//K/VTm-|o<^b NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1085 CALCULATION OF TURBULENT SXPANSIOH PROCESSES By Walter Tollmien Reprint Zeitschrift fur angewandte Mathematik und Mechanik Vol. 6, 1926 -^^^^agp^ Washington Septe.T.ber I9U5 Ur41Vt:HSITY OF FLORIDA DOCUMENTS DEPARTMENT 1 20 MARSTON SCIENCE UBR(\RY RO. BOX 117011 GAINESVILLE. FL 32611-7011 USA ^10 >^ ' ' ITATIOITAL ADVISORY COMMITTEE FOR AERONAUTICS TECaiilCAL.. MEMORANDUM EO. . 108 5 CALCULATION OP TURBULENT EXPANSION PROCESSES* By Walter 'Tollmien On the ■basis of certain -forinulas recent ly established hy " L. Prandtl for 'the turbulent interchange of momentum in sta- tionary flows (^reference 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 = pi' xy ^ du I du dy ! dy (1) where u average velocity in x direction; y coordinate at right angle to x I 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 directed mixing velocity ii u - I — !i, while the transversely dy -^ , discounting a constant of ...dy proportionality included in the more or less accurately known I of formula (l); I i.s no constant - at a wall 1 = 0. The previously cited report by Prandtl (reference l) contains a lucid foundation for formula (l). The present report deals first with the mixing of an air stream of uniform velocity with the adjacent still air, then "Eerechnung turbulenter Ausbr eitungsvorgange . " Reprint from Zeitschrift fiir angewandte Mathematik und Mechanik, vol. 6, 19 26, pp. 1.^12. • . • NACA TM No. 108 5 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. I = ex (2) The constant of proportionality c can as yet he determined only hy comparison with experience; it is the only empirical constant of the theory. In many iii st ance s it will he expedi- ent to introduce T\ = y/x as a second coordinate. 1. MIXING OF HOMOGENEOUS AIR STREAM WITH THE ■ ADJACENT STILL AIR (Two-dimensional 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(Tl) Then the stream fun-ction is = X /. fC-n) d.Tl = xE(Tl) hence V = 8\1; GX -F(Tl) + V, F' (n)^ Q,uantity T is put according to formulae (l) and (2) (3) (4) 1 =c2 x2 du I du dy I 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, NACA,.TM Ho.- 1085.,,. furthermor e : -t ; .Fi(rji) = 1 1^ = OT] (5) a condition that i s , w.hich the continuous connection is secured - S^'COi) = (6) v(TiJ,= that i s , liT\x) =■ Til (7) at the second 'boundary T)g (still air) must he u that is, . F'Cn^) = • (8) and, to assure continuous connection H^i = 0; that is. F"('t|,7 '=''o (9) Since "the pressure, in first approximation, can he assumed to be constant, the equation of motion reads 3x 3y p d y _ Counting y and T) from the still toward the moving air, gives, after introduction of the formulas* , t,h,e, .equati on of motion: ' '"'- " >■■■-■ IF"^+ 2c F" F"'~..,0 (10) *It is readily apparent at this point, that the formula u = f(Ti),,iiecessarily requires an I proportional to x. 4 NACA TM No. 1085 ' ■' which is solved "by F" = or F + gc® F "' = 0. It affords uniform velocity in the one case, variahle velocity in the other. The latter solution obviously applies "between Tlj and Tig, 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 + 2c^ F'" =0 (11) must be solved. For the time being, a new scale for T\ is advisable, so that formula (ll) simT)lifies to F + P'" = (11a) The result then is F = c, e-^1 + C3 e^/^ cos -:^ T] + c^ e^/^ sin ^ T| o S The five boundary conditions define the constants of integra- tion Ci, Cg, C3, and, in addition, the still unknoxirn bound- ary points Til and T;s themselves. The calculation is suitably arranged as follows: Put Tf = T) - Til or T| = Ti + Til so that ", F = di e-^ + dg e^/^ cos ^ tT + ds eV^ sin — T| 2 2 The boundary conditions are: F(Ti) = T],. F'(Tl) = 1, F"(Ti) = for ^ = 0, F'(Ti) = 0, F'«(Ti") = for Tip (I2a-e) 1:ACA TM No. 1085 From (l2a to c), d 1 » ^2t and can be linearly expressed in Tlj, equation (l2d) yields Tlx. expressed "by Tig, and (iSe) finally gives a transcendental equation for Tig, solv- atle by successive approximation. It follows that ri'g = -3.02, Til = 0.981, Tig = -2.04, d^ = -0.0062, dp = 0.987, d, = 0.577 With this F and F' are defined as function of T) . For comparison with experience ,. the