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t I l  NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 17O. 1085 CALCULATION OF TURBULENT EXPANSION PROCESSES* By Walter Tollmien On the basis of certain formulas recently established by L. Prandtl for t'he turbulent interchange of momentum in sta tionary flows ('redfrence 1), various cases of "free turbu lence" that is, of flows without boundary walls are treated in the present report. Prandtl puts the apparent shearing stress introduced by the turbulent momentum interchange T = p12 du (1) xy dy dy where u average velocity in x direction y coordinate at right angle to x 1 mixing length The underlying reasoning is as follows: The fluid bodies en tering right and left through a fluid layer with the time av erage value of the velocity u, at turbulence, have the aver age velocity u + I du or u I du while the transversely dy. Idu dy directed mixing velocity is I discounting a constant of dy proportionality included in the more or less accurately known I of formula (1); 1 i.s no constant at a wall 1 = 0. The previously cited report by Prandtl (reference 1) contains a lucid foundation for formula (1). The present report deals first with the mixing of an air stream of uniform velocity with the adjacent still air, then *"Eerechnung turbulenter Ausbreitungsvorginge." Reprint from Zeitschrift fur angewandte Mathematik und Mechanik, vol. 6, 1926, pp. 112. 2 NACA TM No. 1085 with the expansion or diffusion of an air jet in the surround ing air space. Experience indicates that the width of the mixing zone increases linearly with x, if x is the dis tance from the point where the mixing starts. This fact is taken into account by the formula. 1 = cx (2) The constant of proportionality c can as yet be determined only by comparison with experience; it.isthe only empirical constant of the theory. In many'instances it will be expedi ent to introduce T] = y/x as a second coordinate. 1. MIXING OF HOMOGENEOUS AIR STREAM WITH THE ADJACENT STILL AIR (Twodimensional problem of the free jet boundary) By reason of the limiting conditions for the average.ve locity the formula is preferably expressed with u = f(y/x) = f(M) (3) Then the stream function i:s S= Jf(;) dy (4) = x f(TI) d. = hence S' = + TJ F' Q1) Quantity T is put according to formulas (1) and (2) ..... = c2 xa du du *: : p dy dy The following boundary conditions exist: At the first boundary T (homogeneous air stream), u = constant or by introduction of a suitable scale u = 1; that is, IACA TM No. 1085 Fl (T ) = 1 (5) furthermore = 0 a condition by which the continuous connection is secured  that is, F"(11 ) = 0 (6) Sv(11) = 0 that is, F(TI) = I 7) at the second boundary T12 (still air) must be u = 0'; that is, F'( 2) = 0 (8) and, to assure continuous connection au 0; that is, M"(0.1) = o (9) Since 'the pressure, in first approximation, can be assumed to be constant, the equation of motion reads u + v a = i y ax ay p ay Counting y and T) from the still toward the moving air, gives, after introduction of the formulas*, .t,he.equation of motion: FF".+ c2 F" F' =.0q ,,( 0) *It is readily apparent at this point, that the formula u = f(T)),.ecessarily requires an 1 proportional to x. NACA TM No. 1085 * which is solved by F" = 0 or F + 2c8 F"' = 0. It affords uniform velocity in the one case, variable velocity in the other. The latter solution obviously applies between 'i: and 1g, the former, outside of these limits. In the bound ary points the solutions coincide with discontinuity in F?". In order to determine the velocity distribution in the mixing zone, the differential equation of the third order F + 2c2a = 0 (11) must be solved. For the time being, a new scale for 1 is advisable, so that formula (11) simplifies to F + ?" = 0 (lla) The result then is F = ci en + c2 ef/ cos 3 1 + ca el/2 sin  2 2 The five boundary conditions define the constants of integra