Evaluation of RANS turbulence models for the

hydrodynamic analysis of an axisymmetric streamlined

body with special consideration of the velocity

distribution in the stern region

Mattias Johansson

August 31, 2012

Abstract

This is a master’s thesis provided by the Swedish Defence Research Agency FOI and iswritten for the Center for Naval Architecture at the Royal Institute of Technology KTH. Theobjective is to perform a systematic simulation campaign of the flow around an axisymmetricstreamlined body in order to evaluate several RANS-based turbulence models implementedin the open source CFD software package OpenFOAM. Of most interest is the low velocityzone at the stern and the wake behind the body. The results are compared with experimentalflow measurements and with data obtained by LES, in order to evaluate how accurately theflow is predicted by each turbulence model. The study leads to method recommendationsfor this type of flows. Simulation results for two di!erent bodies are also compared in orderto investigate how the di!erent shapes a!ect the flow. Also included is an overview of theturbulence modeling theory behind the RANS-methods which are employed. The resultsdemonstrate that the turbulence models k! !, RNG k! !, k!" and k!"SST are suitablefor simulation of this class of flows and provide a good prediction of the mean flow aroundand behind the body.

Sammanfattning

Det här är ett examensarbete som tillhandahållits av Totalförsvarets Forskningsinstitut, FOI,och är skrivet åt Marina system på Kungliga Tekniska Högskolan, KTH. Målet är att ut-föra en systematisk simuleringskampanj av flödet kring en axisymmetrisk strömlinjeformadkropp för att utvärdera en mängd olika RANS-baserade turbulensmodeller implementeradei CFD programvarupaketet OpenFOAM. Låghastighetszonen vid aktern och vaken bakomkroppen är av extra stort intresse. Resultaten jämförs med vindtunnelsexperiment samt medsimuleringsresultat som erhållits med LES, för att sedan kunna utvärdera hur väl varje tur-bulensmodell kan förutsäga flödet. Studierna leder fram till en metodrekommendation förden här typen av flöden. En jämförelse mellan flödet kring två olika kroppar genomförsockså för att utreda hur de olika geometrierna påverkar flödet. En översikt av teorin bakomRANS-metoderna som använts är också inkluderat i rapporten. Resultaten visar att turbu-lensmodellerna k− ε, RNG k− �, k− ω och k− ω SST är lämpliga för simulering av dennatyp av flöden och de förutser väl hur medelflödet kring kroppen ser ut.

Acknowledgements

I would like to thank my supervisor Mattias Liefvendahl at FOI for providing me with thesubject for this thesis and taking his time to help me through the whole project, this workwouldn’t have been possible without him.

Contents

1 Introduction 1

2 Methods 2

2.1 Reynolds Averaged Navier-Stokes equations . . . . . . . . . . . . . . . . . . . 22.2 k ! ! turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Near wall treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 The log law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Wall functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Other turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.1 Linear eddy viscosity models . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Non-linear eddy viscosity models . . . . . . . . . . . . . . . . . . . . . 82.4.3 Stress-Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Simulation campaign 103.1 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 The mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Boundary conditions and initial values . . . . . . . . . . . . . . . . . . . . . . 123.4 Simulations and calculated quantities . . . . . . . . . . . . . . . . . . . . . . . 13

4 Results 15

4.1 Validation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Results from the Joubert simulations . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Comparison with LES . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Results from the AFF1 simulation . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Conclusions and discussion 25

Chapter 1

Introduction

Computational Fluid Dynamics (CFD) provides a way to simulate and predict the flow ofa fluid. This is of great interest in many applications such as car design, meteorology, shipdesign and so on. Understanding the complex turbulent flow around a ship or a submarineis crucial when designing the ship. Quantities of interest could be friction and pressurecoe"cients to calculate the water resistance, the boundary layer thickness, the velocity fieldand turbulence levels along the body and behind it. The low velocity zone at the stern is ofspecial interest since it will largely a!ect the propulsor. The turbulent flow at high Reynoldsnumber in an incompressible fluid around a ship is too complex to solve the governing fluiddynamics equations (the Navier-Stokes equation, [9]) directly, this makes it necessary tomodel the turbulence in some way. RANS (Reynolds averaged Navier-Stokes equations) isan approach to model turbulent flows starting from the averaged (time or ensemble averaged)Navier-Stokes equations.

This master’s thesis main objective is to investigate the flow around an axisymmetricstreamlined body using RANS-based methods in the software package OpenFOAM, [1],that is the most widely used open source software for CFD. This axisymmetric streamlinedbody could represent an unappended submarine hull or a torpedo. Studies of this type areimportant when designing a submarine for many reasons such as to lower the resistance or toreduce turbulence levels at the propulsor to lower the noise levels. Simulations on a genericunappended submarine hull with several turbulence models are carried out and the resultsare validated against experiments such as Particle Imaging Velocimetry (PIV) measurementsand other types of simulations, Large Eddy Simulations (LES). This investigation will givean understanding on how well the in OpenFOAM implemented RANS turbulence modelspredicts the flow on models like this. This will in the end give a method recommendationfor simulations of this kind. One of the best performing turbulence model is also comparedwith a simulation on another hull geometry to investigate how the di!erent geometries a!ectthe flow.

The report starts with an overview of the turbulence modeling theory behind the RANS-methods employed in this work in chapter 2. Further on in chapter 3, there are descriptionson the two models used. The meshes, the boundary conditions and the calculated quantitiesis also described there. Chapter 4 presents the results from the simulations and comparethem with the validation data. Conclusions is drawn and discussion on the results is donein chapter 5.

1

Chapter 2

Methods

This chapter includes an overview of the theory behind the methods employed in this work.A complete description is available in [4], [10] and [12].

2.1 Reynolds Averaged Navier-Stokes equations

The basic governing equations in fluid dynamics are the Navier-Stokes equations. Writtenfor incompressible constant-property Newtonian fluids they are

DUj

Dt"#Uj

#t+ Ui

#Uj

#xi= !

1

$#p+ %#2Uj (2.1)

and the incompressible continuity equation is

#Uj

#xj= 0 (2.2)

where xj are the Cartesian coordinates, Uj is the velocity in the xj -direction, $ is thedensity of the fluid, % is the kinematic viscosity, p is the pressure and t is the time. Hereand throughout this report is the Einstein summation convention used.

The RANS (Reynolds Averaged Navier-Stokes) equation is a way to model turbulentflows by averaging (time or ensemble averaging) the Navier-Stokes equations. The startingpoint of the derivation of the RANS equations is to decompose the velocity into its mean,U j , and the fluctuation, uj.

