ikfrL'lV) iS NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED April 19U6 as Advance Confidential Report L6C13 FIELD OF FLOW ABOUT A JET AND EFFECT OF JETS ON STABILITY OF JET-PROPELLED AIRPLANES By Herbert S. Rifcner Langley Memorial Aeronautical Laboratory Langley Field, Va. NACA WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. L - 213 DOCUMENTS DEPARTMENT Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/fieldofflowaboutOOIang 7o9 0^ '0 J NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NAG A ACR No. L.6cl ADVANCE CONFIDENTIAL REPORT FIELD OF FLOW ABOUT A JET AND EFFECT OF JETS ON STABILITY CF JET-PROPELLED AIRPLANES Ey Herbert S. Ribner SUMMARY The flow inclination induced outside cold and hot propulsive jets by the turbulent spreading has been derived. Certain simplifying assumptions were employed and the region near the orifice was not treated. The effect of jet temperature on the flow inclination was found to be small when the thrust coefficient is used as the criterion for similitude. The deflection of a jet due to angle of attack has been derived and found to be appreciable but small for normal flight conditions with small normal accelerations. The average jet-induced downwash over a tail plane has been obtained in terms of the geometry of the jet-tail configuration. These results have been applied to the estimation of the effect of the jets on the static longitudinal stability and trim of jet-propelled airplanes. INTRODUCE A jet, as it spreads by turbulent mixing, is known to entrain outside air in the .mixing zone. Air is thus drawn into the jet and the external flow is caused to incline toward the jet axis. If the jet passes near the tail surfaces of jet-propelled airplanes, the jet-induced flow deviation will affect the stability and trim. This flow deviation and its effects on static longitudinal stability are herein investigated theoretic ally for both cold and ho t jets. The present investigation was well advanced when a Eritish report by Squire and Trouncer on the cold jet (reference 1) became available in this country. The considerable rigor of the British analysis was found to 2 C ONF I DENT I AL £H No . hi C 1 5 b^ Impaired by the use of an idealized cosine velocity itribution in the jet, which produces errors as great as 11 percent. Also, the original version of the present analysis was found to be oversimplif ied in one respect, :;n resulted in comparable errors in the opposite direction. In the present revised treatment, most of the advantages :f simplification are retained, but the basic analysis of reference 1 is used to establish the value of a constant. The approximate treatment given herein permits the representation of the jet-induced stream deviation by a single curve. A comparison of the present analysis for the cold jet with that cf Squire and Trouncer is given in appendix A. Reference 1 does n~t treat the hot jet. e first part of the present pap -v is concerned wit analysis of the flow inclination induced ^uts..de cold and hot jets and the jet deflection due to angle of attack. The last part is concerned with applications to itation )f the effects of the jet on longitudinal stability and trim. The computational procedure is out- lined in detail in the r.umB^eai exaiple (tables I to HE) that little reference to the text is necessar;:. SYMBOLS (For diagrammatic representation of some of the symbols referring to jsts, see fig. 1.) ; thrust T absolute r:tream temperature, deyx-ees s tr e am density o f local jet density to stream density V str velocity incr« ment r i t velocity over stream velocity at " » ^ ) increment cf jet velocity over stream velrcity at point x on let axis CC AL NACA AGR No. L6ci3 confidential t increment of jet temperature over stream tempera- ture at pcint (x,r), degrees t m increment of jet temperature over stream tempera- ture at point x en jet axis t jet-temperature coefficient l— s^-j x axial distance from point at which jet, in accord- ance with law of spreading that holds at sub- stantial distances from orifice, would have zero cross section r radial distance from jet axis R radius of jet boundary at section x j (ttov^I-i /2I„ ^ t 2 / 1 / " x 2 " x l / X 2 = R\ I ST, ' /rrp^I 2 /2l A1/-/I2 V F y ST C ' S wing area Ij_, I p constants of velocity profile] defined in text I 1 ', Ij_ n , I 2 r , and so forth functions of T and T J/Y; defined in text k jet-spreading parameter (taken as O.2I4.O) f jet-spreading parameter (taken as 3*3) K constant (taken as O.3I) \J/ stream function CONFIDENTIAL k CONFIDENTIAL ;;A ACR No. LoCly c jet-induced inclination of flow toward jet axis; with subscript w, wing downwash averaged betv:een jet orifice- and horizontal tail mean iet-inducsd downwash ■ : one.' I local inclination of jet axis to general flow a angle of attack of thrust axis a e angle of attack of thrust axis relative to average flow between ,iet and tail fa - e w , A area of jet orifice b^ an of horizontal tail d lateral distance of jet axis from center of hori- zontal tail z distance of thrust axis below center of gravity Cp irpiane pitching -moment coefficient / pitching :n"r.-ent \ V Ipv^Sc J c wing chord I distance of nacelle inlet ahead of center of gravity; measured parallel to thrust axis Cr airplane lift coefficient / — — — — ^ : power en I ipv^s j unless subscripted i^ Lnci lence of horizontal tail, degrees 5 e elevator angle, degn i s; positive downward elevator hinge-moment coefficient > h / Kinoe oion ent l v — pv~~ x Elevator soan x (Elevator chore;)' CONFIDENTIAL NACA ACR No. LoCl5 CONFIDENTIAL r - h Lv, — " iQ da dC h d5 e C h 5 n„ distance of neutral point behind leading-edge mean aerodynamic chord as fraction of mean aero- dynamic chord An-Q shift of neutral point due to power: positive in forward direction Subscripts : j measured at jet orifice T due to thrust force e due to jet-induced flow inclination 1 due to single jet 2 dud" cc two jets nac due to nacelle normal force measured at zero thrust; defined as power-off condition fixed stick fixed free stick free ASSUMPTIONS The basic assumptions for the cold jet are the same as those for prandtl's approximate treatment of the spread of turbulence (reference 2, pp. l63-l65)« The flow studied is incompressible but the results are con- sidered closely applicable to ail subsonic