m. M. I.EADON 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1325 
 
 CONCERNING THE FLOW ABOUT RING-SHAPED 
 
 COWLINGS OF FINITE THICKNESS 
 
 PART I 
 
 By Dietrich Kiichemann 
 
 Translation of ZWB Forschungsbericht Nr. 1236, June 13, 1940 
 
 Washington 
 
 January 19%nWERSVTY?^FLOB0A 
 
<r^2. 
 
 I M 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1325 
 
 CONCERNING THE FLOW ABOUT RING-SHAFED 
 COWLINGS OF FINITE THICKNESS* 
 
 PART I 
 By Dietrich Kuchemann 
 
 ABSTRACT: 
 
 It is shown how one may obtain, in a simple manner, the forms of 
 ring-shaped bodies from existing tables of functions according to the 
 customary method of superposition of flow due to singularities and 
 parallel flow. A number of examples of the forms and pressure distri- 
 butions of annular source bodies with and without hub body are given, 
 and the inlet conditions of such ring-shaped cowlings are investigated. 
 Furthermore, the annular bodies of finite length are indicated that 
 correspond to Joukowsky profiles for the two-dimensional case. The 
 examples are to give a basis for the design of cross-sectional forms of 
 ring-shaped cowlings and a survey of the flows to be expected. 
 
 OUTLINE : 
 
 I. GENERALITIES CONCERNING RING-SHAPED COWLINGS AND PURPOSE OF 
 
 THE REPORT 
 
 II. METHOD AND RESULTS 
 
 1. The Method 
 
 2. A Ring of Singularities in a Parallel Flow 
 
 3. Annular Source Body with Hub 
 
 h. Ring-Shaped Cowlings of Finite Length 
 
 III. SUMMARY 
 
 IV. REFERENCES 
 
 *"Uber die Stromung an ringformigen Verkleidungen endlicher Dicke. 
 Zentrale fur wissenschaftliches Berichtswesen iiber Luftfahrtforschung 
 (ZWB), Berlin-Adlershof, Forschungsbericht Nr. 1236, Gottingen, 
 June 13, 19^. 
 
NACA ™ 1325 
 
 I. GENERALITIES CONCERNING RING-SHAPED COWLINGS AND 
 PURPOSE OF THE REPORT 
 
 In techniques, annulaj- bodies or ring-shaped cowlings are used 
 mainly for two different purposes: In the cooling problem (cowlings of 
 radial engines, nozzle radiators) and in the shrouding of propellers (Kort 
 nozzle, axial blower). In both cases the cowling serves for regulating 
 the flow and its velocity at the location of the cooling block or of the 
 propeller, respectively, although in a different manner. One speaks in 
 this connection mostly of cowled radiators and shrouded propellers, and 
 links with this designation the following conception: 
 
 For the cooling problem, it appears useful to cool at minimum flow 
 velocities; therefore, a diffuser is placed ahead of the cooling block 
 and the through flow is regulated by an. adjoining nozzle of variable 
 opening. For the propeller, on the other hand, it is desired that the 
 greatest possible mass of fluid be reached and accelerated, and that the 
 propeller operate in a region of increased velocity. Thus the nozzle 
 is arranged ahead of the propeller and the diffuser behind it. The 
 theoretical calculations of the flow conditions are therefore so far 
 based almost entirely on the conception of nozzle and diffuser. 
 
