CO ^ NATIONAL ADVISORY COMMITTEE § FOR AERONAUTICS TECHNICAL MEMORANDUM 1311 CONTRIBUTIONS ^O THE THEORY OF THE SPREADING OF A FREE JET ISSUING FROM A NOZZLE By W. Szablewski Translation of "Zur Theorie der Ausbreitung eines aus einer Duse austretenden freien Strahls." Untersuchungen und Mitteilungen Nr. 8003, September 1944. Washington November 1951 UNIVERSITY OF FLORIDA DOCUMENTS DEPART^ylE^^" 120 MARSTON SCIENCE LIBRARY RO. BOX 1 1 701 1 _. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 13 11 CONTRIBUTIONS TO THE THEORY OF THE SPREADING OF A FREE JET ISSUING FROM A NOZZLE By W. Szablewski PART I.- THE FLOW FIELD IN THE CORE REGION ABSTRACT : For the flow field of a free jet leaving a nozzle of circular cross section in a mediTjm with straight -unifonii flow field, approximate formulas are presented for the calculation of the velocity distribution and the dimensions of the core region. The agreement with measured results is satisfactory. OUTLINE: I. INTRODUCTION AND SURVEY OF METHOD AND RESULTS II. CALCULATION OF THE FLOW FIELD (a) Velocity Distribution in the Core Region (b) Dimensions of the Core Region III . COMPARISON WITH MEASUREMENTS IV. SUMMARY V. REFERENCES VI. APPENDICES No. 1 Calculation of the Transverse Component No. 2 For Calculation of the Dimensions of the Core Region I. INTRODUCTION AND SURVEY OF METHOD AND RESULTS Knowledge of the flow field of a free jet leaving a nozzle is of basic importance for practical application. Investigation of such a flow field is a problem of free turbulence. In theoretical research the following specialized cases of our problem have already been treated: (a) The mixing of two plane jets, the so-called plane jet boundary. These conditions are encountered in the immediate proximity of the nozzle. ''^'"Zur Theorie der Ausbreitung eines aus einer Duse austretenden freien Strahls." Untersuchungen und Mitteilungen Nr. 8003, September 19^^. NACA TM 1311 (b) The spreading of a rotationally-symmetrical jet issuing from a point-shaped slot in a wall, the so-called rotationally-symmetrical jet spreading. This state defines the conditions at very large distance from the nozzle. In considering a free jet leaving a nozzle of circular cross sec- tion, we may subdivide the spreading procedure, according to an essential characteristic, into two different regions: (1) Region where a zone of undiminished velocity is still present (the so-called jet core). We shall call this range, which extends from the nozzle to the core end, the core region. For the immediate proximity of the nozzle the conditions of the plane Jet boundary exist. (2) The region of transition adjoining the core region which is characterized by a constant decrease of the central velocity. This region opens into the region of the rotationally symmetrical jet spreading mentioned above. So far, there exists only an investigation concerning the core region (reference l); it is limited to the case where the sixrrounding medium is in a state of rest. Method and Results In the present paper, the spreading of a jet in the core region is treated for the general case where the siirrounding mediimi has a straight uniform flow field (or, respectively, where the nozzle from which the jet issues moves at a certain velocity thro\igh the surrounding medium at rest) . The theoretical investigation is based (reference 2) on the more recent Prandtl expression for the momentum transport e(x) = Kb(x) Z^^ - ^^^ One then obtains in the rotationally symmetrical case the following equations: Continuity: 5(ru) S(rv) "§x ~5i = NACA TM 1311 Momentum transport: — Su — Su _ ( S__u 1 5u where e = Kb(x) (u]_ - -Uq) u-i = velocity at the jet core ■Uq = velocity of the siorrounding mediiom With reference to the present problem we introduce, instead of r, r - rr T\ = as independent variable. We obtain: Continuity ^6\^^ ^ ^x)(^^ -^ ^;^ ^^^^ ^-r)^ = '' NACA TM 1311 Momentum transport -/ du Su - Su e(x)/5 u bj]' Velocity distribution in the Core Region We limit our considerations at first to small disturbances of the flow fieldj that is, to relatively small differences in velocity ^1 - ^0 small quantity . The partial differential equation for the momentum transport may then be linearized Su / X /Su S u 6x I dr ;^^2 where e(x) = Kb(x) ^1 - ^0 ^1 (it should be noted that by the transformation \i is transformed into the equation (x) this equation 2— — — 5 u Bu 1 Bu = which represents a heat conduction equation.) With r| of r one obtains from the equation of momentum ■0 . instead B u Su ^2 ^ 1 X + Tl X Su X "Sx 6(x) = if .