Flow of gas through turbine lattices

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Material Information

Title:
Flow of gas through turbine lattices
Series Title:
NACA TM
Physical Description:
136 p. : ill. ; 28 cm.
Language:
English
Creator:
Deĭch, M. E ( Mikhail Efimovich )
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Cascades (Fluid dynamics)   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: This report is concerned with fluid mechanics of two-dimensional cascades, particularly turbine cascades. Methods of solving the incompressible ideal flow in cascades are presented. The causes and the order of magnitude of the two-dimensional losses at subsonic velocities are discussed. Methods are presented for estimating the flow and losses at high subsonic velocities. Transonic and supersonic flows in lattices are then analyzed. Some three-dimensional features of the flow in turbines are noted.
Bibliography:
Includes bibliographic references (p. 77-78).
Statement of Responsibility:
by M.E. Deich.
General Note:
"Translation of Russian book: Technical Gasdynamics (Tekhnickeskaia gazodinamika) ch. 7, 1953, p. 312-420."
General Note:
"Report date May 1956."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003807492
oclc - 126896936
System ID:
AA00009200:00001


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IJW A oH 93"1











NATIONAL ADVISORY COMMITTEE FOR AERONAUICS


TECHNICAL MEMORANDUM 1393


FLOW OF cAS THIROUGR TURBINE LATTI~CESJ

By M. E. Deich


7-1. GEOMERTICAL AND GASDYNAMICAL PARAMETERS OF THE

LATTICES; FUNDAMENTALS OF FLOW

THROUGH LATTICES

The transformation of energy in a stage of a turbomachine is a re-
sult of the interaction of the gas flow with the stationary and rotat-
ing blades, which form the guide and impeller blade systems.

The lattices of a turbine in the general case represent systems of
blades of the same shape uniformly arranged on a certain surface of rev-
olution. A particular case of a three-dimensional lattice is an annular
lattice with radial blades arranged between coaxial cylindrical surfaces
of revolution.

In flowing through the lattice, the velocity and direction of the
gas flow are changed, and a reaction force is thereby produced on the
lattice. On the rotating lattices of a turbine this force performs
work; the rotating lattices of compressors, on the contrary, increase
the energy of the gas flowing through them. In stationary lattices an
energy interchange with the surrounding medium does not occur; in this
case the lattices bring: about the required transformations of kinetic
energy (velocity) and the deflection of the flow.

Depending on the flow conditions and the corresponding geome~trical
parameters of the blade profile, three fundamental types of lattices are
distinguished:

(a) Converging flow type: the nozzle or guide (stationary) vanes
and the reaction (rotating) lattices of turbines

""Technical Gasdynamics." (Tekhnickeskaia gazodinamika) ch. 7,
1953, pp. 312-420.








MACA TM 1393


(b) Action or impulse (rotating) lattices of turbines

(c) Diffuser: guide (stationary) atnd working (rotating) lattices
of compressors.

Depending on the general direction of motion of the gas with re-
spect to the axis of rotation, the lattices are divided into axial and
radial types. In certain machine designs the gas flow moves at an
angle to the axis of rotation (diagonal lattices).

The most important geometrical parameters of an annular (cylindri-
cal) lattice are the mean diameter d, the length (height, of the blade
2, the width of the lattice B, the pitch of the blades on the mean di-
ameter t, the chord b, and other blade profile parameters (fig. 7-1).

There exist several methods of specifying the shape of a blade pro-
file. The universal method of coordinates (fig. 7-2(a)) has great ad-
vantages. The methods shown in figures 7-2(b) and (c) are based on the
idea of the mean line of a profile; the mean line may represent the g~eo-
metric loci of the centers of inscribed circles or the centers of the
chords connecting the points of tangency. The mean line is defined by
coordinates, and the thickness distribution about the mean line is then
independently given. For specifying the profiles of turbine lattices,
consisting most frequently of thick, sharply curved profiles with small
pitch, the methods shown in figure 7-2(b) and (c) are inconvenient. The
determination of the fundamental dimensions, the construction of the piro-
file, or its checking require complicated graphical work. The most wide-
spread method of constructing the profile from a small number of adjoin-
ing arcs of circles and segments of straight lines (fig. 7-2(d)) is ar-
bitrary and tedious.

If the ratio of the mean diameter of the lattice d to the height
of the blade 2 is large, the lattice may, for the purpose of simplify-
ing the problem, be considered as a straight row lattice. The shape of
the space between the blades along the height may then be considered as
constant. In the simplest case, assuming that the diameter of the lat-
tice and the number of the blades increase without limit, we obtain a
plane infinite lattice (fig. 7-l(c)).

The passage from the cylindrical to the plane lattice is effected
in the following manner: We pass two coaxial cylindrical sections of
the annular lattice through the middle diameter d and through the di-
ameter d + nd. Assuming ad to be small, we develop the resulting
annular lattice of very small height on a plane. Increasing the number
of blades to infinity, we obtain the plane infinite lattice shown in
figure 7-1(c).








NACA TM 1393


The assumption of plane cross sections, that is, used as the basis
of the investigations and computations of modern turbomachines, was
fruitfully applied by N. E. Joukowsky in 1890. The value of this as-
sumption has been confirmed by numerous experiments.

The geometrical characteristics of lattices are usually given in
nondimensional form. For example, the relative pitch of the profile is
determined by the formulas
t t


The relative height (or length) of the blade,



In certain cases in investigating the three-dimensional flow in a lat-
tice, it is more convenient to define the relative height as




where a2 is the width of the minimum cross section of the passage
(fig. 7-1).

A rectilinear lattice is referred to as a system of coordinates x,
y, z where the direction x is termed the axis of the lattice (fig.
7-1()).All profiles must coincide in the translational displacement
along the axis of the lattice. The pitch t of the lattice is equal to
the distance between any two corresponding points.

For a given profile shape, the shape of the interblade passage of
the lattice depends, in addition to the pitch, on the angle Py, which
is defined as the angle between the axis of the lattice and the chord
of the profile (fig. 7-1(c)). In the practical construction of turbine
lattices, the position of a profile in the lattice is often specified by
the geometrical angle of the exit edge pZn (the angle between the tan-
gent to the mean line at the trailing edge and the axis of the lattice).
In certain cases, for a straight-backed profile, the angle P2n is
measured from the direction of the suction surface at the trailing edge.

In the design of the blade lattices it is necessary, besides satis-
fying a number of structural requirements, to ensure that the given
transformation of energy obtains with minimum losses. A detailed study
of the flow process over the blades of the lattice is thus required.
One of the important problems is that of establishing the effect of the
shape of the blades and of other geometric parameters of the lattice on








NACA TM 1393


the mechanical efficiency over a wide range of Mach and. Reynolds numbers
and inlet flow angles.

The flow process of a gas through the lattices of a turbomachine is
a very complicated hydromechanical process. The theoretical solution of
the corresponding problem of the unsteady three-dimensional motion of a
viscous compressible fluid presents great difficulties. A good approach
to the solution of this problem, as in general to the solution of most
technical problems, consists of the investigation of simplified models
which retain most of the essential characteristics of thle actual process.
Succeeding analyses then develop the effect of secondary factors.

At the present time the most highly developed theory is that of the
steady two-dimensional flow through the lattice of an ideal incompressi-
ble fluid. Such a flow may be considered as the limiting case of the
actual flow in, a lattice at small flow velocities (small Mach numbers,
M < 0.3 0.5) and with small effect of the viscosity (large Beynolds
numbers, Re >104 105)

Within the frame of such a simplified scheme it is possible to es-
tablish the fundamental characteristics of a potential flow in a lattice.
However, the solutions obtainable with these limitations require an es-
sential correction. The effects of the viscosity and of the compressi-
bility must be evaluated by theoretical and experimental methods. The
results of other tests permit evaluating certain features of the three-
dimensional flow in lattices and obtaining the characteristics of the
lattices required for the thermodynamics computation of the stages of the
turbomnachine.

Let us consider several features of a plane potential flow of an
ideal incompressible fluid for the case of the flow over the blades of
a reaction turbine (fig. 7-3). On account of the repeated character of
the flow, it is sufficient to study the flow in a single interblade pas-
sage or the flow about a single blade. In figure 7-3(a) the continuous
curves represent the streamlines V = constant the dotted curves repre-
sent the equi~potential lines &t = constant, normal to the stream~lines.
A sufficiently dense network of these lines gives a good characteriza-
tion of the flow. The velocity c at any point of the flow is equal to

c ~ = -- (7-1)

where S and n are the curvilinear distances along the streatmlines
and equipotential lines, respectively.








RA~CA TM 1393


The differentials may be approximately replaced by finite incre-
ments, and we thus obtain

c = -- ---

If SQ = L67= constant at each point, then aS = an. In this case, the
individual elements of the orthogonal network of lines, ( = constant
and Y = constant, become squares in the limit (as AS 0 anld 1Cn 0).
The flow network of an ideal incompressible fluid therefore is termed a
square network.

At subsonic velocities, the losses in available energy are produced
by the effect of viscosity, by periodic fluctuations of the flow, and by
the high degree of turbulence of the flow. When the velocities are near-
ly sonic or when they are supersonic, the losses are caused by the irre-
versible process of the discontinuous energy transformation. The magni-
tude of the losses determines, to a large extent, the mechanical effi-
ciency of t~he turbomachine.

The bodograph plane (fig. 7-3(b)) provides another important method
of representing the flow. At each point along a streamline or equipo-
tential lines (fig. 7-3(a)) the velocity has a definite magnitude and
direction. When these velocity vectors associated with a given stream-
line or equipotential line are drawn from a common origin and their ter-
mini are connected (fig. 7-3(b)), the corresponding streamline or equi-
potential line is established in the hodograph plane. The streamlines
and potential lines thus drawn also form a square network. This network
may now be conceived to represent a flow in the usual sense. The stream-
lines that originally represented the blades are the boundaries for the
new flow. The new flow itself is produced by a so-called vortex-source
and a vortex-sink. The vortex source is located at the end of the ve-
locity vector c1 (the velocity at an infinite distance ahead of the
lattice). The vortex-sink is at the end of the vector c2 (the velocity
at an infinite distance behind the lattice). The origin 0 and the
termini of c1 and c2 form the velocity triangle of the lattice.
From the equality of the flow rate ahead of and behind the lattice,

clt sin Pl = c2t sin P2

it follows that the projections of the velocities c1 and c2 on the
normal to the axis of the .lattice are equal or that the straight line
passing through the ends of the vectors cl and c2 in the plane of
the bodograph is parallel to the axis of the lattice. Considering the
velocity hodograph of the lattice, we may arrive at the conclusion that,
at points on the suction surface of the blade where the tangent to the
blade surface is parallel to the upstream and downstream. flow directions,
the corresponding velocities should be greater than c1 and c2,
respectively.








NACA TM 1393


Of great interest is the distribution of the velocity or pressure
on the surface of the blade. Figure 7-3(c) shows the approximate dis-
tribution of the relative velocities c = c/e2 and relative p~ressures

7 = (p p2 pc2 = 1 c as a function of the distance S along
the profile. If the magnitude c1 and the direction Pl of the veloc-
ity at infinity ahead of the profile are known and also the position of
the point of convergence of the flow 02 (at the trailing edge), the
flow through a. given lattice is determined. In the case of an ideal in-
compressible fluid, a change in the magnitude of the velocity c1 does
not alter the shape of the streamlines or equipotential lines. Neither
does it alter the relative velocity F or the relative pressure p

At finite distances from the lattice, the field of velocities and
pressures is not uniform. The streamlines (fPor Bl $ 900) are wave
shaped, and their shape is generally different from that at infinity;
moreover, it periodically varies along the cascade axis. In correspond-
ence with the conditions of continuity and in the absence of vorticity,
the mean velocity along any line ab (fig. 7-3(a)) between, two points
steparated by an integral number of periods t of the lattice is equal
to the velocity at infinity. One of the streamlines approaching the
leading edge of the profile actually branches at the leading edge. At
the branching point 01 (also called the entry point) the velocity be-
comes equal to zero and the pressure is at a maximum. Starting from the
branch point, at which S = 0 (fig. 7-3(c)), the velocity along the pro-
file sharply increases. Depending on the shape of the leading edge and
also on the direction of the inlet velocity (inlet angle Bl), the ve-
locity near the branch point may have one or two maxima. At the con~vex
side of the profile the velocity is on the average greater, and, the pres-
sure less, than on its concave side. The general character of the veloc-
ity distribution over the profile may be evaluated by considering the
width of the interblade passage and the curvature of the profile contour.
In particular, a narrowing of the -passage, characteristic of a turbine
lattice of the reaction type, leads to an acceleration of the flow; in
an impulse turbine having approximately constant passage width and curva-
ture, the velocity and pressure change only slightly in the direction of
flow (fig. 7-4); in a compressor le~ttie, the interbladfe passage widens
and the velocity correspondingly decreases (fig. 7-4A).

An increase in the curvature of the convex parts of the blade leads
to an increase in velocity, and vice versa. For a~discontinuous change
in curvature at the points of junction of ares of circles, for example,
the theoretical curves of the velocity and pressure distributions have
an infinite slope. At projecting angles of the profile, the velocity
theoretically increases to infinity, while at internal angles it drops
to zero.








NACA TM 1393


In view of the fact that these characteristics in the distribution
of the velocity can not exist in an actual flow, the blade contours of
modern lattices are designed with a smoothly changing curvature.

Near both the leading edge and a..trailing edge of finite thicknessyl
the velocityr may have one or two maxima; at the actual leading and trail-
ing edges, the velocity must drop to zero. The actual trailing edge is
the point of the-tail where the curvature is greatest. At a large dis-
tance behind the lattice, the direction of flow is determined by the
angle B2 *

Figure 7-5 shows the approximate effect of the inlet angle P1, the
pitch t, and the blade setting angle Py on the distribution of the
relative velocity over a blade of the reaction-type turbine lattice. A
change in the angle Pl (fig. 7-5(a)) causes the branch point 01 to be
displaced along the profile. The design entry angle to the lattice may
be considered as the angle for which the point 01 coincides with the
point of maximum curvature at the leading edge of the profile. In this
case maximums of the velocity at the leading edge are either absent or
are least sharply expressed. With a decrease in the entry angle, the
branch point is shifted toward the concave part of the profile, and the
velocity in the flow around the leading edge sharply increases. The
vector to the exit velocity c2 turns in the same direction as the vec-
tor of the inlet velocity; for example, on decreasing .the angle P1
from its design value, the exit angle P2 increases. It should be re-
marked that the effect of inlet flow angle on outlet flow angle is very
small in conventional turbine lattices. When the pitch t is increased
by a translational shift of the profile (fig. 7-5(b)) while keeping the
inlet flow angle Pl constant, the branch point 01 is slightly dis-
placed toward the concave part of the profile; correspondingly, the
velocity distribution at the leading edge changes somewhat. On the con-
vex side of the blade the velocity increases, while on the concave side
it decreases. The exit angle P2 increases. A change in the setting
angle of the profiles (obtained by rotating them while maintaining the
same pitch a~nd inlet flow angle) changes the exit angle P2. The change
in p2 is practically the same as the change in setting angle (fig.
7-5(c)). On, rotating the profiles in the direction of decrease of the
exit angle P2, the corresponding velocities on the 'profile decrease;
the branch point 01 is displaced toward the concave part of the pro-
file, in connection, with which the velocity distribution at the leading
edge changes in a way similar to that for a decrease of the inlet angle
8l"

1The case of an infinitely thin edge is not considered because it
has no practical significance.








NACA TM 1393


When the static pressure on a profile increases in the direction of
flow (such as in diffuser elements) the flow of a real viscous fluid may
separate from the blade. Excperinene shows that the static pressure is
constant over parts of the profile behind the point of separation. The
features of a flow with separation can be approximately taken into ac-
count in, a so-called stream model of the flow of an ideal fluid. A zone
of constant pressure is assumed to exist in this flow. At the boundary
between this zone and the main flow, the velocity is constant, at the
value which corresponds to the static pressure in the zones. In the
plane of the bodograph, arcs of circles correspond to the boundaries of
the separated zones. The radius of an a~re is equal to the velocity at
the boundary of the zone. Flow separation always' occurs at the trailing
edge of a b~lade. The separated flow region theoretically extends an in-
finite distance downstream of the lattice. For the same inlet and exit
flow angles -the velocity behind the! lattice is greater with separation
than it would be with, no separation. At the boundaries of the separated
flow region, discontinuous change in velocity would theoretically occur.
In the actual flow of a viscous fluid, infinitely large forces would
thicn be introduced which would prevent such a discontinuity from exist-
ing. In a real flow, therefore, the boundaries between the separated
region and the main flow break up into individual vortices which are
carried downstream by~ the flow. The presence cf frictional forces also
causes low pressure regions to exist in the separated region immediately
behind the edges. Beyond this region the flow is rapidly equalized;
this phenomenon leads to an increase in the pressure, decrease in the
exit angle, and losses of kinetic energy similar to the losses in sudden
expansion. The paramnetetrs of the equalizing flow are obtained by the
simultaneous application of the equations of continuity, momentum, and
energy (see sec. 7-7).


7-2. TREORE~TCAL MEHODS OF INVESTIGATIONS OF PLANE

POTENTAL FLOW OF :INCOMPRESSIBLE

FLUID THROUGH A LATTICE

There are two problems in the theory of lattices that have the
grea-test sig~nificane. One of these, termed the direct problem, con-
sists in determining the velocities of the potential, flow field through
a given lattice for a given velocity at infinity ahead of the lattice,
and a given position of the rear stagnation point 02 on the profile.
Of greatest interest is the velocity at infinity behind the lattice.
TIhe determination of these quantities may be considered as the fundamen-
ta;l object of the solution of the direct problem. The inverse problem
is that of theoretically constructing the lattice when th~e flow about it
is either know or easily determined for a given velocity triangle. Of








NACA TM 1393


practical importance is the problem of constructing such a lattice with
a velocity distribution over the surface of a profile which is rational
and which assumes small kinetic-energy losses in the actual flow.

