MAC/lr/h-151^ 2 2 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1393 FLOW OF GAS THROUGH TURBINE LATTICES By M. E. Deich Translation "Technical Gas dynamics." (Tekhnickeskaia gazodinamika) ch. 7, 1953, Washington May 1956 I (2(^ ^"1^ ^ -? 6 T^' NATIONAL ADVISOEY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1393 FLOW OF GAS THROUGH TURBINE LATTICES* By M. E. Deich 7-1. GEOMETRICAL AND GASDYNAMICAL PARAMETERS OF THE LATTICES; FUNDAMENTALS OF FLOW THROUGH LATTICES The transformation of energy in a stage of a tirrbomachine is a re- sult of the interaction of the gas flow with the stationary and rotat- ing blades, which form the guide and impeller blade systems. The lattices of a turbine in the general case represent systems of blades of the same shape uniformly arranged on a certain surface of rev- olution. A particular case of a three-dimensional lattice is an annular lattice with radial blades arranged between coaxial cylindrical surfaces of revolution. In flowing through the lattice, the velocity and direction of the gas flow are changed, and a reaction force is thereby produced on the lattice. On the rotating lattices of a turbine this force performs work; the rotating lattices of compressors, on the contrary, increase the energy of the gas flowing through them. In stationary lattices an energy interchange with the surrounding medium does not occur; in this case the lattices bring about the required transformations of kinetic energy (velocity) and the deflection of the flow. Depending on the flow conditions and the corresponding geometrical parameters of the blade profile, three fundamental types of lattices are distinguished : (a) Converging flow type: the nozzle or guide (stationary) vanes and the reaction (rotating) lattices of turbines "Technical Gasdynamics ." (Tekhnickeskaia gazodinamika) ch. 7, 1953, pp. 312-420. MCA TM 1393 (b) Action or impulse (rotating) lattices of turbines (c) Diffuser: guide (stationary) and 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 I, the width of the lattice B, the pitch of the blades on the mean di- ameter t, the chord b, and other blade profile parameters (fig. 7-l) • There exist several methods of specifying the shape of a blade pro- file. The universal method of coordinates (fig. 7-2 (a)) has great ad- vantages. The methods shown in figures 7-2(b) and (c) are based on the idea of the mean line of a profile; the mean line may represent the geo- metric loci of the centers of inscribed circles or the centers of the chords connecting the points of tangency. The mean line is defined by coordinates, and the thickness distribution about the mean line is then independently given. For specifying the profiles of turbine lattices, consisting most frequently of thick, sharply curved profiles with small pitch, the methods shown in figure 7-2 (b) and (c) are inconvenient. The determination of the fundajnental dimensions, the construction of the pro- file, or its checking require complicated graphical work. The most wide- spread method of constructing the profile from a small number of adjoin- ing arcs of circles and segments of straight lines (fig. 7-2(d)) is ar- bitrary and tedious. If the ratio of the mean diameter of the lattice d to the height of the blade I is large, the lattice may, for the purpose of simplify- ing the problem, be considered as a straight row lattice. The shape of the space between the blades along the height may then be considered as constant. In the simplest case, assuming that the diameter of the lat- tice and the number of the blades increase without limit, we obtain a plane infinite lattice (fig. 7-l(c)). The passage from the cylindrical to the plane lattice is effected in the following manner: We pass two coaxial cylindrical sections of the annular lattice through the middle diameter d and through the di- ameter d + Ad. Assuming Ad to be small, we develop the resulting annular lattice of very small height on a plane. Increasing the number of blades to infinity, we obtain the plane infinite lattice shown in figure 7-l(c). WACA TM 1393 The assiimption 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 fonn. For example, the relative pitch of the profile is determined by the formulas * = b °^ *B = B The relative height (or length) of the blade, *b - b In certain cases in investigating the three-dimensional flow in a lat- tice, it is more convenient to define the relative height as a ag where ag is the width of the minimum cross section of the passage (fig. 7-1). A rectilinear lattice is referred to as a system of coordinates x, y, z where the direction x is termed the axis of the lattice (fig. 7-l(b)). All profiles must coincide in the translational displacement along the axis of the lattice. The pitch t of the lattice is equal to the distance between any two corresponding points. For a given profile shape, the shape of the interblade passage of the lattice depends, in addition to the pitch, on the angle Py, which is defined as the angle between the axis of the lattice and the chord of the profile (fig. 7-l(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 Pgn ("the angle between the tan- gent to the mean line at the trailing edge and the axis of the lattice) . In certain cases, for a straight -backed profile, the angle Pgj^ 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 ensixre 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 MCA TM 1393 the mechanical efficiency over a wide range of Mach and Reynolds niombers 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 \insteady three-dimensional motion of a viscous compressible fluid presents great difficulties. A good approach to the solution of this problem^ as in general to the solution of most technical problems, consists of the investigation of simplified models which retain most of the essential characteristics of the actual process. Succeeding analyses then develop the effect of secondary factors. At the present time the most highly developed theory is that of the steady two-dimensional flow through the lattice of an ideal incompressi- ble fluid. Such a flow may be considered as the limiting case of the actual flow in a lattice at small flow velocities (small Mach numbers, M < 0.3 - 0.5) and with small effect of the viscosity (large Reynolds numbers, Re > lo'^ - 10^). 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 thermodynamic computation of the stages of the tiirbomachine . Let us consider several featxires of a plane potential flow of an ideal incompressible fluid for the case of the flow over the blades of a reaction tiorbine (fig. 7-3). On account of the repeated character of the flow, it is sufficient to study the flow in a single interblade pas- sage or the flow about a single blade. In figure 7-3 (a) the continuous curves represent the streamlines T = constant; the dotted curves repre- sent the equipotential lines # = constant, normal to the streamlines. 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 d<|. dY f^ .\ ^ = ds = - d^ (^-^^ where S and n are the curvilinear distances along the streamlines and equipotential lines, respectively. MCA TM 1393 The differentials may be approximately replaced by finite incre- ments , and we thus obtain Af _ AY "^ "^ AS "^ " Z^ If A^ = AT = constant at each pointy then Z^ " An. In this case, the individual elements of the orthogonal network of lines , ^ = constant and Y = constant, become squares in the limit (as AS -*■ and An -♦■ O) . 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 the turbomachine. The hodograph plane (fig. 7-3(b)) provides another important method of representing the flow. At each point along a streamline or equipo- tential lines (fig. 7-3(a)) the velocity has a definite magnitude and direction. When these velocity vectors associated with a given stream- line or equipotential line are drawn from a common origin and their ter- mini are connected (fig. 7-3(b)), the corresponding streamline or equi- potential line is established in the hodograph plane. The streamlines and potential lines thus drawn also form a square network. This network may now be conceived to represent a flow in the usual sense. The stream- lines that originally represented the blades are the boundaries for the new flow. The new flow itself is produced by a so-called vortex- soiirce and a vortex-sink. The vortex source is located at the end of the ve- locity vector c-L (the velocity at an infinite distance ahead of the lattice). The vortex-sink is at the end of the vector Cg (the velocity at an infinite distance behind the lattice) . The origin and the termini of c-^ and cg form the velocity triangle of the lattice. From the equality of the flow rate ahead of and behind the lattice, c-[_t sin p-[^ = Cgt sin Pg it follows that the projections of the velocities c-|^ and Cg on the normal to the axis of the -lattice are equal or that the straight line passing through the ends of the vectors c-|_ and Cg in the plane of the hodograph 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 c-i and Cg, respectively. MCA TM 1393 Of great interest is the distribution of the velocity or pressure on the surface of the blade. Figure 7-3 (c) shows the approximate dis- tribution of the relative velocities c" = c/cg and relative pressures /l 2 _2 p = (p - PgVp" PCp = 1 - c" as a function of the distance S along the profile. If the magnitude c-|_ and the direction 3-|. of "the veloc- ity at infinity ahead of the profile are known and also the position of the point of convergence of the flow Og (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 c-^ does not alter the shape of the streamlines or equipotential lines . Neither does it alter the relative velocity c" or the relative pressure p. At finite distances from the lattice, the field of velocities and pressures is not uniform. The streamlines (for p-j^ ^ 90°) are wave shaped, and their shape is generally different from that at infinity; moreover, it periodically varies along the cascade axis. In correspond- ence with the conditions of continuity and in the absence of vorticity, the mean velocity along any line ab (fig. 7-3(a)) between two points separated 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 O-j^ (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 = (fig. 7-3(c)), the velocity along the pro- file sharply increases. Depending on the shape of the leading edge and also on the direction of the inlet velocity (inlet angle P>^) , the ve- locity near the branch point may have one or two maxima. At the convex 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 contoijr . In particular, a narrowing of the passage, characteristic of a turbine lattice of the reaction type, leads to an acceleration of the flow; in an impulse turbine having approximately constant passage width and curva- ture, the velocity and pressure change only slightly in the direction of flow (fig. 7-4); in a compressor lattice, the interblade passage widens and the velocity correspondingly decreases (fig. 7-4A) . An increase in the curvature of the convex parts of the blade leads to an increase in velocity, and vice versa. For a .discontinuous change in curvature at the points of junction of arcs 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 a.ctual 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 thickness^ the velocity may have one or two maxima; at the actual leading and trail- ing edges ^ the velocity must drop to zero. The actual trailing edge is the point of the tail where the curvature is greatest. At a large dis- tance behind the lattice, the direction of flow is determined by the angle Pg • Figure 7-5 shows the approximate effect of the inlet angle ^-^, the pitch t, and the blade setting angle p„ on the distribution of the relative velocity over a blade of the reaction- type turbine lattice. A change in the angle P-, (fig. 7-5 (a)) causes the branch point 0-^ to be displaced along the profile. The design entry angle to the lattice may be considered as the angle for which the point 0-]_ coincides with the point of maxim.um 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 eg turns in the same direction as the vec- tor of the inlet velocity; for example, on decreasing the angle P-j, from its design value, the exit angle P2 increases. It should be re- marked that the effect of inlet flow angle on outlet flow angle is very small in conventional turbine lattices. When the pitch t is increased by a translational shift of the profile (fig. 7-5 (b)) while keeping the inlet flow angle p-j^ constant, the branch point 0]_ 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 and inlet flow angle) changes the exit angle Pg. The change in P2 is practically the same as the change in setting angle (fig. 7-5 (c)). On rotating the profiles in the direction of decrease of the exit angle pg, the corresponding velocities on the profile decrease; the branch point 0]_ 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 Pi- The case of an infinitely thin edge is not considered because it has no practical significance. MCA 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. Experience shows that the static pressure is constant over parts of the profile behind the point of separation. The features of a flow with separation can be approximately taken into ac- count in a so-called stream model of the flow of an ideal fluid. A zone of constant pressure is assumed to exist in this flow. At the boundary between this zone and the main flow, the velocity is constant, at the value which corresponds to the static pressure in the zones. In the plane of the hodograph, arcs of circles correspond to the boundaries of the separated zones. The radius of an arc is equal to the velocity at the boundary of the zone. Flow separation always occurs at the trailing edge of a blade. 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 then 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 of frictional forces also causes low pressiore 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 pressiire, decrease in the exit angle, and losses of kinetic energy similar to the losses in sudden expansion. The parameters of the equalizing flow are obtained by the simultaneous application of the equations of continuity, momentiim, and energy (see sec. 7-7). 7-2. THEORETICAL METHODS OF INVESTIGATION OF PLANE POTENTIAL FLOW OF INCOMPRESSIBLE FLUID THROUGH A LATTICE There are two problems in the theory of lattices that have the greatest significance. 