Proceedings of the PRADS2013

20-25 October, 2013

CECO, Changwon City, Korea

Calculation of unsteady inflow conditions for

ruder vibration analyses

Wilfried Abels,

Hamburg University of Technology, Germany

Abstract

During the early ship design stage there is a demand for

calculation methods with a short response time, to be

able to calculate the effects of design decisions. Hydro-

dynamic design tools based on potential flow theory

fulfill these requirements. In the past a lot of work has

been done to develop analytical methods with a high

degree of mathematical modeling. This was necessary

due to the lack of computer power.

Nowadays these theories can be used as very fast analy-

sis tools during the early design stage. The unsteady

lifting line theory of Zwick(1962) is an example for such

a method. The method can be used to calculate the slip-

stream of a propeller in behind condition.

The mathematical description of the theory has been

implemented in Fortran and has been integrated in the

ship design framework E4.Within this framework the

tool now can be used to calculate unsteady interaction

between wake field, propeller and rudder.

Keywords

Propeller; wake, ruder; potential theory; lifting line;

early design

Introduction

The handling of physical/numerical models is a main

task during the development of a technical product. An

engineer needs reliable information to design new inno-

vative products. Resulting from the keen competition

within the shipbuilding industry, there is a significant

interest, in having tools to analyze new design features.

Further, the information must be available in a short

timescale. The concept of “Design in Seven Days”

(D7D) needs software tools, which delivers the needed

information fast and reliable, Krüger(2003). For aft ship

and appendix design the interaction between wake and

propeller is of importance. Beside the question of effi-

ciency, questions of pressure pulse, ruder forces and

vibrations are in the focus. Wrong decisions in early

design can result in difficult and expensive problems. It

can be a significant competitive advantage to have tools

during the early design, which are able to assure a de-

sign decision.

Tools which are able to do so have to be adjusted to the

relevant physical features. Often it is not necessary to

analyze the physic in a global way. For the design tools

this mean, a small qualitative model, which describes

the relevant effects, is much more useful than a more

complex model, which describes many different physi-

cal effects. Beside the argument, that complex models

need more computation effort, which is often not avail-

able, it is very difficult to evaluate the reliability of such

models. As more effects are modeled, more parameters

has to evaluated. From the view of an engineer, this

means more uncertainties. A model with well-known

simplifications and a known reliability can be more

useful, than a more detailed model with a not exactly

known behavior.

This means for the development of tools for the early

design stage, that the relevant physical features needed

for the design process must displayed correctly. The

modeling of physic and a practical analytic description

is important and has to be done with accuracy. In a

second step the model has to be translated in a numeri-

cal description, which can be implemented in software

tools. In the past, especially from begin to middle of the

last century there have been done many high sophisti-

cated analytic investigations in different aspects of hy-

drodynamic theories before powerful computers exist. It

was a time without high computer performance. It was

essential to handle mathematical models by hand with-

out computer support. Consequently there was a strong

focus on using analytic technics to solve problems of

mathematical models. A numerical discretization of

equations was only the absolute last solution and has

been only done in the case of no other possibilities.

Beside the fact, that such theories need a complex math-

ematical description, the great advantage is the low need

in calculation performance.

This feature is nowadays very useful during the early

design. Such methods are able to handle many different

design variations in a short time period. They are fast

enough to be implemented within the short time scale of

the workflow during the early design. Calculation meth-

ods which need more as about one day of computation

time are useless, because it is feasible that the actual

ship design differs already from the start point of the

calculation. The former restriction to methods with low

calculation demands is nowadays a great benefit for the

aim of analyzing many different design variations with-

in a short time.

This paper shows an approach to simulate a rotating

propeller in a given wake. The aim is to analyze effects

of propeller and the unsteady slipstream during the early

design. Up to now, the theory has been only tested for

the simplified propeller (Fig.1), qualitative effects of a

specific wake are taken into account and can be already

used to figure out the influence of wake to the unsteady

lift distributed on the blade.

A significant aspect is to handle the free vortexes down-

stream the propeller. Classical lifting line approaches

can only handle a homogenous inflow, Isay(1964).

Modern full viscous flow solvers based on a three-

dimensional domain, have still problems to conserve the

free vortexes and secondly they are time consuming. A

further approach is the QCM (Quasi-Continuous Meth-

od), which based on a element theory of the blades and

wake panels to model the free vortexes, Streck-

wall(1997) and Abels(2006). A problem accurse if the

wake panels of the propeller and the panel description

of the rudder collide with each other.

