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Fakulteten för hälsa, natur- och teknikvetenskap Miljö- och energisystem

Daniel Ahl

Analys av olika meshfunktioners inverkan på resultatet vid CFD-simulering av en marin propeller

Analysis of how different mesh functions influence the result in CFD-simulation of a marine propeller

Examensarbete 30 hp Civilingenjörsprogrammet Energi- och miljöteknik

Juni 2013

Handledare: Rolf-Erik Keck

Examinator: Roger Renström

Analysis of how different mesh functions influence the

results in CFD-simulation of a marine propeller

Reference : Johan Lundberg

Issue : 01

Date : 2013-06-27

Author : Daniel Ahl

Summary Rolls-Royce Hydrodynamic Research Center (RRHRC) has a long experience in propeller design and

analysis by means of both experimental model testing and numerical simulations. Traditionally,

experimental model testing has been the most common method but with the rapid development of

computer hardware, focus has shifted towards computational fluid dynamics (CFD). CFD offers

advantages such as greatly reduced design and analysis time, direct modification of the design in an

early stage and that a wide range of detailed information can be retrieved and analysed in the post-

process.

There are several parameters that influence CFD results but one of the most important is mesh

discretization setup and fineness of the grid. In general, the accuracy of the solution is governed by the

number of cells in the mesh, where a higher number normally results in a more accurate solution.

However, the outcome of the results is also highly dependent of the mesh functions applied and its

settings. Often, meshes that produce the most accurate results are non-uniform, which means finer

(smaller) cells in regions with large gradients between the nodes and coarser cells in regions with smaller

gradients. It is important to limit the number of cells in the mesh in order to limit computational cost and

solution times.

The objective of this thesis was to study what impact different mesh parameters and functions had on the

results. The results refer to the predicted thrust and torque from a converged numerical solution using

CFD software and validate these forces against model propeller experimental data.

After selecting a base mesh that would act as a template and starting point for all meshes, one

parameter or one function was altered for each new mesh generated. This was performed in a

systematically manner where, initially, each parameter or function was tested for three different values.

This resulted in a matrix of different cases, where the new meshes were generated and numerically

solved, with the solver setup kept identical every time. The predicted thrust and torque from the

converged solution were extracted and validated against the measured thrust and torque from model

scale experiments in cavitation tunnel T-32. The turbulence model used was SST k-ω and due to

constant rotational speed of the propeller the computational domain was set up as a moving reference

frame with periodic boundary conditions and only one out of five blades was modelled in order to save

computational resources.

The most important conclusions in terms of achieving accurate predictions of thrust and torque, was that

the function prismatic layers had the greatest impact. Surprisingly, the results did not improve by using a

higher resolution of the mesh along and around the edge. The overall density of the computational

domain does matter, but the results showed only a weak correlation between an increase in overall

density and improved accuracy. According to this study, it seems far more important to refine the areas

of interest.

The mesh generated in Ansys that gave the best result had an average deviation of 0,9% for KT and

0,3% for KQ when compared to experimental data from the same propeller type in cavitational tunnel T-

32.

Sammanfattning

Rolls-Royce Hydrodynamic Research Center (RRHRC) har en lång erfarenhet av design och analys av

marina propellrar med hjälp av både experimentella metoder och numeriska simuleringar. Traditionellt

har experimentella tester med propellrar i modellskala varit den dominerande metoden men med den

snabba utvecklingen av datorkraft har fokus skiftats mot numeriska simuleringar, s.k. computational fluid

dynamics (CFD). Fördelarna med CFD är bl.a. en kraftigt reducerad tidsåtgång för analys och design,

direkt modifiering av propellerdesignen i ett tidigt skede och tillgången till en mängd detaljerad

information som kan analyseras i post-processen.

Det är ett flertal parametrar som påverkar resultaten i CFD men en av de viktigaste är hur meshen

diskretiseras och dess finhet, dvs storleken på cellerna. Generellt styrs lösningens precision (resultatet)

av det totala antalet celler i meshen, där ett större antal normalt resulterar i en lösning med högre

precision. Dock påverkas också lösningarnas precision kraftigt av vilka meshfunktioner som appliceras

och dess inställningar. Ofta är de mesher som leder till lösningar med bäst precision icke-uniforma, d.v.s.

nyttjandet av finare (mindre) celler i de regioner med stora gradienter mellan noderna och större celler i

de regioner med gynnsammare gradienter. Denna metod används för att begränsa antalet celler i en

mesh i syfte att minska beräkningskostnaden (datorkapacitet) och tiden det tar för att nå en lösning.

Målet med examensarbetet var att studera vilken inverkan olika parametrar och meshfunktioner hade på

resultatet. Resultatet innebär här den uppskattade tryckkraften och vridmomentet från en konvergerad

numerisk lösning i en CFD-programvara och validera dessa mot uppmätta krafter från experimentella

metoder med propellrar i modellskala.

En grundmesh valdes som utgångspunkt för de nya mesherna och en parameter eller en funktion

ändrades för varje ny mesh som genererades. Detta utfördes på ett systematiskt sätt där varje parameter

eller funktion intialt testades för tre olika värden, d.v.s. tre olika mesher per parameter eller funktion som

analyserades. Detta resulterade i en matris av olika fall, där de nya mesherna genererades och löstes

numeriskt. Inställningarna i lösaren var identiska för varje mesh. De erhållna värdena för tryckkraft och

vridmoment från de konvergerade lösningarna jämfördes med uppmätta krafter från experiment med

samma propellertyp i modellskala i kavitationstunnel T-32 på RRHRC. För att simulera turbulens

användes modellen SST k-ω och propellerns rotation simulerades kvasistationärt med hjälp av en

moving reference frame (MRF). I syfte att minska användningen av datorresurser modellerades bara ett

av fem propellerblad genom att utnyttja periodiska randvillkor.

De viktigaste slutsatserna från studien i syftet att uppnå noggranna uppskattningar på tryckkraft och

vridmoment, var att funktionen prismatiska lager hade störst inverkan. Överraskande nog förbättrades

inte resultaten av en högre upplösning längs med och runt propellerbladets kanter, vilket hade

förutspåtts. Den totala densiteten av beräkningsdomänen har en inverkan, men resultaten påvisar bara

en svag korrelation mellan ökad total densitet och bättre uppskattning av tryckkraft och vridmoment.

Enligt denna studie verkar det vara mycket viktigare att förfina de intressanta områdena (icke-uniform

mesh).

Den mesh som var genererad i Ansys och gav bäst resultat hade en medelavvikelse på 0,9% för KT och

0,3% för KQ när de jämfördes med experimentella data från samma propellertyp i kavitationstunnel T-32.

Foreword This essay is the result of my master thesis in Energy- and Environmental Technology at Karlstad

University and was conducted at Rolls-Royce Hydrodynamic Research Center (RRHRC) in

Kristinehamn.

I would like to thank all the personnel at RRHRC for their willingness to help and their effort to make me

feel as part of the team. It has been six very interesting and educative months. A special thanks to Johan

Lundberg, Team Leader of the CAE-group for making this thesis possible and always being there to tutor

and help out when I have encountered problems. Furthermore would I like to thank Urban Svenneberg

for sharing his expertise in the subject.

I would also like to especially thank Rolf-Erik Keck and Jan Forsberg who has been my mentors at

Karlstad University for their help and support along the way.

Detta examensarbete har redovisats muntligt för en i ämnet insatt publik. Arbetet har därefter diskuterats

vid ett särskilt seminarium. Författaren av detta arbete har vid seminariet deltagit aktivt som opponent till

ett annat examensarbete.

Table of Contents

INTRODUCTION .................................................................................................................................................... 1

1 THEORY AND FUNDAMENTALS ..................................................................................................................... 3

1.1 CFD ........................................................................................................................................................ 3 1.1.1 Pre-processing ...................................................................................................................................... 3 1.1.2 Solver .................................................................................................................................................... 3 1.1.3 Post-processing ..................................................................................................................................... 5

1.2 MESH ...................................................................................................................................................... 5 1.3 GOVERNING EQUATIONS .............................................................................................................................. 7

1.3.1 Continuity equation .............................................................................................................................. 7 1.3.2 Momentum equation ............................................................................................................................ 7 1.3.3 Navier-Stokes equations ....................................................................................................................... 8 1.3.4 Moving reference frame ....................................................................................................................... 8

1.4 TURBULENCE ............................................................................................................................................. 9 1.4.1 Turbulence ............................................................................................................................................ 9 1.4.2 Eddies .................................................................................................................................................... 9

1.5 TURBULENCE MODELING ............................................................................................................................ 10 1.5.1 Reynolds-averaged flow equation (RANS) .......................................................................................... 11 1.5.2 Closure problem .................................................................................................................................. 11 1.5.3 SST k-ω turbulence model .................................................................................................................. 12 1.5.4 Y+ ........................................................................................................................................................ 12 1.5.5 Law of the wall .................................................................................................................................... 12

2 METHOD ..................................................................................................................................................... 14

2.1 METHOD DESCRIPTION .............................................................................................................................. 14 2.2 PROPELLER ............................................................................................................................................. 16 2.3 GEOMETRY – COMPUTATIONAL DOMAIN ....................................................................................................... 16 2.4 NUMERICAL SOLVER .................................................................................................................................. 17 2.5 POST-PROCESSING .................................................................................................................................... 18 2.6 CASE DESCRIPTION ................................................................................................................................... 19

2.6.1 Edge size ............................................................................................................................................. 19 2.6.2 Maximum and minimum face size ...................................................................................................... 20 2.6.3 Cylinder ............................................................................................................................................... 20 2.6.4 Prismatic layers ................................................................................................................................... 20 2.6.5 Overall density .................................................................................................................................... 20 2.6.6 y+ ........................................................................................................................................................ 20 2.6.7 Sphere of influence ............................................................................................................................. 20 2.6.8 Edge size curvature ............................................................................................................................. 21 2.6.9 Turbulence test ................................................................................................................................... 21 2.6.10 SST transition turbulence model ......................................................................................................... 21 2.6.11 Shaft simplification ............................................................................................................................. 21

2.7 EXPERIMENTAL METHOD - CAVITATION TUNNEL T-32 ....................................................................................... 22 2.8 RRHRC MESH-TOOL ................................................................................................................................. 23

3 RESULTS ...................................................................................................................................................... 25

3.1 EDGE SIZE ............................................................................................................................................... 27 3.2 MAXIMUM AND MINIMUM FACE SIZE ........................................................................................................... 28 3.3 CYLINDER ............................................................................................................................................... 29 3.4 PRISMATIC LAYERS ................................................................................................................................... 30 3.5 OVERALL DENSITY..................................................................................................................................... 31 3.6 Y+ ........................................................................................................................................................ 32 3.7 SPHERE OF INFLUENCE ............................................................................................................................... 33 3.8 EDGE SIZE CURVATURE ............................................................................................................................... 34 3.9 SST K-Ω TRANSITION................................................................................................................................. 35 3.10 TURBULENCE TEST .................................................................................................................................... 36 3.11 SHAFT SIMPLIFICATION .............................................................................................................................. 37 3.12 RRHRC MESH-TOOL ................................................................................................................................. 38

4 DISCUSSION ................................................................................................................................................ 39

5 CONCLUSIONS ............................................................................................................................................. 41

6 REFERENCES ................................................................................................................................................ 61

