Resistance of cascade of airfoils in gas stream at subsonic velocity

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Title:
Resistance of cascade of airfoils in gas stream at subsonic velocity
Series Title:
NACA TM
Physical Description:
30 p. : ill. ; 27 cm.
Language:
English
Creator:
Loĭt︠s︡i︠a︡nskiĭ, L. G ( Lev Gerasimovich )
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aerofoils   ( lcsh )
Aerodynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
A method of computing the resistance of an airfoil in cascade in a viscous compressible gas stream with subsonic velocity is presented. An essential assumption of the method is that there is only a small degree of nonhomogeneity in the flow at the section downstream of the cascade where the trailing wakes from the individual airfoils merge. The resistance is expressed in terms of the boundary-layer properties at the trailing edge of the airfoil in cascade; these properties can be computed by any compressible boundary-layer theory or can be measured.
Bibliography:
Includes bibliographic reference (p. 28).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by L.G. Loĭt︠s︡i︠a︡nskiĭ.
General Note:
"Report date September 1951."
General Note:
"Translation of "Soprotivlenie reshetki profilei v gazovom potoke s dokriticheskimi skorostiami." Prikladnaia Matematika i Mekhanika, vol. XIII, no. 2, 1949."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779974
oclc - 99486389
System ID:
AA00009230:00001


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f'( rIA-\- 1303







1 (L/ ,c- 7 //- "-



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1303


RESISTANCE OF CASCADE OF AIRFOILS IN GAS

STREAM AT SUBSONIC VELOCITY*

By L. G. Loitsianskii


A method of computing the resistance of a cascade of airfoils
in a viscous compressible gas flow is described.

The case of an incompressible gas is considered in reference 1
and appears herein only as a simple particular case of the general
theory of resistance of a cascade in a compressible gas.

The investigation was restricted to subsonic velocities (that
is, when the local velocity of sound is nowhere reached on the air-
foil surface) because the required assumption of isentropic flow,
that is, the absence of shock waves in any region of the motion,
is valid only under these conditions.

The second reason for the restriction to relatively small
values of Mach number is the possibility under this assumption of
applying a lift formula analogous to the well-known Joukowsky
formula (reference 2) and of thus assigning a definite meaning to
the term "cascade resistance" or, more accurately, the "resistance
of an airfoil in cascade."

The resistance formula can be derived for an isolated airfoil,
as is known, by applying the momentum theorem between two parallel
cross sections of the flow at an infinite distance upstream and down-
stream of the airfoil. In the problem of cascade resistance, dif-
ficulty is encountered, namely, the absence of an external potential
flow downstream of the cascade where the boundary layers (wakes)
from the individual airfoils merge. This essential difficulty,
which is expressed quantitatively in the impossibility of employing
the boundary-layer (wake) equation up to a plane at an infinite
distance, can be circumvented by introducing the plane of merging
of the boundary layers (wakes) and by establishing relations
between the gas dynamic elements in this plane and in the plane
at infinity downstream of the cascade.


*Soprotivlenie reshetki profile v gazovom potoke s dokriti-
cheskimi skorostiami, Prikladnaia Matematika i Hekhanika, vol. XIII,
no. 2, 1949







NACA TM 1303


An essential assumption of the present investigation is that
a small degree of nonhomogeneity of the flow exists in the section
of the aerodynamic wake of the cascade where the boundary layers
from the individual airfoils, considered as layers of finite thick-
ness, merge; the larger powers of the small velocity differences
may then be neglected. The same assumption was made in the investi-
gation of cascade resistance in an incompressible gas (reference 1)
and was subsequently confirmed experimentally. The plane of merging
of the boundary layers is then assumed to be the control surface
required for the application of the momentum theorem and in the case
of the isolated airfoil is taken to be the plane at an infinite
distance downstream of the airfoil. It is evident that when the
relative pitch of the cascade is increased, this plane will be farther
and farther away from the axial plane of the cascade and in the limit,
for a relative pitch equal to infinity, that is, in the case of an
isolated airfoil, will go to infinity. This assumption may evi-
dently be made for cascades with moderate solidities, a case that
corresponds in practice to turbine and compressor cascades.

Any method of calculating the boundary layer in a compressible
gas may be used to compute the characteristic thicknesses of the
layer and to estimate the effect of the compressibility of the gas
on the external flow. The solution of the proposed problem reduces
to a straightforward and direct form that is independent of the
method of computation.