original scale mtist he used - that is, the reduced T| , emploj^ed thus far, must he multi- 3 / g plied by v 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 = is a straight line with angle of inclination •tan (0 19V2c^) For- compar dynamic pr corder at ployed. T ters , the kilograms sumpti on s size of th sure 'is sh pressure . from the c The width i son , a Go essure di the edge he distan dynamic r> per squar of the tw e nozzle own ^s a The unkn onver sion of the mi ttingen mea stribution of the nozz ce from the res sure of e meter . I o-dimensi on In figure dashed line own constan factor for Xing zone i surement (reference 2) of the with an automatic pressure re- le of the big tunnel was em- nozzle edge was 112 centime-^ the undisturbed jet q = 56 t can be presumed that the as- al problem hold good for the 5 the calculated dynamic pres- over the measured dynamic t of proportional ity c follows Tl, which is ^2c^ = 0.0845. s b = v^2c^ X 3.02 3c= 0.0845 x 3.02.x = . 255 x 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 aver-age velocity dis- tribution is very goad. 2. JST EXPANSION AS TWO-DIMSNSI ONAL PROBLEM '•'^isu'slize a v/sll with a narrow slotj which for the study may bo re,-:arded 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 of the variables - that is, x and Ti . By reason of the constant pressure the momentum in x direction must be r u dy = constant J — 00 Putting u = Gp(x) fdl) results in +C0 n cp^'Cx) X / f^dl) dT| = constant Consequently cp(x) = —00 1 ^ u = 4= f(Tl) (13a) X ^ = / -i^ f(T^i)dy = v'x / f(ri)dTi = Vx F(n) (13b) J -^ J V = L- F(T|) + Jl_ F'di) ?! (13c) 2vl^ '/x The eauation of motion can be set up again, which nov;, however, csn be immediately integrated once. This interme- diate integral can equally be obtained direct from the moraen- tum theorem. By marking off a control surface conforming to figure 6, the imptilse entering through the lower boundary is puv, while on the other side the impulse variation KACA TM,.Ko. 1085 i:':y TC ■ T = p C X i-~- acts as outside force, hence the relation exists y uv + -^ Fu^ dy = I ... dx / p From this follows the equation for F(Tl): 2c ' F" = F F ' (14)-; (valid for positive T) , reflected velocitj- distribution for negative ?, ) , With a suitable scale for 'f| the differential equation is simplified to F " = F F (14a) The order of this differential equation can be lowered by in- troducing z = In F; that is, F = e^ as new dependent variable. Then, (z" + z'^)^ = z' whence, after putting z' = Z, finally folloVs thef differential equation of the f ir str rojjder : . ' . Z' = -Z' yz The solution of the original eq\iation then requires only squaring and removal of the logarithms. The following conditions must be satisfied: For T] = (center of jet), v = - that is, F = e^ = 0. Since u ~ F' = z'e^ is not to disappear for T| = 0, z' must be of the same order of o= for Tj =""0'-as e^' is of 0. Hence, for Ti = by suitable scale determination. F = F' = 1 (15a) (15b) 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 T\^ itself then follows from the condition v , , u = 0; that is. z ' = Z = (16) for the boundary Tjj.. Integration of equation (14) gives '(Z -v^ > 1)^/^ n = C - 2 jin (Jz + 1) - In + ys tan-i 3/^-1 (17) The constant of integration C follows from the condition E ' = Z = 00 for Tl = 0: OsC-iyiH at C = JL- ^2 ^3 . . .;, V The condition (16) Z = for Tlj. yields n, = C .1^-3 tan- V- J-)=i^ = 2.412 Compliance with equation (15) i s'-pTedickt ed on a study of the "behavior of equation (14) for T| = 0, Z = oo. As the solu- tion (l7) in this range i.s inconvenient, a new form of solu- tion which applies for fi = 0, Z = oo i s derived. For Z — >t», obviously d^ dTl -Z 2, that is, Z = z' -$> 1 hence In T] + Cj,, so that F' = z' e^ = 6°^; for 11 = 0. Thus the last constant of integration follows from equation NACA. TM No.. 1085'; ■ ' (15b) as Ci = 0; and the asymptot,ic approximation for "n = follows at .