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3r0 7qS /) ? 7?7 T7 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1358 CALCULATION OF THE SHAPE OF A TWODIMENSIONAL SUPERSONIC NOZZLE IN CLOSED FORM* By Dante Cunsolo SUMMARY The idea is advanced of making a supersonic nozzle by producing one, two, or three successive turns of the whole flow; with the result that the wall contour can be calculated exactly by means of the PrandtlMeyer "Lost Solution." PURPOSE OF THE INVESTIGATION The subject matter of this paper is based on the artifice of not letting the expansion waves emanating from one wall of the nozzle be reflected from the opposite side, but of cancelling out their effect by compensating compression waves emanating from this opposite wall. In this way the difficulties attendant upon the intermeshing of the Mach waves are avoided, and it is no longer necessary to integrate the differ ential equation of the hodograph by recasting it as a finite difference equation, the solution of which is necessarily approximate. The tech nique illustrated here is nothing more than a quite direct application of the PrandtlMeyer relationship for flow around a sharp corner. On the basis of this procedure, the calculation of the coordinates of any point on the nozzle contour is independent of that for any other point; that is to say, this way of handling the problem eliminates the lack of precision usually associated with the tail end of the effusor in com parison with the better accuracy at the beginning sections. In addition, what is of the utmost importance, is that it is not necessary to determine the characteristics of the flow throughout the interior of the nozzle, which leads to a tremendous saving in computa tional labor, with no deterioration in accuracy. *"Sul Calcolo in Termini Finiti dell'Effusore di una Galleria Bidimensionale Supersonica." L'Aerotecnica, Vol. XXXI, No. 4, 15 August 1951, pp. 225230. 2 NACA TM 1358 Theoretical Background Let us focus our attention now on the expansion which occurs when a stream turns around a sharp corner (see fig. 1). In this figure 1 the following notation applies: is the Mach angle, is the com plement of the Mach angle, v is the angle through which the flow has turned as it progresses through its turn around the corner, so that related to this angle of the flow vector is the angle 9, which is the angle between the normal to the wall and the radius vector originating at the corner and lying along the points in the flow which have turned through the angle v. Then, upon making use of wellknown theoretical relationships, it follows that S= v + tan X1 = X tan Y + that is, v = arctan tan X9 (1) Now let us employ this PrandtlMeyer relationship in order to determine the streamlines of the flow. Let us denote a small lineal element of a streamline by the symbol ds (see fig. 2), and then we may take dp and pdO to represent the components of this element in polar coordinates. Thus one may write: dp tan X9  = tan t = M pdO X dp tan X9  =  d p X NACA TM 1358 3 and consequently log p + log cos X3 = const. X2 or 1 p(cos xa) = const. and upon selection of the value of 7 as 1.40, one obtains: p cos6  = P (2) One of the walls is thus the straight terminal side of the angle of expansion while the other curved wall is defined by means of equa tion (2) as far out as the location where the angle characterizing the radius vector has ultimately reached the value 'M, at which point the desired Mach number will have been attained. From here on out this wall is also straight and lies parallel to the terminal side of the corner angle. As may be seen from reference to figure 3 the minimum length of effusor which it is possible to have is one which terminates right at the point where the wall stops curving, at which point the very last expansion wave has just been included in the process of executing the total turn. Of course, the depth of the "throat" or critical cross section is the quantity denoted by p4. If one wishes to avoid reliance upon a perfectly sharp corner for making the expansion, the two curved walls (see fig. 4) given by the equations P1 cos  = P (3) \ 6= p2 cos = P2* NACA TM 1358 may be employed. Of course, in this case the depth of the throat or criticalcrosssection is given by p* = p2* Pl*" Naturally, for an effusor configuration such as this, the length becomes longer than in the case of the sharp cornered type. The expan sion waves which originate from the number 1 wall are exactly counter acted or "swallowed up" when they meet the number 2 wall, without pro ducing any reflection. It is just for this reason, even in case one wants to give some other arbitrary shape to the number 1 wall, that the calculations may be carried out in parametric, but closedform, provided the Mach lines are maintained as straight lines. Once it is decided what the shape of the number 1 wall is to be, one may find a value of v which corresponds to the value of B defining the