2 181
Lab. y. Scheepsbouwkunth
ARCHIEF
Technische Hogeschool
NAVAL HYDRODYNAMIC PROBLEMS
SED
BY RHEOELECTRIC ANALOGIES
PROBLEMAS DE HIDRODINAMICA NAVAL RESOLVIDOS ATRAVES DE CRITERIOS DE ANALOGIA REO-ELETRICA
por/by L. MALAVARD (5)
1. - INTRODUCTION
For ten years the Centre de Calcul Analogique
(C C A.) of the Centre National de la Recherche
Sci-entifique has contributed by various works to the study
and to the solution of quite a large number of naval
hydrodynamic problems. This contribution may be
considered very significant since
it has been made
possible by a small team of research scientists using very simple computing equipment. This equipment could seem inadequate for the work to be done in theeyes of the non initiated or of the staunch believers
in computing on large computers.
However, it is not possible to consider these studies
of naval hydrodynamics completely isolated from a
context where rheoelectric analogy is the means which has enabled, and still enables, important developments
in the most varied fields of Mathematical Physics.
And, in this connection, it is convenient to recall that
the first studies carried out in France using the
electrical analogy techniques concerning some hydro-dynamic problems; flows around bodies with or without circulation, Oseen flows (1) s (2), flows with jet streamlines (3), etc.; premise of a budding vocation; a
vocation which became more decisive as from 1958 thanks to the experience acquired by the C. A. C. in
the treatment of problems in incompressible
aerodyna-mics, thin foils, lifting line, lifting surface, cascades, simple helicoidal machines, etc., (4), (5), (6), and
thanks to the introduction by Tulin and Burkart (7)
in 1955 of the linearised theory of cavitations. One of the assets which has assured the success of rheoelectric analogy since its early beginnings has been its rapidity, as well as its ability in solving La-placien field equations. This capacity for computing,
Professor of the University of Paris (Chair of Aviation), France
Diretor of the "Centre de Calcul Analogique" of the C.N.R.S., France
246
APRESE NTAÇÂO
Durante dez anos o Centro de Cálculo Analógico do Conseiho Nacional da Pesquisa Científica de Paris, na França, con tribuiu corn vários trabalhos para o estudo e a soluçäo de urn sem-número de problemas de Hidrodinámica Naval. Tal contribui. cáo deve ser considerada multo significativa, pela relevância dos estudos realizados e pela influéncia exercida na soluçâo de pro blernas fundamentals da Arquitetura Naval.
Tal importante contribuiçâo se tornou possivel, entretanto, graças apenas a urna p e q u e n a equipe de pesquisadores de grande valor que, utilizando equipamentos de cálculo e de experimentaçâo bem simples, deu ao mundo urna sobeja
dernonstra-cáo do péso das qualidades humanas, ncluindo nelas a perfeita aplicaçáo de correta metodologia ementalidade científica, na con-dução de pesquisas teorético-aplicadas.
Corn éste trabalho, todavia, o Professor Malavard, Ilustre catedrático da Unlversldade de Paris e Diretor do referido Cen-tro de Cálculo Analógico, conseguiu, a nosso ver, produzir a tese de major importância para a Arquitetura Naval, do referido time daquela instituiçâo, tendo merecido um significativo sucesso no SÉTIMO SIMPÓSIO INTERNACIONAL DE HIDRODINÁMICA NAVAL, recentemente realizado em Roma, na Itália, de 25 a 30 de agâsto próximo passado.
TECNOLOGIA NAVAL está apresentando, portanto, um trabalho de grande valor, pràticarnente inédito,
together with the experimental character of the
tech-nique employed, makes an ideal means for the practical
worker, engineer or physicist, who remains in contact with a model on which his controlling action may be exercised without any restraint. Nevertheless, for an
intensive and complete use of the method, analog
simulation often requires turning to certain methods of theoretical formulation familiar to the mathema-tician. It is in this way, for example, that the
know-ledge of elementary analytical solutions, the use of
conformal mappind, the analysis of singularities, etc.,
allow the solution of each problem in the most efficient way
From these three given elements cited, experience
acquired in incompressible aerodynamics, the
lineari-sed theory of cavitations and auxiliary analytical data,
naval hydrodynamic studies have been developed as
follows.
1.1. TWO-DIMENSIONAL PROBLEMS (fig. 1)
In 1958, Luu carried out studies on the solution of
the direct problems of supercavitating hydrofoils (8), 9). These studies were the continuation of important
research devoted to the problem of thin jet streams
in aerodynamics (8), (10), (11), (12) and came within the framework of linearised free boundaries.
In 1960 a: research programme was envisaged concerning the effects of the free surface on slightly immersed sub and supercavitating hydrofoils. In the case of small Froude numbers, that is to say a
consi-derable influence of the gravity field effect, is was
possible to proceed easily to their design for imposed
pressure distribution (inverse problem) (13) (14). These
studies took into account the gravity effect on the free surface and on the finite cavity, which, to our knowledge, had not yet been treated. The direct
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VOL. - N.° 3 JULY/SEPTEMBER 1968 245Ci=
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AUXIL!*RY MAL,rC.4. 5OLUTOS
Fïgure i
problem in the case of the immersed flat plate was also solved and allowed a useful comparison with
analytical results (13) (15).
In the case of high Froude numbers and zero
cavitation number, Luu and Fruman published, in
1963, a rheoelectric method permitting the design of
ventilated hydrofoils with arbitrary local pressure
dis-tribution (16). The results obtained agreed with those
of Auslaender (17), published shortly before, and
exten-ded them by the definition of shapes with larger lift-drag ratios.
It have been provedthat the drag of
supercavitating hydrofoils is related to the angle of the spray for downstream, and it seems natural that
these studies lead to the design of base vented hydro-foils with zero drag (13).
The studies of subcavitating cascades had been dealt with thoroughly by Malavard, Siestrunck and
Germain, (18), (19), (20). (21), (22), within the
frame-work of the foil theory. The linearisation used by Luu
in the case of thin jet flap on the trailing edge of
cascades (8) was easily extrapolated to the case of
supercavitating cascades (23) liable to be used in cer-tain types of pumps and turbines.
1.2. THREE-DIMENSIONAL PROBLEMS 1.2.1. HYDROFOILS
The two-dimensional studies on supercavitating hydrofoils led Luu to carry out an analog simulation in the case of finite span wings (24). The experience acquired in the treatment of lifting surface problems thanks to the work of Malavard, Duquenne and
En-selme (25), (26), (27), (28) indeed allowed a very rapid
implementation of the supercavitating problem in
unbounded flow field by the introduction of an inge-nious decomposition of the potential, which will be examined in detail further on.
VOL. I - N.° 3 - JULY/SEPTEMBER 1968
The method used also permitted the design of
supercavitating wings at zero cavitation number near
the free surface (29) . The optimal vortex distribution
over the span was obtained by using the properties of
the potential in the Trefftz plan (30) and by
trans-posing the analog simulation used in the principle of the "lifting line computer" (31) . Finally, in order to allow the comparison of analog results and experiments
in a small highspeed hydrodynamic channel, a special simulation device permitted the design of
supercavi-tating wings with strut and walls effects (32), (33) and calculation of hydrodynamic characteristics ol
flat supercavitating wings. 1.2.2. SCREW PROPELLERS
On the same principle as the "lifting line compu-ter", Siestrunck had conceived, in 1944, an "an.alog propellers computer" for the case of large aspect ratio
blades . This realisation was taken up again by Sulmont
in 1959 who improved it in making it easier to use by introducing a resistance network; and besides he
adapted it to simulate easily hub effects (34).