List of figures Figure 1 – Left: Segregated (SIMPLE) algorithm Right: Coupled solver algorithm ..................................... 4 Figure 2 – Left: Orthogonal quality Right: Aspect ratio (source: Ansys Theory Guide 12.0) ................... 5 Figure 3 – Cell skewness .............................................................................................................................. 6 Figure 4 – Boundary layer over a flat plate (source: CFD – Abdulnaser Sayma) ...................................... 9 Figure 5 – Instantaneous and average velocity in turbulent flow (source: CFD – Abdulnaser Sayma) ...10 Figure 6 – Law of the wall ...........................................................................................................................13 Figure 7 – Workflow of the method .............................................................................................................14 Figure 8 – Rolls-Royce Propeller 1301-B and hub “dummy cigar” .............................................................16 Figure 9 – Computational domain from the side and from the front ...........................................................17 Figure 10 – Computational domain from above and from ISO-view ..........................................................17 Figure 11 – Pressure distribution on the propeller faces ............................................................................18 Figure 12 – Left: Leading edge, Right: Trailing edge .................................................................................19 Figure 13 - Square ......................................................................................................................................19 Figure 14 - From left: Tip inner, Tip outer, Singularity point .......................................................................19 Figure 15 – Tip region .................................................................................................................................19 Figure 16 – Geometry with cylinder ............................................................................................................20 Figure 17 – Edge size curvature .................................................................................................................21 Figure 18 - Geometry using the shaft instead of the “cigar dummy “ .........................................................21 Figure 19 – Computational domain of RRHRC mesh-tool from the side and from thefront .......................23 Figure 20 – Computational domain of RRHRC mesh-rool from above and from ISO-view .......................23 Figure 21 – Result overview .......................................................................................................................25 Figure 22 – Edge size predicted deviation of KT (top) and KQ (bottom) ....................................................27 Figure 23 – Maximum and minimum face size predicted deviation of KT (top) and KQ (bottom) ..............28 Figure 24 - Cylinder case predicted deviation of KT (top) and KQ (bottom) ...............................................29 Figure 25 – Prismatic layer case predicted deviation of KT (top) and KQ (bottom) ....................................30 Figure 26 – Overall density case predicted deviation of KT (top) and KQ (bottom) ...................................31 Figure 27 – y+ case predicted deviation of KT (top) and KQ (bottom) ........................................................32 Figure 28 – Sphere of influence case predicted deviation of KT (top) and KQ (bottom) ............................33 Figure 29 – Edge size curvature case predicted deviation of KT (top) and KQ (bottom) ...........................34 Figure 30 – SST – k-ω transition case predicted deviation of KT (top) and KQ (bottom) ...........................35 Figure 31 – Turbulence test case predicted deviation of KT (top) and KQ (bottom) ..................................36 Figure 32 – Shaft simplification case predicted deviation of KT (top) and KQ (bottom) .............................37 Figure 33 – RRHRC mesh-tool case predicted deviation of KT (top) and KQ (bottom) .............................38 Figure 34 - Pressure distribution of ‘Sphere of influence 3,0mm’...............................................................42 Figure 35 – Pressure distribution of ’3 prismatic layers’ .............................................................................42 Figure 36 – Pressure distribution of ’20 prismatic layers’ ...........................................................................42 Figure 37 – Pressure distribution of ‘Cylinder 0,8mm’ ................................................................................42 Figure 38 - Pressure distribution of ‘y+ 116’ ...............................................................................................43 Figure 39 – Pressure distribution of ‘RRHRC medium’ ..............................................................................43 Figure 40 – Pressure distribution of ‘RRHRC – low’ ..................................................................................43 Figure 41 – Pressure distribution of ‘y+ 78’ ................................................................................................43 Figure 42 - Curves of KT (top) and 10*KQ (bottom) for Edge size case ......................................................48 Figure 43 - Curves of KT (top) and 10* KQ (bottom) for Max and min face size .........................................48 Figure 44 - Curves of KT (top) and 10* KQ (bottom) for Cylinder case .......................................................49 Figure 45 - Curves of KT (top) and 10* KQ (bottom) for Prismatic layer case .............................................49 Figure 46 - Curves of KT (top) and 10* KQ (bottom) for Overall density case .............................................50 Figure 47 - Curves of KT (top) and 10* KQ (bottom) for y+ case .................................................................50 Figure 48 - Curves of KT (top) and 10* KQ (bottom) for Edge size curvature .............................................51 Figure 49 - Curves of KT (top) and 10* KQ (bottom) for SST k-ω transition case .......................................51 Figure 50 - Curves of KT (top) and 10* KQ (bottom) for Shaft simplification case ......................................52 Figure 51- Curves of KT (top) and 10* KQ (bottom) for Turbulence test case .............................................52 Figure 52 - Curves of KT (top) and 10* KQ (bottom) for RRHRC mesh-tool case .......................................53 Figure 53 - 2D fluid element with pressure and velocity .............................................................................54 Figure 54 - 2D fluid element with added pressure and shear stresses ......................................................57

List of tables Table 1 – Matrix of cases ............................................................................................................................15 Table 2 – Inlet velocity at different advance numbers ................................................................................18 Table 3 – Base mesh settings ....................................................................................................................44 Table 4 - Edge size .....................................................................................................................................45 Table 5 - Maximum and minimum face size ...............................................................................................45 Table 6 - Cylinder .......................................................................................................................................45 Table 7 - Prismatic layers ...........................................................................................................................46 Table 8 - Overall density .............................................................................................................................46 Table 9 - y+ .................................................................................................................................................46 Table 10 - Sphere of influence....................................................................................................................47 Table 11 - Edge size curvature ...................................................................................................................47 Table 12 - RRHRC mesh-tool .....................................................................................................................47

1

Introduction

Rolls-Royce Hydrodynamic Research Center (RRHRC) has a long experience in propeller design and

analysis by means of both experimental model testing and numerical simulations. Traditionally,

experimental model testing has been the most common method but with the rapid development of

computer hardware, focus has shifted towards computational fluid dynamics (CFD). CFD offers

advantages such as greatly reduced design and analysis time, direct modification of the design in an

early stage and that a wide range of detailed information can be retrieved and analysed in the post-

process.

The long-term aim at RRHRC is to create a highly automated and accurate analysis tool that is integrated

with their current design tool PropCalc. By making it an automated part of PropCalc enables users

without the skills and knowledge of a CFD-engineer to take advantage of its benefits and thus have more

information at their disposal when making design decisions.

Before being integrated into PropCalc and used in propeller design, the automated analysis tool must be

validated against data from a wide range of different propeller designs in order to verify its accuracy and

reliability. The objective for RRHRC is a tool with a maximum error within 3% of thrust and torque acting

on the propeller at different conditions. However, before an extensive validation work can commence, the

objective must first be reached for one propeller type. There are several parameters that influence CFD

results but one of the most important is mesh discretization setup and density. In general, the accuracy of

the solution is governed by the number of cells in the mesh, where a higher number normally produce

better results, but there is always a trade-off between accuracy and computational cost. However, the

outcome of the results are also highly dependent on mesh quality and how it is generated, such as the

use of unstructured or structured mesh, the use of hexes or tetrahedrals and refinement of certain areas

to mention a few. Often, meshes that produce the most accurate results are non-uniform, which means

finer (smaller) cells in regions with adverse gradients between the nodes and coarser cells in regions with

more favourable gradients. It is also important to limit the number of cells in the meshes in order to limit

computational costs and solution times.

A marine propeller has a complex geometry because of variable section profiles, chord lengths and pitch

angles. To meet the demand of today’s high-speed vessels, the propellers are often operated at high

rotational speeds to deliver the required loads. This generally leads to a higher skewness in the propeller

design, further complicating the geometry. Compared to other lifting bodies there are also additional

difficulties due to strong twisting of the blade, periodicity, stagnation point on the hub close to the

propeller and physical space limitations in where its suppose to operate. This complexity makes it difficult

to design a propeller that meet load requirements and at the same time avoids cavitation and excessive

vibrations. Precision is therefore vital and in order to reach accurate predictions in the CFD models the

mesh has to be of a certain quality and set up in a way that ensures this. This has been one of the main

obstacles in numerical simulations of marine propellers.

The accuracy of the simulations was measured as deviations from experimental data in terms of the non-

dimensional coefficients KT (thrust) and KQ (torque). In a similar study -Validation of RANS predictions of

open water performance of a highly skewed propeller with experiments by Li Da-Qing [1] the predicted

average deviation of KT compared to experimental data was less than 3% and KQ within 5%. In another

study - Influence of grid type and turbulence model on the numerical prediction of the flow around marine

propellers working in uniform flow made by Mitja Morgut and Enrico Nobile [2] two propellers were

subject to a mesh analysis. The results from the first propeller showed an average deviation of 4,1% for

KT and 5,1% for KQ. The meshes of the second propeller had showed better results with an average

deviation of 1% for KT and 4,3% for KQ. Deviations of KT and KQ above 3% cannot be considered

accurate and leaves too much guesswork in the performance of the propeller.

The objective of this thesis was to study what impact different mesh parameters and functions had on the

results. Here, the results refer to the predicted thrust and torque from a converged numerical solution

using CFD software and validate these forces against model propeller experimental data. The data

2

acquired is meant to support the development of the analysis tool and the current mesh approach when

conducting open water propeller simulations at RRHRC.

The method applied was as follows; once a geometry of the computational domain had been established

it was kept identical throughout the study. All new meshes that were generated were based on this

geometry, and for each new mesh generated only one parameter in the mesh settings was altered. The

new mesh was generated and numerically solved, with the solver setup kept identical every time. The

predicted thrust and torque from the converged solution were extracted and validated against the

measured thrust and torque from model scale experiments in cavitation tunnel T-32.

3

1 Theory and fundamentals

In order to fully grasp the contents of this thesis it is necessary with a brief description of the basics in

Computational Fluid Dynamics (CFD), its governing equations and the phenomenon of turbulence which

plays a vital role in practically all engineering applications. Turbulence is by nature chaotic and complex,

making it hard to predict and with the computer hardware available at this date, it is not practically

manageable to fully resolve for engineering purposes. Instead, simplifications and mathematical models

are used of which some are described in this section. Because of the complexity and extent of the

governing equations a more thorough description is given in appendix E.

1.1 CFD

Computational Fluid Dynamics (CFD) is the simulation and analysis of fluid flows, heat transfer or

chemical reactions in a defined system with the aid of computers. The technique can be used for many

different applications, E.G. hydrodynamics of ships, flows in turbomachinery, mixing and separation in

chemical process engineering, equipment cooling in electrical engineering, lift and drag in aerodynamics

of aircrafts, thrust and torque in propeller simulation or pollution distribution in environmental engineering.

With powerful computer hardware becoming less expensive and the software being made more user-

friendly has led to an increase in areas where CFD is used. It’s now a common tool in R&D, design and

manufacturing. Moreover, there are several advantages compared to experimental testing; like reduced

time and cost of new designs or techniques, the ability to study very large systems which is hard or

impossible to mimic in a laboratory environment and the ability to study systems in hazardous conditions

without the risk of personnel injuries or damage to facilities.

Today there exist a number of commercial CFD codes based on different numerical methods for solving the problem, but they all contain three main elements; pre-processing, solving and post-processing. [3]

1.1.1 Pre-processing

The objective of pre-processing is to create and define a model of the problem before handing it over to

the solver for calculation. Normally, it includes the following procedures;

Creation of the geometry that includes the region of interest and the surrounding domain. In this

study the region of interest refers to the propeller and hub, and the surrounding domain refers to

the fluid domain. All together this is called the computational domain.

Generation of the mesh (or grid) which means dividing the computational domain into a number of smaller, non-overlapping cells (sub-domains).

Selection of the appropriate physics model, type of fluid(s) and what kind of material(s) that are involved in the problem.

Definition and specification of the boundary layers.

1.1.2 Solver

There are mainly three different types of numerical solution techniques in CFD; finite difference, finite

element and spectral methods. The most commonly used and thoroughly validated general purpose CFD

code today is the finite volume method, which is a special formulation of the finite difference method. This

study has solely used the finite volume method which will be described in further detail, while the others

are out of scope in this paper.

4

The finite volume refers to the volume that surrounds each node in a mesh and the numerical method

uses conservation of relative properties for all cells in the computational domain. This is achieved by

formal integration of the governing equations in fluid flow. Because the fundamental physical phenomena

in fluid flow, especially turbulent which is the case in almost every engineering application, are complex

and non-linear an iterative solution method is necessary. In order to convert this system of integral

equations into a system of algebraic equations that can be solved by such a method, the discretization

substitutes a number of finite-difference-type approximations such as convection, diffusion and sources.

[3]

Apart from numerical solution techniques there are also different types of solvers. In this study two

pressure-based solver algorithms have been used, one segregated and one coupled referring to their

respective algorithm. The pressure-based solver employs an algorithm which belongs to a general class

of methods, called the projection method. This method obtains the continuity of the velocity field by

solving a pressure equation which is derived from the continuity and momentum equations.

Because these equations are coupled to one another the algorithm must be carried out iteratively, where

the entire set of governing equations is solved repeatedly until the solution is converged.

The segregated and coupled algorithms are presented below.

