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 is
written for the Center for Naval Architecture at the Royal Institute of Technology KTH. The
objective is to perform a systematic simulation campaign of the flow around an axisymmetric
streamlined body in order to evaluate several RANS-based turbulence models implemented
in the open source CFD software package OpenFOAM. Of most interest is the low velocity
zone at the stern and the wake behind the body. The results are compared with experimental
flow measurements and with data obtained by LES, in order to evaluate how accurately the
flow is predicted by each turbulence model. The study leads to method recommendations
for this type of flows. Simulation results for two different bodies are also compared in order
to investigate how the different shapes affect the flow. Also included is an overview of the
turbulence modeling theory behind the RANS-methods which are employed. The results
demonstrate that the turbulence models k − ε, RNG k − ε, k − ω and k − ω SST are suitable
for simulation of this class of flows and provide a good prediction of the mean flow around
and 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 utföra en systematisk simuleringskampanj av flödet kring en axisymmetrisk strömlinjeformad
kropp för att utvärdera en mängd olika RANS-baserade turbulensmodeller implementerade
i CFD programvarupaketet OpenFOAM. Låghastighetszonen vid aktern och vaken bakom
kroppen är av extra stort intresse. Resultaten jämförs med vindtunnelsexperiment samt med
simuleringsresultat som erhållits med LES, för att sedan kunna utvärdera hur väl varje turbulensmodell kan förutsäga flödet. Studierna leder fram till en metodrekommendation för
den här typen av flöden. En jämförelse mellan flödet kring två olika kroppar genomförs
också för att utreda hur de olika geometrierna påverkar flödet. En översikt av teorin bakom
RANS-metoderna som använts är också inkluderat i rapporten. Resultaten visar att turbulensmodellerna k − ε, RNG k − �, k − ω och k − ω SST är lämpliga för simulering av denna
typ 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 the
subject for this thesis and taking his time to help me through the whole project, this work
wouldn’t have been possible without him.
Contents
1 Introduction
1
2 Methods
2.1 Reynolds Averaged Navier-Stokes equations
2.2 k − ε turbulence model . . . . . . . . . . . .
2.3 Near wall treatment . . . . . . . . . . . . .
2.3.1 The log law . . . . . . . . . . . . . .
2.3.2 Wall functions . . . . . . . . . . . .
2.4 Other turbulence models . . . . . . . . . . .
2.4.1 Linear eddy viscosity models . . . .
2.4.2 Non-linear eddy viscosity models . .
2.4.3 Stress-Transport Models . . . . . . .
2.5 Numerical methods . . . . . . . . . . . . . .
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2
2
3
4
5
6
7
7
8
8
9
3 Simulation campaign
3.1 The models . . . . . . . . . . . . . . .
3.2 The mesh . . . . . . . . . . . . . . . .
3.3 Boundary conditions and initial values
3.4 Simulations and calculated quantities .
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10
10
11
12
13
4 Results
4.1 Validation data . . . . . . . . . . . .
4.2 Results from the Joubert simulations
4.2.1 Comparison with LES . . . .
4.3 Results from the AFF1 simulation .
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15
15
16
20
22
5 Conclusions and discussion
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25
Chapter 1
Introduction
Computational Fluid Dynamics (CFD) provides a way to simulate and predict the flow of
a fluid. This is of great interest in many applications such as car design, meteorology, ship
design and so on. Understanding the complex turbulent flow around a ship or a submarine
is crucial when designing the ship. Quantities of interest could be friction and pressure
coefficients to calculate the water resistance, the boundary layer thickness, the velocity field
and turbulence levels along the body and behind it. The low velocity zone at the stern is of
special interest since it will largely affect the propulsor. The turbulent flow at high Reynolds
number in an incompressible fluid around a ship is too complex to solve the governing fluid
dynamics equations (the Navier-Stokes equation, [9]) directly, this makes it necessary to
model the turbulence in some way. RANS (Reynolds averaged Navier-Stokes equations) is
an 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 axisymmetric
streamlined body using RANS-based methods in the software package OpenFOAM, [1],
that is the most widely used open source software for CFD. This axisymmetric streamlined
body could represent an unappended submarine hull or a torpedo. Studies of this type are
important when designing a submarine for many reasons such as to lower the resistance or to
reduce turbulence levels at the propulsor to lower the noise levels. Simulations on a generic
unappended submarine hull with several turbulence models are carried out and the results
are validated against experiments such as Particle Imaging Velocimetry (PIV) measurements
and other types of simulations, Large Eddy Simulations (LES). This investigation will give
an understanding on how well the in OpenFOAM implemented RANS turbulence models
predicts the flow on models like this. This will in the end give a method recommendation
for simulations of this kind. One of the best performing turbulence model is also compared
with a simulation on another hull geometry to investigate how the different geometries affect
the flow.
