mck^^^n\^l 1 Co.'.;'i'^' •-• '''^i ate /h f^ ^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1214 ' \ JET DIFFUSION IN PROXIMITY OF A WALL By D. Kiichemann Translation of ZWB Untersuchungen und Mitteilungen Nr. 3057, December 1943 Washington May 1949 I2^HH ifO N-ATIONAL ADVISORY COMMITTTHE FOE AERONAUTICS TECHi^rICAI■ MEMOPAffiDUM NO. 121^1 JET DI&TBTJBION IN PROXBIETY OF A W.Jlt 3y D. IZucIiomarm SIM'IARY Wlien au:;iliary ^ct ensines are instelled on airframes, as ■(•rell aa in somo neir designs, the jet ensinop. ai.-e mounted in aiich a way that the jet strsaci exhausts in close proximity to the fuselage. This report deals with the behavior of the jet in close proximity to a two-dimensional surface. The ezpei-iments were made to find out vrhether the axially symmotric stroem tends to approach the flat Burfaco. This report is the last of a seiles of fovr partial teat reports of the G-Sttingen progr-Mi for the installation of jet engines, dated Octoher 12, 19^+3 • -his report is the complement of the report on intake in close proximity to a wcJ-l. I. IKIRODUCTIon Considerahle confusion sti!)JL attends the installEtion of turbojet engines as regards the dischar-ging jot, especiallj^ when it comes near other parts of the airplane and interference phenomena a:!:'e possible. If the engine is mounted near to the fuselage, there is apprehension that the jot will adhex^e to it with consequent undesirable heating and possibly also drag increabo. The purpose of the present report is to treat these problems in somewhat gr-eater detail . The feared jet processes are caused by the nearr.esE of the wall. In order to secure more general and fijndamental data, all special wall forms were disregarded and the jet was; meaaiur'ed in the proximity of a flat well. This preclu.de d the p?:'ocesses which depend on the particular pressure distribution at the wall and in the surrounding space. Furtheimore, the work was done on a cold jet, principally on account of experimental facility. The extent to which fundamental phenomena were suppressed by it must be left •■■"Stralilausbreitvng in Wandnahe,'" Zentrale fiir WigsenBchaftlichea Berichtsvesisen der Luftfahjrtforschung des Creneralluftzeugmeisters (ZWB) BerllivAdlershof, Unterguchungpn i;nd MitteilTaif.-en Nr. 3O57, Dncember 3I, 19^3. MCA TM Ko. 1214 to future experiments.^ As vai^iable parameters there remain the velocity of the Jet, for which as criterion tlie iiean velocity v^ in the ezit of the engine model is chosen, and the outer flov velocity Vq| and indeed it suggests Itself to once consider the difference v^^ - Vq and then the quotient v-ji/Vq as significant. Another parameter is the distance a of the exit cone from the wall (that ia, the distance of the point of exit closest to the wall from the wall) , and lastly the design of the fairing "between engine and wall will also play a part. In every case, the thi'ee- dimenslonal variation of the jet do-tmstream from the exit must he measured. II. COKVERGION TO OTHER OPEFiATIM> COUDTl'IOHS In view of the multiplicity of potential variations, it is desirahle to establish simplifying connections. For practical pm-poses it wo^ald he more advantag:eous to he ahle to use