Scientific Bulletin of the Workshop on

Politehnica University of Timisoara Vortex Dominated Flows –

Transactions on Mechanics Achievements and Open Problems
Special issue Timisoara, Romania, June 10 - 11, 2005

DEVELOPMENTS IN THE DESIGN OF SHIP PROPELLER

Mihaela AMORĂRIŢEI, Lecturer,
Naval Hydrodynamic Department
“Dunarea de Jos” University of Galati
*Corresponding author: 47 Domnesca Street, Galati, Romania
Tel.: (+40) 236 495400, Email: mamor@ugal.ro

ABSTRACT bearing forces induced by propeller are related to

The paper presents aspects regard propeller design vibrations.
procedure, which involves theories and underlying The design of screw propeller in non-uniform
assumptions, analytical tools, computational fluid flow behind ship can be carried out in three stages:
dynamics models and model tests, to predict the preliminary design, design and analysis. Once the
hydrodynamics performances of marine propeller in design point is chosen and the main parameters are
non-uniform wake field behind ship. A properly fixed, the problem is to design a propeller to give
design propeller is a compromise between structural specified performances in given conditions. In pre-
and hydrodynamic considerations. The complex liminary design the traditional propeller diagrams
nature of the design and operation of marine propeller are used, and the parameters estimated (diameter,
requires knowledge of basic hydrodynamics, naval number of blades) are a starting point for next stages.
architecture and typical experience. The second step, design, known like “indirect”
problem, can be done using the lifting line theory
KEYWORDS
with correction factors on lifting surface theory and
Propeller, standard series, lifting line theory, the objective is to find the blade geometry for a
lifting surface theory, panel method, RANS specified distribution of blade loading over the radius.
Once the design is completed, the propeller is analysis
NOMENCLATURE
in all operating conditions: this is the third stage,
J [-] advance ratio known like ”direct” problem, and the objectives are
kT, kQ [-] thrust, torque coefficient to find the pressure distributions on propeller surfaces
CT, CP [-] thrust/ power loading coefficient
and to evaluate the hydrodynamics performances of
Q [kN] propeller torque
propeller in off-design conditions. In the design stage,
T [kN] propeller thrust
the performances of the propeller are predicted at
1. INTRODUCTION the design point, which corresponds only to a mean
flow. The real flow in the propeller plane behind
In recent years a drastic increase in power and hull is non-uniform, the velocity changes magnitude
ship speed has been observed for all kind of vessels.
and direction at each propeller revolution, which
This new trend demands propulsion devices designed
causes continues and cyclic fluctuation in blade
to give maximum efficiency and to absorb minimum
loading and pressure distribution [1].
power, with minimum cavitation, noise and vibrations.
The most common propulsion device is the screw The designer must analyses the propeller’s behaviour
propeller, which convert power in thrust and play an in unsteady flow taking into account aspects regard
important role in the interaction between ship and cavitation and fluctuations of unsteady forces and
the main engine. moments arising from operation in non-uniform hull
The design of a propeller operating in non-uniform wakes induced by the propeller and transmitted to
flow behind ship is an iterative process to optimise the hull through the water by pressure effects and
the propeller efficiency with less restrictive constrains thought the shaft bearing.
concerning cavitation, noise, vibrations geometry The analysis of a propeller operating in non-uniform
and strength. The propeller is an important source of flow behind ship can be carried out experimentally
noise and vibration; for the performance of the ship, and theoretically. Taking into account that the experi-
cavitation is related to noise and pressure pulses and mental tests in towing tanks and cavitation tunnel are
132 Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005