The report starts with an overview of the turbulence modeling theory behind the RANSmethods employed in this work in chapter 2. Further on in chapter 3, there are descriptions
on the two models used. The meshes, the boundary conditions and the calculated quantities
is also described there. Chapter 4 presents the results from the simulations and compare
them with the validation data. Conclusions is drawn and discussion on the results is done
in 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. Written
for incompressible constant-property Newtonian fluids they are
DUj
∂Uj
∂Uj
1
≡
+ Ui
= − ∇p + ν∇2 Uj
Dt
∂t
∂xi
ρ
and the incompressible continuity equation is
∂Uj
=0
∂xj
(2.1)
(2.2)
where xj are the Cartesian coordinates, Uj is the velocity in the xj -direction, ρ is the
density of the fluid, ν is the kinematic viscosity, p is the pressure and t is the time. Here
and throughout this report is the Einstein summation convention used.
The RANS (Reynolds Averaged Navier-Stokes) equation is a way to model turbulent
flows by averaging (time or ensemble averaging) the Navier-Stokes equations. The starting
point of the derivation of the RANS equations is to decompose the velocity into its mean,
U j , and the fluctuation, uj .
Uj = U j + u j .
(2.3)
Equation (2.3) in equation (2.2) gives the mean conservation of mass equation for incompressible flows
∂U j
= 0.
∂xj
(2.4)
Inserting equation (2.3) in equation (2.1) and averaging leads to the mean-momentum equation, also called the Reynolds equation or the Reynolds-averaged Navier-Stokes equation
(RANS)
∂U j
∂ui uj
1 ∂p
D̄U j
∂U j
= ν∇2 U j −
−
.
≡
+ Ui
∂t
∂x
∂x
ρ ∂xj
D̄t
i
i
2
(2.5)
Equation (2.5) can be re-written as
$
! "
#
∂U i
∂U j
D̄U j
∂
µ
+
− pδij − ρui uj .
ρ
=
∂xi
∂xj
∂xi
D̄t
(2.6)
where the terms inside the square brackets represents three stresses. The first term is the
viscous stress, −pδij is the isotropic stress from the mean pressure field and −ρui uj is a
stress arising from the fluctuating velocity. The last term is called Reynolds stresses. This
term without −ρ, i.e. ui uj will also be referred to as Reynolds stresses in this report. Since
the equations are averaged the quantities calculated with RANS methods are statistical, [10].
2.2
k − ε turbulence model
The number of unknowns in the Navier-Stokes equations, (2.1), is four, the three velocity
components and the pressure. The Reynolds stress tensor, ui uj , is a symmetric tensor
and 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 components
of the RANS equation, (2.5), and the mean conservation of mass equation, (2.4). This
presents a closure problem and to solve it modeling is needed. This is why turbulence
modeling is necessary. There are many different turbulence models available for RANS and
one commonly used is the k − ε model described here.
According to the turbulent-viscosity hypothesis introduced by Boussinesq, [10], is the
Reynolds stress
"
#
2
∂U i
∂U j
ui uj = kδij − νT
+
(2.7)
3
∂xj
∂xi
where νT is the turbulent viscosity and k the turbulent kinetic energy defined as
k≡
1
ui ui .
2
(2.8)
The trace of the Reynolds stress tensor is 2 k and the term 32 kδij in equation (2.7) is the
isotropic stress. The turbulent-viscosity hypothesis states that the mean rate-of-strain
"
#
∂U i
∂U j
Sij =
+
(2.9)
∂xj
∂xi
via the turbulent viscosity is linearly related to the Reynolds-stress anisotropy tensor, aij ,
defined by
2
aij ≡ ui uj − kδij .
3
Equation (2.10) in equation (2.7) gives that linear relationship by
"
#
∂Ui
∂Uj
.
aij = −νT
+
∂xj
∂xi
(2.10)
(2.11)
In the k − ε model the turbulent viscosity is specified as
νT = Cµ
3
k2
ε
(2.12)
where ε is the dissipation of turbulent kinetic energy and Cµ is a model constant. The k − ε
model is a two equations turbulence model consisting of two model transport equations for
k and ε. The exact model transport equation for k, derived in [10], is
∂k
+ U j · ∇k = −∇ · Tj! + P − ε
∂t
(2.13)
1
1
ui uj uj + ui p! − 2νuj sij
2
ρ
(2.14)
where
Tj! ≡
is the energy flux,
P ≡ −ui uj
∂U i
∂xj
(2.15)
is the production of turbulent kinetic energy,
ε ≡ 2νsij sij
(2.16)
is the dissipation of the turbulent kinetic energy and
"
#
∂uj
1 ∂ui
+
sij =
2 ∂xj
∂xi
(2.17)
is the fluctuating rate of strain. Equation (2.13) however add a lot of unknowns, therefore
the energy flux, Tj! , is modeled with a gradient-diffusion hypothesis as
T! = −
νT
∇k
σk
(2.18)
where σk is the turbulent Prantl number for kinetic energy. Physically this asserts that there
is a flux of turbulent kinetic energy down the gradient of k. The standard model equation
for ε is
"
#
D̄ε
Pε
ε2
νT
∇ε + Cε1
− Cε2
(2.19)
= ∇·
σε
k
k
D̄t
where σε , Cε1 and Cε2 are model constants. The model equation for ε is best viewed as being
completely empirical, [10]. The total number of model constants is five and their standard
values according to [10] are
Cµ = 0.09,
Cε1 = 1.44,
Cε2 = 1.92,
σk = 1.0,
σε = 1.3.
