TECHNISCHE uNIvERS1rFEIT
ScheepshydrOmeCbafliCaArchief
Mekeiweg 2, 2628 CD Delft
Tel: 015-2786873/Fax:2781836
A NEW ThEORETICAL MODEL OF FORMATIONOF VORTEX CAVITATION
T. Koronowicz Ph.D.,D.Sc J.A. Szantyr Ph.D., D.Sc.
Institute of Fluid Flow Machinery Gdañsk, Poland
The paper presents the results of experimental and theoretical research
on the
modelling of the flow around the tips of propeller blades and of the formation of vortex cavitation. A new vortex model of the blade is described, which enables
detailed studies of the mechanism of formation of vortex cavitation.
INTRODUCTION.
A possibly low level of acoustic emission from propellers is one of the primary requirements
for naval ships, research vessels, fishing boats and passenger liners, enabling their proper
functioning. One of the most important sources of this emission is vortex cavitation.
Experimental and theoretical research into the formation of tip vortices has been initiated long ago. A comprehensive review of theoretical models of tip vortices has been presented in
the work of Sarpkaya ¡1101. Papers concerning problems of vortex cavitation are being
presented at almost every conference devoted to cavitation. Analysis of these results leads to
the concLusion that the most practically applicable model is basedon the Rankine vortex or on
its modifications. In most cases these modifications concern accounting for dissipation of
vorticity.
Without questioning this model, which is supported by extensive experimental data, we would like to assess it on the basis of our own research results. In the 1970s WFM conducted
extensive experimental and theoretical studies related to the computational model of the
propeller slipstream based on concentrated tip vortices with cavitating kernels [3,4,5]. This research led to a conclusion that a serious modification of the structure of the bound vortices on the blade itself is necessary for proper modelling of the free vortices in the slipstream. The main assumptions of such a modification were formulated [3,5], but the work on the model, named the Double Layer Lifting Surface, has started relatively recently. Certain results of
research into vortex cavitation based on such a model are presented later.
VORTEX MODEL OF 1,11E DOUBLE LAYER Lwr1JG SURFACE.
Experimental evidence shows that the free vortices are never shed from the blade trailing edge, but that they detach nearer the leading edge and then pass relatively closely to the blade suction surface (this is particularly visible on highly skewed blades). In such a situation a
marked change in the velocity and pressure distributions, but first of all in the bound vorticity
Following the theory of highly skewed wings, Greeley and Kerwin [2] have suggested a propeller vortex model with an additional tip vortex related to the phenomena present at the leading edge of the blade. However, this model does not comply fully with the laws of vortex flow, primarily with the theorem on vorticity conservation (the additional vortex is not related
to the bound vorticity representing the blade itself).
The problem of the formation of concentrated vorticit in the vicinity of a wing is treated by
Thwaites [11], who quotes the work of Betz [1], stating that a flow with strongly concentrated vorticity (with a kernel rotating as a rigid body) may be generated only through the rolling up
of the vortex surfaces and through the process of vorticity accumulation. Experimentally
observed "attachment" of vortices near the leading edge on the suction side of the blade may be
a result of separation of the free vortex surface quite far from the trailing edge and of its rapid deformation. Simultaneously, such separation is never observed on the pressure side of the blade. Ou this side the free vortex surface always separates from the trailing edge. In order to model this phenomenon theoretically, two independent separation points must be introduced on each blade section profile, enabling the starting of two free vortex lines. This particular feature requires the introduction of a so called Double Layer Lifting Surface, with independent bound vortex systems on both sides of the blade, in order to simulate properly the deforming
free vortex system, including the flows with double separation [12].
The lifting vorticity is calculated on the basis of a classical single layer lifting surface theory. This vorticity appears when a blade section operates at a certain angle of attack andior when its mean line is cambered. This vorticity is related directly to lift (hence its name), but it may exist
also when the mean line has a symmetrical S-shape, or when a cambered profile operates at zero-lift angle. The lifting vorticity cannot be identified with the tangential induced velocity, which is related to the total vorticity. The lifting vorticity is only one of the components of the
total vorticity, which may be divided in a way similar to the tangential velocity on a profile:
y -v -sin 9++V,
st(I)
where: J<,, sin p component of the undisturbed flow
V - component induced by sinks and sources modelling finite thickness of the profile
V,. - component induced by the lifting vorticity q) - angle between x-axis and tangent to profile.