To avoid these problems, an unsteady lifting line theory

has been used, which describes the propeller blades as

bounded vortexes. For solving the boundary condition,

an integration of the law of Biot-Savart has been inte-

grated from the bounded vortex to infinite downstream

the propeller. Because this integration has been done

analytical, the whole circulation downstream could be

taken into account without usage of discrete wake pan-

els. This analytic handling of free vortexes allows to

stay compatible with other panel based potential theo-

ries.

An Overview about the unsteady potential pro-

pulsion theory

The unsteady lifting line theory used in this paper was

developed by Zwick (1962). The flow had been modeled

as a potential flow and a 3-dimensial distribution of free

vortexes downstream the propeller. There is an incom-

pressible, source and sink free fluid. The propeller and

wake are described in cylindrical coordinates��̅, ��, ���.

The propeller has a constant rotation of � 0 and it

moves with a velocity � � 0 in the negative direction

of the ��-Axis. The propeller blades are working in a

wake ��� , ��, ��� � ���̅, ��, �̅ � ��� with a period of 2� in�. From view of a cylindrical coordinate system

fixed on the propeller blades, the propeller works within

a local flow ���, � �, �� �0, �, ��. Additionally

the propeller has � blades, the hub is at �� and the tip

at��. The cord length is 2�� with � � 2� � !"#�.

Fig.1: Propeller model, Zwick (1962)

The model of the propeller is from a relative definite

character. The blades are modeled as cylinder sections

and an a constant pitch, Fig.1. In contrast to this, the

wake is modeled with axial and radial components,

Fig.2

Fig.2: axial and radial components of the used wake,

Zwick (1962)

In spite of a simple lifting line, this unsteady theory

could be used to analyze the change of circulation on

the blade due to the wake. This mean, effects resulting

from the wake, could be investigated in a qualitative

way. The propeller is described as a fixed vortex and

free vortexes drain off downstream. The mathematical

model based on an analytic solution of the law from

Biot-Savart for the fixed and free vortexes at the ¼

point at – �� and the flow constraint is solved for the ¾

point at �� . If now & � �&� , &� , &�� � &��, �, �, �� the following equation has to be solved:

� � �� � &� '��� ∙ � ∙ � � �� �&��� 0

)!�+,-,. �� / � / ��� � � ∙ '��� ∙ �� � 2�� ∙ 0 � �0 � 0,� ∈ R 3,4

,5

(1)

Now the propeller can be modeled as a circulation

distribution in the following form:

��, �� � à 78, 2�� 0 � �9à 7��, 2�� 0 � �9

� à 7��, 2�� 0 � �9� 0

(2)

By using such a circulation distribution it is in principle

easy to describe the free vortexes downstream the

propeller. Resulting from the law of conservation of

rotation, the free transversal circulation is defined as:

:à ;8, 2�� 0 � �<:8 =8 (3)

The direction is �0,1, '�8�� . In the same manner the

free longitudinal circulations are defined:

:à ;8, 2�� 0 � �<:� =� (4)

The direction is in this case �1,0,0� . To describe the

induced velocities, the law from Biot-Savart has to be

used:

� Γ2�� (5)

In Principle it is easy to describe the unsteady flow by a

combination of the equations (1) to (5). The

mathematics becomes a bit complicate because of the

cylindrical coordinates and the geometrical description

of the law of Biot-Savart. The mathematical solution in

the way of Zwick(1962) is shown exemplified in

appendix B. To calculate the induced velocities on the

blades, it is necessary to sum up the integration about

the radial axis of every blade and to do an infinite

integration in direction of the helicoid along the free

vortexes:

&? � @ A A :Γ�⋯ �:8 ∙ ��⋯ � ∙ =C=8DEF�

GHIFGJ

KLMNF�

&O � @ A A :Γ�⋯ �:C ∙ ��⋯ � ∙ =C=8DEF�

GHIFGJ

KLMNF�

(6)

The solution has been done by a numerical

approximation of the circulation Γ in the following

form.