easily made static tests (without stream flow) and to compute all phuses \Tlth stream flow from it. Such a process is described in the following: It ie assumed that the general state of I'low (v) results, in first approximation, from the superposition ox' the stream flow (vq) with the Jet flow (v«): V = v» + v^ (1) This implies that the jot diffusion is to depend only on the difference of the velocity in the jet (vp) and outside of the Jet (vq), so that the velocity v in the form (v - Vo)/(vji^ - Vq) for fixed pai^ticles is independent of the operating condition. A cei-tain difficulty is involved in the finding of the location of these particles, that is, to pass from the velocity transforma- tion (l) to the related transfonaation of the cooixlinates . A rectangular system of hody- axes (x, y, z) is used with x in the flcfw direction and the time coordinate t, with x = (plane of exit) for t = 0. The space coordinates of the paa-ticles are functions of the time. Thus for equal time intervals t we get a relation between the coordinates x,y,2 of the particle in the general flow (v,, ^ o) and the coordinates x*,ySz' o^ the flow without streen flow (vg = O) . iThe prohlens of model similarity end reproduction of the hot-Jet in wind-tunnel tests ere discussed in reference 1. IIACA TM No. 121'^ It fvirther is assumed that trensverBe floT/s can be diarecarded, hence that the velocity has the direction of the x -axis, so that y = y* and z = z' . This leaves the connection "between x and t and x' and t to ho determined. V = — ; v» = — (&) dt dt the velocity relr-ition (l) then reads dx dx* ,„x — = + Vo C3J dt dt which, integrated, gives x(t) = x'(t) + v^t {k) In this equation t is yet to 'he eliminated and to he replaced with the aid of (2) hy neans of the velocity v' = v*(x',y, z), which is accessihle to measurement: ^=l\ v«(xSy,.) ^5) In this manner the desired transformation of the cooi-dinates ,ix'/i -.A±ll^_.. (6) v»(2:»,y,z) 'A' follovE from (k) , made dimensionlcsv^ with the diameter d of the exit nozzle and 'the average velocity v^* = v/^ - Vq in the exit nozzle. This transformation states that planes normsJ.- to the x-axis are not maintained, hut that according to the velocity distri • hution v«(x',y,z) the x displacement for the faster particles is less than for the slover ones. In practice '^"'/''"^a' ■'•''^■^^-l'^ ^^ measured, the integration carried out,' and the x correspondin/j to x' caluclated hy (6) . There the ai-ea cf small velocity would cause difficjlties, especially for the points x » 0, ^J j"^ + z'- > d/2, since v' vith x -^O must approach zei'o in a certain way in order that the integi-al ma^'- exist. Moreover, the mmericsJ. eveluation in this ai'ea requires extreme accuracy of measurement. MCA TM ITo. 12lij- A detailed check of the pi-acticatility of these assumptions " was outside the ecope of the present report. A thorougli discussion mth consideration of the transverse motions must "be the object of a special investigation. For the present, these assiAmptions were, after several other simplifications, simply used as "basis for the test program. Since the potential coi'e wl-th its high dynamic pressures and presuma-hly high temperatures is of