-^^0 MCA TM 1311 This equation is a linear partial differential equation of second order of parabolic type. For the plane case I — — > 00) this results in the equation with ^0 ' X \ rQ V ui / therein c = lim and is to be regarded as a function of . — =0 With the boundary conditions taken into consideration, the integration yields with ^0 y2,.c(- 1 - ^ ^1 We now obtain an approximate solution of our problem by generalizing the plane velocity distribution and setting up the following formulation: NACA TM 1311 With ^^(- 1 - ^u ui / X X iL ri ' ""oi 1 ^o/x \ fh ap(x) =±,24--—-^ -^ l^]\z d 2V \ ^1 / /iL\ Jo ^O/Vx \r ^o) 0. we obtain a function which corresponds to the exact solution for small — as well as for large positive ti , thus in boundary zones of the ^0 region of integration as well as in the interior of the region along the jet T\ = 0. If we now consider larger disturbances, the solutions obtained for small disturbances are to be regarded as a first approximation. ""i ■ ^0 For the plane case the solution for arbitrary already ^1 exists, compare Gortler (reference 3)- It is found that, p-arely with respect to shape, even the first approximation represents a very good approximation. The velocity distribution calculated by Gortler still shows an uncertainty insofar as u(ti + a), with u(ri), also represents a solution. This uncertainty here may be eliminated, because for the jet core vanishing of the transverse component v is required. There- with the initial profile of the velocity distribution for arbitrary U]_ - Uq is then unequivocally fixed. ^1 If we limit ourselves, with respect to shape, to the first approxi- mation, the initial profile is I Jo ^1 l'^ W J Jo 2 \ ui where NACA TM 1311 and •^0 For the fiirther development of the profiles starting from this initial profile the regularity found for small dist\irbances is then taken as a basis n 1/^0 \ n* _.^2 W Uq where , . , 1 / e-^* dTi* + 3 1 + T]* = C^TI - 0.36 I \ "^ I + 02 ^1 - ^ For >0, this function is transformed into the approxi- mation function constructed for small di3tu7.'hances . How far it may be considered an approximation in the region foi arbitrary disturbance is not investigated in more detail. The functions appearing in the integral a^ix) , 02(x) resu'.t from the approximation calculation for the dimensions of the core region, carried out on the basis of the momentimi theorem. Calculation of the Transverse Component /^ The transverse component v of the flow is determined from the continuity equation °r V = - -, ;:rT / I ^ + IT j( X ^ - T] ;t~ jdTi respectively, with our approximate function being substituted for u. 8 MCA TM 1311 The integration constant is determined from the requirement that at the jet core the transverse flow component vanishes. In order to avoid complication of the calculation, rectilinear coiirse of the mixing width b(x) is assumed. This assumption proves approximately correct as results from the calculation of the dimensions of the core region. Dimensions of the Core Region The dimensions of the core region (jet core and width of the mixing zone) are calculated according to a formulation of the momentum theorem (u - UQ)vr - -^ f" u(u - n^)r dr (= rrxy) r = r(Kb(x)(ui - Uq))^ indicated by Tollmien (reference k) . The occurring integrals as well as the ^ defining the shearing stress are determined approximately with the course of the velocity distribution assumed rectilinear Is^ , l'±s^)a - n) Then there result for the limiting curve d(x) of the jet core and the width b(x) of the mixing zone two ordinary differential equations of the first order which can be reduced to one equation g = f(x) f y = ^f(x) dx This Integral can be represented with the aid of elementary functions; however, for simplicity its calculation here is perfonned by graphical method. ^ appears as the only empirical constant which results by compari- son with measurements given by Tollmien (reference k) as k = 0.01^76. NACA TM 1311 Comparison vith Measurements In order to carry through a comparison hetween theory and experi- U3_ - uq ment^ a measurement for the case = 0-5 was performed with the test arrangement described in reference 5. The comparison with the theory offers satisfactory results if one takes into consideration that the effective radius of the nozzle flow referring to a rectangular velocity distribution is different from the geometrical radius. II . CALCULATION OF THE FLOW FIELD (a) Velocity Distribution in the Core Region We base the theoretical investigation on the more recent Prandtl expression for the turbulent momentum transport 6(x) = Kb(x) u^x - u^. am where K = dimensionless proportionality factor, b = measure for width of the mixing zone, and u = temporal mean value of the velocity. We have at our disposal, for calculation of the flow field, the continuity equation and the momentum equation for the main direction of motion, which read in rotationally- symmetrical rotation Continuity Momentum transport where ^(ru) _^ S(rv) ^ Q ox dr (2 5 u 1 Su ^j.2 r or e(x) = Kb(x) ^U3_ - UqJ 10 NACA TM 1311 ui = velocity of the issuing jet Uq = straight tmiform velocity of the surrounding medium We may integrate the continuity equation by introduction of a flow potential 't - St - ^^ ru = ■^— rv = - "r— or ox The momentum equation then is transformed into where e(x) .