1. Resistance of airfoil in cascade. Joukowsky force as com-
ponent of total force exerted by incompressible fluid on airfoil. -
For two-dimensional flow of a real fluid, the resistance (or drag)
of an isolated cylindrical wing of infinite span referred to unit
length of the wing is the component of the total force exerted by
the fluid on the wing in the direction of the velocity of the approach-
ing flow at infinity, or, in other words, of the velocity component
of motion of the wing in an incompressible medium.

This definition is invalid in the case of an airfoil in a
two-dimensional cascade, because in this more general case there is
no unique velocity direction at infinity upstream and downstream of
the cascade and there are no considerations by which preference is
to be given to any particular direction for determining in this
direction the resistance component of the total force acting on the
wing. In this case, the problem is to determine what may be termed
resistance.

An isolated .-in;! of finite span is now considered. In this case,
as also in the case of an airfoil in cascade, for each section of the
-rln., in vi. of t-:e presence of vortex systems (films) shed from the







NACA TM 1303


v'n,. and poss.:ng. do;:strean to infin it ., two velocities :"ferent in
*na_.nitude 3nd direction exist at -in-!"r..ty u.strcam rund downstream
of the :;in;. For iiei flo.! about a ting of finite span in -ccordance
. lth the sche.ae of liftinE- lines, the totl pressure force of the
fio4 at ia c-ven section of T-he w-in- is k-nown to be ;.er-endicular to
the velocity of the flow at zhe correspond 'n- point of -he section
under consideration on ..he -1ftin- line. This velocity, which repre-
sents ha-lf the vector sum of the velocities :t 'nfinlt' urstream:! and
do;.nstrea.a of the wing: is ass'.ied at the section considered as the
effective velocity oi flow; the on le cetween the chord of the wing
section and the direction o' tne effective velocity is considered
as the effective angle of attack: a.nd so forth.

For a two-dimensional infinite cascade of airfoils, a similar
assumption is made with the difference only that in the theory of
the ;-ing of finite span the effective velocity differs slightly from
the velocity of the approaching flo-i; whereas in the case of the
cascade the jump is of the same order c-.s the geometrical angle
of attack.

In the aerodynamics of a wing of finite span, the profile dra,,
is the difference between the he.d -esistance, which is represented
by the component of the total force exerted by the real (viscous)
.a.s low' in the direction of the velocity at infinity upstream of
the win., nid the induced dra-, ;ihic' is the comronent in the same
Jirect:on o' the effective lift force.

For a small difference between the directions of the effective
velocity and the velocity of the approaching flow, this Cefinition
of the profile JraE of a win. section differs by small terms of
higher order from the true profile 'ira,-, strictly defined as the
vector difference between the tot-l force exerted by the real flow
on the win; section and the effective lift force for a real fluid.

In the case of the two-dinensional cascade, it is natural to
assume for the profile dr.-s R' the difference between the vector
of the total force R (fig. 1) and the Joukoe:sky force Rj (in
the terminology of reference i) which for an incompressible gas
is given by


R. = pV
cn


(1.1)







NACA TM 1303


acting in the direction perpendicular to the fictitious velocity
at infinity V determined as


V = 1(V + V) (1.2)
m 2 DO am

where Vl. (u, v ) and V?. (ua, v,) are the vector velocities
at infinity upstream and downstream of the cascade, p is the den-
sity of the fluid, and r is the circulation determined by the
equation


r = (v2 1- Vlt (1.3)

where t is the pitch of the cascade.

Introducing the concept of the vector pitch t, which is equal
in length to the magnitude of the pitch t and directed at right
angles to the axis of the cascade downstream of the flow, gives
the Joikowsky force by the following vector equation (reference 2):


Rj = PVm x (t x Vd) (1.4)

where the following vector


Vd = V2 V I (1.5)

gives the vector change of velocity produced by the cascade. The
preceding formulas are valid not only for the flow of an ideal incom-
pressible fluid but also for a viscous fluid.

The profile drag R' is as follows (reference 1):


R' = p't (1.6)

where p' is the pressure loss in the cascade determined by the
equation







NACA TM 1303


P' =(Pl + pVi2)- (P 2c + L V2O
(1.7)

P 2) =2)
=Pl 2 1 PV) (" 2 2

The total force R is equal to the sun


R = R + R' = pVm x (t x Vd) + p't (1.8)

2. Resistance of two-dimensional cascade in real as flo'. at
subsonic velocities. The expression for the total force R of
the interaction of the flow with a two-dimensional cascade at sub-
sonic velocities may be represented in the follo..ingt for'r (reference 2):


R = (Pl F F)t + 'iP (Vl r t( )V v- p'= ( 2.1)

where Pi', Pl and p2', p-. are the pressures and densities
upstream and downstream of the cascade, respectively The e:,uiva.l.ent
expressions

Pl, V t = V2, t (2.2)

evidently express the rate of mass flow per second throu':h the
section of the flow parallel to the axis of the cascade'and eDual
in length to the pitch.

