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c  NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1311 CONTRIBUTIONS TO THE THEORY OF THE SPREADTIG 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 medium with straight uniform 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 RESULT? 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 socalled 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 1944. NACA TM 1311 (b) The spreading of a rotationallysymmetrical jet issuing from a pointshaped slot in a wall, the socalled rotationallysymmetrical 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 socalled 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 1); it is limited to the case where the surrounding 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 surrounding medium has a straight uniform flow field (or, respectively, where the nozzle from which the jet issues moves at a certain velocity through 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) uImax umin One then obtains in the rotationally symmetrical case the following equations: Continuity: + = (r) I1ACA TM 1311 Momenti.um transport: ~ ~ 2 du ou u 1 ou u V =\ u + v = \r2 rEr X d r \^r2 r Tr) whe re E = Kb(x)(ul u0) ul = velocity at the jet core uO = velocity of the surrounding medium (ul > uO) With reference to the present problem Il/////II/i we introduce, instead of r, r r r x as independent variable. Coint inuity We obtain: r Mlkil77q regq/ion ( ox 7\ x ( \ rox/ r + )x x3 +v+ +x= o *11" I.. ... ...... . ro77 4 NACA TM 1311 Momentum transport 1 ou oul ou E(x)/ 1 ou u x 1 + v ,EW 2 rO S +  Velocity distribution in the Core Region We limit our considerations at first to small disturbances of the flow field; that is, to relatively small differences in velocity (ul O small quantity). The partial differential equation for the momentum transport may then be linearized  2 o \ r x = E(x)( + r 2 where e(x) = Kb(x) 1u Ul (It should be noted that by the transformation this equation is transformed into the equation 2 2u Ou 1 2u ; + i x = 0 r r0 which represents a heat conduction equation.) With r = x instead of r one obtains from the equation of momentum u +u 1 xx U x 22 + + r0 Tl (T = x if r0 x 0 NACA TM 1311 This equation is a of parabolic type. equation linear partial differential equation of second order For the plane case ) oo this results in the (x dT2 d xTix  rO with 1 c ul ul) KC r b(x) therein c = lim ix x x =0 r0 With the yields and is to be regarded as a function of Ul UO ul boundary conditions taken into consideration, the integration "oo e ( O )2 o ( ) e ( d(a0O) 0 with 1 a C ul uO 2 c(Ul ul) We now obtain an approximate solution of our problem by generalizing the plane velocity distribution and setting up the following formulation: u ( el d a + + 1 + ul (ul o 1L" 2J 2\ uli +u O ul ro = ( 1 NACA TM 1311 With al(x) = (ui x b ulO) x x 1 Kul uO I x b \rO we obtain a function which corresponds to the exact solution for small  as well as for large positive r, thus in boundary zones of the r0 region of integration as well as in the interior of the region along the jet I = 0. If we now consider larger disturbances, the solutions obtained for small disturbances are to be regarded as a first approximation. u1 u For the plane case the solution for arbitrary already ul exists, compare GBrtler (reference 3). It is found that, purely 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( + a), with u(), 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 ul uO is then unequivocally fixed. ul If we limit ourselves, with respect to shape, to the first approxi mation, the initial profile is u 1 eu\ t2 + 1 ( uO) .. l Ie