are two ways to determine the wave resistance of the ship: pressure integration and wave cut analysis. This section will describe the two methods.

4.4.1 Pressure integration

The pressure integration method determines the wave resistance by integrating the pressure on the hull panels. The pressure on the hull consists of the hydrostatic and the hydrodynamic pressure. For the linear solution the hydrostatic pressure sums to zero and this makes i t possible to integrate only the dynamic pressure to get the wave resistance. For the non linear solutions the hydrostatic pressure does not cancel and thus both pressures need to be integrated. The magnitude of the hydrostatic pressme is often larger than that of the hydrodynamic pressure and this can cause some problems concerning the accuracy of the pressure integration method. The solution to this problem is to use a sufficient number of panels on the hull surface.

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4.4 Determination of the wave resistance 21

4.4.2 Wave cut analysis

The wave cut analysis technique determines the wave resistance by analyzing the wave pattern. Longitudinal or transverse wave cuts can be used but the transverse method is preferred because it puts less demands on the size of the free smLace. The method determines the wave elevation in a number of transverse wave cuts behind the ship. The requirements with respect to the location of the wave cuts are that the wave cuts need to be in a region where the wave pattern is relatively smooth and the distribution of the wave cuts must not be equidistant. The method approximates the wave elevation in each wave cut by the sum of a series of elemental waves. The wave resistance is determined with the result of this approximation. A detailed description of the method can be found in [19]. The advantage of the wave cut analysis is that it is less dependent on the number of panels on the hull. This will make the wave cut method more robust than the pressme integration method for hulls with a complicated geometry (high curvature areas).

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Chapter 5

Verification

Verification is done to determine tfie numerical error and uncertainty witliin the computed solution and is a necessary step iir the validation process. This chapter will start with a introduction to the basics of verification and the available verification methods. Then two different test cases will be described. The first test case is the free surface flow around a submerged sphere and the second test case is the free surface flow around sailing yacht model nr 1 of the Delft series. This chapter will end with the conclusions of both the test cases.

5.1 Basics of Verification

According to [20] the simulation error {5s) is the difference between the truth (T) and the simulated result (S). This simulation error consists of a modelling error (SSM) and a numerical error {6SN)

5s = S - T = 5sM + SsN ( 5 . 1 )

Veriflcation is a process to estimate the numerical error and the uncertainty in that error estimate. There are different ways to estimate the numerical error and the basics will be described here.

The best way to determine the numerical error in the simulation is to compare the result with the analytical solution. Unfortunately the three dimensional non linear free surface potential flow is too complicated to solve analytical and thus are there no analytical solutions available. One of the characteristics of the potential flow is the so caUed dAlembert's paradox which states that the drag of an ai-bitrary body in a potential flow is zero. This is only vahd for non lifting flows and for flows without the influence of the free surface.

Since no analytical solution exists for most aero- or hydrodynamic flow cases a different strategy needs to be used to determine the numerical error and uncertainty. A n method

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5.1 Basics of Verification 23

whicfi is often used is Richardson extrapolation (RE). RE is based on the assumption that the numerical error can be represented by the following Taylor series expansion.

i = i

in which 4>i is the numerical solution, is the exact solution, aj are constants, hi is a parameter which represents the grid size and pj are exponents related to the order of accuracy of the method. The assumption used in RE is that the results are within the asymptotical range, which means that the first term in equation 5.2 is dominant over the higher order terms. This assumption is used to simplify equation 5.2 into the foHowing equation.

- 00 = ah\ (5.3)

There are three unknowns in 5.3 and thus three solutions on different grids are required to determine ^o, « and p. A n extra requirement for using this method is that the convergence is monotonic. For a constant refinement ratio r = = 7^ the numerical error in the solution on the finest grid can be estimated with the next formulas.

ln(£fe3,/E^3J ^gg^ ln(r)

In which e/^gj = S'ha - is the change in solution between coarse and medium grid and

s^^^ = -Shi is the change in solution between medium and fine grid. A constant refinement

ratio is not required but makes RE easier to apply.

The uncertainty in this error estimate is determined by multiplying the error estimate by a factor of safety {Fs). The recommended value for the factor of safety is 1.25 for careful grid studies (when p is close to the theoretical value) and 3 for all other cases.

Uhi ^ FS\5RE^^\ (5-6)

A disadvantage of the generalized Richardson extrapolation based on three solutions is that it can not recognize oscillatory convergence which can lead to wrong conclusions about the accuracy of the solution.