The suggested Double Layer Lifting Surface model is based on a hypothesis that the lifting
vorticity on both sides of a blade profile may be calculated from the relation:
=V,=(V,,-sin +V)
(2)with the assumption that the term (V,,, -sin +Vd) is determined for a symmetrical profile at zero angle of attack, having thickness distribution identical with the actual profile.
Figs. 1, 2 and 3 show the results of calculations for a profile hang NACA16 thickness
distribution and NACA a=0.8 mean line at the angle of attack equal to the ideal angle of attack. Fig. 3 shows the lifting vorticity distribution according to the Double Layer model.
1.5 i -0.5 -1 -1.5 i 0.5 O -1 -1.5
Fig. i Tangential velocities for syrnetncal profile NACA 16. vi = 0.12
Fig.2 Tangential velocities for NACA 16, a=0.8,
t/10.12, f/l=0.05. a=a
L
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 YJI 0.9//
0.5 g r O 0. i 0. 2 O3 0, 4 O 5 O 6 07 08 x/1O, 9 10.25 0.2 0.15 0.1 0.05 o g
f
. - 4.(u)
(b)
Fig. 4 Lifting vorticity in single layer model (a) and in double layer model (b).
Fig.5 Lifting surface model in single layer variant (a) and in double layer variant (b).
u.
/
V
file NACA 16 a = 0.8 = 0.12
0.05
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 xii
Fig.3 Lifting vorticity.
It should be made absolutely clear that the presented mode! is not at all equivalent to the surface vorticity model described e.g. by Martensen [7,89]. The basic features of the Double
Layer model are:
- using only the lifting vorticity
- independent separation points of the free vortices on suction and pressure sides. Figs. 4 and 5 demonstrate the differences between single and double layer lifting surfaces.
The literature often presents theories attempting the modelling of tip vortices being created far from the trailing edge, e.g. in the works of Hoeijmakers [10] and Thomas [10]. Both these approaches are based on the single layer lifting surface, where separation of the free vortices is
allowed only on the trailing edge of the blade. The presence of the secondary vorticity,
independent from the system of bound vortices, brings the theoretical model closer to physical reality. A picture of primary and secondary vorticity obtained from the Double Layer Lifting
Surface is similar to that presented by Hoeijmakers (cf Fig. 6). It is obtained directly, preserving the continuity of bound and free vortex lines. Introduction of free vortex separation on the suction side with simultaneous limitation of separation (due to the non-permeability condition) on the pressure side results in a similar structure of free vortices (Fig. 8). Visually the differences are not so pronounced, but their theoretical and computational meaning is very important. Fig.9 shows the deforming free vortex system calculated behind a rectangular wing by the Double Layer model. Fig. 10 presents the results of a similar calculation performed by means of the single layer model. As may be seen, the pictures of the deforming free vortices
look very different in both cases, despite the identical initial loading distribution.
The algorithm for determination of the deformed free vortex system in the tip region of the blade is based, similarly as in [4], on the vortex model composed of vortex lines (see Fig. 5) of constant intensity zF TTfl3X
( F
- maximum value of bound circulation on the blade, N-number of vortex lines), which remains unchanged during the calculation. In the consecutive steps of the calculation the location of separation points on the suction side of the blade aredetermined, where the bound vortices are shed and transformed into free vortices.
The work on the Double Layer Lifting Surface is not finished yet. Many problems remain, for example the separation of vortices from the pressure side at the blade tip is not properly modelled. Simïarly, a secondary flow in a "pocket' between the free vortex systems shed from
the suction and pressure sides respectively (seen in Figs. 8 and 9) is not fully understood.
Considering the present speed of popular PCs, the application of the Double Layer Lifting
Surface theory in three-dimensional flow as a standard tool for detection of cavitation on
propeller blades is not feasible. Therefore the computer program for detection of tip vortex
cavitation on propeller blades has been based on a two-dimensional analysis of vortex
deformation., with an initial vcrticity distribution corresponding to the Double Layer model and taking into account the impermeability of the blade.
3. DEFORMATION OF TifE TWO-DIMENSIONAL VORTEX SYSTEM BASED ON
DOUBLE LAYER LIFTING SURFACE iIiJORY.