Uj = U j + uj . (2.3)

Equation (2.3) in equation (2.2) gives the mean conservation of mass equation for incom-pressible flows

#U j

#xj= 0. (2.4)

Inserting equation (2.3) in equation (2.1) and averaging leads to the mean-momentum equa-tion, also called the Reynolds equation or the Reynolds-averaged Navier-Stokes equation(RANS)

DU j

Dt"#U j

#t+ U i

#U j

#xi= %#2U j !

#uiuj

#xi!

1

$

#p

#xj. (2.5)

2

Equation (2.5) can be re-written as

$DU j

Dt=

#

#xi

!

µ

"

#U i

#xj+#U j

#xi

#

! p&ij ! $uiuj

$

. (2.6)

where the terms inside the square brackets represents three stresses. The first term is theviscous stress, !p&ij is the isotropic stress from the mean pressure field and !$uiuj is astress arising from the fluctuating velocity. The last term is called Reynolds stresses. Thisterm without !$, i.e. uiuj will also be referred to as Reynolds stresses in this report. Sincethe equations are averaged the quantities calculated with RANS methods are statistical, [10].

2.2 k ! ! turbulence model

The number of unknowns in the Navier-Stokes equations, (2.1), is four, the three velocitycomponents and the pressure. The Reynolds stress tensor, uiuj, is a symmetric tensorand will then add six new unknowns into the system without adding any new equations.The number of unknowns is then 10 and the equations available are the three componentsof the RANS equation, (2.5), and the mean conservation of mass equation, (2.4). Thispresents a closure problem and to solve it modeling is needed. This is why turbulencemodeling is necessary. There are many di!erent turbulence models available for RANS andone commonly used is the k ! ! model described here.

According to the turbulent-viscosity hypothesis introduced by Boussinesq, [10], is theReynolds stress

uiuj =2

3k&ij ! %T

"

#U i

#xj+#U j

#xi

#

(2.7)

where %T is the turbulent viscosity and k the turbulent kinetic energy defined as

k "1

2uiui. (2.8)

The trace of the Reynolds stress tensor is 2 k and the term 23k&ij in equation (2.7) is the

isotropic stress. The turbulent-viscosity hypothesis states that the mean rate-of-strain

Sij =

"

#U i

#xj+#U j

#xi

#

(2.9)

via the turbulent viscosity is linearly related to the Reynolds-stress anisotropy tensor, aij ,defined by

aij " uiuj !2

3k&ij . (2.10)

Equation (2.10) in equation (2.7) gives that linear relationship by

aij = !%T

"

#Ui

#xj+#Uj

#xi

#

. (2.11)

In the k ! ! model the turbulent viscosity is specified as

%T = Cµk2

!(2.12)

3

where ! is the dissipation of turbulent kinetic energy and Cµ is a model constant. The k! !model is a two equations turbulence model consisting of two model transport equations fork and !. The exact model transport equation for k, derived in [10], is

#k

#t+ U j ·#k = !# · T !

j + P ! ! (2.13)

where

T !

j "1

2uiujuj +

1

$uip! ! 2%ujsij (2.14)

is the energy flux,

P " !uiuj#U i

#xj(2.15)

is the production of turbulent kinetic energy,

! " 2%sijsij (2.16)

is the dissipation of the turbulent kinetic energy and

sij =1

2

"

#ui

#xj+#uj

#xi

#

(2.17)

is the fluctuating rate of strain. Equation (2.13) however add a lot of unknowns, thereforethe energy flux, T !

j , is modeled with a gradient-di!usion hypothesis as

T ! = !%T'k

#k (2.18)

where 'k is the turbulent Prantl number for kinetic energy. Physically this asserts that thereis a flux of turbulent kinetic energy down the gradient of k. The standard model equationfor ! is

D!

Dt= # ·

"

%T'!

#!

#

+ C!1P!

k! C!2

!2

k(2.19)

where '!, C!1 and C!2 are model constants. The model equation for ! is best viewed as beingcompletely empirical, [10]. The total number of model constants is five and their standardvalues according to [10] are

Cµ = 0.09, C!1 = 1.44, C!2 = 1.92, 'k = 1.0, '! = 1.3. (2.20)

The equations for k, ! and the specification of %T is what forms the k!! turbulence model.

2.3 Near wall treatment

The flow close to walls are in many ways more complicated than free shear flows, thereforeconsideration is needed for flows close to walls. Considering a fully developed 2D flow closeto a wall, x1 = x is the flow direction and x2 = y is normal to the surface, with thefreestream mean velocity, U, in the positive x-direction. The total shear stress, ( , consistsof the viscous stress and the Reynolds stress

4

( = $%dU

dy! $uv (2.21)

where u and v are the fluctuations in the x- and y-directions. At the wall the no-slipcondition implies that the Reynolds stress is zero, then the viscous stress is the only stressand the shear wall stress is

(w = $%dU

dy

%

%

%

%

y=0

. (2.22)

Close to the wall the viscous stress is the dominating stress in contrary to the free flow wherethe viscous stress can be neglected compared to the Reynolds stress. Close to the wall, %and (w are important parameters and from these appropriate velocity and length scales canbe defined as

u" "

&

(w$

(2.23)

which is the friction velocity,

&# " %

&

$

(w=

%

u"(2.24)

which is the viscous length scale and

y+ "y

&#=

u"y

%(2.25)

which is viscous lengths or wall units. Di!erent layers in the flow close to the wall are definedby y+. At y+ < 5 is the viscous sub layer where the Reynolds stress is negligible, y+ < 50is the viscous wall region where there is a direct e!ect of the viscosity on the shear stressand y+ > 50 is outer layer where the e!ect of viscosity is negligible, [10].

2.3.1 The log law

The mean velocity gradient close to the wall can on dimensional grounds be written as

dU

dy=

u"y#

"

y

&#,y

&

#

(2.26)

where # is a dimensionless function and & is an appropriate length scale much larger than&# . Introducing the function

#I

"

y

&#

#

= limy! "0

#

"

y

&#,y

&

#

(2.27)

can equation (2.26) close to the wall be written as

dU

dy=

u"y#I

"

y

&#

#

, fory

&<< 1. (2.28)

Defining u+(y+), the mean velocity normalized by the friction velocity as

u+ =U

u"(2.29)

5

and using equation (2.25) in equation (2.28) is

du+

dy+=

1

y+#I(y

+). (2.30)

When y+ > 30 the distance to the wall is large enough to assume that the dependence ofviscosity in #I vanishes making it constant, )#1, ) is the von Karman constant and this inequation (2.30) gives

du+

dy+=

1

)y+. (2.31)

Integration of equation (2.31) gives

u+ =1

)ln y+ +B. (2.32)

where B is a constant from the integration. This is the log law where ) = 0.41 and B = 5.2are the usual values of the constants, they alter some in di!erent literature but are generallyclose to these values. Comparing the log law with velocities from direct numerical simulations(DNS) data shows good accuracy for the log law in the region 30 < y+ < 5000, [10].