In [21] a different method similar to RE is derived. I n this method the same basic assumptions are used but the error estimate is based on a lar-ger number of solutions. The parameters

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5.2 Submerged sphere ₂₄

Si<Po, a,p) =

7E(<^'-

- ('/'o + (5.7)

This clearly is a more robust method to determine the numerical error but it is also more expensive than RE because it requires 4 or more different solutions. For the estimation of the uncertainty a safety factor approach is used similar to equation 5.6. I n [21] conservative ways to bound the upper value of the safety factor are given.

A problem with both described verification procedures is that they assume that the error can be expressed by a Taylor series expansion of the grid size. This assumption is probably true for focal flow quantities in simple cases such as two dimensional potential or Euler flows on a cartesian grid. I t is questionable if tins assumption holds for global quantities i n complex flow situations. A number of workshops on this topic have been organized in the last couple of years i n which different CFD codes are used to calculate the flow in two and three dimensional test cases. Often the observed numerical order of accmacy differs from the theoretical value and the discussions about this topic are ongoing.

5.2 Submerged sphere

The verification procedure was started with a saihng yacht hull of the DSYHS but this im-mediately caused a problem. When the hull is left free to trim and sink a change in wave resistance between different cases is not only caused by a refinement of the grid but also by a change in t r i m and sink. An option to eliminate the influence of the trim and sink is to calcu-late the flow around a fixed sailing yacht hull. The disadvantage of this option is that there is no control over the displacement which may not stay constant when the grid is refined. Since the displacement has a lot of influence on the wave resistance a similar problem occurs as for the free sailing yacht case: a change in the wave resistance will be caused by a combination of grid reflnement and a difference in displacement. The solution to the previous mentioned problems is to use a fixed submerged body for the verification procedure. The submerged body used in this test case is a sphere because points on the smface of this geometry are described by a simple analytical expression.

5.2.1 Objectives

The objective of this test case is to do a verification of the wave resistance of a submerged sphere calculated by a non linear free surface potential flow solver (SHIPFLOW)

5.2.2 D e s c r i p t i o n of the test case

The test case is split in two parts: the potential flow around a sphere without and with free surface. For the case without free surface the drag should be zero because of dAlemberts paradox. Different distributions of panels on the sphere are used to asses the influence of

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5.2 Submerged sphere 25

grid refinenrent on the numerical error. The number of panels in longitudinal and transverse direction is equal and goes from 16 to 56. The panels used for this case are so called first order panels: fiat panels with a constant source strength. Since there is no free surface the resistance is determined by pressure integration.

For the case with free surface three different Froude numbers are tested: 0.35,0.4 and 0.45. The reference length for the Froude number is the diameter of the sphere. The distance between the undisturbed free smface and the center of the sphere is different for each Froude number. For Fn 0.35 and 0.40 this distance is equal to 1 diameter while for Fn 0.45 this distance is equal to 1.25 diameter. The reason to increase the distance for the higher froude number is to avoid breaking waves. When wave breaking occurs the free surface is no longer a single valued surface and this will stop the computation.

The origin of the coordinate system is iocated i n the most forward point of the sphere. For Froude number 0.35 and 0.40 the inflow boundary is located one diameter upstream, the outflow boundary 3 diameters downstream and the width of the domain is 2 diameters. For the EVoude number 0.45 the inflow boundary is located 2 diameters upstream, the other dimensions are equal to the Fn 0.35 and 0.40 case. The size of the domain is relatively large compared with a standard case to make sm-e that the calculated results are not affected by the boundaries, (see table 5.1)

The panel distribution on the sphere is the same as for the case without free smface. The number of panels on the free surface in longitudinal direction (on a reference length of 1 diameter) is equal to the number of longitudinal panels on the body. The size of the panels on the free surface in transverse direction is two times the length i n longitudinal direction which results i n an aspect ratio of 0.5. This is done to reduce the number of free surface panels and thus the computational time and costs. Both the panels on the sphere and the free surface are first order panels (flat panels with a constant source distribution). A n example of the free surface mesh is given in flgure 5.1.

Table 5.1: Size of the submerged sphere test case (diameter of the sphere is used as reference length) Froude number 0.35 0.40 0.45

depth 1 1 1.25

upstream boundary -1 -1 -2 downstream boundary 3 3 3 width of the free smface 2 2 2

A free smface computation is converged / stopped when the maximum wave change is below the convergence tolerance for wave height change. The default value is 5.0 • 10"^ but during this investigation the value was set to 1.0 • 10^^ to be sure that the results are free of any iterative convergence error.