Because of the small span ratio of their blades,
naval propellers can only be calculated from theories used in aeronautics by introducing more or less justi-fied empirical correction coefficients. It is only in
:959 that our first efforts were made to apply the
theory of the lifting surface to the helical flows. The many difficulties of solution by analytical and flume-rica! means are well known; they come, for the most part, from the complexity of the flow field to be
con-sidered.
Now the rheoelectric method allows the represen-tation of this flow field, and thus the design of small
span ratio blades become possible by means of
techni-ques similar to those perfected for wings of arbitrary shapes (35). The boundary conditions corresponding
to supercavitating blades can also be imposed without
major difficulty and lead to a correct definition of the lower surface for imposed pressure distribution (34) (36). It should be mentioned that this problem has not yet received any numerical treatment and
this is why the studies being, made at present at the C.C. A. are aiming at the transposition of the analog method into a programming which could be used on
large computers.
In the same frame work Sulmont has studied the
problem of ducted propellers; by making some
assum-ptions of the propeller's nature (infinite number of blades) he has been able to define adapted duct forms
which seem to promise high propulsion efficiency.
To complete and conclude this account we must mention the studies being carried out at the C.C. A.
At present our attention is slanted towards the solution
of the problem concerning the immersed or
semi-immersed bodies which may be so thick that the
linearised boundary conditions relative to the obstaclecan no longer be applicable, however the linearised
free surface is preserved. A two-dimensional study
(37)
has permitted to test the validity of a new
theoretical schema (38) and in forthcoming studies
results in the three-dimensjonal case should be obtai-ned very soon and, from there, the calculation of the
247 wIr4ÇR -WWaCE
wave resistance of a thinck hull will be undertaken.
It would be difficult to sum up completely here ail the publications referenced above. We will just present some of the most signif.icant examples of the
rheoelec-tric method and the most outstanding results of its
use.
2. - THE HYDRODYNAMIC PROBLEM GENERAL EQUATIONS AND BOUNDARY
CONDITIONS
Consider the permanent and irrotatlonal flow of an inviscid, incompressible and heavy fluid with
den-sity p past a supercavitating hydrofoil located at a
depth d beneath the free surface, the velocity far
upstream being V0 . A set of cartesian coordinates,X', y' and z' is chosen in such a way that the positive
directions of the x' and z' axis are respectively those of V0 and of the upward direction. Because the plan form of the wings are generally symmetrical, the field simulation can be limited to a quarter of the space.
The movement is described by the perturbation
velocities potential ' which must fulfil the following
boundary conditions (Fig. 2).
2. 1. On the free surface z = 0, the pressure Po S
constant and thus the equilibrium condition give
2
(1)
x2
that Is a Poisson condition for4; where F=V0 h./gd is the Froude number and g being the gravity force , the
inward normal and 4 = 'i", x =
and n
-V0d d d
are non dimensionai magnitudes.
2.2. Inside the cavity where p = p the equilibrium
of its boundary requires that:
= f
+ F
(Z0-z) (2)which becomes after Integration
(y)+ x + Ji½(zoz) d (3)
where is the value of the perturbation velocities potential at point , z0, for example those of the
leading edge, and the signs + and - relating to the
upper and lower surface of the cavity. The cavitation number a is defined as
a=
P0 -Pc2.3. On the lower surface of the hydrofoil the boun-daiy condition can be given in two ways:
Direct problem. The geometric form of the wing
-(x,y) is given, then the velocity tangential condition permits to write
d
- dx
that is the classical Neumann condition.
(4)
BOTJNDAR( CONDITIONS FOR A SUPERCAVITATING
WING AT y=Ct
& d ,c - _____ d r(Y) -J 248 TECNOLOGIA NAVALDIRECT PROBLEM INVERSE PROBLEM
q_p(y)r!g (x-xy)4-X
Figure 2
2.4. Inverse problem. The pressure distribution over
any local chord p'
p - p
is given. The boundarycondition may be written
"X
- TRVO
2
(z0-Z)
-
f
-
.. C (x,y)±f2(z0-z)This equation can be Integrated to obtain a Dirichiet
condition: (5)
2 r ' is the local lift coefficient at
where C1(y)
V0C(y)a given section y=cte with chord C(y), and F (y) the circulation around this section.
r(Y)=_=$
p' dx',
(6)XL
xtx1+c
x = x1 (y) is the position of the leading edge at the
same section. The function g should be such as
g(x,y) = O and g(x,y)
2.5. On the trailing edge of the wing the
Kutta-Joukowski condition must be respected=0
(7)\SX/ x=xt
Xt = x1 + C is the position of the trailing edge.
2.6. On the plane y = 0, by the symmetry of the
flow, the normal velocity is zero:= 0
(8)\fl/y=o
and at infinity upstream the gradient of t is also zero.
grad=0
(9)2.7. The cavitation pocket must be closed, that is to say that in a section y=cte, on a closed contour sur-rounding the foil and the cavity
x1(y):
--- dx =
(Çj--
ds = 0
(10)7z
Jfl
2.8. The boundary value problem defined by the con-ditions (1), (3), (7), (8), (9) and (4) or (5) is not yet determined since the distributions of the potentials
on the lower and upper surface of the cavity remain arbitrary. In fact, this does not constitute an
indeter-mination since in the inverse problem they are
connec-ted by the known value of Ct (y) in equation (5). In
the direct problem
it may be considered as the
unknown of the problem which fulfils the condition (7). We shall not give the detailed discussion of this
question, but we will rather insist on the methods used
for its solution.
2.9. In the two-dimensional case there exists an
asso-ciated harmonic function , perturbation stream
func-tion, defined by the transformation of the condition
(1), (3), (7), (8), (9) and (10) On the free surface
_=i2(_)
oOn the upper and lower surface of the cavity
(2') +
On the lower surface of the foil in the direct problem '4c=
,1,or
and ¡n the inverse problem
2
_=Cg(x) +--+F (4)
2At the infinity upstream of the field
grad ' = o
The closure cavity condition is now written
I-iC = Ye,
where e and c' are two points placed at the downs-tream top of the cavity on both sides of the slit.
The symbols have the same significanceas in the three-dimensional case, except CL, global lifting
coeffi-cient, and g(x), function which should now fulfil the
conditions rxt
g(x1)=0, g()<0 for x1<<x and
Jxl.
g(E)dl
3. - RHEOELECTRIC ANALOGIES - PRINCIPLES
The principle of rheoelectric analogies is classical
and various publications on this subject (5), (6), (39)
give enough elements on the special technology
requi-red, however, some general ideas are recalled in the
following in order to avoid tedious repetitions further on
The analogy is made between the Laplacian of
the velocities potential (or of the stream function),
and the Laplacian of the electric potential, created
VOL. I - N.° 3 - JULY/SEPTEMBER 1968
in a homogenous and isotropic conductor. The later
is generally constituted by a liquid contained in a
rheoelectric tank confined by boundaries where elec-trodes, of judicious form and disposition, are placed.
The boundary conditions are introduced in a generally
discontinued way, by means of suitable electric set-ups. The two most simple conditions which are very
often found in the problem treated are those of the constant potential, condition (1) of § 2 for F =
or zero normal derivative, condition (8) of § 2, on one or several boundaries. They are conveyed respectively by conductor or insulating surfaces.
