EXPERIMENTAL INVESTIGATION OF SHEAR FLOW IN RECTANGULAR BENDS by Warren Jby B.S.,Massachusetts institute of Technology(1949) Submitted in partial fulfillment of requirements for the Degree of Master of Science at the Massachusetts Institute of Technology (1950) Signature of the Author Department of Aeronautical tngineering,May 19,1950 Certified by Thesis .p e rv is oY Uhkairman, .oepartmental committee on Graduate students Massachusetts Institute of Technology CambridgeMassachusetts May 19,1950 Professor Joseph S. Bewell Secretary of the Faculty Massachusetts institute of Technology Cambridge ,Massachusetts Dear Sir: Submitted herewithin partial fulfillment of the requirements for the degree of Master of Science,is a thesis entitled "Experimental Investigation of shear Flow in Rectangular Bends." Respectfully submitted ACKNOWLEDGEMENTS Grateful acknowledgement is made to the following persons and organizations: a.) to Professor Hawthorne for guidance in this project and for contributions to the theory presented herein, b.) to Jim Hands and the staff of the Gas Turt. bine Laboratory for the use of tools,eguipmentand apace, c.) to F. Lustwerk for help in obtaining required materials, d.) to the Westinghouse Electric Corporation for financial assistance, e.) to M. Demayo for assistance in preliminary investigations, f.) and to ZisfeinFelbeck,Valentine,and others for the use of their thesis equipment and apparatus. ABSTRACT Experiments on Secondary Circulation induced in rectangular bends are discussed in this thesis.All experiments were performed in the 4as Turbine Laboratory at M.I.T. in 1950. a velocity gradient was pruduced upstream of the bends testedand an analysis of the flow in the bend and downstream of the bend was ma&e.Experimental results are presented in the form of velocity distribution "maps". The bends tested had five inch by ten inch cross-sections (the shorter side was parallel to the plane of bend). The radii of turn tested were seven inches and fifteen inches.Angles of turn up to 180 degrees were produced. An approximate theory was derived from the results of the work of Squire and Winter and of vawthorne and is presented in this thesis.This approximate theory corrolated fairly well with the experimental results.byurther work of a similar nature wiil be required to check the accuracy of the theory presented. The results of this thesis indicate that this approximate theory can be applied to design I' problems in an effort to minimize losses due to secondary circulation. Figures S-1,S-2,and S-3 are included here to give an indication of the type of flow patterns encountered in this project. 9Q0 RECTANGULAR STA. R=15 BEND STA. iNS. 2C = 5 INS. 2U = JOiNS. 87 x 50 (1 0 STA. VELOCITY IN F.P.S. 2.C] 2b _L 0 STA G) 1600 RECTANGULAR STA. STA. R = 15iN5 STA. ® 50 aC r 5 INS. zb BEND 6o0 o=10 INS. 808 8f0 mT 50 0 STA. bJ hu- lE VELOCITY IN F. P S. STA. @ STA. o(f) ZI 900 RECTANGULAR BEND EFFECT OF BEND RADIUS 71iN). ca-5Ns. b -t1NS. 3- , 7. AVERAGE PROFI LE 2- T R STA STA. ID STA t - - 0* 0, v (fps.)- 14 q+ t -- a x 2b L2 6 15 INS. I AVERAGE PROFIL E -5ms b -low.5 ~1 vi TABLE OF CONTENTS Abstract Table of contents PO. i p. Vi p. Table of Symbols vii Sections: I. Introduction and Background II. Description of Apparatus III. Test Procedure Discussion of Errors IV. Results V. Theoretical Considerations VI. Conclusions VII. Recommendations VIII. Bibliography P. 2 p. 10 p. 25 p. 33 p. 48 p. 94 p. 113 p. 117 P. 121 Appendices: A. Raw Run Data p. 125 B. Data for C.F. derivations p. 133 C. Data for vorticity caleulations p. 141 Figures : 1-1 to 1-3 pp. 6-8 2-1 to 2-6 pp. 17-23 3-1 to 3-3 pp. 43-46 4-1 to 4-37 pp. 55-92 5-1 to 5-6 pp. 105-111 Y11I TABLE OF SYMBOLS 2a width of bend cross-section Ax cross-sectional area AR Aspect Ratio ( Lb/2a ) 2b height of bend cross-section CF1 correction factor for correcting velocity for variations in Q, and P,.9 GFz correction factor for correcting velocity for vakiations in upstream velocities. CF3 correction factor correcting for incorrect manometer calibration. D outside diameter of pitot tube d diameter of cylindrical tube g acceleration of gravity gradA- total pressure gradient A h height of liquid head A1 axial distance a particle moves in time interval &t . Pamb ambient air pressure Pb stagnation pressure inside settling chamber Pbd b""am )measure& value of stagnation pressure lnside settling chamber. Pbd ref Phead Pind an arbitrary value of Pbd chosen as a standard pressure under a head of liquid measured ambient pressure of outside air Vill P,0 stagnation pressure in flow Pstat static pressure in flow P air pressure at standard temperature and pressure conditions (29.92" Hg ) stp Ptt stagnation air pressure in flow AP E (Pstat"' pressure q b) measured value of the static unction dynamic pressure ( i-47 2 ) dynamie pressure gradient R radius of turn RE Reynolds bumber Ref. Reference in sect. 8 a a. linear distance along the manometer tube a s are moved by rotating flow in time interval At (As) ave average value of .As T measured ambient air temperature T standard ambient air temperature (59 deg.F) A axial velocity 116 Amn minimum velocity in a velocity gradient maximum velocity in a velocity gradient V velocity VII velocity inside settling chamber Vind Vindoorr velocity indicated by the manometer ind corrected for &p error Vmin minimum velocity at a section Vmax maximum velocity at a section Vref indcorr corrected for errors in and Pbd Vrefcorr Vref corrected for error due to changing upstream velocities Vref 3 .7 Vref where Pbd = 3.7" H2 0 ref 340 Vref for run number 340 Vref 4 6 1 1ref for run number 461 Istd Vindoorr corrected for errors in only AV correction on velocity due to Ap error (d) velocity gradient V tangential velocity induced by the seconctary circulation x horizontal coordinate at cross-section y vertical coordinate at cross-section c angle of flow rotation r, circulation B a deviation in coordinates due to the effect of the total pressure gradient on the pitot tube reading e angle of turn e angle of turn when OC - vorticity maximum value of vorticity tat e ) radius from center of circulation air density density of liquid measured air density ?i air density at standard temperature and pressure conditione (.002378 slugs/ft 3 ) stagnation densit y of air angle between the normal to the stream surface and the direction of acceleration £2. initial velocity gradient ( IT 3.1416 ) -7 SECTION 1 Introduction and Background Section I The existence of secondary flows has been known for some time.These flows were first observedin connection with the flow of riversby J.Thompson in 1876.Yen and aowe state: "The lateral currents induced in bends were first observed by J. Thompson in 1876 in a series of experiments conducted in an open channel having a 180 degree bendthe spiral motion being observed by seeds and drops of aniline dyes.I 2 A study was made by BlueHerbertand Lancefield concerning the flow around a bend in the Iowa River.They state: "Current velocities at the bed of the stream are very much less than those near the surface.....&ear the outside bank the water moves from top to bottom...near the inside it moves from bottom to top....(this)combination... results in a helicoidal flow."4 Figure 1-1 is taken from their work and shows definitely the "scouring" action of the helicoidal currents. l.Ref. 31,p.5 2.Ref. 30,p. 714 3.Ref. 3 2,p. 259 4.Ibid 3. Some experimental work has been done by Shukry on the flow of water in channels. Figure 1-2 shows flow patterns taken from his work. In this case, since there are two velocity gradients upstream, we get two distinct spirals downstream. The secondary flows produced by the turning of a fluid are not confined to the field of hydraulics alone. They are found in the field of aerodynamics also. The flow of air in ducts, pipes, and in cascades of airfoils may contain secondary circulation due to initial velocity gradients. it has been estimated that secondary flow may acoount for 50% of the losses in compressor cascades. Most of the analysis of air flow around bends done hertofore has been confined to flow with no initial velocity gradient 3 4 5 . Squire and Win- ter 6 7 were the first to work out the theory for simple flows containing initial velocity gradients. kawthorne has modified this theory to apply to the flow in cylindrical pipes 8 .6ection 5 in this thesis contains the results of Squire and Winter 1. 2. 3. 4. 5. 6. 7. 8. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. 29 33 13 14 21 25 11 12 4%. and the Hawthorne modification cited above. A variety of experimental work has been done on fluid flow in bends. 12 3 4 5 6 7 8 9 10 mere again, the flows were uniform upstream and contained no velocity gradients.A bibliography of this work is presented in reference 19 . Squire and winter II were the first to experiment with a flow containing an initial velocity gradient and to corrolate the experimental results with their theory.They observed the flow in the boundary layer of a wind tunnel and its action as it moved through the turning vanes at the corners. Some work has been done also on the same type of flow by uemayo 12 at vi.I.T. in 19491950 with an apparatus similar to that used by the author. Because the results of Squire and winter and those of Demayo indicated that more work needed to be done on this problem, Professor Rawthorne 13 instituted two simultaneous proects. One project was concerned with ninety-degree bends 1. Ref. 1 7. Ref. 19 2. 8. Aef. 20 Ref. 3 3. Ref. 15 9. fief. 21 4. Ref. 16 10. Ref. 22 5. Ref. 17 11. Ref. 25 6. Ref. 18 12. Ref. 4 13. Westinghouse Professor in Mechanical Engineering at m.I.T. 5. of circular cross-section to be carried out by Eichenberg .The other was concerned with bends of rectangular cross-section and was to be carried out by the author. Fig. 1-3 shows a general picture of the type of flow to be investigated. The two projects were proposed in order to obtain quantitative data on the vorticity in ninetydegree bends , and to ascertain the effect of changing the design variables of the bends (such as aspect ratio, radius ratio). Work done by Eichenberg 2 on 180 degree bends of circular cross-section led to similar work by the author on rectangular 180 degree bends. It was hoped that the results of these propects would either corroborate the present theories or serve as a foundation for modified theories. When the theory connected with this problem is made more adequate and when more is discovered about the problem , attempts can be made to minimize losses due to secondary circulation by more efficient design of ducts and cascade passages. 1. Ref. 26 2. Ibid 6. SECONDARY FIG. I RIVERS FLOW IN DEPOSITED URING" 1925 1933 BOUNDARY BOUNDARY SECT10N TOP AA FIGURE FROM REF. 32 FIG. I TAKEN FLOW 2 FROM VELOCITY LOOKING I REF. PATTERNS 29 MAPS DOWN STREAM 5' UPSTREAM 0' FLOW PATTERNS LOOKING DOWNSTREAM 600 90 150 VELOCITY IN CM/SEC. F FIG. I -3 SHEAR FLOW TH ROUGH BENDS INITIAL VELOGITY GRADIENT.,.. r Low v. HIGH V. HI GH V. 9. SEOTION II Description of Apparatus 10. Section II A schematic diagram of the apparatus set-up is shown in fig. 2-1 . The apparatus can be described as a combination of five units. These units are ; the blower-nozzle, the extension containing the velocity gradient screen, the "exploration chamber, the bend, the extension downstream of the bend. A description of each follows below. The blower-nozzle unit was designed and constructed by Ziefein.A detailed description of it is available in his thesis 1. its function was to pro- vide the required air flow for the experimental work. The rate of flow produced was approximately 1200 cu. ft./min. through an exit cross-section of five by ten inches. The unit contained provisions 2 for measuring the stagnation pressure inside the settling chamber (Pb )* The extension containing the screen served to change the flow from one with a uniform velocity distribution to one with a velocity gradient.Fig. 2-la shows qualitatively how this was accomplished. This unit was designed and constructed by ielbeck and is more fully described in his thesis 3. 