Air-water analogy and the study of hydraulic models


Material Information

Air-water analogy and the study of hydraulic models
Physical Description:
22 p. : ill. ; 27 cm.
Supino, Giulio
United States -- National Advisory Committee for Aeronautics
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aeronautics   ( lcsh )
Hydraulic models   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


The author first sets forth some observations about the theory of models. Then he establishes certain general criteria for the construction of dynamically similar models in water and in air, through reference to the perfect fluid equations and to the ones pertaining to viscous flow. It is, in addition, pointed out that there are more cases in which the analogy is possible than is commonly supposed.
Includes bibliographic references (p. 19).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Giulio Supino.
General Note:
"Report date July 1953."
General Note:
"Translation of "La similitudine aria-acqua e lo studio dei modelli idraulici." L̕ Energia Elettrica (Milano), vol. XXVIII, no. 11, Nov. 1951."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
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oclc - 24079845
sobekcm - AA00006178_00001
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Full Text

A tL'




By Giulio Supino


The author first sets forth some observations about the theory of
models. Then he establishes certain general criteria for the construc-
tion of dynamically similar models in water and in air, through refer-
ence to the perfect fluid equations and to the ones pertaining to viscous
flow. It is, in addition, pointed out that there are more cases in which
the analogy is possible than is commonly supposed.


1. The use in model experiments of a fluid which is different from
that of the original one has been known for some time. One is reminded
in this connection that tests are made in water of bodies which are
designed to operate in air (dirigibles, airplanes, railway cars, etc.),
and the first investigations in this direction were carried out by G. A.
Crocco in 19062; and it is recalled that tests are made in air (that is,
in wind tunnels) on configurations which are designed to operate in

*"La Similitudine Aria-Acqua e lo Studio dei Modelli Idraulici."
L'Energia Elettrica (Milano), vol. XXVIII, no. 11, Nov. 1951.
LTheoretical background for experiments now going on at the
Hydraulics Laboratory of the University of Bologna, sponsored by the
ANIDEL (National Association of Electric Power Companies).
2In particular, one should consult:
Crocco: Dynamics of Dirigible Balloons (Dinamica degli Aerostati
Dirigibili), Bulletin of the Italian Aeronautics Society
(Bollettino della Societa Aeronautica Italiana),
April 1907.
B. Finzi Contini: Concerning an Experimental Method for the Study
of Car Models (Su di un Metodo Sperimentale per lo Studio
dei Modelli di Veicoli), The Engineer (L'Ingegnere),
G. Corbellini: The engineering of rail transportation (Tecnica
dei Trasporti su Rotaia). Rome, 1941, pp. 210-232.

NACA TM 1359

water, or are otherwise subjected to hydraulic loads3. Within this frame
of reference the Hydraulics Laboratory of the University of Bologna has
in progress the setting up of the necessary mechanical devices for the
carrying out of certain research work, and this paper, in the form of a
preview, is intended to present some facets of the possibilities inherent
in experimentation that can be conducted with these two fluids.

2. Some considerations of a general nature are to be discussed
first of all. The theory of mechanical similitude can be developed from
two different points of view. The first of these ways of looking at the
question is of an inductive sort, and it is based on the n-theorem (of
Vaschy-Riabuscinski-Buckingham), while the second viewpoint is deductive
in nature and stems from the differential equations which govern the

Since the import of the two methods of approach each differs from
the other, it is worth while to review them briefly with respect, how-
ever, merely to the hydraulic problem. The expression of the i-theorem
is then as follows:

"Let us assume that the following physical quantities are involved
in the hydraulic phenomena under consideration:

p density of the fluid

I linear dimension which characterizes the body whose behavior
is to be scrutinized

U average velocity of the fluid

f force (drag of the moving fluid on a given length of tube,
thrust of a jet against a restraining wall or of a stream
against an immersed body, drag of the fluid resisting the
motion of an immersed or floating body)

3Tests made in air, in a pressure tunnel, to study liquid flows may
be found in:
G. Remenieras: Predetermination de la perte de charge dans une
canalisation d'eau sous pression 'a partir de celle
mesur4e sur le mgme canalisation parcourue par de l'air.
Third Congress of the AIRH, Grenoble 1949, vol. III,
no. 2.
P. Bourguignon: Predetermination de la perte de charge dans une
canalisation d'eau sous pression 'a partir de cell
mesuree sur le meme canalisation parcourue par de
l'air. Controle de la validity de le methode sur la
galerie d'amenee l'usine Hydro-electrique du Pont-
Escoffier. Ibid. III.9.

