831

1 INTRODUCTION

The availability of adequate mathematical models of

the propulsive ship complex is of crucial importance

in the development of effective ship control systems,

the construction of high-quality simulators, and in the

study of the ship's behavior during maneuvering.

Many works are devoted to the construction and

application of mathematical models of the propulsive

ship complex [1 - 23].

One of the important components of the non-

inertial forces and moments acting on the ship are the

forces and moments caused by the operation of the

ship's rudders, the study of which, in particular, is

devoted to works [24 - 31]. The study of the operation

of ship rudders is based on the processing of

experimental data of model and field tests [2, 3, 25, 26,

28, 29]. Recently, due to the rapid development of

Computational Fluid Dynamics (CFD) methods, the

Reynolds-Averaged Navier-Stokes (RANS) method

has been widely used to solve this problem [27, 30,

31]. Both approaches complement each other and are

used to determine the distribution of hydrodynamic

forces on the rudders, which is the basis for obtaining

mathematical models of these forces with their further

consideration in the general mathematical models of

the ship's propulsive complex. To build mathematical

models of ship rudders at the first stage, it is

necessary to have adequate mathematical models of

hydrodynamic forces and moment on an isolated

rudder in an unbounded flow. Mainly, linear

approximations of the specified forces are known and

widely used [2, 3, 7 - 17], which use only the first

hydrodynamic derivative and sufficiently accurately

describe their behavior at small angles of attack of the

flow on the rudder, but do not take into account, in

particular, the tangent component. Presentations of

the components of the resulting hydrodynamic force

on the rudder are also known [3, 29], which can be

Analysis of Known and Construction of New

athematical Models of Forces on a Ship's Rudder in

Unbounded Flow

. Kryvyi, M. Miyusov & M. Kryvyi

National University “

Odessa Maritime Academy”, Odessa, Ukraine

ABSTRACT: The forces arising on the ship's rudder at different angles of attack in an unbounded flow are

investigated. The components of the resulting force on the rudder are represented in terms of the rudder lift and

drag forces, as well as in terms of the normal and tangential forces on t

he rudder. The well-known

mathematical models of hydrodynamic rudder coefficients are analyzed, and their disadvantages are found.

New mathematical models of hydrodynamic coefficients have been obtained, in particular, the coefficients of

rudder lift and d

rag, which take into account the aspect ratio of the rudder, its relative thickness and can be

applied to any angle of attack of the flow on the rudder. On specific examples for rudders of the NACA series,

the adequacy of the proposed models and their consistency with known experimental studies are illustrated. It

is shown how the rudder lift and drag change, as well as the components of the resulting force for the

maximum possible range of changes in the local drift angle and the rudder angle.

http://www

.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 17

Number 4

December 2023

DOI: 10.12716/1001.17.04.

832

used for a wider range of changes in the phase

coordinates of the ship's motion, but they require

clarification and further development.

The aim of this work is the construction and

numerical analysis of adequate mathematical models

of forces on ship rudders in an unbounded flow,

which would, on the one hand, cover a sufficiently

wide range of changes in the local drift angle and the

rudder angle, and on the other hand, would be

convenient for use in solving various problems of

dynamics of the ship's propulsive complex.

2 VESSEL AND RUDDER SPECIFICATIONS

The geometric and technical characteristics of the ship

and the rudder will be denoted as follows: L – length

of the ship on waterline; B – breadth of the ship on

waterline; T – amidships draft,

– mass density of

sea water, C

b – block coefficient; W=CbLBT –

displacement volume of the ship;

S LT= σ

– area of

the underwater part of the centerplane of the ship’s

hull;

– reduced coefficient of the underwater

centerplane of the ship. On modern ships, depending

on their purpose, different types of rudders are used.