As can be seen in figure (1) the equations solved in step 2 and 3 in the segregated algorithm are solved

simultaneously in the coupled solver. This improves the rate of solution convergence but to a cost of

increased memory requirements of the order of 1,5-2 times. [4]

Stop Yes No

Update properties

Solve velocity components U,V and W

sequentially

Solve pressure-correction continuity

equation

Update mass flux, velocity and pressure

Solve energy, turbulence and other

scalar equations

Solution converged? Stop Yes No

Update properties

Update mass flux

Solve energy, turbulence and other

scalar equations

Solution converged?

Solve system of momentum and pressure-based

continuity equations simultaneously

Figure 1 – Left: Segregated (SIMPLE) algorithm Right: Coupled solver algorithm

5

1.1.3 Post-processing

Post-processing of the solution enables the user to further analyze the model and problem by e.g.

examining pressure and velocity contours, vector plots, 2D and 3D surface plots, track particles and view

manipulation just to mention a few.

1.2 Mesh

In general, the accuracy of the solution is governed by the number of cells in the mesh, where a higher

number normally produce better results. However, the outcome of the results are also highly dependent

on mesh quality and how it is generated in terms of unstructured or structured mesh, the use of hexes or

tetrahedrals, refinement of certain areas and so on. Often, meshes that produce accurate results are

non-uniform, which means finer (smaller) cells in regions with large variations between the nodes and

coarser cells in regions with little variation.

Cell quality is often measured in three quantaties; orthogonal quality, skewness and aspect ratio. Orthogonal quality (see figure 2) is calculated by means of two equations, where the first is the

normalized dot product of the area vector of a face and a vector from the centroid of the cell to the

center of that face

| || | (1)

The second is the normalized dot product of the area vector of a face and a vector from the centroid of

the cell to the centroid of an adjacent cell that shares that face

| || | (2)

This is done for all sides of the cell and the minimum value is defined as the orthogonal quality. The

values range from 0 to 1, with 0 being the lowest quality and 1 the highest.

Aspect ratio (see figure 3) is a measure of how much a cell is stretched in all directions. It is computed as

the ratio between the maximum and minimum value of the distances from; cell centroid to face centroid

and cell centroid to the nodes.

Figure 2 – Left: Orthogonal quality Right: Aspect ratio (source: Ansys Theory Guide 12.0)

6

Skewness is a measure of how close to ideal a face or a cell is, where an ideal square is rectangular and

an ideal triangle is equilateral. The figure below shows ideal and skewed cells.

The quantity of skewness is simply calculated as

(3)

where zero equals a perfect cell and one a degenerate cell. [4]

Figure 3 – Cell skewness

7

1.3 Governing equations

Below follows a brief description of the governing equations, and as mentioned earlier, a more thorough

description is given in appendix E because of their extent and complexity.

The governing equations of computational fluid dynamics are the continuity equation, momentum

equation and energy equation. The energy equation describes the heat transfer and compressibility of a

fluid flow, and since the flows in this thesis are regarded incompressible and without heat transfer, it will

not receive further attention.

1.3.1 Continuity equation

The continuity equation, or the conservation of mass equation, states that all matter is conserved in a flow. The sum of all masses flowing in and out of a given volume per unit time, must equal the change of mass due to change in density per unit time.

( ) (4)

where ρ is the density, is the divergence operator and is the velocity vector in the Cartesian coordinate system. Expressed in Cartesian coordinates it reads

( )

( )

( )

(5)

For an incompressible fluid the ρ term is dropped because of constant density, and thus the following equation is obtained (6) This equation is applicable to both steady and unsteady flow. [4] [5] [6] [8]

1.3.2 Momentum equation

Newton’s second law state that the rate of change of momentum of a fluid particle equals the sum of the

forces acting on this particle. The forces could be divided as surface and body forces, where surface

forces consists of pressure- and viscous forces, and the body forces are gravity-, centrifugal-, Coriolis-

and electromagnetic forces. In an non-accelerating reference frame (inertial) the conservation of

momentum equation is described as

( ) ( ) ( ) (7)

where is the stress tensor is given by

[ ( )

] (8)

where µ is the molecular viscosity and I the unit tensor. The second term on the right describes the effect of volume dilation, and thus, equals zero for incompressible flow. [4] [5] [6] [8]

8

1.3.3 Navier-Stokes equations

The Navier-Stokes equations provide the fundamentals for modelling fluid motion, and are applicable for

both laminar and turbulent flow, although the latter need some modifications. Navier (1832) and Stokes

(1845) both derived the equations of motion for a viscous fluid in a laminar flow by assuming that the

shear rate in a fluid was linear to the shear stress.

The laws of motion that apply to solids are also valid for fluids, which mean that a fluid particle will

respond to a force in a similar way as a solid particle. The difference is that the fluid distorts without limit

whereas the solid return to its initial position once the acting force is removed. The fluid however, will not

return to its initial position when the force is removed.

The Navier-Stokes equations given in the x-direction only, yields

(9)

On the left-hand side, X is the contribution of a body force described in the momentum equation above,

followed by the pressure and viscous terms. The viscous terms express the rate at which a fluid element

is deformed is opposed by the fluid viscosity. The inertial term on the right arise from momentum

changes, and gives a measure of the change of velocity of a fluid particle. [3] [6] [7] [8]

1.3.4 Moving reference frame

The reason for employing a moving reference frame is to make a problem which is unsteady in the

stationary frame steady with respect to the moving frame. If the rotational speed of the moving frame is

constant, it is possible to transform the equations of fluid motion to the rotating frame such that a steady-

state solution is possible. The technique is employed in this thesis and further described in the Method

section. The fluid velocities can be transformed from the stationary frame to the rotating frame as

(10) And (11)

where is the relative velocity, is the absolute velocity and is the velocity due to the moving frame.

By implementing these relationships on the continuity and momentum equation yields,

(12)

( ) ( ) ( ) ( ) (13)

𝜌𝑋 𝜕𝑝

𝜕𝑥 𝜇

𝜕2𝑢

𝜕𝑥2 𝜕2𝑢

𝜕𝑦2

𝜌

𝜕𝑢

𝜕𝑡 𝑢

𝜕𝑢

𝜕𝑥 𝑣

𝜕𝑢

𝜕𝑦 𝑤

𝜕𝑢

𝜕𝑧

Viscous terms Body force term

Pressure term

Inertial terms

9

1.4 Turbulence

1.4.1 Turbulence

Reynolds number expresses the ratio between inertial and viscous forces. The equation describing this is

(14)

where L denotes the characteristic length which is geometry based, e.g. the diameter of a pipe. If the

Reynolds number of a certain flow is below the so-called critical Reynolds number Recrit , the flow is

described as laminar with the viscous forces being dominant. Laminar flow behaves in an orderly fashion

with the fluid particles sliding past each other in a smooth flow. If the Reynolds number reaches Recrit a

series of complicated events start to take place and the flow characteristics change dramatically. The

flow becomes unstable and unpredictable with flow properties varying in a random way and the flow is no

longer smooth. This state is called the transition zone and occurs at or around the critical Reynolds

number. With a further increase of the Reynolds number the flow becomes even more random and could

be described as chaotic. This kind of flow behavior is called turbulent and the dominant forces are

inertial. The three states; laminar, transition and turbulent flow are shown in figure (4) below.

Figure 4 – Boundary layer over a flat plate (source: Computational Fluid Dynamics – Abdulnaser Sayma)

Figure (4) illustrates a flow with free-stream velocity U∞ over a flat plate. As the free-stream approaches

the plate a boundary layer is formed, where the particles closest to the plate has zero velocity because of

friction, the so-called no slip phenomenon. The velocity increases further away from the plate until it

reaches the free-stream velocity far enough from the plate. As can be seen in the figure the flow is

initially laminar but as it moves along the x-axis in a positive direction the Reynolds number increases

with inertial forces becoming more dominant, until it finally becomes fully turbulent with random and

chaotic eddies. [3] [7]

1.4.2 Eddies

Hinze (1975) stated the following definition of turbulence; “Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned.”

10

Turbulence is commonly described by researchers and experts as “eddying motion”, which describes the

local swirling motion observed in turbulent flows. These eddies appear in a wide range of sizes and its

vorticity can often be very intense, which give rise to forceful mixing and high turbulent stresses

compared to laminar flow. Such effective mixing causes high transfer of heat, mass and momentum.

The eddies interact and gain energy from the mean flow, where the eddies with larger wavelengths

interact more strongly and extracts more energy than smaller ones, and thus carries the most energy.

This process is called vortex stretching and is the main physical process for energy transfer in turbulent

flow and provides the energy which maintains the turbulence. Smaller eddies are stretched by larger

eddies but not as much by the mean flow and thus receives most of their energy from larger eddies. This

transfer of kinetic energy from larger to smaller eddies is called energy cascade and is progressively

carried out through the entire range of wavelengths. The limit of how small an eddy can become in a

certain flow is dictated by viscosity, and in the final stage of the energy cascade kinetic energy is

dissipated to heat through the action of molecular viscosity. Thus there exists a balance between the rate

of energy supplied by larger eddies and the rate of dissipation to heat from the smallest eddies. [3] [7] [8]

1.5 Turbulence modeling

An analytical solution of the Navier-Stokes equations can only be closed for some simple laminar flows.

For turbulent flows however, an analytical solution presents a major problem. A direct numerical solution

(DNS) requires a mesh with Re9/4

cells for a sufficient spatial resolution and the CPU-time needed to

solve the problem is approximately Re3 seconds. Because of the complexity it is not yet practical for

engineering purposes to completely solve the Navier-Stokes equations using the instantaneous values.

For example, a fully turbulent flow with a high Reynolds number and a domain measuring 0,1 x 0,1m

could contain eddies down 10 to 100μm. This would require a mesh with 109 to 10

12 cells. For

comparison, the highest number of cells used in this study was 3*107 in a computational domain with the

approximate measurements 0,5 x 0,5 x 1,8m. [3] [15]

In a steady and fully turbulent flow, measuring velocity at just one single point would result in something

like figure (5). By changing position of the measuring point in the turbulent region a similar, but not

identical result would be achieved. This random behaviour makes it difficult to fully describe the motion of

all particles, which if even possible, would demand a vast amount of computer resources as described

above. Instead, the velocity could be expressed by using a mean value with the fluctuations added

to it, as follows

( ) ( ) (15) This technique can also be applied to other properties, e.g. pressure. It’s furthermore useful for numerical

modelling because it allows use of coarser meshes and larger time steps, where the time steps should

be large enough to filter out small variations but still small enough to “capture the physics”. [7]

Figure 5 – Instantaneous and average velocity in turbulent flow (source: Computational Fluid Dynamics – Abdulnaser Sayma)

11

1.5.1 Reynolds-averaged flow equation (RANS)

To implement the technique described above in numerical modeling, the instantaneous values found in

Navier-Stokes equations are substituted by the sum of the average and fluctuating values as described in

equation (15). This results in what is called the Reynolds-averaged flow equations (RANS). A

comparison between the Navier-Stokes equation in its original form (16) and the RANS equation (17)

below clarifies this implementation (x-direction only) [8]

2

2 2

2 2

2

(16)

[

2

2 ( )

] [

2

2 ( )

] [

2

2 ( )

]

(17)

Where U is the mean flow velocity and the added terms – , and – are stress terms

known as Reynolds stresses, where the first is a normal stress and the next two are shear stresses. Also

considering the y and z direction there is a total of three normal stresses and three shear stresses. The

stresses in the y and z direction are straightforward and easy acquired from the ones given above for the

x direction and not given here. They are however available in the appendix E.