It was remarked prev-iously that for the flow of an incompressible
fluid the shape of the streamlines, the shape of the equipotential
lines, and the magnitude of the relative velocities do nnot depend on
the absolute magnitude of the flow velocity. Moreover, for the same
boundaries, the different potential flows of an incomrpressible fluid
may be summed. For example, any flow of an ideal incompressible fluid
through a lattice may be considered as the sum of two or several flows
through the same lattice. In figure 7-6 the flow through the lattice
is represented as the sum of two flows: a noncirculatory (irrotational)
(fig. 7-6(b)) and a circulatory axial (fig. 7-6(c)). In the irrotational
flow there is no circulation of -velocity about the profile, or, in other
words, the lattice does not change the direction of the flow; moreover,
this direction is chosen such that the point of convergence of the flow
is on the trailing edge. In a rotational-axial flow the direction. of
the velocity at infinity is parallel to the axis of the lattice; the
magnitude of the circulation or the ratio Ac2/Alcl = m is chosen such
that the velocity at the trailing edge is equal to zero. Any flow
through a lattice (with the point of flow convergence~ on the trailing
edge) may be obtained by summation of the irrotational and rotational-
axial flows. In particular, the velocities at infinity ahead of anrd
behind the lattice will be equal to the vector sum. The velocities on
the surface of the profile itself will be equal to the algebraic sum of
the corresponding velocities in the irro-tational and rotational-axial
flows. If it is taken into account that the magnitudes of the relative
velocities do not depend on their absolute values in each of these flows,
it is possible to find in a simple manner two important properties of
the flow of an incompressible fluid through a lattice.

First, there exists a linear relation between the cotangents of the
inlet and outlet flow angles of any given lattice. From the velocity
triangle (fig. 7-6(a)), notice that

cot B0 cot P2 Ac2
S = m =constant (7-2)
cot so cot B1 -Ac1

where cot p1 corresponds to the angle B1 assumed in figure 7-6(a).
For a given lattice, the magnitude of the coefficient m can be com-
puted theoretically. For a lattice of flat plates in particular, the
coefficient m is related to the relative pitch t/b and the setting
angle B0 by the equation

ab m 1 73
t = 2 cos Bgare tan 1 + m cot PO + sin POln m(73











As may be seen from the graph of figure 7-7, the coefficient m
decreases with a decrease in pitch, so that the exit angle B2 ap-
proaches the setting angle B0 of the flat plates.

To any lattice of airfoils there corresponds a unique equivalent
flat plate lattice which has a coefficient m of the same magnitude,
and the same direction of the irrotational flow. The equivalent flat-
plate lattice for any inlet angle P1 has the same exit angle P2 as
the given lattice of airfoils. In present-day turbine lattices the
ratio b/t2 of an equivalent plate lattice is not less than 1.3; the
angle B0 is between 150 and 400, and the angle 81 is between 900
and 200. The magnitude of the coefficient m is not greater than
0.015; the anlgle of the velocity behind the lattice therefore differs
from the angle PO for the equivalent flat plates by no more than lo
For present-day compressor lattices this deviation may be as high as 30.

Second, the magnitude of the relative velocity on the profile of
any lattice depends linear-ly on the cotangent of the exit angle. In
fact3
Sc obu cu obu CO eu aC1
c =-= + -=-*- +
C2 c2 c2 CO c2 901 C2
Utilizing the obvious correlations (fig. 7-6a) ob O n

cu
, nc we obtain
sin 82
c =cbusin80+ ui(cot PO cot Bl1)sin P2 (7-4)

As was said, in present-day turbine lattices, 02 130 = constant, the
direction of the velocity behind the lattice differs little from the di-
rection of the irrotational flow for a wide range of inlet angles. Hence,

E I iEbu + u(cot P2 cot Pl)sin 82 (7-4a)

2NACA note: This ratio is written astbinoinate.

3NACA note: obu is the irrotational flow, fig. 7-6(b), and cu
is the circulatory flow, fig. 7-6(c).


[CO sin P27 mel (cot P0 cot P1)
= and =
LC2 sin POJ c2 ese P2


NACA TM 1393








NACA TM 1393


At any point of the profile where cu = O (fig. 7-6(c)) the rela-
tive velocity does not depend on the inlet angle. If the distri-
bution of the relative velocities c is known for two values of the
inlet angle D1, then the distribution c can be computed for angle
B1 with the aid of equation (7-4a).

Of practical significance in the theory of the two-dimensioh~al mo-
tions of an incomprecssble fluid is the mathematical theory of the fune-
tions of a complex variable. Without entering the mathematical side of
this problem, the discussion of which is given in any modern course of
hydrodynamics, we shall nevertheless make use of the important concept
of conformal transformation or mapping.

Conformal transformation may be defined as the continuous geometri-
eal transformation (extension and compression or conversely) of a part
of the plane regionj) in which at each point of the region, the extension
or compression occurs uniformly in all directions about this point. In
such a transformation the magnitudes of the angle between the tangents
to any two curves passing through each point of the region are preserved
as is also the shape of infinitely small figures, as is indicated by the
term conformal transformation. Exceptions may be represented only by
individual (singular) points of the region.

Every orthogonal square network in any conformal transformlation may
go over into a second orthogonal square network. This property explains
the significance of conforma~l transformation in the investigation of the
flow of an ideal incompressible fluid. Any conformal mapping of a region
of flow translates an orthogonal square network of curves cf = constant
and I = constant of this flow into a new orthogonal square network,
which mayi be taken as a network of a second flow in the conformally
transformed region with equal values of the velocity potential and stream
function at the corresponding points. The velocities of flow change in-
versely proportional to the extension at each point of the region.

In this way, the problem of determining the flow of an ideal fluid
reduces to the mathematical problem of conformally transforming the
given region into at simlpler one in which the flow of an ideal fluid is
initially known or else can be easily computed. After finding the con-
formal transformation of the points of the required region, the velocity
is computed by differentiation (c = d~fdS). Several examples of the con-
formal transformation of lattices are shown in figure 7-8.

The above defined equivalent lattice of plates is obtained by means
of such a conformal transformation in which the flow region outside the
airfoil lattice is transformed into the flow region outside the plate
lattice. The infinity of the plane of the lattice of airfoils goes over
without extension or rotation into the infinity of the plane of the plate








MIACA TM 1393


lattice. The pitch of the lattice is maintained, and the rear stagna-
trion point of the flow at the outlet edge of the airfoil goes over into
the given edge of the plates. It should be remarked that the conformnal
transformation is completely determined by the above condition. The
nloncirculatory flow through the airfoil lattice (fig. 7-8(a)) corre-
sponds to the noncirculatory flow through the lattice of plates (fig.
7-8(b)). The singular points, at which the conformality of the trans-
formtion does not hold, are the edges of the equivalent plates. Con-
sidering the corresponding nloncirculatory flows about the equivalent
lattices of plates and airfoils, we note that the length of the equiva-
lent plates, for equal pitch of the lattices, should be greater than the
half perimeter of thie profile. This property permits the parameters of
the equivalent plate lattice to be approximately evaluated.

A clear picture of conformal transformation may be obtained in the
following manner: The flow region of the lattice is assumed to be a
plane in which an ideally elastic film is stretched without friction
over the contours of the profiles and on which is drawn the network of
lines 9 = constant and V = constant of any flow through the lattice.
This film may then be stretched over the contours of any lattice which
can be a conformal transformation of the given one. In the transition
all the points of the film are displaced in a definite manner, both
along the contours and in the flow region. The correspondence of points
in a conformal transformation is thus achieved. The network of lines
4! = constant and I = constant of the flow through one lattice goes
over into the network of the same lines of the equivalent flow of the
other lattice.

Of great significance is the conformal transformation of a lattice
of airfoil profiles into a lattice of circles (fig. 7-8(c)). In con-
trast to the equivalent network of plates, characterized by two param-
eters (tlb and B0)> the equivalent network of circles is determined
by onlyl one parameter, the relative diameterr (density of the lattice)
2r/t = 27. As a result, lattices of profiles corresponding to different
equivalent lattices of plates can have one and the same equivalent lat-
tice of circles. The point 02 in the circle lattice is not uniquely
determined by the relative diameter, however.

An example of the conformal transformtion of the region of flow in
one period of a profile lattice into a bounded region is shown in figure
7-8(d). Infinity ahead of the lattice corresponds to the center of the
circle ("1); the infinity behind the lattice corresponds to a certain
point on the horizontal radius ((*2); the flow lines in a period to a
segment between the points *1 and *2. As in the case of the equiva-
lent lattice of circles, the region of transformation is characterized
by only a single parameter, the ratio of the distance between the points








NACA TM 1393


=1 and *D2 to the radius of the circle. For modern turbine lattices
this ratio is generally greater than 0.99. The points corresponding to
the uniformly arranged points of the profile contour are very irregularly
arranged over the circumference of the circle; the greater part of the
circle corresponds to practically only the leading edge of the profile,
while the remaining part of the profile contour becomes a small are near
the point *2. In a conformal transformation of the type considered (in
which an infinite distance from the origin in one flow field is only a
finite distance from the origin in the other) the displacement of a pitch
ahead of or behind the lattice corresponds, respectively, to a passage
around the point "I or "2. The flow about the lattice is transformed
into a flow of a special form produced by a vortex source at the point
"1 and a vortex sink at the point "2. TIn the regions of the conformal
transformation considered, the lattices are relatively simply determined
by the potential flow of an incompressible fluid.

The problem of the flow about a lattice of plates was first solved
by S. A. Chaplygin (in 1912) and then by the more simple method of
N. E. Joukowsky. Their work laid the foundation for the theoretical in-
vestigations of the flow about bydrodynamic lattices. Approximate meth-
ods of determining the flows about lattices of circles were worked out
by N. E. Kochin and E. L. Blokh. An exact solution was given by G. S.
Samoilovich. B. L. Ginzburg constructed tables of values of the velocity
potential and the velocities on a circle as functions of the central
angle 8 for transverse, longitudinal, and purely circulatory flows
about lattices of circles with values of the spacing 27 = 0.20 0.90
(for circles in contact 2F = 1.0). By summing the flows considered,
any flow through a circle lattice can be obtained (fig. 7-9). The values
of the velocity potentials and the magnitudes of the velocities on a cir-
cle are obtained by summation from tabulated values multiplied by certain
constants, the magnitudes of which are found from the given direction of
the velocity at infinity ahead of the lattice and the condition of zero
velocity at the branch points of the flow given on the circle. By making
use of the solution for the lattices of circles, the solution of the di-
rect problem, that is, the determination of the velocity on the surface
of the blade in the given lattice for given inlet angle, reduces to the
problem of obtaining an equivalent lattice of circles and then obtaining
a conformal correspondence of the points of the blade contour in thne lat-
tice with the points of the circle in the equivalent circle lattice.
The analogous problem of the mapping of the outside region of a single
blade on a circle has been well studied and at the present time presents
no essential difficulties. For a lattice of blades the problem is more
complicated. An approximate solution of this problem has been given by
N. E. Kochin starting from, the known conformal correspondence of a- single
profile and a circle. The method of Kochin, however, is suitable in
practice only for lattices of small spacing.








MACA TM 1393


The exact solution obtained by G. S. Samoilovich may broadly be de-
scribed as follows. First, by one of the known methods, a conformal
transformation is obtained which maps the exterior of a single circle
into the exterior of a single profile (fig. 7-10(a)). Then, from the
condition of conformal correspondence of t'he exterior of the lattice of
profiles and the exterior of the lattice of circles, the spacing of the
equivalent lattice of circles 2Fx (fig. 7-10(b)) is obtained. The spac-
ing 2S depends on the pitch of the profile lattice and the angle at
which they are set. In the example considered, 2F =: 0.85. When the
blades are more closely spaced by decreasing the pitch or rotating them,
the spacing density of the equivalent lattice of circles increases. The
flow is then related to the flow about a unit circle. For determining
the velocity distribution on. a profile there is computed the displacement
function LAB equal to the difference in the central angles of points on
a unit circle and on a circle in the equivalent circle lattice corre-
sponding to the same point of the profile. The displacement function
aB determines the correspondence of points of the profile in thle pror-
file a~nd circle lattices. By maing use of previously computed values
of the velocity potential or the velocity on the circle, the velocity
distribution on a profile of the lattice is determined for any given in-
let angle Pl'

In figure 7-11 a comparison is shown of the experimental and theo-
retical distribution of the non~dimensional pressure over the profile
of a lattice for the example considered with 01 = 900. The experimental
values p were obtained by measuring the pressure in the middle section
of the experimental blades at small air velocities. The scatter of the
test points for different M2 numbers is found to be within the limits
of accuracy of the measurements. There should be noted the characteris-
tic divergence between the experimental and theoretical values of P on
the back of the blade, produced by separation of the flow.

The velocity-at each point of the blade in a lattice differs from
the velocity at the same point of an isolated blade (for equal magnitude
a~nd direction of the velocity of the approaching flow and the same rear
stagnation. point 02); first, because of the difference in the distribu-
tion of the velocity potential on a circle in a lattice of circles and
an isolated circle; and second, because of the displacement of the cor-
responding point on a circle in. the circle lattice.

The use of the method of conformal transformation permits determin-
ing the velocity distribution on a profile of a lattice for any inlet
angle Pl whenever one flow about it is known. Suppose, for example,
there is known the distribution of the velocity potential 9 on a pro-
file of the lattice with pitch. t = 1, for irrotational flow with inlet
angle Pl = 900 and velocity at .infinity el = c2 = 1 (fig. 7-12(a)).








NACA TM 1393


This is sufficient for obtaining the equivalent lattice of circles and
the correspondence of the points of the profile in the lattice with the
circle in the circle lattice. Using the tables of distribution of the
velocity potential on a circle for the corresponding flow about the lat-
tice of circles makes it possible to construct the difference in poten-
tial L#12 at the forward and rear stagnation points as a function of
the lattice spacing with t = 1 and cl = c2 = 1 (fig, 7-12(b)). The
value of a412 in the circle lattice coincides with the same potential
difference in the profile lattice for the single value of the spacing
2r/t characterizing the equivalent lattice of circles (fig.. 7-12(c)).
The conformal correspondence of the points of the profile and the circle
is found by equating the known velocity potentials Q, on the profile in
the lattice with those on a circle in the equivalent circle lattice (fig.
7-13). For determining the velocity distribution on the profile for any
inlet angle B1, it is necessary to determine, by emsploying tables of
flow about circle lattices, the distribution of the velocity potential
Sor velocity ok on a circle in the circle lattice. The proper inlet
flow angle P1 must be used, and the rear stagnation point of the cir-
cle must correspond to the trailing edge of the profile (fig. 7-12).
From the known correspondence of the points of the profile and circle
in the lattices it is possible to construct the velocity potential as a
function of the length of are of the profile, the differentiation of
vbich will give the required velocity distribution over the profile of
the lattice (c = d@/dS). With the described method of determining the
velocity, the number of operations of differentiation is equal to the
number of inlet angles for which the velocity distribution is determined.
Repeated differentiation may be avoided if use is made of the formula
d9 dcp de de
c = dS = de dS = Ck dS

The velocity ck on a circle of the lattice of circles is determined
for any inlet angle with the aid of tables, and the derivative de/dS
is obtained only once from the graph shown in figure 7-13.

If the distribution of the velocity c on a profile of the lattice
is known, then to determine the conformal correspondence it is necessary
first to find the velocity potential

# ~S cdS

where it is assumed that S = 0 (or cp = 0) at the branch ~poinrt.

Practicallyl, for lattices with the spacings that are actually em-
ployed in turbines, the above problem is solved considerably simplified








NACA TM 1393


by the method of conformal mapping of the lattice, not on a lattice of
circles but on the interior of a circle (fig. 7-8(d)). In this case,
it may be approximately assumed that the sink ("2) is situated on the
circle, and the velocity at each point of the profile computed for any
inlet angle Pl by the formula


sin Bi C;
e' = -c
si2 B 1 c

in which the angle in the circle 8 is determined graphically from the
equation
1 8 9\
#= Clt sin &\l cot Bl In sin 2

The primes denote the magnitudes determined for a new inlet angle (8l '
At the branching point the velocity potential #p' = 0 and 9 = 2pl'

The conrverse problem of the theory of hydrodynamnic lattices, as
already stated, consists in the theoretical construction of lattices
satisfying definite conditions. In the construction of theoretical lat-
tices, there is generally given the velocity potential of the flow, and
there is then obtained the shape of the profile that corresponds to it.
The methods of theoretical lattices (like the methods of theoretical
profiles in airfoil theory) permitted determining, in a sufficiently
simple manner, the effect of the individual geometrical parameters of
airfoil lattices of certain special shapes on their hydrodynamic char-
acteristics. A classical example is the previously mentioned dependence
between the inlet and outlet angles for a lattice of plates. Moreover,
the methods of theoretical lattices up to the present time make use of
certain approximate devices for solving the direct problem.

After sufficiently effective general methods of solution of the di-
rect problem have been worked out, artificial devices for constructing
theoretical lattices have to a considerable degree lost their practical
significance. Of some practical interest, however, are those methods of
constructing theoretical lattices that assure obtaining hydrodynamically
a suitable velocity distribution on the profile and correspondingly
small losses of the actual viscous flow of a compressible fluid about
the constructed lattice.

The losses of kinetic energy in the flow of a real fluid (as com-
patred with an ideal fluid) about a lattice may be determined with the
aid of the boundary-layer theory, if the theoretical distribution of
the velocity on the profile is known.








NACA TM 1393


With account taken of what has been said, of all~ possible velocity
distributions, the most suitable hydrodynamicall~y may be considered that
for which the losses in friction. are a minimum and the condition of con-
tinuous flow is satisfied over the entire profile. (See section 7-6.)

Any continuous velocity distribution having a minimum number of
diffuser parts and a minimu velocity on the concave side of the profile
may be considered as practically suitable.

One of the simplest methods of constructing theoretical lattices
that permits satisfying a number of conditions with regard to the veloe-
ity distribution is the method of the bodograph. This method was first
applied to problems of the flow about lattices by N. E. Joukowsky, who
in 1890 considered a case of the flow about a lattice of plates with the
stream uniting at their edges. The possibility of ap-pl~ying the hodo-
graph method for constructing lattices with hydrodynamically suitable
velocity distribution was pointed out by Weinig. A practical applica-
tion of the bodograph method was obtained by L. A. Simonov, who employed.
it for constructing theoretical profiles and lattices.