particular interest in the application, v* = constant was taken equal to v/J = "^A " '"'o' Therefore, ti ^ ^A/v-o - 1 This assumption is, of course, justified only for the region around the Jet axis up to the dissolution of the potentlsJ. core; however, in this ?reglon alone is the assumption of velocity parallel to the X-axis satisfied. In view of the mixing motion, it would pliysically be more logical if a mean velocity mthin the actual mixing zone were regai^dsd as characteristic. The transformation (7)' has the advantage of always permitting measurement in planes where X = constant. III. EXPERIMSriTAX PEOORAM In till tests, the difference v_^ - Vq vras kept constant (= 33 m/s) The first operating condition with zero stream velocity was: State I: Vg = 0; v^v = 33 m/s; v^/vq ="; x = x», the second, with comparatively low stream velocity: State II: Vq = 11 m/sj v_^ = kh m/sj v^/v^ = ),'-j x = I.33 x' and the third, with greater stream velocity: State III: Vq = 33 m/s; v;., = 66 m/s; vpjvo -= 2; x = ?x' . It was found diaring the measurements that the states I and III were inmost cases sufficient for explaining the principal processes. The wall distances themselves were limited to a few values, to a = d (large dj.stance) and a = d/2( small distance); for comparative purposes, data with the jot motor were also measured without a wrO-l (a = CO ) . MCA TM No. 12114- The lining 'heti^en engine and veJ.! was l:ept eapecially slender in several caaes, since Kimzo'a tei^ts had shown the impoi'tance of adeq.riate ventilation "between jet and vail. As contrast to this "good" fa.iiing, a partictilarlj "poor" fairiiif- was used (figi- l) • These cowlings all terrd-nated with the exit plan.e of the po-rer xmit. In one instance, the good version wae shifted haclnvaj.-d h.y Id and cylindrically cut out towai-d the jet, creating a type of "t-annel." A model engine with installed blower wtis used. The measvcements, made in tuimel No. 2, included total pi^essui-e and etatic pi'essi.u.e in y and z direction "ohrougli the jet axit; at various distances X train the exit nozzle. Vr. EE,5TILTS It is foimd that ov:imum of the velocity distribution travels perceptibly away from the wall. The wal-:o flow of the fairing is scarcely noticeable and the boundary layer at the wall also appears to experience no substan-oial vat-iation by the flov;-. On the poor fairing (figs. 3 and 9) the conditions are different. Wliile in state I the jet ntill seems to move a little nearer to the wall than with the good fairing, with stream flow it ceases to move away from the wall and moves into the dead-air region introduced by the fairing. To pet some idea of the form of the jet in the vfffiou^ fairings, figure 10 represents the lines of eQ.ual velocity in a section normal to the flow, as well ae wac possible according to the meaeurements. The good as well as the bad fairir^ show- a form no longer axially sjTametrical, whic-h, however, is flattened out at the wail side in the first case and ovally puJ.led tovrard the wall in the other. The movement of the velocity maxim'om in .iiffei-ent directions is plainly visible. According to this, it might be suspected that the good ventilation of the space between en£;lnc and wall with the good fairing forms a definite air cushion which pushes the jet ai-.