NACA TM 1359

g acceleration of gravity

p absolute viscosity of the fluid

rs nondimensional parameter or group of parameters which char-
acterize the roughness of the walls

rf nondimensional parameter or group of parameters which char-
acterize the shape of the body

The connection between these quantities is to be expressed by a
function of the form

p(p,l,U,f,g,y,rs,rf) = 0

Because this function is not to undergo any modification when the
dimensional units are altered, then the unknown quantities will have to
appear regrouped in such a way as to constitute pure numbers. Now all
the distinct pure numbers which it is possible to construct through use
of the quantities p, 1, U, f, g, 4 (of course rs and rf are
already pure numbers) are the following

N 2U2
N f

N2 -

3 -

(Froude's Number)

(Reynolds' Number)

Thus in place of equation (1) one may write

(Pl(NI,N2,N3,rf,rs) = 0

This relationship is often written in a more simple manner as a
result of the fact that it is possible to rearrange it so that one of

4 NACA TM 1359

the unknown quantities is given explicitly; for example, in the case of
the force f, we have that the relationship

f = pl2U2V(N2,N ,rf,rs) (2')


Suppose now that it is required that some hydraulic phenomenon is
to be studied by means of a model. In addition to the geometric condi-
tions of similitude

rs = rs'1
rf = rf'

which must be satisfied (where primes refer to the quantities related to
the model), it is also necessary to specify the further conditions of
mechanical similarity given by

p p'

U2 U'2
gI gl'


12U2 1'2U'2
P -P'
f f'

If the density of the fluid is p and its viscosity 4 in the
"basic" or original system, and if the model is to be tested in a fluid
for which the density and viscosity are to be p' and 4', respectively,
the preceding conditions (4) constitute a system of relationships in the

U f
-, U, and -
T' U' f'

NACA TM 1359

Let us denote by v = the kinematic viscosity of the basic fluid,
and likewise let v' denote the same quantity for the model system, then
by means of equation (4) the following are obtained

S(/v21/3= k2/3

U = V 1/3 = kl3 (5)

f (p v 2

- ( k2

where k has been set in place of -.

In the case of the other mechanic
tion ratios apply



volume flow


static head

1 parameters the following reduc-

= kl/3

a- 1
Q k5/3

= k2/3




angular velocity

m)( U) = ()k7/3

L p k8/3

= ( )k7/3

S= k-1/3
- kl

p' I '

NACA TM 1359

If air is selected as the fluid to be used in the study of a
hydraulic phenomenon, the value of the ratio between the kinematic
viscosities of water and air, v/v', cannot range out of too extensive
bounds. In fact, for ordinary temperatures the value of this ratio, as
a function of temperature in the two fluids, varies from a maximum of
0.13 at 0 C to a minimum of 0.05 at 300 C, as may be verified by refer-
ence to figure 1.*

On the other hand, if one uses, as a basis of comparison, the kine-
matic viscosity of water at 150, the ratio between the viscosities can
at most vary from the value

V of water at 150 C 1.14 X 10-6
= = 0.0838
v' of air at 0 C 1.36 x 10-5

for air at a temperature of 00 C to the value

v of water at 150 C 1.14 10-6
= = 0.0695
v' of air at 300 C 1.64 x 10-5

for air at a temperature of 300 C.