time consuming and expensive, more sophisticated The average wake velocity over the propeller disc is:
three-dimensional theories were been developed steady R R
and unsteady lifting-surface theory, quasy-steady v = 2π ∫ r vr dr / 2π ∫ rdr (2)
methods, Reynolds-averaged Navier-Stokes (RANS) r r
b b
equations, boundary element methods/panel method. A velocity fields as function of radius and angular
The paper presents aspects regard propeller design position is presented in Figure 1 and 2.
procedure, which involves various theories and under-
lying assumptions, analytical tools, computational fluid
dynamics models and model tests, to predict the
hydrodynamics performances of marine propeller in
non-uniform wake field behind ship.
2. HULL- PROPELLER INTERACTIONS
A propeller fitted at the stern of a ship operates in
water that has been disturbed by passage of the hull’s
ship, which “deform the original streamlines and
causes a retardation of the relative stream velocity due
to viscous actions”[2]. This disturbance behind the
ship is called wake. The wake velocity is associated
with the flow around ship’s hull and it varies in
magnitude and directions. The ratio of the average
velocity over the propeller disc to the ship speed V Figure 1. Curve of constant axial wake fraction.
is named the wake coefficient: w = 1-v/V, and its Transversal velocity in propeller disc
value depends largely of the shape of the hull and on Va/V
the propeller location. The rotation of the propeller 1

alters the pressure and velocity distribution around 0.9

the hull and increase the resistance of the ship. This 0.8

0.7
means that the thrust force T on the propeller has to
0.6 0.4
overcome both the ship’s resistance R and this increase 0.5 0.5
named “augment of resistance” RT. This loss of thrust 0.4
0.6
0.7
is expressed by mean of t = 1-R/T, named thrust 0.3 0.8

deduction factor. The efficiency of the propeller 0.2 0.9

1
operating behind ship is different from its efficiency in 0.1

0
open water. Compared to open water conditions, the 0 60 120 180 240 300 360

propeller’s efficiency behind ship’s hull is affected Unghi

by “relative rotative efficiency” ηR. Figure 2. Axial velocity distribution

The flow behind ship is not uniform over the pro-
eller disc and the inflow velocity to the propeller has A useful presentation of the wake data field exploit
three radially and circumferentially varying compo- the cyclic variation of this patterns and writes for the
nents: an axial component along the axis of the pro- tree components of velocity:
peller, and a tangential and a radial component in va v r m
, = A 0 + ∑ A m cos ( mθ - β m )
the plane of the propeller disc. As a propeller blade V V 1
rotates, a section at any given radius passes through (3)
vt m
regions of very different wake concentrations. These = ∑ A mcos ( mθ - β m )
variations are the cause of unsteady cavitation and V 1
cyclic fluctuations in blade loading and pressure dis- The wake harmonic functions of interest are those
tributions. The tangential velocity components are of multiple of blade number qz for thrust and torque
very important when considering unsteady propeller on the shaft and those at qz ±1 for transverse and
forces, while the radial components are generally small. vertical forces and moments [3].
The axial velocity “v” varies from point to point A distinction must be made between nominal wake
over the propeller disc, function of radius r and angular and effective wake. The nominal wake is the wake
position θ. The average velocity at a radius r is: behind ship’s hull in absence of the propeller. The
2π wake velocities with the propeller operating behind
1
vr =
2π ∫ v (r,θ )dθ (1) ship and developing thrust is named effective wake.
0 Presently, the knowledge of the distribution flow in
Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005 133