(2.20)
The equations for k, ε and the specification of νT is what forms the k − ε turbulence model.
2.3
Near wall treatment
The flow close to walls are in many ways more complicated than free shear flows, therefore
consideration is needed for flows close to walls. Considering a fully developed 2D flow close
to a wall, x1 = x is the flow direction and x2 = y is normal to the surface, with the
freestream mean velocity, U , in the positive x-direction. The total shear stress, τ , consists
of the viscous stress and the Reynolds stress
4
τ = ρν
dU
− ρuv
dy
(2.21)
where u and v are the fluctuations in the x- and y-directions. At the wall the no-slip
condition implies that the Reynolds stress is zero, then the viscous stress is the only stress
and the shear wall stress is
%
dU %%
τw = ρν
.
(2.22)
dy %y=0
Close to the wall the viscous stress is the dominating stress in contrary to the free flow where
the viscous stress can be neglected compared to the Reynolds stress. Close to the wall, ν
and τw are important parameters and from these appropriate velocity and length scales can
be defined as
&
τw
(2.23)
uτ ≡
ρ
which is the friction velocity,
δν ≡ ν
&
ρ
ν
=
τw
uτ
(2.24)
which is the viscous length scale and
y+ ≡
y
uτ y
=
δν
ν
(2.25)
which is viscous lengths or wall units. Different layers in the flow close to the wall are defined
by y + . At y + < 5 is the viscous sub layer where the Reynolds stress is negligible, y + < 50
is the viscous wall region where there is a direct effect of the viscosity on the shear stress
and y + > 50 is outer layer where the effect 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
#
"
uτ
y y
dU
(2.26)
=
Φ
,
dy
y
δν δ
where Φ is a dimensionless function and δ is an appropriate length scale much larger than
δν . Introducing the function
" #
"
#
y
y y
ΦI
= ylim Φ
(2.27)
,
δν
δν δ
δ →0
can equation (2.26) close to the wall be written as
" #
dU
uτ
y
y
, f or
=
ΦI
<< 1.
dy
y
δν
δ
(2.28)
Defining u+ (y + ), the mean velocity normalized by the friction velocity as
u+ =
5
U
uτ
(2.29)
and using equation (2.25) in equation (2.28) is
1
du+
= + ΦI (y + ).
dy +
y
(2.30)
When y + > 30 the distance to the wall is large enough to assume that the dependence of
viscosity in ΦI vanishes making it constant, κ−1 , κ is the von Kármán constant and this in
equation (2.30) gives
du+
1
=
.
+
dy
κy +
(2.31)
Integration of equation (2.31) gives
1
ln y + + B.
(2.32)
κ
where B is a constant from the integration. This is the log law where κ = 0.41 and B = 5.2
are the usual values of the constants, they alter some in different literature but are generally
close 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].
u+ =
2.3.2
Wall functions
Since the profiles of U and ε are steep close to walls allot of the computational effort will
have to be devoted to the near wall region making the computations very expensive. Wall
functions are used to lower the cost of the computations. The idea is to apply a boundary
condition some distance away from the wall in the log law region in order to avoid solving
the turbulence-model equations close to the wall. The wall function boundary conditions
are applied at the point y = yp where y + ≈ 50. The index p will indicate that the quantities
are evaluated at yp . According to the turbulent viscosity hypothesis is
|uv|
=
k
#1/2
"
P
.
Cµ
ε
(2.33)
In the region y + ≈ 50 DNS calculations shows there is a balance between the dissipation
and production of turbulent kinetic energy, also in the overlap region (50 < y + < 1000) the
difference between −uv and u2τ is small, [10]. Then it holds that
−uv = Cµ1/2 k = u2τ .
Near the wall all velocity gradients except
energy then can be written as
dU
dy
(2.34)
can be neglected, the production of kinetic
P = −uv
∂U
∂y
(2.35)
and in the log law region it holds
uτ
dU
=
.
dy
κy
(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
ε=
u3τ
.
κy
(2.37)
By combining equations (2.34) and (2.37) at the position yp the boundary condition for ε is
3/4 3/2
εp =
Cµ kp
.
κyp
(2.38)
The boundary condition of k is set to zero normal gradient. The turbulent viscosity hypothesis gives
−uv = νT
∂U
.
∂y
(2.39)
Using equations (2.39), (2.34) and (2.35), the production term, at yp , is calculated as
1/4 1/2
P p = νT
Cµ kp
κyp
∂U (yp )
∂y
(2.40)
and y + at yp is
1/4 1/2
yp+ =
2.4
Cµ kp yp
.