tion cl, c2, c3, and, in addition, the still unknown bound ary points  and rT themselves. The calculation is suitably arranged as follows: Put T = ) T or = 1 + q, so that F = d, e + da e/a cos r + d3 e1/a sin 3 2 2 The boundary conditions are: F(f) = I1, F'(T) = 1, F(T) = 0 for 1 = 0, F'(T) = 0, F"(Ti) = 0 for T (12ae) NACA TM No. 1085 From (12a to c), dl, d2, and d3 can be linearly expressed in I,, equation (12d) yields 7h, expressed by TIa, and (12e) finally gives a transcendental equation for Tie, solv able by successive approximation. It follows that 11 = 3.02, Ti = 0.981, ,2 = 2.04, di = 0.0062,. de = 0.987, d3 = 0.577 With this F and F' are defined as function of T1. For comparison with experience, the original scale must be used  that is, the reduced Tj, employed thus far, must be multi plied by Y 2c, and the reduced F' by U, the velocity of the homogeneous air stream. The curves of the velocities and of the streamlines are given in the table and in figures 2 to 4. The streamlines are plotted for equidistant values of the stream function. The streamline emanating from x = 0 is a straight line with angle of inclination tani (0.19 V42c) For comparison,a Gottinwen measurement (reference 2) of the dynamic pressure distribution with an automatic pressure re corder at the edge of the nozzle of the big tunnel was em ployed. The distance from the nozzle edge was 112 centime. ters, the dynamic pressure of the undisturbed jet q = 56 kilograms per square meter. It can be presumed that the as sumptions of the twodimensional problem hold good for the size of the nozzle. In figure 5 the calculated dynamic pres sure is shown as a dashed line over the measured dynamic pressure. The unknown constant of proportionality c follows from the conversion factor for TI, which is V2c = 0.0845. The width of the mixing zone is b = J2c x 3.02m = 0.0845 x 3.02.x = 0.255x giving a mixing length of I = 0.0174 x = 0.0682 b NACA TM No. 1085 The relative smallness of I is unusual. The agreement be tween the theoretical and the measured average velocity dis tribution is very good. 2. JET EXPANSION AS TWODIM;IINSIONAL PROBLEM Visualize a wall with a narrow slot, which for the study may be retarded as being linear, through which a jet of air is discharged and mixes with the surrounding still air. As suming, for the first, that the pressure in the jet is the same as outside, the application of the momentum theorem af fords a ready separation 6f the variables' that is, x and TI. By reason of the constant pressurethe momentum in x direction must be +0C u2 dy = constant 00 Putting u = cp(x) f(7) results in +Co p((x) x ( f2) dl = constant Co Oonsequently cp(x) = 1 u =  f(Tl) (13a) = f(T)dy =,x f(T)dn = /x F (k) (13b) v = L F(r) + 1 P'(TI) 71 (13c) The equation of motion can be set up again, which now, however, can be immediately integrated once. This interme diate integral can equally be obtained direct from the momen tum theorem. By marking off a control surface conforming to figure 6,the impulse entering through the lower boundary is pu v, while on the other side the impulse variation NACA TM.No. 1085 p u dy occurs. The turbulent shearing stress 8x.J ap T = pc2 xa j a i y oy acts as outside force, hence the relation exists Y uv + uj dy =  From this follows the equation for F(T1): 2c F"2 = FF' (14) (valid for positive TI, reflected velocity distribution for negative i). With a suitable scale for 'i the differential equation is simplified to F"2 = FF' (14a) The order of this differential eauation can be lowered by in troducing z = In F; that is, F = es as new dependent variable. Then, (z" + z'2)2 = z' whence, after putting z' = Z, finally follows the differential equation of the fir st. order: Z' = Z ./Z The solution of the original equation then requires only squaring and removal of the logarithms. The following conditions must be satisfied: ?or T = 0 (center of jet), v = 0 that