In the development of a method for the detection of tip vortex cavitation on a propeller operating in the hull wake an effective numerical method for determination of the vorticity distribution of the deformed free vortex system must be constructed. It should bea relatively
fast method, as several blade positions must be analysed in one program run. Results of
extensive research on formation of vortices in the propeller slipstream [3,4,5,6] show that the
algorithm for determination of vorticity distribution in the region of tip vortex cavitation
inception may be based on the model of two-dimensional vortex deformation.
In relation to the existing algorithms describing two-dimensional deformation of the free
V
-L
Primaiy
Vortex Primar-v Vortex Secondary Vortex Fig. 6Modelling of the flow around
delta wing by single layer lifting surface (acc.to Hoeijmakers [10])
Fig. 7
Isobars on a delta wing (acc. to Thomas et al [10])
Modelling of the flow around
delta wing by double layer
initiai vorticity distribution as well as the deforming vortex system are modelled using Double Layer Lifting Surface theory
- deformation takes place in the presence of the blade, which introduces appropriate constraints for the deforming vortex system
In practise the bound vortex system, defined as a system of vortex lines of constant intensity,
is divided into two systems on the suction and pressure sides of the blade respectively. The
system on the suction side undergoes deformation from the beginning of the numerical
simulation process, with the permissible direction of deformation above the suction side only, as long as the free vortices remain within the blade outline. The system on the pressure side is subject to a more specific constraint. Namely, only the free vortex line nearest the blade tip is allowed to move above the suction side. In the numerical model it is assumed that the first shedding of the vortex line on the pressure side takes place simultaneously with separation of the first line on the suction side. The motion of the next line on the pressure side towards the
tip and its subsequent shedding is regulated depending on the blade outline and loading
distribution. The method of this regulation is still the subject of continuing research. After
leaving the blade outline both vortex systems can deform freely.
The location of respective vortex lines is indexed by the parameter t, which is defined as the ratio of real time of deformation to the time necessary for covering the distance equal to half of the blade span i (in the case of propeller blade 1 is the distance from the tip to the point
of maximum bound circulation).with the velocity of undisturbed flow (in case of propeller with velocity V=JVo2.(ïz.nD)2 ). The quantity t corresponds also to the ratio of the distance
covered by a fluid particle in real time t to the wing half span:
t s
=110
\=7
(3)The deformation is regulated by the maximum vortex line translation Eps in the respective
calculation step, which is either an input value or it may be determined in the course of
calculation. Selection of this parameter reflects a compromise between accuracy and time of calculation. An example of results obtained from the above described procedure is shown in
Fig.9. If the constraints on the deformation are removed (zero blade span is assumed), the
same procedure produces results completely equivalent to the two-dimensional deformation of the vortex system. An example of such calculation is presented in Fig. 10. Comparison of both
figures demonstrates that the constraints resulting from the impermeability of the blade surface
decisively affect the shape of the deformed free vortex system and in consequence, the
conditions for vortex cavitation inception.
It should be stressed that in the calculation scheme the induced velocities are obtained as for
the Lamb vortex lines, namely:
I
r\
(4)
where:
y -
kinematic viscosity coefficientt - time
This enables analysis of the influence of vorticity dissipation on the shape of the deformed
o.r7 o. +* O.0G3 0. 4 0 3 0.7O t . s s s p u u s
--t*0103
7
*Oj7O
o.g Fig. 9Double Layer Lifting Surface Model
Io
Single layer lifting surface model
4. PRESSIXR E FIELD AROUND A DEFORMING FREE VORTEX SYSTEM.
The information about the geometry of the free vortex system is not enough for prediction of vortex cavitation. The detailed pressure field in the vicinity of suspected vortex cavitation is necessary. Consequently, the program for determination of the free vortex system deformation has been supplemented with a procedure for calculating the pressure field in the region of tip
vortex formation
It is assumed that the free vortex lines coincide with streamlines. This means that in the Navier-Stokes equation xrotV = O. Moreover, only the flows with high Reynolds number are of interest, where viscosity plays a comparatively minor role. In this case the pressure may
be calculated by means of a formula corresponding to the Bernoulli equation, where the
approximation symbol refers to the neglected influence of viscosity:
ipV2+pconsz
(5)The pressure has been calculated in the nodes of the grid defined around the deforming free
vortex surface and then the corresponding isobars have been determined.