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 0„ on the profile. Of greatest interest is the velocity at infinity behind the lattice. The determination of these quantities may be considered as the fundamen- tal object of the solution of the direct problem. The inverse problem is that of theoretically constructing the lattice when the flow about it is either known or easily determined for a given velocity triangle. Of MCA TM 1393 practical importance is the problem of constructing such a lattice with a velocity distribution over the surface of a profile which is rational and which assumes small kinetic -energy losses in the actual flow. It was remarked 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 not depend on the absolute magnitude of the flow velocity. Moreover, for the same boundaries, the different potential flows of an incompressible fluid may be summed. For example, any flow of an ideal incompressible fluid through a lattice may be considered as the sum of two or several flows through the same lattice. In figure 7-6 the flow through the lattice is represented as the sxim of two flows: a noncirculatory (irrotational) (fig. 7-6(b)) and a circulatory axial (fig. 7-6(c)). In the irrotational flow there is no circulation of velocity about the profile, or, in other words, the lattice does not change the direction of the flow; moreover, this direction is chosen such that the point of convergence of the flow is on the trailing edge. In a rotational-axial flow the direction of the velocity at infinity is parallel to the axis of the lattice; the magnitude of the circulation or the ratio Acg/Ac-^ = 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 and 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 rotational-axial flows. If it is taken into account that the magnitudes of the relative velocities do not depend on their absolute values in each of these flows, it is possible to find in a simple manner two important properties of the flow of an incompressible fluid through a lattice. First, there exists a linear relation between the cotangents of the inlet and outlet flow angles of any given lattice. From the velocity triangle (fig. 7-6(a)), notice that cot Pq - cot Pg ^2 cot pQ " ^O't Pi ~ '^'^1 = m = constant (7-2) where cot Pi corresponds to the angle Pi assiuned in figure 7-6(a). For a given lattice, the magnitude of the coefficient m can be com- puted theoretically. For a lattice of flat plates in particular, the coefficient m is related to the relative pitch t/b and the setting angle Pq by the equation -r- = 2 cos Po^rc tan r— r — cot Pq + sin PqIh — (7-3) 10 NACA TM 1393 As may be seen from the graph of figure 1-1 , the coefficient m decreases with a decrease in pitch, so that the exit angle Pg ^P~ proaches the setting angle Pq °f '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 ^i has the same exit angle Pg ^^ the given lattice of airfoils. In present-day tiirbine lattices the ratio b/t of an equivalent plate lattice is not less than 1.3; the angle Pq is between 15° and 40°, and the angle p-, is between 90° and 20°. The magnitude of the coefficient m is not greater than 0.015; the angle of the velocity behind the lattice therefore differs from the angle Pq for the equivalent flat plates by no more than 1°. For present-day compressor lattices this deviation may be as high as Z'' Second, the magnitude of the relative velocity on the profile of any lattice depends linearly on the cotangent of the exit angle. In fact 3 _ c c^u ^u %u ^0 =u . ^^1 c = — = + — = • — + . Cg cg cg Cq cg Ac^ cg ^bu Utilizing the obvious correlations (fig. 7-6 (a)), c-^^^ = and c c„ = -r — we obtain ^ Ac-[_ _ _ sin Pg _ ^ " '^'bu sin Po "^ ^u(cot Pq - cot p^lsin Pg (7-4) As was said, in present-day turbine lattices, Pg * Pq ~ 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, c" « c'-jj^ + c'^(cot Pg - cot p-|_)sin Pg (7-4a) NACA note : This ratio is written as t/b in original text . %ACA note: c^u is the irrotational flow, fig. 7-6(b), and Cu is the circulatory flow, fig. 7-6(c). sin Pg sin Po ,"| TAci ^ (cot Po - cot Pi)"| NACA TM 1393 11 At any point of the profile where c^ = (fig. 7-6 (c)) the rela- tive velocity c" 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 3-|_, then the distribution c" can be computed for angle P-j^ with the aid of equation (7-4a). Of practical significance in the theory of the two-dimensional mo- tions of an incompressible fluid is the mathematical theory of the func- tions of a complex variable. Without entering the mathematical side of this problem^ the discussion of which is given in any modern coizrse 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- cal transformation (extension and compression or conversely) of a part of the plane (region) 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 transformation may go over into a second orthogonal square network. This property explains the significance of conformal 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 ciorves $ = constant and f = constant of this flow into a new orthogonal square network, which may 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 a simpler 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#/dS) . Several examples of the con- formal transformation of lattices are shown in figure 7-8. The above defined equivalent lattice of plates is obtained by means of such a conformal transformation in which the flow region outside the airfoil lattice is transformed into the flow region outside the plate lattice. The infinity of the plane of the lattice of airfoils goes over without extension or rotation into the infinity of the plane of the plate 12 NACA TM 1393 lattice. The pitch of the lattice is maintained, and the rear stagna- tion 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 conformal transformation is completely determined by the above condition. The noncirculatory flow through the airfoil lattice (fig. 7-8(a)) corre- sponds to the noncirculatory flow through the lattice of plates (fig. 7-8(b)). The singular points, at which the conformality of the trans- formation does not hold , are the edges of the equivalent plates . Con- sidering the corresponding noncirculatory 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 the 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 $ = constant and f = 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 axe 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 # = constant and T = constant of the flow through one lattice goes over into the network of the same lines of the equivalent flow of the other lattice. Of great significance is the conformal transformation of a lattice of airfoil profiles into a lattice of circles (fig. 7-8(c)). In con- trast to the equivalent network of plates, characterized by two param- eters (t/b and Po)^ "the equivalent network of circles is determined by only one parameter, the relative diameter (density of the lattice) 2r/t = 2r. 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 Og in the circle lattice is not uniquely determined by the relative diameter, however. An example of the conformal transformation of the region of flow in one period of a profile lattice into a bounded region is shown in figure 7-8 (d) . Infinity ahead of the lattice corresponds to the center of the circle ("",); the infinity behind the lattice corresponds to a certain point on the horizontal radius ("p); the flow lines in a period to a segment between the points ■»]_ and "g. 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 13 '»-]_ and "»2 "to the radius of the circle. For modern tiorbine 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 arc near the point "p. 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 "g • The flow about the lattice is transformed into a flow of a special form produced by a vortex source at the point <*>2_ and a vortex sink at the point "g • '^''^ "^^^ 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 hydrodynamic 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 9 for transverse, longitudinal, and purely circulatory flows about lattices of circles with values of the spacing 2r = 0.20 - 0.90 (for circles in contact 2r = l.O). By summing the flows considered, any flow through a circle lattice can be obtained (fig. 7-9). The values of the velocity potentials and the magnitudes of the velocities on a cir- cle are obtained by summation from tabulated values multiplied by certain constants , the magnitudes of which are found from the given direction of the velocity at infinity ahead of the lattice and the condition of zero velocity at the branch points of the flow given on the circle. By making use of the solution for the lattices of circles, the solution of the di- rect problem, that is, the determination of the velocity on the surface of the blade in the given lattice for given inlet angle, reduces to the problem of obtaining an equivalent lattice of circles and then obtaining a conformal correspondence of the points of the blade contour in the 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. 14 MCA TM 1393 The exact solution obtained by G. S. Samoilovich may broadly be de- scribed as follows. Firsts by one of the known methods, a conformal transformation is obtained which maps the exterior of a single circle into the exterior of a single profile (fig. 7-10(a)). Then, from the condition of conformal correspondence of the exterior of the lattice of profiles and the exterior of the lattice of circles, the spacing of the equivalent lattice of circles 2r (fig. 7-10(b)) is obtained. The spac- ing 2r depends on the pitch of the profile lattice and the angle at which they are set. In the example considered, 2r = 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 A0 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 A0 determines the correspondence of points of the profile in the pro- file and circle lattices. By making 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 p-i_. In figure 7-11 a comparison is shown of the experimental and theo- retical distribution of the nondimensional pressure p over the profile of a lattice for the example considered with p-^ = 90 . 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 Mg numbers is found to be within the limits of accuracy of the measurements. There should be noted the characteris- tic divergence between the experimental and theoretical values of p on the back of the blade, produced by separation of the flow. The velocity at each point of the blade in a lattice differs from the velocity at the same point of an isolated blade (for equal magnitude and direction of the velocity of the approaching flow and the same rear stagnation point Og); 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 p-j_ whenever one flow about it is known. Suppose, for example, there is known the distribution of the velocity potential on a pro- file of the lattice with pitch t = 1 for irrotational flow with inlet angle p,-^ = 90° and velocity at infinity c-i_ = cg = 1 (fig. 7-12 (a)). NACA TM 1393 15 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 ^2.2 ^^ ^^® forward and rear stagnation points as a function of the lattice spacing with t = 1 and c-]_ = Cg = 1 (fig- 7_i2(b)). The value of ^^^2 ^^ ^^^ circle lattice coincides with the same potential difference in the profile lattice for the single value of the spacing 2r/t characterizing the equivalent lattice of circles (fig. 7-12(c)). The conformal correspondence of the points of the profile and the circle is found by equating the known velocity potentials ^ on the profile in the lattice with those on a circle in the equivalent circle lattice (fig. 7-13). For determining the velocity distribution on the profile for any inlet angle 3-]_, it is necessary to determine, by employing tables of flow about circle lattices, the distribution of the velocity potential # or velocity Cj^ on a circle in the circle lattice. The proper inlet flow angle P-i must be used, and the rear stagnation point of the cir- cle must correspond to the trailing edge of the profile (fig. 7-12). From the known correspondence of the points of the profile and circle in the lattices it is possible to construct the velocity potential as a function of the length of arc of the profile, the differentiation of which 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 d# _ d# ^ d£ d£ ^ = dS = dS ' dS = ^k dS The velocity c^ on a circle of the lattice of circles is determined for any inlet angle with the aid of tables, and the derivative d0/dS is obtained only once from the graph shown in figure 7-13. If the distribution of the velocity c on a profile of the lattice is known, then to determine the conformal correspondence it is necessary first to find the velocity potential # = r cdS where it is assumed that S = (or $ = O) at the branch point. Practically, for lattices with the spacings that are actually em- ployed in turbines, the above problem is solved considerably simplified 16 NACA TM 1393 by the method of conformal mapping of the lattice, not on a lattice of circles but on the interior of a circle (fig. 7-8(d)). In this case, it may be approximately assiimed that the sink ("„) is situated on the circle, and the velocity at each point of the profile computed for any inlet angle p-|_ by the formula sin^^ - Pi) o{ ;~§T^=i C = 7-. <--C in which the angle in the circle 6 is determined graphically from the equation 1 /q q\ - cj^t sin pJ- cot Pi - In sin -) $ = - The primes denote the magnitudes determined for a new inlet angle (P-,). At the branching point the velocity potential ' = and 6 = 2p_ . The converse problem of the theory of hydrodynamic 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 hydrodynamic ally 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- pared with an ideal fluid) about a lattice may be determined with the aid of the boundary-layer theory, if the theoretical distribution of the velocity on the profile is known. NACA TM 1393 17 With account taken of what has "been said, of all possible velocity distributions, the most suitable hydrodynamically may be considered that for which the losses in friction are a minimiim and the condition of con- tinuous flow is satisfied over the entire profile. (See section 7-6.) Any continuous velocity distribution having a minimum number of diffuser parts and a minimum 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 veloc- ity distribution is the method of the hodograph. 