Figure 3 shows the three types of boundary
condi-tion, Neumann, Dirichlet and Fourier, and the corres-ponding analog set ups. The Dirichiet condition, poten-tial given ori a boundary, formula (5), is easily given by
the use of potentiometers or of voltage dividers. The
Neumann condition of the (4) type is realised using resistances of a high value R so that, in feeding by a
unity reference potential, the potential on the electrode
be equal or inferior to 0.05. Thus is found.
6n
--where s, represents the surface of an electrode and
K an analog constant. The values of R are
deter-mined by s
-
i is the conductivityo-
K-of the conducting fluid.
The Fourier condition, linear relation between the
potential and its normal derivative, it is frequent in
heat problems and thin jet flap problems (8) or lifting
line problems (2), Considered the set up C of figure 3, the Kirchhoff law permits to write
v_-a0.
--Y.=v
which is comparable to
BOUNDARY COiITlOw ELECTRICAL REPRENTA11cN
Figure 3 249 A- DnC iEUW =iA) R
-ob
Ro A0,R,CfrtEr-FLOW CON5ERVATION L:' B'. '(LM- FLOW CON5CVTON I) tIij IiUi R-- Cl FOURiER R-- FLOW co'ncRvarwoNa V b
.L,J _,L
Çz,_ RR
(9,)
a
provided that
a and
Upon these three conditions, sometimes it is
ne-cessary to impose between the two sides of a slit the
conservation of the flow. In this case the electric set-ups are similar to those of A, B and C, but they
require a transformer which automatically assures
this supplementary condition.
It is evident that the precision of the analog
representation of a problem depends fundamentally
on the electric transposition of the boundary
condi-tions. To go into the detail of the techniques applied
in order to make the systems
described above asaccurate as possible would go beyond the limits of this papel. Nevertheless, it is interesting to note that, even
in the most difficult cases, the elements inserted in
the electric circuit are passive: resistances, potentio-meters and transformers. This process of simulation contrasts with that used elsewhere (40) where active elements, of intricate electronics, are incorporated in rheoelectric experiments which are in themse)ves of
great simplicity.
4. - TWO-DIMENSIONAL PROBLEMS
4.1. SUBCAVITATING HYDROFOIL NEAR TRE FREE
SURFACE
Although the study of the subcavitating hydrofoil is not, chronologically the first naval hydrodynamic problem to be treated at the C.C.A., we believe it is
interesting to begin the review of two-dimensional
problems with this study.
4.1.1. SOLUTION OF TRE DIRECTPROBLEM
Consider an immersed foil represented by its mean
line, T = ( ), near the free surface. The
hydro-dynamics characteristic of the hydrofoil are
determi-ned in solving the following boundary value problem;
on the free surface we have the condition (1'), on the slit LT representing the foil, '7
i
-
, on th,trailing edge, +
+
= 0.
n fl
The electrical simulation of the condition (1') is
performed by the use of negative resistors (40), but their use is not easy and sure. We preferred to use
an indirect method which allows the replacement of
the Poisson condition by a Dirichlet condition. It uses
the fact that for each vortex
distribution connectedto the lifting foil, the ordinates of the free surface, which is in fact induced by these vortex may be conputed numerically by the composition of known (41) elementary perturbations.
The solution of the problem may be obtained for
s» C ís4* s-) CL_2X L V T 'l d 3 Figure 4
a given shape of the hydrofoil by a series of operations.
each one consisting in two stages. First, for arbitrary values of 'i in the linearised free surface, one
com-putes, by rheoelectric analogy, the vortex distribution over the chord of the foil which fulfils the Joukowski's
condition on the trailing edge, without, however, complying with the constant pressure conditiûji at the
freen surface. Secondly, the ordinates of the free surface, which would induce in reality the preceding
vortex, are determined numerically. This allows a new
distribution of potentials on the z axis
and a new
analog computation of the connected vortex. Thecycle of operations is continued until the potentials
on the free surface and the vortex distribution
con-verge simultaneously towards functions which
repre-sent the solution of this boundary value problem. A few approximations are generally sufficient. Instead of introducing an arbitrary free surface into the first analogical approximation, it is easy to introduce the
boundary conditions corresponding to zer6 or infinite Froude numbers. C-=Q5 -0.25 q5 qzs .25 250 TECNOLOGiA NAVAL d .1,0 c ) L 4. - F p°- qr-
VT
(-l'ci
NACA 55 .Q25 nc -oc NACA 65 C1. Ce,,, 2 3 Figure 5 F=.The accuracy of this method was verified by
comparison of analog results to those obtained by Isay
(42) in the case of a flat plate with incidence (figure
4). The application of the rule of reverse flows to free surface flows and finite Froude number (15) permits the useful exploitation of results obtained in
the case of the plate and the rapid determination of
the influence of the free surface on foils of arbitrary shapes (fig. 5). An interesting example of the
possi-Ci. h
CLOO
1.0
bilities of the method is given in figure 6 which shows
for different Froude numbers the distribution of per-turbation velocities on the lower and upper surfaces of a flat plate with flap slighty immersed.
flot plop. wilt, Flop
-°400 *00.20 2C * V0t
fl\
V
(+ ±)o_g (e.) MEAN- LINE SIJBCAVITATING HYDROFtIL Figure 7 VOL. I - N.° 3 - JULYtSEPTEMBER 1968 Figure 64.1.2. DESIGN 0F SUBCAVITATING FOILS NEAR THE FREE SURFACE
The same method may be used to design hydrofoils
with given load and thickness distributions. Two
effects must be then considered separately, the first
corresponding to the distribution of connected vortex
y ( ), that is to say the lifting effect, the second to
the equivalent distribution of sources and sinks, that
is to say the thickness effect. The boundary value
problem is now completely defined and the rheoelectric simulation is very simple.
Figure 7 shows, for different Froude numbers, the mean lines obtained for the NACA 65 pressure
distri-
p-bution. From the linearised theory results and in order to verify them, a hydrofoil and the corresponding free
surface were represented in a rheoelectric tank. By considering the streamlines of this flow shown in
figure 8 it is possible to verify how the Joukowski's
condition on the trailing edge and the free-entry
shock condition at the leading edge are fulfilled. The
lift coefficient computed from the value of the
cir-culation, corresponding to the electric results, is 0,3% higher than that chosen to design the hydrofoil.
In the case of small Froude numbers the gravity field effects on the free surface and on the boundaries
of the cavity may be considered. The rheoelectric method enables us to take them into account with
precision without inasmuch complicating the
compu-ting process. An important simplification is obtaíned by introducing two auxiliary functions and '2
defined by the following boundary conditions: On the cavity ± 34 + and, ± ± = F2 2
On the lower surface of the foil, taking into account the gravity effect,
± -2
-w'
= .j- +
F ('4ì-'),
and, -2-
-j-
=
CL g(x) + FThe first function corresponds to a non lifting
and free of wave resistance effect, as has already been shown (13). The second function represents the lifting effect connected to the expression of the local pressure
distribution. The calculation is made taking as a start
the solution for F = , which is of an easy analog
determination because at the free surface,
251 Figure 8
4.2. SUPERCAVITATING HYDROFOILS NEAR THE FREE SURFACE
4.2.1. SMALL FROUDE NUMBERS
111 02 113 04 Q5 ".'
a.
I.. .u.s
0,2 0,1 L - 0,1 0,2 0,3 -0,00 - 3,09 - 2,S2and the above conditions are of the Neumann type
with flow continuity (fig. 3). From this first solution It is easy to define distributions of sources and sinks and of vortices Induced by the cavity; this permits to
calculate numerically the free surface for finite Froude
numbers. The iterations are then carried out as in
parc.graph 4.1.1.