As was shom (reference 2) also in the case of a co.ipressible
,gas for Mach nlubers not too near unity, the lift force of an sirfoil
in cascade in :!i ideal g.as flow my,' be represented in the form of
equation (1.4), provided that for the density p is taken the arith-
metical mean density pm equal to

Om 1 ('b + P29) (2.3)

The following opproximate expression of the Bernoulli theoreli
is employed:


S- P PmV V = (V2 2) (2.4)
lW 2. m d m 2m Iam








NACA TM 1303


This equation is valid with an accuracy to tenth parts of the square
of the difference of squares of the Mach numbers at infinity upstream
and downstream of the cascade.

In the case of the real (compressible and. viscous) Las,


p1 -P22 V2) + p' (2.5)
PI P2o. m 2 10 '

where p' characterizes the losses in the cascade due to the internal
friction in the gas; an equation may be obtained (reference 2) analo-
gous to equation (1.8)


R = R. + R' = p V X (t x v) + p't (2.6)

where p' is determined by the expression


=l- 2 (V 2 V2) (2.7)
=P P2- 2 Pm (V2o V10D

The problem of determinin. the profile drag force R' ejual to


R' = p't (2.8)

thus reduces to finding the losses p' which depend on the shape
of the airfoil in the cascade and the character of the flow about
the airfoil.

5. Introduction of intermediate plane; relation between gas
dynamic elements in this plane and corresponJin, values at infinite
distance from cascle. In addition to the planes la and 2W
that were empioy,ed in the analysis of the incompressible fluid,
(reference 1) an intermediate plane 2 is introduced for the compress-
ible gas (fig. 2); plane 2 is located where the boundary layers
(wakes) from the individual airfoils iner.e. The hydrodynnr.ic .nd
thermal bou.nn.c.ary L],ve.'s in the wake downstream of the cascade are
here:inafter assumed to have the same pattern.

The followvi,- assumptions with regard to the motion of the gas
near ,i. ne 2 are necessary: By definition of the position of plane 2,
no i'-li'vidual boundary .;,'.e.;s exist in the flow downstream of this
plane; the .eros.vni.lc and thermal u'.i.es of the airfoils are, however,








TACA TM 1303 7


maintained and depressions in the velocity or total-pressure curves
and also depressions or pea';s in the temperature curves result.
A fundamental property of the boundary layer is that the pressure
transverse to the wak-e is the same at all. points o'' a given normal
section of the wakIe; that is, no pressure drop in the distribution
curve occurs in this section. The I."ressure aJon' the wake changes
sharply in the iruiediate neighborhood of Lhe tr'ling edge of the
airfoil and is gradually e:{u-llzed as the distance from the trailing.
edge is increased.

';o sections of the 'ai:e are passed through the point of inter-
section of plane 2 ;ith the 3xis of the a.;a.e; one section lies in
plane 2 and includes the y--xis (f;7. 3), and the second section
lies in plane 2' normal to the axis of the wa,:e and includes the
y '-axis.

The following; minagnitudes re introduced:
o+t
"UU 2,- = -t-



+t




1 P-
CC,

yr.,



v t v,




1 S P2 P
Zp t P n
Yo




'-I


7o+t

vt T
y- I O







NACA TM 1303


which characterize the mean relative deviations of the hydrodynamic
elements of the flow at the points of section 2 of the wake from
the values of these elements at the boundaries of the wake at the
points of intersection of the boundary layers.

Section 2 will be assumed at such distance from the cascade
that the differences u2 u, and also their mean relative
values Au, may be considered small magnitudes, the higher
powers and the products of which may be neglected. Moreover, the
velocities at different points of section 2 are assumed parallel
and in a general direction coinciding with that of the velocity at
infinity behind the cascade. It follows at once that


Au = vA (3.2)

Comparison with analogous mean relative deviations in section 2'
gives the magnitudes


S u u'
Au 1= tJ dy, (t' = t cos P2) (3.3)
U.te T U21 2
O 2

In the subsequent discussion, it will be assumed that, for a
sufficient distance of planes 2 and 2' from the axial plane of the
cascade, all the magnitudes (3.3) and so forth are correspondingly
equal to the L'agnitudes (3.1); that is,


Au' =A ap' A Ap I (3.4)

This additional assumption may be justified as a consequence
of the assumption of a small degree of variation of the gas dynamic
elements near plane 2 and behind it downstream of the flow.