1.5.2 Closure problem

Turbulence modelling aims to solve the system of equations (58-60) in the most accurate way by creating

approximations of the unknown variables based on known flow properties so there exists enough

equations to solve, or as it is often referred to in literature; close the system. There are several methods

available for accomplishing this. However, in this paper there will be just a brief description of the method

called the turbulent viscosity approach. [7]

The turbulent viscosity approach was proposed by Prandtl and is based on the hypothesis that there

exists an analogy between the Reynolds stresses and the viscous stresses. A new quantity called

turbulent viscosity μt was introduced. When implemented with the RANS equations the x-direction

equation becomes

[( )

2

2 ( )

] [( )

2

2 ( )

] [( )

2

2 ( )

]

(18)

The new equation only differs by the fact that the viscosity is replaced by the sum of viscosity and the

turbulent viscosity, (μ+μt). In order to close the problem, the turbulence models have to find the

distribution of μt in the solution domain. The two most common turbulence models for this purpose are

the mixing length model and the k-ε model. The first is the least advanced model that describes the

stresses by using simple algebraic methods to express μt as a function of position. k-ε is a slightly more

sophisticated two-equation model with one equation (k) describing the turbulence in terms of generation

and transportation, which could be explained as the energy of the turbulence. The second equation (ε)

describes the dissipation and transportation of turbulence, and is the variable that controls the scale of

the turbulence. The k-ε model has shown to perform well in free-shear flows with small pressure

gradients but has a poor performance in flows where viscous forces are not negligible and with large

pressure gradient. In such flows, other models such as the k-ω model show superior performance.

12

However, the k- ω does not perform as well in free-shear flow as the k- ε. Modeling the flow of a complex

geometry such as the marine propeller requires a model that can handle both in order to get accurate

results. Therefore, this study has almost solely used the SST k-ω model, which is a modified version of

the k-ω model. In short, the SST k-ω model utilizes the well-performing behavior of the k- ω model close

to the wall in the viscous sub-layer and the logarithmic region, and then gradually shifts towards the k- ε

model further away from the wall. In this way the turbulence model benefits from k-ω ‘s accurate and

robust formulation of the viscous sub-layer and logarithmic regions, as well as avoid its often inaccurate

and sensitive formulation of the free stream. A more thorough description is given in the next subsection.

[3] [7] [8] [13]

1.5.3 SST k-ω turbulence model

The k-ω Shear Stress Transport (SST) model by F.R. Menter is a combination of the k-ω model by

Wilcox and a high-Reynolds k-є model. The SST k- ω model seeks to combine the positive features from

both models by activating them where they perform best, which is in different regions of the flow. The k-ω

approach is employed in the viscous sub-region because it does not need damping functions, and thus

has a higher numerical stability, without loss of accuracy. The k- ω model is also applied in the

logarithmic region where it has proven to be superior compared to the k- є model, especially in adverse

pressure flows. However, further away from the wall, in the wake and free-stream regions the k- ω model

is highly sensitive to the free-stream value of ω and therefore no longer suitable. This is where the k-є

model is employed, which represents a fair compromise of accuracy in wakes and mixing layers.

To achieve this gradual shift from one model to another, the k- є model is transformed into a k-ω

formulation. It will then be multiplied by a blending function (1-F1) and added to the original k-ω model

times F1. The blending function is designed to be one in the viscous sub-layer and logarithmic region,

and then gradually shift to one in the wake region. The equations for this turbulence model reads

[( )

] (19)

2

[( )

] ( ) 2

(20)

With

(21)

where is the Kronecker delta. The full description of all definitions and constants are given in F.R.

Menter – Two-equation eddy-viscosity turbulence models for engineering applications. [13] [15]

1.5.4 Y+

Y+ is a dimensionless quantity which is used to define the height of the first cell closest to the wall in a

mesh. The cell height is based on the viscosity µ, density ρ and the fact that the velocity friction

divided by the relative speed should equal 0,035-0,050. [9]

(22)

1.5.5 Law of the wall

As mentioned earlier in this paper a flow with a large Reynolds number will be dominated by inertia

forces and with a low Reynolds number the dominating forces are viscous. A Reynolds number based on

the distance from the wall y can be written as

13

(23)

where U is the mean flow velocity, y distance from the wall and is the kinematic viscosity. When y → 0 the Reynolds number will also be zero, but before this point is reached, at very close

distance from the wall the Reynolds number will be low and viscous forces dominate the inertia forces.

Considering the other extreme when y →∞ the Reynolds number will just continue to grow and the inertia

forces will dominate.

Closest to the wall the fluid is stationary because of the no-slip condition and far enough from the wall the

fluid has the velocity of the free stream. Between these boundaries the mean flow velocity varies

logarithmically with distance from the wall (see figure 6), which is true for both internal and external flows.

[3] [8]

The law describing this is called law of the wall

( ) (24)

With y+ defined in a more simple way

(25)

Figure 6 – Law of the wall

14

Base mesh Parameter alteration

Mesh generation

Computation in Fluent

Data extraction

Data analysis

2 Method

The objective of this thesis was to study what impact different mesh parameters and functions had on the

results. Here, the results refer to the predicted thrust and torque from a converged numerical solution

using CFD software and validate these forces against model propeller experimental data. The method is

described in detail below, but first follows a brief walkthrough; the workflow was as illustrated in figure 7,

where the base mesh was the starting point for all meshes generated. Only one parameter in the mesh

settings was altered for each new mesh. The new mesh was generated and numerically solved, with the

solver setup kept identical for every mesh. The predicted thrust and torque from the converged solution

were extracted and validated against the measured thrust and torque from model scale experiments in

cavitation tunnel T-32.

The method section consists of four sub-sections where the first is a more thorough method description,

followed by a sub-section with some detailed information about the propeller used as validation. After that

comes a more detailed and in-depth description of the geometry, numerical solver setup, the procedure

of post-processing the data obtained and a description of the mesh parameters and functions used in the

study. Finally, a brief explanation is given of the RRHRC analysis tool and how it was validated and used

as a reference to this study.

2.1 Method description

After initial literature studies and interviews at RRHRC regarding the available mesh functions in Ansys

Mesher, geometry setup of the computational domain, turbulence models and setup of the numerical

solver, a geometry was established that would be identical throughout the study, which means all

meshes created would be based on it. Next, a base mesh was selected which would act as a template

for the other meshes that were to be generated further on. By using the base mesh as starting point and

change one value at time of a specific mesh parameter or mesh function, generate a new mesh and

validate the results, the effects could be analyzed.

The process of selecting a base mesh started by using default values in Ansys Mesher and adding mesh

functions with appropriate values based on the routines regarding propeller meshing at RRHRC, a total

of 8 meshes were generated. Each mesh were loaded into Fluent and numerically solved for twelve

different values of J, ranging from 0,65 to 1,20. The settings in the numerical solver Fluent were kept

identical for all cases and the mesh that generated the best result was chosen as base mesh. ‘Best

result’ refers to the minimum deviation of KT and KQ at J=0,95 and the average error of the same ranging

from J=0,65 to J=1,20 compared to experimental data from Cavitation Tunnel T-32. Settings for this

mesh are presented in Appendix B.

After establishing the base mesh a matrix of cases were set up. The different cases consisted of the

parameters and mesh functions that were to be analyzed. In every case only one parameter or function

was altered, and initially three values were chosen. In those cases where new functions were added, that

is, they did not exist in the base mesh, the three values chosen were qualified estimations. For the cases

where the parameter to be analyzed did exist in the base mesh, the three values chosen were based on

the corresponding value in the base mesh, e.g. 50%, 100% and 200% of the base mesh value.

Figure 7 – Workflow of the method

15

The new meshes were loaded into Fluent using the exact same setup described in section 4.1.3 each

time. The only cases where the Fluent setup was changed, were the ones where the setup in Fluent itself

was object of analysis e.g. different turbulence method. Unlike the base mesh that was calculated for

twelve different values of J, only four were used to calculate the rest of the cases due to insufficient

computer resources.

The predicted thrust and torque from the converged solution were extracted and validated against the

measured thrust and torque from model scale experiments in cavitation tunnel T-32. In propeller theory it

is common practice to express thrust and torque as the dimensionless quantities KT and KQ as defined in

section 4.2. In the validation of the results the KT and KQ for a given mesh was compared to the values of

KT and KQ from the experimental data at the corresponding advance number J, as

(26)

(27)

where exp stands for experimental, sim stands for numerically simulated and i denotes the advance

number; J=0,65 J=0,80 J=0,95 or J=1,10. In order to get an overall performance, the average deviation

of KT and KQ for all four values of J were calculated. The deviations were both positive and negative so

the absolute values were used for a better comparison. The average deviation was calculated as

∑ √

2

(28)

After the entire matrix of cases had been numerically solved, an analysis of the results was performed

and the matrix extended (see table 1) for some cases of particular interest.

Table 1 – Matrix of cases

Parameter / function Nr of

meshes

Edge size 4

Max and min face size 4

Cylinder 3

Prismatic layers 6

Overall density 5

y+ 4

Sphere of influence 3

Edge size curvature 4

Turbulence test 1

SST k-ω transition 1

Shaft simplification 1

16

2.2 Propeller

The propeller used for validation of the results from the different meshes was a Rolls-Royce 1301-B,

designed at RRHRC. It has five blades and a design pitch of 1,45 at 70% of the radius. Full-scale and

model-scale diameters are 2800mm and 254,3mm respectively, which corresponds to a scaling factor of

0,0908. Hub size diameter is 790mm.

2.3 Geometry – Computational domain

The geometry was created in Ansys DesignModeler and consisted of a domain, propeller and hub with

the propeller positioned in the center of the domain and connected to the hub. The flow around a marine

propeller working in uniform flow can be considered periodic with respect to the blades when the

hydrostatic pressure is assumed constant, so in order to save computer resources only one out of five

blades were modeled. For this to function properly and accurately the geometry must have a perfect

match at its periodic boundaries. The use of swept sides instead of sharp corners has proved to facilitate

the matching by avoiding mesh complications. [10] [12]

The B-1301 propeller was imported in DesignModeler as two parasolid (.x_t) files, one for the main body

and one for the tip. In order to avoid mesh problems with a singularity point at the tip of the propeller, a

fine slice of the outermost part of the tip was removed. The tip section of the blade was further adjusted

by splitting the three lines that made up the edges along the propeller, into two squares on each side of

the tip section. This was done to create a smooth transition between leading/trailing edge and the tip

section (see figure 13 and 14)

The hub was also imported as a parasolid file and merged to the propeller, creating a single body. The

hub used was a so-called “cigar dummy” which has shown to be an effective and accurate simplification

of the shaft that has a more complex geometry. To verify this simplification, in one of the cases the cigar

dummy was replaced by a shaft similar to the shaft used when conducting the model scale propeller

experiments (see section 4.7.11). [10]

The domain used was developed at RRHRC and its geometry is more or less a standard when modeling

propellers using just one blade and a utilizing a ‘Moving reference frame’. Both the propeller and hub was

removed by a Boolean operation from the domain and thus creating an empty volume but with the

surfaces still there. The flow direction was negative z, and both the propeller and hub surfaces were kept

stationary with the domain rotating around it. This is called moving reference frame which is explained in

Figure 8 – Rolls-Royce Propeller 1301-B and hub “dummy cigar”

17

the theory section. The rotation was oriented counter-clockwise around the z-axis when looking at figure

(9). [2] [12] [14]

Figure (9 and 10) illustrates the geometry the computational domain that measured 7,3, 2,2 and 2,3

propeller diameters in length, height and width respectively.

2.4 Numerical solver

The numerical simulations of the meshes were carried out in Ansys Fluent with the aim to resolve the

thrust and torque forces acting on the propeller during different advance conditions, J. The turbulence

model used was SST k-ω described in chapter 2.5.3. The interior cells of the computational domain were

defined as water-liquid (ρ = 998kg/m3) and ‘Moving reference frame’ rotating at 993,6 rpm based on the

conditions used during model scale tunnel test. The center of the frame was located at [0 0 0] with

rotation around the z-axis.

The boundary conditions were set to simulate the flow experienced by the model scale propeller in

cavitation tunnel T-32. The boundary conditions were set up as follows; on the inlet boundary the

magnitude and direction of the flow was given as velocity components and the flow was defined as

uniform, with 2% turbulence intensity and turbulent length scale 0,05m. The outlet boundary was set as

pressure outlet with zero static pressure and the same turbulence intensity and length scale as the inlet

boundary. The top shroud of the computational domain and the hub were set as symmetry. The propeller

was given a wall function with no-slip condition and wall roughness 0,5. The four boundaries making up

the sides of the computational domain (l boundary, lower left boundary, r boundary and lower right

Figure 9 – Computational domain from the side and from the front

Figure 10 – Computational domain from above and from ISO-view

18

boundary) were made periodic by means of the ‘make-periodic’ command available in Ansys Fluent,

leaving just the left boundaries, which were set up as periodic.