is, F = ez = 0. Since u F' = z'es is not to disappear for T1 = 0, z' must be of the same order of wo for Ti =0' as ez is of 0. Hence, for J = 0 by suitable scale determination, F = 0 (l5a) FE = 1 (15b) 8 NACA TM No. 1085. Thus there are afforded two conditions through which the z, that satisfies an equation of the second order, is completely defined~ ':" . The boundary point Tir itself then follows from the condition u = 0; that is, z' = Z = 0 (16) for the boundary Tsr. Integration of equation (14) gives ] = 0 2. in C/Z + 1) In (Z / + 1)/ 2 / +3 tan1 2 1i (17) The constant of integration C follows from the condition z' = Z = c for T = 0: :'S a: .v 0 = C 3 i'T at C = 3 2 3 The condition (16) Z = 0 for Tir yields 7Tr = C 'tan1( ) 4 L 2.412 3 3 ^3 Compliance with equation (15) is predictLed on a study of the behavior of equation (14) for TI = 0, Z = co. As the solu tion (17) in this range is inconvenient, a new form of solu tion which applies for TI 0, Z = co is derived. For Z >>, obviously dZ , that is, = z' > hence z > In T + c,, so that F' = z' ez = eC1; for T 0 = 0. Thus the last constant of integration follows from equation NACA TM No.. 1085 (15b) as ca = 0; and the asymptotic approximation for 7 = 0 follows at z = 0.4 % + 0.01 T ., z = in 8 3/ 0.01 . 1 3 3 (18) The quality of the asymptotic approximation (18) is easily ap praised by a comparison with the exact solution (17) in a zone in which both forms of solution are appropriate. The method is as follows: Compute. T)(Z) and hence Z(TI) = z'(TI) by equation (17), thus obtaining z(T)), possi bly by graphical integration, where in the region about Ti = 0, the previously determined asymptotic approximation (18) is taken into account and z' = m. Then the desired functions F = ez and F' = z'ez are obtained by removal of logarithms and multiplication. The solution F = constant joins the justderived solu tion with a discontinuity in ?"' toward ,the outside. In the center (7 = 0), F' acts as 1 0.4 T3/2, which entails a disappearance of the radius of curvature.* For comparison with experience, it is necessary to revert from the reduced to the original quantities as shown in the table. The conver sion factor for Ti is 3 2F ; the letter s, in the table, signifies a characteristic distance from the gap, where the speed in the center of the jet equals U.. According to (13a) the speed at jet center distant x from the gap is then UmW() = Us J ix (See table.) 3. JET EXPANSION AS ROTATIONALLY SYMMETRICAL PROBLEM The corresponding rotationally symmetrical problem, in which a jet of air discharges from a very narrow hole in a wall, is treated in exactly the same manner as the two Prandtl has given a refinement of the theory by which the disappearance of the radius of curvature in the center can be avoided. But, since it would lead too far afield, it is not discussed here. 10 NAOA TM No. 1085 dimensional prob.l'an. ;.:Firs't, the.vaniables x. and 71 are. easily separated again. For, on assuming that the pre.ssure in the jet is constant, +co 27r uay dy = constant 'whence for u Sx1 Putting (fri) 1 dfl= F(Tl) affords F' F' F " u , v XTl x xTl The differential equation for F is again obtained by i.nte gration of the equation of motion or by a second application of the impulse theorem in analogy .to figure 6: .. I i/ c 2F" F' = FF'1 (19) ": With the introduction of a suitable scale for T, the differential equatibo is simplified to (jF I = T1 (19a) By substitution: z ="ln F, 'F = ez there is afforded z1 i+ o.z' .2 , and lastly, after introducing Z = z', the differential equa tion of the first order NACA TM No. 1085 11 Z, = 7 z ZJZ (20) In addition, the following conditions hold for 1 = 0: u may not disappear, while v = 0; that is, F(O) = ez()= O, while = z'e" remains finite and becomes eoual to unity by appropriate regularization. Now a series development of Z(iT) for Ti = 0 can be applied in such a way that these