propeller plane is based on experimentally measure-

ments. The nominal wake velocities are measured at
model scale using Pitot tubes. The concept of effective
wake implies that the influence of the propeller action
on the stern flow is incorporated and the effective wake
can’t be measured. Some total wake distributions
have been measured at full scale using LDV (Laser
Doppler Velocimeter). The radial distribution of
axial velocity components is transformed from the
nominal (without propeller) value for the model to an
effective (with propeller) value for the full-scale ship
by an indirect method based upon thrust (or torque)
identity: making the thrust coefficient KT (or torque Figure 3. Ship propulsion running point [6]
coefficient KQ) in open water and behind ship equal,
at same axial speed and rotation rate. The notations in Figure 3 are: 2 - heavy propeller
Theoretically, the effective full-scale wake distribu- curve (fouled hull and heavy weather); 6 - light pro-
tion velocity field can be obtained in two steps: first peller curve (clean hull and calm water); MP - specified
the nominal wake measured behind the ship model propulsion MCR point; SP - service propulsion MCR
is corrected for scale effect using the tree-dimensional point; PD - propeller design point; PD’ - alternative
contraction method proposed by Hoekstra [4]. To derive propeller design point
the effective wake distribution from the scaled nominal The main particulars characteristics of the propeller
wake field, Huang’s method can be used [5]. The ef- are usually determined by means of systematic pro-
fective velocity field can be obtained by subtraction peller series based on the results of open-water tests
of the propeller-induced velocities from the total carried out on model propellers: Wageningen B-series,
velocities fields behind a ship with running propeller. Gawn, etc. These screw series comprise models whose
The most reliable values of the hull-propeller inter- geometrical characteristics such: pitch ratio, number
action coefficients: wake coefficient, thrust factor of blade, blade area ratio, shape of blade sections and
and relative rotative efficiency will be found from blade thickness are systematically varied. The hydro-
preliminary self-propulsion model tests, in which a dynamic characteristics of standard series are pre-
model of new ship is propelled by a stock propeller sented in the form kT,kQ - J charts. Using the method
with principal characteristic as near as probable final of the multiple linear regression analysis for each of
design. series, polynomial equations for the series hydrody-
namic characteristics are obtained:
3. PRELIMINARY DESIGN Q x y z
KT , KQ = ∑ Ak (z ) k (J ) k ( P/D) k (Ae/Ao ) k (4)
At this stage of the design, the problem is to deter- k
mine propulsive performances of screw propeller and where Ak are regression coefficient and xk, yk and zk
the main characteristics of the propeller to achieve are the correspondent exponents of the independent
the expected performance: diameter D, number of variable J, P/D, Ae/A0. The propeller of optimum
blade z, mean pitch P/D, blade area ratio Ae/Ao. The efficiency can be automatically estimated using com-
preliminary design requires dates from the hull of the puted codes based on standard series.
ship, the main engine and systematically screw series. In general, higher propeller efficiency is associated
The main engine influences the propeller design with a larger propeller diameter and a lower shaft rpm.
through the propeller rpm and delivered power. The It is usually desirable to install the largest diameter
values of total resistance of the ship play a significant than can be accommodated to the hull lines. There are,
role in the selection of the propeller, which must however, special conditions to be considered: the after-
overcome ship’s resistance. Dates regard hull-propeller body form of the hull depending of the type of the ship,
interaction: the effective wake coefficient, the thrust the necessary clearance between the tip of the pro-
coefficient, the relative rotative efficiency and any peller, etc. When the propeller diameter corresponding
restrictions such as a limit of the maximum diameter to optimum efficiency for propeller-ship system is
of propeller may be helpful. larger than can be accommodated, in these cases the
Prior to the preliminary design of a propeller is the propeller diameter selected is a compromise.
choosing of the propeller design point (Figure 3): com- In preliminary design, an important step is the choice
bination of engine speed and power, which depends of number of propeller blades. Propellers may have
upon so called “mission profile” of the ship: cruising three, four, five or more blades. If the number of blade
on long distances at middle speed, running shortly at increases, the optimum diameter and the open-water
high speed, etc. efficiency decrease. From the point of view of effi-
134 Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005