ν
(2.41)
Other turbulence models
The turbulence models used in this work are the k − ε, realizable k − ε, k − ω, k − ω SST ,
RNG k − ε, Lien cubic k − ε, Non-linear Shih k − ε and LRR Reynolds stress transport
models. These turbulence models can be divided into three categories, linear eddy viscosity
models, 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 − ε described in
section 2.2, realizable k − ε, RNG k − ε, k − ω and k − ω SST models.
If the strain rate becomes to large when using the standard k − ε model it will become
non realizable, since the normal stresses then can become negative. To prevent this the
realizable k − ε turbulence model relate the constant Cµ to the main strain rate, Cµ is not
treated as an constant and is
Cµ =
1
A0 + As U (∗) kε
(2.42)
where A0 is a constant, As and U (∗) depends on the main strain rate and are determined
according to [11].
The k − ω model models νT and k in the same way as the k − ε turbulence model, the
difference is that the specific dissipation rate is modeled with ω ≡ ε/k instead of with ε.
This gives the specific dissipation rate as
"
#
D̄ω
Pω
2νT
νT
= ∇·
∇ω + (Cω1 − 1)
− (Cω2 − 1)ω 2 +
∇ω · ∇k
(2.43)
σω
k
σω k
D̄t
7
where σω = σk . The interpretation of ω differs but it can be described as a frequency
characteristic of the turbulence, the RMS fluctuating vorticity or just the ratio of ε and k.
The k − ω model is designed to give better results than k − ε in the viscous near-wall region
and in how it accounts for the effects of stream wise pressure gradients. However it can be
sensitive to free stream boundary conditions.
The k − ω SST (shear stress transport model) works as a combination of the regular
k − ω close to walls but switches behavior to k − ε in the free stream. This is done by
multiplying the last term in equation (2.43) to a blending function that is zero close to the
wall, so the model works as k − ω close to walls, and it is unity far from the wall making
the model correspond to the standard k − ε in the free stream, [10].
The RNG k − ε turbulence model make use of renormalization group theory to get a
modified k − ε model where the equations for νT , k and ε are the same as in the standard
k − ε model. The difference lies in the constants, Cε2 is not constant as in equation (2.19).
In the RNG k − ε Cε2 is defined as
Cε2 ≡ C̃ε2 +
k'
Cµ λ3 (1 − λ/λ0 )
, λ≡
2Sij 2Sij
3
(1 + βλ )
ε
(2.44)
where
C̃ε2 = 1.68, Cε1 = 1.42, Cµ = 0.085, β = 0.012, λ0 = 4.38
(2.45)
are the closure coefficients. Notable is also that the other constants in equation (2.19) is
somewhat altered in the RNG k − ε model.
2.4.2
Non-linear eddy viscosity models
The Lien cubic k − ε and Non-linear Shih k − ε models belongs the category non-linear
eddy viscosity models. The non-linear eddy viscosity models relate the turbulent stresses
algebraically to the rate of strain and include higher order quadratic and cubic terms. They
often give better predictions in reattachment areas where the linear models sometimes fail
to 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 second
moment closure model, the exact equation for the Reynolds stress-tensor is used and each
term of the equation is modeled by it self. The model is more complex than the two equation
models and demands more computational efforts but it corrects some of the Boussinesq’s
approximation shortcomings. The stress transport model includes effects of flow history in
a more realistic way than the two equation models since it automatically accounts for the
convection and diffusion of the Reynolds stress tensor. It also includes effects of streamline
curvature and it behaves properly for flows with sudden changes in the strain rate. The
LRR Reynolds stress transport model belongs to this category of turbulence models. It is
the most widely used and tested stress transport model that is based on the ε equation.
There is however a difference in the dissipation rate equation compared to what it is in the
k − ε model, [12].
8
2.5
Numerical methods
Since the main objective of this work is to analyze the different turbulence models the
numerical methods and their implementation into OpenFOAM will just be briefly described
in 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 a
scalar φ in a given velocity field u. The integral form of the equation is
(
(
d
ρφdV +
ρφu · ndS = 0
(2.46)
dt V
S
where V is a control volume, S is the control volume surface and n is the unit normal vector
of 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 summed
for all of the CVs the global conservation equations is obtained since the surface integrals
of the inner CV faces cancels each other. This is one of the advantages of the finite volume
method since the global conservation is built in from the beginning. At the centre of each
CV 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 nodal
values. The surface and volume integrals are approximated using some appropriate method
for which there are many schemes available. As a result of this an algebraic equation for
each CV is obtained containing values of neighboring nodes. The method can accommodate
any type of grid and is suitable for complex geometries. A detailed description of the FV
method can be found in [4].