The pressure-velocity coupling scheme used a coupled algorithm and the spatial discretization was set up as; least square cell based gradients, second pressure order, momentum, turbulent kinetic energy and specific dissipation rate were all set to second order upwind schemes. The Courant number, under- and explicit relaxation factors were kept as default. In order to change the advance condition J, the inlet velocity was altered using equation (31). The four conditions and their corresponding inlet velocity can be seen in table (2) below. The computations were run for 500 iterations which was considered more than enough to reach convergence of the solutions.

Table 2 – Inlet velocity at different advance numbers

J VA (m/s)

0,65 2,737

0,80 3,369

0,95 4,000

1,10 4,632

2.5 Post-processing

The forces of interest in this study were primarily torque and thrust acting on the propeller. Fluent was set

up to write the values of these forces to different output files. The generated data was organized in what

kind of force, and under which condition the particular force was generated; Thrust_J065, Thrust_J080..

and so on and in the same way for the torque data; Torque_J065, Torque_J080…

Even though the solution was considered converged after 500 iterations there were still some minor

variations of the forces. In order to minimize this inaccuracy the results extracted from the thrust and

torque data-files were based on the average of the last 30 iterations.

Post-processing of the numerical solution was done in Ansys CFD-Post where the most common

analyzes were regarding pressure distribution, shear stresses, y+ and velocity profiles on the pressure

and suction side of the propeller. The figure 11 below shows the pressure distribution for a certain mesh

case.

Figure 11 – Pressure distribution on the propeller faces

19

Figure 15 – Tip region

2.6 Case description

A total of 11 cases and 36 different meshes were analyzed in the systematically manner described in

section 2.1. Below follows a survey of these cases and a brief description about the parameter or mesh

functions analyzed. A full description of the specific values is given in Appendix C.

2.6.1 Edge size

The parameter to be analyzed in this case was the number of cells or “divisions” in the radial direction of

the leading and trailing edge as well as the tip and the propeller attachment to the hub. The function

Edge size was used along with the sub-function Number of divisions which denotes the number of

cells the chosen edge are to be divided in. Figure 12 shows the leading and trailing edge.

The tip region seen in figure 14 was treated by separate edge size settings, illustrated in figure 13 and

14.

Figure 12 – Left: Leading edge, Right: Trailing edge

Figure 14 - From left: Tip inner, Tip outer, Singularity point

Figure 13 - Square

20

2.6.2 Maximum and minimum face size

Face size refers to the size of the cells on the surface of the propellers pressure and suction side. By

using the function Sphere of influence with only the propeller as target geometry and defining the

minimum cell size, allowed easy control over the cell density on the faces of the propeller. The centre of

the sphere was placed at the centre of the propeller and given a radius of 0,2m.

2.6.3 Cylinder

The idea with this case was to create a volume around

the propeller with a high mesh density which stretched

further than the total height of the inflation layers. A

cylinder with radius 1,2 and depth 1 was placed

centered over the propeller as shown in figure 16. The

cell size inside the cylinder was controlled by the

function Body of influence and Element size.

2.6.4 Prismatic layers

The number of prismatic layers was controlled by the function Inflation layers which defines the number

of prismatic layers generated by the mesh tool. The height of the first cell closest to the propeller wall

was given based on y+ calculations. y+ was held constant throughout the case and only the number of

layers were changed.

2.6.5 Overall density

Overall density was controlled by changing the allowed Max and Min cell size for the entire domain, not

just the propeller or the area around it. The cells increase in size as defined in Growth Rate from the

given minimum size until they reach Max size. From there on there are no further growth of the cells and

the remaining domain was just filled up with cells at maximum size.

2.6.6 y+

Here, the parameter to be analyzed was y+ which refers to the height of the first cell closest to the

propeller wall. This is more thoroughly explained in the theory section. The number of Inflation Layers

(Prismatic layers) and Growth Rate were kept constant.

2.6.7 Sphere of influence

This case was very similar to the case called Max/min face size described in 2.6.2. In both, the function Sphere of influence (SoI) was used and the settings were similar apart from that the total volume covered by the sphere was influenced rather than just the propeller. The position of the sphere was once again at the centre of the propeller.

Figure 16 – Geometry with cylinder

21

2.6.8 Edge size curvature

Instead of analyzing the effect of different number of cells in the radial

direction as in case 4.7.1 Edge size, focus was on the number of cells

around the edge of the blade. Figure (17) illustrates the three lines

forming the edge and how the Edge size function was implemented on

each side of the center line. The number of cells was controlled by the

Number of divisions setting.

2.6.9 Turbulence test

The purpose of this case was to test how the turbulence defined in the

inlet boundary condition affected the torque and thrust. Normally when simulating open water

characteristics on propellers at RRHRC, a turbulence intensity of 2% and length scale 0,05m are used

based on experience and CFD simulations of cavitation tunnel T-32. [10] The tunnel has a honeycomb

formed section upstream of the test area to help stabilize the flow. A turbulent vortex entering this

honeycomb would, in theory, be stretched and the size of the eddies would be limited by the tube

diameter. To test the impact of this inlet condition the length scale was reduced by 50%. All other settings

were kept unaltered and the mesh used for this comparison was “20 prismatic layers”.

2.6.10 SST transition turbulence model

Here, the turbulence model was changed from SST k-ω to SST transition which is a four-equation model

with an improved capability to model the transition from laminar to turbulent flow. The reason for testing

this is that several studies suggest there exist a laminar region when testing model-scale propellers. [1]

[11]

2.6.11 Shaft simplification

In order to analyze the impact of the shaft simplification with the aid of a dummy cigar, a shaft similar to

the one placed in cavitation tunnel T-32 replaced the dummy cigar and a new mesh was generated.

Figure 18 below shows the geometry setup. The mesh settings used for analyze and validation was ’20

prismatic layers’.

Figure 17 – Edge size curvature

Figure 18 - Geometry using the shaft instead of the “cigar dummy “

22

2.7 Experimental method - Cavitation tunnel T-32

Open water tests of the propeller are carried out in the RRHRC conventional cavitation tunnel T-32. The

tunnel is used for testing open water characteristics of propellers and water-jets typically used in ferries,

merchant- and warships. As the name implies, the measurement and study of cavitation are also

conducted in the tunnel.

The water is circulated at a chosen velocity by a powerful 250kW axial pump located in the bottom

section of the tunnel. The propeller testing takes place in the test section located in the upper horizontal

part of the tunnel. The propeller itself is a geometrically uniform model of the real propeller but scaled

down to fit inside the tunnel, normally with a diameter of 200-250mm. It’s important that the size of the

model propeller is not too small in order to keep the Reynolds number sufficiently high. The propeller is

mounted in the centre of the measuring section downstream of a long shaft driven by an electric engine.

A non-rotating smooth transition between the upstream shaft housing and the propeller hub are arranged

and the gap between this transition part and the hub is made according to a RRHRC open water

standard.

Propeller thrust and torque are measured using a force balance. The shaft speed is measured using a

pulse encoder, and water velocity in the test-section of the tunnel is measured with a Prantdl tube, which

adjusts for the blockage effect of the upstream shaft. The total pressure in the tunnel, relative to the

atmospheric pressure, is measured upstream of the test-section with a differential pressure transducer.

The rotational speed of the propeller and the velocity of the flowing water are set to match the full scale

ratio.

All measuring equipment is calibrated regularly. Calibration of the thrust and torque measurements is

checked by systematically retesting Kamewa propeller model 600-A, approximately twice a year.

The actual frictional resistant (torque) between the propeller and the torque meter is measured before the

test and compensated for.

At the open water test, thrust and torque is measured at a range of different advance numbers J and by

changing the shaft speed and/or water velocity. Any combination of ahead/astern condition with positive

or negative thrust can be obtained. The data is presented as non-dimensional thrust and torque

coefficients, KT and KQ. Thrust (T) and torque (Q) coefficients as well as advance number J are defined

as:

2 (29)

2 (30)

(31)

Open water data are usually presented as measured, not corrected for scale effects. The level of

accuracy is within ±0,5% at the efficiency peak.

23

2.8 RRHRC mesh-tool

The long-term aim at RRHRC is to create a highly automated and accurate analysis tool that is integrated

with their current design tool PropCalc. By making it an automated part of PropCalc enables users

without the skills and knowledge of a CFD-engineer to take advantage of its benefits and thus have more

information at their disposal when making propeller design decisions. The mesh-tool is currently being

developed at RRHRC so its part of this study is solely to see how well it performs in comparison with the

meshes created in Ansys Mesher. The results and information acquired in this study will hopefully prove

useful in the preceding development work. Below follows a brief description of the geometry, mesh

approach and the case tested. The setup of the numerical solver and post-process procedure were the

same as described in section 2.4 and 2.5.

The geometry used in this method was very similar to the one described in section 4.4. The same

propeller and “dummy cigar” hub were merged together as a single body and after the surrounding

domain had been created, they were removed leaving just its surfaces. The main differences were the

size of the computational domain and the less swept surfaces at the propeller section in the center of the

domain. Expressed in propeller diameters, the measurements of the domain length was 6,9D, width 2,8

and height 2,0. Figures 19 and 20 illustrates the geometry.

The mesh were generated using the RRHRC mesh-tool which is specialized in meshing the difficult

propeller geometry. The mesh algorithm starts by creating a “band” of structured hex mesh on the

leading and trailing edge, including the tip. The pressure and suction sides of the propeller are meshed

Figure 19 – Computational domain of RRHRC mesh-tool from the side and from thefront

Figure 20 – Computational domain of RRHRC mesh-rool from above and from ISO-view

24

with triangles and then an inner domain is filled with unstructured tetrahedrals. Finally the outer domain is

also filled with tetrahedrals but with a coarser mesh than in the inner domain.

The only parameter analyzed in this method was the mesh density which included the surface mesh of

the propeller as well as the inner and outer domain. The tool is, at this writing, created in such a way that

there are different levels of mesh fineness, e.g. coarse, medium and fine which affects the entire

computational domain.

25

3 Results

Firstly, in order to get an overview of the different cases and their impact on the results, the results are

presented as the average deviation of KT and KQ for all four values of J. The deviations were both

positive and negative so absolute values were used for better comparison. The average deviation was

calculated according to equation (28).

After the initial overview follow a more detailed presentation of the results for each case (3.1 – 3.11) with

bar diagrams that show the actual deviation from the experimental data at every J. The curves of KT and

10* KQ are compared to a curve based on test data from cavitation tunnel T-32 and can be found in

Appendix D.

3,7 1,6

1,5

2,6

1,7

2,6 2,7

2,0 2,0

3,7 4,0

5,8

1,1 1,2

2,4 2,0

1,9 2,1 2,0 2,2

1,7 1,7

2,3 3,8

4,9 4,5

0,9 2,1

1,9

2,1 2,1

1,8 1,7

2,0 1,0

2,3 2,2

2,0

3,2 2,1

0,9

0,6

0,6

1,4 1,4

1,0 1,2

2,7 3,0

5,2

0,4 0,4

1,4 1,0

0,9 1,1 1,1 1,2

0,6 0,6

1,3 2,0

2,9 2,7

0,3 1,0

0,9

1,7 1,3

0,6 0,6

1,0 0,5

1,5 0,9

1,0

0 2 4 6 8 10 12 14

RRHRC - MediumRRHRC - Coarse

Shaft

SST k-w transition

Turbulence test

ES Curv 8ES Curv 6ES Curv 4ES Curv 2

SoI 1,7mmSoI 2,0mmSoI 3,0mm

Y+ 116Y+ 103

Y+ 13Y+ 3

Density 9,5MDensity 7,6MDensity 6,3MDensity 2,2M

20 layers15 layers12 layers

8 layers5 layers3 layers

Cylinder 3Cylinder 2Cylinder 1

Face size 0,7mmFace size 0,4mmFace size 0,3mmFace size 0,2mm

Edge size 750Edge size 600Edge size 450Edge size 150

Base mesh

Avg deviation KT (%)

Avg deviation KQ (%)

Figure 21 – Result overview

26

As figure (21) shows, the base mesh and its settings was a good starting point with an average deviation

of 2% for KT and 1% for KQ. 12 meshes gave better results and 27 gave worse results. The text below

describes the functions in the order of how much influence they have had on the results.