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 applying the hodo- graph method for constructing lattices with hydrodynamically suitable velocity distribution was pointed out by Weinig. A practical applica- tion of the hodograph method was obtained by L. A. Simonov, who employed it for constructing theoretical profiles and lattices. The construction of lattices by the method of the hodograph is based on the fact that the region of flow through a lattice of an ideal incompressible fluid is conformally transformed into another region in its velocity hodograph (see fig. 7-3). As has already been said, to the flow about a lattice in the region of the hodograph there corresponds a special flow of an ideal incompressible fluid produced by a vortex source at the end of the vector c-]_ and a vortex sink at the end of the vector Cg (see fig. 7-3). Taking into account that to a displacement by a pitch ahead of or behind the lattice there corresponds a passage around the vortex source or sink, we can determine the flow rate of the source or sink, ^1 ~ *^1^ ^^"^ Pi = - Qg = Cgt sin pg the circulation of the vortex source, T-^ = c-[_t cos p-]_ and the circulation of the vortex sink T2 - - Cgt cos p>2 At the branching point 0-, and the rear stagnation point Og^ 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 c-, = Cg and the contour of the hodograph enveloping these vectors . 18 MCA TM 1393 Let us consider in greater detail the procedure of constructing the stream flow through a lattice (fig. 7-14). It should he remarked that the direct problem of determining the flow through a given lattice (with no rear stagnation points in the stream) has no effective solution, and the method of the hodograph is practically the only one which permits constructing such flows . The contour of the hodograph of the flow through a lattice with convergence point of the stream at the trailing edge (fig. 7-14 (a)) passes through the point c = and through the end of the vector Cg* The arc S^^Sg corresponds to the boundaries of the flows between one infinity and the other in the plane of the lattice. In the case consid- ered of a turbine lattice for a given hodograph, the absence of diffuser parts on the profile may be assured (fig. 7-14(d)). To construct the lattice, it is necessary to find the flow of an ideal incompressible fluid in the plane of the hodograph, because of a vortex source at the end of the vector cj^ with circulation r = c-|_t cos p-j_ and a sink at the end of the vector C2. The flow rate from the source and sink is Q = C]_t sin |3]_ The magnitudes of the velocity and the nondimensional magnitude t (see fig. 7-14(b)) are connected by the equation of continuity (see sec. 7-7) c-j^t sin 3-]^ = (l - T)c2t sin Pg^ where T = -r- For constructing the profile, it is sufficient to find only the distribution of the velocity potential $ over the contour of the hodo- graph by the method, for example, of conformal transformation of the hodograph into the interior of a circle (fig. 7-14(b)) for which the vortex source goes over into the center of the circle and the sink into the point of the circle 0=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 $ = i(r|-Qlnsin|) NACA TM 1393 • ' 19 At the branch point of the flow, d*/de = or cot bJz = T/q, whence The coincidence of the branch point in the hodograph plane with the point c = is equivalent to the conformal correspondence of the point c = and the point 9 = 6q. With the contour of the hodograph arbi- trarily given, the branch point in the hodograph plane will not, in gen- eral, coincide with the point c = 0. The coincidence of these points is assured, however, by a suitable specification of the shape of the hodograph. In the example of figure 7-14, this coincidence was obtained by choosing the length of the segment P of the hodograph plane (fig. 7-14(a)). After determining the velocity potential on the hodograph contour, the profile is constructed by graphical integration of the expression dS = d*/c The accuracy of the computations and of the construction is checked by comparing the given and obtained boundary conditions a. The neighbor- ing profile of the lattice is at the pitch distance t (fig. 7-14(c)). The velocity distribution over the profiles of the constructed lat- tice for given inlet angle corresponds to the given hodograph. The ve- locity distribution for any other inlet angle can be found simply. For this it is necessary to make use of the known conformal transformation of the region of the hodograph on the interior of a circle. Since the hodograph 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 velocity potential fi, 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' = cjt cos p', Q* = c't sin pj With the aid of evident substitutions and transformations we obtain Q _ d*^ d§J_ d$J_ ^ de _ r' - Q' cot -g ^'"ds'^d^'^^'de'cH'^^ _ 9 ^ r - Q cot p Q cot P' - cot — c-^sin p' sml cot P]^ - cot — C]^sln Pi ^-^^(g ~ Plj^l where the primes denote the changed quantities . (*) 20 NACA TM 1393 We emphasize that formula (*), with change in inlet angle Pi, de- termines the magnitude of the velocity on the boundaries of the con- structed flow with "solidified" streams passing off to infinity. Al- though the exit angle Pg 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 (*) permits computing with sufficient accuracy the 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 (*) , 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 p-^ on the exit angle Pg . For computing the velocity distribution for sev- eral inlet angles P' , formula (*) can be applied only once, and then the linear dependence of the relative velocity c/co on cot P-i must be employed . 7-3. ELECTRO -HYDRODYWAMIC ANALOGY The distribution of the velocity potential in a lattice of airfoils for any irrotational flow about it may be experimentally obtained by the method of electro -hydrodynamic analogy (abbreviated EHDA) . This method was first applied to problems of the theory of hydrodynamic lattices by L. A. Simonov. Until 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 analogy between the differ- 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 homogeneous 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 simplest and most widespread method of applying the EHDA 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 (lO to SSiran) . The water is poured into a flat vessel (gener- ally of rectangular shape) of nonconductive material. The electric cur- rent passes between the electrodes 1 arranged at opposite edges of the vessel (fig. 7-15). A small quantity of salt and carbonic acid which is WACA TM 1393 21 contained in the water assiires sufficient conductivity. For avoiding the polarization of the electrodes in the electrolysis of the water, a low-frequency, variable current (generally using a circuit voltage of 110 or 220 volts alternating current) is connected to the electrodes. The blades of the lattice are made of an insulator material, such as paraffin or plastiline. Several blades of the lattice are studied; for all practical purposes, it is sufficient to study five blades. The measurement of the electric potentials in the bath is generally made by the compensation method. To the parallel current-conducting electrodes, a voltage divider (potentiometer) is connected, the movable contact of which is connected, through a zero current indicator (null indicator), to a feeler or probe situated at the point of measurement of the poten- tial. The probe is a thin straight needle moving along the water per- pendicular to its surface. The simplest and sufficiently accurate zero indicators of an alternating current are radio earphones or a speaker connected through a low-frequency amplifier. For the potentiometer, there is shown in figure 7-15 a water rheostat consisting of a long ves- 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 earphones attains a minimum. The advantage of the described compensation method of measurement is the absence of the effect of the apparatus on the absolute value of the po- tential at the point of measurement. Instead of an electrolytic bath, it is possible to use electro- conductive paper. The blade shapes are then cut from the paper. In this case a direct-current source and highly sensitive galvanometers can be used . The electro-hydrodynamic analogy may be conveniently applied to the direct problem in theory of hydrodynamic lattices . It may be used to establish the conformal transformation of a given lattice to the equiva- lent lattice of circles. According to the above described method (fig. 7-12), it is sufficient for this purpose to know the distribution of the velocity potential on a profile of the lattice for any convenient flow about it, as for example, an irrotational flow with ^2_ - ^2 ~ ^^ ' c-j^ = Cg = 1, and t = 1. The magnitude of the measured electric poten- tials (fig. 7-15) must then be divided by the potential drop (measured in the same units) over the distance of one pitch. This measurement must be made at a remote distance from the lattice and certainly not nearer to it than 2t. In obtaining the conformal transformation of a lattice of airfoil profiles into its equivalent lattice of circles with the aid of the EHDA, 22 NACA 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 EHDA method can also be applied for di- rectly measuring the velocity potential and even the velocity itself in any flow of an ideal fluid, including flow with stagnation point at trailing edge. The modeling scheme is indicated in figure 7-16. The exact form of the bounding walls (streamlines 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 and 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 measi:iring 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 EHDA method is employed also for investigating the flow of an ideal gas with subsonic velocities. For this purpose an electrolytic layer of variable thickness or a net- work model is applied. The electrical model in the plane of the veloc- ity hodograph permits obtaining accurate solutions without successive approximations . 7-4. FORCES ACTING ON AN AIRFOIL IN A LATTICE; THEOREM OF JOUKOWSKY FOR LATTICES For determining the forces acting on an airfoil, we isolate a por- tion of the flow, as shown in figures 7-17(a) and (b). The external boundaries of the isolated region are defined by the segments ab and dc, parallel to the axis of the lattice and of length equal to the pitch t. The lines ab and dc, strictly speaking, shoiild 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 P^ and T^. The magnitude of these forces may be determined from the momentum equation. In the direction normal to the axis of the lattice, the change in the momentum is equal to °^(cal - ^as) = *(P2 - Pi) - ^a MCA TM 1393 23 where P^ is the component of the force P in the direction normal to the SLxis of the lattice; the mass rate of flow of the gas per second is determined from the formula " = Pl^al* = P2^a2* Then Pa = *C(p2<^a2 " Pl^al^ + P2 " Pl^ (^-5) The projection of the force P on the axis of the lattice may be ex- pressed by the equation Pu = tPl^alC^ul - =u2) (^-6) The forces P^ and Pg^ refer to a profile having a unit span. Equations (7-5) and (7-6) may be represented in another form by ex- pressing the forces P^ and Pg^ in terms of the circulation T and the flow parameters at the inlet and outlet of the flow. According to the equation of continuity^ Pl^al = P2Ca2 = (^^a where p is the mean density of the gas. The velocity Cg^ is chosen such that ^al + ^a2 It is easily shown that we then have Sp-j^Pg (7-7) The circulation about the profile is equal to r = t(c^i - c^2) C^-s^ since the line integral along the equidistant lines ad and be are equal and opposite. 24 NACA TM 1393 (7-6) After simple transformations, we olDtain from equations (7-5) and ^a = -tfPs - ^1 - P'^a^^al " ^ag^^ (7-9) Pu = pr^E (7-10) We make use of the equation of energy k Pi ^2 Since k ^2 2 k-lp3_ 2 k-lp2 2 2,2 1 al ul c2 =s c2 + c2 2 a2 u2 2 2 ^1 - ^2 = c„ I c a^^al - ^a2 ) + c^(c •ul " '-u2' (7-11) where =ul + =u2 2 and we obtain from the equation of energy k ^a('=al - ^a2) _[P2 Pl\ A'2 - ^)-'- (=ul - Cu2) Substituting this expression in equation (7-9) and taking into account formiila (7-8) we obtain Pa = t P2 - Pi - ^ k /P2 PlV + prcu Pu = pFcj (7-12) (7-13) The force Pa given by expression (7-12) is conveniently represented in the form of a sum of two forces P = P , + AP a al a WACA TM 1393 25 where and ■ ^a = tL - PI - kT-T P^ - ^)j (7-1^ The resultant of the forces P^j and P^ we will denote "by P„ and the over-all resultant force by P (see fig. 7-17). It is evident that or Py = Pu + Pal -*■-*■ -> P = Pu + Pa P = Py + AP^ The force P„ is determined by the formula Py= Vp^ ^ P|l Substituting the values P^ and Pg^-]. '^^ obtain Py = Pr ^cl + c But 2 2 2 ^u + ^a = ^ where c is the mean vector velocity c = =1 + ^2 Hence ^ the expression for P„ in the flow about a lattice has the same form as the lift force of an isolated airfoil: Py = pTc (7-15) 26 NACA TM 1393 The direction of the force P is perpendicular to the direction of the mean vector velocity c. This follows from the obvious equation c P tan 3 = — = ^- ^u -^al 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 P is determined by the rotation of the velocity o vector c by 90 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, P 2\pi ^ P2) 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 Joiikowsky force (P ) and the additional force (APg^) perpendicular to the axis of the lattice: -► -»■ -*■ P = Py + ^a It is important to note that the characters of the forces P„ and zXPg^ are different. Whereas the force P,, depends on the circulation of the flow and becomes zero for F = 0, the force AP^ does not depend directly on the circulation. The force acting on the profile was determined for the general mo- 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 the equations pg = p^^ and pg = p-^ must be valid; hence, /SP^^ = and Pjj = 0. 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 = pTC MCA note: This result is at least partially dependent on the selection of the mean velocity and mean density. WACA TM 1393 27 where p and c are the density and velocity of the flow^ respectively. 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 incompressible fluid about a lattice, it must be observed first of all that in equation (7-14) the second term on the right side is proportional to the change of the poten- tial energy of the flow (with account taken of the hydraulic losses); that is J 2 2 ^2 In this case of an incompressible fluid, p^^ = po = p, and the energy equation gives 2 2 ^1 - ^2 ^ P2t - Pi 2 " p where ^^, is the theoretical pressure in the absence of losses. Hence, ^a = -t(P2t - Ps) The pressure difference pg, = po is equal to the pressure loss in the lattice Pst - P2 =" ^Pn and 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 Ap^^ should not be confused with the pressure difference p - p, ) . In the absence of losses, Ap^^ =s and AP^ = 0. In this case the resultant force is equal to the Joiikowsky force P = Py = pre 28 MCA TM 1393 This result for the lattice was obtained by N. E. Joiikowsky in 1912. 