When solving a stationary problem with RANS the problem can be regarded as unsteady
until it reaches steady state. In order to reach or speed up convergence can it be necessary
to 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 φ on the nth
outer iteration at the typical point P
)
AP φnP
Al φnl = QP
(2.47)
l
where Q contains the terms not depending on φn . The coefficients Al and Q may involve
φn−1 . When solving this linear equation iteratively the changes in φ between the outer
iterations can become to large causing instability. So φn is only allowed to change a fraction
of the would be change according to
φn = φn−1 + αφ (φnew − φn−1 )
new
(2.48)
where φ
is the solution to equation (2.47) and αφ is the under-relaxation factor that
satisfies 0 < αφ < 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 components
with the pressure from the last iteration and then solving for the pressure using a discrete
Poisson equation, [4]. The velocity and pressure fields now satisfy the continuity equation
but 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 momentum
equations are fulfilled within a certain tolerance. The Semi Implicit Method for Pressure
Linked Equations (SIMPLE) is an algorithm doing this. The simulations in this work are
done using the in OpenFOAM implemented SIMPLE algorithm. A detailed description of
the 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 generic
conventional submarine hull described in [8]. It is the unappended hull of that design which
will be referred to as the Joubert model after the author of the design reports. The Joubert
model 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 while
still carrying out all the normal functions for a submarine. This model was chosen because
there 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 SUBOFF
configuration AFF1 [6]. This model will be referred to as AFF1 and is also shown in figure
3.1. The SUBOFF project is one of the few studies of conventional generic submarine hulls
made in open literature and it is meant to provide a way to compare numerical simulations
of axisymmetric hulls with experimental data for CFD users. The AFF1 model used here is
re-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 geometry
is at the bottom.
10
Total length
Diameter
Volume displacement
Length parallel midsection
Length forebody
Length stern
Diameter/Length
Largest angle stern
Length/width
Block coefficient
Prismatic coefficient
L
D
∇
Lms /L
Lf b /L
Ls /L
D/L
αmax
L/D
∇/LD2
∇/Lπr2
Joubert
1.35 m
0.185 m
0.0283 m3
0.451
0.228
0.321
0.137
21.8◦
7.31
0.615
0.783
AFF1
1.35 m
0.157 m
0.0208 m3
0.512
0.233
0.255
0.117
20.1◦
8.57
0.622
0.792
Table 3.1: Main dimensions of the Joubert and AFF1 models.
The differences in the two bodies can be seen in figure 3.2 where profiles of both bodies
are 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 is
the length of the parallel midsection, Lf b is the length of the forebody Ls is the length of
the stern and αmax is the largest angle between a tangent of the body the stern and the
parallel midsection. For AFF1 is the afterbody cap not considered for αmax . The coordinate
system 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 is
in the x-direction and v is in the y-direction. The differences between the two bodies are
mainly in the different stern configurations and the length to diameter ratio but there are
also differences in the bow configurations. The Joubert body is thicker and have a longer
stern. The stern of the Joubert body has a more conical shape with an angle that is getting
larger all the way to the trailing edge while the stern of the AFF1 body is composed of two
parts, it starts of with a conical shape and then turns into an afterbody cap with a round
trailing edge. Joubert have a smoother transition from the parallel midsection to the stern
and the tapering of the tail is also smother. The forebodies are almost of the same length
and 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.
y/L
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
x/L
0.6
0.7
0.8
0.9
1
Figure 3.2: A comparison of the Joubert and AFF1 models. Joubert in blue and AFF1 in
green.
3.2
The mesh
The meshes were created with the OpenFOAM-utility blockMesh where the model is built
in 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 effects are the boundaries in front of and over the
bodies placed one body length away from the bodies and the boundaries behind the the
bodies 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. Each
block had to be graded manually and the shapes of the blocks are visible in figure 3.3. The
cells closest to the body were designed to give an y + ≈ 40 where y + is calculated according
to equation (2.25). The y-coordinate in equation (2.25) is not the global y-coordinate, it
is the distance in the normal direction of the body to the center of the cell closest to the
body. The value of y + varies along the bodies but at the parallel midsection it is 42 on the
Joubert 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 the
no-slip condition, at the inlet is the velocity in the x-direction U∞ = 60m/s and zero in all
other directions. At the outlet and far field the boundary conditions are set to zero normal
gradient. The pressure boundary conditions are set to zero at the outlet and zero normal
gradient at the other boundaries. For all the models using k to describe the turbulence is
it calculated according to equation (2.8) with the assumption that k is isotropic at the inlet
and that the fluctuations of the inlet velocity is 3%, this gives k as
3
(0.03U∞ )2 .