On ocean-going ships single rudders are usually used

behind single propellers, which are located in the

centerplane. The rudders may differ in the type of

projection on the centerplane, namely rectangular or

trapezoidal; according to the method of fastening:

simplex rudder, spade rudder, semi-balanced (semi-

spade) rudder; according to the profile shape: NACA,

CAHI, NEZ, HSVA, IFS, Wedge, etc. A detailed

classification of ship rudders is presented, in

particular, in [29, 30]. Let us dwell in detail on those

characteristics of ship rudders that are used in the

construction of mathematical models of

hydrodynamic forces on the rudder. The main

characteristics include the area of the rudder blade S

that is, the area limited by the contour of the

projection of the rudder on the centerplane, as well as

the relative area of the rudder blade

( )

S S LT

−



For effective control of the ship, depending on the

goals, the relative area of the rudder blade should at

least satisfy the condition [32]



≥+







0.01 0.5

. (1)

When constructing mathematical models of

hydrodynamic forces and moments, it is important to

check the fulfillment of condition (1). For some types

of commercial vessels, Table 2.1 [30, p. 24], shows the

reference ratios of the rudder area to the lateral

underwater area of the ship's hull. The linear

geometric characteristics of the ship's rudders include:

its height h

R , that is, the greatest distance between the

lower and upper edges, as well as the rudder chord b

, that is, the distance between the leading edge (nose)

and its trailing edge (tail). Since the ship's rudder is

not rectangular in plan, for example, trapezoidal, then

the value:

Rc R R

b Sh

−

is used as the average chord.

The relative thickness value

R R Rc

t tb

−



is also

used, where t

R the maximum thickness of the rudder

blade. Modern ocean-going merchant ships usually

use rudder profiles NACA-0012, NACA-0015, NACA-

0018, NACA-0020, NACA-0025, which correspond to

the parameter values: 0.12; 0.15; 0.18; 0.20; 0.25. The

important parameters of the ship's rudder also

include the aspect ratio of the rudder

21 1

R R R R Rc

hS hb

−−

Λ= =

, which shows how many times the

height is greater (or less) than the chord. For marine

merchant ships this figure is

1.5 3÷

, for inland

navigation vessels -

0.5 2.0÷

. Table 1 lists the main

technical characteristics of hulls and ship rudders for

some types of ocean-going merchant ships.

Table 1. Technical parameters of ships and rudders

________________________________________________

Ship KCS, KVLCC2, VLGC, LPG,

Container Tanker Tanker Tanker

ship

________________________________________________

№ 1 2 3 4

________________________________________________

L [m] 230 320 226 147

B [m] 32.2 58 36.6 25.5

T [m] 10.8 20,8 11.8 8.8

B 0.65 0,81 0.72 0.74

R 9.9 14.32 9.65 6.7

Rc 5.5 7.84 4.76 2.87



0.018 0.01687 0.017 0.01484

1.8 1.827 2.027 2.34

R 54.45 115.04 45.934 19.2

________________________________________________

An important parameter of the operation of the

ship's rudder is also the rudder angle

, that is the

angle of deviation of the rudder from the centerplane

of the vessel. The rudder angle is usually considered

positive if it is placed counterclockwise from the

centerplane. The maximum value of this angle is

limited by the design features of the vessel and,

usually, for merchant vessels

max

δ=

. This should

be taken into account when mathematically modeling

the operation of ship's rudders. The operation of the

ship's rudder is also affected by the value of the

dimensionless Reynolds number on the rudder:

R Rc

−

= µ

, where

– the value of the flow

velocity on the rudder,

– the kinematic viscosity of

the sea water. Based on the definition of the number

Re, its value for each specific vessel depends on the

speed of the vessel and is in the range

2 10 10⋅÷

833

3 HYDRODYNAMIC FORCES ON AN ISOLATED

RUDDER IN AN UNBOUNDED FLOW

Consider the hydrodynamic forces on an isolated

rudder in an unbounded flow, that is, on a rudder

that is not affected by the ship's hull and the

propeller. To study the hydrodynamic forces on such

rudder, let's introduce the left Cartesian coordinate

system

** *

DX Y Z

(Fig. 1), where D the point of action

of the resultant hydrodynamic force





on the

rudder, and the axis



is directed perpendicularly

upwards to the horizontal section of the rudder. The

resultant force on the rudder





arises due to

shifting the rudder to an angle

and flow on the

rudder with a velocity



at an angle



to the

axis



. In the introduced coordinate system, the

angle



, which can be considered the local angle of

drift on the rudder, and the rudder angle

, will be

positive if viewed counterclockwise. The angle of

attack on the rudder is called the angle between the

chord of the rudder and the direction of flow on the

rudder, i.e. the difference between the rudder angle

and the local drift angle (Fig. 1):

 

. (2)

If the angle of attack

is positive, then the

deviation of the rudder chord from the line of the flow

occurs counterclockwise, in the opposite case,

clockwise. In the flat Cartesian coordinate system, the

resultant force on the rudder allows expression



 