Figure 9 shows the for mof foils for the same immersion depth, the same cavity length and for
Froude numbers respectively infinity and 3.99, as a function of the parameter CL / . The Results for the infinite Froude number are given as a means of com-parison; it is evident that in this case the hypothesis
of a finite cavity is no longer valid since the cavitation
number tends to zero at the same time.
SUPER(AVITATLNG HYDROFOIL NEAR THE FREE-SURFACE o.#o on dl =--+E ( - $) 4c -g( C - 9(1)-O Figure 9
4.2.2. INFINITE FROUDE NUMBER
On the free surface, the upper and lower surfaces
of the cavitaty, we have
on the lower surface of the foil a Neumann conditions
is imposed,
= CL'
g(x'.
Ato:1o) j
la. u.
_k lohn -9 -ko,t- -'oli
C 26 DL
'lCol1
5UPRCAVITA11NO HYDROFOIL ro Figure 10This makes rheoelectric simulation of a great
simplicity. Figure 10 allows the comparison of the
foils computed for different linear pressure distribu Lions with that fulfilling the two term law of
Thun-Burkat. The comparison of the lift-drag ratio is fa
vourable in the former and shows the advantage of
the rheoelectric method in the exploitation of pressure
distribution hardly accessible to analytical treatment..
If a convenient pressure distribution over the
upper surface of the foil os imposed, it is possible to
design tase vented hydrofoils with zero spray-jet drad.
The depressions thus imposed should be such that
the cavitation formation is excluded upstream of the trailing edge. For this purpose a number o1 must be
defined, fuction of the physical characteristics of the fluid, the vapour pressure, the degree of air dissolved, etc.. Three foils, obtained for different pressure dis-tributions, and presenting the same value of CL
are shown on figure 11.
4.3. HYDRODYNAMIC CHARACTERISTiCS OF SUPERCAVITATING HYDROFOILS IN UNBOUNDED FLOW
These studies were intended to test the fiability
of the analog representation of the singularities which arise in the solution of the direct problem of superca-vitating foils. It is known that near the leading edge
of a hydrofoil, if the slope is finite at the lower surface
and the pressure constant at the upper surface, the
complex perturbation potential, 4) + i , gives a
sin-gularity of - i k zI'I which corresponds to a complex
perturbation velocity u - iv = -i k z'/
. Thepattern of the singularity is given in figure 12; it is
seen that the equipotential line from the uppersurface
of the slit is bending in the leading edge forming an angle of 240°. Analogically this can be obtained by means of an apparatus indicated on the same figure
9(0 T Q ;, Q QQ 4 Cl, BA VENTED HyOForL3 Z22 Figure 12 Figure 11
The electrode representing the upper surface of the
cavity is extended by a small conductor plate placed
with an angle of 2400. In the prolongation of this
plate a proble is installed, by means of which we can control the correct configuration of the equipotential
line thanks to the adj ustement of the potentiometer.
The use of this set-up is successful in
two-dimensio-nal case. Its extension to the three-dimensiotwo-dimensio-nal case is prevented due to the complexity encountered.
Ano-ther method must be used; we wiLl leave its description
to the section devoted to wings of finite span. 4.4. SUPERCAVITATING CASCADE
The studies of supercavitating cascades are of an undeniable praticai interest in the field of hydraulic
machines; pumps and turbines. It has been possible to carry out the design of convergent or divergent cascades constituted by supercavitating foils which
support imposed pressure distribution. In this case,
the rheoelectric method shows the amount of possibi-lities open to the design of supercavitating propellers.
Null rdQ,olO"
Suppose that the foil camber is small, it is possible
to consider, as in the case of
isolated profils, thelinearised flow respect to the velocity far upstream.
The periodicity of the velocities field allows the study
of the function 4, in a bounded strip, figure 13; the
boundary conditions are defined, no longer on a slit
as in the preceding cases, but on the two surfaces limiting the strip. The flow is supposed independent
of the gravity field and the boundary conditions are given by (3), (5), (7) and (10); a supplementary con-dition which takes into account the periodicity of the
fields is conveyed by
VOL. I - N.° 3 - JULY/SEPTEMBER 1968
00
4B
where B and B' are two points
periodically appartupstream of the foils.
....
'..)*--9o.o)
--(°-i) COCVSo, or CI 1 1
L9IJÇDÍJ
T TJLJ i1J
The analog representation of these conditions is extre-mely simple as can be seen in figure 13. The potentials
law on the cavity as well as the potential differences between the lower surface of the foil and the upper surface of the cavity are obtained by means of
poten-tiometers. The closure cavity condition is fulfilled
automatically owing to the transformers which insured
the conservation of the current. Figure 13
I
-t'!ììiIk
15Figure 14
Figure 14 shows the form or the lower surface and
of the cavities obtained for an uniform load
distri-bution over the foil. They depend on the length of
the cavity and on the value of
the lift coefficient253
imposed to each foil. The highest limit of is
dictated by the thickness tolerance which may be
allowed.
5. - THREE-DIMENSIONAL PROBLEMS
We will first consider the problems connected to the calculation of supercavitating wings; then those of the sub and supercavitating screw propeller.
5.1. SIJPERCAVITATING HYDROFOIL WITH
NON-ZERO CAVITATION NUMBER IN UNBOUNDED
FLOW
Consider the inverse problem as it has already
been defined in § 2. In order to simplify the analog
representation it is better to write the velocity
poten-tial in the form
(x,y,z) = X + X + 4)(x,y,z)
The perturbation velocities potential, 4), is then defined by boundary conditions, slightly different from those corresponding to the function4)== -x. These conditions
are: on the upper and
lower surface of the cavity,= O: on the lower surface of the wing the
pressure is higher than or at least equal to the cavi-tation pressure; we thus have the condition ö4)/bx O,
and to define the
distribution of 4) we will have,according to (5),
(y) + 4
C1(y) g(x-x,.y) (5")where g is given, and fulfils the conditions indicated in paragraph 2.4. At infinity we should find the velo-city of the undisturbed flow, hence
= -
(9")2
The closure cavity condition is conveyed in any section
y = cte, by
--dx
O. With the overall boundary conditions we have just established, the ordenates of the lower surface of the wing and of the contour of the cavity are given, if we take, as Tulin suggested,the tangential velocities condition in the form
-
_Y. instead of
dzi by
dx V0+u dx V0
2 -;;:
JX(
--
dxThe rheoelectric simulation can still be simplified
if the potential 4) subdivided into three parts + 4)E2+4)OD
where, and represent two even potential
functions, they characterize the cavitythickness effect 4) represents an odd potential function, it
corres-ponds to the general camber effect. Analogically the
Figure 15
representation of these functions is extremely simple since it amounts to imposing on the plane of the wing,
and outside the wing, the cavity and the wake, a zero
normal derivative condition for an even function or a
constant potential condition for an odd runction. These
three potentials are defined by boundary conditions
such that their sum on each boundary be equal to the
condition of the potential 4). Thus, for example, ori the lower surface of the wing and cavity we will have:
4)EI = A(y)
C1(y)
4)E2
B(y) +
g(x-x1,y). C1(y)
4)0D ± 4 [1g(x-x Y)]
the constants A(y) and B(y) are connected according to the above expressions by,
C1 (y
A(y) + B(y) = 4 (y) +
4Figure 16
Figures 15 and 16 show the shape of two superca-vitating wings: one of rectangular planform with span
ratio 4 and the other with span ratio 4.5 with an
elliptical leading edge and a straight trailing edge.