conditions are satisfied; z must be negative a for fT = 0, in order that ez = 0, which is like In P2; because F'/T; then assumes precisely a finite value. The result is the following development in powers of Ti3/2: Z = 2 + a7 + bi2 + ci7/2 + d5 + e13/2. (21) The coefficients are obtained'by introduction of this formula in the differential equation and comparison of equal.powers: 2 1 IL, 37 a b = 2 d = e = 0.000014 7 245 1715 240100 The convergence is poor on approaching the boundary point 1,r(Z=0), but a development particularly suitable near 1Ir is as follows: Put S= r  and Z= a + b T + c' + d + e 1 + f . and obtain 1 1 3 3 a = = 4 8 Ir 64 1r2' 64 128 Tir3' e 19 133 0.00278 ... 256 x 5 +Ir 256 X 40 1r  Ir T .r 256 x 5 *Hr 256 x 40 By4 Tlr3 T~ r 5 NACA TM No. 1085 The unknown constant of integration Tir is obtained by making the values for Z, as known from the two developments, agree in a certain junction point. It results in ir = 3.4. Quantity F'/Ti acts like 1 0.202 3/ in the center; the outward junction in F again takes place with a discon tinuity in F"'. The conversion factor from the reduced to the actual quantity Ti is I/ca; s signifies in the table a char acteristic distance from the discharge hole for which the speed in the center of the jet is Ug. The computed velocities were compared with Gottingen test data. (See reference 2.) The diameter of the discharge noz zle was 137 millimeters. The velocity distributions at 100 centimeters and 150 centimeters distance from the nozzle edge were used for the comparison. This nozzle distance a may not be put equal to x, in view of the point discharge ori fice assumed in the present calculation; x is rather com puted from a by addition of a constant quantity e which results for example from the fact that for greater a, for which the comparison with these calculations is solely permissible, the central velocity decreases as 1/x. In the present case e = 26 centimeters. Figure 13 shows the theoret ical and the experimental dynamic pressure for a = 100 centi meters, it amounts to 104 kilograms per square meter at the discharge orifice; the agreement of the average values is good, aside from a certain asymmetry of the jet which must have had different reasons. From the conversion factor for I follows c a= 0.063 The radius r of the jet is r =3 2 3.4 x = 0.063 x 3.4 x = 0.214 x The mixing distance I is = ex = 0.0158 x = 0.0729 r. Zimm (reference 3) has made corresponding experimental investigations at considerably lower speed. His findings would yield V7 = 0.080 with a dynamic pressure of 5.1 kilograms per square meter in the discharge orifice. According to it, a slight increase in mixing path by decreasing Reynolds number is likely. NACA TM No. 1085. 4. PREDICTION OF PRESSURE DIFFERENCE So far, all cases had been premised on constant pressure. This first approximation can be improved by analysis of the pressure differjr.ce due to impulse variation on the basis of the computed. s:,cds ar.d stresses. For the first step, start, say, with the second equation of motion, which in the first two cases reads cv av 1 Sm 80y\ 1 8p .u v + v l (a + aay\ lap a x y p \x y / p ay and V 2v 1 (T 1 y(" ) _G t 1 ap 1 2 + v  + aC7 Zf _ ax oy p \ax y ay y p ay in the rotationallv symmetrical case; Cy and at are nor mal stresses, respectively effective in y direction or at right angles to y and x. Then, integrate with respect to Y: Y Y v + d /uvdy 1 Td y 1 p = .