d++ 1+ ul fl \uu I Jo 2 where S= Gn 0.36 (u  ( ulU NACA TM 1311 and 1 a rc Ul uo For the further development of the profiles starting from this initial profile the regularity found for small disturbances is then taken as a basis Se* 2 e^ 0 1+ dj* + 1 1 2 where Cl lr UO) + = a1' 0.36 (u ) + u2 ul UO For  O>, this function is transformed into the approxi U1l mation function constructed for small disturbances. How far it may be considered an approximation in the region for arbitrary disturbance is not investigated in more detail. The functions appearing in the integral al(x), a2(x) result from the approximation calculation for the dimensions of the core region, carried out on the basis of the momentum theorem. Calculation of the Transverse Component The transverse component v of the flow is determined from the continuity equation v = r dr ~S~jx )p (r / 0) respectively, with our approximate function being substituted for u. r 1 or v ulV J (  ul (r \ul ~ / rO^+ 1 xu bu n'rC + i xAx x 71y id x 8 NACA 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 course 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 u) x u( UO)r dr (= rrxy) r = r(b(x)(ui U indicated by Tollmien (reference 4). The occurring integrals as well as the defining the shearing stress are determined approximately with the course of the velocity distribution assumed rectilinear u uO ul uO r 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 dx d f(x) y =f f(x) dx) This integral can be represented with the aid of elementary functions; however, for simplicity its calculation here is performed by graphical method. K appears as the only empirical constant which results by compari son with measurements given by Tollmien (reference 4) as K = 0.01576. NACA TM 1311 Comparison with Measurements In order to carry through a comparison between theory and experi ul UO ment, a measurement for the case = 0.5 was performed with the Ul 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 e(x) = Kb(x) u/ax umin 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 rotationallysymmetrical rotation Continuity (ru) 6 (ryv) T.+ =0 Momentum transport ou 6u 2u 1 6 u + v r = e(x) + ax br ,2 r r where e(x) = Kb(x) (ul uO NACA TM 1311 U1 = velocity of the issuing jet uO = straight uniform velocity of the surrounding medium Ul >uo We may integrate the continuity equation by introduction of a flow potential * ru=a The momentum equation then is transformed into Tr T Vr o? ox orJ x * r2 7x 'Vr +1 o'ot V((/ J o 2' + = x (X) r  +r 7x 7r ( r2 6r2 +1r or r ir ) where e(x) = Kb(x) U l UO if we make the velocity dimensionless by division by ul. According to a method applied by G'rtler (reference 3) we set up for J the expression Sul uO 2 / u l__o (/Ul U 1 t=^(\ L ir *2 +  developing in powers of the parameter _. the potential of an undisturbed flow (ul = uo); thus ar= ru1 Therein V0 is 0= 0 OX rv = T NACA TM 1311 If we enter with this formulation into the differential equation, we obtain Sor ?~ + ul uO+ * /Ul UO 21 + ul ; 7 + \ ui /d or * ( u o 1 0 r + r(x r 2 0 'r ul uO *1 + ul ()r U S1 r 2 If one arranges according to powers of of differential equations for *1, 2j2 (120 ) T5rFl + 2r 1 (2 ~F^[ r 2 = e(x)(r r r " r2 [2 O _r2 S1 r + (i,/ uo) *1 "lul r 6r3 +1 j0 ur uO) 1 r or \ ul o iul uO one obtains a series \uli ( o) 2r2 Sr2 *0 koi; I 61 + 7x (0 1 +  r For *i ul UO) 1 ul or2 63*1 6r3 62*0 . S7t 12 NACA TM 1311 or, taking *0 *0 r = rul  = O into consideration a2 *1 1 G W 13* 62 + 1 * 2x;r / r33 or2 r 7r etc. On the Theory of Small Disturbances In the following, we shall limit ourselves at first to small disturbances of the flow field; that is, relatively small differences in velocity (Ul small quantity). The velocity field is then defined by the flow potential *1. 