7-5. FUNDAMENTAL 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 the procedure adopted for thermodynamic analyses . The energy characteristics permit evaluating the effectiveness of the process of energy transformation in the stages of the tiorbomachines . The 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 and radial forces are as- sumed in the form 2Pu , X [Note: C^ is a coefficient.] and 2P c' = — V C^-i^) kp^M^b where po and Mo are the static pressure and nondimensional velocity behind the lattice. Analogously, the other aerodynamic coefficients Cx and 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 c -!h '"u - pi u 5The possibility of generalizing the Joiikowsky theorem to the case of the flow of a -compressible fluid through a lattice was first pointed out by B. S. Stechnkin in 1944. The exact solution was obtained by L. I. Sedov in 1948. The basis of the approximate theorem 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 adiabatic flow was given by A. N. Sherstyuk . MCA TM 1393 29 ■where P^ is the peripheral force on unit length of the profile corre- sponding to the "ideal" rectangular distribution of the tangential pres- svtre (fig. 7-18). Evidently, for an incompressible fluid (with low in- let velocity) - = (Pi - V^)B = f P .2. c:b The magnitude P^ is determined by formula (7-6); then ^u = B 2Ca( a'^^ul ■u2 ) Noting that Co = '2 sin Pg we obtain finally Cu = and (c^ - c^g) = Ca,(cot 3-|_ + cot pg) 2 sin PgSindS^ + P^) ^ sin P2_ B (7-18) 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. % = %2/^l or, after simple transformations ^T.= 1 - 1m2 2t (-) \P02J k-1 k (7-19) where Pqi^ P02 ^^^ ^^^ stagnation pressvires ahead of and behind the lattice and Mg-j^ is the Mach number behind the lattice in the case of isentropic flow. Formula (7-19) is suitable for determining the efficiency of a com- pressor lattice. The coefficient of losses of kinetic energy is defined by the obvi- ous expression ^p = 1 - ^p (7-20) 30 NACA TM 1393 The real flow at the inlet and outlet of the lattice 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 p-^c-^sin P2_dt = / pgCgsin (Bgdt '-'0 Introducing a reduced flow rate q, we obtain after elementary transformations .t t sin P^.'^'t ~ I ^^'^ Pg*^"^ Vt7~ '■02 For Tq-i_ = Tq2 = Tq = constant, averaging of the equation of con- tinuity gives ^t Jo (p^q sin |3)^p = - / p^q sin p dt 'o The peripheral force is in this case determined from the equation P^ = I PiC^ sin 3-]^cos 3]^dt -I P2*^2 ^-'■'^ Pg*^^^ Pgdt ^0 ^0 or, again introducing the reduced flow rate q, we obtain Pu = ^^^ I pQ-|_q-]_X-]_sin 2P-|_dt - I pQgqgXg^^'^ 2P2dtl yo ^0 J ^NACA note; r P "^0 C ^ = VgR Po t -/TiRT 7 c NACA note : X = — where a- is the speed of sound when c is ^* 5onic . See eq. (7-25). NACA TM 1393 31 Averaging of the expressions under the integral sign gives 1 r^ (p^qX sin 23) = - / PqQ.^ sin 2(3dt Jo From the equation of energ>'^ the temperature of the flow behind the lattice is averaged^ and the following expression is involved: IT n / (PQq^*'^sin p)^p = :^ / p^qX^sin 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- sidered 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 P-]_ and Pg determined by the momentum equation. The average values, by the equation of momentum, of the projections of the velocity'- behind the lattice are equal to kg p* (cgcos P2)cp = "G" % J P02 0, 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 O) in the case of small velocities (incompressible flow), the momentum thickness 5** is computed on the assumption that the velocity distribution in the boundary layer is given by an exponential law. In the work of W. M. Markov, there is shown the satisfactory agree- ment of the experimental data with the computed results . On figure 7-20 is given the velocity distribution in the boundary layer on the convex surface of the blade of a turbine lattice near the exit edge. The character of the change of the momentum thickness &** along the blade of a turbine lattice may be seen in figiore 7-2l(a) and (b), where the experim.ental values of & are also indicated. For comput- ing the layer, the experimental curves of the velocity distribution \^ of the external flow were used. As may be seen from the curves in fig- ure 7-21, the results of the computation satisfactorily agree with the test data . On the basis of the computational results of the boundary layer on the concave and convex surfaces of the blade, the friction loss coef- ficient in the lattice is computed. The fundamental characteristics of the lattice may be expressed in terms of the known parameters of the boundary layer, Sg and Sg ^ which are determined at the exit edge of the bla.de. Denoting as before (see fig. 7-19(a)) by Ug and pg the velocity and density at a point in the boundary layer at the exit edge, and by Ugg the velocity at the external boundary of the boundary layer in the same section (the veloc- ity of the potential flow), we set up the equation for the coefficient ^_ of the friction losses. NACA TM 1393 35 The ~ kinetic -energy loss in the boundary layer may be expressed by the equation Ahr = 2 / P2^2(^20 - ^2)<^y We transform this equation into the form pSn Ahj. = 1 - ^■^ ^''2 3 PO , ^ ,ir~ ^20P2 -^ <3y It is not difficult to obtain PO PO Po (^ k k + 1 1 k '^20J gRTo 1 - k k "T" T ^2 (7-25) since- ^^ (^ - FTT ^2o) Po k-l 1 -^-^^1 k + 1 ^ where ^2 ^2 = ~ ^"<^ >^20 '20 We set &2 -«-»-»• J 1 k - 1 .2 " k + 1 20 1 k - 1 2 ' k + 1 ^2 u. 2_ 2 20, '20 <3y (7-26) TTACA note: This presiimes 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. 36 NACA TM 1393 Then referring to equation (7-25), the energy loss may be written in the form 1 =*** ^0 3 Summing the losses on the convex and concave surface of the blade, we obtain ^^,np= 2iRT^P2 ^20),^+(S2 ^2o)^o^J (^-27) The magnitude & has a concrete physical meaning; by analogy with the momentum-loss thickness 5,5 is equal to the thickness of the fluid layer moving with the velocity UgQ 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 ^^,np ^T = ~^r^ (^-28) where E^ = Gc_ /2g is the kinetic energy of the flow behind the lat- tice for the isentropic process and G is the actual flow rate of the gas through one channel of the lattice, which can be determined by the equation G = Gt. - g (P20^20 - Pz'^Z^^^y + (P20^20 " P2^2)^Qj^y where pgQ is the density at the outer boundary of the layer in the section at the exit edge and G^ 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 G = G, - [(8%o),, ^ (^*-20)^,,]^ (^-29) MCA TM 1393 37 In equation (7-29) ^r «' = (i - ^ 4)'" 1 - k - 1 ^2 1 - IT+T ^20 ^2 1 - FTT ^ "'° dy (7-30) The theoretical flow rate of the gas may be determined by the formula Gt = gP2t^2t^ si'^ P2 = (l - FTT ^t) ■Po , RT^ Xgt^/ sin Pg Substituting expression (7-3l) in equation (7-29) ^ we obtain 1 G = il - |-^ y^l^j Xg^a^t sin P2 " (^^o)^^ " (S%o) bol (7-31) P0_ RTq (7-32) By using equation (7-27), the equation for the loss coefficient (7-28) now assumes the form ?. = -?- (sr^2o)„. + (sr^2o) en bol 1 k-1 1 k - 1 ^2 ^ ^ - n " k + 1 2t ^2t* ^^^ P2 ■ - (S*^20)en " ■ (^*^20)^o2 ^it (7-33) Bearing in mind that G = ^pGt = ^p(i - FTT ^) i_ k-l Po RT^ X2tV ^^^ P2 38 NACA TM 1393 formula (7-33) can be represented in the form ^ ^^ en ^T = 1 (Sr^20),, + ^^?^20)^„^ .^1 - F^ ^itj ^t^ si'^ h or where (H 52 X20) + (H 52 Xgo)^ , ^ en hot , . ^^ = 1 (7-34) / k 1 2\^^ „*** 2 "■ ^ ^^> o From a comparison of formulas (7-33) and (7-34), it follows that the flow-rate coefficient \i-^ is equal to (H*6j*X2o) + (H*5|*X2o), , Up = 1 ^^^ ^2i_ (7-35) k - 1 , 2 \ k + 1 ^2ty k-1 X2-f;t sin Pg where o For an incompressible fluid, there may be obtained from expressions (7-34) and (7-35) (H Sg U20) + (H 82 U2o)^ - ?T = ^^^f^ -^^^ (7-36) UpCj^t Sin P2 MCA TM 1393 39 and ^ = ^ - c^^t sin Pg ^"^"^^^ In this case (for the incompressible fluid), the values of S^* and 62 are determined by the formula given in table 4-1. The magnitudes h*** and H* entering equations (7-35) and (7-36) should be determined for the turbulent and laminar boundary layers individually. It is evident that the values H and H and the magnitudes 5 and S 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 H and H for the turbulent layer using the assumption of an exponential velocity distribution law and for the laminar layer with dp/dx =0. On figure 7-22 (a) and (b) are given the values of H and H for the tiirbulent layer as a function of He and Xo/-, 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 p>-^ and 32. We assume that the velocity distribution on the profile is approximately that shown by the dotted curves in figure 7-23 for all inlet and exit angles. On the convex side of the profile '^cn/^2 ~ -'-•-'- ^^^ °^ "'"^^ concave side ^i,ol/'^2 ~ ^"^ approximately^ [subscripts en and bol denote convex and concave sides, respectively] On this assumption, the density of the lattice B/t for each pair of values of the angles should have a fully determined value (see sec. 7-5): From equation (7-18) -Q 2 sin P2sin(p-]_ + Pg) t ^ C sin p, u 1 where the coefficient of the peripheral force is P _ _ ^u = 2 " " ^cn + Pbo7 pco B ^ 40 NACA TM 1393 p^j^ and V^qI "being the mean pressure coefficients on the convex and concave surfaces of the blade. For the assumed values of c^^^^ and c-^qi V = 1 en m^ -0.21 and p^ , = 1 -m- 0.75 that is, C^ = 0.96. Assixming further that ^20 = ^2t H' •(HH?- „*** cn %ol "2 and Up = 1 we can represent the friction-loss coefficient of the lattice in the form ^T = 2- sin Pg (7-38) On the assumption of the exponential law of velocity distribution in the boundary layer (with exponent n = \/l) , the momentum thickness is equal to [Note: this expression is very similar to that of E. Truckenbrodt; cf. Schlicting, p. 470.] .** = 0.0973 0.37 Re 0.2 3.86 -^ dx (7-39) In expression (7-39), we assume Re = 10 , and to estimate the arc of the profile S on the convex and the concave surfaces we evaluate approximately (fig. 7-23(a)): ^cn - ^ol - ^ B ^l B 3 sin p-]_ 3 sin Pg (7-40) The graphs in figure 7-23(b), where the friction loss coefficient ^ is represented as a function of p, and p , are constructed with lC>For the case of infinitely thin trailing edges the coefficient ^rp is equal to the profile loss coefficient of the lattice. MCA TM 1393 41 the aid of formulas (7-38) to (7-40). The dotted curves correspond to constant values of B/t . Notwithstanding the considerable reservations with which the entire computation was made, the results are qualita- tively well confirmed by experiment. The friction losses depend on 3-j^ and pg^ increasing with de- crease in these with the greatest influence exerted by P]_. For P " Po (in lattices of the impulse type) the curves of equal ^ al- most pass through the normal to the straight line 3i = Pg' "that is, in this case the losses depend essentially on the magnitude of the angle of rotation of the flow equal to Ap = 180° - (Pi + Pg) We may remark that the effect of Reynolds number on the friction loss coefficient in the lattice can easily be determined by computation. 7-7. EDGE LOSSES IN PLANE LATTICE AT SUBSONIC VELOCITIES The eddy losses at the trailing edge constitute the second compon- ent of the profile losses in a plane lattice. The flow leaving the trailing edges always separates. As a result of the separation there is an interaction between the boundary layers flowing off from the con- cave and convex surfaces behind the trailing edge; vortices thus arise which appear at the initial part of the wake . The photographs of the .flow behind the lattice presented in figure 7-24 show the formation of the initial part of the wake . A large influence on the wake is exerted by the distribution of the velocity in the boundary layer at the point where the flows from the convex -and the concave surfaces unite and also by the difference in pressure at these points. Along the initial part of the wake, (includ- ing the region behind the trailing edge where a Karman vortex street is formed with the usual chess arrangement of the vortices) the interaction between the eddy wake and the nucleus of the flow unifies many properties of the flow field behind the lattice. The static pressure of the flow increases and the outlet angle decreases. As a result, kinetic -energy losses arise, analogous to the losses in sudden expansion. The parameters of the equalizing flow can be obtained by the simul- taneous solution of the equations of continuity, momentum, and energy. The control surfaces shown in figure 7-25 are selected. These surfaces are equally spaced, when measured along the lattice axis; and they en- close the fluid involved in the study. The above equations can be writ- ten for the following assumptions : (a) the density of the flow changes 42 NACA TM 1393 little as it moves downstream (from sees. 2-2 to 2'-2')j (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 pCgCt - At)sin Pgn = ^a»P"t sin Pgco or CgCl - T)sin p2n = C2ccSin pg. (7-4l) where At '' = - The momentum equation in the direction of the axis of the lattice gives 2 2 Cgcos PgnP^"*^ " ^■t)sin Pgn = ^g^cos Pg^pt sin Pg^ or, with account taken of (7-4l), we obtain cgcos p2n = cgoocos pg. (7-42) The momentum equation in the direction perpendicular to the lattice axis can be written in the form c|p sin2p2^(t - At) + P2(t - At) + pj^t = c|^p sin^Pg^t + pg^t (7-43) From equations (7-4l) and (7-42) there is easily obtained P2„ = arc tan[(l - T)tan Pg^] (7-44) Equation (7-43) permits finding the increase in pressure behind the lattice Apgoo = — 2r2P si"^ P2n(^ - i^) " ^2«,0 sin P2o„ + (pj^p - P2)'M P^2 MCA TM 1393 43 Taking into account expression (7-4l), we obtain ^200 = i" ^ ^ = [2(1 - T)sin2p2n + Pkp^ (7-45) 2 P^2 For determining the theoretical velocity at infinity behind the lattice, we make use of the equation of energy which for the assumption made pg = pgoo = P ^^Y ^^ represented in the form (^2«) ^ P2=o ^2 P2 5 + = -^- + — (7-46) 2 p 2 p ^ ' where Cgoo is the theoretical velocity in the section 2 '-2'. From expression (7-46), we obtain • \2 1 - Aip„ (7-46a) &)■ The velocity Cg is expressed in terms of cg^ with the aid of equa- tions (7-41) and (7-44), thus c2_^ "2^ = (1 - T)^sin^P2n + ^°^^hn = 1 - t^(2 - T)sin Pgn ^2 and we have 1 - t(2 - T)sin2|32n /C2„\ 1 - t(2 - T)sin^|32n g / ^ The coefficient of edge losses is T sm Pg - Pk f, = 1 - cp^ = ^2 P ^ (7.48) ^kp kP 1 - Ap2a, Formulas (7-45) to (7-48) given here were obtained by G. Y. Stepanovich. 