The function that had the greatest impact on the results was ‘Prismatic layers’. The results show a strong correlation between an increased number of prismatic layers and better results. There was only a slight difference in KT and KQ between 15 and 20 layers. All other meshes except the ones in this particular case had 10 prismatic layers but in the ‘Sphere of influence’ (SoI) case the mesh generator failed to create any prismatic layers, with poor results as consequence. The ‘Y+’ case had some of the best results in the study, where the mesh with the best results had an average deviation of 1,1% for KT and 0,4% for KQ. This case is interesting since the results were predicted to be reversed, with a correlation between a lower value of y+ and better results. Considered the low amount of cells used in these meshes (~2,7M) compared to the other top-performing meshes such as ‘Cylinder 0,8mm’ (~22M), they had the highest accuracy per cell ratio. In the ‘Cylinder’ cases just mentioned, two meshes performed equally well as the base mesh but with a considerable higher amount of cells. But the mesh with the finest cells ‘Cylinder 0,8mm’ had the best result throughout the study with an average KT of 0,9% and 0,3% for KQ. The ‘Face size’ function did not increase the performance significantly, not even the mesh with 31M cells and a face size of 0,2mm. This mesh had an average deviation of 1,7% for KT and 0,6% for KQ. The ‘Edge size’ function applied to the propeller leading and trailing edge as well as the tip, did not have any correlation between a higher resolution of these areas and better results. However, a very good result was reached with the ‘Edge size 600’ case with an average KT of 1% and 0,5% for KQ. A further increase of the resolution in ‘Edge size 750’ did not result in an improvement of the results. In the ‘Edge size curvature’ case the results showed a decrease in accuracy with an increase in the number of cells across the leading and trailing edge. Overall density hardly had any impact at all and there was only a mere difference of 0,3% for both KT and KQ between the meshes with the least and most amount of cells. Between these two, the amount of cell ratio was close to 1:5. The ‘Shaft’ case did have a 0,2% more accurate prediction of KT but 0,3% less accurate prediction of KQ when compared to the ’20 prismatic layer’ mesh and its cigar dummy setup. The ‘SST-transition’ case had an average deviation of 2,6% for KT and 0,6% for KQ. Two meshes from the RRHRC analysis tool was computed and the coarse mesh showed better results than the Medium mesh. This was contrary to what was predicted beforehand, and there is a substantial difference between the two. The Coarse mesh had better results compared to the base mesh for KT but not for KQ, -0,4% and 1,1% respectively.

27

3.1 Edge size

The results show a weak correlation between better results and an increase in number of cells on the

leading and trailing edge as well as the tip section. However, there was a remarkable improvement in KT

and KQ in case ‘Edge size 600’ which outperformed the other settings when looking at the average

deviation for all four J’s, but had a greater deviation of KT at J = 0,95 and J = 1,10. The hypothesis of

better results with an increase in resolution of the edge around the propeller was proved wrong with

‘Edge size 750’, even though it performed slightly better for some J’s. The results showed an over-

prediction of both KT and KQ.

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

ES 150

ES 300

ES 450

ES 600

ES 750

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

ES 150

ES 300

ES 450

ES 600

ES 750

Figure 22 – Edge size predicted deviation of KT (top) and KQ (bottom) for four values of J

28

3.2 Maximum and minimum face size

A finer mesh on the propeller faces did in general produce better predictions of both thrust and torque.

This was true for all J’s when looking at the torque, but the thrust at J = 0,65 and 0,80 had a reverse

correlation. A mesh with cells of only 0,2mm in size proved to be close to the limit of how small cells the

mesh generator could handle for this geometry. A non-successful attempt to create a mesh with an even

smaller face size was conducted. Interestingly, the largest deviation were found at J = 1,10 for both KT

and KQ, but especially for the latter which had a reverse trend with rising deviation with an increase in J.

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

0,7 mm

0,4 mm

0,3 mm

0,2 mm

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

0,7 mm

0,4 mm

0,3 mm

0,2 mm

Figure 23 – Maximum and minimum face size predicted deviation of KT (top) and KQ (bottom) for four values of J

29

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

1,2mm

1,0mm

0,8mm

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

1,2mm

1,0mm

0,8mm

3.3 Cylinder

This function had some of the best results in this thesis but with the trade-off of fairly high element

counts; 9, 13 and 22 million cells respectively. The mesh with 22 million cells had very good predictions

at both high and low J’s. The overall average deviation was 0,9% for KT and 0,3% for KQ where the latter

did not exceed 1% deviation for any J. The largest deviation were found at thrust predictions at J = 0,65,

which was the situation in almost all cases. The mesh with the least amount of cells performed well

overall with an average of 1,9% for KT and 0,9% for KQ but showed high deviation for KT at J = 0,65 and

0,80.

Figure 24 - Cylinder case predicted deviation of KT (top) and KQ (bottom) for four values of J

30

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT 3 layers

5 layers

8 layers

12 layers

15 layers

20 layers

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ 3 layers

5 layers

8 layers

12 layers

15 layers

20 layers

3.4 Prismatic layers

There was a clear improvement of the results with an increase in number of prismatic layers.

The mesh with 20 layers showed good predictions of both KT and KQ with an overall average deviation

percentage of 1,7 and 0,6 for KT and KQ respectively. The largest deviation of the same was once again

found at thrust predictions at J = 0,65.

Figure 25 – Prismatic layer case predicted deviation of KT (top) and KQ (bottom) for four values of J

31

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

2,2M

2,9M

6,3M

7,6M

9,5M

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

2,2M

2,9M

6,3M

7,6M

9,5M

3.5 Overall density

According to the results there was no correlation between an increase of the overall density and better

predictions of thrust and torque. The difference in average deviation of KT and KQ between the mesh with

the least and highest amount of cells, 2,2M and 9,5M respectively, was merely 0,3%. The predictions of

both KT and KQ showed small differences between the meshes throughout all values of J. The largest

deviation were found at J = 0,65 and 0,80 for KT. Even the mesh with 9 million cells was outperformed by

several meshes in other cases with one third the amount of cells.

Figure 26 – Overall density case predicted deviation of KT (top) and KQ (bottom) for four values of J

32

3.6 y+

When looking at the average deviation, there was a correlation between a higher value of y+ and better

predictions. The mesh with the largest deviation had y+ 3 with an average deviation of 2% for KT and 1%

for KQ. The mesh that performed best had an average deviation of 1,2% for KT and 0,4% for KQ. That is

some of the most accurate predictions throughout the study, and to a relatively low number of cells

(2,7M).

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

y+ 3

y+ 13

y+ 103

y+ 116

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

y+ 3

y+ 13

y+ 103

y+ 116

Figure 27 – y+ case predicted deviation of KT (top) and KQ (bottom) for four values of J

33

0

2

4

6

8

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

3,0 mm

2,0 mm

1,7 mm

0

2

4

6

8

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

3,0 mm

2,0 mm

1,7 mm

3.7 Sphere of influence

These meshes had some of the largest deviations in this thesis even though they contained a high cell

count; 6M, 11M and 17M and had a good resolution in the area around the propeller because of the

Sphere of influence function.

Figure 28 – Sphere of influence case predicted deviation of KT (top) and KQ (bottom) for four values of J

34

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

ES 2

ES 4

ES 6

ES 8

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

ES 2

ES 4

ES 6

ES 8

3.8 Edge size curvature

There was no correlation found that a mesh with a higher resolution of the edge produced better results

than a mesh with a lower resolution, which was predicted in advance. Instead, there seemed to be an

increase d deviation with an increase in resolution.

Figure 29 – Edge size curvature case predicted deviation of KT (top) and KQ (bottom) for four values of J

35

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

SST k-w

SST transition

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

SST k-w

SST transition

3.9 SST k-ω transition

By using a more complex turbulence model as the SST k-ω transition that has an additional two

equations led to larger deviation of KT at three out of four J’s, and two out of four J’s for KQ. The mesh

used for comparison in this case was ’20 Prismatic layers’ with and y+ = 3.

Figure 30 – SST – k-ω transition case predicted deviation of KT (top) and KQ (bottom) for four values of J

36

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

2%, 0,050m

2%, 0,025m

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

2%, 0,050m

2%, 0,025m

3.10 Turbulence test

The results showed very small differences, but the fact that there were differences states that all

turbulence had not faded away once reaching the propeller. The largest difference in KT between the two

models was 0,02% at J = 0,65 and the largest difference in KQ was 0,03% at J = 0,95.

Figure 31 – Turbulence test case predicted deviation of KT (top) and KQ (bottom) for four values of J

37

3.11 Shaft simplification

The average deviation of KT for the shaft mesh was 1,5% compared to 1,7% for the mesh with the

dummy cigar, but with the deviation occurring at different J’s. The shaft mesh showed better results for

KT at J = 0,65 and 0,80 but worse at J = 0,95 and 1,10. The average deviation of KQ was more than 30%

larger for the shaft mesh, with less accuracy at three out of four J’s.

Figure 32 – Shaft simplification case predicted deviation of KT (top) and KQ (bottom) for four values of J

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε KT

Cigar

Shaft

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Dev

iati

on

(%)

J

%ε 10*KQ

Cigar

Shaft

38

3.12 RRHRC mesh-tool

The total amount of cells for the two cases were 2,3M (Coarse) and 4,4M (Medium). There was a

reversed correlation between a higher cell count and better results with the Coarse case giving better

predictions. The average deviation in the Coarse case was 1,6% for KT and 2% for KQ, and in the

Medium case 3,7% for KT and 3,2% for KQ. As seen in several cases the largest deviations were found at

low J’s, especially for KT.

Figure 33 – RRHRC mesh-tool case predicted deviation of KT (top) and KQ (bottom) for four values of J

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Erro

r (%

)

J

%ε KT

Coarse

Medium

0

1

2

3

4

5

6

0,65 0,8 0,95 1,1

Erro

r (%

)

J

%ε 10*KQ

Coarse

Medium

39

4 Discussion

The predicted values of thrust and torque in this thesis are in the same range or slightly more accurate

than compared to other studies. [1] [2] In this study, the numerical solution of the “best” mesh had an

average deviation of 0,9% for KT and 0,3% for KQ. The average deviation of all meshes, RRHRC mesh-

tool meshes excluded, was 2,4% for KT and 1,3% for KQ.

In a study of Li Da-Qing [1] with a mesh consisting of 5M cells, the predicted deviation of KT compared to

experimental data was less than 3% within the range of J = [0,1 - 0,8] but with a greater deviation outside

that range. The amount of deviation is unfortunately not mentioned. The predicted deviation of KQ was

less than 5% within the same range. These deviation s are somewhat high compared to the general

results found in this thesis, where the base mesh (2,9M cells) had an average deviation of 2% for KT and

1% for KQ in the range J = [0,65 – 1,10]. However, it is important to mention that the this mesh had an

deviation of 4,6% for KT at J = 0,65. The fact that these studies used different ranges of J makes it

difficult to draw any conclusions.

In another study of Mitja Morgut and Enrico Nobile [2] the results were better documented. Two

propellers, P5168 and E779A, were subject to an investigation of the influence of different grid types. The

amount of cells in these meshes were 1,2M and 1,3M respectively. For propeller P5168, the average

deviation of KT and KQ for J = [0,98 1,10 1,27] were 4,1% and 5,1%. Propeller E779A was more

thoroughly tested with J = [0,65 0,71 0,88 0,95 1,10] which corresponds well to the range of conditions in

this study. The average deviation of KT and KQ found here were 1% and 4,3%.

Interestingly, both the study of Li Da-Qing [1] and Mitja Morgut and Enrico Nobile [2] have higher

deviations of KQ than KT which is conversely to the findings in this thesis.

As mentioned earlier there are many parameters that affect the outcome of a numerical solution, but one

of the most important seems to be the use of a non-uniform mesh and carefully choose which areas to

refine. The degree of resolution to use in these areas is based on the accuracy required and the amount

of time and computational power available. However, even if there is a lot of time at disposal the choice

of resolution must also consider the fact that, at least in this study, 1M cells requires roughly one core

(processor) in order to reach a convergence of the numerical solution.

