1、L -. 4. - *.-I “ A/:”=-+-. . . , .NATIONALADVISORYCOMMITTEEFOR AERONAUTICSTECHNICAL MEMORANDUM 1288THE DIFFUSION OF A HOT AIR JET IN AIR IN MOTIONBy W. SzablewskiTranslation of “Die Ausbreitung eines HeissluftstrahlesinBewegter Luft.” GDC/2460, September 1946.WashingtonDecember 1950 .iII , _Provided
2、 by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Ill- -. Illlttllltlll-, 31176014411855. _ ., .NATIONAL ADVISORY COMMITTEEFOR AERONAtiICSTECHNICAL MEMORANDUM 1288,.-.$=-THE DIFFUSIONP1 The breadth of the mixing region, as well as the velocity and temperature”
3、drop along the jet axis, is calculated. The theory is then compared tith1Pabstis measurements (reference 2). The ratio of interchange of temperature and velocity yielded a factor E=2. Considering the friction loss at the:i nozzle wall, the agreement between the theoretical and the experimentalIveloc
4、ity decrease along the jet sxis can be regarded as satisfactory. Thetemperaturemeasurement along the jet axis appears to be faulty.L,i.,P *“Die Ausbreitung eines Heisslul%strahlesin Bewegter Luft.”I GDC/2460) September 1946.!Iii-.,_ -,Provided by IHSNot for ResaleNo reproduction or networking permit
5、ted without license from IHS-,-,- .A: SMALL TEMPERATURE DIFFERENCESAND SURROUNDING MEDIUM1. Method and Results1. In the first part of the investigationfield in the core was computed.NACA TM 1288BETWEEN JET(reference1) the flowThe investigation included the variation of the curves bounding themixing
6、zones of velocity and temperature as well as the aspect of thevelocity and temperature distribution functions over the mixing zones.In the secondzone adjoining thepart, (the present report), the flow field in thecore is investigated.o,*Y core I transition zone XThis zone, which in the asymptotejet d
7、iffusion is characterizedbythe cross sections of the jet, isin the so-called axially symmetricalthe affinity of the flow processes intermed the transition zone.,., , ,., , , ,., ,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1288.3b24r1)1!
8、1/ /* Io* IItransitionzone xcoreVariation of mixing width of the velocity and the temperature over thenozzle spacingThe following relations are involved:the breadththe breadthIn additiontemperaturetemperaturevelocity ofvelocity ofvelocity ofof the mixing region ofof the mixing region ofrise of the d
9、ischargingthe velocitythe temperaturejetrise of the jetthe dischargingthesurroundingon jet axisjetmediumthe jet on jet axisaxcore I transition zoneVariation of jet velocity and jet temperatiue along the jet axisProvided by IHSNot for ResaleNo reproduction or networking permitted without license from
10、 IHS-,-,-4 NACA TM 12882. In the extension of the theory of free turbulence to gases ofwidely variable densities, the following turbulent exchange quantitieshad been obtained in partI (reference1).T=Q.Onair jetM=Ez2 $by ay for turbulentz21*/ ($ + %*) for turbulent1.,21$1 Q ay for turbulentdiffusion
11、(la)shearing stressheat conduction(lb)(lC)the assumption of constant pressure for the diffusion of a hotin air in motion (axially setrical case) the motion equationswere then obtained:equation of continuity ofa(ri)axequation of continuity of momentumequation of continuity of the heat (energy princip
12、le)with the apparent kinematic viscosityOwing to the continuity of mass, it further yields:momentum(2a)(2b)(2C). ,m. , , , , .,.,., , , 1, IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1NACA TM 1288 5Heat,=- -(3 = temperature rise of jet.)Integrat
13、ionwith respect to r gives then for the transition zone:momentum)- U1 - U1)*ar “uaii-U1C(x)r A(x) is the jet velocity on the jet axis and A(x)the temperature rise on the,jet axis.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1288 7This form
14、ula, carried in the equations gives (if a practicableassumption can be made for the variation of q(r,x) and $(r,x) four,equationsfor the four unknowns:., ,LA(X) 3A(X), hi(x), and bp(x)The transverse component ; still appearing in the equations isdetermined from the equation of continuity of the mass
15、The next problem isThese functions occur inthe equations and in theheat conduction at point1v-however, it then results in inadequaciesat the edge of the profile wherethe fluctuationsand, with them, the apparent kinematic viscosity cancelout, whereas by the new theorem an amount of e constant over th
16、e entirewidth of the profile is involved. On the other hand, the new theorem hasthe implicit advantage of being substantiallysimpler mathematicallyandhence of simplifyingthe calculations considerably.A very practical representation of the distributions,which givesa very good approximation, is(1 ).7.