2.3.2 Wall functions

Since the profiles of U and ! are steep close to walls allot of the computational e!ort willhave to be devoted to the near wall region making the computations very expensive. Wallfunctions are used to lower the cost of the computations. The idea is to apply a boundarycondition some distance away from the wall in the log law region in order to avoid solvingthe turbulence-model equations close to the wall. The wall function boundary conditionsare applied at the point y = yp where y+ $ 50. The index p will indicate that the quantitiesare evaluated at yp. According to the turbulent viscosity hypothesis is

|uv|

k=

"

CµP

!

#1/2

. (2.33)

In the region y+ $ 50 DNS calculations shows there is a balance between the dissipationand production of turbulent kinetic energy, also in the overlap region (50 < y+ < 1000) thedi!erence between !uv and u2

" is small, [10]. Then it holds that

!uv = C1/2µ k = u2

" . (2.34)

Near the wall all velocity gradients except dUdy can be neglected, the production of kinetic

energy then can be written as

P = !uv#U

#y(2.35)

and in the log law region it holds

dU

dy=

u")y

. (2.36)

Since there is a balance between the production and the dissipation and using equations(2.34), (2.35) and (2.36) the dissipation is written as

6

! =u3"

)y. (2.37)

By combining equations (2.34) and (2.37) at the position yp the boundary condition for ! is

!p =C3/4

µ k3/2p

)yp. (2.38)

The boundary condition of k is set to zero normal gradient. The turbulent viscosity hypoth-esis gives

!uv = %T#U

#y. (2.39)

Using equations (2.39), (2.34) and (2.35), the production term, at yp, is calculated as

Pp = %TC1/4

µ k1/2p

)yp

#U(yp)

#y(2.40)

and y+ at yp is

y+p =C1/4

µ k1/2p yp%

. (2.41)

2.4 Other turbulence models

The turbulence models used in this work are the k ! !, realizable k ! !, k ! ", k ! " SST ,RNG k ! !, Lien cubic k ! !, Non-linear Shih k ! ! and LRR Reynolds stress transportmodels. These turbulence models can be divided into three categories, linear eddy viscositymodels, non-linear eddy viscosity models and Reynolds stress models.

2.4.1 Linear eddy viscosity models

The linear eddy viscosity turbulence models are based on the turbulent viscosity hypothesis,equation (2.7). Turbulence models of this type used are the standard k ! ! described insection 2.2, realizable k ! !, RNG k ! !, k ! " and k ! " SST models.

If the strain rate becomes to large when using the standard k ! ! model it will becomenon realizable, since the normal stresses then can become negative. To prevent this therealizable k ! ! turbulence model relate the constant Cµ to the main strain rate, Cµ is nottreated as an constant and is

Cµ =1

A0 +AsU ($) k!

(2.42)

where A0 is a constant, As and U ($) depends on the main strain rate and are determinedaccording to [11].

The k ! " model models %T and k in the same way as the k ! ! turbulence model, thedi!erence is that the specific dissipation rate is modeled with " " !/k instead of with !.This gives the specific dissipation rate as

D"

Dt= # ·

"

%T'$

#"

#

+ (C$1 ! 1)P"

k! (C$2 ! 1)"2 +

2%T'$k

#" ·#k (2.43)

7

where '$ = 'k. The interpretation of " di!ers but it can be described as a frequencycharacteristic of the turbulence, the RMS fluctuating vorticity or just the ratio of ! and k.The k!" model is designed to give better results than k! ! in the viscous near-wall regionand in how it accounts for the e!ects of stream wise pressure gradients. However it can besensitive to free stream boundary conditions.

The k ! " SST (shear stress transport model) works as a combination of the regulark ! " close to walls but switches behavior to k ! ! in the free stream. This is done bymultiplying the last term in equation (2.43) to a blending function that is zero close to thewall, so the model works as k ! " close to walls, and it is unity far from the wall makingthe model correspond to the standard k ! ! in the free stream, [10].

The RNG k ! ! turbulence model make use of renormalization group theory to get amodified k ! ! model where the equations for %T , k and ! are the same as in the standardk ! ! model. The di!erence lies in the constants, C!2 is not constant as in equation (2.19).In the RNG k ! ! C!2 is defined as

C!2 " C!2 +Cµ*3(1! */*0)

(1 + +*3), * "

k

!

'

2Sij2Sij (2.44)

where

C!2 = 1.68, C!1 = 1.42, Cµ = 0.085, + = 0.012, *0 = 4.38 (2.45)

are the closure coe"cients. Notable is also that the other constants in equation (2.19) issomewhat altered in the RNG k ! ! model.

2.4.2 Non-linear eddy viscosity models

The Lien cubic k ! ! and Non-linear Shih k ! ! models belongs the category non-lineareddy viscosity models. The non-linear eddy viscosity models relate the turbulent stressesalgebraically to the rate of strain and include higher order quadratic and cubic terms. Theyoften give better predictions in reattachment areas where the linear models sometimes failto predict the flow correctly, [3].

2.4.3 Stress-Transport Models

In a stress-transport turbulence model, also refereed to as second-order closure or secondmoment closure model, the exact equation for the Reynolds stress-tensor is used and eachterm of the equation is modeled by it self. The model is more complex than the two equationmodels and demands more computational e!orts but it corrects some of the Boussinesq’sapproximation shortcomings. The stress transport model includes e!ects of flow history ina more realistic way than the two equation models since it automatically accounts for theconvection and di!usion of the Reynolds stress tensor. It also includes e!ects of streamlinecurvature and it behaves properly for flows with sudden changes in the strain rate. TheLRR Reynolds stress transport model belongs to this category of turbulence models. It isthe most widely used and tested stress transport model that is based on the ! equation.There is however a di!erence in the dissipation rate equation compared to what it is in thek ! ! model, [12].

8

2.5 Numerical methods

Since the main objective of this work is to analyze the di!erent turbulence models thenumerical methods and their implementation into OpenFOAM will just be briefly describedin this section. OpenFOAM solves the RANS equations using the finite volume (FV) method.We illustrate this class of FV-methods by its application to a conservation equation for ascalar , in a given velocity field u. The integral form of the equation is

d

dt

(

V$,dV +

(

S$,u · ndS = 0 (2.46)

where V is a control volume, S is the control volume surface and n is the unit normal vectorof the surface. The solution domain is divided into polyhedral finite Control Volumes (CVs)and then the conservation equations are applied to each CV. If the equations are summedfor all of the CVs the global conservation equations is obtained since the surface integralsof the inner CV faces cancels each other. This is one of the advantages of the finite volumemethod since the global conservation is built in from the beginning. At the centre of eachCV is the computational node where the value of each unknown variable is to be calculated.Interpolation is used to express the variables at the CV surfaces in terms of the nodalvalues. The surface and volume integrals are approximated using some appropriate methodfor which there are many schemes available. As a result of this an algebraic equation foreach CV is obtained containing values of neighboring nodes. The method can accommodateany type of grid and is suitable for complex geometries. A detailed description of the FVmethod can be found in [4].