6*  Since = ru, the above equation for V1 may be written as follows: r = e() + r e(x) = Ib(x)( l ) 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 r our equation is transformed into 2 2(x) 0 u lu 1 au + x=o 2 IJACA TM 1311 With reference to Reichardt's discussions (reference 6), it is 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 x = distance jet axis), we introduce instead of coordinate transformation yields (,r=const from the nozzle in direction of the r rO r the variable n . This x = )n=const 7u I x = x u x thus the equation (xT + rO) = c(x) x + (xTr + ro) i2 r or, resFectively, for q + r 0 x (rO = nozzle radius, ou = ur ou 1 Tr !F r 7n x NACA TM 1311 2 6 u 1 x u x2  uX e(x) = Kb(x)( ul ") This equation is a linear partial differential equation of the second order of parabolic type. The solution of this differential equation is fixed unequivocally x by the initial condition that for  0 the velocity distribution ro of the plane jet boundary appears. We first derive (for small disturbances) the velocity distribution of the plane jet rim. For )0 we obtain with the expression u(n) the equation ro d2u du x 2 d x x r0 with r ]r=_ u 1 Ud ro Ul b(x) u_ UO Therein c = lim b( and is to be regarded as a function of u. x x u1 =0 r0 With the boundary conditions ul for i  o u uO for rT +0 NACA TM 1311 taken into consideration, the integration yields ul Uo Ul \~u 4 COT  1)  e (01) d(a0o) + 1( + 1 00 2Kc ul ) ul 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 uO l(x)+a2 E ( 2 (x) ia2(x] (] +. 1 vi ( ul U This formulation insures at the outset a reasonable shape of the approximation solution. For al,T2, there immediately result, because of the initial condition, the requirements lim rl(x) = C0 x __ 0 r0 'O lim a2(x) = 0 r 0 O Now the following equation is valid: 1 a u ul = lim x _ ro 1 uol I x with Accordingly, we put al(x) = 1 x _xu=uol x [ul ;x 2012 NACA TM 1311 Furthermore we take care that our approximation statement for small  yields the exact solution. This will be the case when the (x x r0 of the approximation statement agrees with the (x to be calculated x from the differential equation for = 0. ro According to the differential equation: _u e(x S2 Thus (ro\ x o r0 lim E(x) Lu +u 1 +1 = lim + 2  + C 1 o x2 2 x r T or, with e(x) 1 x 2i12 = lim 1 i + + N, 0 x 2a~2 1+ r0 V ro ro x TI X71 C T IJACA TM 1311 We now enter into this equation with our approximation expression; that is, we put (except for a common factor) eU CllTIY+CT F = e y01 2 e 2 r0 + 2)T2] (1'T + (2 ) We then consider the relations a1 = ao lim x =0 rO furthermore, we assume lim a1 x X>0 r = 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 lim u e (0o) 2 (0) xO (O x ro^ r lim x =0 rO0 c2 = 0 NACA TM 1311 for the right side 2,j 1 6 + 0(00 2) F 6 u dU 2 2 lim + X2 20 a (0 e x 20 1 + ( 0 3 2a 1 jO rO ro r0 Equating yields the equation a2'(0) = E 2a022'(0) or respectively, a2'(0) = This results in 02 = 0 for small r This guarantees first of all that our approximation expression for x 0 represents the exact solution. rO If we enter with the approximation expression thus constructed into the differential equation, we recognize immediately that the latter (due to the factor el11+a 2 is satisfied also for  o (and arbitrary I. rO/ Thus our approximate expression with cl, a2 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 n = 0. For i = 0 the differential equation reads ax) (x u x x 2 (To OJ ro NACA TM 1311 If one enters with the approximation expression and considers E(x) 1 x 2a12 one obtains 2 2' %2'12 + a1Cri or 1 1 a2' + a2 o= As solution one obtains x 1 1 rOx 1 r 0 For small one has again ro a2=1O r) We may also write 12 f.