(3.1)
2
The assumption that the fluctuations are 3% of the inlet velocity comes from [2] where it is
estimated that the wind tunnel in which the validation data is measured has this turbulence
level. At the far field and the outlet is k set to zero normal gradient. The initial value of k in
the wall function is also calculated according to equation (3.1). To estimate the initial and
inlet value for ε is it assumed that the ratio of turbulent viscosity to the laminar viscosity
ν is five. Then ε is calculated as
k=
12
k2
(3.2)
5ν
with Cµ = 0.09 as in (2.20) and k from equation (3.1). Equation (3.2) is also used to calculate
the initial value for the ε wall function. The outlet and far field boundary conditions for ε
is set to zero normal gradient. Using the definition ω ≡ ε/k and equation (3.2) are the inlet
and initial value for the wall function of ω calculated as
ε = Cµ
ω=
ε
k
= Cµ .
k
5ν
(3.3)
In all turbulence models except the LRR model are k and / or ω included, the LRR turbulence model includes only ε and the Reynolds stress tensor. In this model the initial data
for ε is taken from a converged k − ε simulation and the Reynolds stress tensor is calculated
by OpenFOAM from the k − ε simulation as well. The pressure and velocity initial data
and boundary conditions are also taken from the converged k − ε simulation. All the models
except the LRR model are started with zero velocity in the initial flow field. The boundary
conditions in the azimuthal directions are set to the OpenFOAM boundary condition wedge
which 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 model
is 1.35 m and the fluid is air, ν = 1.5 ·10−5 m2 /s giving ReL = U∞ L/ν = 5.4 ·106 . Presented
in table 3.2 is the simulation campaign. The OpenFOAM keyword is the word specifying
the turbulence model in OpenFOAM. From here on will the turbulence models be refereed
to as their OpenFOAM keyword in table 3.2.
Model
Joubert
Joubert
Joubert
Joubert
Joubert
Joubert
Joubert
Joubert
AFF1
Turbulence model
Standard k − ε
Realizable k − ε
k−ω
k − ω SST
RNG k − ε
Lien cubic k − ε
Non-linear Shih k − ε
LRR
k−ω
OpenFOAM keyword
kEpsilon
realizablekE
kOmega
kOmegaSST
RNGkEpsilon
LienCubicKE
NonlinearKEShih
LRR
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 recommendation for flows of this type. The main focus is to investigate the velocity at the stern
and behind the bodies. This is done by plotting the axial velocity profiles at several places
of the body and comparing it with experiments and LES data. Other quantities of interest
are the pressure coefficient defined as
cp =
|p − pref |
1
2
2 ρU∞
13
(3.4)
where p is the pressure and pref is a reference pressure that here is the ambient pressure set
to zero, and the friction coefficient
τw
cf = 1 2
(3.5)
2 ρU∞
where τw is the wall shear stress. The turbulent kinetic energy will also be analyses. The
quantities 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 validation data. The quantities will be analyzed in order of complexity and models showing
poor prediction of the flow will successively be removed narrowing down the analyze to the
models that predicts the flow well. The validation data used is described in section 4.1
followed by the results from the simulations on the Joubert model in section 4.2. In section
4.3 is the results from one simulation on the AFF1 model compared with a simulation with
the same turbulence model on Joubert.
4.1
Validation data
Available for validation of the results are data from experiments and other simulations on
the Joubert body done in [2]. The experiments were conducted in the DSTO Low Speed
Wind Tunnel in Melbourne on a 1.35 m long aluminum. The main validation data for the
velocity is Particle Imaging Velocimetry measurements, PIV. These measurements are done
using laser to avoid inserting objects into the flow and thereby disturbing it. The PIV
measurements were done at the stern and in the wake region. A more detailed description
on how the PIV measurements were carried out is found in [2]. There are uncertainties in at
what y-coordinate the lineplots from the PIV starts, they do however give a picture on how
the velocity profiles should look. Measurements of the surface pressure were done on the
model with static pressure taps along the centreline on top of the body. The static pressure
was measured at the taps and the pressure coefficient is calculated as
cp =
ptap − ps
pT − ps
(4.1)
where ptap is the measured static pressure at the tap, ps is the freestream static pressure and
pT 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 reference
pressure pref in equation 3.4. The friction coefficient was measured using the Preston tube
method. This method requires a turbulent boundary layer to work so the skin friction where
the boundary layer is laminar is not correct. It is however possible to determine where the
transition from laminar to turbulent occurs by looking at where it is a discontinuity in the
graph when plotting cf against the axial position. In some of the experiments boundary
layer tripping devises were used. In those cases the tripping devise consisted of either a
circumferential 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 the
experiments are available in [2].
The LES data was provided from [2] and are a type of near wall modeled LES of which
three types have been used as validation. LES is based on separation of the scales in the flow
utilizing spatial filtering. Scales in the flow larger than a certain filter width is resolved and
the effect of the scales below the filter width are modeled. The types of LES simulations used
as 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 Model
Spallart Allmaras here referred to as MMSA. The simulations were done on grids of 18
million 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 number
of outer iterations needed for each model to converge.
Turbulence model
Standard k − ε
Realizable k − ε
k−ω
k − ω SST
RNG k − ε
Lien cubic k − ε
Non-linear Shih k − ε
LRR
Convergence
1000 iterations
3000 iterations
2000 iterations
4000 iterations
1000 iterations
Not reached
10 000 iterations
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 pressure
and 10−8 for the other variables and no relative tolerances was used. The solutions were
considered to be converged when the initial residuals reached a value that didn’t improve
with more iterations and this value was considered to be low enough. Figure 4.1 shows
the initial residuals for each outer iteration as functions of outer iterations for the best and
the worst cases considered to be converged among the simulations. The residuals in figure
4.1 are the extremes and most of the other models have converged much better then the
realizableKE. The LienCubicKE model never reach convergence.
