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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 71. 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 threedimensional 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. 312420. 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. 71). There exist several methods of specifying the shape of a blade pro file. The universal method of coordinates (fig. 72(a)) has great ad vantages. The methods shown in figures 72(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 72(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. 72(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. 7l(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 71(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 threedimensional 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. 71). 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. 71()).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. 71(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 straightbacked 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 threedimensional 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 twodimensional 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. 73). 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 73(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 ~ =  (71) 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. 73(b)) provides another important method of representing the flow. At each point along a streamline or equipo tential lines (fig. 73(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. 73(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 socalled vortexsource and a vortexsink. 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 vortexsink 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 73(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. 73(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. 73(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. 74); in a compressor le~ttie, the interbladfe passage widens and the velocity correspondingly decreases (fig. 74A). 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 thetail where the curvature is greatest. At a large dis tance behind the lattice, the direction of flow is determined by the angle B2 * Figure 75 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 reactiontype turbine lattice. A change in the angle Pl (fig. 75(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. 75(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. 75(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 socalled 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. 77). 72. 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 greatest 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 kineticenergy losses in the actual flow. It was remarked previously 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 76 the flow through the lattice is represented as the sum of two flows: a noncirculatory (irrotational) (fig. 76(b)) and a circulatory axial (fig. 76(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 rotationalaxial 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 irrotational and rotationalaxial 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. 76(a)), notice that cot B0 cot P2 Ac2 S = m =constant (72) cot so cot B1 Ac1 where cot p1 corresponds to the angle B1 assumed in figure 76(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 77, 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 presentday 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 presentday compressor lattices this deviation may be as high as 30. Second, the magnitude of the relative velocity on the profile of any lattice depends linearly 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. 76a) ob O n cu , nc we obtain sin 82 c =cbusin80+ ui(cot PO cot Bl1)sin P2 (74) As was said, in presentday 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 (74a) 2NACA note: This ratio is written astbinoinate. 3NACA note: obu is the irrotational flow, fig. 76(b), and cu is the circulatory flow, fig. 76(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. 76(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 (74a). Of practical significance in the theory of the twodimensioh~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 78. 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. 78(a)) corre sponds to the noncirculatory flow through the lattice of plates (fig. 78(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. 78(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 78(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. 79). 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. 710(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. 710(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 711 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 velocityat 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. 712(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, 712(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.. 712(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. 713). 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. 712). 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 713. 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. 78(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 boundarylayer 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 76.) 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 appl~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. 73). 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. 73). 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. 714). 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 effective 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. 714(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. 714(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. 714(b)) are connected by the equation of continuity (see see. 77) 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. 714(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 71_4, this coincidence was obtained by choosing the length of the segment P of the hodograph plane (fig. 714(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. 714(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 plcot~ 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. 73. ELEC~TROHYDRODYNAMIC 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 electrohydrodynamic 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, 715). 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 lowfrequency, 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 currentconducting 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 lowfrequency amplifier. For the potentiometer, there is shown in figure 715 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 directcurrent source and highly sensitive galvanometers can be used. The eliectrohydrodynamic 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. 712), 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. 715) 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 716. 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 . 74. 