(ul u) 1 fror x, d.x Therewith we. have obtained for small disturbances the following approximation function I. 1 UO 1) ealTI*1 dai +2 ( 1 ) uO S e  1 J+ 1 + + I ul 0 + ul 20 NACA TM 1311 where al(x) = 1 J2(UI uO)b x 1 uO 1 rx a (x) = 2K( d _ 2 2 u x rx r ro To sum up: This function satisfies the differential equation with the initial conditions prescribed for small as well as for large rO positive n; in the interior of the region it satisfies the differential equation along the jet Tr = 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 = 0 is to be regarded as exact solution. On the Theory of Larger Disturbances U1 Uo Let us now consider larger disturbances not a small U1 quantity First, we shall treat the problem of the initial profile. Gortler's calculations (reference 3) showed that even the first approximation for small u1 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 arbitrary requirement that U(0) = 2 However, Gortler points out that with U(rI), V(il), the equations u* = U(q + a), v* = V(q + a) au(Tr + 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((), 'V(n) is the velocity distribution calculated by Gortler which is characterized by (O) = 2 the quantity a must there fore be determined in such a manner that v1 aul = 0 which NACA TM 1311 V1 yields a = . Taking Gortler's calculations as a basis, one obtains u1 in first approximation V1 1ul uO` u0.3 ul a u1 U, thus aa = 0.36 u ,ul uI u0 Therewith the initial profile for all  determined. If we base the shape representation on mation, the initial profile is is unequivocally the first approxi 1" 1 1 + Ul ~Ui 0 I where uI u0 S0 0.36 Ul and 1 U0 = uO) ( ul10 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. U 1 uO Tj* ul F(F 0 e d + 1 + O u 22 where ul uO n* = aln 0.36 l + a2 with the terms al(x), a2(x) determined before. NACA TM 1311 This function therefore yields the initial profile in first approxi mation. How far it may be regarded as approximation in the region is Ul UO not investigated in more detail here. For  0 it is trans Ul formed into the approximate function found for small disturbances. Our approximate function generalized to arbitrary disturbances therefore reads Ui= ll 1 e *2 ul \UIl )0 2\ ul where tI* = a 01  6 ul u 0.36 l + a U l , with al(x) = 1 l, X x 1 2K ul u) 1 ro a2(x) = 2(u i 1 ( 0 The coordination to n is obtained by ul uO ,* + 0.36 u O 1 a1 S1x dx \rO x NACA TM 1311 where a2 1 1 o _ro x _ro 1i 2 a12 x0)J Thus the curves result from one another by similarity transformations. Calculation of the curves requires, furthermore, knowledge of the functions al(x), a2(x). and, respectively, of the mixing width b(x) and the constant r. 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 ul UO Ul = 1.0; 0.8; 0.6; 0.4; 0.2 ul the velocity distributions  ul calculated for = 0 r0 and the core end. In figure 2 the functions al(x) and parameter values named above, as functions Calculation of the Transverse Component a2(x) are plotted for the x of up to the core end. r0 The transverse component V of the flow is determined from the continuity equation + = 0 and, respectively v= r (r dr r 0 NACA TM 1311 r r0 Transformation of r into = results in x v = r) 1 +  n1 x. 1 di ao ') ar l ro + x 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 T determined by the bounding of the jet core (concerning the dimensions of the core region, compare next paragraph). In order to avoid complicating the of the mixing width b(x) was assumed. correct. (Compare fig. 11.) calculation, a rectilinear course This assumption is approximately u For the velocity distribution we substitute our approximate Ul function. The performance of the calculation (appendix no. 1) yields the following final formula. S1 1 uO 1 1 x  t 1' I + ul + ) ul 2 2 rO 2 c1 x 1 1U( uO 1 1 r1 rO u1 2 12 x 2 a, 1 + 1 al x where S e 2 + e[. 