direction of the Mach wave with reference to the yaxis. This value of v locates a point with coordinates xl and yl on the number 1 wall. Consequently, the point lying on the number 2 wall which is marked out by this hray will have the coordinates (see fig. 5): x2 = x1 + p sin 6 ( Y2 = Yl P cos 6 where p = p* (cos 1 Obviously it is necessary to have xl and yl given as functions of 8. Let us follow through on the details in the case where the number 1 wall is taken to be circular, with radius = po. Then the value of 9, used as the independent parameter, locates a point with coordinates (see fig. 6): xl = PO sin V Yl = P* + PO(l cos v) NACA TM 1358 wherein the value of v is given by: v = 8 arctan (6 tan  and the point (xlyl) lies on this 0ray. Likewise let us examine more fully the case where the number 1 wall is parabolic in shape at the start, with a radius of curvature equal to PO at the apex of the parabola (see fig. 7). Taking the equation of this parabola to be x2 y = p, +  2pO then its slope is given by tan v= dy = x (6) dx PO Consequently, since B is taken as the independent parameter, one obtains v = arctan 6 tan  xl = PO tan v (7) PO Yl= P* tan v In those cases where the slope dy/dx has more complicated formula tions (these would be cases devoid of any practical interest for that matter, in view of the extreme simplicity of construction exemplified by NACA TM 1358 the contour of the number 1 wall for the configurations just examined in detail) one may still solve the problem, to whatever degree of accuracy is desired, by working with the isocline system: v = arctan ( tan  dy = tan v (8) dx y = f(x) which gives the slope of the contour applying at each assigned value of the parameter 3. Then, after carrying out this construction, one may find the corresponding values of the coordinates x2 and y2, by means of equations (4). If, however, one insists on having the working section not offset from the axis of the throat, it is necessary to apportion the total angle through which the stream is turned into three pieces. To be more precise, let this total deviation of the flow, corresponding to the ultimate Mach number to be attained, be denoted by vM. Then the three partial turnings of the flow would be, respectively, an upward turn of vM amount VI', a downward turn of amount  = v', and once again an VM upward turn of amount V M = V". With this arrangement one now has an effusor made up of three expansion regions separated by two sections of uniform flow, wherein the Mach numbers attain the values M' and M", respectively. Now let v be the net angle through which the flow has been turned up until the time it has reached a certain location in the effusor; and let 9 be the angle linked to this value of v by means of the Prandtl Meyer relationship, equation (1). In addition, let a be the angle swept out by the Mach wave with respect to the yaxis. Then, in the first expansion region, we shall have that a = (see figs. 8 and 9). The first Mach wave beginning the second region of expansion will be characterized by the angle a = 2*' Y', while within the second region it will be true that a = (29' ') + ( 5') = 8 2v'. The last Mach wave ending the second region will be characterized by the NACA TM 1358 angle a = 9" 2V', and the first wave beginning the third expansion region is then given by a = 24" (6" 2v') = 23" 2v" 0" + 2v' = 8" vM. For like reasons we have that, within the third region, the Mach angle is given by a = (6" vM) + (b t") = b VM. In summary, we have that, in the three expansion regions, the following hold, respectively a= a = 2 2v a = 3 VM Let us consider what the appropriate relationships are, in the case of a tunnel design such as illustrated in figure 8, wherein the sections of tunnel wall effecting the expansions, that is to say, the convex por tions of the wall, consist of circular arcs. For this example the value of v' must turn out to be calculated in such a way as to make the pro jection of the broken line A B C D E F G H L upon the yaxis have the value LM y 2 (ABCDEFGHL) y = 2 where oM and ao are the widths of the effusor at the very end and at the throat. Writing this condition in explicit form, we obtain the expression P0 Scos P2 cos v' + cos6  P2 cos (  3 1 cos v +  cos (I" 2V')  cos  w6 VM 2 Pl(1 cos v')  V = m (10) NACA TM 1358 where v' and v" are the net angles through which turned at the end of the first and at the end of the regions; i.e., tan Xi' v' = 3' arctan VM V"= V' + = 2 tan Xd"  arctan k the flow has been second expansion (11) In fact, one has in particular that: (AB)y = P1 (BC)y = pl cos V' (CD) = CDi cos = p' cos = (DE)y = p2 coB v' (EF)y = p2 cos (EF,y) = P2 COB 2  (FG)y = p" cos (FG,y) = p* cos6 %F6 cos (g" 2v') ( v v (GH)y = P3 cos p cos ' co6  NACA TM 1358 In this case where such circular sections of wall are considered to effect the expansions, it is necessary to utilize equations (5) and (4) (when suitably modified for treatment of the flow in the second and