^ay from the wall. This concept is supported by the conditions in a horizontal section tlirough the jet xn figui-'c 11. MCA TM Nc. 12lil- However, in spite of thei30 dj-ssimilai^ities, the effects are cciiip8j.'atively small. In the projc.imity of a flat wa].l, the possibility/ of a ventilation l"roni the sides is so great as to preclude the existence of jet adherence even vith an extremely poor fairing. 3. Short-Well Distance One noteworthy fact ir; that the aforementioned processes are repeated with the short wall distance, hence ai^e not limited to the comparativelj'" large distance frcaE the wail. As with tlie pood fairing, there is a slirht movement towai'd the wall in the absence of stream flow ana a movement awaj'' from iz with increasing stream flair (fige. 12 end I3) . The fairing with tiinnel extending ba'jkwai-d beyond the afteibod;}'- of the engine \mit is of practical interest for the reason that in many cases it Is the only way to obtain a sufficiently elongated fona. This fairing likewise exhibits no markedly unfavorable behavior. The jet naturall;^ adheres in this case at the tunnel (figB. Ik and i;;) • This tunnel surface was therefore to have no projected area in the flight direction for reasons of i'esistance. Since, however, the tunnel must be adapted to the form of the jet, and this is not known at once for the different cnginoi uirits, difficulties may arise, so that, if at all possible, such a tunnel fairing should be avoided. It is perhaps not immediately comprehensible whj'" in these measui'ements only these fevr generalized types of fairings were investigated axxd the form of the fairing' not fui'ther varied, to est-ablish, for example, which form could be still designated as good. According to the cited results, however, the solution of such a problem does not appear possible at once, since it was seen that geometrical conditions such as the form 01 the afterbody of the engine iinit or the past history of the outer flow ha/e some effect on the phenomena, so that a separation of these pi'oblr.ms from the others for the installation of given conditions seems hardly correct, and the Reynolds number and the tempe.rutui"'e conditions woxild then also have to be teicen in'iio accoimt: Hence the limitation to basic experiments. That the conditions in an installed jet -propulsion urdt are similar in the fundamental phenomena is shotm by the wind-tunnel tests on an auxiliary ti;arbojet niit model mo-jnt^-d below the fuselage on the Heinkel J19 (reforence 2) . So, in order to be absolutely' certain about the jet diffusion for a spocific design, a test on the total model WACA TM Wo. 1214 is probably unavoidable, and judged by past experiences, a water-channel test is best suited for this purpose. Translation by J. Vanier National Advisory Committee for Aeronautics PEFERETJCES 1. Brennecke, H. : Messungen an dem Modell einer Strahlantriebsgondel. Forschungsbericht Nr. 17^3 , 19'<-3' 2. Bauerle, H. and Hildenbrand, H. : Widerstands und Schubanderun3 bei nachtraglichen Anbavi eines TL—Triebwerkes unter den Eujnpf der He i:^19. Untersuchungen und Mitteilungen Wr. 30i)-l, 19i4-3. 