The pressure distribution following the chord of each
section was chosen of the Tulin-Burkat type, and
that of the span circulation is elliptical; the maximum
length of the cavity, in the median section, is three
times the chord. The difference between the sections
of the two hydrofoils and especially the thickening
254
of the rectangular wing at the wing tip, will be noticed. 5.2. SUPERCAVITATING HYDROFOILS WITH ZERO
CAVITATION NUMBER
In the case of high Froude numbers the flow
around the wing is similar to that which has already been studied in the two dimensional case in § 4.2.2.
However, the solution of an important problem should precede the design of supercavitating wings; this
pro-blem is the determination of the optimal distribution
of span circulation.
Luu (30) has shown that this problem is reduced to that of the optimum vortex distribution of the finite
span biplane - constant induced velocity over the
span -, treated in (43) and (44). In these publications are found only global results concerning the lift-drag
ratio and not the vortex distribution
on the span
which is the most interesting feature. Although it is possible to obtain a solution to this problem by ana-lytical methods, it is not inappropriate to indicate that
the rheoelectric method can be utilised advantageously.
Consider the flow observed in the Trefftz plan. The
potential , harmonic function in y, z, is defined by the following boundary conditions:
= O on z = 0, the free surface,
Ö4/z= w, = Ct,
on the wake (z = d, -s
y .
s) ¡ n=0 for y = O, by symmetry.iumuu
dmu
II!! sumum
° 'smu
-.m..
uuuuarnu
uumuuumuR' s
muauumummuuaUuuL
.umuu.u.uuumu.S1
i.u...ummu..umuu.mi
1 Q5 1.0 Figure 17We can see that these are classical conditions;
Dirichlet on the free surface, Neumann on the
simme-try axis and Neumann with flow continuity on the
wake; the analog simulation is immediate. Figure 17
VOL. I - N.° 3 - JULY/SEPTEMBER 1968
shows the distribution of r/sw,,thus obtained, where
is the circulation, s the half spari of the wing, and w the induced velocity, versus y/s for different values
of the parameter 4. These results permit
us to
approach efficiently the solution of the inverse problem
for a supercavitating wing near the free surface.
5.2.1. DESIGN OF A SUPERCAVITATITÇG WING
The boundary conditions on a section y =cte are
the following:
4)=O, for z=0
q= 4)1(y) , for z = -d
, X1 <x < e)=
(y) , for z = -d , x <
< - C1(y) (P =(y) +
2 g(x-x,,y), for z =for z = -d
, x < x <
X +-um r
¡
\
d)I l+ dx-O
X9.Jx1\öfl
1nJ
When is zero, the growth of the cavity thickness
is simulated by a sources distribution over the wing
and the cavity with a density q(x,y) defined by
ci(1
+ =
n 5n X lx
Far downstream, q is reduced to a function which depends only on y. However for the inverse problem
we dispose a certain latitude in the choice of the
sources distribution. In fact, the boundary conditions
a), b) and C) allow that ori each line parallel to the x axis, within the limits defined by the wing span,
the potential 4) is fixed with an arbitrary lever. If we Indicate by k(y) the mean value of 4) and we have
k(y) = 4)1(y) C1(y)
4
It is evident that the distribution of q over the
surface of the wing and the cavity, i.e. the cavitation shape, is directly influenced by the choice of the law
that is attributed to k(y)
In order to facilitate the analysis of the problem, the potential (P is subdivided into two parts, defined
by the following boundary conditions:
a') 4, = O =
+
k2(y), on
z+=d,xi<x<
e') 4
=
4) (y) , (P; = k2(y),on z =-d ,xt<x<'
4)=4)(y)+C,(y)g,, 4Ç=k2(y),
on i=-d,x1<x<xt
Jim d)1
xiln
ln/
255 b') p1 =on z =0
C) 2d' 11m
2
X..wI (+)dx-+0
Ji1 \bfl
bn/
Following the decomposition of the movement, the
function k(y) is also split into two parts: k1 and k2
We see that the arbitrary choice of this function is
supported by p2 and that is completely defined by
its overall boundary conditions. The solution of the
boundary value problem of 2 depends on the choice of 2 (y), which finally amounts to the choice of the
thickness distribution of the hydrofoil. In the most
general way, the choice of k2 (y) is essentially dictated by the structural point of view. The drag coefficient is
available by considering the kinetic energy on the
Trefftz plane.
An example of the possibilities offered by this
method is presented in figures 18 and 19. The planform
of the two wings is trapezoidal, the aspect ratio 4,
the taper ratio is 1/3 and the swept angle back of the line situated at 25% of the chord is 15°30'. The local distribution chosen is constant along the chord
and optimal over the span for the immersion depth
d/s = 0.2. The calculations were made so that in each
section the thickness relative
to the
local chord should not be lower than 1.6% at 10% from the leadingedge. The choice of a lower CL, 0.3 instead of 0.5,
led, in the case of figure 19, to a higher lift-drag
ratio, 9.52 instead of 6.9.Figure 18 Figure 19
5.2.2. DESIGN OF A SUPERCAVITATING WING
WITH STRUT AND WALLS EFFECT
This special study had to be carried out in order
to allow a verification of analog and experimental results. The configuration of the testing channel is taken into account in the calculations by considering
the strut and walls effects. The latter are easily
represented by rheoelectric analogy, since the zero normal velocity on the walls is conveyed by a zero normal derivative of the potential. The introduction of the strut does not complicate so much the problem, devoid of lifting effect, its sections are obtained by
the introduction of an appropriate distribution of
potential on the projection of the strut and their cavity on the y = O plane. The method of solution issimilar
to that described above.
Figure 20 Figure 21
Figures 20 and 21 show clearly the influence of the strut on two wings of the same planform with the
same load distribution. In the first case, where the
length of the strut is equal to the central chord,
considerable thickening of the sections near it is
noti-ced. In the second case the width of the strut is
imposed to 70% of the central chord, the central
section is more thinner this permits to obtain a 25%
higher lift-drag ratio than that of the preceding.
5.2.3. HYDRODYNAMIC CHARACTERISTICS OF A FLAT WING WITH STRUT ANI) W A L L S EFFECT
We have already indicated the difficulties Involved
in the solution of the direct problem. In paragraph
4.3. a method applicable to the two dimensional case
was described; in the three dimensional case at = O
it seems possible, granting a plausible approximation,
to remove these difficulties. With this object, consider
the expression of the drag coefficient CD, CD=
4Vs2 J
where the integral is applied to the slit representing the wake of the wing with strut. Since the leading
edge of the wing is supposed sharpd the suction drag is here zero, this expression, applied to the wing only,
represents the resistance due to the pressure being exerted on the lower surface. We can thus assume
that the resistance of a section of the lower surface is
equal to the contribution of the preceding integral,
at points corresponding to this section in the Trefftz
plan, that is to say that:
fXt
x,oxdx
2This propertj, accurate in the tw dimensional case,
is only approximative i: the three dimensional case;
in adopting it
in the latter case we are at least
assured that the balance of the total resistance will
always be respected. For a fiat wing placed at
inci-dence a the above expression becomes even more
d
simple since = a
dx
C
') =
---a +
The solution of the problem is then more simple. The overall boundary conditions of the function 4> are
imposed as indicated in figure 22. Owing to tle roll
play by transformer, the zero flow condition is auto-niatically fulfilled; the potentiometer, P1 , allows the
Joukowski condition on the trailing edge, and the
potentiometer P2 , serves to regulate by successive
approximations the condition of equality between the
potential difference C (y)
-
4> (y) and thevalue of the resistance calculated at the same section in far downstream. 0,1 A .4 1.5' ci.. oa cj c,.11.5 d/. .0,4 L 'o IIIIIFfITTI or,
J
Figure 23Figure 23 shows the snape of the cavity thus
obtained for a flat wing of a trapezoidal planform, with a strut of the same width as the central chord, for incidence a = 50 The calculated lift coefficient is CL 0.12 and the lift-drag ratio LID = 11.5, for a
VOL. I - N.° 3 - JULY/SEPTEMBER 1968 Upp. r
-Sorfoc,
reduced immersion d/L = 0.4. The high value of the
lift-drag ratio with respect to the foils designed
accor-ding to given pressure distribution, is not surprising
since
for the flat wing the CL corresponding to
a = 5° is very low.