third expansion regions) for getting the solution, but if this circular wall contour is replaced by the curve given by means of equa tions (3), then this case is handled by setting p2* Pl* cos  c6 d' Cos  P2* CO6 cos  46 01 cos (i' 2V') + cos6  f6 cos (B" 2V') + COs 4_6 PI* o}M 0* S cos (M VM) = cos6 M 2 provided it is assumed that IDE = BCJ = pl = * C6 O 5 cos  GHI = IEF = P = c 6. (12) 10 NACA TM 1358 In fact, one has in particular this time that (BD)y = cos ' COB6 6 cos6 _' {6 angle between ED and y = (ED,y) = ' 2v' 02* (EG)y = cs (E,y) 6 '" COB0 angle between EG and y = (EG,y) = S" 2V' angle between GH and y = (GH,y) = o" VM and (HL)y= l* cos6 M f6 wherein the symbol *M stands for 'M VM. Numerical Applications For illustrative purposes let us suppose that one wishes to design a tunnel which will reach a Mach number of M = 2.5 by means of an effusor such as depicted in figure 6, or else with one with three kinks as in figure 8. NACATM 1358 Setting down the numerical values available to us, we get the following listing: M = arcos = 660 25.3' M = 6 arctan tan M = 6 arctan M2 = 1050 327' f6 46 vM = 390 7.4' _M 1 ( =+ M2 OM o f_5= = 6.5919 cos 6_ 6 M 1 /5 +M2 3 = 2.6368 a* M R 6 Let us compute the value of v which applies to the type of circular arc expansion incorporated into the figure 8 design. In this case if we let a = p = 1 P1 = 0.5 02 = 03 = 1 M 0* = o.8184. 2 then, from equation (10), NACA TM 1358 Let the function A(V') represent the left hand side of equa tion (10) (which is legitimate because ', v", and B" are all func tions of a'), and we then propose to find the value of 9' which makes A(') = 0.8184 (13) In order to effect the solution of equation (13), the author made use of the method of approximation by secants (see fig. 10). Let us assume for example that 91' and B2' are two values of the independent variable which when substituted into the expression A(V') renders a result which is too small in the first case and one which is too large in the second case in an attempt to find a value of V' which makes equation (13) hold true. In addition, let 3' be a value of the independent variable which is much more accurate, and which can be obtained by linear interpolation. Then the two secants P1 P3 and P2 P3 will cut the horizontal line A(4') = 0.8184 in two points, which may be labelled S1 and S2. A better approximation to the exact solution of equation (13) is then obtained by selecting V' as equal to a value '4 which falls within the interval defined by the endpoints S1 and S2. After discarding one of the more distant points on the curve (in the case illustrated by fig. 10, the point '1i would be dropped) the whole process is repeated by working with the points P2, P3, and P4 to start with again. We shall try out this procedure by first selecting the value of 9' as = 600. Then one finds that: B' = 600 V' = 11.90 v" = 31.50 " = 94.30 A = 0.59 <0.8184 NACA TM 1358 The value of 6" listed here is obtained by use of the relationship V" = j" .arctan 6 tan " through use of the method of approximation based a tangent. According to this method, if a first which is too large is determined and denoted by accurate approximation to the exact value of 3" *2 = ' V" V" V V"  1 1 dv" 5 on interpolation along approximate solution i then a more is given by 6 + cot2 1 2 if6) where the value given above for of V1l is obtained by an analogous formula to the one v"; i.e., here one uses the relation Vi = *l" arctan (V tan )" 46/ From ' = 600 the value the above derived numerical result it is seen.that the guess is a too small solution for equation (13). Consequently if ' = 700 is now tried, it turns out that: 3' = 700 v' = 16.90 v" = 36.5 8" = 101.70 A = 1.47 > 0.8184 14 NACA TM 1358 Thus the trial value B' = 700 is a too large solution for equa tion (13). Linear interpolation then results in the value for a' of 62.60. Consequently, using this value it results that: 4' = 620 36' v' = 130 5' v" = 320 39' a" = 960 4' A = 0.762 < 0.8184 Now continuing the computation to obtain a better approximation by means of the method outlined above (see fig. 10), one gets that a' = 630 20' v' = 130 26.7' v" = 330 0.4' S" = 960 36.9' A = 0.8183 Consequently the solution of equation (13) is v' = 630 20' with an error which is less than 1 minute. In carrying out the actual construction of the effusor contour it is necessary to point out that, if 0 is taken to denote the direction of the flow with respect to the xaxis, the following sign relationships hold: dP = dv in the first (I) and last (III) expansion regions dB = dv in the central (II) expansion region NACA TM 1358 15 Now perform the integration indicated by equations (14), substi tuting the proper limits, and one obtains 3 =v 0 = 2 v (15) p = v VM in the regions I, II, and III, respectively. If the coordinates of the points B, E, and H are assumed known, then the equations (15) afford the means of determining the coordinates of the point (xl,yi) which lies on the