jets and approximately applicable to supersonic jets. The starting point for the present paper Is a corollary of CONFIDENTIAL CONFIDENTIAL NACA ACS No. L6C13 the assumptions of reference 2, derived in appendix r ' of an approximate differential equation for bl reading of a jot in a ■ fluid. In r< V r nee 1 spreading Df the jet is obtained in a rigorous manner from first ] ciples. It is shown in appe by a suitable c ;e of a constant f t I ..of t expressions can be made to a,-;re-j very closely. const ias been so c;,os:n in b e present anal-sis. On the oasis of experimental data (r .oes 5 anc or a jet in still air, exclusive of the orifice ion, the velocity profile is ass '. to have bhe so: e shape at all sections of the jet. no special con- eral .on is given bhe region - appro: .. el; : i . _ce . . bers in length - in which tr tion occ fr - the i : : • i velocity at the jet orifice to the c 3tic profile o^ I 'ully developed burbule b jet. 'he fore- .: assumption is s\iitable for determining the downw. indi t] ■ ■ ital tail by wing- unted jet mc bors; it is not suitable for determining t 3 flow con itions r the jet crifj.ee. The experimental velocity profile of reference 3 is used. Velocity compon parallel to the j'et axis induced by the jet In the external flow are omitted in the an: ' . This omission effects a considerable . impli- ficatien in that it permits representation of the field of flow outside the jot by a single curv* in a gr h • As is pointed out in reference 1, neglect of tie induced axial flow Implies hoot "... the radial flov, .ch section of the jet [is] independent of t. - flow m oth r sections: this is approximately true v< :o ..hose to the boundary of the mixing region but is quite Invalid at ■ge distances fron jet axis. The actual flow out- jet can be regard :lo: iy equivalent to that proc o ' by a system of sinks along the jet axis, strength sufficient to secure the proper inflow at bhe I ;i of the jet....". The results cc bhis t x - nation are therefore restricted in applicability to the general vicinity cyr the jet. The region is mere pr< ciselj defined sub: ntly in the pr r. For the hot jet the assi sstbility is abandoned for flov; inside the jot but is retained for flov; outside the jet. : ^'l.e perfect-gas law is applied, with the temperature e ;ion at any point in the assumed to bo • >ti to the diffj ie between t local jet velocity and the stream velocity. Such a CONFI AL MAC A ACR No. L6C13 CONFIDENTIAL 7 temperature distribution is known to follow Iron: the momentum-transfer theory when the temperature differences are so small that density changes and heat transfer by radiation may be neglected. This principle will be applied herein without restriction to small temperature differences ana without regard for the divergence from experiment. (See fig. 2.) Because of these simplifying assumptions the analysis of the not jet can hardly be valid quantitatively. The analysis should be valid qualitatively to the extent of establishing whether the effect of temperature on the jet-induced flow inclina- tion is large or small. All AX, YS 13 Cold Jet parallel to Stream V e 1 o c i t y i n j e t . - If all the fluid of the jet is taken locally from the stream, momentum considerations show that the thrust equals the mass flow per second through any element multiplied by the excess of the jet velocity over the stream velocity at the element integrated over the cross section of the jetj that is (see fig. 1(a) for notation), rrR 2 pufvin + ui 2 ) -1 or If + £lil 1 - = o (i where II = «J0 u r_ dr U R R CONFIDENTIAL CONFIDENTIAL 'if AC A ACR No. L6d3 PI fu\ 2 r dr J 2 = J ^ i- If any of the fluid of the jet is not taken from the stream, the thrust F in equation (1) must be replaced by (F - Plight velocity x Added mass per second). The added mass per second contributed by the fuel is negli- gible for air-breathing jet motors. For rockets the added mass per second equals the thrust divided by the jet-nozzle velocity. Aspirator- type jets lie between the two categories. Equation (1) may be solved for the ratio of the peak jet additional velocity U to the stream velocity V in the form U _ Ii V 21- where i + tT- - 1 (2) /Troy 2 I n 2 /2 1? " = R V — ¥ L — V ST,. > and is a nondimensienal parameter. Spreading of jet .- By extension of Prandtl's qualitative reasoning (see reference 2, pp. 163-I65) it is shown in appendix B that dR dx -4 (B2) where k and f are constants that are determined in appendixes A and B, respectively. By use of equation (2), equation ( B2 ) may be written CONFIDENTIAL MAC A ACR No. L6d5 CONFIDENTIAL dR ax 1 + When the new variable 'rro ■5 = x, vV/ 21, - Xi H5s V ST.' is introduced dr; _ dR d£ ~ dx 1 + k 1 / n vr d 'rf + 1 ^ (3) and upon integration r; + 31- £ + 3/2 1 ■r + Vn" + V" ~ !f k£ M Equation (4) provides the lav; of spreading for the jet since R ~ tl and x ~ £,; the thrust F is con- tained in both r\ and £. Near the origin, where the jet additional velocity U is large in comparison with the stream velocity V, v, is small in comparison with unity and equation (4) is approximately r] k£ or R kx CONFIDENTIAL 10 CONFIDENTIAL NAD A ACR No. L6CX3 Thus, near the origin of the jet, the spreading is approximately linoar with the axial distance x. Far from the origin, where the jet additional velocity is snail in comparison with the stream velocity, n is large in comparison with unity and equation (1l) is approximately 2fl 2 or — v * tc = Constant x x That is, far from the origin the let spreads as the one- third power of the axial distance x. Some further com- ments on the spreading of a jet are made in appendix B. For the velocity profile (fig. 2), experimentally found for a jet in a still fluid, I-j = 0.0991 and Ip = O.OL895« ?or greater generality k will be left undetermined for the present. With these values of Ij_ and Io, equation (i|.) has been used to prepare figure 3, which shows the variation of r//ST c ' with kxA/3T c ' . Equation (i|) has also been used with equation (2J to provide the variation of u/v with Ic://st c ' shown in figure k. The point origin of the idealized jet of the present treatment, which is the origin of the coordinate x, is located a distance Xj upstream of the orifice of the actual jet. ' (See fig. 1.) The value of x ,• varies with T c ' but an average value is 2. J orifice diameters. More precise values can be obtained from figure 3 with R interpreted as the orifice radius R,-. J Fl ow inclination .