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5.2 Submerged sphere ₂₆

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5.2.3 Results

The increase in the number of panels leads to a change in the resistance coefficient and a change i n the reference area. The comparison of the different panel distributions is done by the so called drag area which is the resistance coefficient multiplied by the reference area. Values of the drag area for the sphere without free smface can be found in table 5.2.

Table 5.2: Initial drag area of a sphere

nr of panels 16^ 20^ 242 28^ 32^ 36^ 40^ 442 48^ 522 562

Cd-S- 10^ 2.62 1.53 3.68 5.37 6.02 5.45 4.02 -6.11 -7.03 -7.84 -8.54

The resistance of the sphere shows no clear trend with an increasing number of panels. This is unexpected because normaUy the numerical error should decrease with a decreasing cell size. A further investigation shows that the reasons for the unexpected resuft are caused by the mesh generator and the iterative matrix solver.

A closer inspection of the output shows that the corner points of the panels are not exactly on the surface of the sphere. This is caused by the mesh generator which uses splines to approximate the sphere. The difference is very small but it may affect the result of the calculations. To eliminate this the mesh generator is run in 'manual' mode in which the panel corner points are read from an input file (see figure 5.2).

The main reason for the unexpected trend of the drag area is the iterative matrix solver which stops when certain convergence criteria are met. By changing these convergence criteria or using a Gaussian elimination procedure to solve the system of equations, the drag area of the calculation results i n values which are of machine accuracy as can be seen in tabic 5.3.

Table 5.3: Final drag area of a sphere using the correct settings

panels 162 202 242 282 322 362 402 442 482 522 562

Cd-S- 10^^ 0.71 -3.30 -0.62 -3.45 -3.11 1.89 -6.55 7.57 2.52 -1.30 -1.33

During the free surface calculations of the submerged sphere the manual input option and correct convergence criteria for the matrix solver are used. The results for Fn 0.35,0.40 and 0.45 can be found in tables 5.4, 5.5 and 5.6

Table 5.4: Wave drag area of a submerged sphere at Fn 0.35

panels 16 20 24 28 32 36 40 44

Cwpi • 5 • 10^ 1.2359 1.1491 1.0918 1.0431 1.0004 0.96241 0.92837 0.89708 Ciü^c -s-io'^ 0.69117 0.74197 0.77275 0.79028 0.80193 0.81071 0.81836 0.82529

The least squares method is used to determine the numerical order and extrapolated drag area of the submerged sphere. First the least squares method is applied using the results from

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5.2 Submerged sphere ₂₈

panels

Cwpi -s-ioF

Table 5.5: Wave drag area of a submerged sphere at Fn 0.40 16 5.6248 3.9864 20 5.6287 4.0810 24 5.5969 4.1149 28 5.5475 4.1276 32 5.4947 4.1337 36 5.4429 4.1374 40 5.3961 4.1416 44 5.3521 4.1459

Table 5.6: Wave drag area of a submerged sphere at Fn 0.45

panels 16 20 24 28 32 36 40

Cwpi • 5 • 10^ 1.0556 1.0149 0.99121 0.97675 0.96762 0.96145 0.95759 Cw^c • S -10^ 1.0706 1.0980 1.1148 1.1268 1.1364 1.1443 1.1518

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aU the available grids. Then the result on the coarsest grid is removed and the least squares method is applied again, which is repeated until 6 grids are left. The method needs only 4 different grids to work but the reliability increases when more results are used.

m of grids 00 • 10' a • 10' P 5-10' 8 -0.8912 1.791 0.169 1.788

7 - - -

-6 - - -

-Table 5.8: Extrapolation of the wave cut analysis results of a submerged sphere at Fn 0.35 m of grids 00 • 10^ a-lO^ P 5- 10^ 8 8.5321 -2.9048 1.686 2.7919 7 8.5574 -3.1725 1.613 3.0445 6 8.7992 -5.5131 1.092 5.4627 13 12 11 10 X 10 * DA pi O DA wc 0.5 1.5 hi/h1 2.5

Figure 5.3: Extrapolation of the drag area for a submerged spere at Fn 0.35

For Fn 0.35 a negative value is found for the extrapolated drag area using 8 grids. When 7 or 6 grids are used no solution is found in the range 0.01 < p < 5 which means that the

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5.2 Submerged sphere 3 0

wave resistance determined by pressure integration diverges. The wave resistance determined by wavecut analysis converges but the extrapolated value depends on the number of grids included i n the extrapolation procedm'e. The difference between the result on the finest grid and the extrapolated value is between 4 and 7%.