When solving a stationary problem with RANS the problem can be regarded as unsteadyuntil it reaches steady state. In order to reach or speed up convergence can it be necessaryto limit the change of each variable between the iterations. This is called under relaxation.This is illustrated by looking at the algebraic equation for a generic variable , on the nthouter iteration at the typical point P

AP,nP

)

l

Al,nl = QP (2.47)

where Q contains the terms not depending on ,n. The coe"cients Al and Q may involve,n#1. When solving this linear equation iteratively the changes in , between the outeriterations can become to large causing instability. So ,n is only allowed to change a fractionof the would be change according to

,n = ,n#1 + -%(,new ! ,n#1) (2.48)

where ,new is the solution to equation (2.47) and -% is the under-relaxation factor thatsatisfies 0 < -% < 1.

The equations are solved with an iterative method and they are solved sequentially i.e.the equations are solved for each component in turn first solving for the velocity componentswith the pressure from the last iteration and then solving for the pressure using a discretePoisson equation, [4]. The velocity and pressure fields now satisfy the continuity equationbut not the momentum equation. The procedure is then repeated with the new pressure.This is one outer iteration and it is repeated until both the continuity and momentumequations are fulfilled within a certain tolerance. The Semi Implicit Method for PressureLinked Equations (SIMPLE) is an algorithm doing this. The simulations in this work aredone using the in OpenFOAM implemented SIMPLE algorithm. A detailed description ofthe SIMPLE algorithm is found in [4].

9

Chapter 3

Simulation campaign

3.1 The models

There are two geometries considered in this investigation. The main model is a genericconventional submarine hull described in [8]. It is the unappended hull of that design whichwill be referred to as the Joubert model after the author of the design reports. The Joubertmodel is shown in figure 3.1. This shape is the result of design considerations presented in [7]and [8] in which the aim is to design a submarine with minimum resistance and noise whilestill carrying out all the normal functions for a submarine. This model was chosen becausethere are both experimental data and other simulation data available for this model from [2].The second model used is also an unappended generic submarine hull, the DARPA SUBOFFconfiguration AFF1 [6]. This model will be referred to as AFF1 and is also shown in figure3.1. The SUBOFF project is one of the few studies of conventional generic submarine hullsmade in open literature and it is meant to provide a way to compare numerical simulationsof axisymmetric hulls with experimental data for CFD users. The AFF1 model used here isre-scaled to have the same length as the Joubert model used in wind tunnel tests, see [2].

Figure 3.1: The two models used. The Joubert geometry is on top and the AFF1 geometryis at the bottom.

10

Joubert AFF1Total length L 1.35 m 1.35 mDiameter D 0.185 m 0.157 mVolume displacement # 0.0283 m3 0.0208 m3

Length parallel midsection Lms/L 0.451 0.512Length forebody Lfb/L 0.228 0.233Length stern Ls/L 0.321 0.255Diameter/Length D/L 0.137 0.117Largest angle stern -max 21.8% 20.1%

Length/width L/D 7.31 8.57Block coe"cient #/LD2 0.615 0.622Prismatic coe"cient #/L.r2 0.783 0.792

Table 3.1: Main dimensions of the Joubert and AFF1 models.

The di!erences in the two bodies can be seen in figure 3.2 where profiles of both bodiesare plotted together and in Table 3.1 that shows the dimensions of the two bodies in numbers.The total length of the model L, D is the diameter, # is the volume displacement, Lms isthe length of the parallel midsection, Lfb is the length of the forebody Ls is the length ofthe stern and -max is the largest angle between a tangent of the body the stern and theparallel midsection. For AFF1 is the afterbody cap not considered for -max. The coordinatesystem is a 2D Cartesian system where x is the coordinate parallel to the symmetry axis,0 at the bow and L at the stern, y is the perpendicular distance to the symmetry axis,0 at the symmetry axis and D at the parallel midsection. The velocity components u isin the x-direction and v is in the y-direction. The di!erences between the two bodies aremainly in the di!erent stern configurations and the length to diameter ratio but there arealso di!erences in the bow configurations. The Joubert body is thicker and have a longerstern. The stern of the Joubert body has a more conical shape with an angle that is gettinglarger all the way to the trailing edge while the stern of the AFF1 body is composed of twoparts, it starts of with a conical shape and then turns into an afterbody cap with a roundtrailing edge. Joubert have a smoother transition from the parallel midsection to the sternand the tapering of the tail is also smother. The forebodies are almost of the same lengthand the forebody of AFF1 is a little more blunt than the Joubert forebody at very low x/L.The transition from the forebody to the midsection is smoother on the Joubert body.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

x/L

y/L

Figure 3.2: A comparison of the Joubert and AFF1 models. Joubert in blue and AFF1 ingreen.

3.2 The mesh

The meshes were created with the OpenFOAM-utility blockMesh where the model is builtin sets of 3D hexahedral blocks. The meshes are 2D of wedge type with 5% angle. To capture

11

enough of the flow and to avoid blockage e!ects are the boundaries in front of and over thebodies placed one body length away from the bodies and the boundaries behind the thebodies are placed two body lengths behind them. The Joubert mesh, figure 3.3 (a) and (b)consists of 80 000 cells and the AFF1 mesh, figure 3.3 (c) and(d) has 75 000 cells. Eachblock had to be graded manually and the shapes of the blocks are visible in figure 3.3. Thecells closest to the body were designed to give an y+ $ 40 where y+ is calculated accordingto equation (2.25). The y-coordinate in equation (2.25) is not the global y-coordinate, itis the distance in the normal direction of the body to the center of the cell closest to thebody. The value of y+ varies along the bodies but at the parallel midsection it is 42 on theJoubert model and 35 on the AFF1 model.

(a) (b)

(c) (d)

Figure 3.3: The meshes used, (a) and (b) is Joubert, (c) and (d) is AFF1.