ciency, is a preference for a small number of blades. gral along the vortex line. The second way to establish
But the major criterion in the selection of the number this velocity field uses the Laplace’s differential
of blades is vibration considerations. The fluctuations equation, which applies since the flow outside of
of unsteady forces induced by the propeller and vortex space is a potential flow and the problem is
transmitted to the hull through the water by pressure reduced to a boundary problem of a linear partial
effects and thought the shaft bearing are lower for a differential equation [9].
larger number of blades. Not only the vibration A propeller that is rotates in the water induces three
excitation is to be considered, also attention is to be velocity components: axial, tangential and radial.
paid to the resonance structure. The choosing of the Certain assumptions are required: one of them is
number of blades, which has a common factor with that there is no contraction or reduction in diameter
the number of cylinders of the Diesel engine, could of the slipstream [10]. The radial component of the
lead to vibrations problems [7]. The expanded blade induced velocity can be ignored and the other two com-
area is a result of optimization and may be restricted ponents, axial uA and tangential uT must be calculated.
by cavitation criteria (for example Burill diagrams). From the circulation distribution, the induced velocities
The performances of a propeller designed based can be computed based on the Lerbs induction factors:
on the systematic series are insufficient for today’s ua 1 R dG dr0
expectation: the results give good agreement between =∫ ia ;
VA 2 r dro r - r0
shaft power, propeller revolution and ship speed and b (6)
will be used as a starting point for the next stages. uT 1 R dG dr0
= ∫ iT ;
4. DESIGN VA 2 r dro r - r0
b
When using systematic series charts, no account is where VA is a average, nominal axial wake velocity
taken of the variation of the wake over the propeller determined by means of measured axial wake ve-
disc and the propeller is designed to suit average flow locity components Vx (r, θ):
conditions behind ship [8]. Once the main parameters 1 1 2π
are fixed, the problem is to design a propeller to devel- VA = ∫ dr ∫ Vx (r, θ) (7)
oping thrust or absorbing the specified power at given 0π o
rpm, under given conditions. The inflow to the pro- and ia and iT are the Lerbs induction factor which
peller is assumed to vary radially and the objective are expressed in a Fourier series.
is to find the blade geometry for a specified distribution
( )
¥
of blade loading over the radius. The pitch of the ia j, j0 = ∑ Ian ( j) cos (nj0 )
sections can be chosen to suit the average wake at n=0 (8)
( )
¥
each radius (wake adapted propeller) and the shape iT j, j0 = ∑ I nt ( j) cos (nj0 )
of the blade is chosen to minimize cavitation. The n=0
problem can be done using the circulation theory Propeller design using circulation theory is divided
(vortex theory) of propellers: lifting line theory with in two parts. The first part named hydrodynamic stage
correction factors on lifting surface theory. consists on determining the values of non-dimensional
In lifting line theory, each blade of propeller is re- circulation Γ and the induced velocities uA and uT. The
placed by a bound vortex or lifting line, the circulation second stage consists on determining the optimum
Γ of which depends on the radial coordinate r. The blade geometry from the point of view of cavitation
variation of Γ necessitates a free vortex line being suppression at the shock-free angles of attack and
shed from the lifting line. The free vortex line with a strength criteria. Lifting line calculation is always made
circulation distribution (Γ/dr)dr is not acted by forces. in steady conditions. The mean value of the wake
The assembly of adjacent free vortex lines forms a (average wake velocity over the propeller disc) and
free vortex sheet (trailing vortex sheet) helicoidal in the average velocity over one revolution at different
shape. radius are known. The required thrust or delivery
The velocity induced by the vortex system of the power has to be specified.
propeller can be determinate by the law of Biot Savart The diagram of velocities around a blade section at
or by Lapace’s equation. Using the Biot Savart law, radius r is presented in Figure 4, where α is the attack
the velocity vector induced by a vortex line of cir- angle, β the advance angle, βI the hydrodynamic pitch
culation Γ at a point in space is: angle, δ the final pitch angle.
Γ dl x R a
From the velocity diagram, the relation between
VP = ∫ (5) induced velocities uA and uT is:
4π l R 3
uA uT tgβ
+ tgβ i = i -1 (9)
where R is the vector distance between the point and
VA VA tgβ
the vortex vector dl. The problem is reduced at an inte-
Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005 135

NACA66. The thickness distribution must satisfy a

classification society class, a linear variation of blade
thickness is often adopted.
One of the major defects of the lifting line theory
is that the propeller blade is represented by a vortex
line or lifting line. The induced velocities are evaluated
only one point on the lifting line. Since the propeller
blades are like lifting surfaces, have a finite thickness
and operate in a viscous flow, it is necessary to correct
the value of angle of attack and chamber ratio to account
for lifting surface, thickness and viscous effects. Lifting
Figure 4. Velocity diagram surface correction may be made using the factors
For specified values of βI at various radii the values due to Morgan[13]. The factor correct ideal inflow
of dimensionless circulation G are calculated substi- angle and chamber by:
tuting relations (6) in (9). Each of the induction factors α ( r ) = kα ( r ) ×αi + k t ( r ) × to/D
pl
and circulation are expresses in Fourier series and a (13)
set of linear equations can be solved to calculate the fmax (r) = kc (r) fplmax
values of circulation at various radii. where kα, kt and kc are lifting surface correction factors
After the problem of circulation and induced ve- to the angle of attack, for thickness, respectively for
locities is solved, the ideal thrust loading coefficient chamber ratio.
CTi and ideal power coefficient CPi are calculated by The pitch angle will be:
relations:
(r ) = β (r ) + α ( r ) (14)
1 2  1 uT  i