The different functions and parameters analysed in this thesis showed some expected results, but also

quite a few unexpected. The resolution of the propeller edge was predicted to be an important parameter,

but according to this study, does not influence the results significantly. One explanation which needs

further investigation, could be that a too high or too low number of cells on the edge affects the scewness

of adjacent cells and thus gives less accurate values. This deviation could perhaps be avoided by trying

to find an optimum between the number of cells used to resolve the propeller along the edge, across the

edge and the cell face size. Perhaps this was what happened with mesh ‘Edge size - 600’ which clearly

outperformed the other meshes in the same case. A visual surface mesh comparison of the propeller

between ‘Edge size 600’ and the other three meshes in the same case did not support this theory, but a

more thorough analysis is required to make any conclusions.

Furthermore, the overall density (uniform mesh) did not improve the results as predicted. Instead, this

seems to confirm the fact that it is more important to use non-uniform meshes and carefully select and

better resolve the areas with high gradients of pressure and velocity. However, the mesh with the highest

density in that particular case had 9,5M cells which probably are not enough to resolve a computational

domain of that size and still give accurate predictions of thrust and torque.

The high influence of prismatic layers was expected, but not to the extent seen here. The results showed

a clear improvement of the results with an increase in number of prismatic layers but seemed to level out

at around 15-20 layers. There was only a small difference between 15 and 20 layers which could prove

that there is no need to go beyond this number because a “mesh-independency” might have been

40

reached. To investigate this further an attempt to generate a mesh with 22 layers was conducted but the

mesh generator failed to create such a mesh at the current y+ = 3.

Results from the meshes created with the RRHRC mesh-tool showed promising results but the Coarse

case performed far better than the Medium case, which of course, was not expected. An analysis of the

mesh quality (orthogonal quality and skewness) showed no significant differences as when compared to

meshes from Ansys mesher. The average y+ of the propeller was 44 for Coarse, and 16 for Medium. The

latter may have performed better with a higher or lower y+ because an average value of 16 most likely

means that a large area of the propeller faces has y+ values around 10. This is in the middle of the buffer

layer which is somewhat of a problem area where y+ does not correspond to neither the velocity profile of

the viscous sub-layer nor the log-law layer. The meshes generated in Ansys had y+ = 3 except for case

‘y+’ where y+ was the parameter analysed and thus varied.

There is a lot to be done in this area, but to mention a few thoughts regarding further work that would be

of high interest is to test the findings in this thesis using the same settings but with another propeller

geometry. Maybe some of the meshes that resulted in small deviations would fail to do so on a different

geometry, or perhaps there is a mesh setting that works well on a wide range of geometries. Some other

thoughts for further work are:

- Deeper cylinder in the ‘Cylinder’ case

- Increase the number of prismatic layers in the ‘Prismatic layers’ case, but with a lower y+ in order

to avoid errors in the mesh generation

- Try more meshes in the ‘y+’ case that showed very promising results

- Further investigate the differences between a cigar dummy and a shaft similar to the one in

cavitation tunnel T-32.

- The SST k-ω transition case could be tested with an even lower y+ and more prismatic layers.

41

5 Conclusions

The objective of this thesis was to study what impact different mesh parameters and functions had on the

results. The results refer to the predicted thrust and torque from a converged numerical solution using

CFD software and validate these forces against model propeller experimental data. The data acquired is

meant to support both the development of the analysis tool and the current mesh approach when

conducting open water propeller simulations at RRHRC.

The most important conclusions from this thesis;

- The total amount of elements used in a mesh does play a role, however, there are of greater

importance to have a high resolution in areas of “interest” with large gradients and important

physics taking place, such as close to the propeller wall.

- What seems to be most important in terms of accurate predictions of thrust and torque is the use

of prismatic layers. The case ‘Prismatic layers’ shows a strong correlation between higher

accuracy and an increase in the number of prismatic layers. Another case that supports this

conclusion is ‘Sphere of influence’, where the mesh generator failed to implement the functions

Sphere of influence and Inflation (prismatic) layer together. These meshes show a great loss of

accuracy when compared to the others, especially regarding their fairly high element count and

that the mesh is non-uniform, which means that they have a high resolution of the area with

adverse pressure gradients.

- The largest deviations of accuracy was found at low or high J’s, J=0,65 and J=1,10.

- In almost all cases, the predictions of KQ showed higher accuracy then of KT. This is conversely

to what have been found in other studies.

By utilizing the technique ‘Body of influence’ as in the ‘Cylinder’ case showed good results and the mesh

generator managed to create a mesh with both the Body of influence and Inflation layer function. The

mesh that employed these functions along with the finest cells (0,8mm) inside the cylinder, achieved the

highest accuracy of all meshes tested, but the computational time to numerically solve it was high

because of the approximately 22 million cells.

The overall density of a mesh seems to be of little interest, at least in the range tested here. The results

show only a weak correlation between an increase in overall density and improved accuracy.

The resolution of the propeller edge “curvature” seems to have minor effect on the accuracy. The results

had a negative trend with a decrease in accuracy with an increased resolution. The amount of elements

along the edge as tested in ‘Edge size’ showed similar results.

There were some deviation of the predicted thrust and torque when comparing the ‘cigar dummy’ and a

real shaft which need further investigation.

42

Appendix B – Pressure distribution

Figure 35 – Pressure distribution of ‘Cylinder 0,8mm’

Figure 34 – Pressure distribution of ’20 prismatic layers’

Figure 37 – Pressure distribution of ’3 prismatic layers’

Figure 36 - Pressure distribution of ‘Sphere of influence 3,0mm’

43

Figure 38 - Pressure distribution of ‘y+ 116’

Figure 41 – Pressure distribution of ‘y+ 78’

Figure 40 – Pressure distribution of ‘RRHRC – low’

Figure 39 – Pressure distribution of ‘RRHRC medium’

44

Appendix B – Base mesh settings Table 3 – Base mesh settings

Parameter Value

Parameter Value Setting Sizing

Edge size 300 Bias hard

Use advanced size function On: curvature

Edge size 2 300 Bias hard

Relevance center Fine

Edge size 3 100 Bias hard

Initial size seed Active assembly

Edge size 5 4 Bias hard

Smoothing Medium

Edge size 6 50 Bias hard

Transition Slow

Edge size 7 20 Bias hard

Span angle center Fine

Edge size 8 3 Bias hard

Curvature normal angle 2 ° Edge size 9 1 Bias hard

Min size 1.4809E-03 m Max face size 1.1847E-02 m Inflation

Max size 2.3694E-02 m -first layer height 1.18E-05 Growth rate 1.20

-growth rate 1.2

Minimum edge length 6.61E-05 M -layers 10

Inflation Use automatic inflation None

y+ 3

Inflation option Smooth transition Transition ratio 0.272 Maximum layers 5 Growth rate 1.2 Inflation algorithm Pre

Assembly meshing Method None

Patch conforming options Triangle surface mesher Program controlled

Advanced Shape checking CFD

Element midside nodes Dropped Straith sided elements - Number of retries 0 Extra retries for assembly Yes Rigid body behaviour Reduced Mesh morphing Disabled

Defeaturing Pinch tolerance Default (1,3328e-003)

Generate pinch on refresh No Automatic mesh based

defeaturing On Defeaturing tolerance Default (7,4045e-004)

Statistics Nodes 642727

Elements 2903208 Min skewness 6.06E-09 Max skewness 0.998412068 Average skewness 0.247949863

45

Appendix C This appendix contains information about the changes made from the base mesh in each case in and also the total amount of elements in the mesh. Table 4 - Edge size