(22) o Px. p o p L o 0 0 and S y y Y Y Y v + u /vdyv+ iv dy 1 Td 1 0 ox y p ax. P 0 0 10 0 o o o Sy +dyG + d y 1 I (23) F y P y P JO 0 0 which is equivalent to applying the impulse theorem. If, as heretofore, the normal stress, in this case Cy and ut, are discounted, there is obtained I1 Fai' 2/F] i 1 .i 2 F F TdF 2 d'l : p fL the feP j yo for the free jet boundary, NACA TM No. 108.5 x L 4o0 P o for the twodimensional jet expansion +Po d I; 0a o T13 P L P o 0 for axially symmetrical jet expansion. With Pr denoting the pressure at the jet bhun&ary, pm the pressure at jet center and of the homogeneous air stream, respectively, particulari zation of the above formulas yields Pm Pr 0.410(2c8a/a 2 and Pm Pr = 0.248(2c )2/3Um(X) P P and Pm Pr = 0.316(c2)/ U (x) Quantity U indicates the speed of the homogeneous air stream, and Um(x) the central speed at x. In the first and third case, c has been determined, giving Pr = 0.00584 P 2 and U, ^ andPm Pr = 0.0025 p Umg(x) 2 It is apparent that the thus computed pressure differences, being small, do not cause a' substantial .modification of the velocities. When computing the pressure difference with respect to still air, it should be borne in mind that at the jet boundary a negative pressure equal to the dynamic pressure of the. ra dial inflow speed prevails. With po as the pressure in still air WACA TM No. 1085 m p = 0.338(2c) p U = 0.00482 P U2 2 .Pm Po = 0.124(2c.)/ p Um (x) Pm Po = 0.372(c ) p U(x) = 0.00295. U (x) 1 2 Hence there is positive pressure within the jet in the two dimensional cases, but negative pressure in the axially sym metricsl case. This surprising result,which also is at vari ance with a rough impulse consideration, points to a defect in the theory. The necessary extension will be given in the fol lowing. 5. EXTENDED THEOREM FOR THE APPARENT STRESSES The theorem applied up to now to the stresses introduced by the turbulent impulse exchange T = I2 uI u 7 = 7, =' t = 0 T = t is no more than a fist approximation. I n any case, it awn be easily proved that au/by in tihe cases in point is great with respect to Cu.. and ad hence the theorem for the mix ox ox oy ing speed I Cu caused by the speed difference is good. So, i,n a natural generalization of the previous theorem, the stress' tenso'r is p'ut equal to I] (7X X) . (Vv = affinor of v; v7 is the conjugate affinor.) LThis relation is important for the calibration of nitot tubes in a jet discharging from a nozzle. NACA TM No. .1085 The stresses to be newly added here, are, in general, neglected, except r = 2 8U aV and d t = 2 12 8U which are used y ly y y, y for calculating the pressure differences. This portion cancels in the model problem worked out for the twodimensional case because of the employed boundaries, but not for the axially symmetrical case. Here the pressure differences are augmented by the integral y Sy dy so that pm Pr = +0.151(c2) p Um 2() = +0.0012 p Uma(X) 2 and m Po = 0.095(c2)/3 p Ua(x) = 0.00075 P Uma(x) 2 that is, positive pressure within the jet, as in the other cases. Translation by J. Vanier, National Advisory Committee for Aeronautics. REFERENCES 1. Prandtl, L.: Bericht uber Untersuchungen zur ausbebildeten Turbulenz. Z.f.a.M.M., vol. 5, no. 2, 1925, pp.'136139. 2. Betz, A.: Velocity and Pressure Distribution behind Bodies in an Air Current. NACA TM No. 268, 1924. 3. Zimm, Walter: Uber die Str6mungsvorginge im freien Luft strahl. Forschungsarbeiten aus dem Gebiete des Ingen ieurwesens.,. no. 234, 1921. NACA TM No. 1085 17 I M co W o M o co4 o M 1 1 1 1 1 1 1 a;C c7w e C; n0'e Wo Co r 4 ot 00 C 0 o Wo L o M ( LO mo i 7 asoo io'oo ose dd'doois oo O ' t m C> '0 r w m0 o i rCtOO MO cO' O lO OO I 0 Cd tz 0 Q O 0 0D o oooooooecooooooooo 0L 0 0l: 0lO. 0t 0 0 0 0 C 00000000000 0 wI w iiIi+ +C++ W 0 00 00 00o 0 0 00 0 0 4 3 LO 0 01 i 0 M co 4n o eq o c e m Cn 4N 0 u r OS c' O a in i t QQ u ^C e CD C7 Cr Cw c) iN rli I Oi O Siiodd cooTi doc dopdd M ) > 1 1o1o 1 1 1 1 1 I IOl l l NACA TM No. 1085 Figs. 1,2,3,4,5,6,7,8   \ / .,4. z Z'p \ oo P a lr Cr4 4 O 0 ,. cr , . & I NC t 4 10 N9 a a' C > C at / ::; a C iC / 2 c a 1'se a L \ i \ i II 0  I\ 054 / 1\ NACA TM No. 1085 ci I, PC , a $4 a W 0 0 *r .r4 0: 43 * 5=V E1 CM a '~ ~EI r Ist4E40 Figs. 9,10,11,12,13 / 3 r */ 6 .p S* II \ / 3 a P A 4 E* 9+* ii UNIVERSITY OF FLORIDA 3 1262 08106 276 1 
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