- The condition of continuity may be expressed by forming the stream function CONFIDIENTIAL NAG A AOR No. L&C13 CONFIDENTIAL 11 (u + V)r dr Outside the jet this expression is approximately 2 Vr 2 ■1 2 (5) if the small values of u induced by the jet in the external flow are ignored. The angle at which the external flow inclines toward the jet axis is then, for small angles , 1 6\|/ rV ctx r dx U/V d(U/V) R d - (6) The use of tj and £, in place of R and x, respectively, (with ratios of the farm x/g, permitted, however) serves to eliminate the thrust as a separate parameter. When this change is made in equation (6) _ xI l 2 <^n re, d& 2U/V + d(u/v) Tj dt) if x/c, is written for its equal R/r). Then by the use of equations (2) to (I4.) there results finally e = kI l 2 x v ^ 2 - 1 - il} 2 2I 2 rg < H <2fl 2 > 1 -i-| '-p(a/i 2 + 1 + tj) Vr 2 + 1 v J l > (7) in radians, where ti Is related to the independent variable c, by equation (i|) . An asymptotic approxima- tion, accurate to within 1 percent for y\ %. 0.18, is CONFIDENTIAL 12 CONFIDENTIAL WA ACR " . L6C13 p kl ] _ t - x 1 - 2t5 21 ^ 1 + [2f— )t; (7a) -2 If r; is expressed in terms of c, ~, the flow- inclination relation (7) is of the form e = Constant x — x Function of ST, t r 2 x Within the limits of application of equation (7) the flow inclination outside the jet thus is inversely pro- portional to the radial distance r from the jet axis. Equation (7) can be conveniently represented by the r ST>. ' variation of — e with — ±— . The values of the con- x 2 stants k, f , I-, , and l£ therein are determined in appendixes A and E as O.2I4.O, 3. J, O.O99I, and Q.Ql\.S$^ s respectively, for the velocity profile of figure 2. For these values the variation of — £ with — Pr— is given X x d in figure 5* This single cur ve pro vides a ll the n eces- sary information on th e flo w inclination. A typical flow pattern is shown in figure IT. The flow-inclination relation (7) and figure 5> which is computed from it, are limited in application to points reasonably near the jet but well away from the orifice. The first limitation results from the neglect in the computation of the stream function of values of axial velocity induced by the let in the external flow. The second limitation results from the neglect of the transition region between the orifice of the jet and the region of similar velocity profiles. The charts of reference 1, in which these omissions were not made, show that the — variation of equation (7) holds, in general, to *5 percent within twice the jet radius at distances greater than 8 orifice diameters downstream of the orifice. This accuracy should be sufficient for the usual relative position's of the jet and the horizontal tail for wing- mounted jet motors. CONFIDENT! AI NAG A ACR No. L.6cl3 CONFIDENTIAL 13 The foregoing remarks may be interpreted from another point of view. The diameter of the jet orifice does not appear in the equations of the flow analysis, but it has been ascertained that these equations are applicable, in general, for distances greater than 8 orifice diameters downstream of the orifice. The dovnwash induced at the horizontal tail by v. ing jets at a given thrust may there- fore be concluded to be almost independent of the size of the jet orifice up to a diameter about one-eighth the distance to the horizontal tail. For very high ratios of the jet velocity to the stream velocity fii > 30J, r, is very small, and equa- tions (7) and (7 a ) become approximately tan^-i-^, 1^- -^ ^ ( 7b t is small is dropped. Such conditions may occur with rockets at take-9ff and at low speeds. For rockets the mass flaw from the nozzle is not taken from the stream and, as has been stated, the coefficient T c ' must be multiplied by one minus the ratio ©f the stream velocity to the jet exit velocity for use in the formulas. Rocket jets are ordinarily supersonic near the nozzle and the equations are not strictly applicable. Rot Jet parallel to Stream Veloc ity in jet.- The local air density in the hot jet will be some variable fraction a of the density in the free stream. For the present purpose the temperature elevation at any point in the jet will be assumed to be proportional to the difference between the local jet velocity and the stream velocity (see section of present paper entitled. " Assumptions" ) ; that is, t _ TU T ~ V CONFIDENTIAL lk CONFIDENTIAL SACA . LoClJ re t ia a constant. (See fig. 1(b) for notation.) erfect-gas law th a = T + t 1 t 1 + - T i + t H. 2 IT V (3) With the inccr] sration of the density I tor O", the equations for the 3<>ld jet will be modified to to the heat et. The momentum equation will take * c (tf + 111 H : L_ = (9) w L 2 --r7-Ip'K^ ^i u r dr :- p - V = U JO 1 + T U V pi 1 AA r _n_ - ' ~ 1 u U 1 + T U V Com "iso I-,' and Ip' with the :^rresponding quantities foi the cold jet, I-. .i Ig (equation (1)), its the folia Lng a pre i tions : :; intial MAC A ACR No. LoClJ CONFIDENTIAL 15 V V V L 2 12 1 + la 2 U — KT — V 10) > where K is a constant to be determined by substituting values computed by the exact equations in the second of equations (10). An average value over the range of is greatest interest, =t— b 1.2, experimental velocity profile of figure 2 can now be expressed in the soluble form K = O.Jl fcr the Equation (9 i-'-Tti /'V + II 2 W: •n -2 _ from which U V n - KT + 1 1/2 I 2 ? v ? Tf" - KTy 1 + TT 11) where t) is the function of equation (2 ) . R md T c f defined under The jet-temperature coefficient t may be determined from the following considerations if the temperature at the jet orifice is known. Equation (9) &s applied to conditions at the jet orifice (designated by subscript j), across which the velocity will be assumed uniform, takes the f»rm 1 + ll\h. • (16) v/hero [FIDENTIAL NACA ACR No* LoC13 CONFIDENTIAL 19 h n = T 1 , xP ( Li V — — ar *d (13), respectively. --2 If f] is expressed in terms of c, and t means of equation (13), the flow-inclination relat is of the form except alone; and t t in by ion (IS) fidential 20 CONFIDENTIAL NACA r o . LoClJ e = Constant x — x Function o£ As is the care Tor the cold jet, the flow inclination outside the jet is thus inversely proportional to the radial distance r. The effect of the jet temperature is determined by the jet-temperature coefficient t. Equation (18) for the flow deviation about the hot jet has been evaluated for the single value t = O.l 1 ). ST ' The curve of — e against — -— is shown in figure 5> x ° ? X - where e is measured in degrees, alone, with the curve for the cold jet (t = 0). Similitude of Hot and Cold Jets with Applications to Wind-Tunnel Tests A typical value of the temperature coefficient in t = at maximum flight T ' a propulsive jet is From the curves of figure 5 > therefore, the effect of temperature on the jet-induced flow inclination can be seen to be small, provided the comparison is made at the -.