By use of equation (5), it is seen that the corresponding values of
the ratio 1Z/' are, respectively,0.187 and 0.169, to conform with these
just-mentioned values of the ratio v/v'. In the first case, that is,
the scale of the model will have to be 5.35 times larger than that of
the original, while in the second case it will have to be 5.90 times

It is clear that if the average temperature of the flow in which
the original body is to operate as a water model is different from this
150, other values for the 1Z/' ratio will be obtained for the range of
air temperatures under which the experimentation is done; however, these
values will not deviate greatly from the limiting values just adduced

One may thus conclude that it will be useful to employ dimensional
experiments with models in air to large scale whenever the original

*The NACA reviewer suggests that the exponents of 10 in the values
of kinematic viscosity for water and air should be 5 and 4, respectively;
however, the ratios of these values are unaffected.

NACA TM 1359

setup in water is of such small size that it would not permit an accu-
rate examination of its behavior.

On this account dimensional experiments can be employed for the
study of flow from vents, from small valves, etc. where the influence
of gravity and viscosity of the fluid can be taken iito account at the
same time.

This mode of approach to the subject of mechanical similitude has,
as mentioned earlier, an inductive aspect, that is, it is substantiated
only provided that the initial hypothesis is true that the hydraulic
phenomenon under examination depends only on the physical quantities
enumerated in equation (1).

When all these quantities (and only these) come into play in more
diverse theories, then I have already demonstrated earlier that the
analogy between model and original is common to all the theories thus
constructed, and thus the model test and the transfer from model to
original lead to a result which, in a certain sense, does not depend
upon these theories On the other hand equations (3) and (4) are not
amenable to any further polishing up; i.e., once the dependence of the
function pl on N1, N2 and N3, rf and rs is established, no
further importance can be attached to one or the other of these numbers,
nor can any criterion be set up for preferring one over the other. The
elimination of the quantities which may be considered to be superfluous
has to come about through the initial supposition that the phenomenon
does not depend on them. If one or more quantities are cut out then
also one or more of the pure numbers among the group found previously
will be lacking. For example, if one is concerned with a perfect fluid
(for which MI = 0), then the Reynolds number would be missing.

If the liquid is to be uninfluenced by the force of gravity, then
the Froude number would be missing and if one should like to forget about
the density, p, then the numbers N1 and N3 would be lacking. In

this case, the number N1, defined as N1 = would be inserted in
place of N1 and N3, in addition to having N2 still involved.

3. It now remains to ask whether it is possible to extend the range
of usefulness of the air experiments by reducing the dimensions of the
model. The answer is in the affirmative, because it is not actually

G. Supino: On dimensional analysis and the theory of models
(Su l'analisi dimensional e la teoria dei modelli) Two notes in the
"Rendiconti della Accademia Nazionale dei Lincei", Sept. 1, 1947.

b NACA TM 1359

necessary to keep in the Froude number, and upon dropping the pure num-
ber N2, a much greater liberty of choice is vouchsafed. But in order
to study this, it is necessary to study the hydraulic analogy from the
deductive point of view, as applied to a particular one of the analogies.

Let us focus our attention, for example, on the equations describing
the behavior of an incompressible viscous liquid (that is, on the Navier

These Navier equations are constituted of:

(a) The equations of motion:

In addition to the symbols already defined in section 2, the fol-
lowing nomenclature is introduced at this point. The local components
of the forces acting on a unit mass are denoted as X, Y, Z which
are components directed along the corresponding coordinate axes,
respectively; p = pressure, P = viscosity of the liquid, p = density,
du dv dw
and d, d, and d are the "material" derivatives of the velocity
dt dt dt
components, that is to say, they are components of the acceleration of
a little particle of liquid which, at the time t, is located at the
point with coordinates x, y, and z. The symbol A stands for the

x 2



Then the equation of motion is

pX -- +

pY -
PY- +

pZ +

I AU = p --

P Av = p -

l Aw = p -

(b) The equation of continuity

u + (- + =0
6x ov oz

NACA TM 1359

(c) The boundary conditions

In reference to these, there are no uncertainties introduced here
in the case of flow along walls, because it is obvious that the liquid
sticks to them, and thus its velocity is equal to that of the wall, and
also it is seen that the protuberances are reduced in the same proportion
as required by the geometric proportionality factor. The question of
"free surfaces" will, on the other hand, be considered later.