 Recently, in the reports of Horn (reference 1) and Dickmann 
 (reference 2), another interpretation of the physical phenomena, parti- 
 cularly for ship propellers, is adopted and investigated. The cowling 
 is regarded as a ring-shaped wing. For propellers, the shrouding is 
 selected in such a manner that a (negative) circulation is produced which 
 increases the velocity in the interior of the ring. This interpretation 
 has already led to successes in the calculation of the Kort nozzle, and 
 it may be surmised that a similar approach may be used for the cooling 
 problem. Especially the question of regulating the mass flow thereby 
 assumes new aspects. The ring-shaped wing has, compared to the ordinary 
 wing, the advantage that a "two-dimensional" wing theory is sufficient 
 since no tip vortices are present, due to the rotational symmetry. Thus 
 it is necessary to extend the two-dimensional profile theory, which 
 replaces the wing section by so^jrce, sink, and vortex arrangements, to 
 include a ring-shaped wing; however, it becomes evident that difficulties 
 arise in calculating the velocities induced by source and vortex rings; 
 these velocities are no longer simple functions of the distance, but are 
 composed of elliptic integrals in a complicated manner. Dickmann 's 
 theory represents an approximation in which the elliptic integrals are 
 developed into exponential series, an approximation valid only for rings, 
 the diameters of which are large in proportion to the length. In calcu- 
 lating rings of greater length, one must use numerical treatment. The 
 numerical treatment is facilitated by setting up tables of functions for 
 the velocity components of source and vortex rings (reference 3). 
 
NACA TM 13^5 
 
 Comparisons with tests in shrouded ship propellers showed, moreover, that 
 Dictanann's theory of the ring-shaped wing (in which the profile thickness, 
 too, was neglected) cannot come fully into play if the flow conditions 
 at the cowling, which, in many cases, lead to separation, are not 
 completely under control. Thus consideration of the finite thickness 
 of the cowling also seems necessary. Naturally, this influence of the 
 finite thickness is of like importance in the fairing of radiators. In 
 addition, any large incremental velocities at the cowling for high flight 
 velocities must be avoided here. 
 
 The present report is intended as a first step in the direction 
 of determining the influence of the finite thickness of annular bodies. 
 We shall, at first, consider the annular body alone and disregard the 
 propeller or the cooling block itself. In a few cases, the influences 
 of a hub body will be investigated. First of all, we shall find out how 
 the forms of the semibody (single source) and of the Joukowsky profile 
 (source-sink distribution), known from two-dimensional flow, appear in 
 the rotationally symmetrical case where the singularities are distributed 
 on circles. Pressure distributions will then give us information on the 
 flow conditions to be expected for inflow into ring-shaped cowlings. 
 We are not treating the problem of finding for a prescribed cowling the 
 substitute singularity and hence the flow, but the opposite and simpler 
 one of determining the form for prescribed singularities. 
 
 II. METHOD AND RESULTS 
 1. The Method 
 
 We use the method of superposition. Since we want to treat flows 
 free from circulation only, we select as singularities distributions of 
 source and sink rings, and superimpose a parallel flow. We execute the 
 calculation numerically, and take the stream function and the velocity 
 components of the source rings from the tables mentioned above 
 (reference 3). Reference 3 also contains the formulas for the velocity 
 components and the stream function of vortex rings. 
 
 In the composite flow, we look for the streamline which gives the 
 
 body contour. This streamline is characterized by the stagnation point 
 
 lying on it. The thickness of the originating body is linked with the 
 sotirce strength E in the following manner: 
 
 We consider first an annular source body produced by a single source 
 ring in a parallel flow. Designating the cross-sectional aj-ea at 
 infinity by F and the velocity of the undisturbed parallel flow by U, 
 we have, according to the continuity equation 
 
 E = FU (1) 
 
NACA TM 1325 
 
 From reasons of symmetry F is annular 
 
 2 „ 2^ „. 'i + ""2, 
 
 - ,(r,= - r/) - 2. '±^(r, - r,) = 
 
 ?irRd 
 
 with r-^ designating the inner radius, Tg the outer radius, R 
 mean radius, and d the thickness. Thus we obtain 
 
 the 
 
 E = 2nRUd (2) 
 
 The pressure distribution is obtained, as customary, by means of 
 Bernoulli's equation from the velocity distribution at the body surface. 
 