- , ■. = i - 0.4/t\ + 0.01 T]2. . ., z = In Tl - 9.il T\^ / T] 3 s (18) The quality of the asymptotic approximation (l8) is easily ap- praised by a comparison with the exact solution (l?) in a zone in which both forms of solution are appropriate. The method is as follows: Compute.' T| ( Z ) and hence Z(Tl) = z'(Tl) by equation (l?), thus obtaining z(T|), possi- bly by graphical integration, where in the region about T] s 0, the previously determined asymptotic approximation (18) is taken into account and z' = c». Then the desired functions P = e^ and F' = z'e^ are obtained by removal of logarithms and multiplication. The solution F = constant joins the just-derived solu- .■: tion with a discontinuity in F"' toward the outside. In the... center (T) = O), F' acts as 1 - 0.4 T\^'^, which entails a disappearance of the radius of curvature.* For comparison with experience, it is necessary to revert from the reduced to the original quantiti es a s shown in the. table. The conver- sion factor for T\ is V~2c^; the letter s, in the table, signifies a characteristic distance from the gap, where the speed in the center of the jet equals Ug, According to (l3a) the speed at jet center distant s frpm'.the gap is then (See table.) Um(pc) = Us /s V X 3. JBT EXPANSION AS EOTATIONALLX SYMMETRICAL PBOBLSM 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 0"f curvature in the center can be avoided. But, since it would lead too far afield, it is not discussed here. 10 NACA TM No. 1085 dimensional problem . ■J'ir-st , the .vaElable s x. and T] are easily separated again. For, on assuming tha-t the pre.ssixre in the jet is constant, . . +00 v;h e n c e for u 2 TT / u^y dy = constant — OS Putting •u = i f(n) ■ X - f(ri) T| dTl = F(r,) affords u = xTl V = F xT) The differential equation for F is again obtained by inte- gration of the equation of motion or by a second application of the impulse theorem in analogy,. to figure 6: c ^ ^F" - II) = FF' (19) '••-'■-With the introduction of a suitable scale for T\ , di f f erehtial ■ eqtiat ibh is simplified to the F" U-T = F F ' T] (19a) By substitution: there is afforded z = In F, F = e z" + z'^ T1 = z and lastly, after introducing Z t i on of the first order = z th^ differential equa- NACA TM Uo. 1085- 11 Z'' = ^ - Z^ -JT (20) ■ -■ Tl ^ In addition, the following conditions hold for T\ = 0: u may not disappear, while v = 0; that is, F(o) = ez^o)=0, while -^ — = -2 — § — remains finite and "becomes eoual to unity Tl Tl ■ "by appropriate regialari zat ion . ITov/ a series development of Z{T\) for T| = can be applied in such a way that these conditions are satisfied; z must he negative oo for Tj = 0, in order that e^ = 0, which is like In 11 '; "because F'/T; then assumes precisely a finite value,. The result is the following development in powers of T]^/^: Z = $ + ajvi + hTl^ + cTl'^'^^ + dTl^ + eTl^^/^. . . ( 21 ) The coefficients are obtained " by intrbduAtion of this formula in the differential equation and comparison of equal-: power s: a = - 1/2", h = L_, c = ^S- . d = — ^Z— ', e .= 0.000014 . ■■•'.■ 7 :. 245 1715 240100. , ... „ The convergence is poor on approaching the boundary point Tlj.(Z=0), but a development particularly suitable near Tjj. is as follows: Put Ti = Tij. - in and : -. . Z = I Tl"^ + b TJ"^ + c' t[*+ d TJ"^ + I TJ"^ + f ri"'^ . . . and obtain _ 1 |_; •■ 2 - ' 3 •^•- 1 - 5 ' • a = — , t) = _ , c = - -, d = — -, 4 8 Tlr 64 Tlr^ 64 128 T|r^ 7 - 19 133 7 0.00278 ^ e - - - , 1 = _ + 256 X 5 T)r 256 X 40T)j.*-': 1)^ .^ ' T|.j. ^ 12 NACA TM Ho. 1085 The unknown constant of integration T\j. is obtained "bv making the values for 2, as known froiu the two developments, agree in a certain Junction point. It results in Tjp = 3.4. (Quantity F'/r|/'acts like 1 - 0.202 T]^'^ in the center; the outward junction in F again takes place with a discon- tinuity in F"'. The con ver sion factor from the reduced to the actual quantity T\ is Vc®; s signifies in the table a char- acteristic distance from the discharge hole for v/hich the speed in the center of the Jet is Ug. The computed velocities were compared with Oottingen teat 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 l/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 v^hich must have had different reasons. From the conversion factor for T\ follows 2 _ 0.063 The radius r of the jet is r =Vc2 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 yi eld - ^/c^ = 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. NAGa ■"TM No. r085.