17、3/2 2already used by other workerspresented by the functionand(reference 4).Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1288 9Thus 4 = (1 - #)2 (ga)(gb)with q = rlb2. To simplifythe calculation for b2ibl for the case ofoutside air in moti
18、on p+o)j the ratio of the widths resulting fromthe theory of the asymp otic distributionfunctions for outside air inmotion is approximatelyput asb2l= (lo)(See section III.)The case of quiescent outside air (UI = o) is treated later.4. Part A of this report is restricted to small temperature differ-e
19、nces. This case is amenable to calculationand represents the conditionsaccompanying any temperature differences in first approximation.The velocity field is computed by the momentum equations.The equation of conservationof the momentum gives the breadththe mixing region of the velocity bl as a funct
20、ion of the axialvelocity iiAwith the constants dl = 0.133 and el = 0.27, and the subsequentcalculation gives the axial velocity iiA as the describingvariableof the flow field rather than the nozzle distance x. The momentumequation is specializedto r = ro.The use of the foregoing result gives,c. of t
21、he first order for the nozzle distanceon the jet axis GA.of(11)a linear differentialequationx .asfunction of the velocityProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACA TM 1288GAWhen the integration constant is fixed by the postdate - U1=1o -
22、 ufor the boundary of the core XK,with(12)I 1and the constantsa71a = 0.0273, p = 0.0549, 7 = 0.0375, P = 1.9259The length of core xK/ro is to be taken from the theory of the core;K is an empirical constant.In the asymptote the formulas are obtained,/ro+#2Al.A:u,l/2.()y%with the constant Al = 1.972an
23、dz+ .,y/3 2 +with B2 = 0.216.lh figures 1, 2, and 6are shown for the parameterthe functionsvalues= 1.0, 0.75,oThe temperature field is computedby(15)(16)of the velocity mixing fieldand 0.5means of the heat equations.By the equation of conservation of heat, the breadth of the mixingregion of the temp
24、erature b2 is%/% = *with = 0.0786 and el = 0.257.(17)The heat formula contains the empirical constant E, the ratio ofinterchangeof temperature and velocity; according to the measurementsby Pabst (reference 2) and others it is equal to 2.From the heat formula a Bernoulli differentialequation is obtai
25、nedIA - U1or A O as a function of # Unfortunately the quadrature cannoto - 1be carried out. Fixing the integrationconstantby the initial condi-/A- U1ti”n TA o = 1 for u - 1 = 1, the resdt iSoProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12,2C41/2 1
26、2D“o-ul fiA-ul +(0 ) R:?) CC-115:$2 “0 U1)(18)with the functions1- -!and the constantsC =-o.61L.1 = 0.0241 p, = 0.05221 = 0.0066 vz = 0.0253v = o.1, 5 = V4In the asymptote the formulasp, = 0.0876 IJ4= 0.25713 = 0.0196v=p61 ,7 = P2.(19)Provided by IHSNot for ResaleNo reproduction or networking permit
27、ted without license from IHS-,-,-NACA TM 1288are obtained.in agreement with theb2/bl=mathematical assumptionIn additionIn figures 3, 4, 5, and 6 the functions of thefield are represented for the parameter valuesThe integralgration.Uo - U1 = 1.0%temperature0.5(20)mixingoccurring in formula (18) was o
28、btainedby numerical inte-The case of quiescent outside air ul =( O) presents a singularbehavior as evidenced by the fact that the breadth of the mixing regionin the asymptote is representedby a l-near function)bl(x) - x, whilewith outside air in motion bl(x) -X1 3. It is found further that theratio
29、of the asymptotic mixing width obtained for outside air in motionis not applicable here. The theory of the asymptotic distributionfunctions produces, in this instance, an impracticableresult (b2/bl+ =).The asymptotic ratiob2/bl.- 1.33 for Ul=o (21)is obtained from the law of conservation of heat by
30、an approximationmethod.The functions of the turbulent diffusion for the singular caseU1 = O are as follows:velocity field. I1blro=Fl UA.U(T1JO- U1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACA TM 1288with F1 = 2.737x/ro= *F21(:1 )/-l+ro- U1-