44 NACA TM 1393 The nondimensional pressure behind the edges entering equations (7-45) and (7-48) is Pkp - P2 2 P^ and it must be determined from experimental data. With an accuracy up to magnitudes of the second order as compared with T the coefficient of edge losses is expressed by the formula ^kp = - Fkp'f For small velocities, according to test data (see below) Pkp = - 0-1 From the above arguments, it is seen that the edge losses are directly proportional to i. 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, and is quite dependent on the thickness of the edge. The region of in- tensive mixing ends at a distance y = (l.3 to 1.7 t) behind the trailing edge. This is confirmed by the graphs in figure 7-26 in which are given the results of an investigation of the wake behind a reaction lattice according to the data of R. M. Yablonik. Figure 7-26 (a) shows curves of local loss coefficients of the wake at different distances behind the reaction lattice. On figure 7-26 (b) is shown the variation of the coef- ficient of nonuniformity in the flow field behind the lattice. This coefficient is defined by the formula V = 'a .max - c a, mm 2c a,m where c and c . maximum and minimum values of component velocity Cg. in the given section 'a,m mean value of velocity Cg in the same section A detailed investigation of the flow behind the trailing edge of a reaction lattice was conducted by B. M. Yakub. The results of these tests reveal certain effects of the shape of the edges on the flow NACA TM 1393 45 structure in the eddy wake. Measurements of static pressure on both sides of the wake show that there is a considerable nonuniformity in the pressure field along the boundaries of the wake (fig. 7-27). Moreover, the static pressure along the wake boundaries changes periodically. As the flow leaves the concave surface of a blade, its pressure must drop, while on the concave surface it must increase. Further, be- hind the principal edge vortex, the static pressure decreases on both sides of the wake, it then again increases somewhat, and so on. Finally, there is a complete equalization of the field of flow. From figure 7-27 it is seen also that the amplitude of the fluctuations of the static pressure depends on the shape of the edge. By making a two-sided taper (sharpening of the edges b and c in figure 7-27) it was possible to decrease somewhat the nonuniformity of the static -pressure field. The tests showed that a sharpened edge of the type b raises the efficiency of the lattice, as compared with the normal edges, by 1 per- cent and that an edge of type c increased the efficiency by 2.5 per- cent (for a medium velocity of flow). It should be remarked that, not- withstanding their high effectiveness, the forming of very sharp edges of the type c introduces serious difficulties 'because such an edge rapidly deteriorates under actual operating conditions. 7-8. SEVERAL RESULTS OF EXPERIMEIWAL 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. Dzerzhinskii Institute, and in other scientific re- search organizations and institutes. We shall consider as an example several results of an experimental investigation of the effect of the pitch, the blade angle, and the angle of incidence of the flow on the velocity distribution over the profile of an impulse and reaction type lattice. Figure 7-28 shows the velocity distribution over the profile-^^ 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 t = 0.904 on fig. 7-28). In this diffusing region a boundary layer is formed, and its thickness increases and in certain -'-^The local velocities are made dimensionless by dividing them by the vector mean velocity. 46 MCA TM 1393 cases separates . With increasing pitch the nonunif ormity of the flow in the passages between the blades increases; the velocities on the con- vex side increase, while on the concave side they decrease. At high values of the pitch, the flow about a profile in the lattice approximates the flow about a single profile (fig. 7-28) . The effect of the blade setting on the velocity distribution over the profile is shown in figure 7-29(a). The maximum favorable velocity distribution for a given profile is obtained at a setting angle 3 = 50°. In this case both along the upper and lower surface the velocities in- crease more \iniformly. A change in the inlet angle of the flow (fig. 7-29 (b)) greatly affects the velocity distribution along the profile. Large inlet angles tend to impair the flow along the concave surface, while small angles similarly affect the flow along the convex surface. The investigation of an impulse lattice conducted by E. A. Gukasova shows that, similar to the reaction lattice, a change in pitch, causes a considerable change in the velocity distribution along the profile (fig. 7-30) . For all values of the pitch an adverse pressure gradient is found immediately 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 reaccelerate. On the convex surface of the blade be- hind the leading edge, the flow accelerates and reaches a maximum veloc- ity downstream of the part of greatest curvature. We note that, as for the impulse lattice, diffuser regions are formed near the trailing edge of the upper surface for all regimes. With decreasing pitch, the nonunif ormity of the velocity field in the channel between the blades decreases. A similar trend accompanies an increase in the inlet angle of the flow; as (3-]_ 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- since of adverse pressure gradients near the inlet of both the convex and concave surfaces. For inlet angles somewhat higher than the profile angle P^n' "^^^ most favorable general velocity distribution is found. The change of the coefficient of profile losses in impulse and re- action lattices as a function of the pitch and inlet angle may be seen in figure 7-31. The curves show that for each lattice there exists a definite optim\jm pitch for the minimum profile losses. Thus, for exam- ple, for the reaction lattice having the profile shown in figure 7-28, the optimum pitch is tQp-j; = 0.673. For the impulse lattice, tQp.(. = 0.50-0.60. WACA TM 1393 47 In spite of_the favorable velocity distribution, in a closely- spaced lattice (t < topt) the loss coefficient is relatively high be- cause of the greater losses produced by friction. Decreasing the pitch also causes an increase in the coefficient of edge losses. The curves in figure 7-31 show that for all pitches a decrease in the inlet angle (below the optimum) has a sharper effect on the effici- ency than an increase in the angle. An increase in t,^ also noted for P-|_ > P-]_^ for the impulse lattice of large pitch. It should be empha- sized that as a rule the values of the optimum inlet angles exceed the geometric angle of the profile. From the results of the investigations, it can be concluded that the experimental determination of the optimum pitch must be carried out over a wide range of inlet angles . The tests show that the direction of the equalized flow behind the lattice may with sufficient accuracy be determined by formula (7-44). The familiar formula given in the literature for determining the effec- tive (actual) angle of the flow 0,0 Pgg = arc sin -r- (7-49) gives somewhat lowered values of Pg • More closely agreeing values of Pp with test results are obtained by formula P2e = ^^^ sin ^ _ ^^ (7-50) At small velocities tests confirm that for all practical purposes, the outlet flow angles of a reaction lattice depend only slightly on the direction of the flow at inlet, that is, on the angle p-[_ (fig. 7-32). The angle Pg is, however, influenced to a large extent by the pitch and the setting angle of the profile. With an increase in p and t, 13 the angle Pg increases. 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. -'-^Analysis of formula (7-44) leads to the same results. i 48 MCA TM 1393 Immediately behind the lattice, the field of the flow angles is nonuniform; the angles ^2 vary along the pitch (fig. 7-33). The greatest changes in P2 ^^^ foiind 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 Pg"' The non\iniformity 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 nonuniformity 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 tiirbulent flow permit drawing several general conclusions as to the character of the change in profile losses in lattices as a fiinction of the parameters defining the flow regime (inlet angle p>-^ and Reynolds number 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 magnitudes results in the formation of ad- verse press\ire gradients on the profile, the boundary layer thickens, and the transition from a laminar 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 remarked that in impulse lattices the losses as a rule are greater than in the reaction type which are characterized by a more 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 p,-^, t, and Py 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 separation occurs on the profile, the profile losses tend to decrease markedly with an increase in Reg. For nonseparating flow about the profile, the effect of Reg for the reaction lattice is small (fig. 7-34). NACA TM 1393 ■ 49 7-9. FLOW OF GAS THROUGH LATTICE AT LARGE SUBSONIC VELOCITIES; CRITICAL M^ 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 streamlines \|/ = constant and equipotential lines $ = constant remains orthogonal, but it is no longer square. The velocity at any point of the flow is _ d§ _ PO dt '^ ~ dS ~ p dn and, as a result, when A^ = Ai|; = constant, then Z!^/An = p/Pn ^1- '^^ the plane of the hodograph, the network of lines ^ = constant and if = constant is no longer orthogonal. According to the condition of equality of the flow rate ahead of and behind the lattice, we have Cn p-jt sin P-, = Cppgt sin Pp For c-j^ < Cg, the projection of the velocity Cg on the normal to the axis of the lattice (cgsin Pg) becomes larger than the same projection of the velocity c-[_. 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 Mg = cg/ag. An approximate method of estimating the velocity distribution over the profile may be used to establish the characteristic regimes of the flow about the lattice at subsonic velocities. The approximate method is based on the circiimstance that in modern turbine lattices of high solidity the flow between the profiles may be considered as a flow in a channel. The flow velocity in an interblade passage of constant width and curvature (fig. 7-35) can be determined in a particularly simple manner. A comparison with more accurate theory shows that for a perfect gas the velocity distribution across the channel approximately satisfies the equation ^cn c = ^- c^n (7-51) -'-'^The method considered, proposed by A, Stodola, was developed sub- sequently by G. Y. Stepanov. 50 NACA TM 1393 and in particular _ ^cn ""■bo^ " RboZ ''^'^ The velocity on the convex side of the profile c^^j^ can be deter- mined from the equation of continuity r\oi c-^p-^t sin 3i = I cp dR (7-52) en In equation (7-52) it is convenient to transform to the nondimensional functions q and \ q,t sin Pn = ( q. dR *^Rcn Using expression (7-5l), we obtain finally q-^t sin pi = X^^R^^ 1 <^(x) = ^cn^cn^ I (7-53) ^ ^cn "-cn Computation of the integral I for small subsonic velocities gives (the constant of integration is omitted) I-L = mi In i (7-54) where _1^ k-1 mi =(^y For a gas with k = 1.4 we obtain l2 = mi [;osh-l _i_ . (II . 11 ,2^2 ^ 1 ,2^4J ,^|I~^] (,.55) where mg - Yk + 1 MCA TM 1393 51 For k = 4/3 we have I3 = m-L^^ln - + - m2X - j mgX +gm2X^ (7-56) If the computed function I is used, the equation of continuity can be written in the forin-'-^ q-[_t sin (3-|_ —T^ = ^cn^^boZ - ^cn) (^"S-^) where I^^ by equation (7-54) or (7-55) corresponds to X^^^ and Ii^ol corresponds to X, , = R /R _ X boi cn' Doi cn It is possible to apply the process of successive approximations for computing X^^^ by equation (7-57), since the expression in paren- • theses depends little on X . In the first approximation ^ X^t sin p^ (7-58) ^ ^^ - un -"-bol ^^^ determined from cn Co^ ^I't sin p., (7-59) and so forth. For X^^^ < 0.5 the first approximation (7-58) is suffi- cient. The solution of equation (7-57) is conveniently represented in the form of the graph shown in figure 7-36, which gives the magnitude q = q-,t sin &^/(R. , - R ) as a function of R /r, _ for various ^cp ^1 '^1' ^ b)o2 cn cn' TdoJ values of X^,^. A critical value q^^ and a corresponding Xj^ or M^^* denote critical flow in the lattice; that is, a condition where X^^ =1. In the curved channel for which R /R^ - < 1, the graph in figure 7-36 in- dicates that the maximiun flow is attained for some X-j_ > X-yj_ . ''cn X^t sin p^ pproximation, 1^,^^ and cn q-j^t sin p^ ■ (1) (1)" cnLDOi cn _ -'- KACA note: Subscript cn refers to convex side, boZ to concave side. 52 MCA TM 1393 The above described method can 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 7-37 and to determine their diameter and also the radii of curvature R^j^ and 'R^^qJ at the points of tangency of the circles. For computing the velocities \^^ and X-^o2 ■» formulas (7-58) and (7-59) may be used substituting, for example, ^cn = ^cn' ^bo2 " ^cn + ^ or ^bol Ren ^^n ' ^" ~ "' RboZ ~ RboZ ^boZ = ^boZ' ^cn = ^boZ ■ - a. The differences °-'" -^boZ ~ ^boZ ' ^cn ~ -^boZ " ^* "^^^ differences in the values of X^^ and ^tjoZ obtained in each case characterize the error of the applied method. As an example, in figtire 7-37 are compared the :~esults of the exact solution (in the flow of an incompressible fluid) with the results of computations by the described method. The satisfactory agreem.ent of the values of the velocities tha.t is observed also in the other examples attests to the feasibility of applying this method for preliminary computations . Let us now consider flow of a gas through a reaction lattice when the velocities are nearly sonic. For a critical value of Mg = Mg^j. at a certain (critical) point of the profile, the critical velocity is reached. With further increases in Mg , the press\are distribution ahead of this critical point changes little. The pressure distribution behind the point of sonic velocity changes considerably. In the so-called dif- fusing (i.e., for subsonic flow) region behind this critical point, there is an increase in the supersonic velocity. The experimental determination of the critical values Mg^ shows that its magnitude largely depends on the geometric parameters of the profile, the lattice, and the direction of the flow at the inlet. In a l^This m.ethod of com.puting the flow in a channel was based on the approximate determ.ination of the length of the potential line and on the assumption that the distribution along it of the curvature of the stream- lines differs little from, the case of vortex flow. With a certain com- plication of the comiputations, this nethod can be rendered more accurate by the successive refinements in estimating the distribution of curvature. MCA TM 1393 53 reaction lattice for an entry angle p-j^ = 3]^j^^ the values of Mg^^. de- crease with increase of pitch because the local velocities on the con- vex siirface at the points of maximum curvature increase. In figure 7-38 are shown the curves of maximum velocities on the hack of the profile as a function of Mg . From these curves the values of Mg^j. can be deter- mined. For 1 > Mg > Mgjj. 