2  (2 0.36 + F = Fl 1 ) 0 U i 10i C + F 0 NACA TM 1311 25 iI. ={[ e 2 [0 e 02 + 2 (2 0.36 Ul UO)( e2 + e3[02 S + (a2 0.36 U Uo) (F[ 1 Therein ul 7 [] = l 0.36 Uluo + c2) S('ik 0.36 u1 UO + ) F' 2 r e]2 dC[ and FI'[] and F' [0, respectively, signify the values of the error integral taken at the points [] and C]0, respectively. In figures 3 to 7 the distributions of the transverse component for a section )= 0.1 near the nozzle and a section of 3/4 of the core VO ) uI UO length are plotted for the parameter values 1.0, 0.8, 0.6, 0.4, 0.2. Ul Ul uO In the case = 1.0 there are shown, moreover, the distri U1 butions for the sections 1/4 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.) ul UO For small the transverse component becomes very small Ul (note the different scales in the various representations). 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's procedure (reference 1) we take as a basis the theorem of momentum in Tollmien's formulation (reference 4). NACA TM 1311 27 If one marks off a control area in the indicated manner, one obtains in the rotationally symmetrical case U U vr uu uO r dr = (rTXy) u0 = velocity of the medium surrounding the jet. According to the more recent Prandtl expression Txy = Kb(x) (ul U) Thus we obtain, if we, furthermore, take the limits of the mixing zone into consideration U uo) vr 7x I U(u uO) r dr = rab u U O According to the existing conditions we transform (according to Kuethe) with r d(x) S= b(x) Then r d(x) S b(x) r = bq + d S d' b' fx b b If we make, in addition, the assumption that u depends only on T, not on x, there follows rd+b 1 [( u uO 2bb' T(bd' + b'd) dd]d 28 NACA TM 1311 For v we finally insert the continuity equation v = r dr r 1 or V= (b + d) a 2bbl i(bd' + b'd) dd d For approximate calculation, we write for the velocity distribution the sample expression u uO ul ) ul ul r d(x) S= 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 br determining the shearing stress also will probably result in a usable approximation for the central region of the mixing zone. The result is u ul 1 ul uo 8r~ b ul d u u 2ul 2 ( 1 U TLI j 2u( ul) uul ) (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 UO)r dr = const 0 NACA TM 1311 or fd+b ulUl uO)r dr + 2 (U UO)r dr = ul(ul u) ro 2 Ul(U" O ) *1o +J u(u uo)(lb + d)b d = ul (ul UO) IUl u0)(d2 2 21 i 2) b2 1 uu +  r +  dsj + bd 0 / J Q 1\ U0 / . 1 U 'O) 0 I u If one inserts one obtains S r2 2) 2 d2 r ) U Ul (l ) (I _) and carries out the integration, Ul ) + b2 ul"  ulb 1 ul E + bdul ul or 1 ul uo  ul + bd 1( 1 1j M i. i^ S (2) In order to obtain a second equation between b and d, we put r = rO. If one performs the somewhat lengthy elementary calculation, one obtains finally (compare appendix no. 2) bbl uO 1 rO d' 1 + (rO d3 bb  + ro + ul 7 3 b 3 + ul u 1 rO d +3 + 1 (rO d 2 1 (b'd + bd') l u rO d)3 + 2 + ddtro d 1] r .ul uo a'L 1 ul 1d(x) O 2 rg 2 b 2 + d = r + = 0 ul 2) 30 NACA TM 1311 The theorem of conservation of momentum reads in differentiated form. (Compare (1).) bb' uVl + (b'd + bd') ( u O] + dd = 0 By addition of the two equations one obtains bb Ob d + (b'd + bd')L ob d  \(u uo)(ro d + dd' = rOK([I U u ) D\ bl I b Y % u1 ) (3) We now proceed to determine b obtained. We replace b in the second the function which we obtain by solving to b. and d from the two equations equation by the expression for the first equation with respect b d )\ 2 ro a(0 + 1a a2 r0 ko)r/ V1 " where 1 ul Uo a =  0 2 1 ul u10 3a,3 aul Uo A U1 3 a2 = 2kul 1 1 3 u  4 1 ul UO 2 ul NACA TM 1311 b' a d' Substitution then yields d' = i 1uf ul with K (I2R + 01 [)+ al a2(d)] ([ a23o 6) 3 a d a/d d 2 a2 1 1  !