In accordance with the fundamental property of the wake A = 0,
the following equation may be obtained:


Ap = 0


(3.5)







NACA TM 1303


Because of the smallness of the magnitudes u', Apo the
gas dynamic magnitudes in the intermediate plane 2 are easily shown to
be connected with the corresponding values of these magnitudes in the
plane 2oo by relations that are analogous to the case of the incom-
pressible gas.

For this purpose, a segment of a flow tube is assumed between sec-
tions 2 and 2m, where a length equal to the pitch t is taken for the
transverse dimension of the tube in the direction parallel to the axis
of the cascade.

Application of the theorem of the conservation of mass then yields
YO+t yO+t

S pu dy =[ P 2 (P 2 P)] 22 u2 u)]dy= PU t
Y0 YO
Expanding the brackets and neglecting the product (p2 p)(u2 u)
as a small quantity of higher order gives the following equation:
yO+t
S[P 2u2 u2(2 P) p2(u2 u) dy= p2o~u
Y0

From this expression, the following relation is obtained in the
notation of equation (3.1):


2u2(1 a \ u) = PZu2,

or, with the same degree of accuracy,


p2u2= p9,,u(l + Ap + A) (3.6)

The momentum theorem in the projection on the x-axis applied to
the same segment of the flow tube gives
Y0+t y0+t
Spdy +J pu2dy p2-t P2ut =
Yn YO






NACA TM 1303


This equation may be written in the form


yo+t
- p]dy +r
Yo


[2 (P2 P] [u -


(u2 u)]dy


p t + P u 2t
200 z 2= t


If the smallness of the differences p2 p, p2 p, and u2 u
is taken into account, the following expression may be obtained:


p2(l A) + p2u22(l A 2a) p + P2u2m (3.7)

With the aid of equations (3.5) and (3.6) and the same approxima-
tion, the following equation may be written:


P2 + P2ou2ou2(1 'u) = P2o + P2ou2O2


(3.8)


The momentum theorem is now applied in the projection to the
y-axis, which gives
Y0+t
f puv dy = p2,u2'v2Dt
yo


yo+t
S 2- (P2 u2 (2 u
YO


[v2 (V2 vOldy = p2ou v2a,


Rejection of small terms of higher order leads to the equation


p2u2v2(l Ap A A) = pau2-v2


YO







NACA TM 1303


or, according to equation (3.6),


v2(1 Av) = vZm

v = v(1 + A)


(3.9)


The assumption of parallel directions of the velocity vectors in
sections 2 and 2m, with the aid of equation (3.2), yields


2 = u2(1 + A)

V2 = Va(l + ) = 2(1 + AU)


(3.10)


Equation (3.8) then gives


(3.11)


On the basis of equation (3.10), there
tion (3.6)


P2 = P2m,(1 + A )

Finally, from the Clapeyron equation,


p P (P2 P)
S 2 -(2-P) RT = RT2
- p2 ( 23 p)


also follows from equa-


(3.12)


- (T2- ]


or, when the smallness of the differences is accounted for,


P2 P
P2


2 RT2 T2 T
+ P2 = RT 1 -T2
P 2 T2


From this equation, A A = tT is obtained by integration, or,


(3.13)


P2 = P2o


P2 (







NACA TM 1303


Conversely, the same Clapeyron equation in planes 2 and 2cm
yields, by equation (5.11)

T2 P2 = P2 P P2a = P2 A
R(T T2, __
2 p2a P2 P22 P2 P2=
or, by equation (3.13),


T2 T= TB, = A T2- T


that is,


T2 = T2J1 + T)


(3.14)


4. Relation of fictitious wake thicknesses to magnitudes A and
A'. Expression of profile drag in terms of fictitious wake thick-
nesses. The momentum equation in the wake behind the airfoil of
the cascade will now be employed; the equation contains the following
fictitious wake thicknesses defined (reference 3) as integrals over
section 2': displacement thickness 82 and loss-of-momentum thick-
ness 2 **
2'


yO


Yo'+t
82* =




*2 S P
yO'
YO '


1 2dy'
P2V2


(4.1)


)V
2V2


(1- y
v2)


When these thicknesses are connected with.the magnitudes A,







NACA TM 1303


S2 YOI t P2 P2 CV2- (V2 V
2 Y ( 0y2V
YO


P2 p '
dy'


y0'+t'