r
(
CT = 4z ∫ G 1- w ( r )  -
 tgβ V  )
 dr A suitable distribution of skew to define the ex-
i b  A panded blade may be selected at this stage. Theoretical
3 (10)
1 G (1- w ( r ))  u  and experimental data show the advantages of highly
C P = 4z ∫  1+ A  dr skewed propellers compared with conventional pro-
i r tg β  V 
b  A  pellers. The advantages, in general are reduction in
unsteady bearing forces and pressure forces and
Iteratively, the hydrodynamic pitch angle β i is ad-
increased cavitation inception speeds. The reason
justed to match the ideal thrust loading coefficient (or
for the decrease of unsteady bearing forces with
the ideal power coefficient) to the required values:
increasing skew can be found by examining the ship
8T P wake. The purpose of skewing a blade is to allow
CT = i ,C i (11)
P =
i π × ρ × V 2 × D2 i π × ρ × V3 × D 2 each radial section of the blade to enter the wake at
The relation between the dimensionless circulation a different instant, thereby reducing the peak forces.
G and the lift coefficient CL is: The effect of skew on unsteady forces and moments
depends on the wake’s structure; an arbitrary skew
CL × c 2 π × G × cosβi use without consideration of the wake structure could
= (12)
D 1 uT lead to very disappointing results [11]
-
tgβ VA
i 5. ANALYSIS
where c is the chord length of blade section at r Once the design is completed, the propeller is analy-
radius and D-propeller diameter sis in all operation conditions taking into account the
When the final values of Γ, uA, uT, hydrodynamic complete wake distribution. This is the third stage,
pitch angle βI and CLc/D are determined, the geometri- known like ”direct” problem, and the objectives are to
cal design can started. The lift coefficient depends find the pressure distributions on propeller surfaces,
on upon the type of airfoil section, its chamber ratio, to evaluate the hydrodynamics performances of
thickness chord ratio and the angle of attack. The propeller in off-design conditions and to determine
problem is to select a combination of chord length, how the ship’s wake influences the cavitation per-
chamber, blade contour and pitch to match the data formances and the unsteady forces induced by the
from hydrodynamic design. The pitch is chosen propeller and transmitted to the hull through the water
according to the hydrodynamic pitch to attain shock by pressure effects and thought the shaft bearing. Some-
free entry, and the chord length c must satisfy cavitation times, the pressure distribution is taken as an indication
and strength criteria [2],[12]. of the behaviour of the cavitation on the blades.
The airfoil sections generally used in propeller In the design stage, the hydrodynamic performances
designed using the circulation theory are NACA 16, of the propeller are predicted at the design point,
136 Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005

which correspond only to a mean flow. The real where :