Γ∗�#, Q� � @ RST

SFLT �#� ∙ UVSW (7)

RS�#� � @RS,V ∙ )�" �#�XVFM

# � ��Y ; Q � � �

The coefficients RS,V can be calculated by a set of

systems of linear equations:

@[S,\,V ∙ RS,V � ]S,\X

VFM (8)

In equation (8) the Matrix [S symbolizes the effects of

the free vortexes and ]S represents the effects of the

wake. If these coefficients are known, it is easy to solve

these systems of equations. The size of the linear

equation systems is about 10, which is easily handled by

modern computers. In contrast to other methods the

numerical effort to solve the linear equations is nearly

irrelevant.

In a next step it was necessary to transfer this mathemat-

ical description to numerical model implemented. This

system of mathematical equations has been transformed

in a numerical FORTRAN method. The way this has

been done is explained in Abels(2011).

Fig.3: Examples of calculated circulation ^∗�_� at ` � a, bc�defgh�; i � a, jk�efllfm�

The numerical method

The method described above is integrated into the ship

design tool E4. The wake from Zwick(1962) (Fig.2) has

been digitalized. It has been used as input for the pur-

pose of evaluation. The evaluation of the implemented

method has been done with help of the published circu-

lation. The aim was to reproduce them as good as possi-

ble. The results are presented in the Fig.3 and they are

in a good accordance with the original calculations from

Zwick(1962). Theoretical the results should be the same,

but in practice it is complicate to reproduce calculations

done 50 years ago, much information has been lost over

the years. The original data has been taken from small

pictures copied out of an old publication and had been

digitalized. Beside the fact that this procedure contains

an information loss, the data is not complete. As seen in

Fig.2 information is only available at particular radial

points. Everything between has to be interpolated. Con-

sequently the real wake used for the former calculation

is unknown. The number of original computing nodes is

unknown as well as the way of interpolation and the

way of integration carried out by the technical staff.

Next, the method has been compared to the QCM. This

method has been already proved as useful for the pur-

pose of design process. The kt over J calculation has

been done for different blade numbers (Fig.5). The

results show a good coincidence with the QCM if the

blade number is higher. This is not astonishing because

this behavior is already known from the steady lifting

line theorie. To solve the integrals of the free vortexes

downstream it is common to distribute the dedicated

free vortexes homogenously over the pitch of one pro-

peller revolution. This approach has been used by

Zwick for the unsteady method in the same way as it

has been done by the steady methods.

The effect of this approach is that the propeller behaves

similar to a propeller with infinite blade numbers, where

the circulation is distributed homogenously over the

smooth propeller plane. In contrast, the QCM has dis-

crete vortexes for blades and wake panels. The steady

lifting line method use the Goldstein factor to reduce the

overestimated thrust of a propeller with a definite num-

ber of blades, Goldstein(1926). At the moment such an

unsteady Goldstein factor does not exist. To develop

such an unsteady Goldstein factor will be done in the

future.

Nevertheless the classical steady Goldstein factor has

been used to test the effect to the Zwick method. The

Fig.5 shows that a kt reduction is necessary for smaller

blade numbers and that a Goldstein correction is help-

ful.

Calculation of the unsteady slipstream

Beside there are some principle questions resulting from

the mathematical modeling there is now a very fast

method available to calculate an unsteady flow down-

stream of the propeller plane without wake panels with

discrete singularities. The bounded circulation of the

blades as descript in equation (7) can be calculated in

some milliseconds. Carried out at a normal desktop

computer the calculation time is not observable for a

human. After the coefficients no,p are available the flow

can be calculated all over the wake and for any point in

time.

Fig.5: Kt values of QCM, ZWICK and ZWICK with

goldsteinkorrektion for propeller with blade number of

N=4 to N=5

As the mathematical model does not need discrete sin-

gularities downstream the of the propeller plane, the

method can be with other potential theory models. As

discussed in the introduction the future aim is to get a

method for unsteady ruder force calculations. Although

at the moment the direct mathematical connection has

not been realized, it is already possible to do qualitative

investigation of unsteady effects at different positions

downstream the propeller plane. This information can

be used as input for an existing rudder panel method.

By calculating the induced velocities of the fixed and

free vortexes for the relevant time steps and positions a

complete unsteady wake field can be generated. In ap-

pendix A Fig.7/8 the calculated wakes are plotted for

different angles of the propeller at the position y=0.0

and y=0.3*Ra downstream. One can get an impression

of the unsteady effect of the fixed blade vortex at the

propeller blade at y= 0.0. But still at position y=0.3*Ra

the effect is visible.