YO'
y0


V2 Vd
V2 dy'


= t'(A + Au') = (A + Au)t cos p2-


y0 +t'

2 Y
Y0


P2 (P2 -


p)] [ 2- v')] 2 v
P2V2 V2


dy.3)
(4.3)


= t'Av' = tAu cos P02


The profile drag will now be determined; the magnitude p' must
first be found. In equation (2.7), p' is expressed as a small dif-
ference between two large magnitudes and is therefore unsuitable
either for experimental or for approximate theoretical determination
of p'. In order to eliminate this defect, equation (2.'7) is rewrit-
ten in the form


P' = Plm Pz= (PI + P2o) (v22 Vp2)


(4.4)


and the flow is considered between section la and the limits of the
boundary layers that merge in plane 2. In this entire region, the
flow is nonvortical so that the Bernoulli theorem may be applied
without the additional term that accounts for such losses. The fol-
lowing expression similar to equation (2.4) may then be written:


Pl P2 = (Pl+o P2) (V22 2) (4.5)


YO' I+t

= f
y-
YO'


S(4.2)


dy'







NACA TM 1303


Because by equation (3.11) p2 = P2,, the following expression
is obtained when equation (4.5) is compared with equation (4.4):

p' = ( + p (V2 2 -1(p + p2 V) (V2 V2)
p, (P
4 2 2 4
When V2 and p2 are replaced by their expressions in terms of
V2o and p2w, then according to equations (3.10) and (3.12),

p' = l (Po + p P + P+) [2(C + 2A ) ] -


l (p + p32) (V2 2 V,)

S(2_ v 2) A2
= (Pl2 + P2) V 2 + 1 a z -PV(2) v

or, by equations (4.2) and (4.3),

2 p*2i V S* p52* 2
P' = V 2 t c2 3+ 1 2 VL2) cs 52 (4.6)
p m 2c t cos P0 4 P2V 20 t Cos 02.

The following magnitude is now introduced:

H2 = +- A +


which is for the case of the motion of an incompressible gas; the
following simple equation is then obtained:

P' = mV22 + (2 1) p2.(V2 Vl t cos p2 (4.8)

The formula for profile drag is immediately obtained from
equation (4.8),

S= = 2 2 1) 52 (4.9)
R' p't = [PmV2O2 + 14 (H2 1) p2Z(V2G v C (4.9)
2 ~21 cos BZp2+ (49







NACA TM 1303


From this expression, the profile-drag formulas for a cascade in
an incompressible viscous fluid are obtained as particular cases and
for the isolated airfoil in the general case.

For the case of a cascade in an incompressible fluid
(p = constant),

P =P


A =0
P

H2= 1

and equations (4.8) or (4.9) are converted into


, PVZo2c *
P' =
t cos Pg


R, _pV225 2a *
cos P02


(4.10)


which are identical to equations (2.12) of reference 1.

For an isolated airfoil in a viscous compressible fluid,


Plo= 2P = Pm =


V = V = V


02M = 0

Moreover, plane 2 extends to infinity, so that


R' = PoV2 2,(*


(4.11)







NACA TM 1303


Equation (4.11) is the well-known formula of the resistance theory for
an isolated airfoil.

The losses and the profile drag of the cascade are expressed by
equations (4.8) and (4.9) in terms of known elements at infinity
ahead of and behind the cascade and in terms of the elements H2 and
8 ** referred to plane 2, the position of which remains unknown,
because up to the present no reliable theory of the turbulent wake
exists.

A formula will now be obtained for the profile resistance of the
cascade; by the theory of the boundary layer at the airfoil, this
expression makes possible the computation of the resistance of the
cascade, and the dependence of the magnitudes H2 and 82** just
mentioned on the elements of the boundary layer at the rear edge of
the airfoil of the cascade can therefore be determined.

5. Establishment of relations between wake elements in sec-
tion 2 and boundary-layer elements in sections at trailing edge. -
A generalization is given herein of the known device of setting up
relations between the elements of the boundary layer at the trailing
edge of the airfoil and in the wake behind it at infinity, as proposed
for the case of the isolated airfoil in the incompressible fluid by
reference 4.

In this generalization, for the case of the cascade the section
at the trailing edge is connected not with the plane at an infinite
distance downstream of the flow but with plane 2 of the merging of
the boundary layers or, more accurately, with plane 2' inclined to
it by the angle P2 Moreover, the generalization requires passing
to the compressible case.