The construction of lattiees by the method of the bodograph is
based on the fact that the region of flow through a lattice of' an ideal
incompressible fluid is conformally transformed into another region in
its velocity hodograph (see fig. 7-3). As has already been said, to the
flow about a lattice in the region of the hodograph there corresponds a
special flow of an ideal incomrpressible fluid produced by a vortex
source at the end of the vector el and a vortex sink at the end of the
vector c2 (see fig. 7-3). Taking into account that to a displacement
by ~a pitch ahead of or behind the lattice there corresponds a passage
around the vortex source or sink, we can determine the flow rate of the
source or sink,

Ql = Clt sin Bl = 2z = c2t sin B2

the circulation of the vortex source,

r1 = Clt ~cos 8l

and the circulation of the vortex sink

r2 = c2t cos B2

At the branching point 01 and the rear stagnation point 02, the veloc-
ity is equal to zero. Hence, the corresponding points of the flow in
the region of the hodograph coincide with the point c = 0. For con-
structing the lattice, there are given the vectors 01 = c2 and the
contour of the bodograph enveloping these vectors.








NACA TM 1393


Let us consider in greater detail the procedure of constructing the
stream flow through a lattice (fig. 7-14). It should be remarked that
the direct problem of determining the flow through a given lattice withh
no rear stagnation points in the stream) has no effecti-ve solution, and
the method of the bodograph is practically the only one which permits
constructing such flows.

The contour of the hodograph of the flow through a lattice with.
conrvergenlce point of the streamn at the trailing edge (fig. 7-14(a))
passes through the point e =- 0 and through the end of the vector e2.
The are S1 2 corresponds to the boundaries of the flows between one
infinity and the other in the plane of the lattice. In the case consid-
ered of a turbine lattice for a given hodograph, the absence of diffuser
parts on the profile may be assured (fig. 7-14(d)).

To construct the lattice, it is necessary to find the flow of an
ideal incompressible fluid in the plane of the bodogra~ph, because of a
vortex source at the en~d of the vector cl with circulation

r = elt cos Bl

and a sink at the end of the vector c2. The flow rate from the source
and sink is

Q = clt sin F3l

The magnitudes of the velocity and the nondimensionatl magnitude z (see
fig. 7-14(b)) are connected by the equation of continuity (see see. 7-7)

clt sin Pl = (1 7)c2t sin 82, where 7 =

For constructing the profile, it is sufficient to find only the
distribution of the velocity potential 4P over the contour of the bodo-
graph by the method, for example, of conformal transformation of the
hodograph into the interior of a circle (fig. 7-14(b)) for which the
vortex source goes over into the center of the circle and the sink into
the point of the circle 6 = 0. The conformal transformation of a given
hodograph may be determined by some method of numerical mapping: or with
the aid of an electrical analog.

The velocity potential of the flow on a circle is, in the case con-
sidered, expressed by the simple formula

= F -Q In sin








NACA TM 1593


At the branch point 0 of the flow, d9/dB = or cot 90/2 = T/Q,
whence
80 = 2P1
The coincidence of the branch point 0 in the hodograph plane with the
point c = 0 is equivalent to the conformal correspondence of the point
c = 0 and the point 6 = GO. With the contour of the. hodograph arbi-
trarily given, the branch point in the hodograph plane will not, in gen-
eral, coincide with the point c = 0. The coincidence of these points
is assured, however, by a suitable specification of the shape of the
hodograph. In the example of figure 7-1_4, this coincidence was obtained
by choosing the length of the segment P of the hodograph plane (fig.
7-14(a)).

After determining the velocity potential on the hodograph contour,
the profile is constructed by graphical integration of the expression

dS = d&&/c

The accuracy of the computations and of the construction is checked by
comparing the given, and obtained boundary conditions cr. The neighbor-
ing profile of the lattice is at the pitch distance t (fig. 7-14(c)).

The velocity distribution over the profiles of the constructed lat-
tice for given inlet angle corresponds to the given hodograph. The ve-
locity distribution for any other inlet angle can be found simply. For
this it is necessary to make use of the known conformal transformation
of the region of the hodograph on the interior of a circle. Since the
bodograph is, in turn, a conformal transformation of the flow region
about the constructed lattice, the conformal correspondence of its exte-
rior and interior on the circle is known. The change in the velocityi
potential 9, accompanying a change in the direction or magnitude of the
velocity, is obtained in the circle as the change in the velocity poten-
tial of the flow due to a vortex source and sink with the changed
strengths
r' r C't cos p', Q' = elt sin B'

With the aid of evident substitutions and transformations we obtain

d@' dcp' d@' de rl' -Q' cot Z
C' c = C =c
ds d@r de drf 8
T Q cot 2


cot pl-cot~ elsina pl sin-'001

cot Pl cot e lsin Bl sin 2 1 01l


where the primes denote the changed quantities.








NACA TM 1393


We emphasize that formula (4t), with change in. inlet angle Pl, de-
termines the magnlitude of the velocity on the boundaries o~f the con-
structed flow with "solidified" streams passing off to infinity. Al-
though the exit angle P2 evidently does not change and the velocities
at the boundaries of the stream zones are no longer relatively constant,
the previously mentioned change in the exit angle in lattices of vari-
able spacing and the change of velocity near the trailing edge are neg-
ligibly small. With account taken of these remarks, formula (4) permits
computing with sufficient accuracy thet velocity distribution on the pro-
file of any lattice with change in the inlet angle if the velocity dis-
tribution for any one inlet angle is known. The exact solution of this
problem (by obtaining the equivalent lattice of circles) has been de-
scribed. The application of formula (4t), in view of the evident advan-
tage of simplicity of the computations, is justified in practically all
cases where it is possible to neglect the effect of the inlet angle P1
on the exit angle B2. For computing the velocity distribution for sev-
eral inlet angles Pl formula (r) can be applied only once, and then
the linear dependence of the relative velocity c/c2 on cot P1 must
be employed.


7-3. ELEC~TRO-HYDRODYNAMIC ANALOGY

The distribution of the velocity potential in a lattice of airfoils
for any irrotational flow about it may be experimentally obtained by the
method of electro-hydrodynamic analogy (abbreviated EHD1A). This method
was first applied to problems of the theory of hydrodynamic lattices by
L. A. Simonov. Uhtil a general method of solution of the direct problem
has been worked out, the method of EHDA is practically the only one which
permits determining the flow about any arbitrary lattice with sufficient
accuracy.

The EHDA method is based on the formal analogyr between the di~ffer-
ential equations which are satisfied by the velocity potential for the
flow of an ideal incompressible fluid and by the electric potential for
the flow of an electric current through a hom~oge~neous conductor or semi-
conductor. By making use of this analogy, the theoretical computation
of the velocity potential is replaced by the direct measurement of an
electric potential.

The simlest and most widespread method of applying the EBDA is the
following: A flow of an electrical current, analogous to the flow of an
ideal incompressible fluid, is produced in a layer of water of constant
thickness (10 to 25mm). The water is poured into a flat vessel (gener-
ally of rectangular shape) of nonconductive material. The electric cur-
rent passes between the electrodes 1 arranged at opposite edges of the
vessel (fig, 7-15). A small quantity of salt and carbonic acid which is








NACA TM 1393


contained in the water assures sufficient conductivity. For avoiding
the polarization of the electrodes in the electrolysis of the water, a
low-frequency, variable current generally using a circuit voltage of
110 or 220 volts alternating current) is connected to the electrodes.
The blades of the lattice are made of an insulator material, such as
paraffin or plastiline. Several blades of the lattice are studied; for
all practical purposes, it is sufficient to study five blades. The
measurement of the electric potentiaals in the bath is generally made by
the compensation method. To the parall1el current-conducting electrodes,
a voltage divider potentiometerr) is connected, the movable contact of
which is connected, through a zero current indicator (null indicator),
to a feeler or probe situated at the point of measurement of the poten-
tial. The probe is a thin straight~needle moving along the water per-
pendicular to its surface. The simplest and sufficiently accurate zero
indicators of an alternating current are radio earphones or a speaker
connected through a low-frequency amplifier. For the potentiometer,
there is shown in figure 7-15 a water rheostat consisting of a long ve~s-
sel filled with, water. Under the conditions of exact design and horizon-
tal position of the vessel, the electrical potentials are distributed
proportionately to its length and can be measured in fractions of the
applied voltage. To measure the potential, the moving contact is slid
along the potentiometer and the reading of its scale taken at the in-
stant the force of the sound in the ea~rph~ones attains a minimum. The
advantage of the described compensation method of measurement is the
absence of the effect of the apparatus on the absolute value of the po-
tential at the point of measurement.

Instead of an electrolytic bath; it is possible to use electro-
conductive paper. The blade shapes are then cut from the paper. In
this case a direct-current source and highly sensitive galvanometers
can be used.

The eliectro-hydrodynamic analogy may be conveniently applied to the
direct problem in theory of hydrodynamic lattices. It may be used to
establish the conformal transformation of a given lattice to the equiva-
lent lattice of circles. According to the above described method (fig.
7-12), it is sufficient for this purpose to know the distribution of the
velocity potential on a profile of the lattice for any convenient flow
about it, as for example, an irrotational flow with Plr "2 = 900
C1 = c2 = 1, and t = 1. The magnitude of the measured electric poten-
tials (fig. 7-15) must then be divided by the potential drop (measured
in the same units) over the distance of one pitch. This measurement
must be made at a remote distance from the lattice and certainly not
nearer to it than 2t.

In obtaining the conformal transformation of a lattice of airfoil
profiles into its equivalent lattice of circles with the aid of the EHDEA,








MACA TM 1393


the direct measurement of the potential distribution of the flow is con-
ducted for the case of the flow with. no circulation about the blades.
With certain assumptions, the ]ERDA method can also be applied for di-
rectlyr measuring the velocity potential anld even the velocity itself in
anly flow of an ideal fluid, including flow with stagnation point at
trailing edge. The modeling scheme is indicated in figure 7-16. The
exact form of the bounding walls (streamnlines .intersecting branch
points) may~ in principle be obtained by the method of successive approxi_-
mations; practically, however, with this method there may simultaneously
be given with sufficient accuracy the magnitude of the inlet angle andf
the shape of the bounding streamlines. For measuring the magnitude of
the velocity at any point of the flow, a probe 1 is used with two paral-
lel needles placed in a holder at a small distance from each other. One
then measures the difference in potential between the needles in the di-
rection of the straight line passing through them. In measuring the
velocity on the profile, both needles are set on the boundary of the
model in the direction of flow. For measurements in the flow, the probe
is rotated.

In concluding, we may remark that the ERDA method is employed also
for investigating the flow of an ideal gas with subsonic velocities.
For this purpose an electrolytic layer of variable thickness or a net-
work model is applied. The electrical model in the plane of the veloc-
ity hodograph permits obtaining accurate solutions without successive
approximations .


7-4. FORCES ACTING ON AN AIRFOIL IN A LATTICE; THEOREMI

OF JOUKCOWSKY FOR LATTICES

For determining the forces acting on an airfoil, we isolate a por-
tion of the f-low, as shown in figures 7-17(a) and (b). The external
boundaries of the isolated region are defined by the segments ab and do,
parallel to the axis of the lattice and of length equal to the pitch t.
The lines ab and de, strictly speaking, should be at an infinite dis-
tance from the lattice because the flow parameters along these lines are
assumed to be constant. The inner boundary of the region. is formed by
the contour of the profile.

Since the streamlines ad and be are equidistant throughout their
length, the resultant of the forces acting on the surfaces defined by
these lines are equal and opposite. The projections of the force with
which the flow acts on the profile are denoted by Pu and Pa.Th
magnitude of these forces may be determined frolm the momentumn equation.
In the direction nlormnal to the axis of the~ lattice, the change in the
momentum is equal to
m(cal ca2 9t~2 1 "








NACA TM 1393


where Pais the component of the force P in the direction normal to
the axis of the lattice; the mass rate of flow of the gas per second is
determined from the formla

m = plealt 2 ca2t

Then

a, 2 [Pca2 P1cal) + 2 91) (7-5)

The projection of the force P on the axis of the lattice may be ex-
pressed by the equation

Pu= "Pleal(cul cu2) (7-6)

The forces Pu and Pa refer to a profile having a unit span.

Equations (7-5) and (7-6) may be represented in another form. by ex-
pressing the forces 'Pu and Pa in terms of the circulation r and
the flow parameters at the inlet and outlet of the flow.

According to the equation of continuity,

P1cal = P2ca2 = pea

where p is the mean density of the gas.

The velocity ca is chosen such that

cal +.ca2
ca = 2

it is easily shown that we then have
2pl 2
p= (7-7)
P1 2 P

The circulation about the profile is equal to

r = t(cul cu2) (7-8)

since the line integral along the equidistant lines ad and be are equal
and opposite.







24 NACA TM 1393


(76After simple transformations, we obtain fronm equations (7-5) and


Pa. 2 tP 1 pea(cal ca2)]


(7-9)


(7-10)


Pu = plea


We make use of the equation of energy


2
e2
7


k P1
k-1l qi


k P2
k-1 p2


Since


2 2 2
1l Pal + ul


c2 = e2 + c2
2 a2 u2

2 2
cl e
2 =ca(cal- a2) + Cu(cul cu2)

where
Cul + Cu2
cu I 2

and we obtain from the equation of energy


eqcael ca2) k- -


Substituting this expression in equation (7-9) an~d taking into account
formula (7-8) we obtain


(7-12)


(7-13)


Pu = p~ea


The force Pa given by expression (7-12) is conrveni~ently :represented in
the form of a sum of two forces


Pt = Pa + Ci0


(7-11)


- cu(cul cu2)


Pa 9 91 "k p + preu








NACA TM 1393


where


Pal = preu


and


(7-14)


The resultant of the forces Pal
and the over-all resultant force by P


and
(see


Pu we will denote by Py
fig. 7-17).


It is evident that


P~ = P + P~

y u +al


The force Py is determined by the formula

P r= JP2 pal

Substituting the values Pu and Pal we obtain


Py = pr e2~ + C2


But


2 2 2
cu + ea=


where c is the mean vector velocity

cl + c2
C =2

Hence, the expression for Py in the flow about a lattice has the
same form as the lift force of an isolated airfoil:


PY = pre


(7-15)


riPa = 92 k Pl- -








RIACA TM 1393


The direction of the force Py is perpendicular to the direction
of the mean vector velocity c. This follows from the obvious equation
c P
a u
tan. B -= --
Cu Pal

Thus, the Joukowsky force acting on an airfoil in a lattice is
equal to the product of the mean density of the gas and the velocity
circulation about the airfoil and the mean vector velocity. The direc-
tion of the force Py is determined by the rotation of the velocity
vector c by 900 in the direction. opposite to that, of the circulation.

We recall that the mean density p corresponds to the mean speci-
fic volume; that is,




Thus we have established that, in contrast to the isolated profile, the
resultant force acting on the profile in a lattice is equal to the sum
of the Joukowsky force (P ) and the additional force (aP,) perpendicular
to the axis of the lattice:

P = Py + LiPa

It is important to note that the characters of the forces Py and
nAP are different. Whereas the force Py depends on. the circulation
of the flow andd becomes zero for r = O, the force L1Pa does not depend
directly on the circulation.4

The force acting on the profile was determined for the general m~o-
tion of a gas. With the aid of the obtained relations it is not diffi-
cult to investigate the magnitude of the aerodynamic force for certain
special cases. Thus, for example, in passing from the lattice to the
isolated profile it is necessary to increase the pitch of the lattice to
an infinitely large value. At an infinite distance from the profile thie
equations p2 r p1 and p2 p1 must be valid; hence, LYP, = O ancd
Pu3 = In the case of isentropic flow about the isolated profile, the
the resultant force acting on the profile is therefore equal to the
Joukowsky force
P = Py = pfe

4NACA. note: This result is at least partially dependent on the
selection of the mean velocity and mean denlsity.








NACA TM 1393 27


where rj and c are the density and velocity of the flow, res-pectivel~y.
The direction of the force is perpendicular to the direction of the ve-
locity of the approaching flow.

Passing to the case of the flow of an incomrpressible fluid about a
lattice, it must be observed first of all that in equation (7-14) the
second term on the right side is proportional to the change of the poten-
tial energy of the flow (with account taken of the hydraulic losses);
that is,
2 2
Ic- Z 1= el 2C2


In this case of an incompressible fluid, pl n p2 m p, and the energy
equation gives
2 2
C1 C2 P2t p1
2 p

where p2t is the theoretical pressure in the absence of losses. Hence,

aPa = -t 92t -92)

The pressure difference p2t = p2 is equal to the pressure loss in the
lattice

Pit P2 =C apn
and

aPa = -thPn

Thus, in the case of the flow of an incompressible fluid about a lattice,
the additional force is negative and is determined by the losses of pres-
sure in the lattice (the pressure loss apn should not be confused with
the pressure difference p2 1 Pi)

In the absence of losses, npn3 = and LIPa = 0. In this case the
resultant force is equal to the~ Joukowsky force


P = Py = pre








NAA M 1393


This result for the lattice wais obtained by N. E. Joukowsky in. 1912.5


7-5. FUDMNA CHARACTERISTICS OF LATTICES

For evaluating a lattice, energy characteristics are generally in-
troduced. This procedure is different from that used for isolated air-
foils. The need of energy considerations is determined by thie procedure
adopted for thermodynamic analyses. The energy characteristics permit
evaluating the effectiveness of the process of energy transformation in
the stages of the turbomachines. Th~e component forces acting on an air-
foil in. the lattice are expressed in terms of the dynamic pressure of
the flow at the inlet to the lattice or behind it. In the latter case
the formulas for determining the peripheral an~d radial forces are as-
sumed in the forn
2Pu
C' = (7-16)


[Note: C; is a coefficient.]

and
2P
C'= a (7-17)
a kp2M22b

where p2 and M2 are the static pressure and nlondimensional velocity
behind the lattice.

Analogously, the other aerodynamic coefficients Cx anrd Cy may
be determined. These are employed mainly in the computation of com-
pressor lattices.

In choosing the fundamental geometrical parameter of the lattice,
the pitch, it is convenient to employ the concept of peripheral force
determined as the ratio
P,



5The possibility of generalizing the Joukowsky theorem to the case
of the flow of a-compressible fluid through a lattice was first pointed
out by B. S. Stechnkin in 1944. The exact solution was obtained by L. I.,
Sedov in 1948. The basis of the approximate theoremn of Joukowsky for
lattices in the flow of a compressible fluid was proposed by L. G.
Loitsyanskii in 1949. The generalized theorem of Joukowsky presented in
this section for a lattice in an adiabaitic flow was given by A. N.
Sherstyuk.