1. Ref. 1 ,pp. 10-15,21 2. Ref. 3 ,p.16 3. Ref. 2 ,p.lll8 The extension had a cross-section of five by ten inches throughout.The screen used was of the 60 mesh size and of a double layer. The exact shape is given on page 18 in ref. 2 . Typical velocity profiles are given on figures 4-2,4-12, and 4-24 in section 4 of this thesis. it should be noted that the performance of this screen configuration can be sharply impaired by any accumulation of dust that may gather upon it. The so-called "exploration chamber" was designed and constructed by Valentine. A detailed description of it is given in his thesis . it was used by this author as part of the apparatus so that the velocity distribution upstream of the bend could be determined accurately. Its cross-section was five by ten inches throughout. The above three components of the apparatus remained unchanged as a single unit during all tests. it was dis-assembled only when it was necessary to remove dirt from the gradient.producing screen. The relative size and shape of the three bends used is shown in fig. 2-2 . The 15 inch bends were designed by the author and were constructed with the cooperation of the carpenter shop and the Gas Turbine Laboratory . The non-turning sides (i.e,the 1. Ref. 3 ,pp.19,114,119-122 12. top and bottom ) were made of plywood. The turning sides were made of sheet brassproducing a smooth surface as well as a rigid structure. The seven inch bend was used by Zisfein and Valentine and is des12 cribed in their theses .The non-turning sides were made of plexiglass *The turning sides were made of laminated hardwood. Investigations by Demayo on a similar apparatus, but one without the extension downstream of the bend, demonstrated that both mixing and separation phenomena distorted the flow downstream of the bend.This indicated the need for an extension to cut down these effects. Such an extension was incorporated by the author and served several purposes: 1.) It eliminated mixing effects. 2.) It out down separation effects by applying a slight back-pressure. t3,4) 3.) It more nearly simulated the flow of air in a duet. 4.) It served as a channel for investigation of the flow downstream of the bend. 5.) It served to stabilize the flow somewhat by its damping effect. l.hef. 2.ref. 3.nef. 4.aef. 5.Aef. 1 ,p.20 3 ,p.14 18 21 4 t3. This extension was designed and constructed by the author with a five by ten inch cross-section throughout.it is 6.5 feet long. Its top and bottom (the five inch sides) were made of plywood. The first four feet of the sides was constructed of 1/8 inch plexiglass. The remaining part of the sides was constructed of t inch plywood.,with provisions for the installation of plexiglass there also,if required. The plexiglass served three purposes: 1.) It provided a smooth surface. 2.J It made orientation of the pitot and static tubes easy and accurate. 3.) It made visual observation of the flow possible in future experimentation,if desired. The traversing mechanism used was a milling machine similar to that used by Valentine 1 This machine provided motion in three mutually perpendicular directions which could be readily calibrated. An angle iron was bolted to the bed of the milling machine and was used as a pitot tube. support for the By varying the length and position of the angle iron, and bend-extension configuration could be investigated without moving the milling machine. 1. Ref. 3 ,p.ll,12 '4. Built into the bottom of the fifteen izch bends and the extension were slots with moveable slides similar to that used in the "exploration chamber" by Valentine l. The slides were designed to be flush with the inside of the ductand each contained a hole.This hole had a diameter approximately equal to the pitot tube barrel diameterin order to insure a tight fit. This design enabled the traversing of the flow with a minimum of disturbance to the flow. Figures 2-3 and 2-4 show the slot and slide configuration in a fifteen inch bend. Slots were cut into the bottom of the seven inch bendbut clearance limits prevented the incorporation of slides.These slots were kept taped over when not in use. The position of each slot in the bend and extension was given a Station Number. The system of numbering varied from bend to bend. The system is given in the results section before the velocity distribution maps of the applicable bends .The figure numbers are fig. 4-1,4-ll,and 4-23 . Velocities were measured with the pitot tube shown in figures 2-4,and 2-5 * it was constructed by the author out of thin-walled 0.057 inch bypodermic tubing.The nose was slightly rounded off. l. hef 3.,p. 19,122 15. This tubing was wedged into a barrel of .410 inches diameter with a wooden wedge of very small size. The nose of the pitot tube was approximately two inches upstream and above the barrel ,lessening interference effects. To the bottom of the hypodermic tubing was soldered an adapter,to which was attached the tubing leading to the manometer. uuring all tests, the nose of the pitot was placed perpendicular to the plane of the cross-section and was pointed upstream. The manometer used was the one used by Valentinel with a slight modification in the reservoir arrangement. The arrangement is shown schematically in fig. 3-1 in the next section of this thesis. Static pressures were measured by a static probe designed by Valentine and described in his thesis 2, it consisted of a bent piece of .045 inch tubing with two .020 inch holes drilled in opposite sides.Figure 2-6 shows the shape and the location of the holes. This piece of tubing was soldered to a brass barrel to facilitate traversing. In all runs this static probe was traversed manually by removing the slides from their slots. The manometer configuration is shown in fig. 3-1 . The static probe was set up so that it measured the pressure differential Ap. 1. Ref.3 pp. 30,33 (called #3) 2. ibid pp.15,17 16. The temperature in the flow was measured with a mercury thermometer placed through the hole in the slide at each station. Ambient pressure was measured with a mercury barometer located in the room next to the cell where the experiments were carried out. The stagnation pressure in the settling chamber was measured by a simple vertical manometer constructed by the author. The arrangement was somewhat like that shown in fig. 3-1 . All apparatus mentioned in this thesis was located in the center combustion cell on the second floor of the Gas Turbine Laboratory (unless specified elsewhere). 17. FIG. 2 -1 A PPARAT US ARRANGEMENT TOP VIEW - EXTENSION CONTAINING SCREEN SETTLING CHAMBER "EXPLORATIO N \ CHAMBER vs BEND BLOWER-NOZZLE UNIT EXTENSION OF BEND t8. F G. 2-IA SCREEN CONFIGURATION CON FIGURAT ION OF SCREEN FLOW DIREC TION SCREEN VELOCITY DISTRIBUTION DOWNSTR EAM OF SCREEN '9. FIG.2-2 COMPARISON OF BEND DIMENSIONS 15" x 180" BEND RI1" NOTE: ALL CROSS-SECTIONS THROUGHOUT ARE 5"x 10" I5 " x 90' BEND 7"1- 90' BEND R= 15" R a7 20. FIG. 2-3 SLOT MECHANISM 21. FIG. 2-4- PITOT POSITION 22. FIG. 2-s PITOT TUBE 23. FIG. 2-6 0.35' STATIC PROBE PRESSURE jA A .o-.045" .020, DRI LLED HOLE HOLE SECT. AA 24. SECTION Test III Procedure and. Discussion of Errors 25. Section III The manometer configurations are shown in fig. 3-1.The manometer used for velocity measurements is the one labeled g3 by Valentinel. It was first set up to read impact pressure in inches of water by the author as follows: Ah x (Boa.) sin 14.5 where &s is the linear movement of the liquid along the tube, where s.g. is the specific gravity of the liquid used, and where a h water. is the head in inches of The author used kerosene in the reservoir and the value of s.g. was taken as 0.82 Thus: As'= 2.92(Ah) (eq. 3-1) That is,for a pressure of one inch of water,the movement along the tube of the column of kerosene (As' ) will be 2.92 inches. From aerodynamic theory we have: Ptot !t at where: P l.±ef. 3 p.30,03 z2 is total pressure, 26. the density of air, QA is V is the velocity, Pstat is the static pressure. From bydrodynamic theory we have: head where: P 0 a is the total pressure exerted ti9e liquid in the reservoir, Pamb is the ambient pressure in the room, qL is the density of the liquid in the manometer, Combining the above two relationsand assuming Pstaf amb , we get: tggtV2 and: =tge 2(eq. 3-2) This function is plotted on figure 3-la and was obtained from reference 5.1t is based on a manometer using water at 24 deg. 0 in the reservoir. By combining eq. 3-1 and eq. 3-2 ,the author calibrated the manometer to read velocity direetly. Since this calibration depended on the assumption that P stat - .amb ,a correction was l.for simplification in reduction of data 27. Pamb derived to allow for cases where P amb stat This correction chart is plotted as fig. 3-2,and was derived from the chart on fig. 3-la. The author defined V indcorr as follows: Vind corr where: ind+A V (Eq. 3-2a) Vin&corr is the corrected indicated velocity, is the reading of the velocity rind scale on the manometer, &V is a function of &p and V and is plotted on fig. 3-2, aP stat ~Pamb is the true Thus V indoorr velocity at the point being investigated. in order to eliminate the effects of variation in outside atmospheric conditions and the effects of conditions in the blower-nozzle unit, it was decided to standardize all velocities ( Vindcorr in the manner described below. in the analysis that follows the flow is assumed to be incompressible. 1. subject to the errors discussed later in this section 28. To correct for variations in R. due to temperature and pressure Vstd , we note that : (from eq. 3-2) - stp incorr thus: rind Vstd Vind indcorr stp --------- stp stp (eg. 3-3 ) Tind where: st is the velocitystandardized to standard temperature and pressure conditions, , and Tind are actual, ind IP measured conditions in air, ot =.002378 T =.. 9 deg.F , stp Ps slugs/cu. ft. = 29.92 in Hg. To correct for errors in velocity due to errors in the total pressure in the settling chamberthe author defined a quantity Pbd Pbd 2 Pb *amb where: ?bd was measured directly with a manometer, is the total pressure inside Pb the settling chamber 29. Pam is ambient air pressure outside the settling chamber. To standardize the results,the author chose an arbitrary value of Pbd for each bend and called it Pbdref . in most eases Pbdref was the value of Pbd found when investigating station 2 or 3 arom aerodynamic theory we have: Pb =Ptt+ thus: pbd+ Now if we assume then: amb ab Vindcorr2 i stat stat Pbd iA( Vindorr ) *(ie. Ap* Vindoorr thus: indoorrO"b The author now defined Vref by the'following relation: ref Vstd = lbdref Pbd (Eq. 3 -4) 1. see section 4 for station locations 2. all such superscripts are exponents 3. tests showed this assumption to be valid. 30. where: V Pbd is defined in eq. 3-3, is the measured value of Pbdo Pbdref is the arbitrary value of Pb chosen. Combining equations 3-3 and 3-4 ref indoorr Ebdref Tstd Tin bd , we get: Pind std The author further defined a correction factor: bdref Pbd std Pid Tind 2std therefore: Yref indcorr X(cF1 ) (eq. 3-5) Therefore,if we were to take readings at the same location at different times ( allowing the pressure,temperature and Pbd to vary ) , we would expect Vind and Vindoorf to differ in each case. kowever, we would expect Vref to be the same in each case. bow it was found that Vref did not 31. remain the same. This was not the fault of the theory proposed above,but was due to the accumulation of dirt on the velocity gradient-producing screen. Because of the above factthe author made duplii. cate runs: at various stations (usually sta. 2 ) at different times to note the change in velocities (i.e. Vref ) and to derive a second correction factor ( C.F.2 ) to account for this effect.Appendix B gives calculations involved. The author defined C.F.2 as follows: C.F*2 =Vrefideal Vrefactual where: Vref is defined in eq. 3-5. By experiment C.F. 2 was found to be a function of either the run number or Pbd. Whichever was more convenient was used in the calculations.The author further defined Vrefcorr as follows: Trefcorr =Vrefx"G.F2 (eq. 3-6) In runs 100 to 200 an incorrect manometer scale was used( it assumed water to be in the reservoir). 