= Kb(x)l — ) if we make the velocity dimensionless by division by U]_. According to a method applied by Gortler (reference 3) we set up for ^ the expression /u-L - •Uq\ developing '^ in powers of the parameter I ). Therein ^q is the potential of an undisturbed flow In-^ - Uq\; thus ^ = rui ^ = NACA TM 1311 11 If we enter with this formulation into the differential equation, we obtain ^^0 ^ / ^l - ^ \ ^^1 ^ bx \ u-[_ / Br h\ dx Sr U;L - UqN B ^x Ui Bx 5r Sio /^l " ^' ^^1 3x \ u-j^ /' dx s^^o ^ / ui - M ^h ^ Sr2 ^1 / Sr2 1 + — r Bx \ ^1 / ^x Sr I ^1 / ^^ = 6(x)<^ ^^0 _^ ^ ^1 - ^ ^ ^^1 ^ ar3 ^1^1 ^ Bi3 S ig f^l - -UoX ^^1 u. 1 ' S: + . . . 1 + — r Bio _^ / ^i - ^ \ ^^1 _^ dr V u^ / Sr . -:^ Jj If one arranges according to powers of of differential equations for \lf, , Tifp, ^1 - ^0 ^1 one obtains a series For % Bt]_ lb\ \ h\ I^q] h\ (hi. br VSx dr/ Sx Sr \ dr / ^ 2 \Bx , ^ or ^^1 /^^o'\ . 1 ^^1 (^*o\ ^ 1 ^^1 (^^0 , dr \3x / r Sx \Sr ^ ^^ \ :.^2 i "^ r = e(x) r B3^;L ^^^1 1 ^^1 Sr3 Sr2 r Sr 12 MCA TM 1311 or, taking ^ = rui ^ = into consideration 2 Sr3 ar ,^ — 5- r = e(x) r — tt + ~ '^ dxdr \ >^3 ::^ 2 r dr etc . On the Theory of Small Dist\irbances In the following, we shall limit ourselves at first to small disturbances of the flow field; that is, relatively small differences /^l - ^ \ in velocity I small quantity). The velocity field is then defined by the flow potential '^\. Since -^ = ru, the above equation for V-[_ may be written as follows: Su / , ( Su S u r 3- = 6(x)K— + r — - dx \ar ^^2 6(x) = Kb(x)( \^ "^ j Therewith we have attained for small disturbances a linearization of the equation of motion. (it should be noted at this point that by the transformation r [1 - — our equation is transformed into 5 u Su 1 Su _ ;:^,,2 ^ u " 5x ~ NACA TM 1311 13 With reference to Reichardt's discussions (reference 6), it la of interest to point out that this equation is of the type of a heat conduction equation.) In view of the conditions existing in our problem 1 r ^^^-^^^ 1 1 1 1 11 1 1 1 1 1 ' 1 ..+ Hffimm. c ///////////// ppn> X ^~--^_^ Jet core (rQ = nozzle radius, x = distance from the nozzle in direction of the jet axis), we introduce instead of r the variable t| = coordinate transformation yields r - rr This Su _ Su ^ _ Su 1 ^ ^ OT ^ X (15) a; Su ^ _ Su Su 1Q_ ^^4=const ^^Cconst ^ ^ 5^ ^ " ^ x thus the equation Su Su r| ^^^ ^^o)(^ -^i)- ^(^) ^u 1 / s S u 1 Sf or, respectively, for t^ + — ^0 Ik NACA TM 1311 a u ^ ro T1 + _ X + T1 K^ hu X Sx €(x) = e(x) = Kb(x)^ ^^; This equation is a linear partial differential equation of the second order of jjarabolic type. The solution of this differential equation is fixed unequivocally by the initial condition that for — — > the velocity distribution •^ ro -^ of the plane jet boundary appears. We first derive (for small distiirbances) the velocity distribution of the plane jet rim. For — — >0 we obtain with the expression u(t^) the equation d^ du x = X ^0" with X - - JL_n ^1 - ^ ^i b(x) ^1 " '^ Therein c = lim and is to be regarded as a function of Ui With the boundary conditions u > ui for T\ ^ - Uq for Ti > + NACA TM 1311 15 taken into consideration, the integration yields ui \ln\^l j J Q ' ^^' 2\ U1I with Oq = f¥^) Turning now to our problem, we can expect great difficulties in con- structing the exact solution. We limit ourselves therefore to forming an approximate solution. For this purpose we generalize the plane velocity distribution (the initial profile) and set up the following expression u 1 /^ ,\ .,[ai(x)T)+a2(x')l _[^^(x)T,+a2(x3 p -j ~l^fA~l jJ d[^,(x), + a2(x)J + u. This formulation insures at the outset a reasonable shape of the approximation solution. For cr-L,a2 there immediately result, because of the initial condition, the requirements lim o-^ix) = CTq lim a2(x) = JE vO JL_vO "0 Now the following equation is valid: CTq = . = lim H"^) ^^° f¥i^^ ^ 16 NACA TM 1311 Accordingly, we put cri(x) = g(x) _ /^l - ^\b 2cn Furthermore we take care that our approximation statement for small — ^0 yields the exact solution. This will be the case when the [^ of the approximation statement agrees with the (■t~) to be calculated X from the differential equation for — = 0. ^0 According to the differential equation: e(x) o u ^ 5u 2 ^ ^ ^0 ri + — X + Tl X Thus Su ^/x =0 ■0 = lim ^0 :(X) •> S u Su 2 ^ ^' 1 ^0 T\ + X + ^ X or, with c(x) 2ai 3U rn/ — ^ = lim ^0 au ^ ^ (^)^ .^ Su Su 2 2 2 ^ 5^ ^ ''I ^0/ MCA TM 1311 17 We now enter into this equation with our approximation expression; that is^ we put (except for a common factor) Su = e I— oir\+a2 3^ ^u = e 2 _- - 2fa-^T\ + crgja^' du ^0 = e (^1'^ + CTg- We then consider the relations lim X ^0' ^1 = <^0 1 1/2 Kc /U^ - ^\ \ ^1 / lim 02=0 ^0 furthermore, we assume lim ^0 = The last relation signifies that the width b of the mixing zone is, in the proximity of the nozzle, of rectilinear character, an assumption which seems justified considering the fact that we approach, in the proximity of the nozzle, the conditions of the plane jet boundary. We then obtain for the left side of the equation 2 Su ■^0 ro lim x ^JL X ^Q d — ^0 = e •(f^O^) (0) 18 NACA TM 1311 for the right aide lim 7 ^0 Su i ^2- ^_ — d u du 2 2 ^ 2 -^ ^ ^ '^l -( ^)^ (^) 20^3 a^ - 2aQ a^'Co) .