NACA TM 1393 29


where Pu is the peripheral force on unit length. of the profile corre-
sponding to the "ideal" rectangular distribution of the tangential pres-
sure (fig. 7-18). Evidently, for an inlcompressible fluid (with low in-
let velocity)

Pu 1 2 EcB~c~

The magnitude Pu is determined by formula (7-6); then

t 2ca(cul cu2)
Cu = 2
C2
Noting that
Ca
c2 ia2 and (cul cu2) = ca(cot Bl + cot PZ2

we obtain finally
2 sin B2sin(Pl B2 t
C, = (7-18)
u sin Pl B

The most important of the energy characteristics of the lattice is
the efficiency defined as the ratio of the actual kinetic energy behind
the lattice to the kinetic energy that should have been available if
there were no losses,

Sp = H02/H01
or, after simple transformations
k-1

2 1 0
9 = 1 (7-19)

where pOlr P02 are the stagnation pressures ahead of and behind the
lattice and M2t is the Mach number behind the lattice in the case of
isentropic flow.

Formula (7-19) is suitable for determining the efficiency of a co~m-
pressor lattice.

The coefficient of losses of kinetic energy is defined by the obvi-
ous expression


(7-20)


t;P = 1 ~P








RACA TM 1393


The real flow at the inlet and outlet of the Ilattice is nonuniform;
the velocities, angles of outflow, and static pressures vary along the
pitch. The equations of continuity, momentum, and energy must then be
written in integral form. Thus, the equation of continuity for the sec-
tions ahead of and behind the lattice can be written in the form


plelsin Bldt = pgg~sin Bgdt


Introducing a reduced flow rate q, we obtain after elementary
transform~ations6

to9 ti ~t O9
01 1 02 2


For TO1 = T02 TO = constant, averaging of the equation of con-
tinuity gives

(p~~~q si q~p0 sin B dt


The peripheral force is in this case determined from the equation

2 2
Pu 1C1 sin B1cos B1dt pzc2 sin p2cos 82dt


or, again introducing the reduced flow rate q, we? obtain


Pu = ket r90191 1sin01t -f pO292 siln 2802d+


6NACA note:
YI P O C



7NACA note: 1 e where azis the speed of sound when c is
as#
sonic. See eq. (7-25).







NACA TM 1393


Averaging of the expressions under the integral sign gives


(P09X sin 2P)ep $ 09X sin 2Pdt

From t~he equation of energy, the temperature of the flow behind the
lattice is averaged, and the following expression is involved:

(p ~ ~ 2si 0) =p0 2sin pdt


For determining the nondimensional characteristics of the lattice,
it is necessary to formulate the concept of an ideal (theoretical) proc-
ess in the lattice for a nonuniform flow. An ideal process may be con-
sider~ed an isentropic process for which in the section investigated there
remain unchanged, as in a real process, the field of static pressures and
the directions of the velocities. According to another definition of an
ideal process, the angles at the inlet and outlet of the lattice are
equal to the mean of the angles 01 and P2 determined by the momentum
equation.

TheP average values, by the equation of momentum, of the projections
of the: velocity behind the lattice are equal to


(c7cos Bll2; ep ~ pO2422cos P2in' P2df


(c2sin B2 ep = e6 pO242 sin2 2dt

-here C is the flow rate of the ga~s through one channel of the lattice.
The mean socie is then


@2 = ae tan(7-21)
8Z~p arc apO2 2~' 2 in 22d





32 NACA TM 1393


Besides the efficieincly in the -omrputatio~ns o~f a stage, there is
employed a -oefficient of dischargle equal to t~he ratio of the actual dis-
charge to the discharge in the ideal process"


pO2 2sin 02~dt

pO1 92sin P2ect

a~nd a. coefficient of momentum (often termed coefficie~nt of v~elo!city)



9 PO1 2k~sin @2 n3


which is the ratio of the momentum of the flow: Iin the r-eal 9?J ideal
-rrocesses.

The efficiency of the lattice in a nonun~itorn llow~ is co-rpulted b:
the formula


pO4@i "2n~ d

pO1 21 ~isin 8$2 Fp


For an approximate determIination of 9 equation (7-19) ma:: be
used) substituting in it the mean Sy.nam~ic pressure F;ehind the lattice.
In the denomnina~tor of equations (7-21) to (7-24), the functions qF~t
and X2t may be approximately determined from! the pres-sure ratio
pZ~m 901, where the mean static pressure behind the lattice is


p2m pt pJ1 L a",


8The index t denotes that the parameters refer to an ideal proc-
ess in the lattice. [NACGA note: The prime in eqs. (7-22) and (7-23)
and the double prime in eq.- (7-24) are not def'ined in the text. They
denote ideal conditions for which the author claims he uses the index t.
Later in the text he does use t in q2t and h;,t to denote ideal
conditions.]








NACA TM 1393


In working up the results of tests of lattices, the local coeffi-
cients Ci> ~i, and Biare used which are defined for each streamline
by the formulas


pi """"9Wi = and Bi = 2
2t2t X2t



7-6. FRICTION LOSSES IN PLANE LATTICE AT SUBSONIC VELOCITIES

In the flow about a lattice the losses of kinetic energy produced
by friction in the boundary layer and the formation of eddies in the
wake behind the trailing edges are termed profile losses.

The part of the profile losses due to the friction may be evaluated
if the velocity (or pressure) distribution over the contour of the pro-
file is known. The determination of the structure of the boundary layer
formed on the profile, the establishing of the points of transition and
separation of the layer is an important part of the problem of profile
losses in la-ttices. The theoretical and experimental investigations of
the boundary layer in lattices permit determining to a first approxima-
tiojn the losses in friction for the continuous flow about a profile and
finding the thickness distribution of the boundary layer on the profile.

The scheme of formation of the boundary layer on a profile in a
plane lattice is shown in figure 7-19(a). Making use of the graph of
the velocity distribution of the external flow, we follow the character
of the change of the layer on the concave and convex surfaces of the
blade. On. the concave surface behind the branch point the thickness of
the layer at first slightly increases. At the points of increasing cur-
vatures where the velocity of the external flow either does not change
or drops (the diffuser region on the concave surface) the thickness of
the boundary layer increases. At these points of the profile there
occurs the transition of the laminar into the turbulent layer or even a
separation of the layer.

On the converging part of the concave surface where the pressure
drops sharply, the thickness of the boundary layer decreases and attains
minimum values at the point of departure from the profile. On the con-
vex surface, in the direction toward the narrow section, the thickness
of the layer likewise decreases, and at the points of maximum curvature
of the profile it is a minimum.

Along the convex surface in the oblique section, there is noted a
sharp increase in the thickness of the layer reaching a maximum value
at the trailing edge. On this part of the profile (diffuser part of the
convex surface) the flow as a rule has a positive pressure gradient
which may lead to separation (fig. 7-19(b)).








NACA TM 1393


The boundary layer on, the profile may be computed if the velocity
distribution of the external flow is given and the condition of the
boundary layer (whether it is laminar or turbulent) is known. The ex-
isting methods of computing the boundary layer do not take into account
the effect of the turbulence of the external flow and of strong curva-
ture of the~ profile. In designing a lattice, the factor of practically
most importance is the determining of the position of the point of tran-
sition fromn the laminar into the turbulent flow and the conditions of
continuous flow about the profile. As computations and tests have shown~,
the transition point most often, coincides with thle point of minimum pres-
sure on the profile or is somewhat shifted in the diffuser region. In
those cases where the flow is strongly turbulent or when local regions
are formed in which dp/dx > O, in the converging part of the channel,
the transition point may be displaced against the flow.

The computation of the turbulent parts of the boundary layer is
conducted as a function of the character of the velocity potential dis-
tribution. In the converging parts or the parts of constant pressure,
(dp/dx 4: 0) in the case of small velocities (incompressible flow), the
momentum thickness 8** is computed on the assumption that the velocity
distribution in the boundary layer is given by an exponential laiw.

In the work of N. M. Markov, there is shown the satisfactory agree-
ment of the experimental data with the computed results. On figure 7-20
is given the velocity distribution in the boundary layer on the convex
surface of the blade of a turbine lattice near the exit edge.

The character of the change of the momentum thickness B** along
the blade of a turbine lattice may be seen in figure 7-21(a) and (b),
where the experimental values of 6S) are also indicated. For comput-
ing the layer, the experimental curves of the velocity distribution hC
of the external flow were used. As may be seen from the curves in fig-
ure 7-21, the results of the computation satisfactorily agree with the
test data,.

On the basis of the computational results of the boundary layer on
the concave and convex surfaces of the blade, the friction loss coef-
ficient in the lattice is computed.

The fundamental characteristics of the lattice ma~ be expressed in
terms of the known parameters of the boundary lawyer, 82 and Gj whiich
are determined at the exit edge of the blade. Denoting as before (see
fig. 7-19(a)) by u2 and p2 the velocity and density. at a point in
the boundary layer at the exit edge, and by u20 the velocity at the
external boundary. of the boundary layer in the same section (thte veloe-
ity of the potential flow), we set up the equation for the coefficient
(T of the friction losses.







NACA TM 1393 3E


The-kinetic-energy loss in. the boundary layer ma be expressed by
the equation


2 2
P2u2(u20 u2)dy


We transform this equation into the form


2
u2 u2 3 P0
u20 u2u20P2 00dy


It is not difficult to obtain


(k 1 2 \k 1
1 k + 120/
1 -klX


0
PO=gRT O


(7-25)


since e9


\"-k 1 R20
S12
k + 12
1 k -


p2 T
2 0 2


P2
PO


where


u20
20" a


u2
X2 -and


We set


an

-


1 k- X
k +1 20
1 -kl 12


2u u
2~/z 2


(7-26)


NFACA note: This presumes that the static pressure in the bound-
ary layer is that of the mainstream and that the recovery factor within
the boundary layer is unity.


1
nsb, = 2


1 $i
nby 2 1








NACA TM 1393


Then referring to equation (7-25), the energy loss may be written in the
form

1 an~t r0 3
Ah =2 2gRTO u20

Sumrming the losses on the convex and concave surface of the blade, we
obtain
1 0g -sa 3 U20 3727
LlhT rnp 2 gRTO Es2 u20en 2(aH u2 o (-7

Thet magnitude 8 has a concrete physical meaning; by analogy with
we~t seej
the momentum-loss thickness 8 6 is equal to the thickness of the
fluid layer moving with the velocity u20 outside the boundary layer,
the kinetic energy of which is equal to the kinetic energy of the bound-
ary layer.

The coefficient of losses in friction is

= Et (7-28)

where~ E=GO/2g is the kinetic energy of the flow behind the lat-
tice for the isentropic process and G is the actual flow rate of t~e!
gas through one channel of the lattice, which can be determined by the
equation

G =Gt g 92u2 0u eny + (p20u20 02u boldy


where p20 is the density at the outer boundary of the layer in the
section at the exit edge and Gt is the flow rate of the gas through
one channel of the lattice in isentropic flow.

The above expression may be given in the form


t (8 20 n + (8*u20 bRo O (7-29)








NACA TM 1393


In equation (7-29)



9/k+-11 2\


1- 0
- k -1 2
1 L
k + 1


(7-30)


The theoretical flow rate of the gas may be determined by the


formula


1
k k+ 1 pk-1


Ot = o~tc~tt sin 82 = 1 (


PO
RTO ta t sin P2
(7-31)


Substituting expression (7-31) in equation (7-29), we obtain


(8 u20 on


k--


(7-32)


By using equation (7-27), the equation for the loss coefficient (7-28)
now assumes the form


30 n 4 0bol

k-1

)2tt sin B2 20 en z),


C =


1 ---12t


- 20 zoboz 2

(7-33)


Bearing in mind that


1
k-1
k 1 2


P0
RTO 2~ta t sin P2


G = plpGt "


u2
EE-dy
U20


- (8 u20 boJ 0
















~)( k-1




(H 82 h 0) + (H 82 X30
en bol


F~or an incompressible fluid, there may be obtained from expressions
(7-34) and (7-35)

(H***8fHtU 2 o~n+(** u bo'l
(9 = (7-36)
Cppc~tt sin P2


NACA TM 1393


formula (7-33) can be represented in the form


c n 0boZ


5T =


or








where


(7-34)


k- 1 t k 1



H=t
8IH=~~


Stt sin B2


formulas (7-33) and (7-34), it follows that
Cr is equal to


From a comparison of
the flow-rate coefficient


(HC~ 6 12) + (H+ 8 120
en bol


(7-35)


1
k 1 22 -
(~ )k-


2~tt sin P2


where


S6








NACA TM 1393


and
(r 6 u20 + (rPi 6 20)
en bot
Up =1 -C t sin 82 (7-37)

In this case (for the incompressible fluid), the values of (
and 82" are detenninetd by the formula given in, table 4-1.

The magnitudes H#S and entering equations (7-35) and (7-36)
should be determined for the turbulent and laminar boundary layers
individually.

It is evident that the values H and H adtemgiue
8and 8 depend on the velocity distribution in the boundary layer,
that is, on the flow regime within the layer and on the character of the
change in velocities of the external potential flow (the pressure
gradient dp/dx).

N. M. Markov computed the values RJHI and H~ for the turbulent
layer using the assumption of an exponential velocity distribution law
and for the laminar layer with dp/dx = 0. On figure 7-22(a) and (b)
are given the values of H and 8 frtetruetlyra
function ofR n 20 and for the laminar layer as a function of
20*

As an example, we shall determine the theoretical magnitude of the
profile losses in turbine lattices as a function of the inlet and exit
angles 81 and 82. We assume that the velocity distribution on the
profile is approximately that shown by the dotted curves in figure 7-23
for all inlet and exit angles. On the convex side of the profile
c !/'2 = 1.1 and on the concave side ebol/c2 = 0.5 approximately,
[subscripts on and bo2 denote convex and concave sides, respectively].
On this assumption, the density of the lattice B/t for each pair of
values of the angles should have a fully determined value (see sec. 7-5):
From equation (;-18)

B 2 sin P2sin(Pl 2 t
t C sin 81

where the coefficient of the peripheral force is

P,
Cu = 2 = en, +bol
P2








NACA TM 1393


enand yfbol being the me~an pressure coefficients on the convex and
concave surfaces of the blade.

jFor the assumed values of cen and ebol


2
P= 1 -e, = 021
en ~c2


tr2
and pio = 1 -
bol c2


= 0.75


that is, C, = 0.96.

Assuming further that


u20 e c2t


H ~ J =b 2


gnd Cp = 1


we can represent the friction-loss coefficient of the lattice in the
form


5T= 2
Tt sin B2


(7-38)


On the assumption of the exponential law of velocity distribution in the
boundary layer (with -exponent n = 1/7), the momentum thickness is equal
to [Note: this expression is very similar to that of E. Truckenbrodt;
cf. Sjchlicting, p. 470.]


8 = 0.0973


(7-39)


= 105, and to estimate the are
concave surfaces we evaluate


In expression (7-39), we assume Re
of the profile S on the convex and the
approximately (fig. 7-23(a)):


1 B 2 B
Se = Sbo +
en oZ sin pl 3 sin B2


(7-40O)


The graphs in figure 7-23(b), where the friction loss coefficientlO
(T is represented as a function of Pl and B2, are constructed with

10For the case of infinitely thin trailing edges the~ coefficien~t rT
is equal to the profile loss coefficient of the lattice.


3.86- 0.8









NACA TM 1393


the aid of formulas (7-38) to (7-40). The dotted curves correspond to
constant values of B/t. Notwithstanding the considerable reservations
with which the entire comrputation was made, the results are qualita-
tively well confirmed by experiment.

The friction losses depend on Pl and P2, increasing with de-
crease in these with the greatest influence exerted by Bl. For

P1 2 (, in lattices of the impulse type) the curves of equal ST al-
most pass through the normal to the straight line! P1 = P2; that is, in
this case the losses depend essentially on the magnitude of the angle
of rotation of the flow equal to

d@ = 1800 (1 2)

We may remark that the effect of Reynolds number on the friction
loss coefficient in the lattice can easily be determined by computation.


7-7. EDGE LOSSES IN PLANE LATTICE AT SUBSONIC VELOCITIES

The eddy losses at the trailing edge constitute the second compon-
ent of the profile losses in a plane lattice. The flow leaving the
trailing edges always separates. As a result of the separation there
is an interaction between the boundary layers flowing off from the con-
cave and convex surfaces behind the trailing edge; vortices thus arise
which appear at the initial part of the wake. The photographs of the
flow behind the lattice presented in figure 7-24 show the formation of
the initial part of the wake.

A large influence on the wake is exerted by the distribution of the
velocity in the boundary layer at the point where the flows from the
convex -and the concave surfaces unite and also by the difference in
pressure at these points. Along the initial part of the wake, (includ-
ing the region behind the trailing edge where a Karma~n vortex street is
formed with the usual chess arrangement of the vortices) the interaction
between the eddy wake and the nucleus of the flow unifies many properties
of the flow field behind the lattice. The static pressure of the flow
increases and the outlet angle decreases. As a result, kinetic-energy
losses arise, analogous to the losses in sudden expansion.

The parameters of the equalizing flow can be obtained by the simul-'
taneous solution of the equations of continuity, momentum, and energy.
The control surfaces shown in figure 7-25 are selected. These surfaces
are equally spaced, when measured along the lattice axis; and they en-
close the fluid involved in the study. The above equations can be writ-
ten for the following assutmptions: (a) the density of the flow changes








NACA TM 1393


little as it moves downstream (from sees. 2-2 to 2'-2'); (b) the field
of velocities and pressures are homogeneous between the wakes and com-
pletely across the section 2'-2'.

The equation of continuity can then be represented in the form

pc2(t nt)sin P2n =: egpt sin 82m


or


c2(1 T)sin P2n = e~oasin P2=


(7-41)


where


~at
t


The momentum equation in the direction of the axis of the lattice gives

2 2
c2cos P2np(t at)sin B2n = C2,,cos P2.pt sin 82,

or, with account taken of (7-41), wet obtain


(7-42)


C2cos P2n = e2,cos P2*


The momentum equation in the direction perpendicular to the lattice axis
can be written in the form

c2p sin2 2~n(t at) + p2(t at) + pkpnt = C 2p sin28~ 2. P2.


(7-43)


From equations (7-41) and (7-42) there is easily obtained


P2. = are tanC(1 T)tan P2n]


Equation (7-43) permits finding the increase in pressure behind the~
lattice


ap2= 2c22p sin2 2n(1 z) c 22p sin2p, 2* (kp P2)7
pe2


(7-44)








NACA TM 1393 43


Taking into account expression (7-41), we obtain


nP2 = 1 2 [ 2 (1 -1) sin2 2n+ kp ]T (7-45)
2pc2

For determining the theoretical velocity at infinity behind the
lattice, we make use of the equation of energy which for the assumption
made p2 = p2, = p may be represented in the form
(c, 2 2
(c2J~a C2 2 P2
S+ = ~- + (7-46)

where c2, is the theoretical velocity in the section 2'-2'.