32. Rence all values of velocity read were too high, by a factor 1/s.g. . Therefore the author defined a third correction factor to be used on runs 100 to 200 only. C.F.3 g. = .905 (eq. 3-6) In all other runs, C.F' 3.E 1.00 Finally,combining equations 3-2a,3-5,and 3-6, we get: Vrefcorr OF1 x CF2 F3 ( Vindcorr or: Trefcorr OF1 x 0F2 t OF3 x( Vind+AV ). The procedure for a set of runs is outlined below. The apparatus was turned onand the pitot tube was placed and oriented at the proper station. before any run data was taken ti.e. Vind ),readings of ambient pressure ( Pa (Tind j , and 2bd jtemperature were taken.During this time the flow had had time to become stabilized. Each traverse was considered a "run" ,for the relation between the coordinates (y,x ) 33. during any traverse was well determined (due to the rigidity of the milling machine arrangement).In most runs,readings were taken at every i" in each directionas determined by preliminary investigations 1. After data was taken for a few runs , another reading of Pbd was taken. At the end of the runs for each station final readings of ±ebd and Pamb were taken. Then readings of A p were taken with the static probe. During runsfrequent visual measurements of pitot tube position were made and orientation noted. Each of the following paragraphs discusses a possible error in the veloc.ity plotted on the velocity distribution maps.These errors were either due to inaccuracies in measurements of the velocities themselvesor due to inaccuracies in the measurements of the coordinates x and y .in each case the author has underlined either the reason for the error)or the assumption made which leads to the error. The percentage error is estimated whenever it is appreciable. 1. Ref. 4 3+4 Specific Gravity of Kerosene is 0.82. in the apparatus used the actual specific gravity was never determined by measurement. The figure of 0.82 was taken as a weighted mean of values found in the international uritical Tables'. These tables give an absolute variation of from .792 to .869,with the bulk of values listed falling between 0.80 and 0.84 approximately. Pavian 2 gives a variation of .789 to .878. The variation of specific gravity due to temperature variations was neglected because it is small when compared to the uncertainties listed above. Since the velocity depends on N s.g. (eq. 3-2 ) the error in velocity is approximately one- , half the error in the specific gravity (on a percentage basis). The error in specific gravity is 3% ,therefore the estimated error in velocity is S1.15% . The Effect of Yaw on Pitot Tube Readins is veglected . We know that our pitot tube was in yaw, because the flow had a secondary circulation and therefore a component of velocity perpendicular $o 1. ±ef. 6 2. Ref. 7 35. the direction of velocity.The author has calculated this secondary circulation roughly in Appendix C * A measure of the angle of yaw is the quan- tity as/Al (see fig. 4-34a ) where: As is the tangential displacement in the time At Al is the axial displacement in the time &t. Thus the angle of yaw is tan" 1 As/ Al . Reference to Appendix 0 gives the following values of As/ Al : bend ( as/Al)max (As/ Al )ave 15"-90 deg. .36 .20 15"-180 deg. .39 .15 1.32 .30 7"1-90 deg. The corresponding angles of yaw are: bend max.yaw 1511-90 deg. 20 deg. 15"-180 deg. 23 deg. V"-90 deg. 53 deg. Figure 3-3 ave.yaw 11.5 deg. 9 17 deg. deg. gives the results of some testsl run on pitot tubes at 55 ft/see. Oonsidering the relative 1. Ref. 8 36. shapes of the pitot tubesthe author estimated the error to be approximately that of the top one pictured (to be conservative). Thus the average error in the dynamic pressure(q ) is about +6% for the 15" bends and about + 10% for the 7" bend. The error in velocity will be one-half that in the dynamic pressure. Therefore, the estimated error in velocity is +3% on the 15" bends and +5% on the 7" bend,. Error in C.F* 2 - A look at the spread of values used in derivation of C.F*2 (in Appendix .8 )shows a possible error of ±5* in velocity. Blocking Error due to Pitot Barrel. The flow velocity is inversely proportional to the crosssectional area, if we assume the flow to be essentially incompressible. Area of Cross-section is 50 in. 2 maximum height of barrel is 8 in. (at y 9.9") Diameter of pitot barrel is 0.4. in. Thus: Maximum blocking area is 3.2 in2 Blocking Area Ratio is .062 37. therefore: Maximum error is 4 6% Estimated average error is + 3% Error in Velocity due to Errors in Measurement of Ap Appendix A gives measured values ofap.The * maximum value recorded was ± .08 in. ki 20 fig. 3-2 shows the average A look at . V to be ± 2 to ± 3 ft./sec.. The average recorded velocity was 60 ft./sec. (Vind ) . If &p were in error by t 20% ,the error in V would be approximately i 0.5 ft./sec.. Therefore, the estimated error in the velocity is ± 2%. Effects of Uompressibility are Beglected . The maximum velocity encountered (Vrefcorr ) was 87 ft./sec.. Therefore, the maximum Mach Number was ~.08 . ieference 28 gives a zero error at this low Mach Number. aeference 10 gives =.996 ,which means that the error in Q is about + 0.4% the error in the velocity is negligible . Thus, . Apparent Displacement of Lines of Constant Velocity due to the velocity Gradient's Effect on Pitot Tube Readings . Young and Maas 1 have per- formed experiments on the performance of a pitot tube in a velocity gradient. Their results give 1. Ref. 9 38. the function 1 d as a function of D q g where: & is the apparent shift in a line of constant total pressure, D is the outside dia. of pitot, q is the dynamic pressure, (AI)is the dynamie pressure gradient. We know that: 2 Investigation of plots given in section 4 shows: minimum V= 40 ft./see. maximum local , -960 =d) see-1 7 in the seven inch bend ) diameter (D) (at sta. , .057 in. thus: [24 V Cx max. xoung and Maas 1 state that V) from 0.1 to 1.2 , , = .17 in the range of the maximum value of D is 0.30 . lence, the maximum value of 8 in .0012 inches. This error is negligible when compared with other errors. 1. Ref. 9 I 39. Calibration of Manometer Assumes Tube to be Straight . Valentine had previously calibrated the manometer used by the author and a plot given in his thesis 1 shows a linear relation between the pressure and the reading. This demonstrates that the tube was indeed straight.Thereforethere is no error incurred. Errors in Vind , Vind was read to the nearest i0.5 ft./sec. in the range 50 to 9U ft./see. and to the nearest21 ft./sec. in the range 20 to 50 ft./sec. In aduition to the reading error above,an error might have occured due to dynamic lag in the manometer system. However, sufficient time was allowed between readings to let the fluid come to within the reading error before another value was read on the manometer scale.Thusthe lag error was incorporated into the reading error. TXhe estimated error in Vmd is therefore t2o . The Flow in the Apparatus is Assumed to be Steady . Any unsteadiness would probably have been damped out by the manometer fluid-air system.Thus the author assumes no error in readings occurred due to this cause. 1. Ref. 3,p. 33 40. Error due to Uncertainty in Values of x and y. Since there were velocity gradients in the flow, any error in x or y could result in an error in the velocity.(That is, the velocity reading would be correctbut its place on the velocity map would be in error, giving an erroneous flow picture.).The maximum velocity gradient was 960 sea-1 . The maximum estimated error in x or y is 11/8 in. Thusthe potential maximum "error" is±10 ft./see (i.e. error in YVef ). This error seems large, but an analysis of the flow patterns shows that such an error would actually be insignificant in the high velocity gradient producing it. Therefore, such errors were not considered.ihus, the flow patterns can be assumed to be free of this error. The analysis contained in the ten previous paragraphs points to a potential error of 416% and -11% in the velocity. The purpose of this thesis was to obtain data on secondary circulation.The theory used in the calculation of the circulation was approximate anyway, so this error in velocity is not really too high when compared to the errors involved in the theory. 4-I. A look at figures 4-35,4-36,and 4-37 shows a very large spread in the calculated values of . srrors in other measured quantities are discussed below. The largest potential error inAp was due to the angle of yaw between the streamlines and the static probe. Ower 1 states that a 20 degree yaw may give -8v error in static pressure reading. Merriam and Spaulding 2 give -11o error for 20 degrees of yaw.kioweverthese experiments were static tubes in "symmetrical' carried out on yaw (that is, the tube was yawed in a uniform stream). in our case we had a helical flow across the two static holes, The error is probably larger that that predicted abovebut a numerical estimate is difficult to make.kiowever,during measurements of ap ,the probe was yawed from calIy side to side slightly and a very small change in the indicated value of p was noted at mod- erate angles of yaw. There was another error in Ap possibledue to the fact that the sections investigated were not completely isolated from the ambient air. 1. uef. 27 z. Aef. 24 42. This came about because the slides 1 were removed from the slots for the measurements of Ap. The error in Ap estimated from the above considerations is ±30% . we note that an error as large as this would still have only a small effect on the accuracy of the velocities. Pbd was measured with a total pressure probe designed by Valentine ometer to the nearest .Lt ± was read on a man- 0.05 in. H20 .he mean value of Pbd read throughout the runs was 4.0 in. H2 0 . The bdd estimated error in Pbd is therefore There was a small error possible in Pamb because the ambient pressure reading was taken on a mercury barometer in an adjoining room. Ihis error affects the value of k.F.1 ,but is small when considered on a percentage basis. ±hereforeit was neglected. There was a small error possible in the temp- erature measurement, due to conduction and convection effects. xhis,also,was neglected because of its small size when considered on a percentage basis (an error of 5 degrees i is 1 absolute scale of temperature ,F error in the abs. ). 1. see section 2 for a description 2. nef. 3 , p.13,*16 4+3. FIG. 3MANOMETER CONFl GURATIONS VELOCITY PROBE (VIND.) PITOT TUBE AMB. KEROSENE RESERVOIR STATIC V PROBE STATIC PRESSURE PROBE .AMB. WAT +4-. I. I 45. 4-6. FIG. 3-3 PITOT CON FI GURATIO N TO YAW AND TUBE ERROR DUE % ERROR IN - +1t0 ... *0. - I.- -200 20 - -10 YAW PITOTS FOUND IN REF. 8 X ERROR IN V A., -20* V 41ZK2zzzzz PITOT USED AUTHOR BY a -a Ita fta 20' YAW NOTE'. FIGURES ARE NOT TO SCALE RELATIVE TO EACH OTHER 4-7. SECTION Results IV 48, Section IV iigures 4-1 to 4-34 give the results of the velocity measurements taken during this investigation. The results are given in the form of "maps" of lines of constant velocity at the vara ious stations investigated. Figures 4-1, 4-11, and 4-23 give the location of these stations relative to each other. rigures 4-2, 4-12, and 4-24 give typical upstream velocity profiles. ivor purposes of comparison some of these maps are presented together on one pageon fig. 3-1, S-2, and B-3. because the static pressure did not vary appreciablythese lines shown on the maps can be considered lines of constant total pressure. 4e note that frictional effects are appreciable, The minimum velocity changes from 40 ft./sec. to 50 ft./sec . Also, the maximum velocity changes from 87 to 65-70 ft./seo. Thus,a particle with an initial velocity of 70 ft./sec. may end up with a velocity of only 50 ft./seo. herefore , in order to follow streamlines in the flow we must follow the high-velocity and low-velocity contours. 