ES 150 Value

ES 300 Value

Leading edge 150

Leading edge 300

Trailing edge 150

Trailing edge 300

Tip outer 50

Tip outer 50

Tip inner 25

Tip inner 50

Singularity point 10

Singularity point 20

Square 1

Square 1

Curvature 4

Curvature 4

Elements 2.4E+06

Elements 2.9E+06

ES 450 Value

ES 600 Value

Leading edge 450

Leading edge 600

Trailing edge 450

Trailing edge 600

Tip outer 150

Tip outer 200

Tip inner 75

Tip inner 100

Singularity point 30

Singularity point 40

Square 1

Square 1

Curvature 4

Curvature 4

Elements 3.4E+06

Elements 4.2E+06

Table 5 - Maximum and minimum face size

0,2 mm Value

0,3 mm Value

Face size (mm) 0,2

Face size (mm) 0,3

All Edge size Bias soft

All Edge size Bias soft

Elements 3.1E+07

Elements 8.5E+06

0,4 mm Value

0,7 mm Value

Face size (mm) 0,4

Face size (mm) 0,7

All Edge size Bias soft

All Edge size Bias soft

Elements 5.0E+06

Elements 8.5E+06

Table 6 - Cylinder

0,8 mm Value

1,0 mm Value

Element size (mm) 0,8

Element size (mm) 1,0

Elements 2.2E+07

Elements 1.3E+07

1,2 mm Value

Element size (mm) 1,2

Elements 9.3E+06

46

Table 7 - Prismatic layers

3 layers Value

5 layers Value

Prismatic layers 3

Prismatic layers 5

Elements 2.7E+06

Elements 2.7E+06

8 layers Value

12 layers Value

Prismatic layers 8

Prismatic layers 12

Elements 2.8E+06

Elements 2.9E+06

15 layers Value

20 layers Value

Prismatic layers 15

Prismatic layers 20

Elements 3.0E+06

Elements 3.1E+06

Table 8 - Overall density

2,2M Value

2,9M Value

Min size (mm) 2,2

Min size (mm) 1,5

Max face size (mm) 17,0

Max face size (mm) 11,8

Max size (mm) 35,0

Max size (mm) 23,7

Elements 8.5E+06

Elements 8.5E+06

6,3M Value

7,6M Value

Min size (mm) 0,7

Min size (mm) 0,7

Max face size (mm) 5,9

Max face size (mm) 5,3

Max size (mm) 11,8

Max size (mm) 10,6

Elements 6.3E+06

Elements 7.6E+06

9,5M Value

Min size (mm) 0,6

Max face size (mm) 4,7

Max size (mm) 9,5

Elements 9.5E+06

Table 9 - y+

y+ 3 Value

y+ 13 Value

First cell height 1,18e-05

First cell height 5.9E-05

Growth rate 1,2

Growth rate 1,2

Elements 2,9E+06

Elements 2.8E+06

y+ 103 Value

y+ 116 Value

First cell height 7.1E-04

First cell height 5.3E-04

Growth rate 1,2

Growth rate 1,2

Elements 2.7E+06

Elements 2.7E+06

47

Table 10 - Sphere of influence

3,0 mm Value

2,0 mm Value

Element size (mm) 3.0

Element size (mm) 2.0

Elements 6.2E+06

Elements 1.1E+07

1,7 mm Value

Element size (mm) 1,7

Elements 1.7E+07

Table 11 - Edge size curvature

ES 2 Value

ES 4 Value

Edge size curvature 2.0

Edge size curvature 4.0

Elements 2.8E+06

Elements 2.9E+06

ES 6 Value

ES 8 Value

Edge size curvature 6.0

Edge size curvature 8.0

Elements 3.0E+06

Elements 3.1E+06

Table 12 - RRHRC mesh-tool

Beta Value

Coarse Value

Elements 1.7E+06

Elements 2.3E+06

Medium Value

Elements 4.4E+06

48

Appendix D Edge size Maximum and minimum face size

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT ES 150

ES 300

ES 450

ES 600

ES 750

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

ES 150

ES 300

ES 450

ES 600

ES 750

Tunnel test

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT

0,7 mm

0,4 mm

0,3 mm

0,2 mm

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

0,7 mm

0,4 mm

0,3 mm

0,2 mm

Tunnel test

Figure 42 - Curves of KT (top) and 10*KQ (bottom) for Edge size case

Figure 43 - Curves of KT (top) and 10* KQ (bottom) for Max and min face size

49

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT 3 layers

5 layers

8 layers

12 layers

15 layers

20 layers

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ 3 layers

5 layers

8 layers

12 layers

15 layers

20 layers

Tunnel test

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT

1,2mm

1,0mm

0,8mm

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

1,2mm

1,0mm

0,8mm

Tunnel test

Cylinder Prismatic layers

Figure 44 - Curves of KT (top) and 10* KQ (bottom) for Cylinder case

Figure 45 - Curves of KT (top) and 10* KQ (bottom) for Prismatic layer case

50

Overall density y+

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT 2,2M

2,9M

6,3M

7,6M

9,5M

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ 2,2M

2,9M

6,3M

7,6M

9,5M

Tunnel test

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT y+ 3

y+ 13

y+ 103

y+ 116

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ y+ 3

y+ 13

y+ 103

y+ 116

Tunnel test

Figure 47 - Curves of KT (top) and 10* KQ (bottom) for y+ case

Figure 46 - Curves of KT (top) and 10* KQ (bottom) for Overall density case

51

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT

ES 2

ES 4

ES 6

ES 8

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

ES 2ES 4ES 6ES 8Tunnel test

Edge size curvature SST k-ω transition

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT

SST k-w

SST transition

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

SST k-w

SST transition

Tunnel test

Figure 48 - Curves of KT (top) and 10* KQ (bottom) for Edge size curvature

Figure 49 - Curves of KT (top) and 10* KQ (bottom) for SST k-ω transition case

52

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT

2%, 0,050m

2%, 0,025m

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

2%, 0,050m

2%, 0,025m

Tunnel test

Turbulence test Shaft simplification

Figure 51- Curves of KT (top) and 10* KQ (bottom) for Turbulence test case

Figure 50 - Curves of KT (top) and 10* KQ (bottom) for Shaft simplification case

0,17

0,22

0,27

0,32

0,37

0,42

0,65 0,8 0,95 1,1

KT

J

KT

Cigarr

Shaft

Tunnel test

0,45

0,55

0,65

0,75

0,85

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

Cigarr

Shaft

Tunnel test

53

RRHRC mesh-tool

Figure 52 - Curves of KT (top) and 10* KQ (bottom) for RRHRC mesh-tool case

0,15

0,20

0,25

0,30

0,35

0,40

0,65 0,8 0,95 1,1

KT

J

KT

Coarse

Medium

Tunnel test

0,40

0,50

0,60

0,70

0,80

0,65 0,8 0,95 1,1

10

*KQ

J

10*KQ

Coarse

Medium

Tunnel test

54

𝑥

y

𝑣

𝜕𝑣

𝜕𝑦𝑑𝑦 𝑣𝑠

𝑢

𝜕𝑢

𝜕𝑦𝑑𝑦 𝑢𝑠

𝑢

𝜕𝑢

𝜕𝑥𝑑𝑥 𝑢𝑤

𝑣

𝜕𝑣

𝜕𝑥𝑑𝑥 𝑣𝑤

𝑢

𝜕𝑢

𝜕𝑥𝑑𝑥 𝑢𝑒

𝑣

𝜕𝑣

𝜕𝑥𝑑𝑥 𝑣𝑒

𝑣

𝜕𝑣

𝜕𝑦𝑑𝑦 𝑣𝑛

𝑢

𝜕𝑢

𝜕𝑦𝑑𝑦 𝑢𝑛

u,v

Appendix E Governing equations

The governing equations of computational fluid dynamics are the continuity equation, momentum

equation and energy equation.

Continuity equation

The continuity equation states that matter is conserved in a flow. The sum of all masses flowing in and

out of a given volume per unit time, must equal the change of mass due to change in density per unit

time.

Consider a small 2D fluid element as shown in the figure above. To simplify explanation the sides are

labeled n, e, s and w which stand for north, east, south and west. The fluid element is big enough to

ignore molecular movement and molecular structure of matter. However, it’s small enough to express the

fluid properties with enough accuracy by means of the two first terms of a Taylor series expansion. As an

example, the velocity u at faces west and east can be described as

2

and

2

(32)

Figure 53 - 2D fluid element with pressure and velocity

55

The two velocity equations are as follows. Notice that even though this is explained in two dimensions for

easier understanding, the third dimension (z-direction) is included and should be thought of as

perpendicular “into the paper”.

[

] [

] (33)

[

] [

] (34)

Added together results in

( ) ( ) [

] [

] (35)

[

] ⇒ (36)

For a compressible fluid flow in three dimensions the density must be considered and thus, gives the

following equation

( )

( )

( )

(37)

Momentum equation

Newton’s second law state that the rate of change of momentum of a fluid particle equals the sum of the

forces acting on this particle. The forces could be divided as surface and body forces, where surface

forces consists of pressure- and viscous forces, and the body forces are gravity-, centrifugal-, Coriolis-

and electromagnetic forces. Here, the momentum equation is separated into an inertial and a viscous

part for easier explanation.

Inertial term

The inertial term arise from momentum changes, and gives a measure of the change of velocity of a fluid

particle.

The momentum equation expressed from a physical viewpoint in the x-direction, based on figure (53)

( ) ( ) ( ) ( ) (38)

In mathematical terms this yields

[

]

2

[

]

2

[

] [

]

[

] [

]

(39)

[

] [

] [

] [

] [

] (40)

56

[

] (41)

By using the fact that the continuity equation (37) equals zero for incompressible fluids which is the case

here, enables the following substitution

[

] (42)

[

] (43)

Substituting the negative

by

gives

[

] (44)

[

] (45)

Simplifying this equation yields the x-direction inertia

[

] (46)

The inertial term of the momentum equation [3] [7]

[

] (47)

57

Viscous term

By adding shear stresses ( ) and pressure forces (p) to figure (53) result in figure (54) above. In the

same way velocity was expressed as Taylor series, this also applies to the pressure forces. The shear

stresses could be described as a deformation of the square representing the fluid element in figure (54)

into a parallelogram.

The shear stress are defined as follows, where is the dynamic viscosity

(48)

The magnitude of the force resulting from a surface stress (shear) is the magnitude of the stress and area which results in

[

]

[

] (49)

2

2 (50)

𝜏𝑦𝑥

𝜏𝑥𝑥

𝑥

y

𝑣

𝜕𝑣

𝜕𝑦𝑑𝑦 𝑣𝑠

𝑢

𝜕𝑢

𝜕𝑦𝑑𝑦 𝑢𝑠

𝑝

𝜕𝑝

𝜕𝑦𝑑𝑦 𝑝𝑠

𝑢

𝜕𝑢

𝜕𝑥𝑑𝑥 𝑢𝑤

𝑣

𝜕𝑣

𝜕𝑥𝑑𝑥 𝑣𝑤

𝑝

𝜕𝑝

𝜕𝑥𝑑𝑥 𝑝𝑤

𝑢

𝜕𝑢

𝜕𝑥𝑑𝑥 𝑢𝑒

𝑣

𝜕𝑣

𝜕𝑥𝑑𝑥 𝑣𝑒

𝑝

𝜕𝑝

𝜕𝑥𝑑𝑥 𝑝𝑒

𝑣

𝜕𝑣

𝜕𝑦𝑑𝑦 𝑣𝑛

𝑢

𝜕𝑢

𝜕𝑦𝑑𝑦 𝑢𝑛

𝑝

𝜕𝑝

𝜕𝑦𝑑𝑦 𝑝𝑛

𝜏𝑥𝑥

𝜏𝑦𝑥

Figure 54 - 2D fluid element with added pressure and shear stresses

58

This force affects the fluid motion in the x-direction by either accelerating or decelerating it and can be thought of “as packages” entering the fluid element from above or underneath. The normal stress is a function of pressure and shear stress ( ) (51) Where ( ) (52)

The normal stress in the x-direction

( ) (

( ) ) (53)

Substituting the pressure and velocity terms denoted as and with the terms in figure (54) yields

[

] [

]

[

]

[

]

(54)

[

] [

2

2 ] (55)

2

2 (56)

By adding body forces and putting equation (47, 50 and 56) together results in Navier-Stokes equations (x-direction) in two dimensions

(57) For a three-dimensional flow the Navier-Stokes equation can be written as follows [3] [7]

2

2 2

2 2

2

(58)

2

2 2

2 2

2

(59)

2

2 2

2 2

2

(60)

𝜌𝑋 𝜕𝑝

𝜕𝑥 𝜇

𝜕2𝑢

𝜕𝑥2 𝜕2𝑢

𝜕𝑦2

𝜌

𝜕𝑢

𝜕𝑡 𝑢

𝜕𝑢

𝜕𝑥 𝑣

𝜕𝑢

𝜕𝑦

Viscous terms Body force term

Pressure term

Inertial terms

59

Energy equation

The energy equation states that the rate of change of energy of a fluid particle is equal to the rate of work

done to the particle plus the rate of heat addition to the fluid particle. The equation is derived from the

first law of thermodynamics, which is a version of the law of energy conservation.

The work done to the particle comes from surface- and pressure forces, where the surface force is equal

to the force and velocity component in the direction of the force. From figure (54) and looking in the x-

direction the net rate of work done on the particle, can be described as

[ ( )

( )

] (61)

In a three-dimensional flow the total rate of work done on the fluid particle by surface stresses are

expressed in equation (62) below, where the pressure and shear stresses has been separated for easier

understanding.

[

] [

( )

( )

( )

] [

( )

( )

( )

]

[ ( )

( )

( )

]

(62)

The rate of heat addition to the particle can be described as the difference between heat transfer to and

from the faces of the particle. Again, looking at figure (54) in the x-direction from west to east, the

equation describing this is

[

]

(63)

The net heat transfer of the other two dimensions could be derived in a similar way, and by dividing these

equations with the volume, results in the total heat transfer to the particle in three dimensions. The

resulting equation yields

(64)

Fourier’s law of heat conduction states that heat transfer is proportional to the local temperature gradient.

The terms in equation (64) could therefore be written as

(65)

All put together and expressed in three dimensions, leads to the energy equation [3] [7]

[

] [

( )

( )

( )

] [

( )

( )

( )

]

[ ( )

( )

( )

] [

]

(66)

60

RANS

Continuation of section 1.5.1

2

2 2

2 2

2

(67)

[

2

2 ( )

] [

2

2 ( )

] [

2

2 ( )

]

(68)

Where U is the mean flow velocity and the added terms – , and – are stress terms

known as Reynolds stresses, where the first is a normal stress and the next two are shear stresses. Also

considering the y and z direction there is a total of three normal stresses and three shear stresses.

The added terms – , and – are stress terms known as Reynolds stresses, where the

first is a normal stress and the next two are shear stresses. Also considering the y and z direction there is

a total of three normal stresses

(69)

(70)

(71)

and three shear stresses

(72)

(73)

(74)

In three dimensions the complete set of RANS equations are

[

2

2 ( )

] [

2

2 ( )

] [

2

2 ( )

]

(75)

[

2

2 ( )

] [

2

2 ( )

] [

2

2 ( )

]

(76)

[

2

2 ( )

] [

2

2 ( )

] [

2

2 ( )

]

(77)

61

6 References

[1] Li Da-Qing – Validation of RANS predictions of open water performance of a highly skewed propeller with experiments

[2] Mitja Morgut, Enrico Nobile - Influence of grid type and turbulence model on the numerical prediction

of the flow around marine propellers working in uniform flow

[3] H K Versteeg, W Malalasekara (2007) – An introduction to Computational Fluid Dynamics – The

finite volume method, ISBN 9780131274983

[4] Ansys Theory Guide, version 12.0

[5] Shin Hyung Rhee, Shitalkumar Joshi – Computational validation for flow around a marine propeller using unstructured mesh based Navier-Stokes solver

[6] Y. Nakayama, R Boucher - Introduction to fluid mechanics, ISBN 9780340676493

[7] A. Sayma - Computational fluid dynamics, ISBN 978-87-7681-430-4

[8] David C. Wilcox (2000) - Turbulence modelling for CFD, 2nd

edition, ISBN 0-9636051-5-1

[9] DJ Tritton – Physical Fluid Dynamics, ISBN 9780198544937

[10] Johan Lundberg – CAE Group Leader at RRHRC

[11] 23

rd ITTC – Volume I – Chapter 6. Review methods for scale effects on the passive components of

propulsors and for assessing screw propeller scale effects with emphasis on the occurrence of excessive laminar flow

[12] Mitja Morgut, Enrico Nobile – Comparison of Hexa-structured and Hybrid-unstructured meshing approaches for numerical predictions of the flow around marine propeller

[13] F.R. Menter – Two-equation eddy-viscosity turbulence models for engineering applications

[14] Y. Nakayama, R Boucher - Introduction to fluid mechanics, ISBN 9780340676493

[15] J. Blazek (2001) – Computational fluid dynamics: Principals and applications, ISBN 9780080430096