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 flow, as shown in figures 717(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) (75) The projection of the force P on the axis of the lattice may be ex pressed by the equation Pu= "Pleal(cul cu2) (76) The forces Pu and Pa refer to a profile having a unit span. Equations (75) and (76) 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= (77) P1 2 P The circulation about the profile is equal to r = t(cul cu2) (78) 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 (75) and Pa. 2 tP 1 pea(cal ca2)] (79) (710) Pu = plea We make use of the equation of energy 2 e2 7 k P1 k1l qi k P2 k1 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 (79) an~d taking into account formula (78) we obtain (712) (713) Pu = p~ea The force Pa given by expression (712) is conrveni~ently :represented in the form of a sum of two forces Pt = Pa + Ci0 (711)  cu(cul cu2) Pa 9 91 "k p + preu NACA TM 1393 where Pal = preu and (714) The resultant of the forces Pal and the overall resultant force by P and (see Pu we will denote by Py fig. 717). 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 (715) 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, respectivel~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 (714) 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 75. 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' = (716) [Note: C; is a coefficient.] and 2P C'= a (717) 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 acompressible 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. 718). Evidently, for an inlcompressible fluid (with low in let velocity) Pu 1 2 EcB~c~ The magnitude Pu is determined by formula (76); 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, = (718) 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 k1 2 1 0 9 = 1 (719) 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 (719) 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 (720) 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. (725). 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(721) 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 real 9?J ideal rrocesses. The efficiency of the lattice in a nonun~itorn llow~ is corpulted b: the formula pO4@i "2n~ d pO1 21 ~isin 8$2 Fp For an approximate determIination of 9 equation (719) ma:: be used) substituting in it the mean Sy.nam~ic pressure F;ehind the lattice. In the denomnina~tor of equations (721) to (724), the functions qF~t and X2t may be approximately determined from! the pressure 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. (722) and (723) and the double prime in eq. (724) 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 76. 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 lattices. 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 719(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. 719(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 720 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 721(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 721, 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. 719(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 Thekineticenergy 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 (725) 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 (726) 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 (725), 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 momentumloss 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 (728) 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 (729) NACA TM 1393 In equation (729) 9/k+11 2\ 1 0  k 1 2 1 L k + 1 (730) The theoretical flow rate of the gas may be determined by the formula 1 k k+ 1 pk1 Ot = o~tc~tt sin 82 = 1 ( PO RTO ta t sin P2 (731) Substituting expression (731) in equation (729), we obtain (8 u20 on k (732) By using equation (727), the equation for the loss coefficient (728) now assumes the form 30 n 4 0bol k1 )2tt sin B2 20 en z), C = 1 12t  20 zoboz 2 (733) Bearing in mind that 1 k1 k 1 2 P0 RTO 2~ta t sin P2 G = plpGt " u2 EEdy U20  (8 u20 boJ 0 ~)( k1 (H 82 h 0) + (H 82 X30 en bol F~or an incompressible fluid, there may be obtained from expressions (734) and (735) (H***8fHtU 2 o~n+(** u bo'l (9 = (736) Cppc~tt sin P2 NACA TM 1393 formula (733) can be represented in the form c n 0boZ 5T = or where (734) k 1 t k 1 H=t 8IH=~~ Stt sin B2 formulas (733) and (734), it follows that Cr is equal to From a comparison of the flowrate coefficient (HC~ 6 12) + (H+ 8 120 en bol (735) 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 (737) In this case (for the incompressible fluid), the values of ( and 82" are detenninetd by the formula given in, table 41. The magnitudes H#S and entering equations (735) and (736) 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 722(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 723 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. 75): 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 frictionloss coefficient of the lattice in the form 5T= 2 Tt sin B2 (738) 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 (739) = 105, and to estimate the are concave surfaces we evaluate In expression (739), we assume Re of the profile S on the convex and the approximately (fig. 723(a)): 1 B 2 B Se = Sbo + en oZ sin pl 3 sin B2 (740O) The graphs in figure 723(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 (738) to (740). 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. 77. 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 724 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, kineticenergy 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 725 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. 22 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= (741) 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 (741), wet obtain (742) 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. (743) From equations (741) and (742) there is easily obtained P2. = are tanC(1 T)tan P2n] Equation (743) permits finding the increase in pressure behind the~ lattice ap2= 2c22p sin2 2n(1 z) c 22p sin2p, 2* (kp P2)7 pe2 (744) NACA TM 1393 43 Taking into account expression (741), we obtain nP2 = 1 2 [ 2 (1 1) sin2 2n+ kp ]T (745) 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+ = ~ + (746) where c2, is the theoretical velocity in the section 2'2'. From expression ('746), we obtain 2 1 Ap2 (746a) The velocity~ c2 is expressed in terms of c2, with the aid of equa tions (741) and (744), thus c2 and we have (02 1 r( ~si2 2 9p (747) The coefficient of edge losses isl1 7 sin2 62 PkP7(8 klp k1P;; 1 a672* 11Formulas (745) to (748) given here were obtained by G. Y. Stepanovich. RAC TM 1393 The nondimensional pressure behind the edges entering equations (745) and (748) 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 726 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 726(a) shows curves of local loss coefficients of the wake at different distances behind the reaction lattice. On figure 726(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. 727). 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 727 it is seen also that the amplitude of the fluctuations of the static pressure depends on the shape of the edge. By making a twosided taper (sharpening of the edges b and c in figure 727) it was possible to decrease somewhat the nonuniformity of the staticpressure 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. 78. 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 728 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. 728). 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. 