1 1. a more accurate method involves the use of tuftssmoke,etc... 4-9. if we do follow the high-velocity and lowvelocity particles,as suggested, we see a definite tangential (rotational) motion in the flows in all the bends investigated. This denotes the presence of a secondary circulation, We note also that the flow in the seven inch bend is distorted by separation and back-flow phenomena. it should be noted that the initial boundary layer , originally at the edge of the flow, ends up finally in the center of the flow! If this were to happen in an inlet duct , an undesireable flow would be present, in addition to pressure losses due to vorticity. eigures 4-32 ,4-34 , and 4-33 show the variation of the static pressure function where: &p Pstat* P ap/q amb In these plots the value of ap used is the average Ap over the cross-sectional area.Actual measured values of &p can be found in appendix A ,and vary widely ,especially in the bends. A 50. sample variation is as follows: 151-90 deg. bend - sta.6 x (in.) ap (in.H2Q i. -.04 1 -.02 lt. 0 2 21 3 31 4 4j 0 4.01 I.02 4 .03 +.03 +.03 thus +.007 average ihe value of q taken is the mean value of the dynamic pressure at station 2 (i.e. upstream of the bend). In the fifteen inch bends the velocity varied from 50 to 85 ft./sec. across the gradient. i:his gives a mean V9 of 5000 and a qmean of 1.15 in. 2O * -in the seven inch bend the velocity varied from 45 to 65 ft./sec.,giving a qmean of 0.75 in. ki 20 .the author realizes that the mean value of q changes throughout the duct system because of frictional effects. doweverthe error inbp (see section 3) is sufficiently large so that this change in qman can be neglected. Data on &p/q is too unreliable to make any valid conclusions.We appear to get oscillations , 5. perhaps due to the oscillations in the vorticity. indeed, the period seems to be roughly equal to that of the vorticity (see figs. 4-35,4-36,4-37). This data ,inaccurate as it is , is included in this thesis simply because no other similar data is available. The calculations of vorticity given in Appene dix u are plotted in figures 4-,5, 4-36, and 4-37. The vorticity plotted was obtained by an approx- imation, based on the following assumptions: 1.) Viscous effects are neglected and we assume that the velocity of the particles does not change much from station to station.Thus, the rate of movement of lines of constant velocity approximates the rate of movement of the streamlines. 2.) We assume the circulation is centered at(x =2.5 ' y = 2.5) and at (z a 2.5 ; y =7.5 3 3.) We neglect the effect of the cor- ners and assume the flow to be concentrated in two adjacent tubes, five inches in diameter. 4.) We assume area. is constant over the 5.) We neglect errors in the axial velocity (ItL) due to yaw. in actual calculations the author used the formula derived in section 5 52. ( r ) (see also fig. 4-34a) where: is the vorticity, L is the velocity of the particles, Q is the radial distance of the partical from the center of the circulation, As is the motion in the tangential direction in the time At, al is the motion in the axial direction in the time &t. The sign of the vorticity was arbitrarily chosen by the author to be positive when: a.) it causes flow to circulate in a counter-clockwise direction in the upper half ,when looking upstream, or b.) it causes the flow to circulate in a clockwise direction in the lower half,when looking upstream. The large spread in the values of shows that our approximation is indeed only an approximation. viscous effects partially account for this large spread. The curves plotted were faired through the center of the spread of the calculated values.The peaks were placed at the value of E where the normal to the lines of constant velocity (or constant 53. total pressure) was in approximately the same direction as the acceleration.This configuration is denoted by vertical lines of constant velocity on the velocity distribution maps. it is interesting to note that there seems to be no marked difference between values of cal- culated at Q2.5 inches and those calculated at = 1.75 inches. £his means that 9 may indeed be approximately constant over the cross-sectional area. iie see a definite change in the vorticity as the flow goes through the bend. in the 180 degree bend the vorticity builds up, decays ,and(going through zero)builds up again in the opposite direction. All this happens inside the bend. Txhis is of interest because it means that,with this type of velocity gradient, similar bends can be designed so that no secondary circulation would be present at the exit. ience, no energy would be stored in the form of rotation and no losses would occur due to energy unavailability. .ownstream of the bends the vorticity falls off due to viscous effects. ihe results of the tests run with the seven 54- inch bend are probably in error due to the following reasons: 1.) Separation and backflow distorts the flow so that the assumption that the flow is confined in a tube is not valid, 2.) Errors in velocity measurements,due mainly to yawing , are excessive. 55. FIG. 4-1 STATION 15 "x 90' LOCATIONS BEND 30 R=15" NOTE: STATION 11 IS 12" DOWNSTREAM OF STATION 10. PLANE OF BEND 12" 12" 12 SECTION AA Ii: 56. FIG. AXIAL 15 4-2 VELOCITY 90' BEND STATION X= I" YIN. PRO FILES 2 10- IN. 10- 8- 8- X =2"' 642 - 0 20 40 60 I 80 0 I 2p040 6 60 80 VFP.s (DATA FROM RUNS VF,.RS. 200-209) X=3" Y IN. mmmmp 80 X-4" IN. 10- 108 - 8-- 6- 6- 4,- 4- 0 20 40 60 80 VFPS. 0 20 40 60 80 V F.P,S, 57. FIG.4-3 VELOCITY DISTRIBUTION BEND 15"x90' STATION 2 -LOOKING UPSTREAM 450 50 60 65 0 75 80 85 8580 775 65 60 55 '4 5 VELOCITY IN F. P. S. 5"x O" CROSS-SECTION DATA FROM RUNS 200 -209 MAP C- FIG.4-* VELOCITY 15 "x90' STATION DISTRIBUTION BEND 4 - LOOKING UPSTREAM VELOCITY IN FP. S. 5"'i 10 " CROSS- SECT IO N DATA FROM RUNS 210 -217 MAP 59, FIG. 4-5 VELOCITY 15"x90 STATION DISTRIBUTION BEND 6 - LOOKING VELOCITY IN UPSTREAM F. P.S. 5"x10'" CROSS-SECTION 218-226 FROM RUNS DATA MAP 60. DISTRIBUTION 15"x90' BEND FIG. 4-6 VELOCITY STATION 7 - LOOKING UPSTREAM VELOCITY IN F.P.S. 5"x 10" CROSS -SECTION 227-245 DATA FROM RUNS MAP 61. FIG. 4-7 VELOCITY DISTRIBUTION MAP 15 "x 90' STATION 8 - VELOCITY 5" x DATA BEND LOOKING IN UPSTREAM F. P. S . 10'" CROSS - SECTION FROM RUNS 246 -265 62. FIG.4 -8 VELOCITY DISTRIBUTION 15" x90' BEND STATION 9- LOOKING UPSTREAM VELOCITY IN F. P, S. 5"x 10" CROSS - SECTION DATA FROM RUNS 266 - 285 MAP '3. FIG. 4 -9 VELOCITY DISTRIBUTION 15"x 90' BEND STATION 10 - LOOKING UPSTREAM VELOCITY IN F. P. S. 5"x 10" CROSS -SECTION DATA FROM RUNS 286 -296 MAP 64. FIG. 4-io VELOCITY DISTRIBUTION MAP BEND 15 //-9o STATION It - LOOKING UPSTREAM 55 55 4. VELOCITY 5"x10" IN F.P.S. CROSS -SECTION DATA FRO M RUNS 307-315 5. FIG. 4 -u STATION 15 " x I80' LOCATIONS BEND R=is" PLANE OF BEND K 5q SECTION AA 66. FIG. AXIAL VELOCITY 15" x 180' BEND 4-12 STATION PROFILES 2. X =2" YIN. YIN .10----- 642- 0 20 40 60 80 V DATA 0 60 RUNS 80 VF.P.S. F.P S. FROM 461-468 x =3" YIN. 20 40 X 4" 10- 86- 42- I 2 I 0 20 4060 I 80 v-- V EPS. 0 20 40 60 80 VF.P.S. 67. FIG. 4-13 VELOCITY DISTRIBUTION MAP 15 " x 180 ' BEN D STATION 2-LOOKING UPSTREAM 40 50 ~- -53" 50 60 70 80 (=86 82 80 70 60 50 40 F. P. S. IN VELOCITY CROSS-SECTION 5"x 10" DATA FROM RUNS 461-468 68. FIG. 4 -14 VELOCITY DISTRIBUTION 15 " x 180' BEND * STATION DATA 4 -LOO KING UPSTREAM VELOCITY IN F. P. S. 5"x 10" CROSS-SECTION TAKEN WITH 15"x90* BEND (R U NS 210 - 217) MAP 69. FIG. 4 -15 VELOCITY DISTRIBUTION 15"x 180' BEND * STATION DATA 6- LOOKING UPSTREAM IN F.P.S. VELOCITY 5" x 10" CROSS-SECTIO N 15ix90* BEND WITH TAKEN (RUNS 218-226) MAP 70 )ISTRI BUT ION MAP 15"x180' BEND FIG. 4-16 VELOCITY STATION 7- LOOKING UPSTREAM VELOCITY IN F. P. S. 5"x 10" CROSS -SECTION DATA FROM RUNS 469-481 7/. FIG.4 -17 VELOCITY 15" x STATION DISTRI BUTION Ix8O' BEND 8-LOOKING UPSTREAM IN F.P.S. VELOCITY CROSS-SECTION 5" x 10'I 488-497 DATA FROM RUNS MAP 7'~. DISTRIBUTION 15" x 180' BEN D FIG.4 -18 VELOCITY STATION 9-LOOKING VELOCITY 511 DATA X 10" IN CROSS FROM UPSTREAM F. P. S. - SECTION RUNS 498-517 MAP 73. FIG. 4-19 VELOCITY DISTRIBUTION 15" x 180 BEND STATION 10-LOOKING UPSTREAM F. P.S. VELOCITY IN CROSS-SECTION 5" x 10" RUNS 518-536 FROM DATA MAP 74. FIG. 4-20 DISTRIBUT ION MAP 15"x [80' BEND VELOCITY STAT10N 12 -LOO KING UPSTREAM IN F. P. S. VELOCITY 5"x 10" CROSS-SECTION DATA FROM RUNS 537-555 C - - 75. FIG.4 -21 VELOCITY 15" STATION x 1800 DATA BEND 13 -LOOKING VELOCITY 5"x DISTRIBUTION MAP o' IN UPSTREAM F. P.S. CROSS-SECTION FROM RUNS 566-574 76. FIG. 4-22 VELOCITY DISTRIBUTION 15"x 180' STATION BEND 14-LOOKING UPSTREAM VELOCITY IN F. P.S. 5"x 10 " CROSS-SECTION DATA FROM RUNS 580-597 MAP 7T. FIG. 4-23 STAT ION 7"x 90 PLA LOCATIONS BEND 78. FIG. 4-24. VELOCITY PROF I LES 7"/ x 90 BEND AXIAL STATION YIN. 3 YIN. x= I/ 10- 108- 8 - 642- 2 m I I i 20 40 60 80 0 20 40 60 80 V F.PS. DATA FROM VF.P.S. RUNS 343-351 YIN. YIN. 10- 10- 8- 8- 6- 6- 4- 4- 21 1 I 20 40 60 I 1 I 80 VF.PS., 0 - I 20 40 I I 60 80 VF.P.S. 79. FIG. 4 -25 VELOCITY 7"x STATION 90* DISTRIBUTION BEND 3 - LOOKING UPSTREAM IN F. P. S. VELOCITY 5"x 10" CROSS- SECTION 343 -351 RUNS FROM DATA MAP 80. FIG. 4 -26 VELOCITY 7"x STATION DISTRIBUTION 90' BEND 5 - LOO KING UPSTREAM VELOCITY IN F. P. S. 5 "x 10"' CROSS-SECTION 352-360 DATA FROM RUNS MAP FIG. 4 -27 VELOCITY DISTRIBUTION 7"x 90' BEND STATION 7- LOOKING UPSTREAM VELOCITY IN F. P.S. CROSS -SECTION 5"x 10" 361 -379 FROM RUNS DATA MAP 82. FIG. 4 -28 DISTRIBUTION VELOCITY 7"x 90 * STATION 7A- BEND LOO KING UPSTREAM VELOCITY IN F. P. S. 5''x 10" CROSS-SECTION 380-391 RUNS FROM DATA MAP 83. FIG. 4 -29 VELOCITY 7" STATION 90 x 8 - LO DKING VELOCITY 5" DATA x DISTRIBUTION ' BEND 10" IN UPSTREAM F.P.S. C ROSS - SECTION FROM RUNS 394 -405 MAP 8+. FIG. 4 -30 VELOCITY DISTRIBUTION 7"x 90' STATION 9 - BEND LOOKING UPSTREAM IN F. P.S. VELOCITY 5"1 x 101" CROSS -SECTION DATA FROM RUNS 406 - 418 MAP 85. FIG.4-31 VELOCITY DISTRIBUTION BEND 7"x 90' STATION 10 - LOOKING UPSTREAM 50 45 UNEXPLORED VELOCITY 5" x DATA 10" MAP IN FPS. CROSS-SECTION FROM RUNS 419 -430 86. 717fl7F VV 14 Ll I -, -~ " iI r - lit I -41 I I I -1 -- ~- . I .4. 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T I I 1-1 I I -P'- I I' ' i I.I t - I i iii I - I--ii- II_-I- HrI I1 - 4-t_i___. ii _; -1 1 . . IIIIIIIIII 93. SECTION V Theoretical Consiaerations 94. section V The theory presented here is by no means complete. it is carried out only as far as is necessary for application to the experimental results given in section 4 . Squire and winter obtained, the expression 1 a: grad L -2 sin d6 teq.5-1) where: is the secondary vorticity, V is the velocity grad_, is the total pressure gradient, is the angle between the normal to the stream surface and the acceleration, G is the angle of turn. £Wow,by assuming : grad is constantand sink is unity, we obtain: (S-S.)=- 2. .. 1 (2) (eq. 5.2) where: 2 is the initial velocity gradient. 1.jief 11 2..bid 95. hquation 5-2 appears to predict that will build up indefinitely as G increases. hovveverit doesn't, because this relation becomes invalid at the point in = tne flow where .In actual flows can and does become zero. The experimental results plotted on figs. 4-b and. 4-;66 show that 3 increases and then starts to decrease at a rate similar to its rate of increase. it remains,therefore, to obtain some analytical reand ( lation between ,in order to determine the value ofG when # does become zero. nawthorne considers the flow of a fluid in a cylindrical tube 1 and depicts it rotating to an angle om under the influence of the circulation r .iee figure 5-1 for a picture of the flow. lawthorne assumes that the velocity gradient stays approximately constant and that it does not distort.He ob2. : tains the expression - 2 fcoscCO (eg. 5-3 0 where: Um is the maximum velocity in the gradient, %Ab is the minimum velocity in the gradient, 1.Ref. 12 2.Ibid 96. d is the diameter of the tube, oc is the angle of rotation, e is the angle of turn in the bend, r is the circulation, is the vorticity. If we assume to be constant over the area F' then (eq. 5-3a) If we def ine v,the tangential velcoity of the flow due to the secondary circulation,by the following: V = d/2 then dct p = Tr d v and v= Nqow we note that: &, ./4 (eq. 5-4 ) 1 = ( R + d/2 sin- ) d hence: d/2 v dc.. -U(R + d/2 s inec.)d g or or v lkvw .. - .1 /2 40MMW ---d. (R +d/2 sinC) & l.Ref. 12 (eq. 5-5) 97. 5-3, 5-4, and 5-5 ,we get: Combining equations dcc' -U6 (R d/2 sinoc cosc&e (eq. 5-6 ) can be shown to be equal Hawthorne says that to the following dc lv-14 : (R+- d2 sin*c) dn(i i< (eq. 5-7 or: Oc dec (2R/d+s inoc) It in d ( sin ac) looks like the above equation could be d<c solved graphically . kowever , d9 when oc.