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 = l) 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, the ve- locities of the particles leaving the upper and lower surfaces (outside the boundary layer) 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 channel. For a large pitch, on the con- 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 Mp„ are shown in figure 7-39 for P-j. * Pin ^^ ^ function of the pitch for a reaction lattice. It is seen from the graph that for each lattice there exists a pitch t^ for which the critical velocity is reached simultaneously on the back and behind the trailing edge of the profile . In an impulse lattice,-*-' 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 7-40 and 7-41 characterize the effect of the number Mo (and also M-i ) 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- sxxre) are displaced in the direction of the flow. For small angles p^ -'-For the impulse lattice the critical M number is sometimes re- ferred to the inlet velocity. 54 MCA TM 1393 and large niimbers M^, experiment shows the displacement of the branch point 0]_ 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 j the pressure gradient along the convex surface increases. Correspondingly, the flow in the diffusing region on the convex surface also changes. Since the minimum pressure on the profile decreases, the pressure gradient in the diffus- ing region of this surface increases. The pressure changes particularly sharply on the convex surface near the narrow section of the channel. Similarly, but more sharply, the effect of the compressibility reveals itself in the pressure distribution in an impulse lattice. A change in the inlet flow angle at large supersonic velocities in an impulse lattice sharply affects the pressure distribution, particu- larly at the inlet part of the profile (fig. 7-42). 7-10. PROFILE LOSSES IN LATTICES AT LARGE SUBSONIC VELOCITIES The results of experimental investigation permit estimating the change in the profile losses in various lattices at subsonic and near sonic velocities. For M2 < Mg^, 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 Mg ; this behavior is particulary acute when the velocity is approxi- mately sonic. In figure 7-43 is shown the dependence of p on M for a rounded trailing edge. It is seen that with an increase in Mg, the value of p, decreases and reaches a minimiim value at kp Mg = 0.9 - 1.0. With a further increase in Mg, the pressure behind the trailing edge increases. The intensity of the vortice behind the trailing edge and the width and depth of the wake are increased (fig. 7-44). At the same time, for Mg < 1, the extent of the smoothed out part of the flow behind the lattice increases. I MCA TM 1393 55 For an approximate estimate of t^-^ at large subsonic velocities, formula (7-48) may be employed, substituting the test values of pj^ (fig. 7-43). Thus, taking into account the fact that the trailing-edge losses increase, with an increase in Mg, the character of the change of the coefficient of profile losses as a function of Mg is determined by whichever of the above-mentioned factors is the deciding one. In the final analysis, this answer depends on the geometric parameters of the profile and lattice. In reaction lattices the approach to near sonic velocities while Mg < M2^(. does not lead to any considerable increase in the losses if the flow in the interblade channel is without separation. We recall that the resistance coefficient of a single profile sharply increases in the zone of near sonic velocities. In the flow about a single profile, the local shock waves have a considerably greater intensity, and in many cases the flow separates to the im,pairment of the flow. The energ;\^ 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 coefficient increases more rapidly with increase in Mg . Figure 7-45 gives ^-p curves for several reaction lattices consist- ing of different profiles and for two impulse lattices. We note that since the test lattices had different profiles, the dotted curves in fig- ure 7-45 do not characterize the effect of pitch alone. The effect of the incompressibility on the profile losses is more marked for inpulse lattices . The curves in figure 7-45 clearly confirm 1 o this conclusion. 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 im^pulse lattice ^ . In passing to large inlet angles ((3-i_ > 3]_n)j "the losses in the impulse lattice decrease. ^he results of the test were obtained on an apparatus with con- stant back pressure. With increase in the number Mg there is a simul- taneous increase in Reg. As was pointed out in the preceding section, the increase in Reg leads to a lowering of the losses. It may be as- sumed that for Reg = constant the change of ^p as a function of Mg would be somewhat sharper . 56 NACA TM 1393 Detailed Investigations of the flow structure show that an increase in Mg leads to an increasing nonuniformity of the field behind the lattice (figs, 7-46 and 7-47). Analysis of the effect of coir.pressihility on the flow structure in lattices permits drawing the conclusion that the optimum pitch of the profiles decreases as the velocity increases. With decreasing pitch, the 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 Mg . Tests show that for Mg < Mg^j. the compressibility has only a, slight effect on the m.agnitude 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 Po with in- crease in Mg. For Mg > Mg*^ "the mean angle as a rule increases with increase in Mg (fig. 7-48). 7-11. FLOW OF A GAS THROUGH REACTION LATTICES AT SUPERSONIC PRESSURE DROPS In conventional guide and reaction lattices, the flow velocities at the inlet are subsonic; the transition to supersonic velocities occurs in the interblade passages. We will first consider the fundamental prop- erties and structure of the flow in plane reaction lattices for super- sonic pressure drops when Pg/Pos "^ S . The successive change of the supersonic regimes of the flow in a lattice is shown schematically in figure 7-49. In the narrow zone of an interblade passage the critical velocity is established.-^^ Behind the trailing edge the pressure is below critical. In the flow about the point A (fig. 7-49(a)) the pressure drops and the fan of expansion ABC fall on the convex side of the neighboring profile and are then reflec- ted from it. The initial and reflected expansion of waves over expand the flow; that is, the static pressure behind the wave ABC is less than 1 9 The transition surface coincides approximately with the narrowest section of the passage. Actually, as a consequence of the nonuniformity of the flow in the converging part and the effect of viscosity, the tran- sition surface has a certain curvature and is displaced upstream.. NACA TM 1393 57 the pressure at infinity behind the lattice. The further 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 leaving the concave and convex surfaces of the profile approach each other and are then sharply deflected at a certain distance behind the edge. At the boundaries of the initial part of the wake, a system of weak shocks arises which merge with the oblique shock FC, which is formed at the points of discontinuity of the wake. The oblique shock interacting with the boundary layer on the convex on surface of the profile is reflected and again impinges on the trailing wake. Depending on the mean Mj^„ number in this section of the wake, the reflected shock either intersects the wake (Mj^ > l) or is reflected from its boundary (if Mj^-p < l) . Thus, the flow moving along the convex surface of a profile successively passes through the primary and reflec- ted expansion waves and the primary and reflected shocks. The behavior 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 an expansion wave; and the flow about the edge is improved. The streamline leaves the profile not at point D, but at point E (fig. 7-49(3)). On accoTint of the curvature of the wake EF and the rotating of the flow near the point E, there arises behind the expansion fan DLK a system of weak shocks merging with the curved shock FH, 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 a.t the point D. In this case the wake increases. On passing through the system of expansion waves and oblique shocks, the individual streamlines are multiple and variously deformed. On in- tersecting the primary rarefaction wave, the streamline a-a deflects, turning by a. certain angle with respect to the point A (the angle be- tween the tangent to the streamline and the axi^ of the lattice in- creases). The reflected wave som.ewhat decreases the angle of deflection The reflection remains norm.al even at large angles of incidence of the primary shock (eg ->■ ^x.) ) since the interaction of the shock with the boundary 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 sm.all pitch is not observed. 58 NACA TM 1393 of the streamline. On intersecting the primary shock, 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 (and therefore the intensity) of the rarefaction wave in- creases. The angles of the primary, reflected, and edge shocks decrease. The point where the oblique shock FC falls (point C) is displaced down- stream (fig. 7-49 (b)). In correspondence with this, the character of the deformation of the individual streamlines likewise changes. With increase in e^ the mean outflow angle increases . The expansion of the flow within the confines of the lattice ends for a certain relation of the pressures eg = £3* ^°^ flow conditions near this limiting regime, the primary shock is curved and forms a cer- tain small angle with the plane of the outlet section. The exact de- termination of the value eg is therefore difficult. The limiting re- gime may be considered that for which the primary shock falls at the point D of the edge section (fig. 7-49(c)). If eg < ^Q> the expansion of the flow continues beyond the lat- tice (fig. 7-49(d)). The system of shocks at the trailing edge remains essentially as before, but the wake behind the edge is considerably dim.inished. The left branch of the tail shock (the shock FC in fig. 7-49) falls in the subsonic part of the wake of the neighboring profile and 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 regimes separation is observed mainly in lattices with relatively large pitch. It should be remarked that for eg « Gq the separations vanish as a rule. The primary shock falls in the supersonic part of the wake (fig. 7-49(e)). The pressure behind the edge drops, and the separation on the back is eliminated. Thus, a very characteristic property of the regimes eo < eg is the interaction of the primary shock with the wake at the edge. The shock FC passing through the flow field behind the outlet sec- tion sharply decreases the angle of deflection of the flow. This is particularly well marked by the deflection of the wake near the edge. MCA TM 1393 59 The above considered schemes of flow are illustrated by photographs of the flow spectra behind the throat and at the exit from the reac- tion lattice (fig. 7-50) . There is here seen the fundamental system of waves and shocks^ the deformation of the wake behind the edge for dif- ferent regimes, and also the interaction between the waves and shocks with the neighboring profiles and wakes • The flow spectra are given for two lattices: ^ = 0.543 (fig. 7-50) and t = 0.86 (fig. 7-5l) . 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 the regimes eg = 0.288 - 0.258. In figure 7-51 (photographs (a) and (b)) there is clearly seen the vortex structure of the trailing wake and the considerable nonuniformity of the flow behind the lattice. Figure 7-52 gives the pressure distribution behind the throat on. the convex surface of a profile in a reaction lattice for various ratios e„ = p /p_ . The curves show the considerable nonuniformity of the pres- sure on the back of the blade. Behind the throat section (i.e., at the points 2 to 6) the expansion of the flow maj be observed; the pressure at these points is lower than the pressure behind the lattice. The expan- sion ends with a sharp increase in the pressure at those points on the convex surface of the blade where the incident and reflected shocks in- teract with the boundary layer. With an increase in Cg^ "the zones of maximum expansion on the convex surface as well as the sharp increase of pressure in the shocks are both displaced along the back toward the trail- ing edge . In the regimes of limiting expansion ( Eg ~ ^S^ ' ^^^ pressure along the back of the profile continuously drops. The pressure behind the ex- pansion waves at all regim.es Eg ^ ^s decreases as the pitch increases. The effect of the pitch on the intensity of the shocks behind the throat is seen in figure 7-53. The character of the curves Apo/p. ^-^ (Apg is the increase in pressure through the shock wave impinging on the convex surface of incidence of the shock wave) depends on the pitch. With an increase in t the maximum intensity of the shocks at first de- creases and then increases. At the same tim-e, the maxim^um Apg/p-j^ ^^^ shifts in the direction of higher values of Cg. 