Fig.6 the different parts of induced velocities at position

s=0.3; phi=0; y=0.3

More detailed Fig.6 shows the effect of the propeller,

where the different parts of the fixed and free vortexes

are plotted over the time. It is interesting that the in-

duced velocities of the free vortexes (ViQ and ViL)

have no time dependency. But this is also an effect

resulting from the approach of free vortexes which are

distributed homogenously over the pitch of one propel-

ler revolution. This mean only the bounded circulation

on the propeller blade has part in unsteady effects to the

induced velocities. Therefor the time depending effects

of the propeller decrease downstream, which can see in

the Fig.7/8 of appendix A.

Conclusions

By the usage of an analytic unsteady lifting line theory

it was possible to implement a robust and fast method

for analyzing the wake in behind condition of a working

propeller. The problem of free vortexes downstream has

been solved by an analytic approach. Discrete wake

panels with discrete singularities are not necessary, and

the method is open for many future extensions.

The method calculates a small set of coefficients which

describe the distribution of bounded circulation on the

blade with in the time domain. Afterwards the usage of

the law of Bio-Savart allows calculating induced veloci-

ties at every position and points in time.

By taking into account the simplifications within the

theoretical model, there is now a tool available which is

able to help the engineer during the narrow time sched-

ule of the early design. Especially questions about un-

steady effects within a wake downstream a working

propeller can be analyzed in a new way.

References

Abels, (2006), „ Zuverlässige Prognose propellererreg-

ter Druckschwankungen auf die Außenhaut mittels

Korrelation direkter Berechnung”, ISBN 3-89220-

636-8, Schriftenreihe Schiffbau, Hamburg

Abels, (2011), „Modelling physics by usage of a math-

ematical symbolic solver for transferring analytics

theory to numerical First-Principal-Methods”, 10nd

Conf. Computer and IT Applications in the Maritime

Industries (COMPIT), Berlin

Goldstein, (1926), “On the Vortex Theory of screw

Propellers”, Kaiser Wilhelm Institut für Strömungs-

forschung, Göttingen

Isay, (1964), „Propellertheorie“, Springer Verlag

Krüger, (2003), „The Role of IT in Shipbuilding”, 2nd

Conf. Computer and IT Applications in the Maritime

Industries (COMPIT), Hamburg

Streckwall, (1997), “Description of a Vortex Lattice

Method for Propellers in Steady and Non Steady

Flow”, Hamburgische Schiffbau-Versuchsanstallt

GmbH, Report 18/97

Zwick, (1962), „Zur Berechnung der Zirkulation und

der Kräfte eines Propellers im Nachstrom“, Schiff-

bauforschung 14/1962, Berlin

Appendixes A

Fig.7 wake at the propeller plane Yst=0.0 Fig.8 wake at the propeller plane Yst=0.3

Appendixes B

&q∗�#, �, �∗, �� � @ A Γ∗ ;r, stK 0 � �< ∙ u ;#, r, stK 0 � �, �∗ vw∗ <MxFy�

KLMNF� (9)

u��, 8, z, �� � M{t �� ∙ sin�z�, � ∙ !#� z�, � ∙ sin�z���s � 8s � �s 2�8 ∙ cos�z��� s�

(10)

&O∗�#, �, �∗, �� � @ A A :Γ∗�r, C ��:C ∙ �∗�#, r, C �, �∗� ∙ =C=rstEF�

1xFyJ

KLMNF�

(11)

�∗�#, r, z, �∗� � �8�s#'∗�#� � �0,0, # ∙ sin�z��#s � rs 2#rcos�z� �1 � �∗ vw∗�#s � rs � ��∗ vw∗ �s 2#rcos�z������ �sin�z� , cos�z� , 0�

�#s � rs � ��∗ vw∗ �s 2#rcos�z�����

(12)

&K∗�#, �, �∗, �� � A A :Γ�r, C ��:r ∙ )∗�#, r, ψ �, �∗� ∙ =C=rstEF�

1xFyJ

(13)

)∗�#, r, z, �∗� � �r8�s#'∗�#� ��r'∗�r� sin�z� , #'∗�r� cos�z� , r #cos�z��#s � rs 2#rcos�z�∙ �1 � �∗ vw∗�#s � rs � ��∗ vw∗ �s 2#r cos�z����� �cos�z� , sin�z� , 0�

�#s � rs � ��∗ vw∗ �s 2#rcos�z�����

(14)