It may be deduced directly from equations (6), that is to say, with-
out reliance upon the t-theorem) what the conditions are for similitude
to hold in.the gravitational field.

Let us consider, therefore, the third of equations (6), applying to
the original setup. Let the z-axis be directed upwards, whence Z = -g;
then we have

1 ap [ dw
-g + Aw -
P )z P dt

Let p', p', z', etc. be the magnitudes related to the model conditions;
then in this case we have

1 op' ', dw'
-g + Aw' -
p' I z' p' dt'

But it is true that

p' = Op, p' = bp, v' -

z' = Xz, w' = XT w, Aw' =X Aw

so that upon comparison of the equation relating to the original condi-
tions to that relative to the model, it follows that

6X 1
kT = 1
I XT-2

because g remains unchanged in the two cases.

NACA TM 1359

The condition that T-2 = 1 is equivalent to the one denoted by


The condition that kkT = 1 when combined with the preceding condi-
tion leads one to write

kkT(XT-2) = k2T-1 = 1

and this corresponds to the Reynolds number condition.

Finally, the condition 1 when combined with the condition

X-2 = 1 and by making use of the relationship that i = fX-2 leads
one to write

S T -2) 42 = 1
f f

that is to say, we have p const.

It must be remembered that the law of geometric similtiude must be
upheld, and thus equations (3) are valid. Consequently we are brought
back to equations (4). It is evident that equation (7) and the condi-
tions at the wall do not impose any other restrictions.

4. The preceding result does not inform us of anything that could
not also be derived from the t-theorem. But considering that this time
we have within our grasp the very equations with respect to which we want
to deduce the law of similitude, then it is thus possible to intimate
that a deeper study of these equations leads to a reduction, in certain
cases, of the number of imposed conditions. A very simple trick way of
doing this is as follows, which is a well-known device, at least theo-
retically (compare, for example, p. 34 of Pistolesi's Aerodinamica).

Let us return to examination of equation (6). Let the pressure p
be divided up into two portions so that p = pl + P2. After doing this,
we may write the equations of motion which apply in a gravity affected
field as

NACA TM 1359

(pl + P2) du
-+ +ji Au = p -
xbc dt

a(pl P2) dv
+ ') Av = p -

(P + P2) dw
-7- + n Aw = p
dz o-t

If we now substitute for pl the value
with nothing more than the equations

p, = -Tz, then we shall wind up

P2 du
--+ Au = p -
;x dt

P2 dv
+ av = p -
By dt

;2 dw
+ Aw = p -
dz dt

and the law of similitude can be deduced by
ships that it must hold true that

stipulating in these relation-

kXT 1

but without the qualification that these three ratios must necessarily
be equal to unity.

NACA TM 1359

The solution that comes most readily to mind consists therefore in
selecting the Froude number in a purely arbitrary manner and in demanding
that the following two relationships hold

(X-2) = 1 and kk2T-1 1

that is to say, that the N1 and N3 parameters should remain constant;
so that one may write

q1(N,N3,rsrf) = 0

If the function cp1 can be cast into explicit form (let us suppose
it is the drag force which we want to work with) then we get

f = pl2U2*(N3rf,rs)

which is just a way of saying that the drag force, f, turns out to be
a function solely of the Reynolds number and of the shape parameters rs
and rf

Nevertheless the experimental facts will have to be handled in the
following way. First carry out the tests of a hydraulic phenomenon,
directly based on the Navier equations, by means of a model whose dimen-
sions are to be obtained from those of the original configuration through

5It should be remarked that this trick also is of interest from a
conceptual standpoint in that it serves to sever the drag measured in
tubes from the Froude number. In fact, let us write down the expression
for the change in the drag in progressing from one point in a tube to
another as f = 1yi3; then it follows that

u2 (N3,rs) u2
i H (N3r,rfrs) D 2ug

where the symbol X(N3,rs) denotes the nondimensional function which
appears conventionally in all the formulae relating to drag in cylindrical

NACA TM 1359

application of the premise that the Reynolds number is to remain con-
stant. After doing this experimentation, the next step will be to sub-
tract the value pl' = -7'z' from the total gross pressure measured at
each point of the model. It is worth pointing out that the value of pl'
is known at every point because the shape of the model is defined.