From expression ('7-46), we obtain


-2 1 Ap2 (7-46a)

The velocity~ c2 is expressed in terms of c2, with the aid of equa-
tions (7-41) and (7-44), thus

c2


and we have


(02 1 r( ~si2 2 9p (7-47)

The coefficient of edge losses isl1

7 sin2 62 PkP7(-8
klp k1-P;; 1 a672*


11Formulas (7-45) to (7-48) given here were obtained by G. Y.
Stepanovich.








RAC TM 1393


The nondimensional pressure behind the edges entering equations (7-45)
and (7-48) is

Pkp P2
Pkp 1 2
2pc2

and it must be determined from experimental data.

With an accuracy up to magnitudes of the second order as compared
with z the coefficient of edge losses is expressed by th~e formula

Skp PkpZ

For small velocities, according to test data (see below)



From the above arguments, it is seen that the edge losses are directly
proportional to z.

According to test data, the equalization. of the flow behind the
lattice occurs very rapidly at first, and the rate of equalization is a
function of the geometrical parameters of the profile and the lattice,
a~nd is quite dependent on the thickness of the edge. The region of in-
tensive mixing ends at a distance y = (1.3 to 1.7 t) behind the trailing
edge. This is confirmed by the graphs in figure 7-26 in which are given
the results of an investigation of the wake behind a reaction lattice
according to the data of R. M. Yablonik. Figure 7-26(a) shows curves of
local loss coefficients of the wake at different distances behind the
reaction lattice. On figure 7-26(b) is shown the variation of the coef-
ficient of nonuniformity in the flow field behind the lattice. This
coefficient is defined by the formula

c a,max a,mi
2ca,m

where


ca,ma~x Band ca,min maximum and minimum values of component velocity
eg in the given section

ca,m mean value of velocity eg in the same~ section

A detailed investigation of the flow behind the trailing edge of a
reaction lattice was conducted by B. M. Takub. The results of these
tests reveal certain effects of the shape of the edges on the flow








NACA TM 1393


structure in the eddy wake. Measurements of static pressure bn both
sides of the wake show that there is a considerable nonuiformity in the
pressure field along the boundaries of the wake (fig. 7-27). Moreover,
the static pressure along the wake boundaries changes periodically.

As the flow leaves the concave surface of a blade, its pressure
must drop, while on the concave surface it mu~st increase. Further, be-
hind the principal edge vortex, the static pressure decreases on both
sides of the wake, it then again increases somewhat, and so on. Finally,
there is a complete equalization of the field of flow. From figure 7-27
it is seen also that the amplitude of the fluctuations of the static
pressure depends on the shape of the edge. By making a two-sided taper
(sharpening of the edges b and c in figure 7-27) it was possible to
decrease somewhat the nonuniformity of the static-pressure field.

The tests showed that a sharpened edge of the type b raises the
efficiency of the lattice, as compared with the normal edges, by 1 per-
cent and that an edge of type c increased the efficiency by 2.5 per-
cent (for a medium velocity of flow). It should be remarked that, not-
withstanding their high effectiveness, the forming of very sharp edges
of the type c introduces serious difficulties'because such an edge
rapidly deteriorates under actual operating conditions.


7-8. SEVERAL RESULTS OF EXPERIMENTAL INVESTIGATIONS OF PLANE

LATTICES AT SMALL SUBSONIC VELOCITIES

Systematic investigations of the effect of the geometric parameters
of the lattices on the magnitude of the profile losses at small veloc-
ities were conducted in the M. I. ]Kalinin Laboratory, the I. I-. Polzunov
Institute, the F. E. Dzershinskii Institute, and in other scientific re-
search organizations and institutes.

We shall consider as an example several results of an experimental
investigation of the effect of the pitch, the blade angle, and the angle
of incidence of the flow on the velocity distribution over the profile
of an impulse and reaction type lattice.

Figure 7-28 shows the velocity distribution over the profilel2 of
a reaction turbine according to the data of N. A. Sknar. With increase
in pitch, the flow about the back of the profile becomes impaired. Along
a considerable part of the convex surface, the pressure gradient is posi-
tive (see curve for 1 = 0.904 on fig. 7-28). In this diffusing region
a boundary layer is formed, and its thickness increases and in certain

12The local velocities are made dimensionless by dividing them by
the vector mean velocity.








mACA TM 1393


cases separates. With increasing pitch the nonuniformity of the flow
in the passages between the blades increases; the velocities on the con-
vex side increase, while on the concave side they decrease. At high
values of the pitch, the flow about a profile in the lattice approximates
the flow about a single profile (fig. 7-28).

The effect of the blade setting on the velocity distribution over
the profile is shown in figure 7-29(a). The maximum favorable velocity
distribution for a given profile is obtained at a setting angle B = 500
In this case both along the upper and lower surface the velocities in-
crease more uniformly.

A change in the inlet angle of the flow (fig. 7-29(b)) greatly
affects the velocity distribution along the profile. Large inlet angles
tend to impair the flow along the concave surface, while small angles
similarly affect the flow along the convex surface.

The investigation of an. iapulse lattice conducted by E. A. Gukasova
shows that, similar to the reaction lattice, a change in pitch.causes a
considerable change in the velocity distribution along the profile (fig.
7-30). For all values of the pitch an adverse pressure gradient is
found immdiately behind the! leading edge. The diffusing region extends
over the greater part of the concave surface, and only near the outlet
part does the flow reaccelera~te. On the convex surface of the blade be-
hind the leading edge, the flow accelerates and reaches a maximu veloc-
ity downstream of the part of greatest curvature. We note that, as for
the iapulse lattice, diffuser regions are formed near the trailing edge
of the upper surface for all regimes.

With decreasing pitch, the nonuniformity of the velocityr field in
the channel between the blades decreases. A similar trend accompanies
an increase in the inlet angle of the flow; as Bl increases, the flow
on the concave surface accelerates while the flow on the convex surface
slows down. A decrease in the inlet angle is accompanied by the appear-
ance of adverse pressure gradients near the inlet of both the convex and
concave surfaces. For inlet angles somewhat higher than the profile
angle B1nr the most favorable general velocity distribution is found.

The change of the coefficient of profile losses inl impulse and re-
action lattices as a function of the pitch and inlet; angle may be seen
in figure 7-31. The curves show that for each lattice there exists a
definite optimu pitch for the minimum profile losses. Thus, for exam-
ple, for the reaction lattice having the profile shown in figure 7-28,
the optimu pitch is to t = 0.673. For the impulse lattice,
topt 0.50-0.60.








NACA TM 1393


In spite of the favorable velocity distribution, in a closely
spaced lattice (t < topt) the loss coefficient is relatively high be-
cause of the greater losses produced by friction. Decreasing the pitch
also causes an increase in the coefficient of edge losses.

The curves in figure 7-31 show that for all pitches a decrease in
the inlet angle (below the optimum) has a sharper effect on the effici-
ency than an increase in the angle. An increase in Sp also noted for
Pl InP1 for the impulse lattice of large pitch. It should be empha-
sized that as a rule the values of the optimun inlet angles exceed the
geometric angle of the profile.

From the results of the investigations, it can be concluded that
the experimental determination of the optimum pitch must be carried out
over a wide range of inlet angles.

The tests show that the direction of the equalized flow behind the
lattice may with sufficient accuracy be determined by formula (7-44).
The familiar formula given in the literature for determining the effee-
tive (actual) angle of the flow

a2
82e = are sin (7-49)

gives somewhat lowered values of P2. More closely agreeing values of
2ep with test results are obtained by formula
a2
P2e = are sin (7-50)

At small velocities tests confirm that for all practical purposes,
the outlet flow angles of a reaction lattice depend only slightly on the
direction of the flow at inlet, that is, on the angle Pl (fig. 7-32).
The angle P2 is, however, influenced to a large extent by the pitch
and the setting angle of the profile. With an increase in Py and t,
the angle D2 increases.1

Similar results are obtained also for the impulse lattice. In
this case, however, the deviation between experimental and computed val-
ues of the outlet angles increases. According to the data of a number
of tests the outlet flow angle increases somewhat, as the inlet flow
angle increases.

13A~nalysis of formula (7-44) leads to the same results.








NACA TM 1393


Immediately behind the lattice, the field of the flow angles is
nonuniform; the angles 82 vary along the pitch (fig. 7-33). The
greatest changes in 82 are found near the boundaries of the trailing
eddy wake. With increasing distance from the lattice, the flow equal-
izes and the values of the~ local angles approach the mean value 82,*

The~ nonuniformity- of the field behind the lattice depends on the
inlet flow angle. With either a decrease or a considerable increase in
the inlet angle, the nonuniformtity of the flow at the outlet increases.
:Particularly unfavorable is a decrease in the inlet angle.

The results of numerous tests of lattices at small velocities in a
uniform weakly turbulent flow permit drawing several general conclusions
as to the character of the change in profile losses in lattices as a
function of the parameters defining the flow regime (inlet angle B1
and Reynolds numer Re) and of the fundamental geometrical parameters
of the profile and lattice.

A study of the effect of the angle of inlet flow, angle of the pro-
file setting, and the pitch for fixed values of Re shows that, in the
cases where a change in these m~agnitudes results in the formaion of ad-
verse pressure gradients on the profile, the boundary layer thickens,
and the transition from a. lamlinazr to a turbulent boundary layer moves
upstream. As a result, the friction losses increase. In certain cases
the boundary layer may separate in the regions where diffusion occurs,
a circumstance which leads to a sharp increase in the profile losses.
A decrease in the inlet flow angle and an increase in the -pitch. increases
the likelihood of adverse pressure gradients. In this connection, it
should be rema~rked that in impulse lattices the losses as a rule are
greater than, in the reaction type which are characterized by a mnore fa-
vorable (converging) pressure distribution over the profile. The above
considered tests showed that the minimum loss coefficient in an impulse
lattice constitutes about 7 percent, while in the reaction lattice it
is about 4 percent.

Changes in 81, t, and By have an, effect on the magnitude of the~
edge losses.

The effect of the Reynolds number on the efficiency of the lattice
has not yet been, sufficiently studied. The available data, show that a
change in Re has different effects on the profile losses in the lat-
tice, depending on the inlet angle and the geometrical parameters of the
lattice. If sepatration. occurs on the profile, the profile losses tend
to decrease markedly with an increase in, Re2. For nonseparating flow
about the profile, the effect of Re2 for the reaction lattice is small
(fig. 7-34).








NACA TM 1593


7-9. FLOW OF GAS THROUGH LATTICE AT LARGE: SUBSONIC VELOCITIES;

CRITICAL NUMBER FOR LATTICE

The fundamental characteristics of the potential flow of a compress-
ible fluid in a lattice at subsonic velocities is qualitatively the same
as that of incompressible flow. The network of streamnlines JI = constant
and equipotential lines 9 = constant remains orthogonal, but it is no
longer square. The velocity at an~y point of the flow is


dS p dn

and, as a result, w~hen a'P = a# = constant, then nS6n~ = p/pO 4 1. In
the plane of the bodograph, the network, of lines Qr = constant and
Jl = constant is no longer orthogonal. According to the condition of
equality of the flow rate ahead of and behind the lattice, we have

e1 It sin Bl = c2 2t sin P2

For c1 < e2, the projection of the velocity c2 on the normal to the
axis of the lattice (c2sin B2) becomes larger than the same projection
of the velocity cl. The distribution of the relative velocities
c = c/c2, in contrast to the case of the incompressible fluid, depends
on the absolute value of the velocity, or more accurately, on the Mach
number M at any definite point of the flow, for example, on M2 = c2/a2*

An approximate method of estimaating the velocity distribution over
the profile may be used to establish the characteristic regimes of the
flow about the lattice at subsonic velocit~ies. The approximate method
is based on the c-ircumstance that in modern turbine lattices of high
solidity the flow between the profiles may be considered as a flow in a
channel.1

TIhe flow velocity in an interblade passage of constant width and
curvature (fig. 7-35) can be determined in a particularly simple manner.
A comparison with more accurate theory shows that for a perfect gas the
velocity distribution across the channel approximately satisfies the
equation
en
e = -Ri- cen (7-51)


1The method considered, proposed by A. Stodola, was developed sub-
sequently by C. Y. Stepanov.








MACA TN 1393


and in. particular


Ren
CboZ = R~boZ Cp


The velocity on the conlvex side of the profile 0
mined from the equation of continuity


can be deter-



(7-52)


cl01t sin Pl = i o
on


ep dR


In equation (7-52) it is convenient to transform to the nondimensional
functions q and X


q dB


Using expression (7-51), we obtain finally


Xbot
q d = XenRnI I
on,


(7-53)


gives (the


(7-54)


Computation of the integral I: for small subsonic velocities
constant of integration is omitted)

Il = ml In X


where


k + k-1

For a gas with k = 1.4 we obtain


Ir2 = ml o sh1m 222+


51,m '4


(7-55)


where


m2 kl


Ren


qlt sin Bl enRen


1 mX2








NACA TM 1393


For k = 4/3 we have


13 =ml In -+2 mZ2 34 m24 X' m26 / (7-56)

If the computed Function I is used, the equation of continuity can be
written in the forml5

qlt sin Pl
Ren cn Ibol- e) (7-57)

where Ten by equation (7-54) or (7-55) corresponds to hen and Ibo'l
corresponds to 1Lboz = Ren/Roa e n'

It is possible to apply the process of successive approximations
for computing Acn by equation (7-57), since the expression in paren-
theses depends little on Acn In the first approximation

(1) Xlt sin Pl
cn =b (7-58)
R en

Then, in the following approximation, Icn and IboZ are determined from
X(1)
on
,(2) I qt sin P1
en (T1) 1 (7-59)


and so forth. For Xn < 0.5 the first approximation (7-58) is suffi-
cient. The solution of equation (7-57) is conveniently represented in
the form of the graph shown in figure 7-36, which gives the magnitude
qep = lt sin 3l/(Rbol Ren) as a function of Ren/Rol for various
values of X

A critical value Q41 and a corresponding X19 or Mlay denote
critical flow in the lattice; that is, a condition where Ke .In
the curved channel for which Ren/Rot < 1, the graph in figure 7-36 in-
dicates that the maximum flow is attained for som~e X1 1 **w

15NAAnt:Sbcito refers to convex side, bol to concave
side.








NACA TM 1393


The above described method cain also be applied for finding the ve-
locity within an inte-rblade passage of variable width and curvature.
For this purpose it is necessary in the section of interest to inscribe
circles as shown in figure 7-37 and to determine their dia~meter and also
the radii of curvature Ren and R oZ at the points of tangency~ of the
circles. For computing the velocities Xen a~nd XboZ, formulas (7-584)
and (7-59) may be used substituting, for example,

Ron = Ren; RboZ = Ren + a

or
R R'
en on
Rbol Ren = a, -
Rboz Rd~

or Rbol = Rbol, Rn = R oZ a. The differences in the values of Xn
and Xbol obtained in each case characterize the error of the applied
method. As an exa:mplle, in figue 7-37 are compared the results of the
exact solution (in the flow of an incompressible fluid) with the results
of computations by the described method. The satisfasctor.; a,-reemenrt of
the values of the velocities that is observed also in the other e::rple~s
attests to the feasibility of apl.plyingS this method for preliminary
computations.16

Let us now consider flow of a gaEls through a reaction lattice when
the velocities are nearly sonic. For a, critical value of M2l = M24t at
a certain (critical) point of the profile, the critical velocity is
reached. With further increases in M2, the pressure distribution ahead
of this critical point changes little. The pressure distribution behind
the point of sonic velocity obsnges considerably. In the so-called dif-
fusing (i~e., for subsonic flow) region behind this critical point,
there is an increase in the suprsorinic~ velocity.

The experimental determination of the critical values M2w shows
that its magnitude large-tly depends on the geometric parameters of the
profile, the lattice, and the direction of the flow at the inlet. In a

16This method of computing the flow in a channel w~as based on the
approximately determination of the leng-th of the potential line and on the
assumption that the distribution alone-l it of the curva~ture of the stream-
lines differs little from the case of vortex flow. With a certain, com-
plication of the compluta~tions, this method can be rendeljrFd more accurate
by the successive refinements in estima~tintg the distribution of c~urvatLure.








NACA TM 1393


reaction lattice for an entry angle Pl InJ the values of N2, de-
crease with increase of pitch because the local velocities on the con-
vex surface at the points of maximum curvature increase. In figure 7-38
are shown the curves of maximum velocities on the back of the profile as
a function of M2. From these curves the values of M24, can be deter-
mined. For 1 > M2 > M24 on the convex side of the profile, local re-
gions of supersonic velocities are formed, the boundaries of which are
the lines of transition (M = 1) and a system of weak shocks.

Experiment shows that the supersonic zones may arise simultaneously;
in the flow region adjoining the trailing edge and the boundaries of the
wake. Because of the lowered pressure behind the trailing edge, t~he ve-
locities of the particles leaving the upper and lower surfaces (outside
the boundary lawyer) increase. This acceleration may lead to the forma-
tion of zones of supersonic velocity adjoining the boundaries of the
wake. In correspondence with, experimental data obtained at a, small
pitch, the supersonic zones are formed first at the trailing edges them-
selves and then progress to the more curved part of the convex side of
the profile in the interblade cha~nnel. For a large pitch, on the co~n-
trary, supersonic velocities arise first in the channel adjoining the
convex surface of the blade. This is confirmed by the results of meas-
urements of the pressure behind the trailing edges and of the minimum
pressure on the convex surface of the profile in lattices of various
pitches.

The critical values of the number M24t are shown in figure 7-39
for pl I~n as a function of the pitch for a reaction lattice. It
is seen from the graph that for each lattice there exists a pitch 1
for which the critical velocity is reached simultaneously on the back
and behind the trailing edge of the profile.

In an impulse lattice,17 the critical M number is lower than that
of a reaction lattice, this fact is a result of the greater curvature of
the impulse profile. Local supersonic regions in the impulse lattice
may arise, depending on the inlet angle near the leading edge, on the
convex surface and at the trailing edge.