Figure 4.2 shows the axial velocity from the kEpsilon simulation. Visible are the stagnation point in front of the body, two zones with high velocity over the bends at the end of
the forebody and at the beginning of the stern. The boundary layer is growing along the
body and there is a low velocity zone at the stern and behind the body. The RANS models
assume the flow to be fully turbulent everywhere which it is not, the area of most interest
is 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 according to equation (2.25) in which the y-coordinate is not the global y-coordinate, it is the
distance 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 low
y + ≈ 5, grows a bit and the settles on a value around 40, realizableKE and kOmega starts
high, 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 on
the Joubert model.
low until approximately x/L = 0.15 where y + rapidly grows and then settles at 40. It seems
like some threshold is reached there for RNGkEpsilon and LRR, this has however not been
further investigated. The LRR model have a slightly higher y + than the other models along
the body and at the stern. The realizableKE curve is somewhat unsteady in the beginning
and the end of the body. Also notable is that the two k − ω models, kOmega and kOmegaSST
shows different behaviour in the beginning of the body but then has almost identical values
from 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) is
cp plotted against the length of the body. All the models except realizableKE predicts cp
close to the experiments until x/L = 0.9 where they miss the separation occurring according
to the experimental data. The realizableKE model differs from the other models and
the experiments after x/L = 0.7, it is however the only model that predicts a separation
but it is predicted to early. The predicted cp at x/L > 0.4 is somewhat higher than the
experimental cp . Figure 4.4 (b) shows cf along the body. All the models predicts cf within
20 % from the experimental data between x/L = 0.2 to 0.8 except the LRR model which
severely overpredicts it. Since the flow is expected to be boundary layer dominated, a model
that differs 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 and
NonlinearKEShih which underpredicts it and then all of a sudden overpredicts it. This
behaviour is also seen in figure 4.3 where there seem to be some type of threshold for these
models.
(a)
(b)
Figure 4.4: Coefficients cp and cf for the turbulence models and experimental data. The
labels 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 turbulence
models and the PIV-data. The positions of the profiles is shown at the top of figure 4.5.
18
The realizableKE model differ much from the other profiles and will further on not be
considered. As stated earlier is there some uncertainty in how far away from the wall the first
PIV-data is located. At the top four profiles is it shown that the RANS models overpredicts
the 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 larger
x/L. All the models behaves more or less the same there except NonlinearKEShih whom
at places overpredict the velocity with up to 10 % compared to the other models and more
compared to the PIV data. The NonlinearKEShih model shows the same behaviour in the
wake where the other models show very good similarity with the PIV-data where both the
gradient and the magnitude of the velocity are well matched. Due to the poor predictions
by the NonlinearKEShih model is it excluded from the further results. Zooming in on the
profiles at x/L = 0.967 and x/L = 1.08 with the poor performing models removed as done in
figure 4.6 are the small differences between kEpsilon, kOmega, kOmegaSST and RNGkEpsilon
shown. The differences are never larger than 12 %.
y/L
0.1
0.05
0
0
0.2
0.4
0.6
x/L
0.8
1
1.2
Figure 4.5: Velocity profiles for all models and PIV-data along the stern and behind the
body 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, kOmegaSST
and RNGkEpsilon. These models are analysed further by plotting the kinetic turbulent
energy 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. Data
also 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 k
at higher y-coordinates. The two k − ω models shows very similar k-profiles. There are no
k-profiles available from the PIV in [2], there are however RMS data on the velocity avilable
describing 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 kEpsilon model with the poor performing
realizableKE model and the LES data from [2]. In figure 4.8 are cp and cf plotted for the
RANS models, LES models and experimental data. The kEpsilon model is clooser to the
experimental data than the LES models in predicting cp for x/L smaller than 0.9. For
x/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 higher
cp than the experiments. The LES models show much higher cf than the RANS models for
small x/L and then settles in the same level as the RANS and experimental data except for
LDKM which is underpredicting cf very much.
(a)
(b)
Figure 4.8: The coefficients cp and cf plotted for kEpsilon, realizableKE, LES and experiments. The labels are the same in both figures.
Comparing the velocity profiles of kEpsilon and realizableKE models with LES and
PIV at x/L = 0.9 and x/L = 0.967, the prediction of the velocity profiles for the kEpsilon
model are closer to the PIV than the LES models as shown in figure 4.9. The differences
between the LES models are large but the difference 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 the
kEpsilon model show very good agreement with the PIV data. It is in line with the PIV
data differing 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 does
a much better job predicting the velocity in the wake than the LES simulations which are
differing 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 model
is here presented and compared to the kOmega simulation on the Joubert model with the
purpose of investigating how the differences in the geometries affect the flow. The simulation
on 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 size
of 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 0.8 U∞ is the distance away from the
body in the y-direction 20% higher for AFF1 than for Joubert. The wake is however thicker
in 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 Joubert
to the right.