-(t^o^i)' Equating yields the eqiiation a2'(0) = -i 2a^ CTq - 2aQ 02*^0) or respectively. CTg'CO) T^ This results in ao = rr" ( — 1 for small — . for JL ^0 This guarantees first of all that our approximation expression -^ represents the exact solution. If we enter vith the approximation expression thus constructed into the differential equation, we recognize immediately that the latter (due to the factor e is satisfied also for t^ arbitrary — -> 00 ^and ^)- Thus our approximate expression with a^, 02 fixed in the above manner yields a function which corresponds in boundary zones of the region to the exact solution. As to the behavior of our function in the interior of the region, it is found that the function in case of suitable "continuation" into the interior of the region satisfies the differential equation along 1^=0. For T) = the differential equation reads ^0 ^(x) \ X 1^2 ^ Uoy_ NACA TM 1311 19 If one enters with the approximation expression and considers ^ " 2a 2 one obtains "i' ^(t)P^''2<'i'-''i(^) or (T2' + CTg ^0/ 2an As solution one obtains 1 1 X /x \ 1 2 2 /JC r. — ro ^0 y cri \rQ {^) For small — one has again ^0 1 1_ /jc_ oUo ^^2 =t^t;i7 We may also write lU-i - ^2 = 2f "\ 1- M 1 _x ^r, " ^(f)' \m i^) Therewith we have obtained for small dist\irbances the following approximation function t = ^ (^ -) /f ^ ^-^"""^^ <"^- "^)/ K- ^1 20 NACA TM I3II where a-^{x) /-(^)l X ^^''^H^]jt^cm To sum up: This function satisfies the differential equation with the initial conditions prescribed for small — as well as for large positive T\; in the interior of the region it satisfies the differential equation along the jet t^ = 0. Therewith we have constructed an approxi- mate function which in boundary zones of the region of integration and in its interior along the Jet t] = is to be regarded as exact solution. On the Theory of Larger Disturbances U^ - Uq Let us now consider larger disturbances not a small ^1 quantity . First, we shall treat the problem of the initial profile. Gortler's calculations (reference 3) showed that even the first f ^1 - ^\ approximation I for small j represents, purely with respect to shape, a very good approximation. This applies, however, only to the shape of the distribution curve - not to its position. The velocity distribution calculated by Gortler is unequivocally fixed by the _ Uq^ - Uq „ arbitrary requirement that u(0) = ^ . However, Gortler points out that with u(ri), v(r|), the eqioations u* = u(t) + 3.), V* = v(ri + a) - au(Ti + a) also represents a system of solution. But this remaining uncertainty is here eliminated by the fact that for the jet core the transverse component v must vanish as follows from the continuity. If u(t]), v(t^) is the velocity distribution calculated by Gortler U-^^ - Uq which is characterized by u(o) = -x , the quantity a must there- fore be determined in such a manner that v-, - au-j^ = which NACA TM 1311 21 ^1 yields a = — . Taking Gortler's calculations as a "basia, one obtains ^1 in first approximation Un 1 - ^ Ui 0.36 thus aa = - 0.36 ^1 - ^ ' ^1 Therewith the initial profile for all ^1 - ^0 ^1 is unequivocally determined. If we base the shape representation on the first approxi- mation^ the initial profile is u 1 N {n\^l '-f u, al.fi.^ where ^ = CTqTi - 0.36 ^1 - ^0 ui and ^0 = 2kc ^1 - ^ ^1 For the further development of the profiles in the core region, starting from this initial profile, we take as a basis the regularity found for small disturbances. ^1 \^Wi / Jo 22 MCA TM 1311 where Tl* = CJi^Tj - 0.36 + CTg with the terms a-|_(x), OgCx) determined before. This function therefore yields the initial profile in first approxi- mation. How far it may be regarded as approximation in the region is ^1 - ^ not investigated in more detail here. For >0 it is trans- ui formed into the approximate function found for small disturbances. Our approximate function generalized to arbitrary disturbances therefore reads ^1 V«\ui /Jo 2^ ui/ where with ■t\* = a^ - 0.36 + cTg a.{x) = '-1 ■ "0\b U-, yx /U-, - u, .(-lf(H^)(l o\ 1 r^o /jL\/Y,/x The coordination to t) is obtained by ^1 - ^ Tl * + 0.36 NACA TM 1311 23 where '^0 °2 1 J. ^1 (^V^ it °i 22 /_^\ j^ ^0 y Vx Thus the curves result from one another by similarity transformations. Calculation of the cixrves requires, furthermore, knowledge of the functions ct-l(x), cT2(x) and, respectively, of the mixing width b(x) and the constant k. These quantities result from the approximate calculation (carried out with the aid of the momentum theorem) for the dimensions of the core region. Figure 1 contains for the parameter values ui - Uq Ui = 1.0; 0.8; 0.6; OA; 0.2 u X the velocity distributions — calculated for — = and the core end, ^1 ^0 . In figure 2 the functions a-^(x) and ^2^^^ ^^^ plotted for the X parameter values named above, as functions of — up to the core end. ^0 Calculation of the Transverse Component The transverse component v of the flow is determined from the continuity equation 5(ru) _^ ^(rv) _ Q ox Sr and, respectively V = -rl{^f> ^*° 2k MCA TM 1311 r - Tr Transformation of r into t) = V = - 1 ^0 T) + Tl + °V^i results in ^^U ^ ^r^^ (-^^0) The Integration constant is determined from the requirement that in the jet core the transverse component v must vanish. As the lower limit we .choose accordingly the r\ determined by the bounding of the jet core (concerning the dimensions of the core region, compare next paragraph) . In order to avoid complicating the calculation, a rectilinear course of the mixing width b(x) was assumed. This assumption is approximately correct. (Compare fig. 11.) u For the velocity distribution — we substitute our approximate function. The performance of the calculation (appendix no. l) yields the following final formula. V ^1 ro\\M^i / 1 1 2 a^^\^0 l^]A^^^ 2 Oi II 1 /l^ _ 1 ' X ^0 \ ^Jt \ ^1 1 1 fl0\i I 1 ^ ,^^^j 2 „ 2Vx 2 a. where 1 I / = 11> = (¥.' + F-i ' NACA TM 1311 25 {111} - [a .