The examples showing the calculated pressure field in different phases of deformation are presentedin Figs. 11 and 12. Already the first calculations have shown that both the location and mechanism of the formation of vortex cavitation differ markedly from the widespread description of this phenomenon. On the basis of these results the following statement has been
formulated:
- conditions suitable for inception of vortex cavitation (i.e. presence of a region where pressure drops below the critical level, thus enabling explosive growth of nuclei) exist only in
the initial part of the deforming free vortex system
This statement is based on the fact that only in the initial phase of deformation a small kernel with a high concentration of vorticity is created. This kernel is surrounded by a
significant pressure drop (cf. Fig. 11). In the following phases of deformation further parts of the free vortex surface are woven onto this kernel. This leads to expansion of the low pressure region with simultaneous reduction of the minimum pressure (cf. Fig. 12). In the previous
models of vortex cavitation the region in which conditions for explosive growth of nuclei
existed has extended along a substantial length of the already formed vortex (in the form of the
Rankine model or as a cavitating kernel with thin vortex layer [3]). This was possible because of the assumed high concentration of vorticity within a limited tube-shape area of unchanging or slowly changing diameter. In view of the above presented results this region is relatively
short and it often does not extend beyond the trailing edge.
This relatively small region near the wing tip acts as a "factory of nuclei", where a rapid growth of bubbles takes place. Further on these bubbles enter a region of higher pressure and
their size is reduced, but not to the initial value (difference in time scale between vaporation and diffusion plays a role here). These bubbles are moving in a specific pressure field resulting
from the deforming vortex system. They may enter into oscillations or they may join together,
forming the cavitating tip vortex.
The cavitatmg kernel of this vortex reacts to pressure changes in its vicinity, but its size
depends first of all on the quantity of nuclei produced in the "factory" near the wing tip. This is
the second qualitatively new statement concerning vortex cavitation formation. It may be confirmed by a simple experiment conducted in a cavitation tunnel during research on vortex cavitation behind wings. Namely, a cylindrical obstacle of 6 mm diameter has been placed perpendicularly in the way of a filly formed cavitatmg vortex kernel of 1 mm diameter. The kernel has quickly grown to about 6 mm and then stabilised its size. Behind the obstacle the kernel has disappeared and only single nuclei have managed to cross it. Such a behaviour may
be explained only by the existence of a "nuclei factory" at the front end of the cavitating vortex.
5. COMPUTATIONAL MODEL OF litE CAV1TATING TIP VORTEX.
The starting point of the analysis of tip vortex cavitation inception consists of the pressure field generated by the deforming tip vortex and the nuclei content of the oncoming flow. It
follows from experimental evidence that the nuclei are sucked into the centre of the vortex. The quantity of nuclei sucked into the vortex centre depends on the initial nuclei size, the
s
)
It
6.96 8.968 8.978 8.968 8.996 1.888 1.818 i.8
136
8 958 8.966 8.978 8.968 8.998 1.888 1.818 1.826 £.C3
Explanation to figures.
x,y - ordinates in the systems linked to the wing, - half span ofwing,
t' - time related to i = I/V0
V0 - velocity of the undisturbed flow,
Ap
P - 0.5p2
Fig. li
Pressure field defined by isobars in
the vicinity of a deforming free vortex surface for t 0.01
C values
0.0 - 0.6 0.6- 1.2L\\\
1.2-1.8
///A
1.8-2.4
P\HH 2.4-3.0
above 3.0 Fig. 12Pressure field defined by isobars in
the vicinity of a deforming free vortex surface for z0 = 0.08
density of nuclei distribution in water and on the pressure gradient near the vortex centre. For
calculation purposes the pressure field in the vicinity of the vortex is converted into an
axisymmetric one. When the pressure in the vortex centre drops below the critical value, which
depends on the initial nuclei size, an explosive growth of nuclei results, which may lead to
cavitation inception. The behaviour of nuclei in the variable pressure field may be studied using
the Rayleigh-Plesset equation, which is strictly applicable to isolated spherical vapour-filled
bubbles only:
=r--+__ - +-+----
L.p d2r3"dr'\
2S 4vdr
p
dt22.dt)
pr
r
dt (6)where: p - pressure difference across bubble boundary
r - bubble radius
S - surface tension
This equation may be solved separately for each initial nuclei size using e.g. the
Runge-Kutta method, assuming an appropriate time step and initial conditions:
=0
(7)dt
Inception of the tip vortex cavitation is decided when the bubbles reach a size visible witha
naked eye (it is assumed that r0.0002-0.00025 [m] is sufficient). However, acoustic emission from pulsating bubbles may be present even earlier. The frequency of this pulsation may be assessed by summing time steps of integration of equation (6) during one full cycle of bubble
oscillation.