 In order to obtain a hub body, we superimpose on this flow that of 
 a single source on the symmetry axis. Let the latter have the strength 
 Eq. The thickness d^ of the originating hub body is linked with Eq 
 
 and U by the relation 
 
 Eo=^U (3) 
 
 Thus we may write in this case the equation (2) 
 
 E = 8 ^ E^ (2a) 
 
 For a given value of Rd/d^ the ratio of the two source strengths is 
 fixed. If, in addition, we express d^ as a function of the radius r' 
 
 of the source ring, for a given value of d-^/r' the quantity Rd/r ' 
 characterizes the ratio of the source strengths. 
 
 In a further series of examples, we place a distribution of source 
 and sink rings on a coaxial circular cylinder (stream surface of the 
 parallel flow) . We then obtain, if the combined strength of all source 
 and sink rings is zero, a closed profile, a streamline, which is in a 
 flow without circulation (ca =0). We select the intensity of the 
 distribution in the form 
 
 ^lllf^ . p ^ (^) 
 
 E = E' i±—-^j^{l + 2 ^ 
 
MCA TM 1325 5 
 
 (x = coordinate in the direction of the symmetry axis, r' radius of the 
 source and sink rings). For the corresponding two-dimensional problem, 
 
 we obtain a Joukowsky profile-'- from this source sink distribution. The 
 method of determining the contour corresponds in every detail to that 
 used for the single source ring except for the fact that one now must 
 integrate over the entire distribution. 
 
 2. A Ring of Singularities in a Parallel Flow 
 
 As a preliminary remark to this section, we want to show as an 
 example the appearance, in the rotationally symmetrical case, of a flow 
 about a circular cylinder, obtained (as is well known) by superposition 
 of a parallel flow on a plane dipole. The result is shown in figure 1 
 which originated by superposition of a dipole ring and a parallel flow. 
 One sees clearly the deviation from the two-dimensional analogue, also 
 
 from that of a flow about a sphere in the proximity of a wall . It is 
 particularly striking that the two stagnation points no longer lie 
 diametrically opposite each other. 
 
 The following figures show the conditions for a flow about a source 
 ring. 'Figure 2 gives the contours of annular source bodies of various 
 thicknesses. The straight line d/r' =0 is the streamline of the 
 parallel flow which passes through the location of the source ring 
 (r = r'). The contours of the bodies of a thickness different from zero 
 do not lie symmetrical to this straight line, and again the stagnation 
 points do not lie directly ahead of the source ring. Since for all 
 x-values the same mass flow, which at infinity has the velocity U of 
 the undisturbed parallel flow, must flow through between this stagnation 
 point streamline and the axis, this streamline (the interior of the body) 
 rises, for large positive x-values, behind the narrowest place of the 
 cross section again to the same distance from the axis it has had 
 infinitely far ahead of the body. The mean radius R of the annular 
 body, that is, the arithmetic mean of outer and inner radius at infinity 
 is therefore larger than r'. In figure 3 the ratio R/r' is plotted 
 against the thickness. From this state of affairs it follows that the 
 "mean camber line" of the body is not a coaxial straight line, but a 
 line pulled inward toward the leading edge. In practice, where smooth 
 entrance flow is desired, one always uses this "lip" for ring-shaped 
 cowlings even though conditions there may be slightly different, for 
 instance, due to the presence of the cooling block. For the body shapes 
 indicated, the pressure distribution on the outside is given in figure k, 
 that on the inside in figure 5- A negative-pressure region with slight 
 
 Compare F. Keune (reference k) . 
 ^Compare Prandtl-Tietjens, Hydro-and Aeromechanics, Vol. 2, page 125. 
 
NACA TM 1325 
 
 pressure increase results on both sides shortly behind the stagnation 
 point so that for such body shapes probably no separation phenomena will 
 occur. Figure 6 shows how contour and pressure distribution of the 
 annular source body differ from that of the two-dimensional semibody-. 
 It is noteworthy that the pressure distribution on the outside is hardly 
 different from that of the two-dimensional semibody whereas on the 
 inside, for understandable reasons, a greater negative pressure prevails. 
 