5.3. MARINE SCREW PROPELLER
The usual aerodynamic theories of screw propellers,
are particularly effective in the case of air screw.
They do not solve satisfactorily the problems set by
the marine propeller. These methods do not permit
the analysis of two important factors, one of a geometric
nature, the low aspect ratio of the blades, the other
of a hydrodynamic nature, the cavitation phenomena.
Indeed the first factor destroyec the fundamental
simplification of the classic theory where, the blade section is substituted conveniently by an equivalent
lifting line. The second factor requires a precise knowledge of pressure distributions on the blades, the only means of foreseeing or avoiding cavitation, which
a too general theory is unable to provide. Only the
theory of the lifting surface applied to the propeller is able to find an answer to these two questions.
5.3.1. THEORY OF THE LIFrING SIJRFACE OF TRE SCREW PROPELLER
The transposition to this problem of the linearised lifting surface theory, well known in the case of wings, is made without difficulty.
Consider that the propeller's blades, assumed mf
j-nitely thin, induced only small perturbation in the
relative f lo resulting from the uniform velocity V in the negative direction of z-axis and the angular velo-city V0 around this axis, and lie on a helicoidal flow
surface of the non disturbed flow, figure 24. For a
p-blades propeller with maximum radius R, the
per-turbation velocities field is periodical in space and the
study is thus confined to a region between two
heli-coidal surfaces deduced from one to another by a rotation of a 2vIp angle. The flow field is defined
by the following boundary conditions of the
pertur-bation velocities potential 4,'.
5.3.1.1. On the blade surface, the zero normal
velocity condition written, for radius r
In
_2+E2 i±(,.)
where, 4> = 4>' /(')R ,
= V0/R,
E - nR, i
is respectively the slope of the upper or lower surface
of the blade at given reduced radius, E , and curvilinear
abscissa along a chord, 'r , i the normal directed
towards the flukL
The tangential velocities are connected to the
pressure by
I4>+
I4I':
-where, C(E,'r)
- 1
is the local pressure coefficient.
Cp (E,'r) (12) 2+ i2 p_ -p 257 Oler 0,2
This expression can be integrated with respect to
'r, which brings us to a condition similar to that of
paragraph 2.4.
As in paragraph 2, there are two problems:
Direct problem: i (,'r) is given, which amounts
to giving the form of the blade, or
Inverse problem:
= C1 () g (-r) is
given.
5.3. 1.2. On the trailing edge the Joukowski
con-dition is conveyed by
conveyed, according to the linearised Bernouilli
equa-5.3. 1.3. The pressure continuity in the wake is tion, by a potential difference
4 ()
=-
, whichdepends only on E , between the two sides of the
helicoidal free vortex sheet.
5.3. 1.4. At far downstream, the potential presents,
as in the lifting line theory, the helicoidal symmetry
of p-order; the blowing of the propeller implies the existence of induced velocities in the axial
and
tan-gential directions; if, then, the reproduction of the field is limited at infinity by a surface perpendicular to the axis, conditions on the normal derivative to thissurface must be respected.
Acording to conditions (11) and (12) it is evident
that the function4 is defined in the present case by
conditions resembled to those exposed ìn paragraph
2, except the factor 412 2 which is taken into
account here. It is simple to see that the equilibrium
condition of a cavity with constant pressure p is given by UI*IIlltIflhtipi 1 UItI1tftUiiì,,, I1Th14oe*i Itw:'i -. -"
where. u = I) -P/! ,v
is the usual cavitation number. Having this, there is ho difficulty in expressing boundary conditions cor-responding to a supercavitating blade.
5.3.2. ANALOG SOLUTION
The rheoelectric solution of this problem needs
build a special tank . The electrolyte fills the volume
contended between two helicoids: the tank is tims
made up of two helicoidal surfaces, figure 25, covered
with electrodes. the radial angle between them being
217F; one represents the lower surface of the blade
as well as the lower surface of the cavity and/or the
lower side of the free vortex sheet, the other the upper
surface of the blade and/or the upper surface of the cavity as well as the upper side of the take of the
adjacent blade. The two helicoidal surfaces are pro-longated, following a radial direction, to a sufficiently large radius such that the perturbations are negligible,
small electrodes placed on these surfaces and
symme-trically short-circuited assure the potential continuity. At upstream of the blade the symmetry can however be assured more simply by materializing two surfaces
a period apart passing through the axis. A central
core, and a flat sector perpendicular to the axis, close
the tank.
In order to give an idea of the importance of this realization we will just say that it is made up of 160 small helicoidal components moulded in resin, each one contains 20 electrodes. Certain of these
compo-nents are removable in order to represent better the geometry of the blades and the cavities. A total of
3600 electrodes have to be suitably fed for each
cal-culation. For this purpose there is an electric set-up
which consists of about 250 transformers, 200
poten-tiometers or voltage dividers and interconnecting units allowing information to be collected on about 250 points
of the lower and upper surfaces of the blade.
The geometry of the helicoid is characterized by
the speed ratio
/
which here is equal to 6 .6 cm/rad (or 4.8 cm per revolution).5.3.3. SUBCAVITATING PROPELLERS
The rheoelectric installation described is especially
interesting for the solution of the inverse problem
because of the possibility of regulating and controlling
precisely the pressure distribution on each section of
the blade. To illustrate this aspect we describe the
different stages of a complete propeller design which
permits a useful experimental verification in free
water and in a cavitation tunnel.
The characteristics
of the proppeller were the
follcwing:
Advance velocity V0 = 7.25 rn/sec
Number of revolutions
n= 3.75 f/sec
(t = 25.36 rad/secj Blade radius R = 1.20 m Advance coefficient - V0 = 0.256; Vo = 0.805 nD 258 TECNOLOGIA NAVAL Figure 24 Figure 25Thrust Thrust coefficients Figure 26 T 7200 Kgr CT= 2T = 0.119 R KT= T = 0.147 pu D'
Developed area ratio 0.40
Number of blades 3
The conditions imposed
were of two kinds: a)
hydrodynamic - pressure distribution on the blades as regular as possible and higher than the value of thesaturated vapour pressure, optimal span circulation
distribution; b) mechanical - span thickness distribu-tion ensuring everywhere a sufficient mechanical
re-sistance. These conditions can be ensured
indepen-tly sirlce it is possible, as for the case of wings, to divide the problem into two parts, one referring to the
determination of the lifting effect of an infinitely thin surface; the other to the calculation of the thickness effect entirely free of lifting.