 

R YX X Y

P P P Pi Pj

, (3)

where



and



are the unit vectors in the given

coordinate system. The values of the longitudinal

and transverse

components of the resulting force

on the rudder are to be determined. Specifically, these

components are taken into account in the general

mathematical models of the dynamics of the ship's

propulsion complex. Usually, the components

and

are represented in terms of the magnitude of

the lift and the drag on the rudder, or the magnitude

of the normal force

and the tangential force

on the rudder. Let's establish a connection between

the indicated parameters, for this we consider two

more Cartesian coordinate systems associated with

the rudder:

 

L DD

and

 

N DS

. The first of these

systems is formed by turning the system

 

X DY

through an angle

β,

the second - by turning it

through an angle

counterclockwise (see Fig. 1).

Figure 1. Forces on an isolated rudder

The resultant force





in the created coordinate

systems allows the following expressions:

− in the coordinate system

 

L DD



  



 

R LD L D

P P P Pi P j

, (4)

− in the coordinate system

 

N DS



  



 

R NS N S

P P P P i Pj

. (5)

In formulas (4) - (5)



and



are the unit vectors

in the corresponding coordinate system.

Representations (3) – (5) and the transformation

formulas when the coordinate axes are rotated make it

possible to express the values of the longitudinal and

transverse components of the resultant force





through the components

(, )

and

( ,)

 = β⋅ + β⋅





= β⋅ − β⋅





sin cos

cos sin

X RL RD

Y RL RD

P PP

, (6)



= δ⋅ + δ⋅





= δ⋅ − δ⋅





sin cos

cos sin

XNS

Y NS

PPP

P PP

. (7)

It is not difficult to establish the following

relationship between the components

(, )

and

( ,)



= α⋅ + α⋅





= α⋅ − α⋅





sin cos

cos sin

D RN RS

L RN RS

PPP

. (8)

When studying hydrodynamic forces on a ship's

rudder, dimensionless hydrodynamic coefficients of

transverse and longitudinal forces, lift and drag, as

well as normal and tangential forces on the ship's

rudder are usually introduced:

834





 

 

222

2 22

222

YXL

YR XR LR

RR RR RR

DR NR SR

RR RR RR

PPP

CCC

vS vS vS

CCC

vS vS vS

(9)

At the same time, the dimensionless coefficient of

the resultant force can be expressed as follows

22 2 2

 



  



( )( )

R YR XR

LR DR NR SR

C CC

CC C C

. (10)

Taking into account representations (9), relations

(6) - (8) will be rewritten as follows



= β⋅ + β⋅





= β⋅ − β⋅





sin cos

cos sin

XR R LR R DR

YR R LR R DR

CCC

C CC

, (11)

∗



= δ⋅ + δ⋅





= δ⋅ − δ⋅





sin cos

cos sin

XR NR SR

YR NR SR

CCC

C CC

. (12)



= α⋅ + α⋅





= α⋅ − α⋅





sin cos

cos sin

DR R NR R SR

LR R NR R SR

CCC

. (13)

The ratio



L LR

D DR

is called the

coefficient of hydrodynamic quality of the rudder

blade [2, 3], and the inverse value



is called

the coefficient of the inverse quality of the rudder

blade. We will also introduce the coefficient of

tangential force

, which determines the

effect of the tangential component of the force on the

rudder. The coefficients k

, k

and k

are functions of

the aspect ratio of the rudder blade. Let's establish the

relationship between the coefficients k

L, kD and kS, for

this, by dividing the second equality in (13) by the

first, we get

α− α

α+ α

cos sin

sin cos

RS R

, (14)

From here we get the following representation

α− α

α+ α

cos sin

sin cos

RL R

. (15)

Denote by

ϕ=tg

, where

respectively, the angles between vectors



and the resulting vector





, then from relations

(14), (15), we obtain the representation

=− α +ϕ =− α +ϕctg( ), ctg( )

L RN S RD

. (16)

Coefficients k

L, kD and kS make it possible to

represent expressions (11), (12) and (13) as

(sin cos )

(cos sin )



= ⋅ β+ β





= ⋅ β− β





XR LR R D R

YR LR R D R

CC k

; (17)

(sin cos )

(cos sin )

XR NR D

YR NR D

CC k



= ⋅ δ+ δ





= ⋅ δ− δ





; (18)

(sin cos )

(cos sin )