- thrust coefficient T ' The thrust coefficient thus a suitable criterion for the similitude of the flow fields about hot and cold jets of the type for which all the flow from the exit is supplied from the inlet. (For a constant throttle setting the coefficient t increases as T c ' decreases, but this variation does not invali- date the conclusion.) 3e c a ; s e of the typic al \ rmal jet twice the e xit velo same thru st from tn from the ex it is s u of the ho b jet, how half that c f the c o cold jet th e mass f occur with a hot ie ' ate the pro. reduced density the hot jet from a t motor will have of the order of cold jet that develops the city of e same size orifice, If polled from the inlet. all the .ow ;ver, will be of the order of The mass flow one- id jet. For model testing with a low into the nacelle inlet tnat would t should be simulated in order to flow about the nacelle. The mass ' ITS^TIAL NACA AGR No. L&CIJ CONFIDENTIAL 21 flow in the cold jet can be made equal to that In the hot jet by reducing the orifice of the cold jet to such a size that the product of air density and orifice area is the sane for both jets. In wind-tunnel tests at the Anies Aeronautical laboratory of the NACA (unpublished) the scale-size orifice of the cold-jet model was restricted to an annulus by means of a faired plug. If some of the fluid of the cold jet is supplied from a source other than the inlet of the nacelle-, as in the case of an aspirator jet, the mass flow Into the inlet is less than the mass flow from the exit, and the foregoing relations do not apply. In this case simula- tion of the proper mass flow into the inlet is possible without reduction of the size of the exit from the scale value. With an aspirator jet, however, the jet-induced flow inclination at a given thrust will be too small for the reasons explained in the analysis of the cold jet. (See section entitled "Cold Jet parallel to Stream.") Effect of Inclination of Jet Axis Gene r al rem arks.- The effect of inclination of the jet axis to the general flow must be considered in estimations of the jet-induced downwash at the tail plane. If the jet behaved like a rigid body the incli- nation would give rise to an interference similar to that between the fuselage ^uid. the horizontal tail. Vertically above the jet there would be a slight down- wash, and on either side, a slight upwash. Averaged across the tail, the net effect would be negligible. The jet actually approximates a rigid body in that it tends to maintain its shape and direction in spite of any inclination to the main flow. There Is an appre- ciable progressive deviation, hov/ever, from the initial direction toward the stream direction that can be obtained from momentum considerations . This deflection alters the distance between the jet and the horizontal tail, and therefore the jet-Induced downwash. Determination of je t defl e e tion .-Let be the local inclination of the jet axis to the general flow, and let a e be the Inclination of the thrust axis. On the basis of momentum considerations, the following approximate relation for the fractional angular deviation of the jet is derived in appendix C: CONFIDENTIAL 22 COHFinENTIAL NACA AQTi No. L6C13 8 V - L l / v 1 - f- = (C3) ° e 2 + (a ll + 6K T |)| +l2 (|) ; The variation of 1 - — with kx//&T ', for the cold a e /V c jet (T = 0) and the hot Jet (t = 0.1s) is given in figure 7* 'The effect of jet temperature is seen to be negligible . The change due to jet deflection in the radial distance r from the jet axis to the horizontal tail is given by ^~(x - x,)(l - ±) (19) ■7.3 V A a eA v Ar av where x - x ,• is the distance from the orifice to the horizontal tail and (1 - -— ) is the average value of □ V e /av 1 - — between the let orifice and the hinge line of a. the horizontal tail minus the value at the jet orifice. In this application the general flow in the region of the jet is affected by the wing downwash so that, in straight flight, a e = a ~ e w in degrees, where a is the inclination of the thrust axis to the free stream, and e is the downwash clue to the wing averaged over the length x - x • . In accel- erated flight the curvature of the flight path contributes an additional increment to a e . The jet deflection Ar is evaluated in table III of the numerical example, along with various other CONFIDENTIAL NACA ACR No. ^6013 CONFIDENTIAL 25 quantities, end is shown to be no more than 15 percent of r. On the basis o£ these computations the jet deflection appears to be small f©r straight flight and for flight with small normal accelerations. Cn the other hand, the average angular deviation of the jet is an appreciable fraction of the angle of attack. The fractional angular deviation fl - — J is O.2I4. V s ' &v greater for the several conditio] example. (See tables I to III,) or EFFECT OF JSTS ON LONGITUDINAL STABILITY AND TRIM Average Downwash over Tail Plane Consider a general point y along the span of the horizontal tail, with y = directly above the jet. (See fig. 8.) Let the angle subtended at the center of the jet by the length y be 9 . The jet-induced flow Inclination has been shown to be Inversely proportional to the radial distance from the jet axis; therefore, if the inclination at y - is e, the inclination at y is e cos 9. The downwash at y is the component of this normal to the tail plane e cos^G. The unweighted mean downwash angle over the tail plane is therefore b t pd+-r 2 p, c = i b t o-d+f- e c o s - 9 dy 2 °t T CONFIDENTIAL fACA ACE .'