flow is not uniform, the inflow velocity to the pro- ξ = x - fcosφ
peller has radially and circumferentially varying
θ = φ + f/rsinφ (16)
components, as a propeller blade rotates, a section at
any given radius passes through regions of very ρ=r
different wake concentrations and may therefore and (x,r,φ) are the coordinates of the helical line and
give rise of unsteady loading.
(ξ,ρ,θ) are the coordinates of the mean line, φ -
The analysis required a detailed geometrical descrip-
pitch angle and the pitch of the helicoidal surfaces is
tion of the propeller, the effective wake distribution
and the operational conditions of the propeller. The P = 2πr tgφ = 2π/a.
purpose of analysis is to study the propeller’s behavior The boundary condition at the lifting surface is that
in steady and unsteady flow and the objectives are: the velocity component normal to this surface is zero:
∂ F* JJG
• calculation of the open water characteristics;
• calculations of pressure distribution on propeller ∂t
(
+ U'× grad F * = 0 ) (17)
blades operating in uniform flow or in a radially
where U' is the deviations of the main flow and it
varied circumferential mean flow;
• calculation of pressure distribution on propeller
has three components: an axial component: - U + uo
blades in various blade positions and cavitation +up , a radial component: - vo +vp and a tangential
prediction; component - wo +wp . Subscript o indicate disturbances
• calculation of the unsteady forces and moment
present in main flow and subscript p indicate disturbances
acting on propeller shaft (bearing forces); due to the hydrodynamic action of the lifting surfaces.
• calculation of the hull pressures fluctuation. Equation (17) can be written:
∂F *
It will be clear that unsatisfactory results from these
calculations could lead to a new iterative design cycle ∂t
(
+ U + uo + u p ) ∂∂Fx* + (vo + vp ) ∂∂Fr*
(18)
(with a changed propeller geometry). 1 ∂ F*
Circulation theory, RANS and Panel methods can (
+ wo + w p )r ∂φ
=0
predict the open water performances of propeller very
accurately. The quasi-steady methods are still used Starting from this equation, a relation between the
for calculations of unsteady hydrodynamic propeller geometry of the lifting surface, the kinematic distur-
forces. The quasi-steady method propose by Sasajima bance of the fluid motion and the pressure jump distri-
[14] seem to be a practical prediction method for bution over the lifting surface can be established. The
bearing forces. problem is to find the pressure distribution when the
An important step in propeller analysis is to find propeller geometry is given. The integral equation is
the pressure distribution on propeller blade, problem, transformed in a set of linear algebraic equations easy
which can be solved using steady and unsteady to solve.
lifting-surface theory in two ways. In one method: New orientations in analysis of propeller in un-
”mode function method” the lifting surface model, steady flow are CFD methods: panel methods, RANS
stationary or instationary, is solved analytically or methods. The flow around propeller can be derived
numerically by finding from the boundary conditions from the equations of motions using boundary condi-
the coefficients in the expressions in the integral equa- tions. In a viscous flow the equations of motions are
tions. The “vortex lattice method” and the panel method called Navier Stokes equations and the boundary
employ singularity distributions over the blades and conditions at the wall is the no slip conditions. In an
from the boundary conditions of no penetration, the inviscid flow the effect of viscosity can be neglected
strength of the singularities is determined [7]. and when rotation is also neglected the equations of
A development of the unsteady lifting-surface motions become simpler: the Laplace’s equation. In
theory and the numerical solution of the mathematical that case the boundary condition is that the flow is
model are presented by V. van Gent [2]. Some as- tangential to the wall.
sumptions concerning the schematic representation The circulation theory neglects the effects of blade
of propeller are made. The thickness of the blade and thickness and the prediction of the pressure distribu-
the presence of the hub are not taken into account. tion of the leading edge is not valid. This problem is
The geometry and the positions of each lifting surface overcome in surface panel methods. The common
are approximated by the projection of the blade contour description of equation of motion is Laplace’s equation
on a helicoidal surface with constant pitch. The that assumes that the flow is a potential flow. A dis-
mathematical formulation of the mean line section tribution of singularities is placed on panels on the sur-
of the lifting surfaces is: face of hub and blades. The boundary condition of
F* = θ + ω× t - ( )
- zR / ( ρ tgφ ) - f / ( ρsinφ ) = 0 (15)
tangential flow is satisfied on the panels. With the as-
sumption of incompressible, inviscid and irrotational
Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005 137