DR NR R D R

LR NR R D R

CC k



= ⋅ α+ α





= ⋅ α− α





. (19)

According to Figure 1, the angle between the

vector of the resultant hydrodynamic force





and

the vector





is equal to the sum of the rudder

angle

and the angle

. This makes it possible to

represent the dimensionless coefficients of the

transverse and longitudinal forces of the rudder

through the dimensionless coefficient of the resulting

force



= δ+ϕ





= δ+ϕ





sin( )

cos( )

XR R N

YR R N

. (20)

When building a general mathematical model of

the ship's propulsive complex, dimensionless

hydrodynamic coefficients of the transverse and

longitudinal forces

and

on the rudder are

used. To determine them, you can use the following

three approaches:

1. use representations (11) or (17), if mathematical

models for rudder lift coefficient C

LR and drag

coefficient C

DR are known (or instead of CDR,

coefficient of inverse quality of the rudder k

D);

2. use representations (12) or (18), if mathematical

models for the coefficients of normal force C

NR and

tangential force C

SR on the rudder are known (or

instead of C

SR, coefficient of tangential force kS);

3. use representations (20), if mathematical models for

(, )

or for

( ,)

are known.

4 ANALYSIS OF EXISTING MATHEMATICAL

MODELS OF HYDRODYNAMIC COEFFICIENTS

ON THE RUDDER

When building mathematical models of ship rudders,

it is important to know the critical value of the angle

835

of attack on the rudder

, i.e., the angle of attack at

which the stall occurs on the rudder blade and its lift

is sharply reduced. The set of values of angles of

attack

α ≤α

R Rk

is called pre-critical, and the set of

values

α >α

R Rk

is called supercritical. The angles

±α

are actually the angles of the maximum value

of the rudder lift coefficient C

LR, while the angle of

maximum efficiency has a slightly smaller value, due

to the increase in the rudder drag coefficient C

DR. It

should also be noted that the value of the angle

for the rudder located in the propeller slipstream

increases on average by

15 20÷



. The angle

depends on the technical characteristics of the rudder,

in particular its aspect ratio, to determine its values

using experimental data [2, 3], we obtain table 2.

Table 2. Dependence of the critical value of the

angle of attack

[deg] from the aspect ratio

________________________________________________

0.5 0.75 1 1.25 1.5 2 3 5

45.5 41 36 30.5 25.5 23.75 19.5 16.7

________________________________________________

Based on these data, with the help of regression

analysis, we will get the following representation

0.356

29.6824 .

−

αΛ⋅=

Rk R

(21)

Experimental studies [2, 3, 26] show that the

rudder lift coefficient C

LR and the rudder drag

coefficient C

DR depends to varying degrees on the

shape, thickness of the rudder, Reynolds number, but

most of all on the rudder aspect ratio

and the

angle of attack of the flow on the rudder

. In

particular, for small to critical angles of attack, a linear

approximation of the lift coefficient is used, i.e., a

Taylor series expansion along the angle of attack:

()

(0)

(

α ) (α ) α (0) (0) α (α )

LR R R R R R

С f ff o







    



In this case, only the second term is used. The first

is equal to zero:

() ()0 00

С

, due to physical

considerations, therefore

LR LR R

СС





. (22)

To designate the hydrodynamic derivative









, one can use the formula [2, 3]:

′

+Λ+

(23)

or Prandtl's improved formula

πκ

′

+Λ

(24)

where



is the coefficient for spade and semi-

spade rudders of marine vessels.

For a wider range of the angle of attack, the

following formula is known [2]

sinα sin α cos α ,

′

=⋅+ ⋅



LR LR R d R R

СС С

(25)

where



for conventional rudders and

1.2 1.5

С = ÷



for rudders with rounded side

projections.