3.3 Boundary conditions and initial values

The boundary conditions for the velocity are set to zero velocity at the body due to theno-slip condition, at the inlet is the velocity in the x-direction U& = 60m/s and zero in allother directions. At the outlet and far field the boundary conditions are set to zero normalgradient. The pressure boundary conditions are set to zero at the outlet and zero normalgradient at the other boundaries. For all the models using k to describe the turbulence isit calculated according to equation (2.8) with the assumption that k is isotropic at the inletand that the fluctuations of the inlet velocity is 3%, this gives k as

k =3

2(0.03U&)2. (3.1)

The assumption that the fluctuations are 3% of the inlet velocity comes from [2] where it isestimated that the wind tunnel in which the validation data is measured has this turbulencelevel. At the far field and the outlet is k set to zero normal gradient. The initial value of k inthe wall function is also calculated according to equation (3.1). To estimate the initial andinlet value for ! is it assumed that the ratio of turbulent viscosity to the laminar viscosity% is five. Then ! is calculated as

12

! = Cµk2

5%(3.2)

with Cµ = 0.09 as in (2.20) and k from equation (3.1). Equation (3.2) is also used to calculatethe initial value for the ! wall function. The outlet and far field boundary conditions for !is set to zero normal gradient. Using the definition " " !/k and equation (3.2) are the inletand initial value for the wall function of " calculated as

" =!

k= Cµ

k

5%. (3.3)

In all turbulence models except the LRR model are k and / or " included, the LRR turbu-lence model includes only ! and the Reynolds stress tensor. In this model the initial datafor ! is taken from a converged k! ! simulation and the Reynolds stress tensor is calculatedby OpenFOAM from the k ! ! simulation as well. The pressure and velocity initial dataand boundary conditions are also taken from the converged k!! simulation. All the modelsexcept the LRR model are started with zero velocity in the initial flow field. The boundaryconditions in the azimuthal directions are set to the OpenFOAM boundary condition wedgewhich acts as a cyclic boundary condition.

3.4 Simulations and calculated quantities

The simulations are carried out with an inlet velocity, U& = 60 m/s, the length of the modelis 1.35 m and the fluid is air, % = 1.5 ·10#5 m2/s giving ReL = U&L/% = 5.4 ·106. Presentedin table 3.2 is the simulation campaign. The OpenFOAM keyword is the word specifyingthe turbulence model in OpenFOAM. From here on will the turbulence models be refereedto as their OpenFOAM keyword in table 3.2.

Model Turbulence model OpenFOAM keywordJoubert Standard k ! ! kEpsilon

Joubert Realizable k ! ! realizablekE

Joubert k ! " kOmega

Joubert k ! " SST kOmegaSST

Joubert RNG k ! ! RNGkEpsilon

Joubert Lien cubic k ! ! LienCubicKE

Joubert Non-linear Shih k ! ! NonlinearKEShih

Joubert LRR LRR

AFF1 k ! " kOmega

Table 3.2: A list of the simulations carried out.

The objective of the campaign is, as stated earlier, to give an objective method recom-mendation for flows of this type. The main focus is to investigate the velocity at the sternand behind the bodies. This is done by plotting the axial velocity profiles at several placesof the body and comparing it with experiments and LES data. Other quantities of interestare the pressure coe"cient defined as

cp =|p! pref |

12$U

2&

(3.4)

13

where p is the pressure and pref is a reference pressure that here is the ambient pressure setto zero, and the friction coe"cient

cf =(w

12$U

2&

(3.5)

where (w is the wall shear stress. The turbulent kinetic energy will also be analyses. Thequantities analyses are the ones for which validation data are available.

14

Chapter 4

Results

In this chapter the results from the simulations are presented and compared with the val-idation data. The quantities will be analyzed in order of complexity and models showingpoor prediction of the flow will successively be removed narrowing down the analyze to themodels that predicts the flow well. The validation data used is described in section 4.1followed by the results from the simulations on the Joubert model in section 4.2. In section4.3 is the results from one simulation on the AFF1 model compared with a simulation withthe same turbulence model on Joubert.

4.1 Validation data

Available for validation of the results are data from experiments and other simulations onthe Joubert body done in [2]. The experiments were conducted in the DSTO Low SpeedWind Tunnel in Melbourne on a 1.35 m long aluminum. The main validation data for thevelocity is Particle Imaging Velocimetry measurements, PIV. These measurements are doneusing laser to avoid inserting objects into the flow and thereby disturbing it. The PIVmeasurements were done at the stern and in the wake region. A more detailed descriptionon how the PIV measurements were carried out is found in [2]. There are uncertainties in atwhat y-coordinate the lineplots from the PIV starts, they do however give a picture on howthe velocity profiles should look. Measurements of the surface pressure were done on themodel with static pressure taps along the centreline on top of the body. The static pressurewas measured at the taps and the pressure coe"cient is calculated as

cp =ptap ! pspT ! ps

(4.1)

where ptap is the measured static pressure at the tap, ps is the freestream static pressure andpT is the reference total pressure, defined as the static pressure plus the dynamic pressure,measured in the freestream. The total reference pressure is equivalent to the referencepressure pref in equation 3.4. The friction coe"cient was measured using the Preston tubemethod. This method requires a turbulent boundary layer to work so the skin friction wherethe boundary layer is laminar is not correct. It is however possible to determine where thetransition from laminar to turbulent occurs by looking at where it is a discontinuity in thegraph when plotting cf against the axial position. In some of the experiments boundarylayer tripping devises were used. In those cases the tripping devise consisted of either acircumferential strip of silicon carbide grit that was 3.0 mm wide or wire of diameter 0.2

15

mm or 0.5 mm. The trip was placed at x/L = 0.05. More detailed information on theexperiments are available in [2].

The LES data was provided from [2] and are a type of near wall modeled LES of whichthree types have been used as validation. LES is based on separation of the scales in the flowutilizing spatial filtering. Scales in the flow larger than a certain filter width is resolved andthe e!ect of the scales below the filter width are modeled. The types of LES simulations usedas validation are the Localized Dynamic kinetic energy Model here referred to as LDKM,the One Equation Eddy Viscosity Model here referred to as OEEVM and the Mixed ModelSpallart Allmaras here referred to as MMSA. The simulations were done on grids of 18million cells and more information can be found in [2].

4.2 Results from the Joubert simulations

The results from the Joubert simulations are presented here. Table 3.2 shows the numberof outer iterations needed for each model to converge.

Turbulence model ConvergenceStandard k ! ! 1000 iterationsRealizable k ! ! 3000 iterationsk ! " 2000 iterationsk ! " SST 4000 iterationsRNG k ! ! 1000 iterationsLien cubic k ! ! Not reachedNon-linear Shih k ! ! 10 000 iterationsLRR 7000 iterations

Table 4.1: Outer iterations needed for convergence in the Joubert simulations.

The tolerance for the residuals in the simulations was set to 10#10 for the pressureand 10#8 for the other variables and no relative tolerances was used. The solutions wereconsidered to be converged when the initial residuals reached a value that didn’t improvewith more iterations and this value was considered to be low enough. Figure 4.1 showsthe initial residuals for each outer iteration as functions of outer iterations for the best andthe worst cases considered to be converged among the simulations. The residuals in figure4.1 are the extremes and most of the other models have converged much better then therealizableKE. The LienCubicKE model never reach convergence.