The g-raphs shown in figures 7-40 and 7-41 characterize the effct
of the number M2 (and also M1) on the pressure distribution over the
profile for the two fundamental types of lattice. With an increase in
M2, the absolute values of the pressure coefficients increase. The
characterist-ic points of the pressure diagram (-points of minimum pres-
sure) are displaced in the direction of the flow. For small angles 01

17For the impulse lattice the critical M- number is sometimes re-
ferred to the inlet velocity.








P.ACA TM 1393


and large numbers Ml, experiment shows the displacement of the branch
point 01 along the concave surface of the profile.

The effect of compressibility shows up more markedly on the convex
surface, where the pressures change more rapidly; the -pressure grad-ient
along the convex surface increases. Correspondingly, the flow in t~he
diffusing region on the convex surface also changes. Since the minimum
pressure on the profile decreases, the pressure gradient in the ciffu~s-
ing region of this surface increases. The pressure changes particularly
sharply on the convex surface near the narrow section of the channel.
Similarly, but more sharply, the effect of the compressibility reveals
itself in the pressure distribution in an impulse lattice.

A change in the inlet flow angle at large supersonic velocities in
an impulse lattice sharply affects the pressure distribution, particu-
larly at the inlet part of the profile (fig. 7-42).


7-10. PROFILE LOSSES IN LATTICES AT LARGE SUBSONIC VELOCITIES

The results of experimental investigation permit estimating the
change in the profile losses in various lattices at subsonic and near
sonic velocities.

For M2 < M2 with increasing flow velocity, the effect of the
compressibility on the losses due to friction depends on the one hand
on the change in the pressure distribution over the profile. Increas-
ing the velocity increases the diffusion on the convex surface and,
hence, increases the losses. On the other hand, increasing the veloc-
ity changes the velocity distribution within the boundary layer itself;
and this tends to decrease the losses.

The investigation of the wake at large subsonic velocities shows
that the pressure behind the trailing edge drops with increasing value
of N~j this behavior is particular acute when the velocity is approxi-
mately sonic. In figure 7-43 is shown the dependence of -ykp on M2
for a rounded trailing edge. It is seen that with an increase in M2,
the value of 7k decreases and reaches a minimum value at
M2 "" 0.9 1.0. With a further increase in M2, the pressure behind
the trailing edge increases. The~ intensity of the vortice behind the
trailing edge and the width and depth of the wake are increased (fig.
7-44). At the same time, for M2 < 1, the extent of the smoothed out
part of the flow behind the lattice increases.








NACA TMY 1393


For an approximate estimate of 5kp at large subsonic velocities,
formula (7-48) may be employed, substituting the test values of pkp
(fig. 7-43). Thus, taking into account the fact that the trailing-edge
losses increase, with an increase in M2, the character of the change
of the coefficient of profile losses as a function of M2 is determined
by whichever of the above-mentioned factors is the deciding one. In the
final analysis, this answer depends on the geometric parameters of the
profile and lattice.

In r~eactio~n lattices the approach to near sonic velocities while
M2 < M2+ does not lead to any considerable increase in the losses if
the flow in the interblade channel is without separation.

W~e recall that the resistance coefficient of a single profile
sharpl: increases in the zone of near sonic velocities. In the flow
about a single profile, the local shock waves have a considerably greater
in~tensity~, and in many cases the flow separa-tes to the impairment of the
flow. The energy losses in the local shock waves of a lattice are not
large, and they~ evidently do not appreciably increase the loss
coefficient.

In a reaction lattice, thanks to the converging flow, the local
shock~ waves within the channel do not, as a rule lead to separation. In
those cases where the flow separates at supersonic velocities, however,
the loss coeffricient increases more rapidly with increase in M2.

Figure~ 7-45 gives Sp curves for several reaction lattices consist-
ing of' different profiles and for two impulse lattices. We note that
since the test lattices had different profiles, the dotted curves in fig-
ure 7-45 do not characterize the effect of pitch alone.

The effect of the incompressibility on the profile losses is more
msrked for inpu~lse lattices. The curves in figure 7-45 clearly confirm
this conculusin.1 It should be emphasized that, for large velocities, a
change in the inlet angle has a particularly marked effect on the loss
coefficient in the impulse lattice (p. In passing to large inlet angles
( 1 InB1), the losses in the impulse lattice decrease.

18The results of the test were obtained on an apparatus with con-
stant b~ack, pressure. With increase in the number MZ there is a simul-
taneous increase in Re2. As was pointed out in the preceding section,
the inrel~;ase in Re2 leads to a lowering of the losses. It may be as-
samned that for Re2 = constant the change of 5 a.s a function of M2
would be somewhat sharper.








AkCA 'PTM 1303'


Detailed investigations of the flow structure show that an increase
in M2 leads to an increasing nonuniformilty of the field behind the
lattice (figs. 7-46 and 7-47).

Analysis of the effect of compressibility on the flow structure in
lattices permits drawing the conclusion that the optimum pitch of the
profiles decreases a~s the velocity increases. With decreasing pitch,
the1 nonuniformity of the distribution of the flow between the blades is
reduced.

Of practical interest, is the change of the flow direction behind
the lattice as a function of M2. Tests show that for M2 4 M2+ thie
compr.r- ibili ty ha~s only a slight effect on the magnitude of the mean
angle behind the lattice. For the majority of reaction lattices, there
is first noted a certain decrease and then an increase in P2 with in-
crease in M2. For M2 > M24, the mean angle as a rule increases with
increase in M2 (fig. 7-48).


7-11. FLOW OF A GAS THROUGH REACTION LATTICES AT

SUPERSONIC PRESSURE DROPS

In conventional guide and reaction lattices, the flow velocities
at the inlet are Subsonic; the transition to supersonic velocities occurs
in the interblade passages. We will first consider the fun!-dsmental prop-
erties and structure of the flow in plane reaction lattices for super-
sonic pressure drops when

PZ2 02' >

The successive change of the supersonic regimes of the flow in a
lattice is shown schematically in figure 7-49. In the narrow zone of
an interblade pa~ssa~ge the critical velocity is established.19 Behind
the trailing edge the pressure is below critical. In the flow about the
point A (fig. 7-49(a)) the pressure drops and the: fa~n of expansion ABC
fall on the convex side of the neighboring profile a~nd a~re then reflec-
ted from it. The initial and reflected expansion of waves overexpand
the flow; that is, the static pressure behind the wave ABC is less than

19The transition surface coincides approximately with the narrowest
section of the passa~ge. Actually, as a consequence of the nonunifojrmity
of the flow in the converging part and the effect of viscosity, thne tran-
sition surface has a certain curvature a~nd is displaced upstream.








NJACA TM 1393


the pressure at infinity behind the lattice. The fulrther development
of the flow depends to a considerable extent on what -pressure is estab-
lished behind the trailing edge or AE. The bounding streamlines of the
gas leavingE the concave and convex surfaces of the profile approach each
other and are then sharply deflected a~t a certain distance behind the
edge. At the boundaries of the initial part of the wake, a system of
wJeak shocks arises which merge with the oblique shock FC, which is
formed at the points of discontinuity of the wake.

The obilqu~e shock interacting with the boundary layer on the convex
surface of thE profile is reflected20 and again impinges on the trailing
wake. Depending on the mean Mkp number in this section of the wake,
the reflected shock either intersects the wake (Mkp > 1) or is reflected
from its boundary (if Mkp < 1). Thus, the flow moving along the convex
surface of a profile successively passes through the primary and reflec-
ted exp~ansion waves and the primary and reflected shocks.

The behaavior of the bounding streamlines in passing off the edge
depends essentially on the ratio of the pressures at the point D to the
pressure behind the trailing edge. If the pressure of the flow at D is
greater than that behind the edge section, then there is formed at the
point D ant expansion wave; and the flow about the edge is improved. The
streamline leaves the profile not at point D, but at point E (fig.
7-49(a)). On account of the curvature of the wake EF and the rotating
of the f'low~ near the point E, there arises behind the expansion fan DLK
a system of welak shocks merging with the curved shock PH, which arises
at the point of turning of the boundary of the wake F. The system of
the two shocks FC and FH forms the trailing shock of the profile.

If on~ passing through the system of waves, the pressure of the flow
near the point D is below the pressure behind the edge, a shock arises
at the point D. In this case the wake increases.

On passing through the system of expansion waves and oblique shocks,
the individual streamnlines are multiple and variously deformed. On in-
tersecting the primary ra~refa~ction wave, the streamline a-a deflects,
turnin-g b:, a certain angle with respect to the point A (the angle be-
tween the tangent to the streamline and the axis of the lattice in-
creases). The reflected wave somewhat decreases the angle of deflection

20The~ reflection remains normal even at large angles of incidence
of the primarjr shock (e2 -, ), since the interaction of the shock with
the bounrdaryi layer on the convex surface occurs in the zone of negative
pressure gradients (the effect of the reflected rarefaction wave).
W-ithin a wide range of velocities, the separation of the layer in lat-
tic-es with relatively small pitch is not observed.








NACA TMI 1393


of the streamline. On intersecting the primary. shick-i, the streamline is
sharply deflected in the opposite direction (the angle of' the stream-
line with the axis of the lattice decreases). In passing through the
reflected shock CP, the angle of the streamline with axis of the lat-
tice again increases.

With an increase in the pressure drop through the lattice, the
flow spectrum behind the minimum area section changes; the intensity
and character of arrangement of the rarefaction waves and shocks change.
The extent (a~nd therefore the intensity) of the rarefaction, wave in-
creases. The angles of the primary, reflected, and edge shocks decrease.
The point where the oblique shock FC falls (point C) is displaced down~-
stream (fig. 7-49(b)). In correspondence with this, the character of
the deformation of the individual streamlines likewise changes. With
increase in E2 the mean outflow angle increases.

The expansion of the flow within the confines of the lattice ends
for a certain relation of the pressures e2 = (;S. For flow conditions
near this limiting regime, the primary shock is curved and forms a cer-
ta~in small angle with the plane of the outlet section. The exact de-
termination of the value eS is therefore difficult. The limiting re-
gime ma~y be considered that for which the primary shock falls at the
point D of the edge section (fig. 7-49(c)).

If "2 < ES, the expansion of the flow continues beyond the lat-
tice (fig. 7-49(d)). The system of shocks at the trailing edge remains
essentially as before, but the wake behind the edge is considerably
diminished. The left branch of the tail shock (the shock FC in fig.
7-49) falls in the subsonic part of the wake of the neighboring profile
a~nd deforms its boundary; the pressure behind the edge increases. The
intensity of the shock increases at the point D', and in certain cases
separation of the flow occurs on convex surface of the blade (point D').

The wake behind the edge is greatly weakened. In such regi;mes
separation is observed mainly in lattices with relatively large pitch.
It should be remarked that for E2 << tS the separations vanish as a
rule. The primary shock falls in the supersonic part of the wake (fig.
7-49(e)). The pressure behind the edge drops, and the separation on
the back is eliminated. Thus, a very characteristic property of the
regimes (F2 ( S is the interaction of the primary shock with the wakEe
at the edge.


The shock FC passing through the flow field behind the
tion sharply decreases the angle of deflection of the flow.
particularly well marked by the deflection of the wake near


outlet see-
This is
the edge.








NACA TM 1393


The abov~e conscidered schemes of flow are illustrated by photographs
of the flow spectra behind the throat and at the exit from the reac-
tion lattice (fig. 7-50). There is here seen the fundamental system of
waves and shocks, the deformation of the wake behind the edge for dif-
ferent. regirres, anld also the interaction between the waves and shocks
with the neig-hboring profiles and wakes.

The flow spectras are given for two lattices: -f = 0.543 (fig. 7-50)
and I = 0.86 (fig. :-51). The photographs show that in the lattice of
small pitch the flow is void of separation for all regimes. In the lat-
tice of large pitch (t; = 0.86), separation of the flow on the back of
the profile occurs for thel regimes E2 = 0.288 0.258. In figure 7-51
(photog-raphs (s) and (bl)) there is clearly seen the vortex structure of
the trailing- wake and the considerable nonuniformity of the flow behind
the lattice.

Figu-~re 7-52 Sivest the pressure distribution behind the throat on.
the convexr surface~ of a profile in a reaction lattice for various ratios
c2 2 0?/P 1. The~ curv:sS show the considerable nonuniformity of the pres-
sure on the back of the blade. Behind the throat section (i.e., at the
poilt~s 2 to tEl th~e e:-:pnsricon of the flow may be observed; the pressure a~t
these points is lowecJr than the pressure behind the lattice. The expan-
sion ends with a sharp increase in the pressure at those points on the
convex- urf'ace of the blade where the incident and reflected shocks in-
teract with thte bo-undar:, layer. With an increase in e2, the zones of
maximrum ex~pansion on~ the~ convex surface as well as the sharp increase of
pressure in1 the Enxks~!i~ are~ both displaced along the back toward the trail-
in, ed~ge.

In the regimes of limiting expansion (E2 = ES), the pressure along
the back of the profile continuously drops. The pressure behind the ex-
pansion weaves at all reg~imes E2 b S decreases a~s the pitch increases.

The effect of the pitch on the intensity of the shocks behind the
throat is seen in figure? 7-53. The character of the curves np2 Fi,min
(ilp, is the increased in pressure through the shock wave impinging on
the conveix surface of; incidence of the shock wave) depends on the pitch.
With an increase in thle maximu intensity of the shocks at first de-
creases and then incr-ases. At the same time, the maximum aP2 Pi~,in
shifts in the direction of higher values of 82.

The detailed investigation of the flow in the sections behind the
lattice shows that the distribution of the angles and the static pres-
sures is -.ery nonuniform. In figure 7-54(a) is shown the distribution
of the local Flnglpe of deflection P2i 82n -over the pitch of the lat-
tice f'or twlo regimels. The upper curve corres-ponds to the flow conditions








NACA TM 1393


shown in figure 7-49(c)(e2 P S). Aheadl of the primer:. shock, thec floui
deflections are influenced by the expanision wavers; the angles of the
streamlines slightly decrease. At 7 = 0.4 there is a sharp decrease
in P2i due to the primary shock. At 3i > 0.4 the local angles vary
less sharply up to x = 0.9.

From figure 7-54(b) it is seen that the distribDution of the static
pressures over the pitch is likewise ve'ry nonuniforml. The static pres-
sure varies with the system of wa-ves and shocks traversing the section
investigated.

A large effect on the spectruim of the flow behind the lattice is
exerted by the setting angle of the profile (i.e., the angle at the
exit). With a change in the angle a2n the gleometrical parameters of
the section behind the throat vary. For the same pressur-e drop in the
lattice (E2), the arrangement of the fundamental system of waves and
shocks in this section of thE lattice varies.

With increase in P2n the leng~th of the wall of the section BD
(fig. 7-49) is shortened (the pitch is unchanged); the relative effect
of the primary expansion wave increases; the angle of deflecttion in-
creases with increase in P2n'

The equalization of the flow behind the trailing edge for M2 > 1
occurs at greater distances from the lattice than, for M2 < 1. The vari
ation of the distribution curves of pO2 #01 along the pitch as a func-
tion of y for M2 = 1.58 is shown in figure 7-54(c).

We note that the equalization of the flow at supersonic velo~eities
is accompanied by a. decrease in the static pressure behind the trailing
edges.

Supersonic reaction lattices are often used as nozzle lattices (for
"2 < Es)(fig. 7-55). The interblade passages of such a lattice form
supersonic nozzles. At design conditions supersonic velocities may be
obtained in such lattices without any essential deviation angle of the~
flow. On the~t other hand, expansion ma~y arise in the overhaneg section
of the lattice at design conditions. The expansion wav.e is formed as a
result of the lowering of the pressure behind the trailin: edgeF. In the
flow about the trailing edge, a~s in the subsonic lattice, a second shock
at the trailing edge arises. Thus, the same general system of shocks
and expansion waves, although they are weaker, is manintained. also for
the nominal opera~tingS regime of the supersonic lattice.








NACA TM 1393


For the off-design regimes (E2~ < 2comp), the fundamental system
of waves and shocks is organized in a manner similar to that shown for
lattices with converging channels. If, however, the ratio of the pres-
sures e2 becomes larger than the computed one, the shocks are moved
upstream into the interblade channel, the same way as they are in the
one-dimensional supersonic nozzle. It should be borne in mind that,
for the same value of s2, the shocks in the channels of the supersonic
lattice are somewhat weaker than in the Laval nozzle and are situated
near the outlet section.

The flowi structure in a supersonic reaction lattice is shown in
fi,-ure 7-56. At increased pressures behind the lattice, a system of
two obliqule shocks is situated within the channel (fig. 7-56(a)). With
an in~crease- in pressure behind the lattice the shocks move toward the
outlet section (figs. 7-56(b), (c), and (d)). Near design operation
(figs. 7-56(e1 and (f)) primary and reflected shocks intersect on the
convex su~rfae; behind the lattice a. trailing-edge shock may be seen."

The pressures distribution over the profile (fig. 7-57) agrees with
the flo;J picture. At regimes where the relative pressure e2 is greater
than computed, the pressure rises through the system of shocks. It is
characteristic that there is no transverse pressure gradient in the chan-
nel between the blades of a supersonic lattice.2 The velocity field be-
hind a supersonic lattice possesses very great nonuniformity for e2 < E~c



7-12. IMPULSE LATTICES IN SUPERSONIC FLOW

When the velocities are practically sonic a X-shaped shock is formed
on the convrex side of each profile of an impulse latti~ce. This system
of shocks of small curvature merges to form the boy wave for the ne-igh-
boring profile (fig. 7-58(a)). Lamediately behind each bow w~ave the
flow is subsonic. This scheme of flow evidently can take place only in
the case in which the flow accelerates behind each bow wave and then
reaccelerates to the velocity M:1 ahead of the following shock.

There ac-ele~ration of the flow occurs in the expansion waves form-
ing in the flowJ about the leading edges. As the velocity of the oncom-
ing flow increases, the bow becomes curved and moves toward the inlet
edges of the profiles (fig. 7-58(b)). It mayr be assumed that for veloc-
ities. corresponding to the flow scheme in figure 7-58(b) the flow behind

21It may be assumed that the tip losses in such lattices are small
even with small blade heights.








rjACA TM 13933


the shocks will be turbulent. Because the effect of profiles is comrmu-
nicated upstream in the subsonic region, a nonuniform velocity distri-
bution is established behind the leading shock. The velocities vary
periodically in ma~gnitujde and direction along the la~ttice.