The momentum equation for the wake behind a body may easily be
derived from the general equations of the plane boundary layer in a
compressible gas

Vs yV' dp _r
Vs + Vn --= d- +
+s n 0ds

a(pVs) ((PVn)
+s n = 0 0







NACA TM 1303


where for the longitudinal (coordiante s) and transverse (coordinate n)
projections of the velocities, the symbols Vs and Vn are used in
contrast to the velocity projections u and v connected with the
axes Ox and Oyj p is the local density, T the friction stress, and
p the pressure on the outer boundary of the layer.

By rewriting the system (5.1) in the following form, according
to the second of equations (5.1) and the general Bernoulli equation,


S(pVVs) + (pVsVn) = pVs
ds an ds

a(pVs) a(pVn)
as an = 0


where p and Vs denote the density and the longitudinal velocity
at the outer limit of the boundary layer. Both sides of the second
equation are then multiplied by Vs to yield


(5.2)


- (pVV s
as


a ,dVs
+ 6 (OVsVn) 0V --d O
an as


The first of equations (5.2) is then subtracted term by term
from the equation just obtained; the resulting equation is then inte-
grated, which gives


s pv(Vs Vs3 + n(s V s PVs) = --
anan ds an

along the normal to the section of the wake, which is considered either
infinite in the usual sense of the theory of asymptotic boundary layer
or finite, as is assumed in the theory of the finite thickness layer.
In either case, the following relation holds:


d+,5
d-r


dVs +W 5
pVs, Vs) dn + d (v pVs) dn = 0
s s sr


The following expression is then obtained:

OVs2 V + 7V V 1
s sVs Vs ds


V dn = 0
TVs1







NACA TM 1303


By expanding the parentheses and introducing the notation of
reference 4,





(5.3)

\, 6



the required momentum equation is finally obtained.

2 dV 1 dp\ 1 dV ,



differs from the corresponding equation for the incompressible gas
1
only in the term T-1 d/ds (and, of course, in the definitions of
the magnitudes 6* and 8**). If the momentum equation for the
incompressible gas is considered for the case of axial symmetrical
motion, the term T-ld1/ds, which expresses the effect of the variable
density of the gas, may be taken equivalent to the term that takes
into account the transverse curvature.

In addition to the momentum equation, the heat equation is con-
sidered; it can be easily set up by a method analogous to the preceding
method from the known heat equation of the boundary layer.


PVs C) + i+ =a (5.5)

where a = c p/X is the Prandtl number.


i = JCpT



3(n a 2







NACA TM 1303


The value of q is given in the case of the laminar boundary
layer; equation (5.5) holds also for the turbulent layer, but in this
case q would be expressed in a different form.

The so-called temperature of adiabatic stagnation T* is now
considered; it is given by


s = T 2 (5.6)
T =T+2Jc


By means of the continuity equation, the following system of
equations may be set up:


s (pVsT') + n (pvT*) = 1
JCp
(5.7)
T (PVsTi) + 3 (pV-T*) = 0

In the second equation of the system, the stagnation temperature
T* at the outer limit of the boundary layer, which is constant
(because the external flow is isentropic), is taken under the sign of
the derivatives in the continuity equation.

Subtracting one equation of the system (5.7) from the other and
successively integrating over the cross section of the wake gives

+40, 5
dJ pVs(( "- T*)dn = 0

(5.8)
+0, &
S pVs(T* T*)dn = constant
-, &S

The fictitious thickness of the wake is now introduced


J+C pVs T
v T.,
-a.S







NACA TM 1303


which may be termed the thickness of the energy loss; equation (5.8)
may then be rewritten as


-pT*e = constant (5.10)

Equation (5.4) is again considered. After each side is divided
by 5**, the expression is integrated along the wake from section k
at the trailing edge of the airfoil to plane 2', previously defined.
The result is


(k)
In (IT(*in ds (5.11)

The notation of equation (4.7) is used for the ratio of the ficti-
tious wake thicknesses,


H (5.12)
8**
and it is noted that equation (5.11) is integrated to completion
if the magnitude H is replaced by some average value; for example,
the following relation may be set up:


S= Hp = (H2 + k) (5.13)

By this simplification, the following expression is immediately
obtained:


\6kn + Hk) nV

or finally,

52 POk /k 2 + (H2+Hk)
k V2 (5.14)

This equation connects 62** and 5k**, but does not explicitly
contain 52*; the exponent on the right contains the magnitude H2,
which is equal to the ratio 62* /2**. From equations (4.8) and (4.9)
previously derived, equation (5.14) serves as one of the equations







NACA TM 1303


for expressing the two unknowns 52*" and H2 entering in the
equations for the losses in the cascade and the resistance in terms of
the elements of the boundary layer at the trailing edge of the airfoil.