 3. Annular Source Body with Hub 
 
 The following examples show the body shapes resulting if, aside 
 from a source ring in a parallel flow, in addition, a three-dimensional 
 single so'orce is assumed on the symmetry axis. We then obtain a ring- 
 shaped cowling with a hub. The most essential result of these calcu- 
 lations is the fact that noteworthy differences appear according to 
 whether the hub is inside or outside of the ring. Figure 7 shows in the 
 heavily drawn lines the streamline pattern about the hub body alone, 
 the well known three-dimensional semibody flow. If we put into this flow 
 a source ring so that its x-coordinate is equal to or larger than that 
 of the single source, there results the annular body cross-hatched 
 throughout. The third streamline from the axis, in the undisturbed flow, 
 forms its conto'or (fig. 7)- This stagnation-point streamline of the 
 composite flow is plotted as a dashed line. Far ahead of, and far behind 
 the annular source body, it approaches the previous undisturbed stream- 
 line. If one now places a source ring of the same strength at a certain 
 distance ahead of the single source in such a manner that again the same 
 streamline of the undisturbed flow becomes the body contour, an annular 
 body results which, it is true, at infinity has the same thickness and 
 the same distance from the axis as the previous one, but an entirely 
 differently formed front part. The form of this body is plotted in 
 figure 7 partly cross-hatched and shows at the entrance a thickening 
 caused by the stagnation due to the hub body, in contrast to the annular 
 body behind the hub, the form of which does not differ essentially from 
 the afore-treated annular bodies without hub. What additional conclusions 
 may be drawn from this state of affairs will be seen later. The annular 
 bodies with outward protruding hub will now be investigated somewhat 
 more closely. Such bodies of various thicknesses are plotted in figure 8. 
 The thickness d^ of the hub is kept constant at I.38 r', and the 
 x-coordinate of source-ring and single source is equated. The rearward 
 widening which occured before may again be observed; the form of the 
 hub body also is slightly changed with increasing thickness of the cowling. 
 The pressure distribution over the outside of the body (fig. 9) does 
 not essentially differ from that of the source-ring body without hub, 
 even though the pressure increase behind the minimum is somewhat reduced 
 due to the hub, so that the flow conditions on the outside appear slightly 
 more favorable with respect to separation in the presence than in the 
 
NACA TM 1325 7 
 
 absence of a hub. In contrast, true to expectation, on the inside 
 (fig. 10) a considerably higher negative pressure results than without 
 hub; however, these conditions improve greatly if the cowling is extended 
 beyond the hub. Figure 11 shows again, more accurately, the resulting 
 
 body shapes for the specieil position of the ring r' = d^/2j x' = -6,^/2, 
 
 figures 12 and 13 show the pressure distributions over the outside and 
 inside of the body surface. It is a particularly striking fact about 
 these pressure distributions that the suction peaks are almost entirely 
 eliminated, due to the incremental excess pressure ahead of the hub body. 
 For the body of the thickness d/r' = 0.5, one no longer has any suction 
 peak at all. When this case will arise depends, of course, on the 
 special position of the source ring with respect to the single source; 
 for other positions, it could be attained for lesser thicknesses of the 
 cowling. Thus, one may assume that for cowlings of this type, an inflow 
 particularly free from losses is insured. 
 
 The pressure distributions on the hub body itself are represented 
 in figure l4 for both cases. It can be seen that on the hub body, too, 
 the effect is more favorable when the cowling is extended beyond the 
 hub. 
 
 All these results apply, at first, only to the special case where 
 no circulation develops about the cowling; however, they will permit a 
 survey of the conditions to be expected in the presence of a circulation 
 as well, since the flow which is free from circulation may always be 
 regarded as the initial state. Thus these results may offer a basis for 
 the construction of ring-shaped cowlings. 
 
 k. Ring -Shaped Cowlings of Finite Length 
 
 Whereas only ring-shaped bodies extending to infinity have been 
 treated so far, one may cause the body contour to close again by a 
 distribution of source and sink rings on a coaxial circular cylinder in 
 a parallel flow. We choose the distribution given by reference k which 
 results, for the two-dimensional case, in the Joukowsky profile drawn 
 in dashed lines in figure I5. The corresponding annular body (drawn 
 solidly in fig. 15), however, has a quite different cross section. In 
 the region of the sources, the mean camber line rises; in the region of 
 the sinks, it drops off again. Thus the profile obtains a camber with 
 the maximum rise at about l/h of the length counted from the leading 
 edge. Profile forms of various thicknesses are plotted in figure I6. All 
 of them show the characteristic criterion of these annular -body profiles. 
 The camber increases with the thickness of the profile. 
 