. . L6C13 or f *>t -d + — d + -1 2 -l .;an e b t V tar." 1 — I (20) Lifting-line theory suggests that an average weighted according to the chord would provide the most accurate values of tail lift. An Jhted average over, say, 0.$ of the tail span would appear to approxi- mate t 15 jondition. The curves of figure 8, accordingly, have b ' repared from equation (20) with 2 .yt^ sub- stituted for bx. The curves give the variation of T/c with r, b+- and 2.c./c^ where T is now the effec- tive nean jet-induced downwaah across the tail plane, e is the fie* inclination at a radius r from the jet, and r/b-j; ar.d 2 : ,: T locate the jet axis relative to the tail plane, as shown in figure ?. The curves apply to a single jet, and the lown asb is additive for several jets . Pitching-Moment Increments Due to Jet Operation General consideratians . - At a given angle of attack, operation of the jet motors will, in general, change both pitching moment and the lift coefficient. Confusion will be avoided if t^.e changes in pitching m.ement and lift coefficient are initially Dbtai as functions ">f the r-off (zero thrust) lift coefficient Cr r- » which • is a ] . function of angle of attacks The several pitching-moment increments due to jet operation are dis- cussed in the following paragraphs. Each increment is to be re arded as a function of Cr^« The increments are given for a single jet and are to be multiplie i by the :.' . fc : r of jets. Pitch:.; .:; moment contributed by iirect thrust.- If the thrust axis of the jet passes a distance z below center of gravity the thrust will cc~- te an Itching moment, which is In coefficient " NAD A ACE No. LbCl3 CONFIDENTIAL 25 AC. = - T ' mt V c c The thrust coefficient T c ' ordinarily will be known as a function of the power-on lift coefficient C^. In order to obtain T c ' as a function of the power-off lift coefficient Ct„, use can be made of the known LiQ' relation between Ct and a together with the relation C T - Cr = O.T c ' Li i~>Q u where Ct and Ct are measured at the same ana;le of attack a and a is taken in radian measure. A "cut- and-trv" orocedure may be used and a curve of Ct against Ct can be obtained at the same time. ° ij Pitching moment con tributed by jet-i nduced downwash .- It has been shown that a jet induces outside itself an axially symmetric flow field. The Inclination e (meas- ured in degrees) relative to the thrust axis at the point (x,r) (see figs. 1 and 3) for a given thrust coefficient T c ' , can be determined from figure S. A small deflection Ar experienced by the jet A'hen inclined to the general stream can be determined from .equation (19) and figure 7 and used to correct r and then e. The ratio of the value of average downwash over the horizontal tail T to the value of e is given in figure 6 as a function of the geometry of the jet-tail configuration. The -pitch ing -moment coefficient contributed oer jet by the jet-mcuced downwash is then, for the stick fixed, dCL _ AC m = - — ^ e, (21) me fixed dl t 1 If the stick is free and if the jet unit Is mounted under the wing so that the horizontal tail is well away from the orifice, expression (21) becomes CONFIDENTIAL 26 CONFIDENTIAL I . -. I 313 /dC_ dC,„ CVA ac c -f E - 2L Q 1~ (22) E iree \ dl t If the orifice i3 near the horizontal tail, as when the jet issues from the rear end of the fuselage, the horizontal tail will be in a region of curved flow. If the value of (V is negative, the elevator will tend "0 to float downward to conform to the curvature. This downfloating tendency will add a stabilizing or negative amount to the value of the stick-free pitching -moment increment given by equation (22). The change, could be substantial for a closely . balanced elevator (Chg near zero y ; the magnitude of the change will depend on the type of balance. In addition, the hinge-moment charac- teristics might be modified by an effect of the jet on the boundary layer of the elevator. The charts of the present paper (figs. 3, 1+, ~j , and 7) are not valid within a distance of approximately 8 orifice diameters downstream of the orifice, and ref- erence 1 should be consulted for the flow in this region. Equation (21) for the stick-fixed pitching-moment incre- ment will be approximately valid provided £ is evalu- ated at the three-quarter-chord line of the horizontal tail. Pitching, momen t contributed by na celle normal force. - The air taken In at the nacelle inlet is turned through an angle (the angle of attack of the thrust axis) in becoming alined with the jet axis. This turning of the air gives rise to a centrifugal force acting upward at the inlet. The force, wfhich is negligible compared with the wing lift, equals the mass flow per second throu the nacelle multiplied by the stream velocity and the sine of the local angle of attack. The contribution to the airplane pitching-moment coefficient is (Mass /sec) I sin (a - e) a.Cw, = (2$) ~pV3c CONFIDENTIAL N,iCA ACR No. LoClo CONFIDENTIAL 2 •7 where Z. Is the lever ara from the inlet of the nacelle to the center o£ gravity of the airplane and -e " is the upwash Induced by the wing at the nacelle inlet. The upwash -e can be estimated from figure 5 °? reference j. This upwash is large only when Z/c in equation (23) is small, end its neglect therefore introduces small error in the moment . Pitching moment contributed by boundary-layer removal.- The suction and other effects of the jet may tend to remove some of the boundary layer on adjacent surfaces. The pressure distribution would be somewhat altered. In some Instances flow separation may be inhib- ited, which w->uld result in rather large changes in pressure distribution. In case fl$rw separation on the wing is suppressed, an increased downwash will occur at the tail with a consequent positive pitching-moment increment. The determination of the moment changes due to these several effects must be left to experiment. Any change in the fuselage pitching moment due to boundary-layer removal with tail on may possibly be dif- ferent from such a change with tall off because of the . interference between the horizontal tail and the fuse- lage. For this reason the comparison of tests of models with tail on and with tail off .nop not necessarily yield the part of the power-on pitchin; -moment change that can be attributed to the jet-indue raw ash. Neutral -Point Shifts Due to power T'ne power-on curves o£ C ra against Cr