The following formula can be used for the rudder

drag coefficient [3]:

= + ⋅ +⋅



sin α sin α

DR D D R d R

С СK С

, (26)

where

(0.0221 0.0023lgRe)= −



СC

. The coefficients

()=





C Ct

and

(

Λ)

are determined using graphs

[3]. Note that at

α0=

, as a rule, the value of the drag

force coefficient is chosen approximately:

0.014≈

There are also other representations [29] for the lift

and drag coefficients of the rudder, also obtained on

the basis of experimental data processing:



    



Λ (Λ 0.7)

2 sinα sinα sinα cosα

(Λ 1.7)

LR R Q R R R

СC

(27)



  



sinα

DR Q R D

С CC

(28)





0.075

2.5

(lg Re 2)

The authors of the model recommend taking the

value of the constant

equal to one:

C 

Within the framework of the second approach [7,

8], a linear approximation of the normal force

coefficient takes place. Under the assumptions of

smallness

, the dependence is represented not by

the angle of attack

, but by its sinus

sinα

, i.e.

sinα

NR NR R





. (29)

Empirical representation is used to calculate the

hydrodynamic derivative of the normal force:











6.13Λ

Λ 2.25

dС

. (30)

It should be noted that in this case the value of the

tangential force on the rudder is neglected:

C 

Using the methods of direct numerical modeling

RANS [30], representations (22) and (29) are

somewhat improved by taking into account the

relative thickness of the rudders





( sin α)

Λ 2.25

LR L R L

С sc

. (31)

 



( sinα)

Λ 2.25

DR D R D

С sc

. (32)

 



( sinα)

Λ 2.25

NR N R N

С sc

. (33)

836

The constants in representations (31) - (33),

depending on the relative thickness



, are given in

Table 4.2 [30, p. 89] for some types of ship's rudders.

Studies of representations for hydrodynamic

coefficients on a ship's rudder have shown significant

shortcomings of existing models. In particular,

representations (22), (29) and (31) - (32) can be applied

only at small to critical drift angles, while

representations (31), (33) can be considered more

accurate, where the relative thickness of the rudder is

taken into account. In representations (25), (26) and

(32), the properties of the oddness of the lift coefficient

and the parity of the drag coefficient of the rudder are

violated, so they cannot be used for negative values of

the angle of attack

on the rudder. In

representations (27) and (28) there are non-

differentiable functions, such as

sinα

. Therefore, for

a general description of the behavior of

hydrodynamic coefficients, they are acceptable, but as

the right-hand parts of the differential equations of

the dynamics of the propulsive complex, their use is

incorrect, since in this case the phase coordinates of

the ship's motion will be discontinuous functions.

5 CONSTRUCTION AND ANALYSIS OF NEW

MATHEMATICAL MODELS OF

HYDRODYNAMIC FORCE COEFFICIENTS ON

THE RUDDER

To eliminate the indicated shortcomings of the

existing mathematical models of hydrodynamic force

coefficients on the rudder, we will search for the lift

force coefficient as follows

′

= ⋅ −Λ⋅

sinα ( ) sin α

LR LR R R R

СС

, (34)

where

Λ()

the function of the rudder aspect ratio

is not yet known, which we will determine based

on the following considerations. Representation (34)

must reach a maximum at critical values of the angle

of attack, so the condition must be fulfilled

αα

cosα 3 ( ) sin α cosα 0.

R kR

LR kR R kR kR

dС

′

≡ ⋅ − χΛ ⋅ ⋅ =

(35)

From here

′

χΛ

α arcsin

6( )

. (36)

Equating this expression and representation (21),

we find the expression for the unknown function

−

′

Λ⋅

Λ=

2 0.356

()

3s 9.6i

4n( 82 )

(37)

After substituting (37) into (34), we obtain

−



′



⋅



=⋅−





2 0.356

sin α

sinα.

3sin (2 )9.6824

LR LR R

СС

(38)

For the hydrodynamic derivative

′

, after

summarizing expressions (23), (24) and (31), with the

help of regression analysis, we obtain the following

representation, which depends both on the aspect

ratio and on the relative thickness of the rudder:

Λ ⋅η

′

+Λ



()

2.25

R LR

, (39)

η =− ⋅ + ⋅+

 

( ) 50.503 ( ) 11.123 5.638

LR R R

t tt

Similarly, for the hydrodynamic derivative of the

normal force on the rudder, the following

representations can be obtained.

Λ ⋅η

′

+Λ



()

2.25

R NR

, (40)

η =− ⋅ + ⋅+

 

( ) 54.123 ( ) 12.512 5.451

NR R R

t tt

To calculate the hydrodynamic drag coefficient on

the rudder, it is proposed to use the following

dependences on the powers of the sine of the angle of

attack

sin α sin α= + ⋅ +⋅



DR D D R d R

С СK С

, (41)

where

(0.0221 0.0023lgRe)= −



СC

; for the coefficients

()=





C Ct

and

(Λ)

, the following expressions

were obtained by regression methods based on

experimental data [2, 3].