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 very nonuniform. In figure 7-54 (a) is shown the distribution of the local angles of deflection p " P? over the pitch of the lat- tice for two regimes. The upper curve corresponds to the flow conditions 60 MCA TM 1393 shown in figure 7-49(c)(e2 => Cg) . Ahead of the primary shock, the flow deflections are influenced by the expansion waves; the angles of the streamlines slightly decrease. At x = 0.4 there is a sharp decrease in Pgi due to the primary shock. At x > 0.4 the local angles vary less sharply up to x = 0.9. From figure 7-54 (b) it is seen that the distribution of the static pressures over the pitch is likewise very nonuniform. The static pres- sure varies with the system of waves and shocks traversing the section investigated. A large effect on the spectrum of the flow behind the lattice is exerted by the setting angle of the profile (i.e., the angle at the exit). With a cha.nge in the angle Pp„ the geometrical parameters of the section behind the throat vary. For the same pressure drop in the lattice (cg)' '^^^ arrangement of the fundamental system of waves and shocks in this section of the lattice varies. With increase in p the length of the wall of the section BD (fig. 7-49) is shortened (the pitch is unchanged); the relative effect of the primary expansion wave increases; the angle of deflection in- creases with increase in Pgn' The equalization of the flow behind the trailing edge for Mg > 1 occurs at greater distances from the lattice than for Mg < 1. The vari ation of the distribution curves of Pj-,o/pq-i along the pitch as a func- tion of y for Mg = 1.58 is shown in figure 7-54(c). We note that the equalization of the flow at supersonic velocities is accompanied by a. decrease in the static pressure behind the trailing edges. Supersonic reaction lattices are often used as nozzle lattices (for Eg *~ %)(fig- 7-55). The interblade passages of such a lattice form supersonic nozzles. At design conditions supersonic velocities may be obtained in such lattices without any essential deviation angle of the flow. On the other ha.nd, expansion may arise in the overhang section of the lattice at design conditions. The expansion wave is formed as a result of the lowering of the pressure behind the trailing- edge. In the flow about the trailing edge, as 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 maintained also for the nominal operating regime of the supersonic lattice. 1 MCA TM 1393 61 For the off-design regimes (cg < ^2comp^' *'^® 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 eg becomes larger than the computed one, the shocks are moved upstream into the interblade channel, the same way as they are in the one-dimensional supersonic nozzle. It should be borne in mind that, for the same value of Eg, 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 flow structure in a supersonic reaction lattice is shown in figure 7-56. At increased pressures behind the lattice, a system of two oblique shocks is situated within the channel (fig. 7-56(a)). With an increase in pressure behind the lattice the shocks move toward the outlet section (figs. 7-56(b), (c), and (d)). Near design operation (figs. 7-56 (e) and (f)) primary and reflected shocks intersect on the convex surface; behind the lattice a trailing-edge shock may be seen.' The pressure distribution over the profile (fig. 7-57) agrees with the flow picture. At regimes where the relative pressure eg 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. The velocity field be- hind a supersonic lattice possesses very great nonuniformity for e„ < e (fig. 7-57(b)). 2 * 7-12. IMPULSE LATTICES IN SUPERSONIC FLOW When the velocities are practically sonic a \-shaped shock is formed on the convex side of each profile of an impulse lattice. This system of shocks of sm.all curvature merges to form the bow wave for the neigh- boring profile (fig. 7-58(a,)). Immediately behind each bow wave 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]_ ahead of the following shock. There acceleration of the flow occurs in the expansion waves form- ing in the flow about the leading edges. As the velocity of the oncom- ing flow increases, the bow becomes curved and moves toward the inlet edges of the profiles (fig. 7-58(b)). It may be assumed that for veloc- ities corresponding to the flow scheme in figure 7-58 (b) the flow behind ■ It may be assumed that the tip losses in such lattices are small even with small blade heights . ^^ NACA TM 1393 the shocks will be turbulent. Because the effect of profiles is coiranu- nicated upstream in the subsonic region, a nonuniform velocity distri- bution is established behind the leading shock. The velocities vary periodically in magnitude and direction along the lattice. For a certain sufficiently large value of M^^ the right branches of the shocks merge forming a continuous wavy-shaped shock (fig. 7-58(c)). The left branches of the bow wave are turned into the concave surface of the profile. With further increase in the velocity M-^ the angles of the branches of the bow waves decrease; the shocks approach the inlet edges of the lattice. In certain cases at the inlet to the interblade channels there is fonned the system of shocks shown in figure 7-58(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 7-59 there are clearly seen the changes in the shape of the bow waves that accompany increases in Mj. The pressure distribution over the profile at supersonic velocities (fig. 7-60(a)) shows that for M-^ ^ 1.5 the velocity over a large part of the concave surface is subsonic. For M-^ > 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 M^^ toward 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 struc- 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 ^2.n ^^^^ ^^ effect on the structure and, in particular, the intensity of the bow waves. Ahead of an impulse lattice consisting of profiles of small cur- vature (large angles of the inlet edge 3ln) an over-all wave-shaped MCA TM 1393 63 shock is fornied instead of the system of shocks shovm in figure 7-58(b). The shape of this wave ahead of a lattice of plates for various inlet angles is seen in figure 7-61. Since the formation of such a shock ahead of the lattice is possible in the case where M-isin p,-, > 1, the number M-^ corresponding to the type of shock considered increases as P-|_ decreases. 7-13. LOSSES IN LATTICES AT NEAR SONIC AND SUPERSONIC VELOCITIES The above considered properties of the flow of a gas in plane lat- tices of different types at large velocities permit an analysis of the behavior of the over-all characteristics of lattices accompanying a change of velocity of the flow (M-j^ or Mg). Figure 7-62(a) shows curves of the loss coefficients for reaction lattices as a function of Mg and the inlet flow angle p-]_ . Figure 7-62 (b) gives similar curves for impulse lattices. The curves show that^ depending on the entry angle ^ the pitch, and profile shape, the loss coefficient of reaction lattices may increase or decrease in the region of transonic velocities (O.S^Mg <1.2). A marked increase of the losses in a lattice occurs at supersonic veloc- ities (Mg > 1.2). The value of Mg for which this increase is ob- served decreases as the pitch is increased. The loss coefficients of supersonic lattices increase very sharply with an increase in Mg and reach a maximum value when the relative pressure in the lattice is nearly critical (Mg ~ l) . With a further in- crease in Mg, the coefficient ^„ decreases. The losses in a super- sonic lattice are a minimum near the computational (design) value of Mg. For Mg -* Mg^Qjjj the loss coefficient increases with the velocity. From, a comparison of the loss curves in a reaction supersonic lat- tice (fig. 7-62 (a)) with those in a one-dimensional supersonic nozzle, it can be concluded that the variation of ^ with Mg 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 main influence on the effectiveness of such lattices at off- design regimes. The lowering of the losses in the lattice for Mg ^ 1 is explained by the fact tha.t at such regimes the wave and vortex losses go The data presented in the present section refer only to lattices of definite geom.etric parameters . 64 NACA TM 1393 decrease and then (for small Mg) entirely vanish (the interblade pas- sage works as a Ventiiri tube). As in the case of the single nozzle, the losses in a supersonic lattice at the design and off-design regimes vary as a function of the passage parameter F^ /f . With an increase in this parameter, the losses for design operation decrease somewhat and increase for Mg < M^^^^^. 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- fective for the case where such turbine lattices will always operate near design conditions. The points of intersection of the ciorves (the points A and A' in fig. 7-62 (a)) permit establishing ranges of rational application of the two types of lattices compared. The losses in an im.pulse lattice at subsonic velocities increase with increase in the velocity more sharply than those in reaction lat- tices, and they reach maximum values for M^^ = 0.8 to 0.9 (fig. 7-62(b)). A further increase in the velocity leads to a certain lowering of the loss coefficient. Thus, in the zone of near sonic velocities Mg = 0.9 to 1.3 the coefficient ^p of an im.pulse lattice decreases and becomes a minimum at Mi = 1.2 to 1.4. For Ml > 1.4 with increasing ve-, locity, ^ again increases. ^^ The lowering of the loss coefficient in an impulse lattice at small supersonic velocities is explained by the improvement of the flow about the inlet edges and on the convex surface of the profile. For Mg = 0.7 to 0.9 flow separations are formed near the inlet part and on the convex surface of the profile; the points of minimum, pressure and separation are displaced downstream when supersonic velocities are achieved since the flow in the channel is converging behind the bow waves (fig. 7-60). Also change in the inlet angle has a particular effect on the magnitude of the loss coefficient at supersonic velocities for im- pulse lattices. For inlet angles less than p,-^^^ (a "blow" on the concave 23 NACA note: Area ratio, see fig. 7-62 (a). 24 The 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 ^ on M m.ay vary. MCA TM 1393 65 surface of the profile) the loss coefficient increases. The mean angle of the flow behind the lattice increases with an increase of velocity at supersonic velocities (deflection behind the throat) . 7-14. COMPUTATIOW OF ANGLE OF DEFLECTION OF FLOW IN OVERHANG SECTION OF A REACTION LATTICE AT SUPEE SONIC PRESSURE DROPS There exist several methods of determining the angles of deflection of the flow behind the throat of the lattice. The most widespread meth- ods of com-putation are based on the one-dimensional equations of flow. Assuming that the field of flow in the sections AB (fig. 7-63) and EF (chosen at a large distance behind the lattice) is uniform and neglect- ing the losses in the lattice up to section AB, the equation of contin- uity may be written in the form ABpgCg = EFp2„C2„sin Pg™ or; bearing in mind that for very thin trailing edges AB = W sin p2n = * ^^^ P2n we obtain We divide both sides of this expression by p^.]_a^-|_; then ^02 ^2^^^ P2n = V ^ ^^^ P2CO Taking into account that Pg^ = Pg^^ + 5, where S is the angle of inclination of the flow in the overhang section, we arrive at the equation 6 = arc sin f ^ . ^ sin p2n\ - Pgn (7-60) .^12- P02 In the above equation q„ and q„^ are easily expressed in terms of the pressure ratios p„/p^, and p /p- . 66 NACA TM 1393 For a reaction lattice^ with P2oo/P02 '^^*' ^^^ flow parameters in the section AB will have their critical values when qo = 1. For a supersonic lattice = F /f, *' 1 < 1. By ignoring the losses^ Bais rela- ted the flow at section AB to that at DH in a form sim.ilar to that of form.ula (7-60) 'sin 3p_ ' P2n (7-60a) & = arc sin in /'li^L^lnX With account taken of the losses, formula (7-60a) can be written as POl S = arc si P02 sin P2n P2n that Replacing qg^ ^7 ^2t ^^^ ^v ^^^ taking into account the fact PQl P02 - 4t^v -^ 1 k k-1 k - 1 \^ k + 1 2t(l - U 1 - k + 1 1 \^ '^2t we obtain after transformations ^ _1_ k-1 & = arc sin W 1 - FTT 4(1 - ^p) k k-1 sin p2n ^2t\i - FTT ''St ^ \L] Vl - ^T - P2n (7-60b) Whence, it follows that with constant value of the theoretical outflow velocity ^2t^ ^^^ angle of deflection increases with an increase in the losses. According to equation (7-60a), the angle of deflection 6 de- pends not only on the outflow velocity and the losses but also on the angle P2n- Formula (7-60) holds only for eg ^ eg; that is, up to the point for which the primary expansion wave impinges on the convex surface of the blade. The angle of deflection corresponding to the limiting expan- sion over the convex surface of the blade is a.pproxim.ately determ.ined by the relation ^S = °toS - P2n NACA TM 1393 67 where a^g is the angle of the characteristic coinciding with the plane AD. The pressure in the outlet section of the lattice for the regime considered may be determined by the formula 2k eg = e^(sin 3o„) k+1 (7-61) In fact, since sin(|3p^ + 5o) = sm sm °4nS Psn M2S ^123 we have (%) sin 32n = k+1 Solving this equation for Bq, we arrive at formula (7-6l). Making use of the known relation between ix^ and e and substitut- ing in the particular case ^2 ~ ^S' ^® obtain arc sm Vk + lZ (sin p2n) 2(l-k) k+1 k+1 Pzn (7-62) For the one-dimensional case of infinitely thin trailing edges and straight convex and concave surfaces, the exact solution may be obtained by simultaneously solving the equations of continuity, moment\im, and energy. By the equation of energy, Pg ^ k - P2 ^ 1 ^2 P2» k - 1 ~2 + 2 P2» From the condition of continuity, P2 ^, ^^^(^Zn + ^^ P2co h ^^" P2n c_ k 2 ^S NACA TM 1393 I Substituting this expression in the equation of energy, we obtain 2 k + 1 1 P2« ^2« '5i^(p2n + ^) k - 1 « 2k k P2 Xg sin p2n "^ 2k \\^ I (7-63) We write the equation of momentum using the component in the direc- tion of the trailing edges in the form 2 2 pgcgt sin p2n + P2* ^^^ hn = P2»<^2=o* ^i" 32»cos 5 + pg.t sin pg^ or Since we obtain P2^2^«(^2«c^°s 5 - Xg) = P2 - P2o ^2^2^* = ^2^*1^1 = ^^^^2^01 X2«> /^P2 - P2» ^ A 1 ,, ,,v _ — = + 1) ^ (7-64) If in the section AB the parameters are critical, then [l^V ' £* / J cos ^ _ ^ Xg I k V^ e* y ' "^1 cos & The last expression together with equation (7-63) gives whence k ^2" tan S = ^ n V ^ k + 1 /-, ^00 N"^ k ^ ^^+ n k + 1 - — (7-65) NACA TM 1393 69 Approximately, for 6 < 10 , we obtain S = k + A^-tJ tan 3 2n 2k 2»/ eg-A (7-65a) The above accurate solution obtained by G. Y. Stepanov permits de- termining the wave losses in the lattice. The coefficient of wave losses is expressed by the formula ^b = ^ ^it or after substituting for Xo ^0 k + 1 k-1 k 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 curvature with stra.ight, infinitely thin trailing edges (fig. 7-64 (a)) and set up the boundary conditions at the point where the stream- lines coming off the two sides of each plate merge. The streamline 1-1 m.oving along the convex surface of the plate intersects both the primary and reflected expansion waves, while the streamline 2-2 coming off the concave 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 X-|^ = 1 (fig. 7-64(b)). The velocity of the streamline 2-2 after passing through the primary expan- sion wave is determined by the vector Xg^ while the velocity of the streamline 1-1 after passing through both the primary and reflected waves is determined by the vector X^ . The boundary conditions near the point A for two merging streamlines of gas are the