 -%y x' one designates the distance of the source ring from the 
 single source in x-direction. 
 
NACA TM 1325 
 
 This result becomes significant if one wants to obtain a circulation 
 by means of the cowling. If one wants to produce the circulation by 
 giving the profile a camber, one must not (for Ca = 0) start from the 
 
 symmetrical section but from a section cambered in the indicated manner. 
 Thus, the profiles for producing a positive circulation (reduction of 
 the velocity in the interior of the cowling) must have a large positive 
 camber whereas negative circulations are obtained with uncambered 
 profiles. From these results there follows a new two-dimensional inter- 
 pretation for the well-known phenomena on annular -shaped cowlings so 
 far explained by a one-dimensional nozzle theory. 
 
 III. SUMMARY 
 
 The present report represents a first step in recognizing the 
 influence of the finite thickness on the flow about a nnulaj-- shaped 
 cowlings. It is shown how one may obtain in a simple manner the forms 
 of ring-shaped bodies from existing tables of functions according to 
 the customary method of superposition of flow due to singularities and 
 parallel flow. Knowledge of these forms is important for evaluating 
 the entrance phenomena into ring-shaped cowlings, particularly for the 
 fairings of radial engines, for ring-shaped radiators and shrouded ship's 
 propellers. A series of examples show the cross sections of such 
 annular source bodies as well as their pressure distributions without 
 and with the presence of a hub body. It is shown that, among the forms 
 investigated, the entrance of cowlings slightly thickened at the leading 
 edge with a hub body within is particularly favorable and free from dan- 
 gerous separation. Furthermore, the profiles of annular bodies of 
 finite length are given to which Joukowsky profiles correspond for the 
 two-dimensional case. These profiles show a camber dependent on the 
 thickness, with the maximum rise at about \/h of the length counted 
 from the leading edge. All calculations have been performed with the 
 assumption that the flow about the body is free from circulation; how- 
 ever, they may serve as a guide for the flow conditions to be expected 
 also in case of the existence of a circulation or of disturbance bodies 
 (cooling block, propeller). 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
NACA TM 1325 
 
 rv. REFERENCES 
 
 1. Horn, F.: Beitrag zur Theorie ummantelter Schiffssclirauben. Schiffbau 
 
 XLI, 19^0, p. 2 and p. I8. 
 
 2. Dickmann, E. E.: Grundlagen zur Theorie ringformiger Tragflugel (frei 
 
 umstromte Diisen) . Ing. Archiv, Bd. XI, Feb. 19^, pp. 36-52. 
 
 3. Kuchemann, D.: Tafeln fur die Stromfunktion und die Geschwindigkeits- 
 
 komponenten von Ouellring und Wirbelring. Jahrbuch 19^ der deutschen 
 Luftfahrtforschung, p. I 5U7. 
 
 k. Keune, F.: Die ebene Potentialstrbmung urn allgeraeine dicke Tragfliigel- 
 profile. Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 3-1 26. 
 (Available as NACA TM 1023.) 
 
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 NACA TM 1325 
 
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NACA TM 1325 
 
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Ik 
 
 NACA TM 1325 
 
 Annular source body 
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 Annular source body 
 
 Two-dimensional semibody 
 
 Figure 6.- Comparison between form and pressure distribution of the 
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NACA TM 1325 
 
 15 
 
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16 
 
 MCA TM 1325 
 
 Fi^re 8.- Annular source bodies of various thicknesses with hub outside. 
 
NACA TM 1325 
 
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NACA TM 1325 
 
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