For Fn 0.40 no solution is found in the range 0.01 < p < 5 which means that the pressure integration results diverge. The wave resistance determined by wave cut analysis converges and the results of the least squares method can be found in table 5.9. The difference between the result on the finest grid and the extrapolated value is less than 1%

Table 5.9: Extrapolation of the wave cut analysis results of a submerged sphere Fn 0.40 nr of grids 00 • 10' a • 10^ _P 6-10^ 8 4.1456 -0.28128 3.988 0.03368 7 4.1474 -0.39102 3.581 0.1470 6 4.1581 -1.3218 1.936 1.2206 5.8 5.6 5.4 5.2 x 1 0 • * DA pi O DA w c 4.8 4.6 3.8 L 0 0.5 1 1.5 2 2.5 3 hi/h1

Figure 5.4: Extrapolation of the drag area for a submerged sphere at Fn 0.40

For Fn 0.45 both the wave resistance determined by pressure integration and the wave resis-tance determined by wave cut analysis converge. The pressure integration results converge to a value which differs less than 3% from the result on the finest grid. The wave cut analysis results show convergence but the extrapolated value depends on the number of grids included

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in the extrapolation procedure. The difference between the extrapolated value and the result on the finest grid is between the 5 and 8%. Although both methods to determine the wave resistance converge, there is a significant difference between the extrapolated values.

Table 5.10: Extrapolation of the pressure integration results of a submerged sphere at Fn 0.45 nr of grids 00 • 10' a • 10^ P 5-10^

7 0.93644 2.0555 1.920 2.1147 6 0.94014 1.7203 2.121 1.7445

Table 5.11: Extrapolation of the wave cut analysis results of a submerged sphere at Fn 0.45 nr of grids 00 • 10' a • 10* P 5-10* 7 1.2125 -0.61608 0.907 0.60707 6 1.2558 -1.0431 0.595 1.0395 X 10 1.25 1.15 1.05 0.95 hi/hi

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5.2.4 E x p l a n a t i o n of the results

Although the drag area of the sphere without free surface reduces to machine accmacy this does not mean that the numerical error is zero. Because of the symmetric mesh tlie error cancels when the pressure is integrated. This makes i t difficult to determine the quality of the panef distribution on the sphere. The difference between the mesh generated panels and the manual input is so small that it can be neglected i n practical cases.

The grid refinement shows that the calculations for the Fi-oude numbers 0.35 and 0.40 diverge when the wave resistance is determined by pressure integration. The convergence is around second order for R'oude number 0.45. There are a couple of different reasons for this behavior. The wavelength (A) of the waves generated by a ship or submerged body depends on the Pi-oude number and can be calculated by the following formula:

A = 2TTFn^Lwl (5.8) This formula can be used to determine the number of waves per unit length (n).

This formula shows that the number of waves decreases when the Proude number increases. The finest grid for Fn of 0.35 and 0.40 have 44 panels per unit of length which results i n respectively 33.87 and 44.23 panels per wavelength. The finest grid for the Froude number of 0.45 uses 40 panels per unit of length which results in 50.89 panels per wavelength. So although the grid for this case is coarser than the ones used for the lower froude numbers, it has a better resolution of the fiow featmes (waves). This higher resolution may be the reason for the better convergence of Fn 0.45.

A requirement of both Richardson extrapolation and the least squares method is that the grid needs to be systematically refined. This requirement is not totally satisfied here. The number of panels is systematically refined but this is not the case for the location of the collocation points. The corner points of the first order panels are exactly on the surface of the sphere but the collocation points are a small distance inside the sphere (since the panels are flat). This distance becomes smaller when the number of panels increases. The result of this is that a reflnement of the paneis results in slightly different shape of the submerged body which may affect the resistance values and the extrapolation.

Another reason which may explain the better convergence of the Fn 0.45 is the location of the inflow boundary. A ship does not generate waves upstream of the bow which makes it possible to put the inflow boundary at 0.5 times the waterline length upstream of the bow. The disturbance caused by the sphere is so large that i t does generate waves upstream of the body. To minimize the influence of this effect the inflow boundary is put further upstream than for the conventional case. For Fn 0.35 and 0.40 the inflow boundary is located one diameter upstream of the body while for Fn 0.45 the inflow boundary is located two diameters upstream

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5.3 Delft 1 ₃₃

of the body. A further increase in the upstream distance for Fn 0.35 and 0.40 may improve the convergence. Another solution is to use a more slender submerged body with a smaher volume such that upstream waves are avoided.