For the first part, the load distribution, constant on 8/10 of the chord and decreasing to zero at the leading and trailing edges,
and the optimal span
circulation distribution have been chosen; the shape of each section and the velocities distributions on the lower and upper surfaces are determined. If the de-pression created by this velocities is important it is
possible to change the load distribution until that
acceptable levels are reached. Concerning the
thick-VOL. I - N.° 3 - JULY/SEPTEMBER 968
ness effect, the calculation process is similar: the form
of the lower and upper surfaces corresponding to a
non lifting foil in a helicoidal flow and the velocities
distributions connected to them, are determined. Once these two operations are completed, the blade sections
are deduced by composition of the counter and
thick-ness effects. Figure 26 shows the form of different
blade sections for the values of the parameters given
above. Table I gives the expected values of the drag and torque coefficient and the values obtained
in a
test in free water carried out in the Paris Bassin des Carènes.TABLE 1
It may be seen that the agreement between these
values is satisfactory except for the propeller efficiency
which is lower than estimated. The difference seems to be due to an underestimation of the friction
resis-tance in consideration of the Reynolds number imposed by the test conditions. The experiments in a cavitation
tunnel shown, as was expected, that for the design value of A the propeller work without cavitation on the upper surface of the blades, except very near the
tip blade (0.95<E<1) where the end vortex is attached. These satisfactory observations seem to prove that two important objectives have been attained, pressure
control and adaptation conditions,
and that analog
calculation is likely to bring an effective solution to the theory of the lifting surface of the propeller.5.3.4. SIJPERCAVITATING PROPELLERS
It seems superfluous to state the boundary
condi-tions and the analog equipment necessary for this
study. Taking into account expression (13) we can refer to paragraph 4.4 where the case of
supercavita-ting cascades was treated; the boundary conditions
row being imposed in the intersection of the straight
cylinders = Ct, with the helicoids.
Notice, however, the advantages of the method of the lifting surface applied to supercavitatingpropellers,
compared with approximate calculation methods at present in use. Firstly, the blade and cavity countours
are correctly represented, which allows, for a given
blade form, the study of the influence of the cavity
form on the performance of the propeller. Secondly,
the cascade phenomenon and that of the interaction of the cavities is
taken into account durang the
calculation without being obliged to introducecorrec-tive terms.
Various propellers have been designed by this
method. This first propeller calculated was tested in the cavitation tunnel of the Paris Bassin des Carènes.
The results obtained did not confirm the theoretical estimates. This discordance does not seem to be due to a fault in the theory, verified in the subcavitating 259
KT KQ .1)
Expected values 0.147 0.0258 0.730
Table 2
for an advance coefficient A = = 0.261 and a
cavitation number = 2(P0-P) /pV
= 0.4, and
various blade and cavity forms. 5.3.5. DUCTED PROPELLERS
The advantages of ducted propellers compared to
ordinary propellers, for certain speeds coefficients, have been known for a very long time. However, very
little study has been devoted to the improvement of
the working conditions of the nozzle itself and to the
increasing the efficiency of the propellernozzle system.
The analog method (47) (11) offers possibilities of
calculation for this type of device and allows a consi-derable improvement of their efficiency and the f
ulfil-ment of the characteristics required of them.
In order to conduct the study we are evidently
obliged to admit certain hypotheses; we assume that:
the downstream flow is made axial by a straighteners,
the propeller is approximated by an actuator disc
(infinite number of blades hypothesis)
with, in its
passage a discontinuity of constant pressure (constant circulation hypothesis) . Consequently the flow is axi-symmetric and its study can be limited to a demi-plan meridian. This first simplification, although necessary,
is not suficcient since, in order to represent the flow correctly, we must necessarily know the velocities
dis-N' 2 HELICE 'o 2
q 02 00 o.' O
Figure 27
continuity surface which escapes from the trailing
edge of the duct,
free boundary with equilibriumcondition which imposes equality between the pressure jump and the difference of the square of the velocities
on each side of the jet stream. The difficulty in repre-senting such a condition requires the use of a
linea-rised schema where the boundary conditions are
imposed on a straight semi-indefined cylinder of
ge-nerators parallel to the unperturbed velocity,'image
of the duct and of the discontinuity surface.
The flow can then be defined by means of the
perturbation velocities potential, harmonic revolution
function, easily represented by the electric potential
of a tank with a inclined bottom.
3 Because of the many conditions which must be satisfied in order to improve the hydrodynamic func-tioning and the efficiency of the nozzle, it had seemed
useful to take as a starting point a given duct form, a form which was redefined during the calculation on
the basis of the results obtained; the process was
indeed greatly facilitated by the consideration of
seve-ral elementary potentials which revealed the
interac-tions of the propeller and of the hub on the duct. This method permits to show the sole played in the increasing of the efficiency by two effects: on the
one hand, the downstream divergence of the mean line of the duct, connected to the increase
of the
velocity in the plane of the actuator disc, and which facilitates and improves the functioning of the latter; on the other hand, the "adaptation" condition, imposed
during the design, on the nose of the duct, reduces
the risks of flow separation on the inner surface of
the nozzle and, consequently, encourages its efficiency,
as well as that of the propeller. We must point out nevertheless that the widening effect of the nozzle can be obtained by the blowing effect on the trailing
edge, an effect also studied by rheoelectric analogy by means of analog hypotheses (11).
This method has been used to carry out the calcula-tion of a combined propeller-rectifier-nozzle with the following characteristics:
CT 0.0855 0. 0868 0. 0579
CQ 0.0357 0.0338 0. 0225
TI 62.3 % 67.1 % 67.1 %
Thrust T = 19000 Kgr
Diameter of the propeller D = 2.58 m
Length of the nozzle L = 1.87 m
260
TECNOLOGIA NAVAL case, but to an unrealistic choice of speed coefficient.
Three propellers have recently been calculated and one of them should be tested very soon in the DavidTaylor Model Basin. Figure 27 shows one of these propellers,
designed for an optimal span circulation distribution
and pressure distribution on each section of the blade such that, at the leading edge a very localised infinite
pressure encourages the starting of cavitation (beha-viour of flat plate foil), and the most heavily loaded
part of the foil is near the trailing edge (high lift-drag ratio criteria in the two dimensional case). The cha-racteristics of these three propellers are summed up in Table 2, not taking into account the friction
resis-tance.
o o., 02
Advance coefficient Thrust coefficient A 0 = 0.696 n.D
KT=
T =0.258 p n2 D4of which 0 056 corresponds to the thrust provided by the nozzle and 0.202 to that of the propeller itself.
A model of the ducted propeller given in figure
28 has been tested in the Paris Bassin des Carènes.
The results obtained were very encouraging since the
thrust of the nozzle corresponds well to that expected,
as does the efficiency of the overall propeller, 66% experimental instead of 70% theoretical. On the other
hand the total thrust was only attained to within
16%. In any case, with regard to the so-called nozzle itself, the study showed the advantage of the method
of calculation used: if improvements should be sought,
they concn rather with the calculation of the fan
and of the straighteners. In supporting this argument
it may be noted further that the comparison of the
efficiency of this nozzle with nozzle n.° 9 of
Wagemin-gen fitted with a propeller K 4.55, and which was
considered as giving the best performances, is in favour
to the former. With a practically equal diameter, the
gain of efficiency is of around 5 to 13% higher accor-ding to the power.
6. - STUDY OF FLOW AROUND THICK BODIES
In most studies already described the bodies are
supposed very thin.