Thus one will have at hand the quantities p2' = p' p1', and from
these values the law of similarity will permit the determination of p2 6
The pressure existing at the corresponding point on the original body will
be given by the relation: p = p2 yz.

In all those cases where the value of the pressure is of no conse-
quence to the experiment, no corrections of any sort are called for.

It may be further remarked that if the viscosity has but a minute
influence on the hydraulic phenomenon under study, then the equations
which govern the flow are those of a perfect fluid, and upon subjecting
these equations to the same trick, one will arrive at the conclusion that
the law of similarity is completely at our arbitrary disposal (provided
that the pressures are computed by means of the correction factors pre-
sented above, after the similarity ratios have been selected and fixed).

5. Even when one takes into account the viscosity and stipulates
therefore the constancy of the Reynolds number, the artifice being
exploited here renders it possible for the dimensions of the model to
be reduced from those applying to the original case. Set k = -

S= -- (the symbol X used here is the reciprocal of the one employed
in the previous section) and let a = -

When this is done it, of necessity, will follow that _- = 1. Now
if k lies in the range 0.07 5 k 0.08 and if X is taken as 10,
then it will be possible to obtain the desired result, namely, a reduc-
tion in the size of the model to 1/10 of the original, by stipulating
that a should equal -- to 8; that is, by demanding that the model
1000 1000
be tested at velocities of from 140 to 125 times those experienced by
the original configuration.

From the ratio p = = 8X2 -2 it follows that p2
p 82 -2

NACA TM 1359

This condition is certainly quite severe, but it is much less so
than the one which might have been imagined at first thought. Suffice
it to say, in this regard, that velocities in excess of 100 m/see can
be reached and surpassed in aerodynamic wind tunnels7.

On the other hand, when the Reynolds number has a large value it
is not necessary to insist that the two Reynolds numbers applying to
the original and to the model be definitely equal. It is sufficient to
require that the model Reynolds number be merely large enough to render
the drag independent of this N3 parameter itself.

In contrast, the creation of a water analogy to the scale of 1 to 10
obliges one to seek (in order to obtain the same Reynolds number) a
velocity which is ten times greater than in the original. But the diffi-
culty inherent in reaching a fluid flow velocity of 7 to 8 m/see in a
laboratory model is much worse than that of trying to produce a velocity
of 100 m/sec in a wind tunnel. Besides, the experiments on models oper-
ating in an air stream can be performed with very limited apparatus
(inasmuch as such apparatus will have to withstand relatively weak forces)
and thus, in addition, the dimensions can be enlarged. Such advantages
will make the carrying out of the experiments a yet more simple task.

Finally, when one concentrates his attention solely on the inertial
drag forces, the dimensional analogy may be carried out for the case of
a perfect fluid field. In this case, regardless of whether one does or
does not take into account the Froude number, one can take advantage of
sizably larger changes in velocity by employing air-test models.

6. A single difficulty continues to bedevil the experimenter, and
this blemish is not easily eliminated. This troublesome point arises in
the case when any free surface or surfaces are present. The substantial
difference which exists between the free surfaces present in the two
cases can be explained, from a physical point of view, as follows: The
jet discharge of the air stream occurs in air and the discharge of the
water jet takes place likewise in air in the usual case, so that the
density p is continuous across the free surface (while only the velocity
is discontinuous there) for the former jet, but in the latter setup both
the velocity as well as the density are discontinuous.