For a certain sufficiently large value of MI the right branches
of the shocks mterge forming a continuous wavy-Eshape shock (fig. 7-EF(c)) 1
The left branches of the bow wave are turned into the concave surface
of the profile. With further increase in the velocity Mythe angles
of the branches of the bow waves decrease; the shocks approach the inlet
edges of the la~ttice. In certain cases at the inlet to the interblade
channels there is formed the system of shocks shown in figure 7-58(d).
In the sy-stem of intersecting and reflected shocks the pressure
increases.

The envelope of this system of curved shocks lowers the velocity
of the flow to a subsonic value. Supersonic velocities arise again as
a result of the expansion on the convex surface. The flow about the
trailing edge here occurs with the formation of the known system of ex-
pansion waves and shocks. Only for very large supersonic velocities at
the inlet does the flow remain supersonic over the entire extent of the
interblade channel.

The above considered schemes of formation of shocks at the inlet
to an impulse lattice are confirmed by photographs of the flow. In fig-
ure 7-59 there are clearly seen the changes in the shape of the bow
waves that accompany increases in Ml'

The pressure distribution over the profile at supersonic velocities
(fig. 7-60(a)) shows that for MI 4 1.5 the velocity over a largely part
of the concave surface is subsonic. For M1_ > 1.12, the velocities are
supersonic at all points on the~ convex surface. The point of minimum
pressure on the back in the overhang section is displaced with increas-
ing MItoward the outlet section of the lattice.

The investigation of the flow behind an impulse lattice at super-
sonic velocities shows that the distribution of static pressures, ve-
locities, and losses over the pitch is very nonuniform.

A change in the inlet angle of the flow greatly affects the strue-
ture and intensity of the bow waves, the pressure distribution over the
profile, and the flow distribution between the wakes behind the lattice.

The form of the inlet edge of the profile and angle B1n have an~
effect on. the structure and, in particular, the intensity of the bow
waves. Ahead of an implse Ilattice consisting of profiles of small cur-
vature (large angles of the inlet edge P1n) an o~ver-all wave-shaped












shock is f~rormed instead of the system of shocks shown in figure 7-58(b).
The shape of thils wave ahead of a lattice of plates for various inlet
angles is seen in figure 7-61. Since the formation of' such a shock
ahead of the lattice is possible in the case where Mlsin Pl > 1, the
number M1 colrresponding to the type of shock considered increases as
Pl decreases.


7-13. LOSSES IN LATTICES AT NEA SONIC AND SUPERSONIC VE~LOCITIES

The above considered properties of the flow of a gas in plane lat-
tices of different types at large velocities permit an analysis of the
behavior of the over-all characteristics of lattices accompanying a
change of velocityI of the flow (M1 or M ).22 Figure 7-62(a) shows
curves of the loss coefficients for reaction lattices as a function of
M2 and thle in~let flow angle P1. Figure 7-62(b) gives similar curves
for impulse lattices.

The curves shiow that, de-pending on the entry angle, the pitch, and
profile shape, the loss coefficient of reaction lattices may increase
or decreaise in the region of transonic velocities (0.8 4 M2 4 1.2). A
marked increase of the losses in a lattice occurs at supersonic veloc-
ities (M2 > 1.2). The value of M2 for which this increase is ob-
served decreases as the pitch is increased.

The loss coefficients of supersonic lattices increase very sharply
withr an increase in M2 and reach a maximum value when the relative
pressure in t~he lattice is nearly critical (M2 a 1). With a further in-
crease in M2, the coefficient (p decreases. The losses in a super-
sonie lattice are a minimum near the computational (design) value of
M2. For M2 > comp the loss coefficient increases with the velocity.

From a comparison of the loss curves in a reaction supersonic lat-
tice (fig. 7-62(a)) with those in a one-dimensional supersonic nozzle,
it can be concluded that the variation of f(p with M2 is qualitatively
the same in both cases. It follows that the shocks in the interblade
passages and the separations and vortex formations 'associated with them
have the m~ain influence on the effectiveness of such lattices at off-
design regimes. The lowering of the losses in the lattice for M24 1
is explained by the fact that at such regimes the wave and vortex losses

22The data presented in the present section refer only to lattices
of definite georretric parameters.


N~ACA TM 1393








PJACA TM 1393


decrease and then (for small M2) entirely vanish (the interblade pas-
sage works as a Venturi tube). As in the case of the single nozzle, the
losses in a supersonic lattice at the design and off-design regimes vary,
as a function of the passage paurameter F1 Fp.23 With an increase 'in
this parameter, the losses for designl operation decrease somewhat and
increase for M2 < M~com~p,

Comparison of the losses in different reaction lattices leads to
the conclusion that in a wide range of velocities, lattices with con-
verging interblade channels possess a higher effectiveness than super-
sonic lattices. Evidently supersonic lattices are suitable for applica-
tion in the range of large supersonic velocities, but they are only ef-
feetive for the case where such turbine lattices will always operate
near design conditions. The points of intersection of the curves (the
points A and A' in fig. 7-62(a)) permit establishing ranges of rational
application of the two types of lattices compared.

The losses in an impulse lattice at subsonic velocities increase
with increase in the velocity more sharply than those in reaction lat-
tices, and they reach maximum values for Ml = 0.8 to 0.9 (fig. 7-62(b)).;
A further increase in the velocity leads to a certain lowering of the
loss coefficient. Thus, in the zone of near sonic velocities
M2 = 0.9 to 1.3 the coefficient Sp of an impulse lattice decreases and .
becomes a minimum at Ml J 1.2 to 1.4. For Mi > 1.4 with increasing ve-
locity, Sp again increases.24

The lowering of the loss coefficient in an iapulse lattice at sall
supersonic velocities is explained by the improvement of the flow about
the inlet edges and on the convex surface of the profile. For
M2 = 0.7 to 0.9 flow separations are formed near the inlet part and on
the convex surface of the profile; the points of minimum pressure and
separation, are displaced downstream when supersonic velocities are
achieved since the flow in the channel is converging behind the bow waves
(fig. 7-60). Also change in thel inlet angle ha~s a particular effect on
the magnitude of the loss coefficient at supersonic velocities for im-
pulse lattices. For inlet angles less than, p1n (a "blow" on the concave

23NACA note: Area. ratio, see fig. 7-62(a).

24The data presented refer only to the given lattice. With a change
in the shape of the profile and the pitch, the character of the~ depend-
ence of Sp on M may vary.








NACA TMI 1393


surface of the profile) the loss coefficient increases. The mean angle
of the flow behind the lattice increases with an increase of velocity
at supersonic velocities (deflection behind the throat).


7-14. COMPUITATION OF ANGLE OF DEFLECTION OF FLOW IN OTE=RKANG

SECTION OF A REACTION LATTICE: AT

SUPERSONIC PRESSURE DROPS

There exist several methods of determining the angles of deflection
of the flow behind the throat of the lattice. The most wides-prea~d meth-
ods of computation are based on the one-dimensional equations of flow.
Assuming that the field of flow in the sections AB (fig. 7-63) and EF
(chocsen st a large distance behind the lattice) is uniform and neglect-
ing the losses in the lattice up to section AB, the equation of contin-
uityI mayJ be written in the form

ALBp2c2 = EFp2,c2,sin P2,

or, bearing in mind that for very thin trailing edges

AB = EF sin P2n = t sin P~n

we obtain

p~c2sin $2n 2 *ZCZ.sin P2m

We diivide b~oth sides of this expression by pwl 41; then

02.
qZsin P0 = Q2 -O sin Bp

Taking into account that P2 = 2n + 6, where 8 is the angle of
incl-ination of the flow in the overhang section, we arrive at the
equa tion

6 = a~re sin -220 P10sin P."n ~2n (7-60)


In the above equation q2 and q2= a-re easily expressed in terms
of the pressure ratios p2 0O1 and p2. PO2'




1




66 N.ACA TM 1393


For a. reaction lattice, with pB./r~ 02 n the flow parameters in
the section AB will have their critical values when q2 = 1. For a
cup~ersonice lattice q2 1 ~I < 1. By ignorinr g the losses, Eais rela-
ted the flow at section AB to that at DH in a form sim~ilar to that of
formula (7-60)
6 "ascsi(sin P2.
are si 02(7-60a)


With account taken of the losses, formuls (7-r30a) c-an be csritt~en as

z,1 P01P0 ;
8 =~ are sin 1--- sin 62n


thtReplacing q2, by X2t and (p and taking into account the fact


P01 Ik 1 2 k-1 1 k+1E(
p0 2 "2thp I 8 1
k + 1 t

we obtain after transformations
1 -2
r \k 1 2
8~ =~ ar sin sin 02 -

k -1 2 -
(~ k-1l



Whence, it follows that with constant value of th~e t~heoretical outflow
velocity X2t, the angle of deflection increase with an inicrease in the
losses. According to equation (7-60a), the angle of defle-ction 5 de-
pends not only on the outflow velocity and the~ losses but also on the
snJle P2n*

Formula (7-60) holds only for E2 ~ S, that is, up to the point
for which the primary expansion.wave Impinges on the convex surface of
the blade. The angle of deflection co;rresponding t~o the~ limiting expan-
sion over the convex surface of the blade is spprox:im~ately determined by
the relation


6S = amS ~ '2n








NJACA TM4 1393 67


where agl is the angle of the characteristic coinciding with the plane
AD.

The pre-ssur~e in the outlet section, of the lattice for the regime
considered maly be determined by the formula
2k

"S = E,(sin P2n k+ (7-61)

In fact, since

1 sin P2n
sinl(02n + S) = sin CamS = MZ 2


we havie 1

kS sin p2n 2 S;



Sol-.inrl this equation for ES, we arrive at formula (7-61).

M*aki~ng uise of the known relation between at and E and substitut-
ing in the particular case 82 = c;S, we obtain



k + 1762
6S = are sin 2(1-k) P2n -2
k+1 2
(sin P2n) k + 1

For the one-dimensiona~l case of infinitely thin trailing edges and
straight convex and concave surfaces, the exact solution may be obtained
by SlimultaneOUSly, so3lving the equations of continuity, momentum, and
energy.

By the equation of energy,

P2 k 1 C2 P2. k 1 c2
-- + +
p2 k 2 p2m k 2

From the conditions of continuity,

p2 X12 sin(P2n +6
"2. "2 sin Ptn








68 NACA TM 1393


SubsYtituting this expression in the eqution of energy,r we obtagin


sin(a2n 6
sin P2n


k + 1 1 2=a 2= ,
2k k p2 h2


k -1 2.m
S2k Ag


(7-63)


We write the equiation! of momentum using
tion of the trailing edges in the form


the componenit in the djireC-


2 2
P2eZt sin P2n + 2t sin P2n = p0mc2omt sin 02=os 8


+ p2mt sin 02n


PZ2c22(X2O.cos 6 X2) = P2 P2=


Since


PZ2ic2 2 @,la = keqq201


we obtain


2.
X2


2t 2- P 13
ke q2 Z01 cos 8


(7-6~4)


If in the section A~AB the parameters are critical, then

= 1 --- +
E2 k sq cos 8

The last expression together with equation (7-63) gives


=O


k + 1 -


whence


E2e 2o p
E4 ot 2n


Ran 2
I4


k + 1
k L


k 84cot 02n


tan 8 =


E2.
k + 1


(7-65)


( E-fT







NACA TM 1393


Approximately, for 8 4 100, we obtain



S 2k (7-65a)



The above accurate solution obtained by G. Y. Stepanov permits de-
termining- the wave losses in the lattice. The coefficient of wave losses
is expressed by the formula


h2t

or after substituting for X2



b = 1 -k-1
k +1 k 28
k 11 2 cos

For computing the flow behind the throat of the lattice, the method
of characteristics may be applied. We consider a lattice of plates of
small c~urvatiure with straight, infinitely thin trailing edges (fig.
7-64(a)) and set up the boundary conditions at the point where the stream-
lines coming off the two sides of each plate merge. The streamline 1-1
moving along the convex surface of the plate intersects both the primary
and reflected expansion waves, while the streamline 2-2 coming off the
concaveF surface intersects only the primary waves. In the plane of the
hodograph the region of the flow in the section AB is expressed by the
point corresponding to the end of the vector X1 = 1 (fig. 7-64(b)). The
velocityr of the streamline 2-2 after passing through the primary expan-
sion wave is determined by the vector 12, while the velocity of the
streamline 1-1 after passing through both the primary and reflected waves
is determined by the: vector X3. The boundary conditions near the point
A for two merging streamlines of gas are the conditions that the static
pressures are equal and the velocity vectors are parallel. These condi-
tions are satisfied if the oblique shocks K1 and K2 are formed at the
point A, the direction of these shocks shown in figure 7-64(a). If the
angle 81 is small, the primary shock K:1 may be considered as a char-
acteristic, while for computing the edge shock K2 the method of char-
acteristics may be used. We here neglect the wave losses in the shocks.
It is evident that the direction of the shock K2 coincides with the








N~ACA TM 1393


normal to the epicycloid of the second family; at the point d located at
the center of the segment be. With this simplification of the problem,
the wake (which for an infinitely thin edge is considered to be between
the streamlines 1-1 and 2-2) in the immediate neighborhood of the point
A has the direction of the vector \2 (the dot-desh line in fig. 7-64(a)).
The velocities and other parameters of the flow for the remaining stream~-
lines are determined after computing the interaction of the primary, and
reflected exp~ansion waves.

The entire region of flow behind the throat can be divided into
three zones (fig. 7-64(a)): I the zone of influence of the primary
expansion wave (for th7e lattice considered, this region transforms into
a point), II the zone of interaction of the p~rimary, and reflected
waves, and I~II the zone of influence of the reflected wave (in the
plane of the hodograph, this zone corresponds to the characteristic of
the second family bc).

The region of interaction of the primary and reflected waves of
rarefaction (zone II) may, be computed once for all, using the m~inimumi
value of the angle BP,min_ yo to 100. For anl. other angle B2n 2P,min
the computation of the flow downstream of the throat is carried out in
the following ma~nner. Wle draw th~e xr-axis at the g-iven angle to CB (f-ig.
7-64(c)) and find the mean pressure in the section AP = t


pS tl Sidx


characterizing the regime of the limiting expansion. For all regimes
q > p2 01 PS 01 the zone of interaction IT will be bounded by the
broken characteristics, for example, ABi', AB", AE~"' .. (fig. 7-64(c)).
To each value of the pressure drop in the lattice corresponds a fully
determined position of the points E', B", B"'. .. Carrying out sue-
cessively the computation of the flow for different positions of the
characteristics AB', AB", etc., we establish the distribution of thre
pressures (velocities and local angles) over the pitch AB in the zones
II: and III and obtain the mean pressure behind the lattice


P2,cp=J P 92dx

In this way the computation of the local parameters of the flow be-
hind the lattice is conducted for the entire group of possible flow re-
gimes in the lattice, and the relation is established between the posi-
tion of the points B', B", etc., and the pressure drop in the lattice.








NACA TM 1393


The mean angle of deflection of the flow for a given regime may be
obtained from equation (7-21):

2~ gi i~sin2lPz 2n i)dx
tan(PZ + 8 ) =
Qrd 9 sin 2(Pzn i i)dx

where qi, X and Bi are, respectively, the local reduced flow rate,
nondimrensional velocity, and angle of deflection.

Fo3r a. lattice of profiles with finite thickness of the trailing
edges, the computation of the flow in the overhang section by the method
of characteristics is considerably more complicated. In. this case it is
necessaryl to k~now how the pressure varies behind the trailing edges as a
function of the geometrical parametersr of the profile and the lattice
and of the flow regime. Such a relation

pkp
90-f(M2^~Tt

can be established only experimentally. Then, replacing the actual lat-
t-ice by a lattice of planes, the trailing edges of which serve as the
sources of disturbances uniformly distributed in the field of sonic (or
supefrsojnic) flow, the intensity of the expansion waves may be found.
From the boundary conditions at the edge the system of additional expan-
sion wav~es and shocks is determined.

An important advantage of the method of characteristics is the pos-
sibilit/l of constructing the spectrum of the flow on the convex section
behind the throat and at different distances from the lattice and of de-
termining the nonuniformity of the field of velocities and pressures in
different sections.

A comparuis-on of the most widespread and accurate methods of comput-
ing the deviation angles with test data (for two reaction lattices) is
shown in figure 7-65. It is seen from the latter that, for lattices of
small pitch and consisting of profiles with thin trailing edges, formula
(7-65) and the method of characteristics give results that satisfactorily
agree withi experimcent. For small values of 8(s2 >, 0.35) the equation of
continuity, (7-60(b)) may likewise be used for approximate computations.
For lattices of large pitch, only the method of characteristics gives re-
sults which are in good agreement with experiment.








NIACA TM 1393


7-15. CHIARACTERISTIC FEATURES OF TERiEE- DIMEN]ST IOAL

FLOW IN LATTICES

As was already pointed out, the ring lattices of turbomachines con-
sist of radially arranged blades of finite height (length). The shape
of the interblade passages of the lattice varies in the radial direction.
In condensation turbines, lattices with variallie height blades are used.
The guide lattices are always shrouded. Rotating lattices are sor~etimes
designed without shrouds.

These construction features of real lattices have an important in-
fluence on the flow. The phenomena observed in three-dimensional lat-
tices are not taken into account in the analyses of two-dimensional flow.
On the basis of test data we shall a~nalyize the special features of three-
dimensional flow in a straight row of lattices.

In these lattices secondary flows are formed near the tips of the
profile on the convex surface of the blade. The cause of formation of
secondary flows in the interblade channels of a lattice is the viscosity
of the gas and the transverse pressure gradient arising from the curva-
-tnue of the channels.

Because of the increased pressure on the concave surface of the
blade, the gas particles flow toward the convex surface of the blade
(fig. 7-66(a)). For sufficiently high ratios 3/a2 (see fig. 7-66(bs)),
the secondary motion of the gas over t~he concave surface is only achieved
with difficulty, because the particles Icust move over a long path over
which there is friction. Such a flow from the concave surface to the
conv~ex surface of the neighboring profile is possible only in the bound-
ary layer along the end walls bounding the channel. The peripheral flow
of the gas in the boundary layer therefore starts on the concave surface
at the tips of the profile (near the end walls) and continues over the
end walls toward the convex surface of the blade. As experiment shows,
there occurs a nonuniform distribution of the pressures over the height
of the blade; the pressure is lower on the concave surface near the end
walls, while at the tips of the convex sulrface of the blade the pressure
is higher than it is in the middle section. Along the end walls of the
channel, the pressure drops in the direction from the concave to the con-
vex side of the blade. At the tips and along the convex surface of the
blade, the boundary layer flowing from the end walls encounters the
boundary layer moving along paths parallel to the end walls. As a re-
sult, near the tips of the blatde and on the convex surface rapid growth
of the boundary layer occurs; the thickness of the layer increases
sharply. I~n the majority of cases this leads to a local separation of








NIACA TM 1393


the la:yer and therefore to the formation of vortices.25 At the same
time, because of the motion of the boundary layer from the concave sur-
face to the conve.;x surface of the blade, compensating flows are fo~rmed
at the blade tips which are directed from the convex surface toward the
concave surface. These flows, together with the boundary layer separa-
tion on the convex surface, form vortex regions (trailing vortex) near
the ends (butt faces) of the channel walls.