The last explanation of the poor convergence of the wave resistance determined by pressme integration may be found in the calculation of the influence coefficients. The calculation of the influence coefficient between two panels depends on then relative distance. A panel within close range is treated exact (flat panel with constant source strength), while a mid range panel is approximated by four point sources in the panel corner points and a far field panel is treated as a single point source. Refinement of the panels on the submerged body and the free surface will lead to a rapid increase in far field panels. The error in this approximation is small for a single panel but it may have a lot more influence when it is summed over a large number of panels (which happens when the pressme is integrated). This may explain a decrease in the accuracy of the results when the grid is refined, but it does not explain why the results are so poor for Fn 0.35 and 0.40 whfle the resuhs for Fn 0.45, on a simflar grid, are satisfactory.

The wave resistance determined by wave cut analysis converges for all the three cases but the calculated order of this method shows some strange values. For Proude number 0.35 and 0.45 the order is lower than the theoretical order while for Fn 0.40 the order is higher than the theoretical value.

The high value of the order for Fn 0.40 may be caused by the coarse grids which are outside the asymptotical range. The order becomes closer to the theoretical value when the two coarse grids are removed from the least squares method.

The low order of the wave cut analysis method for Fn 0.35 and 0.45 is probably caused by an inaccuracy i n one or more of the steps taken to determine the wave resistance. The basics of the wave cut analysis are explained in section 4.4 and i n reference [?] and consist of the following steps: a spline through the control points, a redistribution of points on the spline, a discrete fourier transform, a least squares approximation and a summation. The numerical order of these steps is unknown and may deteriorate the numerical accmacy of the final result.

5.3 Delft 1

Verification of the wave resistance for a sailing yacht hull which is free to trim and sirüc is not possible. As explained i n section 5.2, the change in resistance, when the grid is refined, is a combination of a numerical error and a change in attitude (trim and sink). Another reason which makes verification impossible is the requirement that the solutions need to be i n the asymptotic range. One of the assumptions used in the derivation of the non linear free surface potential flow method is that the free surface can be described by a single valued function. This assumption makes i t impossible for the method to flnd a solution when wave breaking is present. This limits the grid refinement because there are always regions close to the hull where the waves are breaking. This leads to the following problem for the verification: a solution can be found on a coarse grid, but this solution is probably not in the asymptotic

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5.3 Delft 1 ₃₄

range, while for a fine grid in the asymptotic range no solution can be found due to wave breaking.

Experience with the method has shown that a minimum of 20 to 25 panels per wavelength are necessary to get a solution which gives a good representation of the large wave scales. The number of waves per unit of fength depends on the Froude number and can be calculated by the following formula.

" = 2 ^ ; ? (5.10) This test case will investigate the influence of the most important parameters on the results

within the practical range of 20 to 30 panels per wavelength.

5.3.1 Objectives

The objective of this test case is to investigate the influence of the different parameters on the wave resistance. The parameters which will be investigated are: the hull panels, the convergence criteria and the free smface panels.

5.3.2 D e s c r i p t i o n and results of the test case

Huh nr 1 of the DSYHS, which is the parent hull form of the flrst part of the series, will be used for the investigation. The investigation will start with the calculation of the potentiai fiow around the hull without free surface, the double model. Then the free surface is included and the influence of convergence criteria and free surface panels are investigated. To keep things clear the results are given immediately after the description of the tests.

Panel distribution on the hull

A way to judge the quality of the panel distribution on the huh is to calculate the resistance without a free surface, the double body resistance. This double body resistance should be zero according to dAlemberts's paradox but the calculated resistance value wiU not be zero because of the discretization error. The increase in panels should give a decrease in the discretization error and thus a decrease in the resistance. Second order panels, parabolic panels with a linear somce distribution, are used to represent the hull. The advantage of the second order panels is that less panels are needed, compai'ed to first order panels, to give an accurate representation of the hull. 16 different panel distributions are checked: all possible combinations of 30, 60, 90 and 120 panels in longitudinal direction and 5,10,15 and 20 panels in transverse direction. The results can be found i n table 5.12.

The mesh of 60 by 10 panels is used to investigate the influence of stretching the mesh i n longitudinal direction. Since the highest pressure is found i n the bow and stern it is expected

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Figure 5.6: Mesh of Delft 1 30 by 5 panels and 90 by 15 panels

Table 5.12: Drag area of Delft 1 for different nr of hull panels

Cd--fy- 10^ nr of transverse panels nr of longitudinal panels 5 10 15 20

30 4.253 6.698 7.970 8.679 60 0.5426 2.314 3.191 3.585 90 -0.5394 1.188 1.775 2.077 120 -1.077 0.7495 1.172 1.328

Trend validation of SHIPFLOW based on the bare hull upright resistance of the Delft Series