7Nevertheless it is important in the research we are doing to refrain
from using velocities greater than about 80 to 90 m/sec in order to keep
from surpassing by any appreciable amount the 25 percent-of-sound-velocity
borderline. With regard to this subject the reader's attention is directed
to a recent paper by J. W. Ball, entitled: Model Test Using Low-Velocity
Air, Proceedings of the American Society of Civil Engineers, June 1951,
Vol. 77, No. 76.

NACA TM 1359

Pursuing this line of thought further it is seen that when the
water jet discharge takes place in water (as for example, for flow
across a completely submerged orifice) then there is no difference
between the two cases, and the results derived as applying in air hold
in addition for the case of the water flow, and vice versa (N3 being

equal in the two cases).8

But in general, when talking about an air jet immersed in air, the
free surface constitutes an interface which separates the flow into
two distinct regions A and B; in the former the fluid is in motion,
while in the latter it is at rest.

If we now restrict ourselves here to consideration of perfect fluids
in irrotational steady flow, it follows that the actual pressure head is
constant throughout the whole fluid, or

P V2
z + + = const.
7 2g

(with the flow vector being assumed directed along the z-axis) and this
relationship applies validly right on the interface which separates the
two regions.

Now since, p, y, and z are continuous across the surface while
VB is zero, then it follows that

VA = a const.

In other words, the velocity is constant on a free surface in steady
flow (it being taken for granted that the free surface is interpreted
as the interface between the air to air regions), and thus it follows
in addition that the pressure head is also constant.

If one confines himself to examination of two-dimensional motion,
then V is constant on the interface between the two regions, and in
addition the stream function 4 is also constant9.

8See, for example, the results of Zimm: Ueber die Stromungs Vorgange
im freien Luftstrahl. Forschungsarbeiten VDI, Heft 234, 1921, and those
results obtained by Citrini concerning the spreading of a liquid jet in
L'Energia Elettrica, 1946.

9The velocity V and the stream function 9 are constant in the
two-dimensional case even if the motion is rotational.

NACA TM 1359

It is quite different in the case of a free surface between air and
water. For this type of flow, the quantities p, g, and z are again
continuous across the interface, but y is discontinuous. For this
reason one may write that (for irrotational flow of a perfect fluid)

(AA2 BVB) + ZA 2B = 0

and as a result of dividing through by yA and remembering that VB = 0,
it follows that

-+ Z = z -
VA 2+
2g 7A

If one now drops out the term 7 inasmuch as the specific weight
of air is about 1/800 of that of water, then the simple result

+ z = 0


Thus, in this case, now p = const on the interface; so that,
whereas the pressure head is constant on the interface (or free surface)
between air and air, the pressure is constant on the interface between
water and air. The two cases become identical only when the free surface
is a horizontal planelO

7. In the case of steady two-dimensional flows one gets r = const.
in both situations. On this account, if one can set up an air-to-air
model for a steady irrotational two-dimensional flow wherein the interface
is in geometric analogy with the corresponding interface of the original
water-to-air setup, even if only along a very small section of such an
interface, and if the velocities are moreover in geometric similarity,
then the analogy is vouchsafed to hold throughout the flow field. In

10If the velocity VA is very large along the free surface, then
the influence of the change in height z is very small; in this case
also, therefore, the analogy between the air and water regimes of flow
is approximately admissible.

NACA TM 1359

fact, in this case the functions r and 0 exactly correspond along
the piece of the free surface. But once having the 4-function given
equal to the 0-function or having given the r-function and its normal
derivative, thence, by means of a well-known theorem of Kirchhoff, the
*-function is determined throughout the whole flow field11.

As it has been remarked, this correspondence has to be established,
however, on a free surface or on an element of surface immersed in the
fluid not on a fixed wall, where the condition of no-slip at the wall
contravenes the irrotationality of the flowl.