In this way, at the convex surface of the blade near the tips, a
vortex pair arises consisting of two vortices whose direction of rota-
tion does not coincide with the direction of the circulation about the
profile.2 The vortices rotate toward one another in correspondence
with the direction of motion of the gas in the boundary layers at the
plane walls (fies. 7-66(b) and (c)) and induce a field of velocities
normal to the streamlines of the primary flow (fig. .7-66(d)), which leads
to a certain in-crease in the outlet angle of the flow from the lattice.

In the photogra~phs of the wakes of the flow (fig. 7-67) there is
clearly seen the secondary flow of the boundary layer on the end walls.
Behind the points where the vortices arise, the secondary flow of the gas
continues to be associated with the boundary layers on the plane walls
and the convex suirfa~ce of the blade; the vortices are enlarged toward
the outlet section. On account of the growth of the vortices, their
exes arrange themselves with a certain inclination to the plane valls.

At small ratios 2/a2, the vortex regions are propagated over the
entire section of the channel forming a vortex pair characteristic of
curved chan~nels of square section. The over-all vorticity of the flow
sharply, increases.

25
"Depending on the shape of the profile and of the interblade chan-
nel and also on the flow regime in the lattice (inlet angle, M2 and
Re? numbers) the separation of the boundary layer on the convex surface
mayI not occur. Tests show that separation does not occur for large in-
let angles and small flow velocities.

cIn connection with the question as to the mechanism of formation
of secondary flows in the lattice, it should be remarked that attempts
to make use of thle theory and computation procedure of the induced drag
of a wing of finite span for determining the tip losses in lattices did
not give any essential results. The tip losses in a lattice and the in-
dulced drag of a wing have a different origin. It is sufficient to state
that the tipF phenomena in a lattice arise from the viscosity of the
fluid, whereas the formation of trailing vortices from the tips of a
wing of finite span are not directly connected with the viscosity; the
tip vor~tices ojf a wing should exist for the flow of an ideal fluid also.








rlACA TM 1393


The experimental investigastions confirm the occulrrence of separationj
and vortices in a row of straight blades. The distribution of the dyna-
mic pressure and the static pressure over the height of the btlade near
the convex side in the nucleus of the flow and at the concave side in
the narrow section of the channel in figure 7-68(a) shows the character-
istic variation of these parameters in vortex regions. In the vortex
zone pg. and p. decrease; this decrease does not appear in the nuc-
leus of the flow or a~t the concave surface. In the outlet section of
the lattice the~ picture of the distribution of 17 and p. remains
qualitatively the samne (fig. 7-68(b)). The zones of reduced .alues of
pgi are displaced from the plane walls. The dips in the curves are
more marked.

The separation of the boundary layer on the back of the blade and
the formation of vortices are a source of considerable loss of energy,r
particulary for relatively small blade heights. The change in the geo-
metric parameters of the straight lattice and, in particular, of the re-
lative height and pitch affects the mazgnitude of the tip losses.

With decrease in the height 1a the vortex regions approach each
other (fig. 7-69(a)) and are slightly shifted toward the side walls.
The strength of the vortices, within definite limits of the change of
2a, practically does not change. Only for la42is there a notice-
able increase in the effect of the vortices in the nuclear flowc (the val-
ues of fiOi decrease). For lattices of height fa < 1.7, the entire
flow in the channels is vortical and the pressure of' complete stag-nation
in the nucle-us is lowered.

From this it follows that the absolute m~agnitude of the losses in
vortical regions does not change with decrease in the height of the
blades up to certain limits. The relative losses change in invlerse pro-
portion to the height Ia. With increase in the pitcht of the profiles
(fig. 7-69(b)) the strength of the tip vortices increases, and there
occurs a certain displacement of the zone of maximum losses away from
the end wa~lls.

A large effect on the tip losses is exerted by the curvature of the
interblade channels. As the curvature increases the losses increase.
This trend is particularly intense for lattices of small height.

The flow regrime, that is, the inlet flow angle and thl Re2 and M2
nunmbers, has an effect on the ma~gnitulde of the tip losses. With an in-
crease in the enitr.' angle of the flow (fig. 7-70) the strength of the sec-
ondary flows decreases. We may note that at large ent~r: angles the trans-
verse pressure gradient in the interblade channels decreases. At the same












time th-ere is a lowering of the intensity of the secondary flow of the
boundar la er toward the convex surf ace of the 'jlade, the thickness of
the la:er on the back decreases and the vorticity losses decrease. An
increase in the velocity of the subsonic flow in the lattices leads to
a decrease in! the tip losses, a fact which is evidently explained by
the decreasec in thickness of the boundary layer.

The in-estigation of the three-dimensional flow in lattices of
straight blladies qualitatively shows the same change of the mean (aver-
aged over the pitch) loss coefficients near the end walls for all lat-
tices. With an increasing distance from the end walls, the loss coef-
ficienrt sharply decreases at first and at a small distance reaches the
minimum value beyond which it again increases. In the zone of lowering
of 5F there is found a decrease in the thickness of the boundary layer
on the convex surfaces and of the depth of the end dips. The character
of the variation of (p over the height for short blades for different
velocities is seen in figure 7-71. The curves in figure 7-71 show
clearly, the decrease in (p with an increase in M2 for M2 < 1.

In correspondence with the above-mentioned effect of the curvature
of the channels and the inlet angle, a certain relation must exist be-
tween the profile and tip losses. In lattices with large profile losses
there are found also increased tip losses.27 From a consideration of
the scheme of formation of the tip losses in a straight lattice it fol-
lows that the measures taken to decrease the transverse pressure gradi-
ent in the interblade channel and therefore in lowering the strength of
the periplhersl flows in the boundary layers greatly decrease the tip
losses. Of' geat importance is also the character of the velocity dis-
tribultio~n ov~rr the height at the entry to the lattice. With a nonuni-
form velocity- distribution over the height at the entry the tip losses
increase. In this connection it should be remarked that the use of
overlap28 in the real lattices of turbomachines leads to a sharp in-
crease in the tip losses.

In cylindrical lattices, the character of the tip phenomena changes
somewhat. Because of the change of the pitch of the profile over the
radius and fthe occurrence of a radial pressure gradient, the symrmetry of
the v.orticfes arrangement is disturbed. Both vortices are displaced
along the radius from the easing toward the hub of the annular lattice.

ITt is assumed that all fundamental geometrical parameters of the
lattices compared remain the same (pitch, setting angle, and height of
bladeE).

2B-~y overlap is meant the difference in heights of two neighboring
lattices. As a rule the height of the rotating lattice is chosen to be
greater thani the height of the guide lattice.


NACA TM 139j3








N\ACA TM 1393


The intensity of the upper vortex: thereby increases, while that of t~he
lower decreases (fig. 7-72). The radial pressure gradient in an annular
lattice is the cause of the additional losses of energy since the pe-
ripheral flows in the boundary 1s'.Er are increased by such gradient.

In conclusion it should be emphasized that, for lattices with small
relative height, the value of the optimum pitch must be dete~rmine~d after
s-counit, has been taken of the tip phenomena. The opti~mum~ pitch may de-
crease in comparison with that of a plane lattice.


Classification of the Losses in Lattices

The results of theoretical and experimental investigations consid-
ered in this chapter of the flow of a gas through turbine lattices per-
mit classifying the energ;, losses in lattices according to the following
scheme:


A. Profile losses (in the plane lattice-)

(1) Losses by friction in the boundary layer on thep profile

(2) Vorticity losses by the separation of the flow on the profile

(3) Vorticity losses behind the trailing edge (edge losses)


B. Losses in three-dimensional lattices (in addition

to those of group A)

(1) Losses produced by friction at the bounding valls of the lat-
tice over the height and due to peripheral (secondary) flows in
the boundary lawyers

(2) Losses in the thickened layers on the back of the blade and
vorticity losses due to separation of the layer at the tips and
the formation of vortices


C. 'Wave losses (in addition to those of groups A and B a~t

near sonic and supersonic velocities)

In the general case, for the investigation of lattices of turbine
stages under actual conditions, there are added the losses arising from
the unsteadiness of the flow and the heat losses whenn cooling is
employed).








NACA TMPJ 1393


As was stated above, only the friction losses in the lattice can be
determined byl computation at the present time. The theoretical methods
of computing a potential flow through a lattice and the semiempirical
methods of computing the boundary layer permit solving this part of the
problem with satisfactory accuracy. The total losses in a lattice can
be determined only experimentally. The physically evident connection
between the geometrical parameters of the profile and lattice and the
magnitude of the losses does not at the present time have an exact matrh-
ematical expression.


Translated by, S. Reiss
National Adv~isory Committee
for Aeronautics


BIBLIOGRAPHY

1. Kochin, N. E.: Hydrodynamic Theory of Lattices. GITTL, 1949.

2. Simuonov., L. A.: Computation of the Flow About Wing Profiles an~d the
Construction of a Profile from the Distribution of the Velocities
on Its S~urface. PMM, vol. 11, no. 1, 1947.

3. Sir-o~nov, L. A.: Applica~tion of the Electrohydrodynamic Analogy to
the Computation of Hydroturbines. Nauchnve zapiski ESMMI, vol. 6,
1940.

4. Samoilo-.lich, G. S.: Computation of Hiydrodynamic Lattices. Prikladnaya
matematika i mekhanika, vol. 14, no. 2, 1950.

5. Cinzburg~, E. L.: Computation of the Potential and Velocity of a
Plane Parallel Flow About a. Lattice of Circular Cylinders. Trudy
TsKTI, no. 18, 1950.

6~. Zhulkoivskii, M. I.: Computation of the Nonvortical Flow About a Lat-
tice olf Profiles with Variation of Pitch and Angle of Setting.
Teploperedacha i aerogidrodinamika, no. 8, 1950.

7. Blokh, E. L.: Investigation of a Plane Lattice Consisting of Theo-
retical Profiles of Finite Thickness. Trudy TsAGI, no. 611, 1947.

8. Loitsianskii, L. G.: Resistance of a Lattice of Profiles in a. Gas
Flow with Su~bsonic Velocities. Prikladnaya matematika i mekhanika,
vol. XIII, no. 2, 1949.

9. Deich, M4. E.: The Problem of the Tip Losses in the Guide Vanes of
Steam Turbines. Sovetskoe kotloturbostroenie, no. 6, 1945.








NACA TM 1393


10. Ma~rkov, N. M.: Computation of the Aerod:.namic CharacteristieS Of 8
Plane Lattice of Profiles of Axial Tuzrbormachines. Mashgiz, 1952.

11. Markov, N. M.: Ex~pertmental Investigation of the Boundaryl Layer in
the Channel of a Reaction Turbine. Sovetskoe Xotlotiurbostroenie,
no. 6, 1946; Kotloturbostroenie, no. 2, 1947.

12. Stepanov, G. Y.: Eyrdrodynamic Investig~ations of Tulrbine Lattices.
Obsornyi b:,ulleten aviamotorostroeniya no. 4 and 4, 1949.

13. Povkh, I. L.: The~ Effect of the Pitch on the Aerodynamric Ch~aracter-
istics of Turbine Profiles in a Lattice. Kotlo turbos troen ie ,
no. 6, 1948.

14. Povkh, I. L.: Computation of Efficiency and Resistance of Lattice
Profiles LPI, 1952.

15. Sknar, N. A.: Experimental Investrigation of Rea~ction Lattices of
Profiles. Teploperedach~a i aerogidrodinamika, no. 18, 1950.

16. Gukasova, E. A.: Experimental Investigation of Impulse Lattices of
Profiles. Teploperedacha i aerogidrodinamika, no. 18, 1950.

17. Dromov, A. G.: Effect of Periodic Variation of Flow in a Turbine
Stage on the L~osses of Lapulse Blades. Izvestiya VTI, no. 1,
1948.

18. Y~akub, B. M.: Investigation of the Flow Behind the Outlet Edges of
Nozzles. Izvestiya VTI, no. 11, 1948.

19. Gurevich, Kh. A.: Effect of Pitch and Angle of Attack on, the Aero-
dynamic Characteristics of Impulse Turbine Profiles. Trudy LPI
im. M. I. Kalinina, no. 1, 1951.

20. Rodin, K. G.: On the Tip Losses of Energy in. Tubne Blade Lattices,
Trudy LPI im. M. I. Kalinina., no. 1, 1951.

21. Yablonik, R. M.: Some Results of Simulta~neous Tests of Two Turbine
Lattices. Trudy LPI iml. M. I. Kalinina, no. 1, 1951.

22. Kirsanov, V. A.: Investigation of the F~low in Lattices of Turbine
Profiles for Large Subsonic Velocities. TsKTI, 1952.

23. Stechkin, B. S.: Axial Compressors. ind. VVIA imn. N. E. Zhukovskorgo,
1947.











NACA TM 1393 79











P uO





"E *


*t-a












*rlr








Y1













*r(
r-Il







NACA TM 1393


(a) Equipotential lines and streamlines.


(b) Hodogrcaph of velocity.


30m 0 \ / SA

(c) Distribution of relative velocities sad of
pressure coefficients over the profile.

Figure 7-3. Flow of ideal. incompressible fluid through
reaction latice.








NACA TM 1393


(a) Profile of lattice.


(b) Hodograph.


0Sa 01 Sea

(c) Distribution of relative velocities
over profile.

Figure 7-4 (A) Flow of an ideal incompressible
fluid through compressor lattice.




























(a) Profile of impulse blade.


(b) Hodograph.


Sao


(c) Distribution of relative velocities
over profile.

Figure 7-4. Flow of ideal incompressible fluid through
impulse lattice.


NACA TM 1393







NACA TM 1393


(a) Inlet angle Pl*


9I
(b) 1Pitch t.


(c) Angle of setting B .
Figure ?-5. Effect of inlet angle, pitch,
and setting angle on relative velocity
distribution over profile of lattice.








NACA TM 1393


(a) Sum of flows.


(b) Noncirculatory flows.


Act


~t
dc2


(c) Circulatory-axial flows.

Figure 7-6. Flow of ideal incomrpressible fluid through
lattice of brlades.








NACA TM 1393


90" #o" 30. 200 0 8


. 6 .7 8 3 1. 0 1. 1 1. 2 1. 3 1. 4 1. 5

Figue "-. ependence of' the coefficient m = 102/10 oln pitch
and setting ang-le of lattice of plates. t\-A note: Th-e stalacies
should obvliousl;, b-e t instead ofi t
t bn







86 HACA TM 1395













a)l







(o o






(c) (a )







Fiur 1-.- xmle f ofrmltrnfrmtono atie








NACA TM 1393


C~Itr





(a) Transverse flow.
~C\ T~I~Cllong


-C
21ong
(b) longitudinal flow.


ZCccir




(c) Circulatory flow.
4CI SIY1 = Ecllong+CeClcire

CI Sin # = Alc ltr







CG2= Actr + Boclong + Cociro = 0

(d) Sum of flows.

Figure rl-9. Flow of conformal transformation of lattice
of circles.








NACA TM 1393


Figure 7-10. Computation of a lattice
by the method of G. S. Samoilovich.


Figure 7-11. Comparison of the theoretical and experimental
pressure distribution over a latiee prof ile.














































Figure 7-13.
the points
C1 = 1, pl


- Determination of the correspondence of
of the profile with the circle (t 1,
= 90o)


18 90 0


(C)


_ II


1


HACA TM 1393


7cJ~;i-


f


4


Y


Figure 7-12. IDetermination of an equivalent lattice of circles.


scn 0 s.









NACA TM 1393


I\ bP
P

(b!


6, Scr Sdor
(c) (d)
Figure 7-14. Construction of flow stream.


Figure 7-15. Scheme of electrical model of
flow without circulation. Mesuremnent of
potential.
1, electrodes; 2, source of alternating
current; 3, potentiometter (vater rheosvtat);
4, zero indicator (radio phones); 5, unit
of potential.








NACA TM 1393


Figure 7-16. Scheme of electrical model of flow
with circulation. Measurement of velocities.
1, probe with two needles; 2, amplifier; 3, rectifier;
4, galvanometer; ----, equipotential lines.








NACA TM 1393


















cVJ


(a) Turbine (converging) lattice.


(b) Compreasor (diffused lattice.
Figure 7-17. Forces actin on profile in lattice.









NACA TM 1393 93








je,
\ *89-( -







\\ o


I~ (D
1( 15
1-S
6. r
97d





7 oeo




3 P
3P
rl YI


La CJ -- c

\O
0 0 0
5-1 9
ife 2
*r-


















a ,C
rLP




so

N, 0 O Oo
a Oa c0
re~c .sM

a~q *M a C










Sa,

r-- o n
rN -0
I ,00















O -1 P




,oo
*C


Sa,
----I 6
d
- 'Z0?PU
*(


1 1 I I I I 9


NACA TM 1393








15ACA TM 1393 95




















I+ P





rCS I- o
csI op I my p I 0
Ecs c ct a
r-




o


ao Lr)
h' I


c3 ct












1. 1





so,, a Gsc.,


B/- = 6 L1 & Z2
go~I I Ib I
?2~~ 7 j
0~II


,003



2.0

20 p *... ..
CO r .. -----T


HACA TM 1393


71 80 7 g


169 26


JO7 ~ Up C


Figure 7-23. Computed magnitude of the
in turbine lattices as a function of pl


friction loss coefficients
and 132 *







SNACA TM 1393


(a) Mt = 0.565


(b) Mi =- 0.773


(c) M2 = 0.940


Figure 7-24. Spectra of the flow of air through
a reaction lattice at supersonic velocities.
Relative pitch of profiles t = 0.860, inlet
angle of profile 184n = 15052' (visualization
of trailing wake) .











98 NACA TM 1393










9+




ss I Ic oo
/~~ id g >
/i d *





/' p ., a ,O)1



ra~~ I II
Y'~~ --9"X







0c
r: F( I I I I* -






C1 cr ~N




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