The second equation is obtained by use of equation (5.10), which
may be rewritten as follows:


P2V2T2 *2 = PkVkTk k

or, because of the isentropic character of the motion outside the
boundary layer, T2 = Tkj the expression then becomes


P2V2z2 = PkVKdk (5.15)

In this equation a new unknown quantity e2 appears to enter;
because of the small degree of nonhomogeneity of the fields of hydro-
dynamic elements in planes 2 or 2', however, this term can actually
be expressed in terms of the previous unknowns. When the small degree
of nonhomogeneity and the formulas relating the elements in planes 2
and 2m (derived in section 3) are accounted for, equation (5.14) and
then equation (5.15) are transformed. By equation (5.14),


SP( 2-(H2+Hk v 2 +Hk)



P2I V- 2 k) ((1 +[2- + (H(5.16)


Pk 2 + 1 (H2 + H


Because in this section everything will be expressed in terms of
the unknowns 2"** and H2, equations (4.2) and (4.3) are applied;
the following expression is then obtained:







NACA TM 1503


pk v 22 k2 (H2 -) 2 -



S2 2 +) j


or

82 k ** Pk V 2 k2 1 1+Hk
PzpYM(Cz~


-2) ]


For a first approximation, the subtrahend in the brackets on the
right may be neglected in comparison to unity to obtain


2**= S Pk2 /VL 2 2
-P2c M20-)


(5.18)


The second fundamental equation (5.15) is similarly transformed.
With the chosen degree of accuracy,


+c, 8
e2 ~f Pv5
p2V2


p2V2(T2*
2 P T2*


dn G PVs(T2* T*) dn
SP2V2T2
- W158


dn =
^=
*Vs/fl


T2* T*
2 dn
T2 *
*4


2**


(5.17)






NACA TM 1305


Therefore,

e2 (T2 T)dn
2 p~8 T2 + V22/2Jc~


+ V22/2JC, Vs2/2Jcp dn
-w,6 T2 + V22/2Jcp


2 -f T2 n +
1 + V22/2JCpT2 T2
2 C72jc roJ


'V22/2Jcp -[v22 2V2(V2 V]l/2Jcp dn i
T2 + V22/2Jc 1 + v22/2Jc T2
-oDb r


V22/2JecP t
S2
T22 + V2/J t

-(I
Jc T2


2
T20J+ 20.
ST2- J T2 2m

A T20 a
2W f


or, by equations (4.2) and (4.3),


-2TV (H2 1) 2**
SJCpT2w T2 82


(5.19)


The term e2/t', which by equation (5.19) is proportional to
52**/t', is a small quantity of the first order; with the assumed order
of approximation, equation (5.15) may be transformed into


[pv2 9J V 2 (2 -1)
pT2=


T- 82 = Pk k
T2oJ


[ 2 9
p2M (1 I(2 a T2 PkVkT2 k
P


(5.20)







NACA TM 1303


The system of equations (5.18) and (5.20) gives the required
system of equations for determining the two unknown magnitudes 52*
and H2 as functions of the parameters of the boundary layer, of
the external flow near the trailing edge of the airfoil, and of the
density, velocity, and temperature at infinity behind the cascade.

For the solution of this system of equations, it is noted that
the unknowns are readily separated if equations (5.18) and (5.20) are
divided one by the other. This procedure yields


h Va 2 V2nm/Jp (H2 1)T3,



the Mach number M2 is introduce


S=V2
2 Ca2


V2~
1 2* t;;


S22 (k 1)M2=2 (H2 1

\V, J(k 1)M2 + 1


142H
_l+ f ^


ed at infinity behind the


V2aw
4 (k l)J5pT2


l) 6k /V2 1
8k Vk


This transcendental equation in H2 may be solved by one of the
approximate methods in any concrete case. According to equation (5.20),


S**= 1+ 1)M 22 PVk 8k
1 + (k 1)M2 2 e pV2


(5.22)


The terms 52** and H2 referring to section 2, the location
of which is unknown, are thus eliminated and expressed in terms of
the magnitudes k** and ek either measurable or computable by
any method of boundary-layer theory and in terms of the velocity,
density, and temperature at the outer limit of the boundary layer
near the trailing edge of the air foil and at infinity behind the
cascade. The terms p' and R' may then be obtained with little
difficulty by equations (4.8) and (4.9).


or, when
cascade,


(5.21)


+ k







NACA TM 1303


6. Approximate formulas for computing losses and resistances in
cascade. At these relatively small subsonic Mach numbers considered
herein, the nonisothermal character of the flow in the wake behind
the airfoil of the cascade can occur mainly through the heating or
the cooling of the surface of the airfoil and not through the internal
transformations of kinetic energy into heat.