In figure 4.12 is cp and cf for the two models shown. Figure 4.12 (a) shows smaller
pressure gradients along the body for Joubert than for AFF1. Joubert also has a smaller
absolute 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 are
very similar. AFF1 has a higher maximum value of cf which is shown in figure 4.12 (b).
The effects of the different 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: Coefficients 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 and
x/L = 1.08 in figure 4.13 is it shown how the low velocity zone is bigger for AFF1 when
still on the body and it is the reverse in the wake. Considering the size of the wake to be
two times the y-coordinate where the velocity is 0.8 U∞ at x/L = 1.08 is the wake behind
Joubert 25 % bigger than the AFF1 wake. The wake size divided by the diameter of each
body is for Joubert 0.37 and for AFF1 0.34. The diameter of Joubert is 17 % larger than the
diameter of AFF1. The k-profiles shows smother curves for Joubert but also the maximum
value of k is larger for Joubert.
23
Figure 4.13: Axial velocity and turbulent kinetic energy profiles obtained with kOmega along
the 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 derived
from the Navier-Stokes equations and the kEpsilon turbulence model is presented. Also
included is some theory on near wall flow and wall functions. A systematic simulation
campaign on the Joubert model is carried out in order to evaluate different turbulence
models. A comparison of the flow on the Joubert and AFF1 models is done by comparing
the results of simulations with the kOmega turbulence model on both bodies.
The results from the simulation campaign on the Joubert hull shows that the turbulence models kEpsilon, kOmega, kOmegaSST and RNGkEpsilon are agreeing well with the
experiments while the results of the other models are showing that they are not suited for
simulating flows of this type. The four best models even predicts a flow closer to the experiments than the LES simulations which are done on meshes with 18 million cells compared
to the 80 thousand cells used in these simulations. This is a comparison of a 2D mesh and
a 3D mesh, 80 thousand cells on a 2D wedge mesh with 5 degrees angle would give 5.76
million 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 well
with a more complex model such as an appended submarine hull.
The conclusion is however that kEpsilon, kOmega, kOmegaSST and RNGkEpsilon are
suited for simulations like this. The prediction of cp for those models are very close to
the experiments as are the predictions of cf . The prediction of the velocity also shows
good agreement with the experiments. Especially in the wake where they are only a few
percentages from the PIV data. Notable is how well they predict the velocity at x/L = 1
where the flow is very hard to predict. The convergence of kEpsilon and RNGkEpsilon is
faster than for the two k − ω models, the residuals for kOmega and kOmegaSST are however
smaller and they perform fewer iterations for each pseudo time step. This results in that
the method recommendation for simulations like this is either one of the turbulence models
kEpsilon, 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 the
results of this study where realizablekE were one of the worst performing turbulence
model. The geometry used in [5] is quite different from the geometries used here but the
flow in both cases are flows with significant impact of the boundary layers so the different
results are somewhat surprising. This may suggest that further evaluation of these RANS
turbulence models is needed.
The comparison between the flow around Joubert and AFF1 shows no surprises. The
wake 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 gradients
which is visible when comparing cp for the two bodies. The minimum of cp was also slightly
smaller for Joubert than it was for AFF1.
Further developing of this work could be to perform a mesh convergence study for simulations without wall functions. This was attempted but no convergence was reached for those
models. A parametric study of the stern to investigate how the angle and other parameters
of the stern affects the flow would also be of great interest. The investigation could include
other turbulence models. Other things of interest to further look in to is the theory and
implementations of the turbulence models to understand why they behave different. The
models in this study showed sensitivity to the inflow conditions on the turbulent quantities,
this sensitivity also needs to be further investigated.
26
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Hydrodynamics.
[3] M. Casey and T. Wintergerste. Best Practice Guidelines. ERCOFTAC, 2000.
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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:
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.
1
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.
deeper methodological knowledge in naval architecture.
2
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:
ability to, from a holistic perspective, critically, independently and creatively
identify, formulate and deal with complex issues,
3
I estimate my Feisel-Schmitz level: 4
I learned a lot of this in the design course.
an ability to create, analyze and critically evaluate different technical solutions.
4
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.
ability to plan and, using appropriate methods, carry out advanced tasks within
specified parameters and to evaluate this work.
5
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.
ability to critically and systematically integrate knowledge,
7
8
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.
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.
ability to engage and contribute in teamwork and cooperation in groups of varying
composition.
10
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:
ability to make assessments in the main field of study, taking into account relevant
scientific, social and ethical aspects,
12
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.
awareness of ethical aspects of research and development work
13
14
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.
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.
ability to identify need for further knowledge and to take responsibility for
continuously upgrading personal knowledge and capabilities.
15
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.