conditions that the static pressiores are equal and the velocity vectors are parallel. These condi- tions are satisfied if the oblique shocks K-|_ and Kg are formed at the point A, the direction of these shocks shown in figure 7-64(a). If the angle 5-, is small, the primiary shock K-[_ may be considered as a char- acteristic, while for computing the edge shock Kg 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 Kg coincides with the 70 NACA TM 1393 normal to the epicycloid of the second family at the point d located at the center of the segment be. With this simplification of the problem, the wake (which for an infinitely thin edge is considered to be between the streamlines 1-1 and 2-2) in the immediate neighborhood of the point A has the direction of the vector Xg (the dot-dash line in fig. 7-64 (a)), The velocities and other param.eters of the flow for the remaining stream- lines are determined after computing the interaction of the primary and reflected expansion waves. The entire region of flow behind the throat can be divided into three zones (fig. 7-64(a)): I - the zone of influence of the primary expansion wave (for the lattice considered, this region transforms into a point), II - the zone of interaction of the primary and reflected waves, and III - the zone of influence of the reflected wave (in the plane of the hodograph, this zone corresponds to the characteristic of the second family be). The region of interaction of the primary and reflected waves of rarefaction (zone II ) may be computed once for all, using the minimum value of the angle Pg inin= '^° ^° 10°- ^°^ ^^''^' other angle Pgn^ ^2 min the computation of the flow downstream of the throat is carried out in the following manner. We draw the x-axis at the given angle to CB (fig. 7-64(c)) and find the mean pressure in the section AB = t t PSidx chara.cterizing the regime of the limiting expansion. For all regimes e > p /p > p /p_ the zone of interaction II will be bounded by the "IT c, Ul o Ul broken characteristics, for example, AB ' , AB", AB'" . . . (fig. 7-64 (c)). To each value of the pressure drop in the lattice corresponds a fully determ.ined position of the points B', B", B'" .... Carrying out suc- cessively the computation of the flow for different positions of the characteristics AB ' , AB", etc., we establish the distribution of the pressures (velocities and local angles) over the pitch AB in the zones II and III and obtain the mean pressure behind the lattice ^2,cp=ir P2i^ 1 -H )>5 o3il 1 o t- \ vA T ) ^ r-- ^ ^ ^ ^*-^or*.^ / C=1 • » ^,=90'^ i \ 1 J 1 1 1 A f \ 1 ^ ^ 1 2 .J ft .5 (c) Figure 7-12. - Determination of an equivalent lattice of circleB. 270 m 15D 90 30 330 no -> ■0- \ ^ ^ ,4- \ r^O.^ / \ \ — -y- — — ( \ \ / \ >4 Cii 'cular / ! J \ Pr ofi J \ \ /I \ / \ \ V / 1 1 c \ / \ 7 ^1 1^ \ 1 '~ '" \ / /-- "" ^ 1 I ; 02 \ /», «i. ~n- 02 \V 1^ s, 'en S- 210 210 150 90 30 330 270 ff> Figure 7-13. - Determination of the correspondence of the points of the profile with the circle (t = 1^ ci = 1, Pi = 90°) . 90 RA.CA TM 1393 Figure 7-15. - Scheme of electrical model of flow without circulation. Measurement of potential. 1, electrodes; 2, source of alternating current; 3, potentiometer (water rheostat); 4, zero indicator (radio phones); 5, unit of potential. HA.CA TM 1393 91 Figure 7-16. - Sctieine of electrical model of flow with circulation. Measurement of velocities. 1, probe with two needles j ?., amplifier; 3, rectifier j 4^ galvanometer; , equipotential lines. 92 HA.CA TM 1393 PVP, (a) Turbine (converging) lattice. (b) Compressor (diffuser) lattice. Figure 7-17- - Forces acting on profile in lattice. MCA TM 1393 93 ^ — ^ rt f^ 5 •H M 4J P< %t •H (D a w - • o (1) 0) o o ^ ^1 g.^ o PR 94 WACA TM 1393 0) o 0} HTtr #» o Pi o o \^ ^. "N ^ ^ «5, n^ -P 'O dj pi Pi e o 2 >^ g (u dj > H <)H P3 -H O O w o o CO -H I Q) (U -P • ^ -H CO H o tQ H t- t^ > Pi Cd -H Q) 03 O ^ Pi W p] ^1 ? o o :3 5D rH O -P •H Pi o 05 r^my •C" KSq ^ n ^ :^;.^ ^^ N S-L >-.' \ \ 1 ^1^ _«— 1 «o l^ "^i ^ c •H Xi *> O o -P H PJ <^^ (D > o u e^ ■P H 03 (C> Ch iH O Si 0) Vr» Xi • P^Td Xi Ql Pi 4^ ^ u pi o O Ch ^ O '*» 1 0) >. t- •H >! \ ^ ' ^R^ C3 + \ W * W 1 * w ^ Wl CM c\J u c3 Pi g H U O 05 H O * I m (D -d +> •H I <.\i -p •H 05 Pi pR c:) T! t*5 pi -p ^ pR 96 KA.CA TM 1393 60 50 UO 30 20 10 1.1 c 1.0 0.5 B/t=1.6 l^ 12 n .03 1.8/ I 1 / / 1 \ 1 1 \ 1 1 1 .Oii •// / / 1 / / r < y .c 1 — .06X^ ^ ^-^ ^^ W \ -— ^^^^ ^ fe *^~"~~" ~ 72 ^T A^ / / «- — "^L ^^•-.^ /; 1.0 .8 .6 W 20 .W VZ7 50 (b) 60 70 80 90 Figure 7-23. - Computed magnitudes of the friction loss coefficients in turbine lattices as a function of p^ and ^2, • MCA TM 1393 97 (a) Mg = 0.565 (Id) Mg = 0.773 (c) M„ = 0.940 Figure 7-24. - Spectra of the flow of air through a reaction lattice at supersonic velocities. Eelative pitch of profiles t = 0.860, inlet angle of profile Pg^ = 15^52 ' (visualization of trailing vake) . 98 KA.CA TM 1393 =>^^ / 1 / J a ^^ ^ ^ »-~.- •H m ^ •H A ^H o Q) •H ■!-> ■P p 01 I^H ^ ^ 4-> o d •H c:3 g 0) m 43 0J~-^ O en f>» tH 3 CO •H 0) II d hO -H 3 fl ■P !>5 •A •H •H '^a o (-1 (D pi •H m rH 'H tH h (D 4J o 4-> ^^ +J (1 H ^ n ■H •H 01 1=1 •H c 43 o o to +J •H •H ,D tin +J P ■g tH 01 10 p] J> ni •H •H o ^< rH ^ 01 rH (D > i ■" , — ^ ,Q 1 CO (U «\i (.0 ■p ri 0] 0) 1 ^X. t^ -rt • 0) Q cd •H u ■P 01 ■P Pi 01 H — ^ ^TjS \A. lS^ 3 } ^^^=4-. \ . -■■, — 1 — 1 N X \ //A y r \ s ^\\ /^ /I' 1 1 ^ 1 — \ — ' r^~ 1 \\v v^^^^ i .,, J. 1 1 \^ L_ ^^ 1 1 1 1 ^ 11 &^ ^^ T^ i ! ■- 1 ! i 1 ! 1 1 1 i 1 ; 1 f¥15 IB 17 fB W 80 21 ZZ 23 2^ 25 26 27 28 E9 i 2 3 U 5 6 7 8 ^ V ff 12 f3 fV Convex surface Concave surface (a) Setting angle. i n 15 15 17 18 19 20 31 2223 2V 2526 27 23 29 1 2 3 V 5 6 7 8 9 10 11 12 l3 IH Convex surface Concave surface I (b) Flow inlet angle. 1 ■ Figure 7-29. - Effect of -blade setting and flow inlet angles on the velocity '■ distribution over a profile in a reaction lattice. 102 NACA TM 1393 J,i C 1 trr r- • A ^^SS^iP' )^. -nuc? IS 12 ^ 4_ fe i5 'P,-36'V7', t-asja J fc^r^ r^*' ^ sj ^ ^ -BrJS''/7'.i- 0737 f^ i ^ ^ h ■^ \ A ii r L^- ►— 7 H^l^. J <— X — K — "*^ ^^ir^ / / v, ■ _^ 11^ ^ ''■■■? -1 -r Y 1 : 1 11,11 ! 1 1 1 1 i M 1 1 f8 W Zf 23 25 27 29 31 33 35 37 33 Convex surface (a) 5 7 3 f1 13 f5 17 13 Concave surface 30 2.8 20 '/ '.2 J '/ c i ti '^. ,'3s* , 1 \ ^■'44=?=f^ ^m I 1 r i^iA ^ TT .-_ 1 "My 1 1 V i I 1 te IS 21 23 25 27 29 31 33 35 37 33 f 3 5 7 9 ft 13 15 1? 18 Convex surface Concave surface (b) Figure 7-30. - Effect of pitch and inlet angle on the velocity distribution over a profile in an impulse lattice. i MCA TM 1393 103 Beaction (TsKTI tests) Impulse .(TsETI tests) Impulse (MEI tests) Figure 7-31. - Variation of coefficient of losses in impulse and reaction lattices as a function of the inlet angle p, and the pitch t. 35 JO 25 t^DJOii H /* A« • rz" • // — . 1 — — -1 — — — ■ r- —r ho — 30 25 20 ■-0.5^3 A" > — w. r^ t=1 By fommla (7-^^) &° 50 6L 1 7L i Si 7 9l 7 WO p; t--0.5^3 Pj^ 72^30' Ju -A ^ •"'^ -♦-J "■v^ + dJ \ / z cO 2 4 6 , 8 f.o Figure 7-32. - Dependence c5f the mean outflow angles on the inlet angle for two values of the pitch of a reaction lattice. 'Figure 7-33. - Variation of local angles Pg "with pitch x = x/t. 104 WACA TM 1393 JO o w o •H -P a o •H -P U 03 0) o p< > o -P cd i-l <0 o p< 03 > o en m 0) ^1 ft O Fl O ■H •P P! ,Q •H ^1 4J ■H 110 NACA TM 1393 0) 1 a o > O tH (D C •i-i ■P ■P OS 0) •H o ;^ o ■rt -P ■P 03 CO MCA TM 1393 111 A At point/ oAt point^ +At point J -./ ,2 A .6 .8 1.0 ;,2 11 Figure 7-43. - Dependence of pressure coefficient behind trailing edge p, on Mo number. Figure 7-4* • - Effect of Mo on shape and dimensions of trailing vortex wake. 112 NACA TM 1393 ,P^=9(f- .2 .If J JB 1.0 Figure 7-45. - Loss coefficient of impulse and various reaction lattices as function of Mo. TTACA TM 1393 113 >H^ v^ r <«s \ ^ JD »^ ««s Ci «i' ^^j* -•> II II >l ' « ^ A \ \\ \ ^ \ > \ 1 V \ ■» > \ 1 H • ^ V sss5 ^ f •^. < ■»*. 5a Ti u • (1) d) t> o C) •H P CQ P (1) 01 [Q H OT O Pi H O H -r) ■P ri O ol 01 (I) to ^ fl) H 01 ^ t:) 01 f^ •H •s ri C/J a) (D ,0 •H -P H •H • C) c> O 11 >^ •\ CQ o fl) Pi ^ 05 -p tu CQ CQ •H •H pl -P o pi •H ," -P •H c> M Pi -P J3 C/J ffH •H M OJ 1 CQ 05 CD A -^ o 1 +J l>-'H p< a: ^1 a) §)-p •r Fh 114 NACA TM 1393 'hi '^ -ir -^ -a- 3=1 <^ «L- e- ^-r- •^^^^t-^^^'rr (T — ' 1 ■ c ^ (a) (b) 2B 2it 20 IB 12 ^n . > K J ^ /■ »^ k F^*^- H ^ ^ / 1/ ^ ^ *^i f ./ .« .^ ^ /.^ (c) 50 t, 4- ^ 7 A 10 / / \ \ y / \ & 30. V w\ , ^/ / V) jj ^' 2D' V (W 7 ID J r ^ a / n 1 4 .S (d) .-» f.O Figure 7-47. - Distribution, of local pressure^, velocities, angles and losses as a function of Mg at the distance y = 0.1 behind an impulse lattice . 20 15 10 1 Impluse t-0.52 Reaction t=0,38 —I I I 1 1 — A/c .V .6 .8 10 Figure 7-48. - Dependence of the mean angle of the flov ■behind a lattice on Vi^- MCA TM 1393 115 Figure 7-49. - Scheme of outflow from a reaction lattice at atove-critical pressure drops. 116 NACA TM 1393 t 'I ^^^^m "^^"^ mf ^B M ^ ^^^Msst^ m^ ^B^^f^^nBbB 1 ^^P n ^ kL. ^ I "^ pi ' 05 o (D ■H rH d •H O •H (U « 1-4 (D p] •V\ O to •H ^ -P LO O • 05 O di U II •§. l-P g ^ o 1 -p •H P ft > © o > H •H tH -P 05 M H •H (D ca w - CVl • o in o •HO in Pi in 1 O rH L CQ U II 0) O ^ P* ^ p p( (n: bo w ca ^ NACA TM 1393 117 u -P 05 -p to O ^ c8 m P< o iH © •H ^^ tS H <0 « >cJ o •■ 13 -H w ^ fi in coo SJ to Ln o u H O 03 p( II 1 pi m a o •!:( ax Ln pi 1 S (D t- H o •H ;■« 5 118 NACA TM 1393 Figure 7-51. - Spectra of air flow through reaction lattice at Bupersonic velocities. Relative pitch of profile t = 0.86; exit angle of profile 32ii = 15°52' . MCA TM 1393 119 (c) Figure 7-51. - Concluded. Spectra of air flow through, reaction lattice at super- sonic velocities. Eelative pitch of profile T = 0.86; exit angle of profile P2n= 15°52'. 120 NACA TM 1393 ./ Pi 1 /^ i^ i .>\ .^ \y\ K\ L / .6 / ^^..^ J ^ ^^/( / ,^ Y '^ F^ ^ / / Y L5 1 1 \ // .5 ' /v J / "X \ \ T// 1 / ^ y ^ i \ F .U ^ 1- 7 I rf Ir V .3 ' ~^'. V I b r } \ S/ 1=0. 705 .2 u \ / ! 1 0.662 . \ \ / Z .630 C "Ir^ \ N Y 3 .5 Si \\ / ., / \ V / 6. .U90 .1 V y 7 .¥J0 8 .402 ■ \ y >^ ^ 9 .533 W .297 11 .264 ^i It .. '45 ' / 2 3 « i i y ' i Surface (b) Figure 7-52. - Pressure distribution along convex surface of profile downstream of the throat at above-critical pressure drops. MCA TM 1393 121 S .8 .7 .6 .5 ,1 .3 .2 .1 Ap 2 1 *-^, ^imln. // \ \ 1/ L\ \i= 0,705 ^> \N \ \ \ > ^0,61^ ^ \ \ ^0.5^3 s \ k \ \ s \ i ez .5 .6 Figure 7-53. - Rise of pressure in the system of shocks on the convex surface of a blade as a function of the pressure drops in the lattice. 122 NACA TM 1393 D.ii J E^ 1 Pot \ ^ 2=0.2''t 1 ^ r K K _ (^ K. 1 / r \^ r nL/ 1 \ l\ li-" p^n^ \ 7 V- ' 1 1 \ J _ T- T 1 F .8 1.0 .S KC. Figure 7-54. - Distribution of the flow parameters over the pitch of a reaction lattice at supersonic velocities. (b) Figure 7-55. - Scheme of flov spectrum at the exit from a supersonic reaction lattice at the computed regime. NACA TM 1393 123 xi J i 1 f* /J fi ff »f /; Z? 25 2S SO ^ *y tS *5 ^J9Tfnj3Si29P 10 J. — ^- ^ ^ -^ ■> \ ^^r:::^;:^ir^ / ■ .8 \ L -^f^rr^^^:^ ir«^ / \ ^ - — ^ V -> \ 5^ N \ y ^ ^ X \ s \ \ / / /^ tf." ^ N N k N / \ ^ / t / ^ ^ \ \ "^ 7^ V N ^ y y I ^^ 1 L .^^L • /JI7 W KO SO U W ^ so iO *0 JO u Convex'^ surface W 20 iO *i M U TO U to Concave surface (a) Pressure distribution over profile. 2 A J JB 1.0 ;.2 (b) Velocity distribution over pitch. Figure 7-57. - Pressure distribution over a profile and velocity distribution over the pitch for a supersonic lattice at various regimes. NACA TM 1393 125 » Id (0 o •H ■P -P 0} H -P H g m u o o I o CO CO I •rl 126 MCA TM 1393 (a) U-^ = 0.79. (b) M-]_ = 0.89. (c) M-^ = 0.98. (d) Kj_ = 1.02. (f) Ml = 1.42. Figure 7-59. - Air flow spectra through Impulse lattice at near sonic and supersonic velocities. NACA TM 1393 127 CO <1> •H ■P •H > (3 O u O o •H +^ ^ •H ^1 -P (Q •rW O U 3 to CQ 0) o CD 0) •H 128 KACA TM 1393 Figiire 7-61. - Spectra of supersonic flov about a lattice of plates; Mo = 1.42. MCA TM 1393 129 .8 .9 1.0 l.t 12 13 /.« /.5 t.B (Td) Inrpulse lattices. Figure 7-62. - Loss coefficient as a function of K^^K.) . 130 MCA TM 1393 Jigure 7-63. - Determination of the angle of deflection of the flow behind the throat of a lattice. Figure 7-64. - Computation of the angle of deflection behind the throat by the method of characteristics. NACA TM 1393 131 20 15 W H2<^Pzn o TMck trailing edge - A Thin trailing edge 0.614 • Lattice of large pitch, t = 0.860 — - By formula (7-60, b) -.- By formula (7-65) + Computation "by method of charai^teristics and Figure 7-65. - Experimental and computed mean angle of deflection of a flow behind the throat of reaction lattices. 132 KACA TM 1393 ^^ 7\- /J Back I— a^*| ^^ Pressure --' diagram J (a) Streamline on plane wall and back. --y^l— r ^y^ h (b) Peripheral flows In i boundary layer at ends of blade, thickening of layer on back of blade. ^-^^^ Axis of vortex \ \ '/////////////^ ^Criy- v./ w> ^7777777777777777: (c) Axes of vortical braids, vr (d) Velocity field induced by vortical braids . Figure 7-66. - Scheme of formation of secondary flows in the inter- blade channel of a lattice. MCA TM 1393 133 Figure 7-67> - Wakes of tip vortices in interlDlade channel . 134 NACA TM 1393 - .2 - ./ Pa .9 w + J * .2- .2 -J (a) (b) ./ -^ .2 Figure 7-68. - Distribution of flow parameters over height of lattice for Mg = 0.78. MCA TM 1393 135 1.0 Poc 1 M.2 1 ,r\ , _. . — _ f\ .../. ---. I ■\ \ .3 lA if ^ 2:0 1 \ \ \\ \ \ i \ 1 I 1 1 ^, \ ^7.7 'A i 1 1 -\ V3.0 1 \ n l\. \ i J\ \ 1 / \ 1 , V V t ,0 I) —^ / \ ^ J V. \ ■y Vw \J v^ \j ,/ ^ -17 L-Lp 9 n 1 i a. ^'^ .0 C.U 7/7 / c J.U 02 .5 1.0 IS 2.0 3.0 J.5 (a) ID Poi 'f\ ,--' L-J V / 1 7 i H~- t^O.Si^ ' r — i=0.8bV \ / i-«.^ 1 z ..? (b) in is Figure 7-69. - Effect of geometric parameters of lattice on distribution of dynamic pressure over height on convex surface of blade. 136 NACA TM 1393 ■-'// ///////////////////// ^ // // '^ "?»- •p • c en a) u •H (l> H !-i O tn tH O 0) PI t3 O cri •H H +J ," 3 ," 4-> •H ^H ^H O V -fl CO CO •H H ItH O ' +J 1-1 ^ t-- •H (1) t-^ 0) ^1 u 0) a !> o - -H Q) tH P tn CJ •a oj CO O T3 CO c S 5 2 Is _ -- _ u I* O J= x: H CD a. ex. 2 CQ 5 r^ -HO rt Q < « 2 H ■ t-H CM CO ^ ^dB CU rt m rt 2 ax: S t, o C 01 C T3 3 — to > o . OT OJ O S^ ° C CS -^^ t. O o o -3 ±3 .2 gg t: "• S-S " •art tn I o 3 O CO £ S =« ■ IS- zs a .§ M 0) .-y o 2 "- 1; . •a > S u' « S ™ s ■ ™ ■§ "* m to -c o a c CO 0) g S-d 1 o S !3 •a o u < T3 CD CO d 2 u 01 ■^ ? >, tn u< n1 C) >> (J s o CO -g .2 •o o u a S . rt O u <1J O JS •g CO -a" T* 1^ c^ (U a H o ii c X . — ' J3 3 and E Deich NACA Techn ch. 7, H 1-1 C^J CO cH ^aa OT u CJ si H c < o J OJ W <; 2 i! K > ■< c M O x: O (u — CU CTl 01 X! 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