3. Kuchemann, D. and Weber, J.: Uber die Stroaiung an ringformigen Verkleidungen. XI. Mitteilung: Wandnahe Einlaufe. Untersuchungen und Mitteilungen Nr. 3O5I, 19^+3. NACA TM No. 12l4 / /' \ v. \ \ y ^ 7 T S3 1 \ /"■ ji^gr/ CD U 10 NACA TM No. 12lh ■» ^ J?$N I 1 ^ i ^ Kj ^ '^ \ } r / ■ 1 i-i c 1^ J y j J iL (0 ^ 1 / f U ^--^ 1 / n / J i 1 / /x ^ • / r 1 f ^i > o^ y^o 1 / / / W) IT" ■:^ H (•-" i^ c lo / / «>c ^ rf^ ^ r X > f a / o /, f J / / / / / 1 r ;f >° i / f / » < 1 1 5 / / o <4 A » > i , ^ f 4 J, J >>: 19 '*' 1 i 1 f% V U II II 11 !!, ^ ^ ^ ^ ^ ^ X H K ^ K -WO) o 3 « n -< o a n -w) £ £ „ r \ \ N T \ ^. «a \ \ 1 ^ o o ' >*N >.N >»N >>N >>N \ \ 1 \ 1 >* X + o« <•< as OO 1 1 1 \\ \ 1 \^ i \ eg CD • 1-4 NACA TM No. 12l4 11 12 NACA TM No. 1211^ 1 ' 1 1 C ^ 1 1 / ^^ 1 1 / i ^^ <:^ ^ / / / ' ^ / / / J 1 1 / 1 / / / /i V\ I— / / / / y ^ / ,/ V ^^. 1 ■| V 1 /y ) \ ^*^x. 4 . i-< C *J -H as ■C *J a^ m o a> (. u o o 1 1 03 ^ 1 00 Zi f>i ^ ■^4. \ \ \ ' o ^ « lO ^ < < ^ 1 ^ " 1 S >•* X o < ° \ > V »< \ \ \\ \ \ v c > \ 1 hiD •.-I NACA TM No. 12l4 13 to ci < * 5 ^ \ i •s \ 9 ^' x ^^ ^ ^^ C> c < * i ^ r 1 ^ X »■ 5 » * i *• n «M V 3 0) £ *-> -^ (0 » (A t- Cm > - « ,, - -c £ 5 ^ x: ~ c * c c tfl -•-) t. 3 « ^ p. t- - -u VI •H b. > \ ^4^ II •a S ^^^ 5~" 1 c» c < i 1 i > > i V > II 4" > < « 1 C3 CO II II N 1 >° > ^^ ' m^ 7' — > Q < * >• « > >' < * > .0 i 0) 5 1 »■ "t ■ «M ^ V. 1 K s^ ^^ — -t- .^___ >* \ < ^ < b s C > •i a) ■ LTD CD •r-H Ik NACA TM No. 12l4 ;3 + > s. ^> 1 / i i f H _ /^ ■ ■ ^ s 4 t •♦■ ^ ^ V ^m / 1 1 1 1 • J I • I-"* 1 « t jjl/ i ^ H 1 • 1 ^ 1 \. 1 1 1 rf ^m y: . / *.>S ^ .» ^ y »o J- ^ ^^ ''^ T o' n • 1 >< y / y m / !:? 3 o 3 I f > :^ J ? r * "4 1 4> <> 1 . 9 i < i f i i o 1 1 • 1 1 1 \ 1 \ flH \ \ a > N i\ ^ ,. ^ s=4 h^ ^ ( \ "0 •M r^Q. "^' • \ u^ ? *" ^ > A V k \ 1 » 3' -^ o ,1 * » r i;* ., „ . , <0 II \t u\ \ \ 1 £ M ^ ^ ^ ^ H M H M 1 < H * V o „ « « g 13 • -H ^* P > ^ 1 m m \ IH _ 1 1 1 1 1 1 1 1 A O >,N >.N >.N XN II 1 1 \^ v° 1\ > 5 » + o» << OB oe \ \ \ 1 1 \ 1 ^^ J|^ >w CD 0) •rH NACA TM No. 12lU 15 ■t S ; ♦ ^^S ^ / H ^ : ^ r o'.. ■^ " fcv N^^vs ^ / f f I Si' 1 1 ; / V y V 1^-1 > 1. o ; / ^/ ''^, r \ * 't.. 1 : /. S ^^^J ^ ^tS -^^^ -^ ^.^ ^ ^ ^ i ^■^ \r *-* f .^ 9 7 N \ *-<; ^ r*^^ / / « > A 7 » / / / / t? s? 1 / /J f 1 / / i » 1 1 L^*" 1 \ fj 0( i o < r 1 > O N 1 * ■ s C 1 » >' >^, n ^^ \; V V X 1 \ 1^ \ -s ^ A- \ » > V 1 1 *■ ^ 5^^ ^ K * ^< X L \ N^ o' J— s s 1^ V \ X ^^— - \ V \ pi ^^^ \ \ ^ > v 5 & o ■H n u II ^ \ \\ A .- > > l„t > ^ \ .41 1? «. M 5 c •0 1-1 © o ■ \ \ ? 3- " 1 >" i — -3 > . i . - ^\ v 1 1 • V > 1 1 1 s > 5 \ \, i - -1 ^ %^ ^ 0) ha •rH 16 NACA TM No. 121 1^ 4 • \ i <^ ^ ifj / M W^ N. "^,-^ \l I 1 ] B f / 4 ^ ^ ffl \ ^ o S +. "■ K J ^ V \ ^ i f / >> 4) sN^ ^ t^ > f 0) 1 -K 1 _ » 1 «f ^ /// if y tA 1 ' 1 ■ >. k w * J > 1 tS 1 J. / / Ia 1 ■/ n> / k r" Q> « * y /jOl L • , 1 */" iJ It > / U 3 o r e > Tr- 1 r 1 i 1 i, C i 2 a 1 I < » J; »■ r 1 O 11 c T \ 1 1 ' ' ^ k 4 V i\ 1 4 • \ ^v- A I I 1 N ■( V ^^? k V ^- II c w» ^^ ^an ^\ ^*« •\ 1 l-H 5* ^ b. li \ 3 \ !