for various elevator settings should be parallel like the power-off curves. The shift in neutral point due to power is therefore f dC m \ / iC m \ x u 'Power en x 'power off in units of the wing ciiord. The derivatives are evalu- ated it any convenient elevator setting for the stick- fixed condition and at any convenient elevator tab setting for the stick-free condition. CONFIDENTIAL 28 CONFIDENTIAL NACA ACH No. L^C13 From the earlier discussion it foll©ws that oxpres' si«ns »f the form An r = dc i-o or An = P d dAC m °L are not quite correct , ..where AC,,, is the sum of the several incremental moment coefficients of the preceding paragraphs multiplied by the number of jet units, Ct_ is the power-off lift coefficient, and Ct is the power-on liJ.'t coefficient. Since C r - Cr is small, however, either of the two equations 13 a good first approximation. The exact neutral-point shift is slightly dependent on the position of the power-off neutral point. Numerical Example and Discussion Specifications for a hypothetical airplane propelled by twin wing-m«unted jet motors are given in table I. Detailed computations of the effect of the jets on longitudinal stability and trim are given in tables IT and III. Any moment resulting from boundsry-layer removal that may be caused by jet action is not considered. The computations cover a range of lift coefficients and both cold and hot jets. The m^re important factors cal- culated are the mean jet-induced downwash angle over the horizontal tail; the changes in. the pitching moment with the stick fixed and with the stick free due to this down- wash, to the direct thrust moment, and to the nacelle normal force; and the corresponding shifts in the stick- fixed -and stick-free neutral points. Table II is a suggested short method of computation. The method is approximate in that the effect of jet deflection due to angle of attack is neglected, the variable distance x ,• is taken as li-.bR-j, and the effect CONFIDENTIAL HAG A ACR No, L&C13 CONFIDENTI AT, 29 of temperature Is neglected except in specifying the mass flow per second through the nacelle. Table III gives the detailed computation without these approxi- mations. The maximum influence of the variation in x.i on the jet-induced flow inclination is found to be 1 percent. The maximum influence of both x,- and incli- nation of the jet axis on the mean jet*-induced downwash is found to be 7 percent. The jet deflection does not exceed 15 percent of the distance from the jet axis' to, the horizontal tail. The close agreement between tables II and III suggests that the detailed computation of table III may be dispensed with In many cases. Comparison with Experiment The present method has been used to estimate the stick-fixed pit chihg -moment increments due to jet opera- tion for a twin-jet fighter-type airplane that has been tested in the Langley .full-scale tunnel. The unpublished experimental values are compared witn the estimated values in figure 9* The flaps-neutral curves (fig.-9( a )) show a discrepancy in trim, but good agreement in slope. The flaps-deflected curves (fig. 9("°)) show good agreement in both slope and trim up to a lift coefficient of 0.6, but above Cf - 0.6 the experimental curve diverges markedly from the rather straight estimated curve. This diver- gence is probably associated with some suppression by jet action of separation at the nacelle inlets that was indicated by tuft studies carried out during the- tests. On the whole, the agreement between the estimated p itching-moment increments due to jet operation and the experimental Increments appears to be sufficient for design purposes. A number of further comparisons with experiment will have to be made before the accuracy of the method of estimation can be established. CONCLUSIONS An analysis has been nade of the field of flew about a jet and the effect of jets on the stability and trim of jet-propelled airplanes. The following conclu- sions include an allowance for the limitations of the simplifying assumptions employed: CONFIDENTIAL 30 CONFIDENTIAL NACA ACR No. l6013 1, The jet-induced flow inclination varies very nearly inversely aa the radial distance from the .let axis hin the region between the Jet boundary and twice the radius of the jet boundary at distances greater than 3 orifice diameters downstream of the orifice. 2, The effect of jet temperature on the jet-induced flow inclination is small when the thrust coefficient is used as the criterion for similitude. 3, The deflection of the jet due to angle of attack is small for straight flight and flight with small normal acceleration. The angular deviation of the jet, however, is an appreciable fraction of the angle of attack, i^.« The downwash induced at tat. horizontal tail by wing jets at a given thrust is almost independent of t. size of the jet orifice up to a diameter about one-eighth the distance to the horizontal tail, 5. The radius of a jet varies almost linearly with axial distance near the orifice and varies approximately as the one-third oower of the axial distance very far from the orifice . 6. The equations for jet-induced flow inclination may bs applied approximate l~j to rochet jets if the thrust coefficient is multiplied P. one minus the ratio of stream velocity to jet-nozzle velocity, 7. The influence of wing ;*•: ts on longitudinal sta- bility and trim may be estimated with sufficient accuracy for desian purposes op an approximate method that neglects the effects of jet deflection, size of the jet orifice, jet-induced boundary-layer removal, and most of the effects of jet temperature, Langiev Memorial Aeronautical Laboratory Nat onal Advisory Commit tee for Aeronautics Langley Field, Va, CONFIDENTIAL IT AC A ACR No. L^Clj CONFIDENTIAL 31 APPENDIX A COMPARISON MTH THE ANALYSIS OF SQDTRE AND TROUNC3R The flow-inclination charts of Squire and Trouncer (reference 1) differ from figure 5 of the present paper by amounts from to 11 percent when the flow is .meas- ured at the jet boundary 6 or more orifice diameters from the orifice. Figure 5 i- s believed to be more nearly correct within its region of application because of the use of an experimental rather than an idealised velocity distribution in the jet, although the treatment is less rigorous otherwise. A' detailed comparison of the