728). The effect of the blade setting on the velocity distribution over the profile is shown in figure 729(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. 729(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. 730). 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 731. 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 728, the optimu pitch is to t = 0.673. For the impulse lattice, topt 0.500.60. 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 731 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 (744). The familiar formula given in the literature for determining the effee tive (actual) angle of the flow a2 82e = are sin (749) gives somewhat lowered values of P2. More closely agreeing values of 2ep with test results are obtained by formula a2 P2e = are sin (750) 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. 732). 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 (744) 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. 733). 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. 734). NACA TM 1593 79. 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 circumstance 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. 735) 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 (751) 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 (752) cl01t sin Pl = i o on ep dR In equation (752) it is convenient to transform to the nondimensional functions q and X q dB Using expression (751), we obtain finally Xbot q d = XenRnI I on, (753) gives (the (754) Computation of the integral I: for small subsonic velocities constant of integration is omitted) Il = ml In X where k + k1 For a gas with k = 1.4 we obtain Ir2 = ml o sh1m 222+ 51,m '4 (755) 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 / (756) If the computed Function I is used, the equation of continuity can be written in the forml5 qlt sin Pl Ren cn Ibol e) (757) where Ten by equation (754) or (755) 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 (757), since the expression in paren theses depends little on Acn In the first approximation (1) Xlt sin Pl cn =b (758) R en Then, in the following approximation, Icn and IboZ are determined from X(1) on ,(2) I qt sin P1 en (T1) 1 (759) and so forth. For Xn < 0.5 the first approximation (758) is suffi cient. The solution of equation (757) is conveniently represented in the form of the graph shown in figure 736, 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 736 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 interblade passage of variable width and curvature. For this purpose it is necessary in the section of interest to inscribe circles as shown in figure 737 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 (7584) and (759) 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 737 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 socalled 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 largetly 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 length of the potential line and on the assumption that the distribution alonel 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 738 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 739 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 graphs shown in figures 740 and 741 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 characteristic 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 gradient 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. 742). 710. 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 743 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. 744). 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 (748) may be employed, substituting the test values of pkp (fig. 743). Thus, taking into account the fact that the trailingedge 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 abovementioned 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 separates 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~ 745 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 745 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 745 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. 746 and 747). 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. 748). 711. 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 749. 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. 749(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. 749(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 aa deflects, turning 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). Within a wide range of velocities, the separation of the layer in lat tices with relatively small pitch is not observed. NACA TMI 1393 of the streamline. On intersecting the primary. shicki, 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. 749(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. 749(c)). If "2 < ES, the expansion of the flow continues beyond the lat tice (fig. 749(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. 749) 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. 749(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. 750). 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 neighboring profiles and wakes. The flow spectras are given for two lattices: f = 0.543 (fig. 750) 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 751 (photographs (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 752 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 boundar:, 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? 753. 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 incrases. 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 754(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 corresponds to the flow conditions NACA TM 1393 shown in figure 749(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 754(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 waves 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 pressure 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. 749) 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 754(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. 755). 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 offdesign 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 onedimensional 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 756. At increased pressures behind the lattice, a system of two obliqule shocks is situated within the channel (fig. 756(a)). With an in~crease in pressure behind the lattice the shocks move toward the outlet section (figs. 756(b), (c), and (d)). Near design operation (figs. 756(e1 and (f)) primary and reflected shocks intersect on the convex su~rfae; behind the lattice a. trailingedge shock may be seen." The pressures distribution over the profile (fig. 757) 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 712. IMPULSE LATTICES IN SUPERSONIC FLOW When the velocities are practically sonic a Xshaped 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 neigh boring profile (fig. 758(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 acele~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. 758(b)). It mayr be assumed that for veloc ities. corresponding to the flow scheme in figure 758(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 wavyEshape shock (fig. 7EF(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 758(d). In the system 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 759 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. 