> V « s ■H *« o * a •n II II II II <0 II \, i \ \ bad fa /s; X x/d x/d x/d X \ \ \ \ o 1 ■c " O o II ♦J n *) ^ X \ 4 1 en II u h \ \ J= II >.N \\ I \ l" £ - o X + <■< am ( D • V\ \ p^ f^ / I / Vi ^ 3 W s. ^ \ ^ > / r 7 om 1 v 1 / jt X 4 ? N «f^ lcb= r *" J^ ^^* " )* i — — o-= >i ^ y ^ /* / >> > fl * y ^^ ^ i / y r S? - o I. 1^1 / r 1 1 J /^ ^ I - *~ > i' 1 m 1 » < ^ c * 1 t 4 ^ > [&I Piv'* ll \ ;\ w \ ^ \ 1 ^^ ^\ '^, X^ \ \ w »w ""^ ^ ^ •^ \ ;^ ^ ^ 3^ ^ N e§ N ^ >T^ o ID-W 3 o o ■ BJ= » 1 ■~o- --^ X3 ^ S \ V '^ ^ •^ w ] 1 1 C K OO ■H « O CM OO "H ^ II U II H M V s \^ X 1 ^ 1 V, \^ 1 1 4h M -O t3 ■« 13 \ \ ^ \ ■O .. K M M K a) « \ \ \ ^ A <■ • N V — a \ B •0 M o c woo - > \. \ \ V^ \ \ ■ -H \ V ^ 1? ) w ., « «l \ II II II II 1 > ^ J / >>N >>M >>N >«N 2 „ x+ o» <■< a* t \ c ^ \ \ ^ — < _ \ ►^ A ^ \ V CD CD •i-i 18 HACA TM No. 1211^ Lines of equal velocity In jet (y» z - plane normal to approved flow) With wall, a = d; good fairing State III x/d = 8 '^/V^ ' /. Exit IT /Vo =0.9 Wall With wall, a = d; bsuJ fairing State III x/d = 8 V/V s ho Exit Wall Figure 10. NACA TM No. 12li+ C> -* op <)0 "* ^ 19 -.?j 20 NACA TM No. 121^ p •o / 1 ; 1 *< « f> \ / / / 1 \ s V ^ u / < .<^ ^ in ^ V ^ 1 /[/ / / / / / ^ M (^ ^ \ / [1/ f ' / \ >K^ vvfi V \ \ ^>^ 4 \9 u 4) p. 1 ;^ I* / 77 r JJft / / 1 1 r tr / / > 1 1 1 1 r4 ^ f * 1 C i f — — K J «D V" ^' 1 «i \ V / Ik / ♦' 1 i f - > 1 * il I ;^ •~™ 4* o « 4 ?T B ' 1- < 1 O - » _ 3 O x: * _ «3 _ in I 1 V If \ 1 > I 1 1 4^ ^v V ] \ \ < < H 1 k 1 1 / \ ^^ \ ^ < u s \s ^^'> \ 3 \l 1 1 «f ■^ > Q ^ -N. A ^ c 2 X 1. I II T3 ^ ■a ^-^ ^ X X > o if < . E C ° " « 2 ti qj . .^ \ -J u d « tl -H ^ :>* 1 1 1 « o f a iH - II 11 < X X ■ K KB C C 1 II II N >,N >iN ■4 as oo V' KS \ \ ^< \ 1 A ^^ M N \ V (0 \ s>^ * \ \ \ \ f ^ CM i-H 0) f-l •rH NACA TM No. 121^4- 21 1 »<» > « 1 1 1 1 4 1 \ \ •>> ,^ ^ i i L M ^ ^ V ^ \ J •»•, V ^ ^ IJ / ^ ■ f ^ ^ / v.- J ] . 1 o // r f J 1 ' ' 1 / 'j ^ r \ \ ^ ^ ^ ^ •»^ ^ "->rJ t * ^>s ^ { c ^ ^ / / f. \ ^ * N s \ ^^ v_ X 1 * -.^4^ ^ ss. *5 ^ \ ^W. v> ^ ^ s >^ V \ ? 3 ■out wall 1 T) • 8 H « II X «« 5 ! j: 99 O 00 3 II 11 ■ ^ ^ ^ M M M C o II II II ^N >>N >>N ^ ^ ^> \ \ S m t N \ ^ \, 1 1 \ \ s > \ 1 1 > ^N J"^ 2v^ \ 1 » >» \\ CO i-H 0) .1-1 22 NACA TM No. 12lU \ / 1 // r «o \ ^ 1 ' J / 1 *c ^ , J' I / 1 1 ■* , J J i 1 ^Tn- v y }] rJ KOn^: + 1^ d s •v S. / • 1 1 1 1 ' i/i i / // / s. 1 -T ' : s / lj\l / i 5 1 " r 1 jl;; *^ Vl _*- -••^ ♦ / / >. '^^ if _ ^^ y w. r ^ /. ^ Tf 'J 11 « t x^ / / 16 f_ /< / ^ j i§ i? h f I'l I ?» ^ ir « ii 1 1 « B ■ 1 4 k o c fe, \ 5. jl * 3 k 1 1 'la 1 1 K i « II ' X i \ ^ ^ bK < \ \ 1 1 V t*^. >^ ^ \ \ lo 1 1 f 1 o «H * o II - M 1 1 1 o SS^ "* 1 ' „ 'h ^ ^ ^ "^ , -s s r ,. s » >.N >»N >»N >»f > O - . \1 ^K ^ s ^V \ \ 1 V^\l s \ V^ <^ \ * . Bo! \ 1 X f o* < < Ul 1 ■i-i HACA TM No. 121^+ 23 J 1 1 v V «n i »» \ 1 . ' «< .. ? 1 * 1 1 • 1 1 , 1/ ' ■. • > *4 ^ ^^ \ ^ M W" ^ -0 \ ^ / 'l ^ tx ^Jv^ J. _. A i /l 1 ^ I I "as^ ^ V V »» / < vl n < ( J r • % A / / ^ % 4i, 4' .'< *■ > ') ^i ^^ "^-^ 4 ► **^r ---« -*»^ / ■-^ 'II > /' r X .--- f^ ^ r^ /f •-" w I' r f, f 1 jAj £j ia ;s' TLA ' * c < 1 V ' I < h c < Jr < < 1 • r '■\ih k^ K o 1 V9i 'Via *V ^ V N ^.. > > ^ '^!^ s V '^^ 1 ^ fH g 4^ ^S ■^^ .^ 2a^ k ^ m cJ b 0) a S ^* ^ rr^ 1 If - L M H ^ ^ 5 3? •< •< M ^ ^> ^ i 1 d t 9 s, ' \- s 4 i 1 With wall) o/d - 1 1 9 - ? . . . . . . E >>N >kN >>N g ■ • ~ \