-"' . a, e-C^O^ . 2{., . 0.36 ^)(e-"' . e'^O^) , ft" i . („, . 0.36 "^ - "° 2 V"2 u- 1 ■in"i'no)i Ft V. + Fi Therein [] = [a^Ti - 0.36 ^^^ + (^2 Clo =(^i'^^- 0-36 ^1 - ^ ^1 ^2) ^1' = V^. e ^J dC] and F-] 'pi ^^"^ ^1 ' n > respectively, signify the values of the error LJ LJq Integral taken at the points £} and -CJq^ respectively. In figures 3 to 7 the distributions of the transverse component for a section ( ::;- ) = 0.1 near the nozzle and a section of 3/^ of "the core ^^ " ^ = 1.0, 0.8, 0.6, O.U, ^1 ^0 length are plotted for the parameter values 0.2. U^ In the case ^ Un = 1.0 there are shown, moreover, the distri- hutlons for the sections l/k of the core length and the core end itself. (Remark: The transverse components calculated for the core end seem to yield too small values of the approach flow; the reason is that the poor approximation of the velocity distribution, an essential charac- teristic of the Prandtl expression, in the boundary zones takes the more effect in the calculation of the v component the more one approaches the core end.) U;L - Uq For small , the transverse component becomes very small (note the different scales in the various representations) . 26 NACA TM 1311 (b) The Dimensions of the Core Region The dimensions of the core region are defined by the limiting curve of the jet core d(x) and the width of the mixing zone b(x) or, respectively, the outer limiting curve of the latter b(x) + d(x). According to Kuethe ' c procedure (reference l) we take as a basis the theorem of momentum in Tollmien's formulation (reference h) . NACA TM 1311 27 If one marks off a control area in the indicated manner, one obtains in the rotationally symmetrical case (u - Uq) vr - -r- ox u(u - Uq) r dr = (rTxy) Ug = velocity of the mediiom surrounding the jet. According to the more recent Prandtl expression T^y = *^^U) (ui - Uq) ^ Thus we obtain, if we, furthermore, take the limits of the mixing zone into consideration (^ - ^) 5 vr - r— ox ,d+b ufu - UQJr dr = rKbrn-^ - Uq) ^u According to the existing conditions we transform (according to Kuethe) with r - d(x) ^ = b(x) Then r - d(x) ^ = b(x) r = bri + d If we make, in addition, the assumption that u depends only on t), not on X, there follows >d+b ^, _d_ dTl u(u - Uq) - Ti^bb' - T](bd' + b'd) - dd' *1 28 NACA TM 1311 For V we finally insert the continuity equation V = - — ^uV ^^j^^ or V = 1 p Su (Tib + d) Jq ^ - Ti^lDb' - Ti(t)d' + b'd) - dd'JdTi For approximate calculation, we write for the velocity distribution the sample expression u - uo /ui - UQ ^1 -' (^^)(^ - -) r - d(x) ^ = b(x) This expression, which may be regarded as a first rough approximation for the velocity distribution, will probably lead to not too large errors for the integral calculation. The value -v- determining the shearing stress also will probably result in a usable approximation for the central region of the mixing zone. The result is ^1 1 ^ b ^ Ui _d_ dTi uT - ^ Un ^1 ''Ui (1) We now put r = 0. The momentum theorem is then transformed by integration into the form of the theorem of conservation of momentum >b+d u^u - UQJr dr = const MCA TM 1311 29 or -d(x) nd+h _ ^l(% - Uq)t dr +J u(u - UQJr dr = u-^(^u-^ - Uq) ro i(^)(^^-^oV^=/;t("^).^.-^ ^1 ^/U - Up ^ ^1 dri = If one inserts one obtains u f^l - ^\ ^1 ui l(l - r\) and carries out the integration, 1/.2 1 I'd z \ , , '^ 'Ui ^\ 1 _^ ^ 1 Un /12 ^1 + bd zr + — - = ^1 3 ^i or 3 ^ _ 1^1 - "^ + bd ^ _ 1|^^1 - ^ 3\ ui ^2 2 + d = rp (2) In order to obtain a second equation between b and d, we put r = rQ. If one performs the somewhat lengthy elementary calculation, one obtains finally (compare appendix no. 2) ' ' ■' - ^:)--^! bb'- ^1 - ^ ul 1/^0 - ^ (b'd + bd') dd' ^V~b" Ul - Uo\ ^1 ipo - a-' + — 3V b u 1 1/^0 - ^\^ 1 z\—v- ^z + — if^o - ^ 2\ b > + = - rnK 7^1 - ^ \ Ul 30 MCA TM 1311 The theorem of conaervation of momentum reads in differentiated form. (Compare (l).) bb 1 1/ ^1 - ^\ 3 " 6l ui / + (b'd + bd") + dd' w- By addition of the tvo equations one obtains ^1 bb 1/^0 - ^ 3\ b - 1(")(^) + (b'd + bd') i(^y Ui + dd'.' ■0 /Ui-Uq\ 'oi—^ir) (3) We now proceed to determine b and d from the two equations obtained. We replace b in the second equation by the expression for the function which we obtain by solving the first equation with respect to b. = a. m*i ^1 - ^2(^^ where 3 1/ ^1 - ^ \ ' 3\ u^ J " ^ 1 . 