fluid, the flow field around a propeller is characterized A numerical procedure to solve the boundary integral
by a perturbation velocity potential Φ, which satisfies equations (20) is presented in [16].
The panel methods are useful for calculation of
∇ 2φ ( x, y, z, t ) = 0 (19)
blade pressure distribution in steady and unsteady
Considering a surface S composed of the propeller flow and for prediction the cavitation characteristics
blade surface SB, hub surface SH and wake surface of propeller. This method allows the calculation of
SW, a constant source and doublet distributions are minimum pressure at the leading edge and handles
used to write the perturbation potential φ (P, t ) at any the root and the tip better. A very dense grid with small
point P(x,y,z,t) on the boundary surface: panels is necessary at the leading edge.
The effect of viscosity on the flow around propeller
∂  1  blades can be taken into account using numerical solu-
2πφ ( P ) = ∫∫ φ (Q, t )   dS -
S ∂n Q  R ( P, Q )  tions of Reynolds Averaged Navier Stokes (RANS)
(20) equations. The solutions of Navier Stokes equations
∂φ (Q, t ) 1 make it possible to calculate the flow in those regions
-∫∫ dS
S ∂n Q R ( P, Q ) which are dominated by viscous effects: the tip vortex,
the hub vortex and separation along the leading edge [19].
where Q(x’,y’,z’,t) is the source point where singu- The Reynolds-Averaged Navier Stokes equations
larity is located and R(P,Q) – distance between point are:
P and Q. The flow around propeller has to be derived
from the motion equations using boundary conditions ∂vi
=0
as follows: ∂x
i
• the kinematic boundary conditions on SB and SH (25)
( τij - ρv'i v'j )
is that no flow across blade and hub surface : ∂ vi ∂ vi ∂p ∂
ρ + ρv j = ρ×F - +
∂φ (Q, t ) ∂t ∂x ∂x ∂x
= -  VW ( x', r', θ' - Ωt) + Ω × r ) × n Q (21) j i j
∂n Q These equations are formally identical with Navier
Stokes equations valid for laminar flow with the
• the wake surface is assumed to have zero thickness.
The normal velocity jump and the pressure jump exception of the additional term τij (Reynolds stress
across SW is zero, while a jump in potential is tensor), which represents the transfer of momentum
allowed. due to turbulent fluctuations. Empirical models are
necessary to describe the effects of turbulent: the so-
∂φ +(Q, t ) ∂φ -( Q, t ) p + = p - pe S (22) called turbulence models. The K-ε turbulence model
= , W
∂n Q ∂n Q is one of the most employed two equations and it is
based on the solution of equations for the turbulent
where φ ± p ± are the value of potential and the kinetic energy and the turbulent dissipations rate [21].
pressure on the wake surface (on the upper and In some applications (turbo machines, propellers),
lower side). the control volume is rotating about some axis and the
A Kutta condition must be imposed at the trailing equations are solved in a rotating frame of reference.
edge. This is a physical condition that the velocity at The relative velocity is introduced:
the trailing edge of the blade should be finite. The W = V -ω×r (26)
Kutta condition was developed: the pressure same at
the two control points of the upper and lower panel and Coriolis and centripetal terms must be included
adjacent to the trailing edge: in source term..
To solve the Navier Stokes equations the boundary
ΔpTE ( r, t ) = pTE ( ) TE ( r, t ) = 0
+ r, t - p- (23) condition at the wall is the not slip conditions. For
the homogeneous inflow around propeller blade the
The solution of equation (20) is the perturbation
rotational periodic boundary condition can be use.
velocity potential. The perturbation velocities are
The advantage of application of viscous flow method
obtained by taking the derivatives of the velocity po-
is the possibility of taking into account the interaction
tential over the surface V ' = ∇φ . Adding the tangential
between the wake field of the ship and the propeller
component of the relative inflow VI (x,y,z,t) to the inflow. The RANS codes are important to the investi-
perturbation velocity, the total velocity on the surface gations of hull-propeller interactions problems. In
S is obtained. Applying the Bernoully’s equation in RANS calculations the notion of effective wake is also
unsteady flow, the pressure on propeller surface is: no longer necessary, the flow can be calculated from
1  2 2  ∂φ ( t ) the far upstream, even including the flow around the
p ( t ) = p0 + ρ  VI ( t ) - V ( t )  - ρ (24)
2   ∂t hull. RANS codes required computational grids in the
138 Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005

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9. Lerbs, H.W. (1952) Moderately Loaded propellers with a
dynamic performances of a propeller behind ship are Finite Number of Blade and an Arbitrary Distribution
usually determined through model experiments: open of Circulation, Annual meeting of the Society of naval
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bearing forces measurements. The designed propeller Design Method, Annual Meeting of The Society of
should be tested in model scale in towing tanks and Naval Architects and marine Engineers New York
in cavitation tunnel. A dummy model (shorted ship 11. Cumming R. A, Morgan W.B. Boswell R.J (1972) Highly
model) or grids are installed in the cavitation tunnel Skewed Propellers, Annual Meetingof The Society of
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12. Dumitrescu H., Georgescu A, Ceanga V., Popovici J.S.
that of a full-scale ship wake.
(1990) Calculul Elicei, Editura Academiei Romane
The tests are time consuming, expensive and ad- 13. Morgan W.B., Silovic V., Denny V., (1968) Propeller
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Using a Surface Panel Method, 22th ITTC Propulsion
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Committee Propeller RANS-Panel Method Workshop,
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6. CONCLUSIONS
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