=O goes to 00 and,hence, any graphical solution would not be accurate . in addition to this fact, the experimental evidence shows that the gradient distortsso the exact theory lf we go back to formula (oes not apply. 5-2 , we see that we should like~nowto find the value of where 9 or reaches its maximum. A combination of 1. itef. 12 98. equations 5-4 and 5-5 gives: 2 Vol (eq. 5-8 ) R+d/2 sin .c) dw. d9 By combining equation 5-7 with 5-8 we get: "in .1 -4 d. (i+-sin.c) 2R (eq. 5-9 ) is a maximim when o(= and we note that define and 0 the point where -4 d - . if we 0 as the values of and is a maximum (ct: ) ,then : F! - - at & 2R V'in ( and combining this with equation 5-2 9 -U -m n in ( ( d M but: ICU VWk -U6 by definition, so: 2 F in + 99. You will note that equation 5-9 does not predict a linear rise in the value of S. however, for the purpose of this thesis the author assumes the value of to be approximately correct and uses equation 5-2 as a first approximation. Thus from 0:0 to eO * : - 2 and from 9z9to i:3Q : These functions are plotted in figures 5-la , 5-2 , and 5-3 for different values of U1 and same values of 'UL for the applicable values of on the plots the sign of d/2R . Note that has been arbitrarily cho- sen initially positive by the author for convenience, Jote also that a change inlh. changes values of and 94 slightly , but changes the value of I leav- ing the bend considerably. The above theory was derived for the case of a flow in a circular pipe. This author tried to derive a comparable theory for the flow in a rectangular duct and found that the problem required the 100. use of superimposed image vortices around the outside of the channel to satisfy the boundary condit-ions at the four walls. A derivation of this kind is extremely involved and is not correct because the actual flow distorts, thereby invalidating one of the original assumptions. Therefore , in order to corrolate the experimental results with a theory, the author has chosen the approximation cited in section 4 . This approximation is based on the fol' lowing assumptions: 1.) We assume the rotation of the flow to center about the points x=2.5 and y= 2.5 ; and x= 2.5 and y--7.7. (these coordinates are in inches) 2.) We neglect the effects of corners and assume the flow in the rectangle to be approximated by two adjacent circles of diameter = 5 in. ii.e. d=2a ). 3.) We assume j to be constant over the cross-section. 4.) We neglect the small change in the velocity of particles from one station to anotherhence the rate of turning of lines of constant velocity approximates the rate of turning of streamlines. 5.) We assume the flow in the upper half to be unaffected by the flow in the lower half. 1.NOTE:this type of approximation works best for rectangles of AR= 2 101. T1he above theory is designated "pipe theory" by the author. The calculation of the vorticity ( from the velo- city distribution maps is carried out by the formula derived below : Al at where: 'L. is the velocity of the particles, A 1 is the axial distance moved, A t is the time interval considered. how we know: as V = V thus, a=s( = 1 ( where: v is the tangential velocity induced by the circulation, As is the are distance the particle rotates in time &to. i02. fOW 2 Tr( where: rl is the circulation, is the radius from the center of the circulation. But ,by definition : thus: 2 The values of vorticity calculated in Appendix Q are plotted on figures 4-35, 4-36, and 4-37 is interesting to note that the value of to be independent of the radius Z * it appears ,and hence appears to be constant over the area of the flow. The author realizes that the limited results of this investigation d.o not prove that the area *iowever , is constant over we now have some grounds for assuming it to be constant(as a first approximation). 103. In calculations on the fifteen inch bends the lower half of the flow was investigated because the velocity gradient upstream of the bend seemed to be more nearly linear. see figures 4-3 and 4-13. On the seven inch bend the top half of the flow was used because the lower half was not investigated by the author (due to poor gradient shape upstream). A comparison of the values of calculated from the tests and the values predicted by the approx" imate ("pipe") theory is given in figures 5-4,5-5, and 5-6. T!here is remarkable agreement,especially on the fifteen inch bends, viscous effects tend to damp down the amplitudes of oscillation ,as would be expected.Howeverthe period of oscillation seems to check with the theory. A look at the velocity maps shows that the value of e oc reaches r at approximately predicted by the theory. more experiments on 180 degree bends of different radius ratios should be made before this approximate theory is accepted for engineering purposes. ihe results of the seven inch bend do not corrolate weli. This is due mainly to the fact that the flow separates from the inner wall and J0+. so distorts that it no longer can be considered to be flowing in a cylindrical tube. In addition to this separation phenomenon ,the results of section 3 point out that errors in velocity measurements were probably excessive. 105. FIG. 5-i SHEAR FLOW TH ROUGH BENDS INIT IAL VELOCITY GRADIEN T, -. r LCN V. HIGH V. F HIGH V. 106. 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I 'I. j i ,, I i "I 4 I It ; '- ' + I ; i " - 5, , I T T -'- r - i 4-11- f- ill tt-f , r ! - t i - t : i, I iI ,11 L I ! l I i ++i I I I j - ': i I . I I i I : . 1r I 1 . -A. + I IL, , - I, . I"! .1 i' , . , I - ! -11 i - kI 11 I, I - d ' . I .. vI T, ., I I L . ti. ; - 4 t q- - f I f I -T i , - F. i-t- 4 1 ! 11 I5 - , I - ; . -. ; - , ! i1 .1.11 1 , 1, I , - ,- , I . i I -i ,I I I-1 - , I! j I ii ; I I , I -! -- i I t I 11I I t -- -- -.1 I f -4 4- - -, - 4- I - f t1, .i- * " I ,I 4, il, I ! I !1 " ! - -T . -1 -i I , ,r I , --r F -1 T - f. -,+ 4" -, - It f I , ,- -- T IT - - - I i" 1 ; . ; - , -*- --- : , 11 ; I I :; I , I 1, - Ft I - ;- i I: : . ; ' I q -:-' - ,- t, - - 11 t- i ] - 11 " i-i I- I- 4 -" 1; -I j . I - - "I -I - - f- -- -t 1-I - I11 - -. -- -t .11,- -,-: -1 -.L- - 1 4-i ,-- i -,,- I - . - 11 -I TI I-,,- I 4 I, - !I [" i, I I -i _-7I - - -C - . :-. I - + t -TI!i "I - 11- tr- F .... I +- F I I 4-1- I ,I-t - ,4 f I I A , I , ,,h II ! I , i I , , ' i -4 1 , L -I 7 -1 ,- I -, I , , 1 , ; i I U , III4 - Ii j 11 t I I I- L I I I 1 , , j , iI I 1 I I 4 --; " , , ,, i t -i - , - - - , i ; , oi ; , ITI . . ! i : , . . ': , -i- , " 11 , - -, , I ! - . . . ..I J.; I V I 11-- t 1,I +.- - , - , : , II I I, I , .1I! I I: I i11 i t .. 1; ! i I I ! t ; I1 .t - - I ; I , I f ! ; - , I f , 11 I f I vI I , , , : I -, 4 - i I I , ; I , , i : .i : I I ; . , i ; i f t Ii , 1L 1 i ,! 11It I I : I I: I - , i I 1 t -, I , 4 , I I -it i ! - I; t II I L 14 I I i -, - , 11 I , I - I , i r 1 I! 1 1 T I II, i . i : I i f- - j t- - I I t ; II ; , I It I 4- ; ,f 11 - , t t - T, -I , - I I ,- I ; I ! I1 11.11 I - - ; , ; I I i I r i , I : 1 1 I, , , :i-" .- ...... IJ- . , ,--. .. I.. -: 1,.i- 4 I-1TI , -F -;-, LT , -1 t t R , 7i -1 f f . 11 11 r I1 , I i 4i - j- - - -. I -, , -I' - I - -I I, , T, i 1, - 7-.- - - I j , ,: .I I t ;- , - -I-- - I r - - 1 -! , ,I ,,,, , I . I .. : L .. I- ; ; j ;I , I i 1 I I i:, I- i r ", - I i ,; ,1 ; !, t A , r , - T I 1 4 1 , 1 4 , i I "I . i , I . , , 14, ) I! .; -.I : . I II I; ,, r ;, ,' !-17 , : , * I , i , I II 4 1 . "I'll 1 ; + I- I T ,," ,-- . T, - - I II 7, 4 , : , I , . -, I I I I i i . . 1 I- , i! i+ + 14 I I - , ! 1 ; i 1, 1, ; t I . ; , I , , . II ,, , , 1 1 I - - ; 1 i , I , . j i . ! i 1 1 I II : , . I ; .. . 1, - I ; 1 . , i ; . Ii t I , T I I : I I-;, 1' I - I ( ! . . , 4 i . ; f I , I f I I j , j ; -t I- r - i ,I- , I 11 4 i I I ii - -H- -I I I 4 I- j 1 ,I I t ; I i k" 4 I r 1 , i 1; f -I 4 I I , II I I ,1 : ,- IJ T i i : . I i II I - "I ... .,."* : - - 1. [tH! i --tT I I ! , , +" ,--I1" II r I 11Ati l''_41 't '_1t t 1, 11I!!T-, , 11 I 1 I -,- , ,I i , ,4 . II I I - ,,: , 14 - I I - - I , , , T-4- f r f - H+ -- I I I I . ,"I , , -i, 1 'i , - I 4--'I 4-1 I-- I - f - .,'I ,". I ; I - i I 11 II I - i i Ii t I -, I I , f i , L - i Li I . -I - T - -- I " 1 ! i- I , ! JI I- Ii " -11 -I i - 1 i!, :if T, ;- Iil I Ij 1i-I r,1-1, ,L 11 I t- r , -' 1-1 j i 1 , - - ., . . I- I 1, t I .-" : , II 1 I . .! - [I,i , 11 I I ., ' 1 -1: I --4 - 4 ,; j -, I I I , t 1! , , - I , ,_ - - -f i I Ii , - . , I- -4- [ "I .. 7-. t , - I ; I i Ii . - , -i- I I- - , ------ 4- I --' . t [ I --* - i-, I , - -! i1 - I : , i !, -4 .- . iI -, ... , I1 If - 1 1 I ,i I t- -A -I- - : t I1 1. - Ii I : I - 1 t ; . 11 , I TI j ! " , - ! 1- I , I1 , ! ; ,r!- t - , I I I , .. - II- I , . -t i 1 11 ,: -,, I i i - - - ! I--H . 1 .- rr - - I.- , - . I , - - -- .. : - L 1 --,, -1;- -- I, - ,- , , ,: ... .. ,4 .- . t -t - t. 1, .11 - i -, .., I I t - . ",-; , F, IVT ,, -, - I i- -4'-;0 ,-tr- - 1 1. - - -I -.1 -p , t 7rj - - 11 -L- LL .. 11-1 +. i Ii- -- - . - . --I -, - - --t-+- -1- 4-,- -4- . -"- 1 - . - I I i -1- T.r t + . - .-I-. L. I'll, . I'll : 1_,1 i, - I . ; I _- - -- I ... 'K --- I ,,-, , ; I , I 4 . -- , . II : , J-.r ., : , ! , I ; i ; , : - -"7 I fI I I TI -- iI, ;-; I I I . 1, / 1 4 T r . 7 - , , , -, , , -.;..;- r - - r I -L. 1, , , 1 ., I.1 I I; ; r , , . I I i , j I A I I , I , 1 1 . ; II 1i . . , , _ 1 I IN ; 7 , - ; . ; I , I N , I ,+-4 - , t7 I ls I) t ! , , I- . . . , - - .+- 1 _N , .: 1 ; I i 1 i : , 4 : ,,^ ,0, -, ;; I I , , . ; It 4T 1 , . I : I I I1 , I -- , t !v . I , -.4-i ; I . I , , V I I I .I r . . I I + , .I 1-1- : I, ! , , ,-, , . I.. . , . . I , I,* . I . I '. I I I / ; VF , . , : I ; ,t ! I . , ': I: I-, -.I / , I I , I , f - I : . ! : I r II . - . , 77 -- : I , , : I I - I I I! I . . II 1 , i , . , , I. . , , I 1 I 1: , I 7% ! N,,, , I , , , 11 , L, ; ! ; , , , , -, ! I I *j V -tt I : ,' , Ik, 4 I N I I I II. 1 . I ,I . . --., I 1 , , , I: . ---, I - . I - -- . I L - I : - I I , i I . I I . , 1 I .I . - , . I - , I .- I... -- .I) , , .1I Ir I . .: 1, I I ..I : . , , ., -I . . , r t . i ; . , I I iI a I I I I I., ! .. I i I ,I , I I ;! : ii : .1--I I , I - -, , i, i, ; i I i , -1-1 ji 1 . , -4-+ - '-I 7 -, 'J, _ i I 1-1 ; i 1 -t ! , . 11 ! , . I , - + - ! !. !: . . . ., . , . i ! 11 1 III;III i , -- I i I , . I I , I I, 4 . 1 I . 7 I, . . .1 , I II . i : I i . I I. , - 1 , , :- i ; 1 ; b iI ., . . . I " 1: -. . , ! i I I . . I. + 11, 11 1 . , - 'A -I : I, , II -- ; ! - , .; I: I . -; . I 1 I I : ,i , t I , I . I 1I II , , 11 ; . . , ., I ; i , , " , I , ,, ,. I j I .. -I I I . : A , : I 1. I I . . i p D- I , : j , II - ..... , I. I. . , , ;--.t . , : 1: iL .I I 1 : , I , I I 7 . , . ! , I ; 4 : I: ., II I , I , , ; , r . : -- I II : i Ii , I : - -, ,, , , . II ; 7 I : 7 , ; -r . . e . . : .. I , . i i . , 11 I r , I ,, - ,1 I, j I" F% -- I ' ' r I ; ! , I I. . , . ! I - : . I . II I I . 41 , i -i I ; ,-.11,. 1 I ! . I I1 , II I I . .I. - , . :- , , , ' 7 4 I : I I ; - - L- I I - z : -4 I ; _-I , - t i i I - I I ,- ! I ; , i, 14 - - f I ! -1 11 ; I I - I 4 11I 1 , , I ; " , 1 ! 1 : i - : 1 !- I- i I -i , -7-f I , -, I I ,: f : ! 1 i 1 , I 1.1:, I '. I : , -1 T-i J -I 1 ! I , - - TI I ; - , i .-i I 1 ! I I , , -, , - -t , I _ , , i I t , , _I -I 4 ,1 ",- , L4 , I "- - , i i f i t ., , 1 ,- - - t ft ,-,,-' , , I , f , I, t -I ; . I-iT- , 11- I- I - , 4- - 1 ,- I , I-4 ,1 I , : , I I 'I f I tI .- I . . , , : Ii I Ii I.. . ) - I, I- - t-14 1 J-1i. .11 - , i - - -I 1-I 1 -, +-I - . ,- - . -.. - . -j -- -I -1I I 14-f- -T+ , I Ij - - - -.. --T ! 4-1I I i- ,I -1I -1 - -- .. J.----- , --, 4 1- + - , L --, f '. i - f - t -1-i - - " -1 i - --- 1 -.- + : I i -,- I ;_. I I I !! I 1.. T- - T L - -,- ., I -- (-. ii, L I -.I , ,-- - - IT -, '.. - TI , i I i - I , --- i - T - -. ,j , -, I Ii I - - 4-t-I - , I.[-H -i - - I , : - 1- I11 i i ;L -I -- - I-I..-.- 1 , - - - - - , f1- T , I I - . - - i --.-- t 7 I-41-- - II .. I-- + -i --I- i I f 1, , i I I I t I -1. i .. . - - . i I 1 -1 I I I , t, - IL , -jt " -1 . - I : I - ,,-r.i TI -1 , ITI -I I - ! I- . , 1i - - I+ L . -fl , I - , , 1,[ 4 i-i--1 1 i . . i i ; , 4- , ,- 1 I- I - , j H- -i-, , ,. t-LI . ; - i I--,, ,k -L - t -I . . ; ! LT 1 , 1 ,1 - ; I; I- - T-; , I'll : - II 1 T f T - '- 1 - ,! I i-,, I I . t - Ii , i _j ,-. - I - 11 1 i , f- 7 : -. I -i i I_ - , L- i -1-I I . 