1.36 4.09 29.36=−+





С tt

. (42)

0.856Λ 0.188Λ= −

D RR

. (43)

Figures 2 and 3 show the dependence of the

hydrodynamic derivatives

′

, respectively,

on the aspect ratio

at constant values of the

relative thickness, and on the relative thickness



constant values of the aspect ratio. In both figures,

solid lines show the value of the derivative

′

dotted lines show the value of the derivative

′

Figure 2. Dependence of derivatives

′

from

837

Figure 3. Dependence of derivatives

′

from



In fig. 2 black, red, blue, green, and yellow lines

are obtained, respectively, at values of



0.12; 0.15;

0.18; 0.21; 0.25. In fig. 3 black, red, blue, green, and

yellow lines obtained, respectively, at values of

0.5, 1.0; 1.5; 2.0; 2.5. The three-dimensional figure 4

shows the general picture of the change in the

behavior of the hydrodynamic derivative

′

, from

the change in the values of the rudder aspect ratio

and the relative thickness



. The obtained

numerical results confirm the adequacy of the

obtained mathematical models (38) and (39), and their

consistency with the known results [27, 30].

In particular, it was confirmed that hydrodynamic

derivatives

′

and

′

do not have significant

differences, but they significantly depend on the

rudder aspect ratio

and relative thickness of the

rudder



, which must be taken into account when

building mathematical models of hydrodynamic

forces caused by the operation of ship rudders. Figure

5 shows the dependences on the angle of attack on the

rudder

for the lift coefficient

obtained using

the proposed mathematical model (38) (continuous

lines) and using the mathematical model (31) (dotted

lines) for the NACA-0020 rudder. At the same time,

the black, red, blue, green, and yellow lines are

obtained, respectively, at values of the rudder aspect

ratio

of 0.5; 1; 1.5; 2; 2.5.

Figure 4. Dependence of the derivative

′

and



Figure 5. Dependencies

on the angle of attack

The given results show that at small values of the

angle of attack of the flow on the rudder

α 15

≤

both models agree well, but at angles

α 15

significant differences are observed. According to

model (31), the lift coefficient continues to increase

with the increase in the angle of attack on the rudder,

which does not correspond to the known [2, 3, 25]

results of experimental studies. According to the

proposed model (38), (39), the coefficient

for all

rudders reaches a critical value, after which the stall

occurs on the rudder and the lift of the rudder

decreases. Moreover, the greater the rudder aspect

ratio, the smaller the value of the critical angle of

attack. These results are fully consistent with the

results of known experimental studies.

Figure 6 shows a general picture of the behavior of

the lift coefficient

depending on the change in

the angle of attack

on the rudder and the rudder

aspect ratio, surface 1 corresponds to the proposed

mathematical model (38), (39), surface 2 - model (31).

Figure 6. Dependence

and

838

Figure 7. Dependence

LR DR

СС

and

Figure 7, using the proposed mathematical models

(38), (39) and (40) - (42), shows the dependences of the

lift coefficient

and the drag coefficient

the rudder on the change in the angle of attack

the NACA-0020 rudder at different values of the

rudder aspect ratio

. Calculations, in particular,

show that when the angle of attack increases, the

coefficient

increases, and when

α 15

the

drag of the rudder becomes commensurate with the

lift and it cannot be neglected. It is also noticeable that

the drag of the rudder reaches its maximum value at

aspect ratios

(1.8;2.2),Λ∈

and the lift increases when

the aspect ratio of the rudder increases.

The obtained results are in good agreement with

the experimental studies of the lift and the drag of the

rudder. Figures 8, 9 show the dependence of the

hydrodynamic coefficients, respectively, for the

components P

x and Py the resulting force on the

NACA-0020 rudder for a wide range of changes in the

local drift angle

and the rudder angle

. The

results are obtained using representations (11) and the

obtained dependencies (38), (40). At the same time,

surface 1 in both figures corresponds to the rudder

with aspect ratio

0.5Λ=

, surface 2 – with aspect

ratio

Λ=

Figure 8. Dependence



and

Figure 9. Dependence



and

The results of the calculations show the adequacy

of the obtained mathematical models of the

hydrodynamic coefficients of the resultant force on

the rudder for the maximum possible area of change

in the local drift angle and the rudder angle. The

obtained results make it possible to evaluate the

influence of the drift angle

and the rudder angle

on the behavior of the coefficients



and



In particular, the lift and the drag of the rudder are

greater if the local drift angle and the rudder angle

have different signs than if these angles have the same

signs. This means that it is easier to turn the ship in

the direction of the ship's drift than in the opposite

direction. In addition, the component

of the

resultant force





on the rudder, at drift angles

greater than the rudder angle, will have a negative

sign, that is, the force



acting in the direction of

the ship's movement. The dependence of both forces

on the rudder aspect ratio is also noticeable.