Figure 4.2 shows the axial velocity from the kEpsilon simulation. Visible are the stag-nation point in front of the body, two zones with high velocity over the bends at the end ofthe forebody and at the beginning of the stern. The boundary layer is growing along thebody and there is a low velocity zone at the stern and behind the body. The RANS modelsassume the flow to be fully turbulent everywhere which it is not, the area of most interestis however the low velocity zone at the stern of the body where the flow is fully turbulent.

In figure 4.3 is y+ plotted for all the turbulence models along the body calculated ac-cording to equation (2.25) in which the y-coordinate is not the global y-coordinate, it is thedistance in the normal direction of the wall to the center of the cell closest to the body.There are three types of behaviors of the curves, kEpsilon and kOmegaSST starts at a lowy+ $ 5, grows a bit and the settles on a value around 40, realizableKE and kOmega startshigh, around 130, and then also settles at 40 after x/L = 0.2, RNGkEpsilon and LRR are very

16

(a) (b)

Figure 4.1: Initial residuals as a function of iterations, (a) kOmega and (b) realizableKE.

Figure 4.2: The mean axial velocity distribution, obtained with the kEspilon simulation onthe Joubert model.

low until approximately x/L = 0.15 where y+ rapidly grows and then settles at 40. It seemslike some threshold is reached there for RNGkEpsilon and LRR, this has however not beenfurther investigated. The LRR model have a slightly higher y+ than the other models alongthe body and at the stern. The realizableKE curve is somewhat unsteady in the beginningand the end of the body. Also notable is that the two k!" models, kOmega and kOmegaSST

shows di!erent behaviour in the beginning of the body but then has almost identical valuesfrom x/L = 0.2 all the way to the stern.

In figure 4.4 is cp and cf plotted together with experimental data. In figure 4.4 (a) iscp plotted against the length of the body. All the models except realizableKE predicts cpclose to the experiments until x/L = 0.9 where they miss the separation occurring accordingto the experimental data. The realizableKE model di!ers from the other models andthe experiments after x/L = 0.7, it is however the only model that predicts a separationbut it is predicted to early. The predicted cp at x/L > 0.4 is somewhat higher than theexperimental cp. Figure 4.4 (b) shows cf along the body. All the models predicts cf within20 % from the experimental data between x/L = 0.2 to 0.8 except the LRR model whichseverely overpredicts it. Since the flow is expected to be boundary layer dominated, a modelthat di!ers this much from the experiments is then expected to not predict the flow well.The LRR model will therefore not be considered further on. Between x/L = 0.1 and 0.2

17

Figure 4.3: Plott of y+ for all the models.

are the predictions of cf also good for all the models except for RNGkEpsilon, LRR andNonlinearKEShih which underpredicts it and then all of a sudden overpredicts it. Thisbehaviour is also seen in figure 4.3 where there seem to be some type of threshold for thesemodels.

(a) (b)

Figure 4.4: Coe"cients cp and cf for the turbulence models and experimental data. Thelabels are the same in both figures.

Figure 4.5 shows the axial velocity profiles at x/L = 0.867, x/L = 0.9, x/L = 0.933,x/L = 0.967, x/L = 1, x/L = 1.033, x/L = 1.08 and x/L = 1.1 for the turbulencemodels and the PIV-data. The positions of the profiles is shown at the top of figure 4.5.

18

The realizableKE model di!er much from the other profiles and will further on not beconsidered. As stated earlier is there some uncertainty in how far away from the wall the firstPIV-data is located. At the top four profiles is it shown that the RANS models overpredictsthe velocity close to the body and underpredicts the velocity approaching the free stream.The top four profiles in figure 4.5 shows clearly how the boundary layer grows with largerx/L. All the models behaves more or less the same there except NonlinearKEShih whomat places overpredict the velocity with up to 10 % compared to the other models and morecompared to the PIV data. The NonlinearKEShih model shows the same behaviour in thewake where the other models show very good similarity with the PIV-data where both thegradient and the magnitude of the velocity are well matched. Due to the poor predictionsby the NonlinearKEShih model is it excluded from the further results. Zooming in on theprofiles at x/L = 0.967 and x/L = 1.08 with the poor performing models removed as done infigure 4.6 are the small di!erences between kEpsilon, kOmega, kOmegaSST and RNGkEpsilon

shown. The di!erences are never larger than 12 %.

0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

x/L

y/L

Figure 4.5: Velocity profiles for all models and PIV-data along the stern and behind thebody with an illustration of where the positions of the profiles are located on top.

The models agreeing best with the experimental data are kEpsilon, kOmega, kOmegaSSTand RNGkEpsilon. These models are analysed further by plotting the kinetic turbulentenergy at two cross sections, x/L = 0.9677 and x/L = 1.033 in figure 4.7. The RNGkEspilon

19

(a) (b)

Figure 4.6: Velocity profiles for the best performing turbulence models and PIV data. Dataalso found in the corresponding graph of figure 4.5 + zoom.

predicts larger k at low y-coordinates and then crosses the other curves to predict lower kat higher y-coordinates. The two k ! " models shows very similar k-profiles. There are nok-profiles available from the PIV in [2], there are however RMS data on the velocity avilabledescribing how much the velocity fluctuates.

(a) (b)

Figure 4.7: Turbulent kinetic energy profiles for the best performing turbulence models.

4.2.1 Comparison with LES

Here is a comparison made of the well performing kEpsilonmodel with the poor performingrealizableKE model and the LES data from [2]. In figure 4.8 are cp and cf plotted for theRANS models, LES models and experimental data. The kEpsilon model is clooser to theexperimental data than the LES models in predicting cp for x/L smaller than 0.9. Forx/L larger than 0.9, the kEpsilon model do not predict the separation suggested by the

20

experimental data, OEEVM and MMSA. The RANS models does show a little bit highercp than the experiments. The LES models show much higher cf than the RANS models forsmall x/L and then settles in the same level as the RANS and experimental data except forLDKM which is underpredicting cf very much.

(a) (b)

Figure 4.8: The coe"cients cp and cf plotted for kEpsilon, realizableKE, LES and exper-iments. The labels are the same in both figures.

Comparing the velocity profiles of kEpsilon and realizableKE models with LES andPIV at x/L = 0.9 and x/L = 0.967, the prediction of the velocity profiles for the kEpsilonmodel are closer to the PIV than the LES models as shown in figure 4.9. The di!erencesbetween the LES models are large but the di!erence between kEpsilon and realizableKE

is larger.

(a) (b)

Figure 4.9: Velocity profiles for kEpsilon, realizableKE, LES and PIV.