0.001
(8)
Then the acoustic pressure emitted by a single bubble may be modelled by placing a single oscillating source in its centre, having the intensity described by the following relation:
q1
=-,r(r
_,3)
cos(2irf t)
(9)The linearized form of the Bernoulli equation may be used for the calculation ofpressure pulsations:
p
8x
8t
(10)4irR
where: 1 - potential of the oscillating source
Usually the acoustic emission of the cavitating tip vortex starts with a fairly high tone (2-3 kHz), the frequency of which is gradually reduced with falling pressure. At a certain stage the line of oscillating bubbles is linked into one cavitating kernel, which then grows in diameter with further reduction of pressure. A fully grown cavitating vortex kernel behaves more like a
6. FINAL REMARKS
The basic assumptions of the theoretical mode! of formation of vortex cavitation were
established in the I 970s. However, the detailed research on this subject has started in the i 990s
and at present it is still continued. Many details remain to be clarified and new problems still appear. But even now it may be concluded that the Double Layer Lifting Surface model is the most suitable one for realistic simulation of the flow near wing tips, with particular application to vortex cavitation. As is the case of any theoretical model, it does not account for all
phenomena present in the flow in this important region. These neglected phenomena may be described numerically by a complete solution of the Navier-Stokes equation, which became
possible with the arrival of supercomputers.
A vortex model of the propeller may seem rather obsolete, especially with spreading
application of surface panels methods and Navier-Stokes solvers. However, in our opinion the
vortex model will for many years remain the most practically effective tool for design of marine propellers and for analysis of different aspects of propeller operation.
REFERENCES
i Betz A. :Wie ensteht em Wirbel in einer wenig zaehen Fluessigkeit, Naturwissenschaften
37, 193, 1950
2 Greeley D.S. & Kerwin J.E.: Numerical Methods for Propeller Design and Analysis
in Steady Flow, Trans. SNAME vol. 90, 1982
Koronowicz T.: Theoretical Model of the Propeller Slipstream Based on the Concentrated
Form of Vorticity with Cavitating Kernels... (in Polish), Bulletin of the WFM No. 58/956/79
Koronowicz T.: Deformation of the Vortex Surface in Close Vicinity of a Wing of Finite
Span (in Polish), Bulletin of the IFFM No. 67/942/79
Koronowicz T.: Theoretical Model of the Propeller and its Slipstream Taking into Account
the Deformation of the Free Vortex System, Proc. Symp. on Advances in Propeller
Research and Design, Gdansk, February 1981
Koronowicz T.: Modification of the Vortex Model of a Marine Propeller (in Polish), Proc. 10th National Conf on Fluid Mechanics, Gdansk, September 1992
Lewis R.I. & Ryan P.O.: Surface Vorticity Theory for Axisymmetric Potential Flow Past Annular Aerofoils and Bodies of Revolution with Application to Ducted Propellers and
Cowls, In. Mech. Eng. Sci. , vol. 16, No.6, 1974
Lewis R.I.: Keynote Address: the Vorticity Method - a Natural Approach to Flow
Modelling, Trans. of Inst. of Machine Construction, Wroclaw Technical University,
Series 46, No.9, 1986
Martensen E.: Die Berechnung der Druckverteilung an dicken Gitterprofilen mit Hilf von
Fredholmischen Integraigleichungen zweiter Art, Arch. Rat. Mech. Annal. 3/1959
Sarpkaya T.: Computational Methods with Vortices, The 1988 Freeman Scholar Lecture,
Journ. of Fluids Engineering, March 1988, vol.111/5
il. Thwaites B.: Incompressible Aerodynamics, Oxford 1960