convex portion of the wall contour and which is related to an arbitrarily selected fixed value of the param eter 0. In addition, let the former equations (4) be modified to read X2 = xl + p sin a Y2 Yl T p cos a (16) p = p* cos where the + sign in the second equation holds just for the central II region of expansion. These expressions, together with the equations (9), allow one to compute the coordinates (x2,Y2) of the corresponding point lying on the concave portions of the wall. Working with the illustrative case depicted in figure 9, let the starting data be selected, for example, as P* =1 = Pl* = 0.3 P2* = P1* + P. = 1.3 16 NACA TM 1358 Upon substitution of these values into equation (12), one will find that B(1*) = 0.8184 (17) Continuing the computation by an entirely used previously, it is found that analogous procedure as ' = 630 5.7' v = 130 19.6 ' v" = 320 53.3' 9"= 960 26.2' In order to carry out the actual drawing of the effusor shape (see fig. 9) the relationships given as equations (9) are again employed and the results applying to the three convex portions of the wall will be given by, respectively: x1 = XB + P1 sin a yl = YB P1 cos a xI = XE + P1 sin a Yl = YE + Pi cos a xl = xH + Pl sin a l = YH P1 cos a where PI* P = cos6 ,F6 NACA TM 1358 The determination of the concave portions of the wall is carried out in an analogous way, except for the mere replacement of the sub script 1 by the subscript 2. CONCLUSIONS The advantages accruing from the methods expounded here for design of the various kinds of effusor lie essentially in the great degree of precision with which it is possible to draw in the wall contour. A blemish of the method suggested for design of the effusors possessing three successive expansion regions (see figs. 8 and 9), which must be acknowledged, is that the length is longer than what would be found necessary if the effusor had been calculated according to the usual hodograph method, although the latter would be less exact (see fig. 11). The lengths of effusor, when nondimensionalized by reference to the throat depth, have the following magnitudes in the cases exemplified by figures 8 and 9, respectively: = 12.2 P* (18) = 14.2 P* The same ratio in the case of the effusor represented by figure 11 has the value 9.8, but it will be cut down to only 4.9 if the effusor shown in figure 11 is considered to be merely one half of a complete symmetric tunnel. This defect in the effusors described here ceases to exist in the case of the skewed effusor designs first mentioned in this paper. These offset designs can be fruitfully employed nevertheless where there is restricted room for the setup, because it should not entail a very great deal of trouble to incorporate a suitable compensating kink in the sub sonic part of the tunnel. A promising compromise design can be obtained by use of an effusor having two expansion regions (with equal and opposite amounts of turning). With this configuration it is clear that the working section will be lined up parallel with the axis of the throat section, except that it NACA TM 1358 will be offset laterally (see fig. 12). The length of the effusor and its lateral displacement will have the following sizes, respectively 1 9.7 P* (19) =1.4 P* As is evident, this displacement s/p. is not of formidable size, while on the other hand the length of the effusor has been somewhat reduced, in comparison with the cases illustrated in figures 8 and 9. Despite all that has been said, it is still worthy of note that in the case of effusors designed for Mach numbers which are only slightly greater than unity; that is, for effusors whose lengths do not exceed that of the working section, the effusor designed with three expansion regions can still be employed to good purpose. Such an effusor designed to produce a Mach number of M = 1.2 is illustrated in figure 13. Translated by R. H. Cramer Cornell Aeronautical Laboratory, Inc., Buffalo, New York NACA TM 1358 Ml I ~ I ~77772777 ( / // / \ I Figure 1 dp / //. a M=i Z1 1 7 / / Figure 2 20 NACA TM 1358 Figure 3 p NACA TM 1358 21 / / Figure 4 22 NACA TM 1358 SY2) Figure 5 NACA TM 1358 (x2,y2) Figure 6 24 NACA TM 1358 Figure 7 NACA TM 1358 25 bcm b , I I I / / / / / I, I 1 ^ / I / I / I / / I I / / iI I ,A cia Q J i 26 NACA TM 1358 NACA TM 1358 A (0) = 0.8184 Figure 10 A(,') NACA TM 1358 P4 NACA TM 1358 Figure 12 / \\ M =1.2 I I/ I \ // \',, II \'.// 0\_' Figure 13 NACALngley 130U 1000 r, " d o u Ga E u Oaz:w e z~~d a a .ou3 5  0 0 ONC .e. A pZ a ! Sino Z Wc m zo u *06 a 0 Z m T z z uaE> Lu[ n^o3 " o .^^^"i's C2 r 2 o i c mN aU 0 N_ C~. 0m :Cu'~ zNz . 0 03 0 o W u N 1 C <.= S E Lwz a 3 U C' S .0 0 z z 0 LL < : : i oZU 0C .6 E 'ID O .^ Q .I 0 3 1 I NW 0 NUi o r a C0. g cd m L.WL I a E~; Cu iu Cu Ei1 0 W 0 6. E ci 0 LPP W 5 3 > o. a o cdU u 10,' 0 ' aC L. CLr 5. 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S.. ~ .0 * 3 4)ak A .20 a's^ Se S4 a w i~ 0 ^ * ^ m r UNIVERSITY OF FLORIDA 3 1262 08105 785 2 