1. I . -1 I.- ij I : r , I --I i - I I . . + -- I-' -,I 1, I - 4, ,, -I" 11-j.. . I,-, - I-- I , ; I I: I, I - , -1" ;. L" -L : I t I . -,j : i -I , : I I I, I I ; , 4 I j Ii I I I , , I i1 1;,!4, , L 4" L I . -- -1 I ..I 'I 1 i i " ' -1 I. I I - . I I I I ; ,, j I ,: ,,! !; . -11 I I 1, ; i 1 1 , 1! '. -1 J !, L . , , I-!-! -,: ,, . I , ! -1 , , - I : 1*- ! - 'I,-- - 4. I - t1 L Ij - , -1. -- -- ,- - -. -1, , .., I 4-t- - -- I ,Z , -- I I 1, 14 ,' 1I 1I II L' I , I I i I I1- I _ ' . i - I i. 4 I 1 I I ---- T-, L .1 I I ! I 1 : 11 "1 I ) 1. . , I I I . . .. . , , "I,,, I .00 , : 1I - ' I. -11 Ir I . I a ; L: , -, , -, , I ., ., : .4 - -1 --,,+ T -- H I -- -- ' t 1 I !-1, r - . ; I II . I , I 77-. " j- ^ -A S I 11 ; . 11 i I I , ; i .I [I I I - -f - t -- I i- 1 ; I I I; L I, i i . , i ,' t I I I - I - ; - - I I 1 , I I ,, ; , - I : I . i I, 11 . ,I : : I; i 1 , 1-i : , . I . . . 11 , - - :1 : , . tI . : ! 1 1, , ; . . , ,1 I4 I -. - I , ! 1 . - I I t t I ; . j I I . -"-I i : I , 1 , , . . I , - i -i I - , Ij :- -j :, I II I i 41, 1. . i , I" I , ! ,- L ! 4 --. 4I I1 r Tt A f - i, It , : I , 4 I j- II --- ; , II I . J 1," . . , 1 , , , ; i I , I , , , ;1 1 : , -; , I j , . L i I I-i , 1'.1i :11- I- , -, , , ,I , , , I !- , III I . iI I I. i I I I I k : i I I -1 , , I i" I . . . I: I 1- , : , I : I I ; , i , I -, -: . - . - ,- , ,. 1 - . I . ;; , ; , I, 1 I t . . , -4 I I , ! . i. I. I I-;-- I- - -i ; I I I I ; -'- "I-1 , ! , , , : , I . i. - " 4: * , i I ,, A- , i -i , " , . s-, r , ,_; , I i , I . -U ! I- I : I - ,1 -. , !.i , 1 + i i I II i ; I i i I , j ; , I - ! : * . i I . I . ; I : , , , . II . , _ - N . , - i , , I ., . I . -77 , I I II - , I 'I : I - , , -I , , , , ,, , , : I - : I T . -11 ! 1, : I I , , . f ' -,-- I ! I I , -- -+--- i , + 41 1 I I . I 1 I ! 1 1 1, i II 1 ! - 7 f -, I- --. I 11 - t-1 II --, . - i -! - - - -r-,-; !- - -f - , -1 , -T --- f i 1 i I'll - - 11- - -.11,-- .- I -- !!- . L ,- - I4, I . - -I- 107. JOB. 109. (10. T _ Ill. I. . I. I :- - - ,i l - -1. _ - -1 ;_ ! -I-H - L. I -- I . I I, - -t tj __1 I4- - , I 1 -F '- L , - -I 1, --;-t,- i- I ! t -_ .; -, - _ - II i,ir - I-.- 7 - I I -r , - t. I T_11 - ! I I 1. I -1, I i : 1 ! , I I i t i1 i1 iII t -I I I- I , -, f t , , ! , 1 . t t , ,. I I !, t -" ; i . I j , - i i - 1- , TI , , - .- - .... - - - 4 -_ I--f-Ift II -, -- I F-- !- , 3, ,1 ; I ; - , i , - i-t 1, , ,, i I I.; I- L , ,_ -11 , t1 - tI ,1, I . I , . t - -I I t , i - I---, I I , , ,I, : , I : -, , I- - , , I ,, 1 ! : - -4- , 7 I , .I . -. I , . I_, I r 1 1,. III , , -t I- j I- I -i _ 1 , , I , - I, , I , I - _i -- - I I - - -i i . , - , I t j-1 R I rl; i 1 t i , I j ! f I I , I I I -7-t- 11 1_1 I , I i ; -_ T II i ii . - , i t _t I i : , ; .- .I r ,, : - j 1--' -4 t -_t I _ I i , i , T_ I I - . 11 , . i_ I , . , . 1I - - ,;i, -,1 11 I -,j ;. t. 1[ , 11 .1 t, , i 4 f -- f : I , I - i , , . - I, LI-. I. 1:I I L i, t -1 - - i I- L-1 !i IIri I -,, L ' '1,1 i' It 4 - I- - I t t- I , : 1 I , 1: t , , I I I I ; 1I ._I= 1 i: TI I i I I 11 : , ; , : L !0 1 - I *- f ,_ l -P - ;- - i , i- ,_ .. . I ,t ,I; -, 1I'!ff L - .L _ , . , , 7 ;,t-i: ,II _ . ; , 77 : , I . , , I . I- : t , t , .I ; : t :. 1 , U I I I , II F '- I. I tt r .*r f I 1 Ne 41v' 1: 1 , . - I I . . I , 1 t i=t; . -T . I i , I 7 7 -, __ , , , , , -" - , i ,1 ; 11 I II i t _ _ I, , ,+ "! _j 1 I- _. ,_: , ,I . i : 1-1 t I -4 4 - I I, t I I . , I 4 - Ii , I 1 ,i 1 . i , - ; . 7 . I I _j . I- , I , II - , i : , 1. I. t t I, i 1 1 - ; . j . , I I I I I , I . 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I t_ I : t , .. . ,( i_ W,- * 1 ."." i, 17- It , I 11 ,rlt'' I 'l . ! _ !. -i- i-TI L !- I 4 -: '_i I I'll I- , I 1 - 71 , 7 , i_ I i I , , !- - I I I i - . , -, I . .-i _1 I- ,I _ ;. !-- I . : _+ 11i !_ 11_-, !! --i 'T 5 ,- ''I , 14 - I Ii , _ it I I , , II ,I I I _ I . - , - -I 1 4 i , I ! ' I Rt - T, I - , I 1 .:, I t I I I L ; i I " 1 1 .11i . .1 f- I - I , ,- ,T __- -, I- 1 1 ll -, . -1. I . , .__ I 1r- - I- i _4 4 - . T_I ,- - : -1-- I- - I, A I I, , 11. II. , - , - - 1:LI-;- -. - i34.- i T_ ,-, -1 , , -4,11 t . r ,! _r -, t -- I't - I !- i 1 i , T -; -*,-I I . r- 1 f 4 - !J I :; - I- __- --I __- I -- 112. SECTION VI Conslusions "13. ;ection VI It is clear that secondary circulation is undesireable.The results of this experiment indicate that it may be possible in the future to reduce secondary circulation(or eliminate it) by careful design of the bend configuration and/or by careful control of upstream flow conditions.Therefore,there exists an optimum radius of turn for any given cross-section and angle of turn.Uonverslythere exists an optimum angle of turn for any given cross-section and radius of turn. The approximate theory presented by the author( called "pipe theory" ) seems to corrolate with the experimental evidence fairly wellconsidering the possible errors in the measured quantities. roecause of this there seems to be no present need for the derivation of a more exact theory. If further experiments show this approximation to be greatly in errorthen,of course, a new,modified theory will be necessary. The results of the calculations show that is approximately constant over the crosssectional area. 114. The data presented in this thesis applies only to bends of R/dZ 3 and AR *2 and having similar initial velocity gradients (the data on the 7 inch bend is not reliable). The effects of changing AR,RN, and gradient configuration are not predictable by the theory presented.because of this,more work should be done with rectangular bends containing similar gradients upstream. !he effect of AR, RN, k/d should then be eval-. uated and the theory modified,if necessary. Bo attempt was made to follow the path of any particle in order to determine the position of the streamlines.T±he results presented in the form of velocity maps cannot be used to indicate the positions of any streamlines.The use of tufts,sparklers, or smoke 'will be required. for accurate studies of the movement of the streamlines. The use of screensto produce velocity gradients, seems to be satisfactory.±t is also noted that screens are relatively simple to design and install. 1 Velocity measurements were sufficiently accurate to be used in corrolating any 1. rief. 2 15. approximate theory. in the future,however, more careful velocity measurements should be made, with special care being made to correct for the blocking effect and to eliminate the yaw error bee section 3 for a discussion of the sources of error. ihe static pressure data presented here is not reliable. it may be used only as a qualitative guide. ihe results of the seven inch ninety degree bend are to be used for comparison purposes only.±he errors present in measurements and in the application of the theory are too large. 1. nef. 23 1(6. SECTION VII Recommendat ions 117. bection VII The following recommencaations are made to persons continuing the investigation of secondary circulation on the same apparatus( or on similar apparatus).It is recommened that: 1.) other radii be investigated,keeping the same cross-section (5"1 x 10") 2.) tests be run with different radii,using a cross-section of 4" x 10" *This can be accom- plished easily on the present apparatus. 3.) in future tests,readings be made at 15 degree intervals in the curved sections and at 6" intervals in the straight sections. 4.) tuftssparklers,or smokel 2 be used to get more accurate information on streamlines. 5.) yaw meters be used to check caiculated values of circulation (i.e.- ). 6.) a pitot tube be used which is relatively insensitive to yaw3, 7.) a pitot-static tube be used to obtain simultaneous total pressure and static pressure readings 1. Ref. 7 2. Ref. 1 3. def. 23 118, 8.) provisions be made to prevent the clogging of the velocity gradient screen. The following recommendations apply to experiments using new apparatus.it is recommended that: 1.) different velocity gradients be investigated. 2.) larger Aspect Aatios be investigated. 3.) different methods of producing velocity gradients be developedl2 The following recommendations apply to theoretical considerations.It is recommended that: 1.) pressure losses due to seconiary circulation alone be calculated.This data should then be corrolated with that presented in ±eference 19. 2.) effects of viscosity on vortex decay3be experimentally determined by using the bend extension or one similar.very little experimental work has been done on this prublem. buch data would be helpful in modifying the 1. Ref. 4 2. hef. 15 3. Ibid II9. theory to account for the effects of viscosity. 3.) the existing theory be modified if experimental results shov it to be greatly in error. I20. SECTION VIII Bibliography 121. Section VIII iiumbers appearing in footnotes throughout this thesis refer to the sources listed below. 1.) M.B. Zisfein , M.S. thesis entitled, "An Investigation of vorticity and Losses in albows." - 1948 2.) D.F. .kelbeek , iv.S. thesis entitled, "An investigation of litot Tube aeasurements in a Steep xransverse Total Pressure uradient."1949 3.) P.J. Valentine ,M.S. thesis entitled, "Experimental investigation of Air Flow in ±ectangular iinety-Degree bend's."1-1949 4.) M.J. Demayo ,.. thesis entitled, "An Experimental investigation of ;hear .dlows in Elbows. "1950 5.) Instrument sectionAeronautical Engineering epartment, MAassachusetts institute of Technology - inotes on 16.33 instrumentation iaboratory, Experiment iio. 4. -1949 6.) International Uritical Tables ,Vol. 2, section on petroleum sources edited by B.H. Leslie, and '.0. Geniesse -1927 7.) H.C. Pavian ,"Experimental Aerodynamics" - 1940 8.) H. Kumbruoh , "Pitot->tatie wubes for ±etermining the velocity of Air -1925 (NACA Tech. iviemo. 303 ) 9.) A.D. Young and ..N. Maas, "The Behavior of a Pitot Tube in a Transverse Total kressure Gradient." (H and M No. 1770 ) -1937 10.) H.W. Emnons ,"Gas iynamics wables for Air." -ciro. 1948 12. 11.) vi.R. Hawthorne ,unpublished paper entitled, "A Preliminary "ote of shear flow in ends.,-1949 12.) W.R. hawthorne ,unpublished paper entitled, "Shear elow in a Bent uircular .ipe : an Approx. imate solution for a inear Initial velocity uradient." - 1950 13.) M.H. Vavra , "Steady rlow of Bon-viscous, iWlastio rluids in Axially 6ymmetrical .,hannels." , Jour, of Aero. so.,oivarch 1950 14.) W.R. Dean ,"Fluid iotion in a Uurved uhannel." Proo. ioyal 6oo. of London ,J6l 1211,928 15.) J.R. Weske ,"Experimental investigation of the velocity oistribution iownstream of 6ingle Duct zends." -1948 (NACA Tech. liote 1471 ) 16.) J.R. Weake ,"Pressure Losses in uucts with compound Blbows." -1943 (NACA War Aept. W-39 17.) Stuart,varnerand aoberts ,article in Heating,Piping and air Qonditioning ,iept. 1942 18.) L. vvirt ,"New Data on Design of Elbows in Duct Systems." ,General Electric ieview vol. 30 ,1927 19.) A.G. Anderson ,"Fluid Flow Diversion ; A Summary and Bibliography of Literature.",Project Report bo. 1 ,St. Anthony Falls ydraulic Laboratory at the University of Minnesota -1947 20.) J.R. Zwickel ,"Flow of Air and Its Dist- ribution Through Ducts." , Heatin entilation, February and miarch 1939 21.) R.D. Madison ,article in Aeating and ventilating ,July 1936 22.) J.R. Henry , "Design of kower Plant Installations;Pressure loss Characteristics of Duct Components." -1944 ( NACA War xiept. L-208) 23.) G. Kiel, "Total ead Meter with small sensitivity to Yaw."-1936 j NACA Tech. Memo. 775) 1z3. 24.) K.G. Merriam and B.R. Spaulding, "Comparative Tests of kitot-6tatic Tubes."-1935 ( NACA Tech. iiote 546 25.) Squire and winter ,"Secondary Flow" Report #Aero 2317 ) , (RAE 26.) H. Eichenberg ,uncompleted thesis on shear flows in circular pipes (at MIT) 27.) E. Ower ,"Design of Pitot-Statie Tubes.", -1925 (R and M 981 ) 28.) 0. Walchner ,"Effect of Qompressibility on Pressure neadings of a Prandtl Pitot Tube at Subsonic Plow Velocities." -1935 (NACA Tech. hote 917 29.) A. Shukry ,"Plow around bends in an upen Flume." , Proc. of American Society of uivil Engineers , vol. 7b, -1949 30.) C.H. Yen and 4.W. Howe ,"The Effects of Qhannel Shape on Losses in a vanal Bend.",Civil Engineering ,vol 12 1942 31.) J. Thompson ,"On the Origin of Winding of ±ivers in alluvial Plainswith memarks on Flow Around bends in .ipes." ,Proc. Aoyal Soc. of London, Vol.