analyses follows. Squire and Trouncer present a relatively rigorous treatment by the momentum- transfer theory of the develop- ment of a round jet in a general stream moving parallel to the jet axis. Full consideration is given to the region, approximately 8 orifice diameters in length, in which transition occurs from the uniform velocity at the jet orifice to the characteristic velocity distribution of the fully developed turbulent jet. The present analysis ignores the transition region entirely. Use is made of Squire and Trouncer' s analysis to correct the value of a constant in an approximate equation for the spreading of the jet. (See appendix S. ) The equation is derived from the qualitative considerations of reference 2. In the analysis of reference 1 the values of axial velocity induced by the jet in the external flow are first neglected in determining the stream function, as has been done in the present analysis. Squire and Trouncer, nowever, use the result to determine a system of sinks along the jet axis from which the stream func- tion (or, more accurately, its x-derivative ) is reevalu- ated. This procedure effectively restores the missing axial-velocity increments. Examination of the computed flow-inclination charts of reference 1 in conjunction with the values of — Z P-X in tables II to IV therein c 2 aUl * x shows that this refinement is unnecessary within twice the jet radius at points 8 or more orifice diameters downstream of the orifice. This range should cover the CONFIDENTIAL 32 CONFIDENTHf, ;- .. . - . - usual relative positions of the jet and the horizontal tail for wing -mounted jet motors. Determination of Jet -Spreading Parameter k only questionable point in the analysis of Squire and Trouncer is the use of a cosine-velocity dis- tribution for reasons of mathematical simplicity, rather an the experimental velocity distribution that was used in the present analysis. The general development of the jet (from considerations of mass flew) is affected only slightly by a moderate change in the velocity pro- file. (See reference I.) The determination of the liar spreading of the boundary r£ the jet by means of the experimental data of reference 1, however, is quite sensitive to the shape of the profile. The determination may be made as follows. A jet issuing from a small ori- fice in still air is known to spread conically. According to reference 1 the cone en which the velocity is equal to one -half the velocity on the jet axis at the same section has a semiangle of 5°« With Squire and Trouncer's cosine-velocity profile therefore 0.5R = x tan 5 C r = 0.175X or k = 0.175 (Al) With the experimental velocity profile of reference J used herein (fig. 2), O.365R = x tan 5° R = 0.2J+0x k = O.24O (A2) This valiie is 37 percent more than the value for the cosine profile. CONFIDENTIAL NACA ACR Wo. LoCl3 CONFIDENTIAL 35 Effect of Velocity Profile on Flow Inclination The flow inclination about the jet is in turn dependent on the spreading of the jet. If r> is expressed ~-2 in terras of c , equation (7) is of the form x tlx. l 2 x Function of 3T ' k 4 / 5 respectively, where riDESTIAL 11ACA ACR Fo, L6C13 CONFIDENTIAL 57 nl '0 u r dr U R R = O.1I186 2 ( J 1 3) O.OR78 J P 1 /u\ 2 r dr ' R ~R~ 0.0661 b 2 = 2(j 2 1 T \ 2 J iJ = O.OI4.76 J, = J- c r 2 u r ,J H — = o.oqiL p 1 L J 0^ e.3* = 0.06 R R b* = J. = 0.0911. b ii = 2J 2 " 5" J i = °-°933 5 8 The numerical values apply to the cosine-velocity dis- tribution adopted by Squire and Trounce r. (The symbol c in the equation for b r is used by Squire and Trpuncer and is distinct from the wing chord c of the present report.) Elimination of du/dx between equations ( B3 ) and (b'.|) gives dR dx -b s u(i 1 v + 2i 2 u) ~( 2I 1 V + 2 l2 U )( b 3 V+ b k U ) + f 1 ! 7 + 2l 2 U )( b i V + b 2 U ) (35) If this equation is put into the form of equation (32), the constants therein are TIC k = 8 bi. - br CONFIDENTIAL 58 CONFIDENTIAL HAD A ACF. No. l£cVj f = ^ - b i . h(*> + b ^ v = f /T& = u ^ - + + 'O JLJU (b6) For the values of the constants that apply to the cosine- velocity profile of Squire and Trouncer (given under equation (Bl).)), an average value for f is 2.6. With this value the approximate equation (32.) agrees with the more exact equation (B5) within 1 percent over the range from — = 1 to — - co. V V For the experimental velocity profile that was used herein (fig. 2) the constants are I = 0.0991 b 1 = O.OI51U I 2 = 0.0)4895 b 2 = O.OI76I}. ,T L - 0.0701 b^ = 0.0701 j 2 = o.ol;59 bi = 0.0527 Insertion of these values in equation (B6) gives an average value of 5-3 ^or f. With this value the approxi- mate equation {B2.) agrees with the more exact equa- tion (B5) within 2 percent over the range from ^ = 1 to iL = o. The value f = 3-3 has been used In the V computations of the present paper. CONFIDENTIAL NACA ACR Ho. LwClp CONFIDENTIAL 59 APPENJIX C DEFLECTION OF IDEAL J^T INCLINED TO STREAM Let a p be the inclination of the thrust axis to the general"!" low, and let 9 be the inclination of the jet center line at a distance x from the fictitious point origin of the jet. It is required to determine 1 - jet the fractional change in the direction 'of the a. The momentum relations for the components of the thrust parallel to and perpendicular to the stream are, for small values of a e , r pi a(V + u)u 2nr dr JO SrrR^pv 2 i,« + f^ 2 i,i V 1 vyy L 2 (Cl) PR PR cuT = P ! a(V + u) 2 6 2rrr dr + P I Vt3 2irr dr Jo ^0 The first integral of a e T is the cross-wind momentum of the mass flow in the jet; the second integral is the cross-wind momentum of the disturbed outside air com- puted from the additional apparent mass of the jet. The expression reduces to a e T 8 2rrR 2 py 2 2 - v + 2 |v + ($' L 2 .1 J (C2) Solving equations (01) and ( C2 ) simultaneously gives CONFIDENTIAL kO CONFIDENTIAL NACA ACH No. L'CIJ 1 - ±- ? - T I + ^- T In accordance with the main text put v -_Ji 1-1- 1? TT X 2 2 ^2 u 1 + — ^KT H I II V - " Z } V 5 ^2 U 1 + £-KT ii (Strictly speaking, the values of K should be different in each expression.) Then 2 + 2 + ( 2ll + 6KT ^V + h Q< 1 - f : u e CONFIDENTIAL NAG A AC A No. L0CI3 CONFIDENTIAL l[.l REFERENCES 1. Squire, H, B,, and frouncer, J,: Round Jets in a General Stream. R. & M. No. 19?