In order to verify this fact, equations (5.21) is employed.
The following notation is introduced for briefness:

k 1 2 =
2 cOB


H2-1= -
2

3 + Hk=
2


(6.1)


Equation (5.21),
assumes the form


which is transcendental relative to C, then


m1 + m k V2 \~
1+ m 2k k


The unknown magnitude
of the small parameter m.
does not exceed 0.1)


C is now expanded into a series in powers
(For air the value of m at M2 < 0.7


c = C0 + Glm + C2m2 + ...


(6.3)


Substituting this series in equation (6.2) gives


- + ( l)m ...]( m ...) = '~zpcO
(6.4)

or+- 1)m ...= 2 +C..
- CO+ (I+CO-Cj)m- +C1 a M
k Y


(6.2)






NACA TM 1303


By equating coefficients, the following equation is obtained:

2. = I 1, v' 0
k 2 ,vk/


(6.5)


for determining C0. Because of the assumption previously made on
the small heat transfer from the surface of the airfoil in the cascade,
the quantity CO is considered small for M2, = 0. The following
equation, accurate to small quantities of the second order, is then
obtained:


- Ak [1 + C 1 k


(6.6)


From equation (6.6),

Ak
S- 1 + Ak in (V2JVk)


The ratio VTJ/Vk generally differs little from
the natural logarithm of this ratio is small so that
may be written without great error.

By equating the coefficients of m to the first
equation (6.4), the expression C1 = 1 + O = 1 Ak
with the same degree of accuracy.

The approximate equation is then


(6.7)


unity;
C = -


(6.8)



hence,
Ak


power, in
is obtained


c = Ak + (1 + Ak)m rm Ak


where


(6.9)


- C = A CkO
Vk )






NACA TM 1303


Equation (5.22) is now employed and in the new notation has
the form

2, 1 + m PkVk k
2 (m-. ) pan (.

According to equations (6.11) and (6.7),


2 PkVk k
P V2 2A
2M2a k


S(5+Vk)
k P V2


(6.10)


(6.11)


If it is assumed that at the trailing edge
(6.11) and (6.7) assume the form


Hk = 1.4, equations


& *- k V k 3.2
2 7 .2


Ak I V f \ 2.2
k 2 ** k vk )


(6.12)


From equation (4.8), an approximate formula for the losses is readily
obtained:


P LPi22 + p V22 2)1 *
pV= Y2 + p i E 2V2 -V 2.2*
[Pm 2 W 2 2w 2c 100 t cos 02w


P= mV22 P C )
P2w V 20/


3.2 6 r 2
s + (m Ak) 2~ 2
t cos P2 LPm V;


and therefore a corresponding approximate formula for the resistance
differing from the right side of the previous equation only in the
factor t.


2






NACA TM 1303


The further possible simplifications of equation (6.13) are
connected with the choice of devices for computing the characteristics
of the boundary layer at the surface of the airfoil in the cascade
and for taking into account the effect of the compressibility on the
external flow.


Translation by S. Reiss
National Advisory Committee
for Aeronautics


REFERENCES

1. Loitsianskii, L. G.: Resistance of a Cascade of Airfoils in a
Viscous Incompressible Fluid. Prik. Mat. i Mek., T. XI,
No. 4, 1947. (Translation available on loan from NACA
Headquarters.)

2. Loitsianskii, L. G.: Generalization of the Joukowski Formula to
the Case of an Airfoil of a Cascade in a Compressible Gas
Stream with Subsonic Velocities. NACA TM 1304, 1950.

3. Loitsianskii, L. G.: Inverse Effect of the Boundary Layer on
the Pressure Distribution on the Surface of a Body in a Real
Fluid. Prik. Mat. i Mek., T. XI, No. 2, 1947.

4. Squire and Young: Computation of the Profile Drag of a Wing.
Collection of articles on the problem of Maximum Speed of an
Airplane. N. Oborongiz, 1941, pp. 100-126.

5. Loitsianskii, L. G.: Resistance of a Cascade of Airfoils in a
Gas Flow with Subsonic Velocities. Prik. Mat. i Mek.,
T. XIII, No. 2, 1949, pp 171-186.






NACA TM 1303 29


Figure 1.


Figure 2.


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NACA TM 1303


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