1/ ^1 - ^ ^1 = i m - ^ \ 2\ ui I 1 - a^ = -K^) 1/^1 - ^ 3V u. 1 ^1 - ^ 2 Ut NACA TM 1311 31 b' = /a. a^d K a^d^ Subatitution then yields d' = ^. - ["^h = - K ^^1 - ^ Ul with ^1 = ^-(^flj^K^) ^oi^; + y^i - ^2yrQ ^0 - :(^) f *k d V l;)^r lo — ^2 r, CP'T^ ^o(^) ^\ —,3 6 (=0 ^ ^rj We obtain — as a function of — a(x) = . i 1 W-i--olfi-^^=)^<^' ^1 X ^0 i— J: / 1^1 / _d_ f, - "i;"o f^ld^i- The evaluation of the integral could, in itself, be carried out by analytical method since the integrand is built rationally in — and ^0 32 NACA TM 1311 a square root. However, the breaking up into partial fraction which has to be done in this proced\ire is very troublesome. Hence it is advisable to perform the evaluation graphically. For — =1 the integrand ""0 nite expression tt. The limiting value is -i- 1 - "0 '' -'—^rti assumes the indefi- ao + fl ""0 If — was determined, analytically or graphically, as a function of — > b(x) results from ^0 b d /d The relation db dx/ X =0 = --'^)to*g- ^0 lim ^. -l^^^;^)^. (which by companion with measurements on the plane jet boundary may serve for the determination of k) also is of interest. The symbol k appears as the only empirical constant. With the measTired results on the plane jet boundary with zero outer velocity (given by Tollmien (reference k)) as a basis, there results with ^] = 0.255 r^ MCA TM 1311 33 0.255 = - k(-3) on^ K = 0.01576 Example s : In figure 8 the dimensions of the corresponding core region are represented for the parameter values ■^1 - W) n ^ 1 -^ = 1.0, 0.8, 0.6, O.k, 0.2 U-|_ ) } } } Figure 9 contains the core lengths — , figure 10 the mixing hv ^0 1^1 - Uq widths — at the core end as functions of . i-O ^1 Figure 11 shows the mixing widths — for the various parameter u 1 - Uq X values of as functions of — . ^1 I'D ^0 Figiire 12 represents the angle of spread of the respective mixing region c = \-r-\ ^0 Figure I3 represents CTq = as a function of ^1 U]_ - Uq U;L - Uq , with Tollmien's value c = 0.255 for = 1 being the U;L VL-i ^ defining quantity. Figure ik shows for the mediiai at rest I = l) the quantity K /db\ ^ ^1 '1 as a function of c = hr~ • Figure 15 shows CTq = —r= as a \^^/iL=o \J2kc function of c = (^— ) ^0 3^^ NACA TM 1311 Figure 16, finally, contains the limiting value lim ->1 ^1 " ^ ^1 necessary for calculation of the integrand in X r^ K- /^ ^ Ui _d_ ^^0 ui - up f2 "I III. COMPARISON WITH MEASUREMENTS Measurements on a free jet issuing from a nozzle and spreading in moving air of the same temperature do not exist so far. case In order to test the theory by experiment, a measurement for the Ui - Uq ^1 =0.5 was performed at the Focke-Wulf plant. The measurements were carried out with the test arrangement with the 5 millimeter nozzle described in reference 5* A certain experi- mental difficulty was experienced in producing temperature equality in the two Jets; it was achieved by regulation of the combustion chamber temperature with the test chamber pressure p, and the probe pres- sure p kept constant. However a perfect agreement of the jet tem- peratures could not be accomplished inasmuch as the temperature measure- ment performed with a thermoelement is rather inaccurate in this low region. The test data were: Outer jet: Static pressiore p, = -100 mm Hg (Measured relative to atmospheric pressure) Room temperature Barometer reading to = 20 Pq = 75I+.5 mm Hg MCA TM 1311 35 Inner jet: Total pressure Pg = 3^0 mm Hg (Measirred relative to atmospheric pressure Stagnation temperature t^ = 59 The evalxiation of the measured values was made according to the adiabatic K-l and the efflux equation ^1 ^IJT^^gRT. '^1 ' - [Tj ' K-1 with constant static pressure assured in the mixing region. Due to the imperfect readability of the thermoelement which, as mentioned before, is too rough for smaller temperature differences, it was impossible to measure the distribution of the stagnation tempera- tures over the mixing region. For the evaluation a linear drop of the stagnation temperatures along the mixing width was assumed. For the outer jet there results o — t = 9 '^A ~ '^^'^ meters per second for the jet issuing from the inner nozzle o — t. =13 ^i ~ 302 meters per second The inner jet therefore has, compared to the outer jet, an excess O U-j_ - Uq temperature of k . For the velocity ratio the result = 0.5 was obtained P. + Figure 17 shows the total pressure distribution Ps ^k Pb Pb Pb ^ Pb/ central 36 NACA TM 1311 made dimensionless with the central value, for the various test sec- tions. The section near the nozzle which still shows the character of a turbulent pipe flow is represented in figure I8. Figure 19 shows, in addition, the variation of the total pressures along the jet axis. Flg\ires 20 to 22 contain the corresponding representations for the velocities made dimensionless by the velocity \i-^ of the jet issuing from the nozzle. As to the comparison with the theory, it must be noted that the velocity distribution at the exit from the nozzle is not rectangular, as assumed in the theory, but that it represents