In this case the linearisation
hypothesis are valid. Nevertheless, *hen the relative thickness of the bodies is important it is not possible
to simplify the boundary conditions over their surface. Thus, if we consider the perturbation velocities poten-tial 4) = x, harmonic fonction in x, y, z, it is
convenient to write, for the whole surface limiting
the body, the tangential velocity condition by
14)
= f (X1, y1 , Z1)
In
where f(x1, y, z1) is a known function of points
M(X1, y1, z over the surface . This function depend on the local slope of the body and its motion.
The body can be slighty immersed beneath, or traverse, the free surface. In general it is possible to
simplify the equilibrium condition of this surface by supposing that the perturbation induced by the body
is not very important. The linearised boundary
con-dition on the free surface is
still true and can be
writen in the same form that in paragraph 2,
+ K0 =0
IX2'
far upstream we have the condition.
hm
grad4) = o
X.
This problem can then be solved according to the method that has been described in § 4.1.1 and 4.2.2.
The 4) fonction, solution of the given boundary value
problem, can be considered as the sum of elementary
VOL I N.° 3 JULY/SEPTEMBER 1968
potentials induced in the field by a convenient source
distribution. We can then write
4) = f
q(x, y1, Z1) (+
dV,
where q is the singularity density of the simple layer potential at points (x1, y. z1) of the surface , dc is
an elemental surface around this point; l2 and ilr
are the singular and regular parts of the potential
of an immersed source of unit strenght. In the
two-dimensional case, these two parts are given by the
following classical expressions: Re
Rei-
[vÇ
dKie°1
where = x+iz,
= x +iz
and K0
--The function is also divided in two parts, and
4
s+4)r
correspond to the singular part of 1s that is
I
J V
Its value is the same on both sides of , while its normal derivative are discontinues. The difference
between its normal derivatives, i
- j
,repre-/I
Ö\
\In
önJsents the source flow q. In the present case the
super-scripts + and - correspond to the external and
internal domains defined by .
The potential fonction must then satisfy the
following boundary conditions:
a)on
z=0 ,--=0
In (14)b) for xl -* = o (15)
C) on V
(o4:
.)==
q(xj,yj,zj) (16)with the external normal derivative given by
C -
-*---=
f(x,y1,z) _Jradiìr.nd
(17)6.1. COMPUTATION PROCEDURE
The analog simulation of the above boundary con-dition is very simple, nevertheless two different rheoe-lectric tanks are necessary, one to represent the actual
flow fild outside the body, the other to represent the
field inside the body.
r.,
S.ction do ¡a Iuyr.
¡ ,. 7 r.2ij 16 r. J r., 4,5 45 1 o R.diql# G i. g...r,Ir,c.
F'
'.5 r.. o 7- 5 3- 2 s Figure 2 r .31 0G. 30.s 5G. iOU Casque $ochar, du r.dr.ss.urNeGa Peur ¡os 30db as du r.dr.ss.ur soul Iangl.
ditfl-. 'ICa,Sfla pr.,,tG A 2 2 5 6 's.', '3.54 sr S %J 44., 45% 5,2771 a' N/Zn I. Intl.. 0.!Z7 1.5750 2.530 Bras du r.dr.s.sei,r Sur Io tuyère 300 3 513.3 4 .3075 5.50p7 O 5143 7 524.4 5.53,,
'
' 115 i. ,,...E
-ISO 0t
',
504 5 GG.Gr.IOis. JOtS-
UI
I,, I=
..-t __
L-=0;
-__,_7
TTu..;
---
i!uiiu!iÌì!I!
u..
--ZOU O.n.r.fis. Il, E 308.5
I"
i.% 5% ll4 SecSions lSèIgc. 47.3, Ins I tI 4. ..rbs .4 ?000. 55%The computation procedure is the following:
- for the initial iteration it is suppose that
= b n
that is to say that the regular part
4 of is neglected. It correspond to the solution of
the external field problem for a zero Froude number.
The potential distribution c$ (x ,y1 ,z1) on each point of is then obtained.
2.° - these values are then introduced on to the corresponding points of the internal domain. The
measure of the normal derivatives
--- gives a first
plausible distribution of (b+
q(x,y1,z)= (---
-\bfl
bnWith these values it Is possible to compute numerically, for a given Froude number, the normal derivatives over
due to the regular part, i.e. the normal derivatives
induced by the free surface, grad . . qd
£ grad 1r
. n- qda.30 -
Introducing this integral into equation (17)gives a new corrected distribution of q- which are
ön
then imposed on the surface of the external flow
field. Hence, a new distribution of the c potential
is obtained and permits to continue the procedure by step number 2.
This iteration procedure is repeated until the
con-vergent values of are obtained.
6.2. APPLICATION OF THE METHOD
To test the validity of this method the above
computation procedure was applied to the case of an immersed circular cylinder beneath the free surface.
The results obtained was in good agreement with
those computed from the analytical solution by Have-lock (46), figure 29.
Presently the works carried out at the C.C.A.
concerning the study of three-dimensional flow fields with free surface. They permit to obtain the pressure
distribution over a
thick hull and the wave drag
attached to it over a wide range of Froude numbers. The proposed hull is represented in the rheoelectric tank by 240 electrodes, that is to say that the velocity tangential condition is satisfy on 240 control points over its surface. To solve numericall this problem it
was necessary to solve a 240 x 240 matrix at each
iteration.
We hope the aforementioned studies may permit
to give a valuable contribution tothe problem of thick hulls and wave resistance, one of the most importants
of naval hydrodynamics.
VOL I - N.° 3 - JULY/SEPTEMBER 1968
Figure 9
7. - CONCLUSION
The purpose of this paper was to give a glance of the possibilities of the rheoelectric analogies in the field of theoretical naval hydrodynamics probleir :..
The examples depicted were chosen toillustrate these
possibilities and can be summed up in the following way: sub and supercavitant hydrofoil problems with
or without free surface effect; supercavitating cascade design ; hydrodynamics characteristics and optimum
design of finite span wings with or without free surface,
strut and wails effects; design of sub and
supercavi-tant marine-screw propellers; and finally, a tentative method to solve the problem of thick hulls.
The rheoelectric analogy is a very suitable method of study for these hydrodynamic problems, because most
of them can be considered as potential flows defined
by the Laplace equation. The rheoelectric tank is,
indeed, a praticai and effective method of simulation of this harmonic functions. The knowledge of their boundary conditions is sufficient to realise the
simu-lation. Therefore it is not necessary afterwards to
look on an explicit
analytical formulation of theproblem.
Some of the given examjles shown that to obtain the best results of the rheoelectric method it is
con-venient, very often, to modify the theoretical statement
of the problem in view to simplify the electrical
set-ups of the boundary conditions and get a good fiability. This approach can suggest a new way to build
theore-tical models applied to hydrodynamic problems. This
models are elaborated in fact to simplify the analogy.
On the contrary these models would prohibitively
complicate the duty of the mathematicians who may
try to treat them by analitical or numerical methods. It is noticed that the rheoelectric methods utilised the possibilities of the numerical analysis and the use
of computers in view to facilitate the preparation of
the data and the exploitation of the results. They
are employed too in the establishment of new hybrid
analogy-digital methods. The solution of the thick
hull problem shows clearly this interconnexionbetween numerical and analog computation.
Many kinds of problems can be treated directly
numerically, but our experience shows that before a large and scrupulous programmation it is interesting
to check the validity of the theoretical
model byrheoelectric simulation. The physical nature
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