\ro o) + 1 ~,2 O 0 Sro 2 x d We obtain as a function of  rO rO d(x) 1 1 K ul Uo) \ ul / d \ u1 SUl u f2) d(d) ul UO f d ul rO1 The evaluation of the integral could, in itself, be carried out by analytical method since the integrand is built rationally in and ro fl= 1 2 = 32 NACA TM 1311 a square root. However, the breaking up into partial fraction which has to be done in this procedure is very troublesome. Hence it is advisable to perform the evaluation graphically. For = 1 the integrand f UO f nite expression The limiting value is nite expression . The limiting value is (f l uO ) 1 2 assumes the indefi 1 (a,) ao d  was determined, analytically or graphically, as a function rx b(x) results from b d+ a, ( 2 80 O 0+ al 8r2 The relation K (ul u aO + a2 =  a, +   lim f1 o1 r0 1 (Ul U)f (which by comparison 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 measured results on the plane jet boundary with zero outer velocity (given by Tollmien (reference 4)) as a basis, there results with (db) Wj=_o r^A = 0.255 lim d pl rO If x of , r0 ro IACA TM 1311 1 0.255 = K(3) 0.185 K = 0.01576 Examples: In figure 8 the dimensions of the corresponding core region are represented for the parameter values ul uo = 1.0, 0.8, 0.6, 0.4, 0.2 ul xk Figure 9 contains the core lengths , figure 10 the mixing bk rO ul U widths at the core end as functions of ro Ul Figure 11 shows the mixing widths b for the various r0 ul Uo x values of as functions of . ul ro parameter Figure 12 represents the angle of spread of the respective mixing region c = x r 1 Figure 13 represents UO = as a function of Ul u Ul ul uO with Tollmien's value defining quantity. defining quantity. c = 0.255 ul UO for = 1i uI being the Figure 14 shows for the medium at rest Ul = ) the quantity K as a function of c = bx Figure 15 shows Or = as a fu0 (db rO function of c b *dro rO NACA TM 1311 Figure 16, finally, contains the limiting value lim lul UO r necessary for calculation of the integrand in d x 1 1 fro ul UO f d = ____ r fl U]_U\ d\ rO ul uo 1 u)l r0 ul 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. In order to test the theory by experiment, a measurement for the ul uO case = 0.5 was performed at the FockeWulf plant. Ul 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 pk 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 pressure pk = 100 mm Hg (Measured relative to atmospheric pressure) Room temperature tO = 200 Barometer reading PO 754.5 mm Hg NACA TM 1311 35 Inner jet: Total pressure Ps = 340 mm Hg (Measured relative to atmospheric pressure Stagnation temperature t, = 590 The evaluation of the measured values was made according to the adiabatic T1 =T2( and the efflux equation ul K gRT2 1 ( 2; ] 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  tA = 9 uA = 151 meters per second for the jet issuing from the inner nozzle o  ti = 13 ui = 302 meters per second The inner jet therefore has, compared to the outer jet, an excess o UI uO temperature of 4 For the velocity ratio the result = 0.5 uI was obtained. P P+ Ps + Pk PB PB Figure 17 shows the total pressure distribution , s central B + PB Ientral 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 18. Figure 19 shows, in addition, the variation of the total pressures along the jet axis. Figures 20 to 22 contain the corresponding representations for the velocities made dimensionless by the velocity ul 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 turbulent 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 ul which is equiva lent to the existing momentum distribution. That is, we calculate the effective nozzle diameter from the equation 2 r effect. 