^, British A.R.C., 2. Prandtl, L. : The Mechanics of Viscous Fluids, Spread of Turbulence. Vol. Ill of Aerodynamic Theory, div. G, sec. 25, S», P. Durand, ed,, Julius .Springer (Berlin), 1935, pp. 162-175, 3. Fluid Motion Panel of the Aeronautical Research Committee and Others: Modern Development's in Fluid Dynamics . Vol. II, oh. XIII, sec. 255* S. Goldstein, ed., The Clarendon Press (Oxford), 1933, p. 596, fig. 236. I., Corrsin^ Stanley: Investigation of Flow In an Axially Symmetrical Heated Jet of Air. NACA AC A No. 3L22, 191+3 • 5. Ribner, Herbert S.: Notes on the Propeller and Slip- stream in Relation to Stability. NACA ARR No. Ii+I12a, 19i|li.. CONFIDENTIAL 1*2 CONFIDENTIAL &A AOR No . L6C13 TABLE I SPECIFICATIONS FOE NUMERICAL EXAMPLE Twin wing jets 3, square feet 275 R., foot O.H r, feet 3 x - x- (to hinge line of horizontal tail), feet ... 3 d, feet 3 b t , feet 12 l/c 0.5 z/c 0.1 dCa/dit - -°3° dC m /do e -0.015 °ha/ C h 5 0-5 T c ' per jet 0.l6c L J ?eam temperature t-, °F . lh-1 Stream temperature ?, °F abs 53° Jet temperature minus stream temperature t-, °F . l^t-30 NATIONAL- ADVISORY COMMITTEE FOR AERONAUTICS CONFIDENTIAL NACA ACR No. L6C13 43 CONFIDENTIAL table h SHORT APPROXIMATE COMPUTATIONS FOR NUMERICAL EXAMPLE [jet deflection neglected and x. taken as 1+.6R,; jet temperature neglected except In atep 1}J Jet Cold Cold Cold Cold Remarks Step ( assumed) ^\Plap deflec- ^vtlon, deg 1+5 1+5 Olven Parameter\^ 1 C L 0.5 1.0 1.0 2.0 Given 2 V .08 .16 .16 •32 Olven 3 ST e >/x 2 • 227 • U55 .1+55 .909 3 -, — - x atep 2 X 2 1+ h .222 .1+20 .1+20 .750 Prom fig. 5» by use of step 3 (curve for t = 0) 5 1, deg • 75 1.38 1.38 2.1+6 Jet-Induced downwash angle at section of horizontal tall vertically above Jet (step 1+ x i"\ 6 r/b t .25 .25 .25 •25 r and b c given In table I 7 2d/b t • 5 • 5 •5 •5 d given In table I 8 TA .?6 .526 .526 .526 From fig. 8 by use of steps 6 and 7 9 « 2 » deg • 77 1.1+5 1.1+5 2.59 Mean Jet-Induced downwash angle over horizontal tall for two Jets (2 x step 5 x step 8) 10 iCm ) x a top 9 1} • 5U • Jl • }1 .21* • 21* p Average of curve of 1 - — between values of kx/ytaT c ' given by atepa 7 *nd 12, respec- Q tlvely, mlnue value of 1 - — for step 7 ll* o, dag 5-7 10.3 -} 1}.0 1}.0 Olven 15 C w . dog 2-5 5-1 10.0 15-1 15.1 Wing downwaah, estimated 16 o.. d.g 1.2 5-2 -10.} -2.1 -2.1 Averege Inclination of flow relative to the Initial direction of the Jet uli (atap 11* - atap 1; ) 17 »r, ft -.06 -.25 .1*5 .07 .07 Jet deflection at horizontal tall due to Inclina- tion to the atreaai -(x - xj) k atap 1} .tap 16"] L 57.} J 18 r, ft 2.91* 2.77 }.1*5 }.07 5.07 Dimension r (fig. 7) corrected for Jet deflec- tion (}.00 + atap 17) 19 f» dag .71* 1.1*9 1.20 2.1*0 2.27 Jet-Induced flow Inclination at point of horl- iontal tall vertically above Jet /.tap 9 * •*'? 1X ) V atap 18/ 20 r/b t .21*5 • 2J1 .288 .256 .256 Step l8/b t 21 2d/b t • 5 • 5 • 5 .5 • 5 d and b t given In table I 22 ?A .522 .502 • 570 • 555 • 5}} From fig. 8 with the use of atepa 20 and 21 2} 7 2 » *»8 .77 1-55 1.97 2.56 2.1,2 Mean Jet-Induced downwaah over horizontal tall, for two Jeta (2 x atap 19 x atap 22) ■ 77 1.1*5 1.1*5 2.59 Approximate value from table II Tor comparlaon Pros 1 this | joint th » procedure of table II la followed. CONFIDENTIAL RATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NACA ACR No. L6C13 Fig. I- Z u o u. z o u I iS <5 o 3 > p ° 2 § < z 2 5 o u Id o u. Z O U 0> G *> 14 lM 1) C •H B .-H • m tt) rH n O nl -Q ,0 e >> H «-> *J 01 -H ••-3 *J C HI Ol -C ►3 C >j 01 a 0> o> UJ k< in 3 +^ nl ■ Li □ 0) u P. 01 1= N 0> *J aj *j ■a 3 c ^H a) a m >> A ♦j al -* O m o -H H ID a> > -H —I 0) H c O ■1 b 0> a E 01 0) u 3 bo Fig. 2 NACA ACR No. L6C13 or /.O ><3 JL U 4- .Z O O NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS C( )NFID ENTI AL \ . N \ \ \ \ , V \ > \\ \ b^ \ c ONFI DENT , IAL . ^ r 2. V .2, .4 _ZL .(5 R .3 AO Figure 2.- Velocity and temperature profiles for a round jet in still air. (a) Experimental velocity profile adopted for the present report. Replotted from reference 3 with r/R taken as the value therein divided' by 2.74. (b) Experimental velocity profile of figure 20 of refer- u ence 4 fitted to curve (a) at TT 0. 5, (c) Theoretical cosine velocity profile of reference 1, (d) Experimental temperature profile of figure 20 of reference 4 to same r/R scale as curve (b). NACA ACR No. L6C13 Fig. 3 \ \ \ \ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS \ A V \ \ \ \\ \ \ \ _l \ \ \ < ■ 1- - \ > \ -1 ' <- - Z- a u. z Q UJ ■ Q - \ \ \ \ z o o L \ \ \ \ \ \ '-N \ \ \ \ O o o A \\ \ A \\ v\ \\ v v U 1 1 1 1 1 ^ \ 1 \ \ v s \ vO Q ^ VO *o N cvi Q> Si o a o > •P a; ■•-* a; xi E cd ID >5 o o (1) > E i t-. D bo NACA ACR No. L6C13 Fig. 5 h z UJ Q O v" J'J ^o >■ i- a 2 ° z 2 o > a* 5< -■§ < "- Z UJ 2 " P t- < X < I — I Z UJ q l. z o ^ s * II 71 4 \t - ' - -< h 2 u C u 2 C t 1. \ 444- '» T] * 444- T -'. • ; 4 4-h 1 \D 1 \ / 44 1 is > \ / _\V T i! \ / ,' ' ^ 1 i | r-tti i- - Jn f 1 1 f A ttti 1 "tfj 1 1 r] \ i titi 1 M 1 1 ] ii r r[ r J4 1 f \ \ it-L. _-. 1 | y - . 1 1 ftttt 111 I i i \ 1 Hi f / / \ 1 444- 1 / \ I 111 r l-«w / \ I H 14 P ; r -- ti 1 / \ Hi J I / ; 1 1 / i 1 | 1 1 \ ~i E-U-4- / / \ p \ rtti / / \ / \ \ 1 ttti j / \ \ \ titer T ' i f 1 tiit" 4- -^ J ' / / \ 4-ittt- 1 1 / / \ \ -iut 1 , \ 1 1 1 * \ 1 - '.' C\ * r s nv 1 M "3 4 . *J.; * ° J ! , , 1 ^ J\ LJ 1 1 1 < 2 bJ O L. z o u NACA ACR No. L6C13 Fig. 7 II i ii i i i 1 i 1 i il ii i\ -1 < H Z UJ 9 u. z o o n .<_ h- UJ Q iZ- z o - o- \ \ A a \\ \\ v \ N it n <0^ \ ^ \ \ \ \ k \x s, \ k \ ^ ^>^ ^ CK °0 N K) ^ \ *) <\ O Q> <0 ^O ^ ^ ^ U > H a 2 ° 1 £ o l< ■or ^2 ^ 8 o CO <& o ■p o ho C cfl o •4-> 0) •a u 3 bo C < I a> u 3 bo Fig. 8 NACA ACR No. L6C13 J_ ^ 11 \\ , 4> 1 w r 11 1 \ \ ^^ \ "* \ \ ' xC^S^v >» \ - \ \ s \ \ \\ h ^ jO ^ U > P a 2 z 2 o Z UJ 2£ o o 2-<0 * ^o ^ *3 Cu O ■"-J 3 •H CD ■O J3 d «3 kl >t id fl ■^ Xi * 0) V ■d 3 cu X) u c 3 •H T3 C ■H IHJ \U J= n d C s o c •H s o •H i-i u > fl o bo fe. CQ ^> M ^ NACA ACR No. L6C13 Fig. 9a <3 CO u > »- a 2 ° z £ o > a I< -> 9 < "- CD D T3 <-> • C rH 0) 0) -H > a D •H H o to o i-. iJ E a; CD O 5 CD E 1 o ft -^ to rH C -a CD ■H CD c rC .j c O cd 3 ■^ u ♦J — 1 1 & * X3 CD c > ^H c 1-J 3 •H 1 c^ Tj CD ■o CD +» a> CD Li r-j CD Q •a 0) T3 H U c o o I en 0) u 3 bo UNIVERSITY OF FLORIDA UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT GAINESVILLE, FL 32611-7011 USA