Further downstream in the wake at x/L = 1.003 and x/L = 1.1, figure 4.10, does thekEpsilon model show very good agreement with the PIV data. It is in line with the PIVdata di!ering only a few percent. The closer to the body the better is the prediction, which

21

is shown in figure 4.10 and can also be seen in figure 4.5. The kEpsilon simulations doesa much better job predicting the velocity in the wake than the LES simulations which aredi!ering a lot from the experimental data.

(a) (b)

Figure 4.10: Velocity profiles for kEpsilon, realizableKE, LES and PIV.

4.3 Results from the AFF1 simulation

The results from the simulation with the kOmega turbulence model on the AFF1 modelis here presented and compared to the kOmega simulation on the Joubert model with thepurpose of investigating how the di!erences in the geometries a!ect the flow. The simulationon the AFF1 model converged after 2000 iterations and y+ $ 35 on the parallel midsection.Figure 4.11 shows a comparison of the velocity at the stern of the two simulations. The sizeof the low velocity region is a little bit larger at the body (x/L < 1) in the AFF1 simulation.At x/L = 0.933, at the point where the velocity is 0.8U& is the distance away from thebody in the y-direction 20% higher for AFF1 than for Joubert. The wake is however thickerin the Joubert simulation but Joubert has a larger diameter.

(a) (b)

Figure 4.11: Velocity fields at the stern obtained with kOmega. AFF1 on the left and Joubertto the right.

In figure 4.12 is cp and cf for the two models shown. Figure 4.12 (a) shows smallerpressure gradients along the body for Joubert than for AFF1. Joubert also has a smallerabsolute minimum value and the two dips are located more towards the middle of the body

22

than they are for AFF1. At the parallel midsection of the two bodies both cp and cf arevery similar. AFF1 has a higher maximum value of cf which is shown in figure 4.12 (b).The e!ects of the di!erent stern configurations is clearly shown in both plots in figure 4.12,at x/L > 0.7 is the Joubert curves smooth while AFF1 has large gradients.

(a) (b)

Figure 4.12: Coe"cients cp and cf plotted for AFF1 and Joubert, obtained with kOmega.

Comparing the axial velocity and k profiles at x/L = 0.867, x/L = 0.933, x/L = 1 andx/L = 1.08 in figure 4.13 is it shown how the low velocity zone is bigger for AFF1 whenstill on the body and it is the reverse in the wake. Considering the size of the wake to betwo times the y-coordinate where the velocity is 0.8U& at x/L = 1.08 is the wake behindJoubert 25 % bigger than the AFF1 wake. The wake size divided by the diameter of eachbody is for Joubert 0.37 and for AFF1 0.34. The diameter of Joubert is 17 % larger than thediameter of AFF1. The k-profiles shows smother curves for Joubert but also the maximumvalue of k is larger for Joubert.

23

Figure 4.13: Axial velocity and turbulent kinetic energy profiles obtained with kOmega alongthe stern and behind the bodies for AFF1 and Joubert.

24

Chapter 5

Conclusions and discussion

Included in this work is some theory of turbulence modeling, the RANS equation is derivedfrom the Navier-Stokes equations and the kEpsilon turbulence model is presented. Alsoincluded is some theory on near wall flow and wall functions. A systematic simulationcampaign on the Joubert model is carried out in order to evaluate di!erent turbulencemodels. A comparison of the flow on the Joubert and AFF1 models is done by comparingthe results of simulations with the kOmega turbulence model on both bodies.

The results from the simulation campaign on the Joubert hull shows that the turbu-lence models kEpsilon, kOmega, kOmegaSST and RNGkEpsilon are agreeing well with theexperiments while the results of the other models are showing that they are not suited forsimulating flows of this type. The four best models even predicts a flow closer to the exper-iments than the LES simulations which are done on meshes with 18 million cells comparedto the 80 thousand cells used in these simulations. This is a comparison of a 2D mesh anda 3D mesh, 80 thousand cells on a 2D wedge mesh with 5 degrees angle would give 5.76million cells in a 3D mesh. So the resolution is considerably lower in the RANS-models.The studies done in [2] is however showing that the RANS models wouldn’t perform as wellwith a more complex model such as an appended submarine hull.

The conclusion is however that kEpsilon, kOmega, kOmegaSST and RNGkEpsilon aresuited for simulations like this. The prediction of cp for those models are very close tothe experiments as are the predictions of cf . The prediction of the velocity also showsgood agreement with the experiments. Especially in the wake where they are only a fewpercentages from the PIV data. Notable is how well they predict the velocity at x/L = 1where the flow is very hard to predict. The convergence of kEpsilon and RNGkEpsilon isfaster than for the two k ! " models, the residuals for kOmega and kOmegaSST are howeversmaller and they perform fewer iterations for each pseudo time step. This results in thatthe method recommendation for simulations like this is either one of the turbulence modelskEpsilon, kOmega, kOmegaSST and RNGkEpsilon.

Another study on RANS-turbulence models done in [5] is recommending realizablekE

for flow problems with significant impact of boundary layers. This is contradicting to theresults of this study where realizablekE were one of the worst performing turbulencemodel. The geometry used in [5] is quite di!erent from the geometries used here but theflow in both cases are flows with significant impact of the boundary layers so the di!erentresults are somewhat surprising. This may suggest that further evaluation of these RANSturbulence models is needed.

The comparison between the flow around Joubert and AFF1 shows no surprises. Thewake behind Joubert is slightly wider than the wake from AFF1. The Joubert body does

25

however have a larger diameter. Joubert was designed to produce small pressure gradientswhich is visible when comparing cp for the two bodies. The minimum of cp was also slightlysmaller for Joubert than it was for AFF1.

Further developing of this work could be to perform a mesh convergence study for simula-tions without wall functions. This was attempted but no convergence was reached for thosemodels. A parametric study of the stern to investigate how the angle and other parametersof the stern a!ects the flow would also be of great interest. The investigation could includeother turbulence models. Other things of interest to further look in to is the theory andimplementations of the turbulence models to understand why they behave di!erent. Themodels in this study showed sensitivity to the inflow conditions on the turbulent quantities,this sensitivity also needs to be further investigated.

26

Bibliography

[1] OpenFOAM version 1.6. www.openfoam.com, 2012.

[2] B. Anderson, M. Chapuis, L. Erm, C. Fureby, M. Giacobello, S. Henbest, D. Jones,M. Jones, C. Kumar, M. Liefvendahl, P. Manovski, D. Norrison, H. Quick, A. Snow-den, A. Valiy!, R. Widjaja, and B. Woodyatt. Experimental and computational inves-tigation of a generic conventional submarine hull form. In 29th Symposium on NavalHydrodynamics.