-- ,karch 1876 32.) F.L. Blue,.j.K. kierbertand R.L.Lancefield, "Flow Around a River Bend investigated." ,Civil Engineering ,Vol. 4-1934 33.) W.R. Hawthorne ,remarks given at a seminar of the ivechanical Engineering Department of m.I.T. in imay 1950. 124-. APPE1DIL A 125 APPENDIX A DATA - VI ND. RAW 15"x 90 MAR. 15, 1950 I5"x 90* 13END MAR. 15, 1950 STA. 8 P=.0Q3 N.H,0 Pa= 3.75 IN. 8 6 7 5 60 -7S - I±5'4 (6 5 7b - s 4 46 - 431 '7 4 48Fo48 17 7 70r 15 719 8 7x 7592.5'7 S AP 10 29 F9P86 7 2.513.0 2. 1.011.5 .5 T=81' - RU N.s F t. CF718 9 3 *3 - - - Sr . t0 S-9 551 175 7/29 (s 4.0 3.5 -7 7 ' 5 4.5 '74 &')B. ; 73 *.s 3 8 54 5 &JO . l r1 7457.37(. 4S 8. 574 732- S4 S SS .70.77. 5.o 2.5 '2. - I 2 - - '.- 64 51 ' . 49 _ ___ _ 51 - (05 7+ 67 8~ 1 'f 64 53 77 &8.;5 - 1 4 _ 1.3 3 8( 171 7S so ?3 i - 5 4(. S56 So 90 'I s 7~ 7 ~9 5751 84.5 o.5 M 9 -7M 71.5 L5 54 t.49479 5to SA09 -.5(4 9 5l +4 5 1 .C 171 71-S _S 4'f S3 + 4 4y5 47 0 T 7 0 2. .5 2 7 49 0FP s-L 45 AR27/0ST. 45 .5 '9 18. S SS 7 72-1. ( 8 47 to0 SIS -4 E- - 70 21. L 57" 57 -S $5 1 7 8i / 5-+ 77 _-S_5+ ' 1 431St. 74 17 581047 70 0 TDi8 819 .8 70 87.5- 1g44eS(85.5 ,2-7-4+1, 8 4 S7 7 7 5 2 222 223 72 419 22022 78/ 38 04 03 847-1 43 0-t04-&060Bs 3+.5 40 845 952.027530 5 .60 3 96 - _ 1 .5 4.0 3.0.3. CS7 1 -7rS 157 3 . . +.01+.02 1 49 f-/F77.( T7C2 1 1 1 43 (C 3r2 42,S 47 40to 62 1 (---f .3.583 5 1 146 P 5(,-7 3 8 7F 2 t0f-75 -7C77F41 3 I7 47 -,9 2 - 43 4 (0 0 l.0 47 213 214 25 21 0 8+.5.2. 0 00 8896 1.82.5i57.5815 212 02 -. 04 5.6 (.3t E isM5 1B55 2 -. = 98 STA47Is f5+ o4-0206 .9 s 5 7. "9 t-43 9. 490 4 3i4F'779;t9 b'1+3 8. 423 '40 (o 59 3 3, 3 10 183. 5 5 4.5 7I1& -o. "7 55 8 18 46 - 3.0 3 - L 0 0 0 0 2.5 18 1.3 0 4 829 F Go 4(/ 2324 1122 65, s G 4 t r.o0 3757 t7.0 t 5+t 02 70 20910 2s SS Z .9 02 -. - - - 43 (64 O 91 54 s 14 - 2.5 - -. 15 7 4 - - -4 8 71 4 525a 1 91 45 00F79 .0 - 2a--MAR531422+75, 89-8-90671 571.01.5o2.0 '7 - +1. If7 425208017208 RUNNB32087202720 53 9.5 172 17 - 1S1 57 5 5. - - 4.5 4.0 B -9 79 7 7 5CF.=-.0-70 G'£. r7 "7 (2. 5-7 -S 177187 - N1 oF1140C M Sq 5r 3 7t-L 47S5418047 5S772.s CF,=. 5s__4 "56 5- SD S, f85 6 59 £9 - 05. % ll (lI - 7+ 9 5 ~r e 0D IN4 HG.zT(2 42A /. 051 720. 62o0 32504 t-to (7 45 -3 2. 7 5 6 &2 1 41 9.520 51 (1 0 53 737-2222 3.57 90 7.is 69.8 SZ. 59 -. MARS.290STA.52 58 2 1 SD 99 -- 1 > 75 7+54.7 73 79 G 5 .9 t.5 -7 82 s*9 53 - 170 171 20 8 0 .x 49 .- 57 9*3 75 -7-4-2. 207 206 205 - q3S5+ s- g4C - 8 795 0 6 6 8 R4.A2 7 q7 (.' 55 r.0 67 . -5- - 4 3 S7 4 y .s 59 s -o' 70-0K68T(. 44.3-T _- _ - RUNNO.2037204 M - 75-4. 34 5% 58 67 4.7 (0 C 70 S6 9 8 - - - - 168 169 167 7 6 5 1 - - - - - - - CFa= 1.06 152 164 165 166 6.5 2 STA. 9 -3.78 EST. CFg.905 pa CF,= .970 CF 3-. 90 5 8 7 6 go AP=.00 (61 S- '75 - I' P0- -. 2-5- CFL= 3 4 5 - 1 -. - - -- r5 61 5.0 8 2 3 4- 9.5 9 74 - - - 1.0 1.0O4 134 135 136 137138 13 140 141 142 143 144 145 146 147 148 149 1501 51 RUN NC 133 47- 5 PabV3.7 CF, =.975 CF 3=.90 5 C.F2 =l.04 C.F =.975 STA, 10 AP= .00 H0 15"x MAR. 18, 1950 2 38 j- 47 4 S+ .~iz 7 0-' 7S £7 && 0L 4 7 47J 5 4C -+.524 24 0 '(1.. E 38 17 4o 5S+s7 47 (7 - 7. 1Fl 77f(.4 9 . 47 1 E 70 7) 7 7L & 7 - A -3 126 APPENDIX RAW A VI ND. D ATA- 5'x 90* MAR.27) PA:= 30.10 1950 T= 810 F F,=.970 C RUN NO. 227 22B 229 9.5 9.0 8,5 59 (67 2 2.5_ 233 234 235 236 237 238 239 240 241 242 8.0 7,5 5 7.0 6.5 6.0 5.5 5.0 4.5 3.5 3.0 2.5 2.0 S'2 50 33 47 '7 35 42 4.0 45 46 19 f9 5D _(__ 3 45. ((70 79 S. ~71.5 57 5? 59 5-/ 59 55, 57 7j -49 (. (70,49-4.59 it 7 79 & 4I9 40 -Z 73 ~73.5 5 59 . 7._71.5-D71 1-7 o -7 57 44 57.5 o7 7- -2 .7 -79 70 37 47. 1 4 1 4 S T=81* 249 250 251 252 253 4.0 3.5 3,0 2.5 2.0 1.5 C5 -77 (.7 (-7 gu 50 46 4-7 4 6. % 45 5L Z.5 1 46 __ 3.14 4.s( 4_L 5. -0 5L( 4-7 _____ (.7 r1. 4. 7+ - j 13 . -1 254 255 256 1.0 0.5 5.5 70.5 7 L 71 75 54 7,74 -/l.4 54 73 77,5 j74 76 5D 1 i6 46 1 (.9 7 ,/ 71-- 5_ C F 272 273 274 275 276 40 35 30 25 20 1 5 10 0.5 77 7f- 73 4f (o z(.9 5.5 (,3 45 57 42. (1.5 42.5 o 53 S'7 _ S5 4.s 5 4.7= 5--7 5 -3 5s .5 r9 76- 3 & 44 (64z2 1 5 S 39 41 47 5 19 5" S*O 3 47 ho 53 5o .L i5 <u -4L -45 48 49 7 7T 46. s 49 46 5fo5 s 55 -------- 569 6.0 6.5 7.0 (,4 (0.. .2 4-& 54 -74 7f 78 260 261 262 7.5 8.0 8.5 (6.5 7/ 71 77 6. 4 . . 7 .74 41 (4. 579'o " 54.5 55 . . ''/. IL 53 I s 4i s 5z+ 55 5259.5 2 s7 17 45 '14.5 99 44 I 4s.s $5 +7 4L It.5 7. 263 264 265 9.5 9.9 7 ---... j - .49g < 70 L '2 -L 7/ 76 -+97 4 .3 (.- 7' 40 --- - 277 278 7 77 6.0 (, C) 280 282 6.5 7.0 7.5 8.5 59 sLi5 .o 53 So 57 51 ±.48 40 55o 7 5 5 7 46 51,5 57 / g4 _ . 067 57 53 945I 11 3 So_ s? 4L 4-1 95 34 41. 40 4. 53 sI 47 281 9.0 9.5 9.7 8.0 (1_____ j 3 .5 57 4s _- .71 24 5f 5-? 52 _ 1 285 49.5 9_ 49 _s 284 40 5T ,0f 52 SS7 55 s5 6 283 -46 7 '5 64- _1 _ _1 60. (9_7 6.e it s .5 98 _.5"_ G,- LFIG. . - ,7 - 279 45 99.s %5 V3 5.9 259 (..3 (.-3 4. 'Li 4 5 1'. 515 is555I 5-. 44 4 .43 49 T. 1 .07 271 2 4 =- 270 2.5 2.5 *6 44c 269 (7 258 57 ' -0 268 70_7o7 257 4-. 267 "7 _0 -2. STA. 9 L P= 0 Pet= 3.89 CF,= .945 50 -8 / 77 174 'T 70 15"x 90* MAR.28, 1950 T=80 0 F P -= 29.57 RUN NQ 266 G3 a2 . 42. . 7/) 7176 Pe,3.8 8 .4- (.4 7 Lv S3 C7 06 -) tA712-744L 4o 71 7 LP--O F -S157 SM (.7 1e .00.5 S. 5~/. 2 4. 7/ 1.5 5o 51 ggrG 4 24245 STA. 8 0 L.. o -7 - 10,5 1; 53 - (.i .6 72. 7 3 57 C3 50, So f-7 LF5s 55 15' iL 4Lm st iq. 5. 4(0 1947 4-7 48 49 52 S'/ 70 ( 51eS7 5+ 3 (.o CF.=L 248 63 .40 452 - 5 s 1--7 5 3o 5 773 5+ +18 51r 243 90 * CF=955 5.04.5 T 4) 5-1 47 tS L If_'t 1 5"x 247 4 1- _f-S_ -7Z. Pwr~30.05 4v 4 9 MAR.27) 1950 RUN NO. 246 0.0 1.06 232 73-7 S' C Fa= 231 (of 3 7 AP= 230 - 75 (.3 BEND STA. PaD=3.87 A- 127 A APPENDIX DATA -VIND RAW 90* 10 15"x 90* o15"x MAR. 28,1950 STA. P4m= 2 9. 5 8 P:= 3.95 AP 0.0 CF=.9 32 CF,= 1.09 T=82*F RUN NO. 286 28T 2e8 MAR,29,1950 T481* STA. F PO=3. 251+ 295 254 2 9 PA,= 98 151 x900 MAR.29, 1950 T=82 0 F .7 4 302 297 298 299 300 301 309 3O( -10S330 56 19 5 19 195 ~ 1V 50 _5.3 1 0 - 4L..s -46 - -m -- - - - ____So -ME 2 STA. PA -29.96 AP CF, =z.9 35 CF,= RUN NO.336 337 338 A P .03 .02 .02 __-= 41 9~- 44 .~ r 1.9 7 --5 E 344 345 346 347 348349350 .02 .02 .02 .02 -.05 -. 040.0 +.02 .02 .02 .02 2.0 2.5 3.03.54.0 15 E1E1. 9.,s 342 343 4.0 3.5 0 5-1 49 467 _ O.so 0.2.5~ 1 20 1 +0 2-7 - 45 - - 39 v t.( 145. 58 TF 61 45 8 51.45 5 ss 5 '57 4.7 -IF( 44 4+7 . L 49 b (-1 ( .2' 5 (499 -11 -7 . 4 75..5 son - 49.5 45 g5 5/ 9-5 + - - +6 o50 49 to 5 0 49 46 5d 495 E3F9E53 7.E55E57.E E E9 5 9 6 7.S 9 1 5 f 64 74 + .9 6 159 10 2 4 57 / 50 4-f 4 4,. - - - - f. 4 1 - c9 - - 4+9 - ~ 1E 1-9 +9 1.01 E *.5A 4 -- 4 - n9 358 359 360 .08 .08 .08 ,08 - 4. 4.5 49 *? 53 EE3 60 32. - - SI - 5, it 47 EE si -t%5.o 1 5 57 '? 57 6 97 44.. (1zE07 S8.I F? 69.5. (.7 4 4* 4. 59 46 o + 44 55 s1-59 4 5 ,50 5. 4 70 tl l. 47 _S S7+ST. 5* 53 4.577 3 6o 47'(T M & (1- 43 s ;-+ 5 J i 6 f8 ME 5 r 5 49 n 3 46 9 4 -_..._ -__-_11__ - 357 - 0 2 (9 .95to 9s 66 49 -- + - CF 2 : 4- 4 4. - , - E 5'9 53 E0 1 E61.5E 5 E6.8 4 4 9. 56 8 '73 - ( s6 o 1+ (. 63 47 70 (.954573 44 1f 54 '.457-71 4 f-t 43 4(? 1- 4G.5 7T CL L 5 -7 4- 73 41.5 57 7z. -8 . to (9 4 " 49 . 714 <.+ 5 S+.s 5 j.51 . ST 41 T 9 G5 51577~ 1.4.s* +9-5,Fl f (0 40 (0 D 555 513 so 1 g IMEM~(,EE E ,E 5,4 5,1 MM F Z.1;3 M T 1 52 I2 47 4L 2 2. 13 5;7fF2.. st' V3 5.2 *51+9 4 41 44V. -L 49 73 PAhE STA, 5 30.33 EQ.. 2.5 - - o57 50 C + EE .07 .03 2.0 1.5 1.0 0.5 11- 7 , -4 / 71 5 (-3 6.5 C3 4. c4 0 '(.45.4.s5.1j51 + so 4-2 5T3gggg44gg 43 7 7+.5 ?.5 ?.1.572 75 /4 7+ 69 9 4. .1 1 14 0.0 +.03 .04 - 05 EEEE1ME'L51E49-S . 49.5 50 - 49ETE ES 51 . 73 E - F775EsFEb5E,9 /-40 G5 ~ S~C. 2.5 5 c 72 63 5.9 s 4 7.5 7 3 65.5 '?7.5 6. 71 . ,5 ,5 5.E - ,1 5 ~o~~ g 7 , - 70. i7 45- -7 4.5~ - (.c 48 -~1n 4 4. - 56 69 s 45 SSo 41 6C 3.I 40 51+ 49 60 3 55 55 4; 4 ~,51 51 5+If 5" 5SF'I0 2.s . S/ 53 -5 5o 53 2.5 5/ IIF1.3 1.0~~ +7.6 0.19 +4 3- 46. 0 50 43 o 5EM _-EMMEMM - 1 355 356 352 353 354 +4~ - - - - 38 .03 351 - 4S Y7 3+ 3o 47 51 +4.'; 9 1 5 +5 .z 55g7Sq. 5r5 3 54 51.5-+9 9 91 s;i 9 .9 . 195 S3 5o 5 2 E E-7. 5ES ,a 4+ 48 rM 31 9+ 1.5 1.0 05 4. 5 . 7 .149. 5 7 5.s1.t r2 5*5 5- EE E 47S +9 CF,=.91 5 CFz: 1.005 _, 340 341 3.0 +9 6 PBD r 4.36 339 2 2.5 I 5-5- T=78*F PAs = 30.30 CF,=.910 1.20 MAR. 3 STA. 31)1950 0 T=78 F Pa)= 4-.20 MAR. 5 7'"it90* 31)1950 7"90* 30) 1950 Tm78 0 F Po= 4.07 -E -ME - -EEEEEMEM -EE - -M E- S!454.1 5 .5+9. s S 5 F 57 57 S"77 - 5 5 _ _ _5 -E 15"x 90* MAR. 1 3 0 Y3SSo 27-60 s E 5C.S. EM 1_9 41-.15 05,4 6 5 93 1 1 5/ 57.5 5 So 4r+ 5(55. 60D 57.5 57 51 <60 5.7 5 /. ' 51 sLI 521 54-5 50 50 53 5-7 -5-157 53 G3 52. s3 sl -I 5.0. 5 '1 1.I 54.S 5o 57 54 T.3 154 5 si15+ 513 S4SM5E.E E1EEE5MEE -741__s_-3K1 5-1TF1 49 1 -EE 51 5D 54 - 54,T 5 3 9 8 - - -- 4 7 6 5 53.5 3 312. u 4. 54 4 o50 5 . > 554. S.4& 159 f 4(t 57 54 E+7 E4 M1 52 sfz.11s5 5E1 EE5E4.E5 5E9.5E 55 140 1+9 5s[ 52- 153 +-7 .54 3 _ o3 r 44 48 50 52.5 3 3 53 1.5 45 5-9 '93 50 G73 47 44.5.49 J. 5 53 -49 s-Ez.1 EE M5E3EME53EEE51 151. 1 45~~ 1_ 53 5 50 .5,jj 5-7 55.51 5 555* S 45 z s 310 3 2 1 5.5 6. G65 7.0 7.5 8.0 8.5 9.09.5 9.8 5.0 454.03.73.22.82.41.51.00.7 53.5 -3 . 5 57.547.9 515 S'9 (0 .5 (.0 5-3 5j 59o(47s3.5 5" 53 59 GO Go 53 r/ I (Z. ' 7 4/ 4/ 7 QI(3 . / SS*75354 S1 52. 51 '? 62 7o z 6o6.5 54 5G-5 51 5Zs 5/ 53 1-125 CF2 30 3D9 '307 =5.0 .s = 0.0 CF; -. 921 1.= 2 CFa: STA.I1 PAM= 29.75 PBD=4.0 O0P AP =0.0 CF;=.925 233 292 233 290 291 10 S9 44 49 44 - - '-~~ - - -- - -FIG. A-3 128 A APPENDIX RAW DATA-VIND. x 9 0 * BEND 7" MAR, 31.)1950 STA. T=75 0 F P,: 362 RUN N. 361 5.0 4.5 4.0 -30 -7 -3 - -s -3 / -25-5 0 2.5 25 2 443 41 (.0 .z 47 (4 31 , . ___ 9,5 3 46 s1 50 49 3.+ 92.. i .4 L 51 C, 43 4.s- 5- +9 $9.5 5"+ .5. 52 + 386 387 4-2 +8 41 31 49 38 49 4845 49 5 i 7.7 57 (r 15~ /0 / 3o 39 .9 5I 7 1 3 4& (.0 (7 7 C7 4G 43 ? (z (__6 (15 STA 4.5 A P%= S. io 5 o10 10 -.01 -- os 2.1 1 271 3 1 c -. o = 9.5 0 0 0 31 0 4C G 6 ~ - +T557.±14 s 9.0 8.5 8.0 45 7.5 +6 4 f.-x -+/1 + 42. o 40 39 39 T-'5 5+5553 19 59 I 5s09 2 59 2.5 5-9 3 - 4 -- 5z 47 44 +.5 47 44 - T8 5 S___W 413 0 7.0 6.5 4 +C. +/ 41 ~ 5t +X5 2.5". 5-+1 _-j v to 5 9.5 /P 6/ 41.5 43 _ 3.o =1.5 =9.5 9.0 55 -7 72 5 57 1 0 59 5& 8.5 S+L i f9 7.5 S 52 4C 55'2. 50 vs - _s. I 4. .. - 'C+s £ . 49) 41 414 41 5 416 417 418 0 0 0 0 5.5 4? 5.0 4.5 4.0 47 +8 51 3.5 45 44 47 +1-8 1 SO 45 4 % S 4$45 4' 44 j4+4-++ 4 43 43 .+2.5 '1 9 I 4.5 f-+ 4L 4 ± 47 +Z+2 47 97,8 420421 - - 4-03 404 6.0 5.5 5.0 4.0 41 3 41 3/ 3/ 35 2.9) 3123 zj 32. 4 34 31 30 +- o 3I-a35 3(. 44 41 42. 31 3+ e 7 3 42. - 7 _ 4.0 4.5 5.0 5.5 6.0 16 49 So 6.5 49 53 o 49 47 -7 44 4C 45 43 43.5 43 4+ 4. +- -q47 9 42. So 4 4, -72 51 '17 14 +1 47 4L 4543 4-2. 4L.5 45 45 +31 +3 43 +14-3 -1 45 45 4,71 +5 4+ 4-3 44 - I. 42. 5K43.5L43 7.0 P. 0 =4,30 C-F=r.0S5 24-30 C=9 426 427 42 8 429 7.5 - 8.0 8.5 42 1-5 L 42 42 4-z. 47 4, j. 45 45 4 42 41.5 44' 4 43 4+ 45 14 143 1 - 9.0 9.5 49 47 47 . So 53 2 47.5 +7 -7 44-+144 430 - 5o /. o il S3 % 4.-1 43 4 STA3 4- (- 42 o (- 7"x90 - - 43 ____i -7 e0 AP=0 - - 3__1 STA.10 422 423 42442 - 32. 70 =- 29.97 I.5 __5f. 41 4 419 ~ +,o1 900 x PA, - 5 59 .03 F2= 7 1 T7 T i 402 3633 § z. 1 - STA. 8 Pa: 4.26 &P= OC 43 39 4r 71 +.8 44 - -- 03 x 90* 7.0 6.5 45 +2 5 2. (4 63 - _ __ 55 43 +413934 3-f 4--L 39 T= 81* F CF; =.900 6.0 C. 4 50 'S 4(1 142.9 3_T 2. LC3 4 .2 7 0 +f 47 49 4 5C+.s 57 -4- 0 +0 4- 4-7 4,l950 PAma =30. 00 8.0 .45 C 5949 f9 +C 5.5/--55C. 3 S0 5/ 17 S 5s l 48 57- s4CC3 _ 50 394 395 396 397 398 399400 401 0 _ +, -. 0 T V1- 4.5 ?31 3 s.+ 4-6 5o So 57 __/J _ -. 08 4 9 +5 CF,=.910 393 -9/49 75+e -s _ _ + o o 4o.5 4f11F 0-5 41 4- LI+ I +2. 12. ±S'2.1151+94 9.0 7" +4.41 39.5 53 53 3.S _ 19 ri 8.5 %4 APR.4,1950 PD= 0 8.0 7" CF, r1.0 37 +.02 7.5 z7 APR. T=79"F STA. 9 RUNNl 406 407 408 409 410 411 412 .02 379A 5f 56 (og_ 3 s - 75-i153 CF; =.910 6P 379 3 49 5,4.5 CF.925 5.0 49 + 4-+ 4)1950 Pa-= 29.92 T= 77 0 F 378 53 3 55 53 7" x 90* APR. 377 5 13 ; 55 "x 90* MAR.31 5.5 5. 3 +0 -s .±.+.1 4 (o 49 45 376 7.0 -2o 36 -30 ___3f 388 389 390 391 392 6.0 6.5 '2 1 15 '27/S 22. 33 3 6.5 375 7 7A Pero= 4.32 7.0 - 55 9 90* 7.5 8.0 841 50 5 3 F9 I (2- 383 384 38 4 51 54 59 57 6.0 so 49 5V 91.- Go 6 69 0o 5.5 374 373 -30 -30 7 25 -20 1? 27 4f 47 _ CFzr .025 T4 4274538 t+8 - +5 49s 4 2.5 85 _ 372 C-5, _ 51 STA. j3 40 350 5 t 5 _$7r So il £3 -9 f5- _ ___ 8.5 9.0 49 47 49 50 I6 50 4 5i I.02 F:- 371 0.5 3 9 47 5 PeS30. 