0 uo)r dr l u0) 2 u0)r dr 0 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. = 0.95rgeom. Whereas the plotting over n = r geom, lets the test points appear x r effect. as still lying on one curve, the plotting over T = eff x results in a stagger of the velocity distributions with increasing  ro toward negative T. This stagger toward negative q expresses the immediately obvious fact that the isotacs of the flow field are curved toward negative n (toward the jet axis). x Xk Figure 23 contains the theoretical curves for 0 and  ro r (the core end); in addition, the test points of the sections x = 10 mil limeters and x = 45 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 surrounding 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 xk = 22.0reffect, whereas the measurements along the jet axis (compare fig. 22) result in about xk = = 20. 1rffct. x K0.945 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 = Kb lmax uminl for the momentum transport. The continuity equation and the equation of momentum are at disposal for calculation of the velocity distribution. In case of limitation to small disturbances Iul O small quantity, where ul is jet exit velocity, u0 velocity of the surrounding medium the equation of momentum may be linearized r x = E(x)r + r L An approximate solution is constructed which is characterized by the fact that in boundary zones of the region as well as along the jet n = 0 in the interior of the region it has to be regarded as exact solution. (ul u0 initial For arbitrary disturbances arbitrary > 0 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 course of mixing width is assumed. The dimensions of the mixing region (limiting curve of the jet core d(x) and mixing width b(x)) are approximately calculated from the theorem of momentum u u(vr (u uo) r dr (= rxy) r = rKb(x)(ul UO) r under assumption of a rectilinear course of the velocity distribution r d over T, where t = b In order to test the theory by experiment, a measurement was ul uO performed for = 0.5 with a 5 millimeter nozzle. In order to U1 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 ( u \r effect. ul ul "O) 2 = u2 uO)r dr 0 The agreement between theory and experiment is satisfactory. NACA TM 1311 V. REFERENCES 1. Kuethe, Arnold M.: Investigations of the Turbulent Mixing Regions Formed by Jets. Jour. Appl. Mech., vol. 2, No. 3, 1935, pp. A87A95. 2. Prandtl, L.: Bemerkungen zur Theorie der freien Turbulenz. Z.f.A.M.M., Bd. 22, 1942. 3. GSrtler, H.: Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen NNherungsansatzes. Z.f.A.M.M., Bd.. 22, Nr. 5, Oct. 1942, pp. 244254. 4. Tollmien, Walter: Berechnung turbulenter Ausbreitungsvorgange Z.f.A.M.M., Bd. 6, Heft 6, Dec. 1926, pp. 468478. (Available as NACA TM 1085.) 5. Pabst: Die Ausbreitungheisser Gasstrahlen in bewegter Luft. UM 8003, 1944. 6. Reichardt, H.: Uber eine never Theorie der freien Turbulenz. Z.f.A.M.M., Bd. 21, 1941. 