[3] M. Casey and T. Wintergerste. Best Practice Guidelines. ERCOFTAC, 2000.

[4] J.H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics. Springer,2002.

[5] Eric Furbo. Evaluation of RANS turbulence models for flow problems with significantimpact of boundary layers. Technical report, FOI, Swedish Defence Research Agency,2010.

[6] Nancy C. Groves, Thomas T. Huang, and Ming S. Chang. Geometric Characteristicsof DARPA SUBOFF models. David Taylor Research Center, 1989.

[7] P. N. Joubert. Some Aspects of Submarine Design Part 1. Hydrodynamics. DSTOPlatforms Sciences Laboratory, 2004.

[8] P. N. Joubert. Some Aspects of Submarine Design Part 2. Shape of a Submarine 2026.DSTO Platforms Sciences Laboratory, 2006.

[9] Horace Lamb. Hydrodynamics. Cambridge, 6th edition, 1932.

[10] S. B. Pope. Turbulent flows. Cambridge, 2000.

[11] Tsan-Hsing Shih, William W. Liou, Aamir Shabbir, Zhigang Yang, and Jiang Zhu. Anew k-epsilon eddy viscosity model for high reynolds number turbulent flows. ComputersFluids, 24(3):227–238, 1995.

[12] D. C. Wilcox. Turbulence Modeling for CFD, 3rd edition. DCW Industries, Inc., 2006.

27

Personal Reflection on Program-Level Learning Objectives

INSTRUCTIONS: Please consider the list of intended learning outcomes for the master program (and specialization in your civilingenjörsprogram) and reflect on your status in relation to them. Your task is to

• Estimate your proficiency using the numbered levels according to the Feisel-Schmitz taxonomy (at the yyy inside the table). See description below the table.

• Write a few lines on each outcome to indicate your status (at the Xxx inside the table). Try to indicate what learning activities you have been engaged in that made you climb the taxonomy.

Program Learning Objectives

The main objective of this program is to educate skilled engineers for industry and research institutions. The field is broad and multi-disciplinary with strong emphasis on systems engineering. A naval architect needs a variety of skills, knowledge and abilities to contribute to the complete processes of design, implementation and operation of marine vessels/systems which can be very large and complex systems, as well as deep understanding in some subjects. The program offers specialization within the predefined profiles Lightweight Structures, Fluid Mechanics, Sound & Vibration, Management, and Sustainable Development, as well as the possibility to individually tailor the profile. The subject hence is attractive also for students who are not devoted to work in the maritime sector and relevant for careers also in other fields.

Knowledge and understanding: A Master of Science in Naval Architecture shall demonstrate:

1

broad knowledge and understanding in naval architecture, scientific basis and proven experience, including knowledge of mathematics and natural sciences, substantially deeper knowledge in certain parts of the field, and deeper insight into current research and development work.

I estimate my Feisel-Schmitz level: 4

I belive that I through the education has got a broad knowledge of naval architecture and a deeper insight in hydrodynamics and fluid mechanics through my masters thesis.

2

deeper methodological knowledge in naval architecture.

I estimate my Feisel-Schmitz level: 4

The courses in Naval Architecture gave me much knowledge here.

Skills and abilities: A Master of Science in Naval Architecture shall demonstrate:

3

ability to, from a holistic perspective, critically, independently and creatively identify, formulate and deal with complex issues,

I estimate my Feisel-Schmitz level: 4 I learned a lot of this in the design course.

4

an ability to create, analyze and critically evaluate different technical solutions.

I estimate my Feisel-Schmitz level: 4

Same here, the design course made me realize how hard this could be and I have

a good ability to do this.

5

ability to plan and, using appropriate methods, carry out advanced tasks within specified parameters and to evaluate this work.

I estimate my Feisel-Schmitz level: 5 This has been a key part in many courses in KTH and I think I have done well.

6

skills required to participate in research and development work or to work independently in other advanced contexts so as to contribute to the development of knowledge.

I estimate my Feisel-Schmitz level: 5 My thesis included research and I learned a lot about it there.

7

ability to critically and systematically integrate knowledge,

I estimate my Feisel-Schmitz level: 5

This was an important part in all courses in Naval architecture and I did well in most courses there.

8

ability to analyze, assess and deal with complex phenomena, issues and situations, and to model, simulate, predict and evaluate events even on the basis of limited information.

I estimate my Feisel-Schmitz level: 5

This has been a big part of my masters thesis.

9

ability to develop, design and operate products, processes and systems taking into account people’s situations and needs and society’s objectives for economically, socially and ecologically sustainable development.

I estimate my Feisel-Schmitz level: 4

Not my strong part but I understand the importance of economics and such.

10

ability to engage and contribute in teamwork and cooperation in groups of varying composition. I estimate my Feisel-Schmitz level: 5

I work very well in groups.

11

ability to clearly present and discuss conclusions and the knowledge and arguments behind them, in dialogue with different groups, orally and in writing, in national and international contexts

I estimate my Feisel-Schmitz level: 5 I consider myself to be a good presenter in both Swedish and English.

Judgment & approach: A Master of Science in Naval Architecture shall demonstrate:

12

ability to make assessments in the main field of study, taking into account relevant scientific, social and ethical aspects,

I estimate my Feisel-Schmitz level: 5

Doing my thesis for the Swedish defence made me think a lot about and discuss social and ethical issues making good at this.

13

awareness of ethical aspects of research and development work I estimate my Feisel-Schmitz level: 5

Same thing here, doing my thesis for the Swedish defence made me very aware of the ethical aspects of the work.

14 insight into the potential and limitations of technology and science, its role in society and people’s responsibility for how it is used, including social and

economic aspects, as well as environmental and work environment aspects.

I estimate my Feisel-Schmitz level: 5

I have good understanding in this area.

15

ability to identify need for further knowledge and to take responsibility for continuously upgrading personal knowledge and capabilities.

I estimate my Feisel-Schmitz level: 5

I do not consider my self to be finished with my education, doing the thesis made me realize that there is much more to learn and I am eager to do it.

Feisel-Schmitz taxonomy

The Feisel-Schmitz taxonomy of educational objectives is used to describe the level of proficiency after participating in a course or program expressed in measurable observable formats (instructional objectives). The numbers range from the lowest level (1) to the highest level (5).

5. Judge (värdera): To be able to critically evaluate multiple solutions and select an optimum solution.

4. Solve (lösa problem): Characterize, analyze, and synthesize to model a system (provide. appropriate assumptions)

3. Explain (förklara): Be able to state the outcome/concept in their own words.

2. Compute (räkna typtal): Follow rules and procedures (substitute quantities correctly into equations and arrive at a correct result, Plug & Chug).

1. Define (återge): State the definition of the concept or is able to describe in a qualitative or quantitative manner.