2 5 RUN N. 380 381 382 2 31 1950 C0F; =.9 20 = 1.0 31 C3 43 T =75* 0 F X=.5 1.5 to Ft 7"1x MAR 2.0 31 C 370 369 2.5 _5+ + (3 4 368 367 =-.917 F, 3.0 _1 _ Co C 366 5Z-5 _5 34 41 4.31 .9 to _7+: (.5 PAma= 30.30 365 -20 Vz 57 8 7 -2 z7. _5(_ ?3 9 4$ 4.5 3.5 -3 S_ L ( 3.5 364 363 7 431 8 58 I5*52 _ _ St. 5/ SI 4-9 5r/4 4'/. 46 <5 L ./ + FIG. A - 4 129 APPENDIX A RAWDATA-VIND. 15'' APR. 13,, 1950 T=75*F x 180* BEND 2 STA PA,s=29.8I PaI = 3. 30 0.5 3.5 9. 44 8.,5 5(63 8 1.0 464 465 3.0 1.5 2.0 : 35 +1 53 s.+ 303 So 5'o 55 .2- 48.5 :1.00 0CF; , ,.9 80 466 467 4.0 4.6 4C -7 7o 7(.9 80 S/ 81 18 I.E e62- (.I 7 . S- £3 3 s (.Z 71 -71 77 7'77 4. 7.9 + &5 2.-8.9 '6 190s-7 /.t 79 &8 5.9 5- 3.0 81 (8. .&L -. 01 S-78t3 7) .5 8 -75 -7(4 -77 9S 78 3 __ 4. , 9 5~7 f U7 S so 5 2 t 7Z9 _ _ 73 _-__ 15, _-9 APR. 14)1950 T=72 0 Pm 9.5 9.0 (.5 45 5c4 ,L 67.5- 41 36 (3 2. 73.5 8.5 472 473 8.0 7.5 145/ V8 5 S~5 5-t 15 5-8 34 Z,5 s- 4 4(.. (9 5-6 63 -7 79,5- 80 -76 '7z.-S RN N 9S- 40 63 47.5 13.5 4 -L.j o 1-5 9 1--. 68 473 9,5 9.077 8.5 474 475476 7.0 6.5 5/41--0 s 52.13 4-9 CS s (o , 42 I/ 4 44P/-- 8.0 7.5 40 39 ,9 Cl 475 47 7.0 6.5 -71( f 41 (ao 53 -T-9 43 +5 ____t _ 43 45 4-41- 7 41 45' 44 47.53 +8 ~7-5 +4.5 o45 s 44 46 4544 .44 41- C+.9 413 +7 +39 il 6.__5 _ co ? 4.r 53 , 57 75 / 57 3 '8-o7 4-2 3j 35 52,5 S'7.5 44 7 4651 56 45 7~42 + ( 4. (9, _ 7 _+ .5_ +,of5+ 4.o+,48 53 .- _ +,1 5 __: .7.5 _s .5 4 1ar, 43 +. -o1 o.o _ 57,5 _r_ ( +3 4-5 + 98 479 480 481 482 483 484 485 486 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 3- 9 5 /5 _ 50(__ (_5_ 3 '-4 4z -0 ___ 754 5.5 +-7 1 1 -,6 6.0 35 - OF -Z1.010 47747 49 - 7 CF -. 990 RUN NO. 469 470 471 0.5 - .9 + 4C F lB D =.3.2 I'D -7/ - .47 79 75. 1.0 497 I80~ STA. 7 15 1 x .5 51 44 1S (7 .S . +9 496 71 4C.5 +5 4-3 7! _ 7 49 4-6 t5 9 1.5 -DS C.4-5 50 7.+3 5'5' 5-0 3 4-95 4 Z S 5 4-7 3 -- ,53 5______ 8t_ 4.5) 4 8~7.75'+ 4 -44,5 __70 _ -77 5 (7.5 -7(.465 -73 2.0 -7'5_ jj 7. 62 2..5 49 4 5's -7 7Z9 85FR3 6+5 8 t s1- 7D7-1 493 45-+Tq'I5- L-----7 IT9 4,S 73 Y9 .I 492 1- -14 - CF: 1.02 5 f ( 5(A f, 5(, (A44+94347.9..5(+._40_o57 55T g -0 5 5. 3.5 so 15 44-7A± 82 1.5 57 I.0 .() 70.5 -71 491 '6392.s 6 -~79 1 3 5 '7 84678 9 81 7 - Z7.5 4a 7 756-79 490 7=s45 4.0 . 77 - 7 5(.Z .1.o 2.s 10 ( .55 70 '7/ AP - '3 .,o S +.o . 5 $o 4.02. 153 - 35o F '3 4,5 468 488 489 54f.7 55 (5 7.5 '7 CF 462 463 STA. 8 Pem=29.82 Peg 3.30 CF =.9 85 RUN NO. 461 180" 15' x APR. t4, 1950 T=72 0 F3 6.- It -4o 40 ( 47 7 45-- +0 54''-.010o 40'4 0 1-Z. 3-9 3 (5' 39 140 5-C~ 53 / 34,394-40I4,9 4 Go f 33 4/ 38 9750.-STA.8.7.8 35 3: +1 1+ .5-2. 1 4 f8 4S 4 9 so 5___ (7 48 47 43 48 701 76. 7 54. 75 50 52. s - 5 4.9 59 535.9 43 5 o &45' 482 3.0 483 2. 52, 9 53 '.0 54( 57. 5 48 9,g 64e- _4o sr.F.G. - 700 -7o 7+± 484 485 46 2.0 1. 5 77.0 __ 0 487 o4-0 4A 0-7, - AP PENDI X 130 A DATA - VIND. RAW 15''x 1804 APR. 14, 1950 STA.9 T= - F P 9b:::3.31 P - 2 9.83 CF, =.980 C = 1.04 501 498 499 500 RUN e'6 45-5 4 5 2.0 4 2.5 68 o 58 7.0 6.5 6.0 -47 42.5 47 4~7 +1 13 56 43 52. 5/ 41 49 4-7 44 8.0 AL..._5-7 55 - 6" . 503 504 50 5 506 7.5 50 -4 8.5 9.0 9.5 - 502 0) 119 3 .5 7 4. 78 79 4.0 75 7 7 43 7 / 42. 9 ,f -72- _355 6c5 72-.. '71. C F, 51 519 520 521 M 9.5 - 9.0 s- - 8.5 50 r.3 57 Z+915' 8.0 13 -9, -49 -+9 RU N -5 1.0 1, S3 5 'j- (5 ('+.. 2. 2-- 5-1 5 Go .3 3.0- ~ i/ 2.5 4( 45 17 7.5 47 7.0 e4 6.0 - -f 7 .- (. 6.5 455 . 5 4g + J3 1+<.5 51 50o 55.5 5-C, 41 (t 515 3.0 2.5 1.0 5'+ 4+9 2.0 5f 9 1.5 - 6'Z.3 43 +5 41 .- ( 45 '4-7 49) 47.5 4-7.5 52.53 5 - -5 (.0 1-5 45 1+*2. 524 2. 3.0 5 59 149. +.01 66 70 (a7 re8.5 41 7oK #-.,0, 7/ 4104 0735 SS5 55 72.75?.354 _ 530 $31 4.0 31 4c 3.5 4Y4 3.0 1-5, 47 JK 45. 526 527 528 529 5.5 + i3-4 5.0 +'? 43 4.5 1-3 -f3.5 419 '1-5 +-7 48 -452S9'47.5 53 59 51L..... r90 5 43LE 6 7.3 47 (o &I43(at 533 532 te__0_7 17 . 5'1 515<, 533 534 53 5 2.0 1.5 0 47 44 4-9 47 47.15 -100 53 4. (1 . 6 C44.5 E 4 5 4.0...418. 50 7 4 ~71a5 C . '1+4! .o.To 4 (044 56 45 15 18 50 5,0 53 (.06.7 570.5-' - f 4-' G. 7 I4 CS 73 - ,os 5475 47 STA. 12 PAMS= 30.0 15 x 1800 PBD= 3.36 APR. 15, 1950 T=74 *F .OL .03 10 PAms = 30.0 0 5 51 4.0 -7o 0 4.0__ (l1 7C) 49 (7 73 -70 5-7 4G1. 5(0 63. .1 A ________ (3 jjM 0.5 4(05 1 43 51 4.I 5{ 52. 516 C.7 CF. 35 44.5 -7 %S 7 514 STA. AP10 oL 525 4- . 4.5 6- (.0,5 (1375 1 46 71.5 7o2. 48 15,x518.0 Pav = 3.3 6 524 40 -7 j $13 +1,5 45 18 55 3 (2. 43 4 &1 67 75 47 7 53 _ 55L Iz2 48 5 9 9 _ 7_ 47 50 522 '523 5,( 1-7 5 5 2 .L5 55 53 2.75 4.5 45 54 .U 2 ~ - 4-,-7 - 5'0 'f9. 5 45(.s G-s5 49 4(09 2,5* k +7 47.5 47 47 (.3 7 15 46 48 '1 I q APR.215.1950 T=73 OF -3,6- 47 81 14-5 5 .948 73 4.0 45 3._5_-_+_(_2- 53 3.5 4.5 41 4 43 _5' 3.5 5.0 512 5H1 510 4 45 42 14547 o 5-9 5.5 47 +3 13 1 43 44 507 508 509 &P-= 0.0 _ _____ RUN 538 539 537 9.0 9.5 . 6 .5 2.0 2.5 50 5o.5 5.57 5' 595(.5 5(,5s 3. 81-6~ 5 172 542'54-3 544 6.5 6.0 541 8.0 1I - 7.0 7.5 8 49.5~-7 4.50 52 53 4 2.5 52.56 55 57 S7 (f 7+ 4 _ 4.5 5.75 59 5Z. 57.5. ' 43 _ F _C 545 546 5.0 5.5 +75 ' -4.57 52 - 5. 9.59o3 2.5 5C3 27 M 5 3 ( 57 5 753 C 55 3 34 il 6 (549 I 50515_ 73 57_5_ 5_ 545 4.5 IF.5 4.0 537 +3 549 550 551 3.5 3.0 47- 2.5 2.0 4z 4z. 45 47 50 I51 47 53 4+ (.1(4.- -7 f ( 456 251 49 (4. 14 is ~ 4-9.5 .55. (o il,5, .59. 45 46.5 59 51 _ __ __ 1, 4 4T 43 G7 ( 1 G5 (5.5 .70,5 CA GI . _ 547 2A- ~70 59 =1.07 495 5s .5 5.7 553.5 5.5 2 (5 &747 . 5 5A- 52. 55 .3 C9 5t(.5 C5 _ ____CF_.980 540 .5 9 57 53 (S5- 3.0 4-o 8.5 47 __ s 553 552 45 14 _ _ _ _ 5 4 0.5 1.0 1.5 9 +- 4 o 3 64 4~4,- l-s 5.1 41 4 45 7- i7 5 3050 5 8A66 S 571F+9G.53 SC.03W 60t. -4 (5.5- 585 5 . 5 53 I C . 4 . 7 131 A APPENDIX RAW DATA- VIND. 180* 15 " x APR. 14, T=730 F CF; =.975 557 558 559 560 9.0 8.5 8.0 7.5 49 so 51 5- 57 59 51.5 53 51 55 52 54 SO.- 2 51 53 RUN NO. 556 9.5 + '1-+ _ Z=.5 51( 1-7 /-1. 1.5 -8.s 5 /.o 2.0 49.5 So 2.1 3.o 3.5- 51 4.0 5 548 59 59 57 5'.5 Co 'I (;.iS 565 7.0 6.5 6.0 5.5 5.0 4.5 57.5 57.5 59 5 8 C.5 C (.3 65.5 4 0 (03 ,2 . CA.+ 40 41 sC, 5 C 54 57 .s 564 2- ;5 55 55.5 55 _9 57 ,CV5 67 C+ 45 6,2.5 17 C7,s CS.5 4Z 4,7 45 4/ 57 45.5 44.5 -T 5.LS-2.5 s- 1 ss- STA.2APR. Pet -= 30.0 RUN NO. 575576 577 -.02 .02 .02 578 579 1.57 69 2.o 2.5 3. ~7+ -70 2 -77 ~-0 o 7-7 70.S ;8 '1Z.s5 ?&, 70.1; ~-7b 9.o *7o 77 582 583 584 58 5 586 - 3 2 53 9.5 -3 -7.5 i 2 61- .5 9.0 tS 5o5,t5 - 8.0 7.5 5584-2. 5 57 C (4 CS 4' 53 5 r. t.3 4L 49 5l 5C . (.5 44 Ct.5 59 4/ 43 o 6? 55 51 55.!g5 e f5 5- 74&(.S 5$ +4 5t 5Z 53 53.5 .- +. . C 9 5 1 2.5 5f 5 -. 5.0 5771 4 5 5.5 45 57 5.5 53.5 6.0 .5-G 5~C 4257D 5 1f9 - 57 55 ,S4 9.S ~71 ~79' C9 - 40 49 5 51 , Si/ 5 2.5 2.0 1.5 40 5/ 5! 5s 54 3 __ -7 49 5 52. '5 -L.5 4? 5z. 49. 52 J7 1.0 0.5 47-. t-1 +27 43 44 +1+ 5 1*+ 43 45 4. ___- so1- (o___2 se 7 T2. 41 59 is - 591 592 593 594 59 5 596 3.0 58 2.5 55 4" 2.5 (0 54 5 C1 57 52. 5/ 4/ 57.5 51 ro 46,5 is.1 59 5C 51. 9 47 / +-5 47 19 47.5 42. 5 53 f 7 57.5 6..5 53 5'f (- '7 i f. 5'3. 5'2. 49.5- + 52.5 53 5-. +7 .53.5 Ti.575.3 5t 5 2.0 55 3.5 161 4.0 4.5 &(4.5 41 42.5 1/ 5' 0 597 - - - - 1 s/49 574 +9 S' 2 51.57 5'S 57 _5-8 573 47 -5 4 588 589 590 .0 4C 5$15 2I 2C4 %C..s ;3 5 44 9-9 15 6.5 7.0 S .3 587 572 STA. 14 P*:mg29.9 5 AP-0.0 _CF= 1.11 EST. - - - 5 '73 757C 3-5 580 581 571 180* 18) 1950 T=76* F .02,02 6 5 4 -. 5 75 174.5 (C5 -7o 8.5 7 o ( 0CT 55 _CF=.965 = -x. 41 =3.41 CF;=.978 P (o Z. 5"* Pao= 34 1Pa2 A 59 4C 3.0 570 2-$57 44 6 47 15"x 180* APR.l5,l950 T= 74*F 5 6CL (.5 5-7 569 3.5 C. 56.5 95 56 6! 63 + 4.0 l.5 .3 5.5 55 51 568 566 567 563 az 55 5 CF 2 z=.105 562 561 (2.5 STA. 13 P 6 =30.00 1950 PD-3.41 P O=0 1.5 5755 z. s 1-9 +9 - 9 17 S' 4 3- -. 4- 45+. 4.4 45 7 40* 4- 1.0 4 44 4fC +. FIG. A -7 132. APPENDIX B - -7 +,-+ rt-t -4 - - +- -7L -El--LL - r4- -T -j - -+ .L --. - -+--- - - - - - + - - 4- I _.i 12H-- -F- - r -l -, -- 1 -4 - - - --... -- * T -e--- 4~ --' - '- -.- - {~ - - . - + - i+ + +---- _L - - -T- rr 1 - - 1 T -I - t - 1 T -t--t- .. -- 4 .- -7 - --.- - i -+ -L--t - K -- - --- - - - 7 *. L- -- - I -1.7-7- - +r - - - - t+ r - -i -7 - - - - -4-1-1 71-7 - -- .. -T - --. - 4 - 1 -- - - -~~j . t- - - - - . r - -7 -- - 'T -T F4.B. 1- 134-. B APPENDIX WORK -- 5- x SHEET 180' BENDY=3 Yx=2 RUNS 4 +CI-4L6 .0 4t-4C8 g 69 7s4~ ; 58 -7-7 65 5s -7&. 168.5 53.5 (.9 ( ~79 ~74 5(. 5.s 1 44. 77 -i 59 +9. q o70 (~ 76 73 5-7 4-D.5 X=2.0 s9 so.s1 X= 3.0 s9 X=4.0 X=4,5 '*f61-+r8 96 57S-579 Xc1.s *I-446 77 26.6 76 7.5 2- X= 1. Z. 46 Y=6 Y= 5 X=0.5 51 S-579 +CI-In (-37.s S-- X=1.5 RUNS 5S-S-r 55C.5 57 X-0O.5 X= -S-573 Y=4 -78.5 80 so 575-579 44.s ALL FIGURES FAGR ES 4* ______ARE X= 2.0 82. X=3.0 873 X-4.0 9-7(. X=4.5 -7 *7' 78 VREF. 71.5 (IN F.P.S.) 27 49 73 47. 5 Fl G. 8-2A 135. FI G. B.- 2. 136. v 4 -- I I - I~ 44- 1 11 I-il 2 - 1 14 - 2 vY62& ~T i t I, 47-4 4t4- 4 t T L,4 4 L I- 1 t ,i I-v ~I i~.i I~I Ff7V777 rT]77 i t I~if7ti1l 11 -~---~-1-~-t t -~ j4j 71- - - - -~~ j i i Ifi T . Ir- -all -t-Y I-I-i TT 'lit IIT~i-'I - -T ~{j1 I,' 4A T-: L I A I ~4L 1~ FIG. B- 3 1-37, FIG. B-4 13. FIG. B-5 139. FIG. B-6 140. APPEBDIX C ?4'. APPENDIX CALCULATIONS VORTICITY 15"x 90* BEND ( BOTTOM) (i-) O (F7:)A . Pos 5o 0 (60 TO 9, (a-5 1.00 9.8 12.5 /.t0 12.Q .60 lo.6 76 .5o 3.42 S.5 .40 +.S0 72.5 2.9 23. 1-74 .11 .35 1.15 8(0 S T7 1 2.25 so STA (se- (A 0 .75 70 .50 s0 4 80 1.75 (.0 .75 60.2 8" 70 . 70 -79 2.5 50 3-3 go 40 1--7 1. 5 '2o STA, 4 870 1.6 go j. i To 0 6 2 1. 2.-7 (0 2 2s.4 2.5 _ __3 __3_ 210 so A t = 7.5 " 0 3.5 54 ' .5 .1i44 t 70 o .-71.&5 - -- . _ ---_ '. o~ /__ ?S/ '2 __ 9'5 12.0 FI-G.C-I 142. APPENDIX C CALCULATIONS VORTICITY 15 "x 90 * BEND (rr. Pos. 'LA.(pps.) ta> a 3.25 Go_ _ - 51.7 5 7-2.9 18.4 13F 02. o 2.2 21. 2 5- to 70 -L-- -70 ST A,o . . 2.0 1. 4 f.so T AV'e (se Ce' (A s),tv o 1. -7 )~.1 4 70 . . .92.0 J4..o 21~0 2.8 19.9 4 10 - So STA. 1.9 To To 1_ __ So 18 5o _ 0_ -60__ 1.-o _ ._ _ Ill 12.I _ __ _ 7-5 _ _ 5 . /.5 2.0 1.75 4 Z. 4.1 100 11.9 79 7010.0 ISO T.o STrA. 0 s~ 4s 50 To 9 -- - A- s 12 . ,144 159 teo 1158 ---- ------ 3.75 220ao . 15 '1-l3 4-7 /.0 01+.1 15 F1 G.C -2 143. APPENDIX VORTICITY CALCULATIONS 15"x 180* BEND BOTTOM (-r.) PC S. STA. .2.o8 'IA. (mps.) A s (I.) 80 l,/ 70 1.o 4 t. 0 6.7o7 50 _7 .7s ._ 1.1 so so 5 19. I1o5 (_ -_7_ 2.0 '7 /0 ^o1.0 7o .a (to 4.V5 -47 4 S'0 '- o .5 .45 -7 t 70 4-1.7 - 1.9 -_4._s -_-_ 47 3.'2 . -. 2-2. __ 2.3Z. . -. -Is, SS - .4 50 70.208 4+5 .201B 0 -. 50 S TO9 1Ist -O *7o S1rA '8.3 6.o_ o 1. ~3 70 '7To 8 /Us 12.3 40 STA /86t--s 3.07 50.o 2, 0 -- 1.85 2.2 50 Luc') -- 2.1 STAZo-70 3 IP'(Pr:) (a s).. _ _ -02 _ -2.3 1 -0 to f60 -14.L -__ 0--4 5o -S -- s 3. _ -45 FIG. C-3 I.9 24%__x 144. C APPENDIX CALCULATIONS VORTICITY 15x 180 BEND BOTTOM 5) f 0s. 70 -70 -(.70 -/.zs (.S - (W4 -2.0 ~ -9 '.48 - - - -7e I(bee') &. (.g .zo6 ~*7 - -25-- 5TA. '.5 -'.0 9 To 10 ~ -0 4-.1 -. -/.75 8" S 81 (D0 .14(. g55 so - .7o 55- .75 - .0 - I.z. 83 -12 --- ~- -% sl 5 -4. . - 4.68 So (5 - 1l4o 0.5 1.5 - -. 0 so -/06 -5.5 o. .- 9.0 -4 .208 S 'A. I0OTo 12. 5o - .75 - /./ -. -2.o -14 75 55v 'At, 1 z2" -4.9 /-15 55 - I. / - 2.55 ~ 1' 5 40 too - 2. 5 A - 1. o -9.9 - FIG. C-4 9 145. APPENDIX C VORTICITY CALCULATIONS 15"xl80* BEND BOTTOM Po s,. (FT.) A (r. p. s.) A 6 (m.) 2. 50 (rs), (FT 9.1 s- ~i .Z08 -2-0, (00 S TA. TO -( 9.0 -(. -c (0o - s - 45 - I e) ,-') _ I.2 + .- -- -0. (.0 -. 1 (-0 -Il / 15 - .72 "x 9 4 1/0 - -7 -3 -- ( .4__4_ - 7 .- 2.5 So .0 - _ __ _ _34. _ -1.2 5. -6 3 -1.2 -5.3 -89 BEND TOP PO S. "Q _ .208 I' (& s) A S -_ _5_o __5 (.2S7 STA. G0 3 5O _ .__5 39 2.30 17 4.os 30 0 . V; To 5o 50 '5 S.35 5_5___._3___4._3 5 0 7 15 2.2 33 4.3555. FIG. C - 5 _ 146. C APPENDIX CALCULATIONS VORTICITY 7"x 90" BEND TOP I po S. I (Fr.) 1 11 1 r(;-Ec)., It (. .45 AS (iW.) 7.0 .0 (sece-') 570 3.0 . zos 4.2 3.4f. 50 3.0 55 5 2, 18~ STA. 5 70 7 1 1.8 3.0 2.15 '0 4 la 1___ ___ 4o 50 390 50 Z.o . Jgg 24.5 2. 2..25 37b [4 Z-0 19 .Z08 STA. 7 To 200 2.0 4o .3 7A A t-- .~7 S .4 49 40 ------- L. 0 '# I .o - .11s . 14 . Its 5Jo 2. 5 so) -40 I 1--J I FIG. C-6 '4-7. APPENDIX VORTICI TY CALCULATI ONS 7 "x 90' BEND TOP 5. _____ Cvr.) U (rpr.) As 0mw) +o ST A. 7A .. g 3C~ r____ -b 5 o f (As ?-.-t 5 1% 12.1 90 It A5 -(.5 45 Z. .AIS.5 5o .o 1.0 +o ,I 2oS To 10 B- L.. +0 .144 If 99 s.0 i.. ,15 +0 144 2.4 30 5 _ _ _ - 1.6 4o 7.9 S 45.( 05o Zo sTA. 50.9 8To 9 S~o /0-9 .47 043 _ _ -1.0 _ _ _ .%Q0 g4.0 -d 5 4-o ,7g 2,3 /02. l'3.4 . J.'5. 45. 50.i .10 -35 5-3 S7.s FIG. C-7 _