8. The conclusion from what has been set forth above is the fol-
lowing: The laws of hydraulic analogy can in general be established
starting out from an inductive or a deductive origin as basis for con-
sideration. In the former case one makes a list (hypothetically) of
all, and only, those factors which are involved in the phenomenon. The
similarity conditions are deduced then by means of the i-theorem, and
these conditions are validated through means of other well-established
theorems. In the second case (the deductive one) the physical factors
which are involved in the phenomenon appear in the equations which govern
the theory, and with these as a basis the law of similarity is formulated
(without it being necessary to bring up the i-theorem in any shape or
manner). These equations also serve to suggest some very useful tricks.

In the case of the mechanics of fluid flow these tricks have led us
to establishment of the fact that experimentation performed in air on
hydraulic phenomena has a quite wide field of application. Summarizing
the results of sections 4, 5, 6, and 7 we can declare that it is possible
to carry out experiments in air on hydraulic phenomena when one of the
following conditions is conformed to:

1. The Froude and Reynolds numbers analogy is maintained (in other
words the gravitational and viscous effects are taken into account),
when a model is made which has a larger scale than that of the original.

2. The Reynolds number analogy is maintained during a reduction in
scale (relegating to an insignificant role the influence of gravity or
taking it into account by means of the artifice mentioned in section 4).

3. An analogy based on arbitrary ratios when the liquid can be con-
sidered to be a perfect fluid (taking into account the effect of gravity
by means of the artifice mentioned in section 4).

11See Kirchhoff: Mechanik, pp. 187-188; and Almansi: Rendiconti
della Ace. Naz.dei Lincei, Sept. 1, 1907, pp. 865-867.
12G. Supino: Sul Moto Irrotazionale dei Liquidi Viscosi, Rendiconti
della Ace. Naz.dei Lincei, Serie VIII, Vol. VI, p. 615, et seq., Sept. 1,

18 NACA TM 1359

The similarity relationship is applicable in all cases where free
surfaces do not enter the problem. When these surfaces are present the
cases in which similarity can be brought into play are discussed in
section 6.

Translated by R. H. Cramer
Cornell Aeronautical Laboratory, Inc.
Buffalo, New York

NACA TM 1359

Bibliographic Notes

The publications concerning dimensional analysis, similarity laws,
and the theory of models are so numerous that it is not possible to list
them all. We wish to record here merely those of most importance and
those most easily obtained. Thus we first mention the work of

P. W. Bridgman, Dimensional Analysis, New Haven, 1932; also translated
into German as Bridgman-Holt, Theorie der physikalischen Dimensionen,
Leipsig, 1932.

In addition to this basic test we wish to add the papers of:

F. Marzolo, Alcune Considerazioni sui Modelli Idraulici, Giornale del
Genio Civile, 1917.

G. DeMarchi, Omogeneita, Similitudine e Modelli Idraulici, Rendiconti
del Seminario Mat. e Fisico di Milano, 1933.

E. Pistolesi, Omogeneita, Similitudine, Modelli, "Fondamenti Teorici"
of the Atti del II Conv. di Mat. Applicata, 1940.

A quite extensive bibliography will be found in:

U. Puppini, Idraulica, Bologna, 1947, pp. 617-618.

While the less recent research work is cited and partly paraphrased
in the book:

Hydraulic laboratory Practice, New York, 1929.

Moreover one should consult the various notes of the late Prof.
E. Foa, contained in the issues of L'Industria, from 1928 to 1929, and
in Politecnico for the years 1929 to 1930 (these articles advance a point
of view that is a little different from that brought out in sections 2
and 3).

Other works, in particular the most recent one of J. W. Ball, are
referenced in the notes appended to the bottoms of the pages.

NACA TM 1359

1* I I.


0 0





E 0
E o

Figure 1

50 10 15 20 25
Temperture, t, deg Centigrade


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NACA TM 1359


0 20 40 60 80 100

Figure 2.- Froude number analogy.

NACA TM 1359


AV 0


U' .

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Um~ox~ 1.00 m/s
0 50 7 100

Figure 3.- Reynolds number analogy -=-1 (The symbols
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