the profile of a tiorbulent pipe flow. (Compare fig. 21.) Hence it proves necessary to introduce the conception of the "effective diameter" in contrast to the geometric diameter. We define the effective nozzle diameter as the width of the rectangular velocity distribution of the amount uq_ which is equiva- lent to the existing momentum distribution. That is, we calculate the effective nozzle diameter from the equation r P°° ^1(^1 - ^) — ^f^^^^ = J Up - u^Jr dr . with the integral, which according to the theorem of conservation of momentum represents a constant, to be extended over an arbitrary cross section. In our case the integration over the cross section near the nozzle yields ^effect. = 0-9^5r geom. r — r Whereas the plotting over t) = lets the test points appear r - -""effect, as still lying on one curve, the plotting over t\ = - results in a stagger of the velocity distributions with increasing — ^0 toward negative t\ . This stagger toward negative t] expresses the immediately obvious fact that the isotacs of the flow field are curved toward negative t\ (toward the jet axis). X ^^ Figure 23 contains the theoretical curves for — = and — ""O P (the core end); in addition, the test points of the sections x = 10 mil- limeters and X = U5 millimeters were plotted. The agreement appears NACA TM 1311 37 to be good as far as the velocity gradient and the orientation in space in the central mixing region are concerned; the agreement in the transitions toward the jet core and the siirrounding medium is less satisfactory. Deviations in these transitions are essential charac- teristics of the more recent Prandtl expression, but are caused here probably mainly by the approximation character of our developments. For the core length there results according to the theory a value of x, = 22.0r„^-p„„-u, whereas the measurements along the Jet axis (compare fig. 22) result in about xv = — 7^ = 20. Ir „„ , . K 0.9^5 effect. It has to be noted that the experimental determination of the core end is affected by some uncertainty. IV. SUMMARY The spreading of a free rotationally symmetrical jet issuing from a nozzle represents a turbulent flow state. The theoretical investigation is based on the more recent Prandtl expression e = Kb lu^jg^^ - Uj^^^j^ I for the momentum transport. The continuity equation and the eqioation of momentum are at disposal for calculation of the velocity distribution. In case of limitation to /u;L - Uq small disturbances I small quantity, where ui is jet exit velocity, uq velocity of the surrounding medium J the equation of moment-urn may be linearized ' 2-\ ^U / n/Su S U) An approximate solution is constructed which is characterized by the fact that in boundary zones of the region as well as along the jet T) = in the interior of the region it has to be regarded as exact solution. ^1 ~ ^0 For arbitrary disturbances ( arbitrary > O) the initial profile which corresponds to the velocity distributions of two mixing plane jets is determined by the fact that the transverse component in the jet core vanishes. The regularity found for small disturbances is taken as a basis for the further development of the profile from this initial profile. 38 NACA TM 1311 The transverse component of the flow is determined from the con- tinuity equation, with the use of the approximate function for the velocity component in the main flow direction. For simplicity a linear covirse of mixing width is ass\amed. The dimensions of the mixing region (limiting c\irve of the Jet core d(x) and mixing width b(x)j are approximately calculated from the theorem of momentum i ■ . ~v r 00 (u - UQJvr - ^ / u(u - u^jr dr (= rr^y) = rKb(x)(ui - Uq) ^ \inder assumption of a rectilinear course of the velocity distribution ^, r - d over r\, where t^ = — r — • In order to test the theory by experiment, a measurement was U-L - Uq performed for =0.5 with a 5 millimeter nozzle. In order to ui carry out the comparison with the theory, the conception of the effec- tive nozzle diameter Is introduced which complies with the deviation of the effective velocity distribution for an issuing jet from the rectangular velocity distribution 2 ■J Q The agreement between theory and experiment is satisfactory. NACA TM 1311 39 V. RETFERENCES 1. Kuethe, Arnold M. : Investigations of the Turbulent Mixing Regions Formed by Jets. Jour. Appl. Mech., vol. 2, No. 3^ 1935^ pp. A87-A95. 2. Prandtl^ L. : Bemerlaingen zur Theorie der freien Turbulenz. Z.f.A.M.M., Bd. 22, 19^+2. 3. Gortler, H.: Berechn-ung von Aufgaben der freien Tiirbulenz auf Grund eines neuen Naheriongsansatzes. Z.f.A.M.M., Bd.. 22, Nr. 5^ Oct. 19^2, pp. 2U4-25^. k. Tollmien, Walter: Bereclm\mg turbulenter Ausbreitungsvorga'ngo Z.f.A.M.M., Bd. 6, Heft 6, Dec. I926, pp. 14-68-^78. (Available as NACA TM IO85.) 5. Pabst: Die Ausbreitungheisser Gasstrahlen in bewegter Luft. UM 8003, 19^^- 6. Reichardt, H. : Uber eine neve Theorie der freien Turbulenz. Z.f.A.M.M., Bd. 21, 19^1. Uo MCA TM 1311 APPENDIX NO. 1. CALCULATION OF THE TRANSVERSE COMPONENT V = - G ^ ^)^ ^ + — MX x:: - Ti ^JdTi or ' X '^ V = (-?)l! (^), ^0 Tl + X '"^ Bu P, f^O -■^y ^ We substitute r" -cr ^ if^.X r^e-LJ d[].il.