- 13 4. PREDICTION OF PE3SSURE DIFFSRENGS So far, all cases had been premised on constant pressure This first approximation can be improved by analysis of the pressure differences due to impulse variation on the basis of the computed spoods and stresses. For the first step, start, say, with the second equation of motion, v/hich iin the first two cases reads and u 91 + 0X ay ~ p Vdx, y 3y ~ y > P ^y in the rotationally symmetrical case; ffy and a^ are nor-r mal stresses, respectively effective in y direction or at right angles to y and x. Then, integrate with respect to- ys y y and' a 7 P y y . y + _9_ /uvdy+ /^dy-i-^ / T d y - 1 3x . / , / y p 3x .J PL ° ^ y ; ■ -/ : - . a y i3 1^ P . J (22) (23) whi'chi^s equivalent to applying the impulse theorem. If, as heretofore, the n'ormal stress, in this case are discounted, there is obtained Or. md Q>^ , '"■2 F F ' n -■ F .1^1 s. J 11 = - 2 / F'^ n d ri - - ■ r, 1 P for the free jet boundary, 14' NACA TM No. 1085 ^1- X ? ' !'■'■ Tj' - L ':L:f> 5Tn -J- ^ 1 '- - a for the two-dimensional, jet expansion.' and 2 F g'T 'n r , x^H J, ""j x3Tl3 2^^ - 1 r " dTl = - I p P L . for axially symmetrical jet expansion. With pj. denoting the pressure at the jet ""boundary, pj^ the pressure at jet center and of the homogeneous air stream, respectively, particulari- zation of the above formulas yields — ~ = 0.410(2c^) U^ and — — = 0.248(2c^) U„^(x) P P and - m 'm - P r _ .0.315(c^)' 7c U.^Cx) Q,uant ity and U^(: case , c U indicates the speed of the homogeneous air stream, the central speed at x. In the first and third has "been determined, giving P„ - P^ = 0.00584 £-^ and Po - Pr = -0.0025 P Uin^(x) It is apparent that the thus computed pressure differences, being small, do not cause a su'bstantial modification of the velocit i e s . > . 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 Pq as the pressure in still air JIAC^L TM,; No-. 1085 15 Pffl ~ ^0 Pm - iPo = 0.338(2c^) p U^ = 0.00482 H-^i- 2 3/3 2 0.l24(2c. .,) p Ujh (x) Pm - Po = -0 3.72(c^)^^^ p U,^(x) = _0.0029&P-£-^ (x) Hence there is positive pressure within the jet in the two- dimensional cases, hut negative pressure in the axially sym- metrical 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 he given in the fol- lowing, . , - ■ 5. EXTENDED THE0S3M FOR THE APPARENT STRESSES The theorem applied up to now to the stresses introduced ty the turbulent impuipe exchange .:,..■. " - " T = I ay Bu By' ^x = ^y = ^t = ° is no more than a first approximation. In any c.ase, it' ■can- be easily proved that du/By in the cases in point is great with respect to ^, . -^ and -^,- ,hen-ce the theorem for the mix- ox dx Sy " ' ing speed I ' ou I dy caused by the speed difference is good Bp, ix a natural, generaji sat ijon of .the previous theorem, the stress'' t'ehso'r is put equal to I' :3u 3y (\7 v + vv ) ( Vi = affinor of v; vv is the con jugat e ^f f inor . ) ^This relation is important for the calibration of pitot tubes in a jet discharging from a nozzle. 16 NACA TM Ho. 1085 The stresses to be newly added here, are, in general, neglected, 2 except (7=2 1' ^ and a. = 2 l^ Sy ^ -^ — v/hich are used oy I y for calculating the pressure differences. This portion cancels in the model problem worked out for the two-dimensional case "because of the employed boundaries, but not for the axially symmetrical case. Her6 the pressure differences are augmented by the integral y n 0", dy so that and Pj^ - p^ = +0.151(0^)^/^ p U^2(x) = +0.0012 P ^m^^^^ Pm - Po = 0.095(c^)^'^'' p u/(x) = 0.00075 L^^^Sll m that is, positive pressure within the jet, as in the other cases . Translation by J. Vanier, National Advisory Committee for Aeronautics. HEFSRSNCES 1. Prandtl, L.: Bericht iiber Unt er suchungen zur ausbebi Idet en Turbulenz. Z.f.a.M.M,, vol. 5, no. 2, 1925, pp. 136-139, 2. Beta, A.: Velocity and Pressure Distribution behind Bodies in an Air Current. MACA TM Xo . 268, 1924. 3. Zimm, Walter: Uber die St r omungsvorgange im freien Luft- strahl, ?or BChungsarbei t en au s dem G-ebiete des Ingen- ieurwesens, no. 234, 1921. KACA TM No. 1085 17 9> 09 o 5 ^ ^ '- i?» N «^ ■^ •^^ "^^ "=. 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