760(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~verall waveshaped shock is f~rormed instead of the system of shocks shown in figure 758(b). The shape of thils wave ahead of a lattice of plates for various inlet angles is seen in figure 761. 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. 713. 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 overall characteristics of lattices accompanying a change of velocityI of the flow (M1 or M ).22 Figure 762(a) shows curves of the loss coefficients for reaction lattices as a function of M2 and thle in~let flow angle P1. Figure 762(b) gives similar curves for impulse lattices. The curves shiow that, depending 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. 762(a)) with those in a onedimensional 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 offdesign 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. 762(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. 762(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. 760). 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. 762(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). 714. 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 widesprea~d meth ods of computation are based on the onedimensional equations of flow. Assuming that the field of flow in the sections AB (fig. 763) 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 inclination of the flow in the overhang section, we arrive at the equa tion 6 = a~re sin 220 P10sin P."n ~2n (760) In the above equation q2 and q2= are 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 (760) 6 "ascsi(sin P2. are si 02(760a) With account taken of the losses, formuls (7r30a) can 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 k1 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  (~ k1l 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 (760a), the angle of deflection 5 de pends not only on the outflow velocity and the~ losses but also on the snJle P2n* Formula (760) 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 pressur~e in the outlet section, of the lattice for the regime considered maly be determined by the formula 2k "S = E,(sin P2n k+ (761) 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 (761). 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(1k) P2n 2 k+1 2 (sin P2n) k + 1 For the onedimensiona~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 (763) 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 (76~4) If in the section A~AB the parameters are critical, then = 1  + E2 k sq cos 8 The last expression together with equation (763) 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 (765) ( EfT NACA TM 1393 Approximately, for 8 4 100, we obtain S 2k (765a) 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 k1 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. 764(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 11 moving along the convex surface of the plate intersects both the primary and reflected expansion waves, while the streamline 22 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. 764(b)). The velocityr of the streamline 22 after passing through the primary expan sion wave is determined by the vector 12, while the velocity of the streamline 11 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 764(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 11 and 22) in the immediate neighborhood of the point A has the direction of the vector \2 (the dotdesh line in fig. 764(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. 764(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 xraxis at the given angle to CB (fig. 764(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. 764(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 (721): 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 90f(M2^~Tt can be established only experimentally. Then, replacing the actual lat tice 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 comparuison of the most widespread and accurate methods of comput ing the deviation angles with test data (for two reaction lattices) is shown in figure 765. It is seen from the latter that, for lattices of small pitch and consisting of profiles with thin trailing edges, formula (765) and the method of characteristics give results that satisfactorily agree withi experimcent. For small values of 8(s2 >, 0.35) the equation of continuity, (760(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 715. 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 threedimensional lat tices are not taken into account in the analyses of twodimensional 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. 766(a)). For sufficiently high ratios 3/a2 (see fig. 766(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. 766(b) and (c)) and induce a field of velocities normal to the streamlines of the primary flow (fig. .766(d)), which leads to a certain increase in the outlet angle of the flow from the lattice. In the photogra~phs of the wakes of the flow (fig. 767) 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 overall 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 768(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. 768(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. 769(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 stagnation in the nucleus 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. 769(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. 770) 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 there 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 inestigation of the threedimensional 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 771. The curves in figure 771 show clearly, the decrease in (p with an increase in M2 for M2 < 1. In correspondence with the abovementioned 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. 772). 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 scounit, 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 threedimensional 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. Siro~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 * *ta *rlr Y1 *r( rIl 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 73. 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 74 (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 74. 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) Circulatoryaxial flows. Figure 76. 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 angle of lattice of plates. t\A note: The stalacies should obvliousl;, be 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 rl9. Flow of conformal transformation of lattice of circles. NACA TM 1393 Figure 710. Computation of a lattice by the method of G. S. Samoilovich. Figure 711. Comparison of the theoretical and experimental pressure distribution over a latiee prof ile. Figure 713. 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 712. IDetermination of an equivalent lattice of circles. scn 0 s. NACA TM 1393 I\ bP P (b! 6, Scr Sdor (c) (d) Figure 714. Construction of flow stream. Figure 715. 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 716. 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 717. Forces actin on profile in lattice. NACA TM 1393 93 je, \ *89(  \\ o I~ (D 1( 15 1S 6. r 97d 7 oeo 3 P 3P rl YI La CJ  c \O 0 0 0 51 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 723. 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 724. 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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