^^ ^1 \f^\^l I Jo 2\ u^y C] = fa3_Ti - 0.36 ^1-1^ + a u. 2 ^u/u-^ = ^6 -)=■"'&'<- ''^"==^ "Sri ^\^i MCA TM 1311 41 If we agsume a rectilinear coiirse of the mixing width bCx), we have a-]^' =0. We then obtain X 1 1/^ 3^\ , ^1 " " / _^foj M^i " / ^"2 -CJ e TidT] + / e'--' dT] ^ -c:= -^1 e Ti dT] + — / e T] dT) (a) The evaliiation of the integrals yields e d"n \-0 ,, .A. Te-" d(aiTi) =-i- Te-^J d [J n = (c^in - 0.36 ^^4^ + a^) thus e dTi= — / e dLJ -T] 'G "0 = ^1 - ^0 a^(.,j - 0-36 —^ ^ a^ k2 NACA TM 1311 rL] po ri:] r-[]o r n where thus (b) "e-" .„ = fy. [] G .r-^2 \/ir ^ error Integral e '--' T dT] [J = a^Tl - 0.36 — ^ + Or 1 U-, - U(-, :j + 0.36 ui - ^2 ^= ^^ + F, ■ '\ This results in U]^ - Uq e LJ T^ dT) = J^ / e '--' i d(a-,Ti) ^ij <^1 ■ */•■"' n •" - (:^^^^/--"' « MCA TMI3II h3 Ti OJ ? OJ OJ I — t3 OJ 0) -— . o OJ € OJ H + l^OJ ? o I OJ OJ k-^' ^i\ ^1 > 1 1/ ^0 - ^ 2 " 2\ b -H^; ro - d^ 1/^0 - ^ :?\ ^2l b ^/^i - M ^i\ ^1 '^0 - ^^ I 48 MCA TM 1311 If we insert these expressions into the equation of momentiain and order, we obtain bb' 3\ u 1 + - 6\ U;l 6\ u-| 2 K - ^ \^/^o - P~ /, J Y \ 5 ."f I X c I II XT J ^|:5~ =- L ^ V 7 c >?k^ J / :i!:r ^ ^ T >^ 1^ I. C «0 d ^ / 51 ci o o XI •iH o o 0) > gj % Si I * ^ / •^^i L iy =>|:5- ^ ^ 1--^ c > <^ ::^ -^ < 3 q / / d cs NACA TM 1311 51 i 0,' y^K ivi^ i SO (i L - U^ _ - I*'' 60 L. — a 4 pn — Q 6 3 m 10 20 30 40 SO 60 X • a r^' 1 — 1 ,r>; 1 1 A ^ 2 -* U.-U. ' 1 If "0 / / / 0£ 1 r 1 i 7 r ) / / / a .« jl V / ^ y \l / ^ ^ ^ — °-^ ^ ^ —- 10 20 30 40 SO 60 Figure 2,- Auxiliary quantities for calculation of the velocity- distribution. 52 NACA TM 1311 11 c CVJ Ol (0 C3) (6 m: ■^k? "Mi? "^if 5 "^ . ) ^;? C) 1 1 1 \ 1 M y \ 1 - a " \ / \\ \ / Jo // / / f c 2 } ' / y / ;3 // / y / 1 3 ;3" J^ y y /^ //, y ^ V/ / V/ / i 7 ^ ^- i;^ ;3 - r 111 > s 2 cs C ^^ \\\ ^^ \ s \ \ \ •^ c NACA TM 1311. 53 i\ £>• _ fn \ 11 I Hi! D / f CN - 0' ^ ) *» a i y / i - ci ■ y -^ ^ ^ ^^ y ^^ ^ ^^ ^ ^ y^^ ^ -^ ( /^ \ V -.;.|;J ^ ^ X ^ 1 \ d a o •1-1 -3 • i-H u w • rH o a o o (D w > Eh ha •r-i 54 NACA TM 1311 11 1 £ s^ ") " Ci " 1 5" " a ' 1 y / 3 J y <3 ^ -^ :=== =^ '^ / >^ '- \ ^^ -i- hb~ "<= ^ ^. ' op N: ci 1 _ 1 MCA TM 1311 55 ^ - t CO " Ci C) Ci i ^K^ _ 'V . CS n 1 < 5 c ;3 cs \ 1 ;5 i J J ^ ^ (^ f — — 1=" 3 — 3^ 5 C3 a 1 " ci ■ 1 o ■l-l •1-4 u +-> w •1-1 T3 o a o o 0) ra I CD % NACA TM 1311 c- ro ^ ^ " "^ ■» / / ' z* / CM >o 1 t\J 1 1 d r^ o\ \ ,3 \ 1 3 1 ;3 \ >o 0' /^ ^ ^r" . ^ 'O ^*.^ IN P ^ •» (0 "* - ip>l; 3 3 2 c c S 1-1 J t i c i < 3 lO d NACA TM 1311 57 ■ / \ 1 1 \ I I \ 1 \ / \ / \ /' c 1 5 5 3 D « 3 § c 5" 3 3 / • / \ 1 \ 1 \ 1 \ i \ \ / \ * / \ \ 1 1 \ i t i \ \ 1 \ \ \ 1 / / \ \ \ i t P ■\ \ \ 1 1 1 / \ \ \ \ • 1 1 /* / s \ \ \ \ /' 1 1 /' y \ \ \ 1 \ \ 1 / 1 1 1 / / / / V, N, \ \ \ \ \ / i 1 / ^ \ s- \ \ \ \ .', ' / / / X \. \ > \ t * //. / s \ ^\^ 1,'A // -« ' , X >\\i W Ttl.O. - -< i*^ ; ^ C ^4 » :i ^ ^ s i c )* f\j •o ^ »? CO * ^ ^^ , • rt ■* •Ed (U ^1 0) 10 u •0 (U A - -tJ «M t\J ^ w fl .r^ w a CO Q) CM a a 1 00 (U ^ hD •iH ^ \ 30 \ \ \ 20 \ s. \ X 10 n 0.2 0.4 0.6 0.6 1.0 Figure 9.- Gore lengths. MCA TM 1311 59 • ^^ r^ 2.0 ^ ^ ^ ^ ^ 1.0 n 0.2 0.4 0.6 0.8 to ^r^c V, Figure 10.- Mixing widths at the core end. 60 NACA TM 1311 ! \ \ ^ \ \ \ \ \ \ y \ \ \ \ » \ '\ S. ^ \ \ \ \ \ \ y \ V \ =» N N, \ \ >, \ > \ \ N k. \ y \ ^ \ «w \ s \ \ s II V ~N \ \ \ \ \ 3 3 ■^ -~^ \ X \ \, \ y """"^ "^ \ \, \ \ ^ ^ :\ \^ A _ nl ^^^ l«^s. ^ ■0|l^ 1 N o N s f \J < 3 5 kI ^ *» o M a o > bO C! 2 NACA TM 1311 61 C = o.z 0.1 1 / J / / J 1 / /c --0.0) 576 / / / / / A A / / y / y y 0.2 0.4 0.6 0.8 1.0 t<;-U, lif Figure 12.- Angle of spread of the mixing region. I 62 NACA TM 1311 t „, ■ 1 1 V K- 0.01576 'Z K C 6.0 v \ \ 4.0 \ N \, 2.0 s ^- . 0.2 0.4 0.6 0.6 1.0 U, -Uo Figure 13.- Parameter value of the plane jet rim. NACA TM 1311 63 0.02 y u 1 '0 y y y y V / y 0.0 J y y y y^ y / y .y ^ a ^ y 0.1 0.2 0.3 'J--.0 '0 Figure 14.- k as a fimction of c. 6k NACA TM 1311 26 oTP K c 24 20 16 12 V \ \ \ Ui-Uo . . \ u. \ < \ 1 \ \ \ \, v^ 0.7 0.2 -m Xdxj 0.3 h - -0 Figure 15.- an as a fimction of c. NACA TM 1311 65 r § 00 o s Ci Ci o •So 0) u o o 0) -a o •i-H w ^ o o «r-i o -^ o ^1 o -I— » CD •r-t ci ^-1. ee NACA TM 1311 r E 1^ X * + ? t- « (^ w 4+0 5 M ■ 1 Hll! •'1'? + > 8 > 4 1 *1 ? a + X 1 X 4l A •a + □ i *o 1 s 1 ^ if ) I 1 :5 :? D i-,^ X < •6 : C < + y\ X 1 ^1 a" -l-^' IT •PI 1 e "^ ci i 00 1^ >- J «0 NACA TM 1311 67 X E II ><■ X ^" X X X ^ X X X 11 L ( X X i 1 1 X u- ) t c 1 > c 1 :5 X X X X c U > < .^ cf 2? IQT "- i >ie M X S X X X u- ) 1 X 1 IS" >c > t X X < -^ < '0 — 1 ^1' ^ S n n U 1 C t i 1 5 > < O ^ X 10 5 > a □ « L 1 ■° i X ^ ^' i .I5 c > ^^^ :s^ ■ d ^ 1^^ r i ^<|v? 1 X M f * X >." vf ft C) Ci C) NACA-Langley - 11-6-51 ■ lOUO ■e ^ ; ^ 3 o a S > ::S 2=S z tsi flj _ w 3 ^ 3 o ,-M e J3 o u S S i «n U ^ tM P 1-< ^ N M H 5 < u •< < « O 0, z zo < C . 3* s i ^ i> rt hg Ik m Hii •< 0) ZN CO "g (u ^1 So.s I-" N g O ^ CO H to CQ m "3 ^ o .5 oh"" V M ° L, u c <^ ^ r •" O U M ■ Cd = QJ