40 NACA TM 1311 APPENDIX NO. 1. CALCULATION OF THE TRANSVERSE COMPONENT (1 )(X 1 d n d r0O 1 + f x = +  \ x 1T dTl r. S1] 2da _g'p dT1 + a 3k r0 d] + ITk l k We substitute C = (an1 1u  7u/u1 "srr 0 Uo  0.36  ul + 2) e ]2 l' ( + a) + a2] 1(/uo vr. uj  i) e]2 p[] Jo []2 e  1) d[l +  = (1 ro ^k X ) I i c LIl NACA TM 1311 If we assume a rectilinear course of the mixing width b(x), we have ai' = 0. We then obtain SV 1 1 x e 2  1)2 T l) 2.. f Ir TI T 1 d i + ' k e2 d + 1 _l/uO 1 +  x 1 1) 1 S' _ 2 _ 2 ro 2l e I x  The evaluation of the integrals yields e Tnk d le 2 dlI  i e2 d(ail) alJ e 2 1 fe1 d[J a1 [] = (a  0.36 ul uO + c2) uI Se dTI = 1] e k [0 []o ~TI ) 0.36 Ul 2 d[a e 2 1 e dq e 2 thus NACA TM 1311 po .p0[ 0H O I e"2 d[] = [[ 1  []2 e daC error integral d (F1 +F 1 2 I 1 .41 J e112 k ul uo C] = al 0.36 u + 2 U1 Ul u0 Tr (2 U + 0.36 This results in Ul 10 1 r+2 0] + 0.36 U a e1 d(alqi a1 f Q1 = 12 1e' 2 *= f2 0.36 u u l a 2 [I 2U1 r e0 / I C []0 where F 1 2 ro F = L 1:K 10 thus []2 e re2 e & = J 11'"0 0o + " Jo NACA TM 1311 0 . H rr r4 0 Cu HI4 iO I II CU ,n 0 0 Cu CI 0 JCU 0 So II OJ O I 4 + Hl tIcyi Cu 4 (O I 0 rC '0 I b ,O I Cu n 4) 0 m, cu t  Ib bt cu CO I (1 o   cu 0 ID r ICU cu F '^ F F~ Cuj Cu I HI H I CU 4) HI II CU Cu 44 NACA TM 1311 4 + 4; Sa + a) r4 cu Sao u cu cu  +1 cu b Io ri lOJ m b , + 4d 0 cu n Cu o c u fcu 0cu r4 + (D C n b 0 4 CWu + U C I0 , + 0 0 Cu r cu m r  + P HIu 4, 1 0 u pY ~ NACA TM 1311 APPENDIX NO. 2. FOR CALCULATION OF THE DIMENSIONS OF THE CORE REGION The theorem of momentum with r = rO reads U u0 r=r0 drb Lr0 r=r0 r r ^ul ul Sul d Wth the coordinate transrmtion b e obtain With the coordinate transformation r = we obtain (r=rO O d+b O )= rod 1 b ro 0 d u/ul 2b dT lbb  T(bd' + b'd) dd] d Sbb' T(bd' + b'd) dd] dT rod b We substitute u O O (u Tu0 = ul'Ul) NACA TM 1311 d [u uOJ] dl ul ul uJ = 2(1 U ul i (i n)  uOc uO) uIV ul 1(u1 kuo) This results in ( U_ uo(r Ul Ul r=r0 ro = (l uo ( S 1 ( ro U ) uo [b b ul TI T)2 Sd + 0 (bd' + b'd) i dn + dd' "0 "0 1 r0d b di] (1 q) + 2 ulO / 2 = rO ul ) 7 = (i} UA ul / Sd+b Sro NACA TM 1311 If one evaluates the integrals, one obtains u uO )r ul U1r=r S(ul u~O 2 ( 0 bb' ,ro d 3 ul b 3 bd' + b'd .2 ) = bb' (ro d 2 dd( ro d + t^_^ 1 rod b 2(1 T)dT + (b'd + bd') 1 r d (1 i))dT + b UO1 uo\ 1 U1\ ul r0od b  11  1(rO d 1 ro d 2 4\o^ul 1 dTb Od b bb' 2 7l1 2 [r z 1 frod b .(1 T)dT + L (i d + (b'd + bd' + (r Ud + ui b 7ui I j 2 u i 1 0 10 d lur  / dd'tul 2 b 2u"1 uu uOl U Ul \ U1 1 1 dT + 1 Jr0d b . 47 S uUr uI ul ) TI dh + X jr d+b 4b NACA TM 1311 If we insert these expressions into the equation of momentum and order, we obtain bb 'O r d 3 ul uO ro d b ul b u b 3'\ u ) ,kb I 1 1 uO Ill d 2 /r d 4 b  (u u 2 '6 ui ^ 1 u(ul UOrO d3 3ul ul b d 2 l2 UO 11 uOI 2\ ul r d b ) 1(u+1 Uo or d)2 + (bd' + b'd) r ) 2 2  u u o O b d \ Uo 3( ul ) 2(u1 uO ro d3 3\ b 3\ ul b UO (Iul UO 1 uoi uu u(r d 2 2 u\ U A \ au v u ) [ 2 (ul U \2r d ul b ( (ul uO)(rO d2 uO(u uo u b ul ul UO lui u0) (rob d b ^ / l Uo . ul o 2 u )  r ( u1 1o uoC 3 ui\ ul ) NACA TM 1311 which finally leads to 1 1 (r d  + + b 3 (b'd + bd') l O)o Ld) + .1 dd' rd) = r l Uo ) Translated by Mary L. Mahler National Advisory Committee for Aeronautics + Iro d i71 NACA TM 1311 o9  c i I 17 1.A I I I I 1i i 0 ci o / 0 * U2 ci loo > J T tLo a 0/ ^zz~z ^5^_ NACA TM 1311 t K. ** ___ .4. 0 10 20 30 40 so 60 Sax    a    0 f0 r0 00 40 SO 60 TO __ H           oo   _/^ 9o / / dzz / / o., /jy t / o 10 20 30 40 SO 60 x Figure 2. Auxiliary quantities for calculation of the velocity distribution. NACA TM 1311 0 4 C1I ( o 0 o __ .r1 NACA TM 1311 54 NACA TM 1311 NACA TM 1311 55 C4 0 le *r  _   I I o O a N o 0 ,3 ____ _.__ 56 NACA TM 1311 C" t. 0 .0 S1 1 U Sa d ,UD 4> U' NACA TM 1311  II lot.   I s 0 "i ,, r) 0 I I I o S\ S i c      _    / I I .I NACA TM 1311 0.2 04 0.6 0. 1.0   Core lengths. Figure 9. NACA TM 1311 59 o  0 0 0.2 0.4 0.6 0.8 1.0 l v Figure 10. Mixing widths at the core end. 60 NACA TM 1311      0  \  * __ V _L  ^^ S ^ 3 : NACA TM 1311 0 0.2 0.4 0.6 0.8 1.0 U U1 Figure 12. Angle of spread of the mixing region. NACA TM 1311 Figure 13. Parameter value of the plane jet rim. 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