6 CONCLUSIONS

The existing and proposed new effective and

convenient mathematical models for the lift and drag

of the rudder in an unbounded flow for a wide range

of changes in the local drift angle and the rudder

angle and which take into account the rudder

thickness and its aspect ratio are analyzed and

proposed. On the basis of these models, a numerical

analysis of the behavior of the longitudinal and

transverse force on the rudder was carried out for the

maximum possible range of values of the local drift

angle and the rudder angle.

The obtained results will make it possible to build

general adequate mathematical models of the ship's

propulsion complex, in particular, they are decisive

for determining the forces caused by the operation of

the ship's rudders, taking into account the influence of

the propeller and the ship's hull.

839

REFERENCES

[1] Pershytz R. Y.: Dynamic control and handling of the

ship. L: Sudostroenie, 1983

[2] Sobolev G.C.: Dynamic control of ship and automation of

navigation. L.: Sudostroenie, 1976

[3] Gofman A. D.: Propulsion and steering complex and ship

maneuvering. Handbook. L .: Sudostroyenie.1988.

[4] Miyusov M. V.: Modes of operation and automation of

motor vessel propulsion unit with wind propulsors.

Odessa, 1996.

[5] Kryvyi O. F.: Methods of mathematical modeling in

navigation. ONMA, Odessa, 2015.

[6] Kryvyi O. F, Miyusov M. V.: Mathematical model of

movement of the vessel with auxiliary wind-propulsors,

Shipping & Navigation, v. 26, pp.110-119, 2016.

[7] Inoe S., Hirano M., Kijima K.: Hydrodynamic derivatives

on ship maneuvering, Int. Shipbuilding Progress, v. 28,

№ 321, pp. 67-80, 1981.

[8] Kijima K.: Prediction method for ship maneuvering

performance in deep and shallow waters. Presented at

the Workshop on Modular Maneuvering Models, The

Society of Naval Architects and Marine Engineering,

v.47, pp.121 130, 1991.

[9] Yasukawa H., Yoshimura Y.: Introduction of MMG

standard method for ship manoeuvring predictions, J

Mar Sci Technol, v. 20, 37–52pp, 2015. DOI

10.1007/s00773-014-0293-y

[10] Yoshimura Y., Masumoto Y.: Hydrodynamic Database

and Manoeuvring Prediction Method with Medium

High-Speed Merchant Ships and Fishing, International

Conference on Marine Simulation and Ship

Maneuverability (MARSIM 2012) pp.494-504.

[11] Yoshimura Y., Kondo M.: Tomofumi Nakano, et al.

Equivalent Simple Mathematical Model for the

Manoeuvrability of Twin-propeller Ships under the

same propeller-rps, Journal of the Japan Society of Naval

Architects and Ocean Engineers, v.24, №.0, p.157. 2016,

https://doi.org/10.9749/jin.133.28

[12] Wei Zhang, Zao-Jian Zou: Time domain simulations of

the wave-induced motions of ships in maneuvering

condition, J Mar Sci Technol, 2016, v. 21, pp. 154–166.

DOI 10.1007/s00773-015-0340-3

[13] Wei Zhang, Zao-Jian Zou, De-Heng Deng: A study on

prediction of ship maneuvering in regular waves, Ocean

Engineering, v. 137, pp 367-381, 2017,

http://dx.doi.org/10.1016/ j.oceaneng. 2017.03.046

[14] Erhan Aksu, Erkan Köse: Evaluation of Mathematical

Models for Tankers' Maneuvering Motions, JEMS

Maritime Sci, v.5 №1, pp. 95-109, 2017. DOI:

10.5505/jems.2017.52523

[15] Kang D., Nagarajan V., Hasegawa K. et al:

Mathematical model of single-propeller twin-rudder

ship, J Mar Sci Technol, v. 13, pp.207–222, 2008,

https://doi.org/10.1007/s00773-008-0027-0

[16] Shang H., Zhan C., Z. Liu Z.: Numerical Simulation of

Ship Maneuvers through Self-Propulsion, Journal of

Marine Science and Engineering, 9(9):1017, 2021.

https://doi.org/10.3390/jmse9091017

[17] Shengke Ni., Zhengjiang Liu, and Yao Cai.: Ship

Manoeuvrability-Based Simulation for Ship Navigation

in Collision Situations, J. Mar. Sci. Eng. 2019, 7, 90;

doi:10.3390/jmse7040090

[18] Sutulo S. & C. Soares G.: Mathematical Models for

Simulation of Maneuvering Performance of Ships,

Marine Technology and Engineering, (Taylor & Francis

Group, London), p 661–698, 2011

[19] Kryvyi O. F, Miyusov M. V.: “Mathematical model of

hydrodynamic characteristics on the ship's hull for any

drift angles”, Advances in Marine Navigation and Safety

of Sea Transportation. Taylor & Francis Group, London,

UK., pp. 111-117, 2019

[20] Kryvyi O. F, Miyusov M. V.: “The Creation of

Polynomial Models of Hydrodynamic Forces on the Hull

of the Ship with the help of Multi-factor Regression

Analysis”, 8 International Maritime Science Conference.

IMSC 2019. Budva, Montenegro, pp.545-555 http://www.

imsc2019. ucg.ac.me/IMSC2019_ BofP. pdf

[21] Kryvyi O., Miyusov M.V.: Construction and Analysis of

Mathematical Models of Hydrodynamic Forces and

Moment on the Ship's Hull Using Multivariate

Regression Analysis, TransNav, the International

Journal on Marine Navigation and Safety of Sea

Transportation, Vol. 15, No. 4,

doi:10.12716/1001.15.04.18, pp. 853-864, 2021

[22] Kryvyi O. F, Miyusov M. V.: Mathematical models of

hydrodynamic characteristics of the ship’s propulsion

complex for any drift angles, Shipping & Navigation, v.

28, pp. 88-102, 2018. DOI: 10.31653/2306-5761.27.2018.88-

102

[23] Kryvyi O. F, Miyusov M. V.: New mathematical models

of longitudinal hydrodynamic forces on the ship’s hull,

Shipping & Navigation, v. 30, pp. 88-89, 2020. DOI:

10.31653/2306-5761. 30. 2020.88-98

[24] Kryvyi O. F, Miyusov M. V., Kryvyi M. O.:

Mathematical modelling of the operation of ship's

propellers with different maneuvering modes, Shipping

& Navigation, v. 32, pp. 71-88, 2021. DOI: 10.31653/2306-

5761.32.2021.71-88

[25] Molland A.F., Turnock S.R.: Wind tunnel investigation

of the influence of propeller loading on ship rudder

performance. Technical report, University of

Southampton, Southampton, UK, 1991

[26] Molland A.F., Turnock S.R.: Further wind tunnel

investigation of the influence of propeller loading on

ship rudder performance. Technical report, University of

Southampton, Southampton, UK, 1992

[27] Molland A.F., Turnock S.R.: Marine rudders and control

surfaces: principles, data, design and applications, 1st

edn. Elsevier Butterworth-Heinemann, Oxford, 2007

[28] Ladson CL (1988) Effects of Independent Variation of

Mach and Reynolds Numbers on the Low-Speed

Aerodynamic Characteristics of the NACA 0012 Airfoil

Section. Technical report, Langley Research Center,

Hampton, Virginia, USA, 1988

[29] Bertram, V. Practical Ship Hydrodynamic, 2nd ed.;

Elsevier Butterworth-Heinemann: Oxford, UK, 2012; p.

284.

[30] Liu J.: Mathematical Modeling of Inland Vessel

Maneuverability Considering Rudder Hydrodynamics,

2020. https://doi.org/10.1007/978-3-030-47475-1_4

[31] Shin Y-J, Kim M-C, Kang J-G, Kim J-W. Performance

Improvement in a Wavy Twisted Rudder by Alignment

of the Wave Peak. Applied Sciences. 2021; 11(20):9634.

https://doi.org/10.3390/app11209634 (CDF)

[32] Veritas D.N.: Hull equipment and appendages: stern

frames, rudders and steering gears. Rules for

Classification of Steel Ships, pp 6–28, part 3, chapter 3,

section 2, 2000