31、 U1with F2 = 0.1347.With the nozzle distance PO as independentvariablebl()K=%+KG25-r.()!iA._21=uo -u 1l+e(i-:)with G1 = 2.737 G2 = 20.321(/ )G2 Gl = 7.425Temperature fieldb2powithiiA- U1 1/2H2+(l-H2)(% -Ul)1tiA-ul1 uo - 1)ZA190= ()/ 2+p-!;(:y7(23)(24)(25)(26)(27)HI = 3.245 H2 = 1.123Provided by IHSN
32、ot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,NACA TM 1288For infinitely great nozzle distances,blfro KG#roandwith K1 = 3.644(EA)-U1 *1% 1% - U1 K 2 (x/ro)/b2 r.1%fiA-ul()o - 115the formulas are,. (28)/ nAoX.K A-%2U.Uo 1(a)(30)(31)with K2 = 0.793.All formulas sti
33、ll exhibit the empirical constant K. It isembodied in the apparent kinematic viscosityand represents a measure for Z/b, the ratio of mixing length to mixingwidth, which is regarded.as constant for the individual jet sections;K and Z/b are to be considered functions of the characteristiclength xro. B
34、ut in view of the far-reaching similarityof the profiles,the dependence on x/r. throughout almost the entire transition zone isexpected to be slight, as the measurements seem to confirm. For example,Tollmiens investigations (reference5) at the plane jet boundary (theseconditions prevail in the immed
35、iatevicinity of the nozzle mouth!)give 2/b = 0.068; for the axially symmetrical jet diffusion (theconditions encountered at very great distance from the mouth of the .nozzle), he obtained 2/b = 0.073 (where, for reasons of continuityat transition from the core to “thetransitionzone, b in the latter,
36、as done here, is to be put equal to the jet radius).Also of interest are the distributionfunctions (section III). Asalready pointed out, the empirically obtained function (1 - q3/2)2 givesProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,16 NACA TM 12
37、88a very close approximationof the experimental distributioris.The theorygives the following asymptotic distributionsfor u + o:Velocity distribution ,.whereby with(32)As reflected by the previously obtained results concerning the structureof the mixing zoneso = 1.811 (33)Ifor = r bl.Utilizing in cor
38、respondencewith the asymptotic behavior of themixing width (bl bmx X1/3) the Coordimte V* = rwith“0 0“427ibU1 2/3() o(uoT )- U1 1/3UoIn the case of quiescent outside air U1 =( o) the asymptoticvelocity distributionfunction is(p. 1 r/x(34)(35). 8- -. ,-,I mm mmImmmm I= mm I m IIn ImllImIIm I IllIlmlI
39、IllProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1288with17anduO = (1/K)X 0.0784For the temperature distributionthe important relation(36)is obtained. This result is already indicated in Reichardts report(reference6).The asymptotic distribu
40、tionfunctions are represented in figures 7and 8.Now, the theory is compared with the experimentalresults. Theformer contains two empirical constants E and K, E being equal to 2.In the first part of the investigation (reference1) the constant Kfor the core had been determined on the basis of Tollmien
41、s study at theplane jet boundary (reference5). The result was K = O.O1O6 on the basisof the heat equation, but the thus defined value is somewhatuncertainowing to the doubtful flow losses as a result of the friction at thenozzle wall and theassumption valid exact only for small velocity andtemperatu
42、re difference that the breadth of the mixing regions oftemperature and velocity which served as b“asisof the calculationsact as r E : 1.Quantity K is determined next for asymptotic conditions. Tollmien(reference5) obtained b = 0.214x for the diffusion of a jet issuingfrom point source, as against bl
43、 = K x 20.321 X x according to thecalculationsby equation (28). The comparison gives K= 0.0105. Whenthe determinationof K is based upon expertientalresults of otherstructures of the diffusion field, such as the gradient of the asymptoticdistributionfunction or the decrease of velocity and temperatur
44、e alongthe jet is, the foregoing value of K is almost exactly reproduced.The calculationshave been based on the valueK.= O.O1O (37)I . ., . . . . -.- - .-. . - . - .,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-18 NACA?!M1288For asymptotic conditi
45、ons, no incidental flow loss due tofrictionat the nozzle wall needs to be considered. (The flow loss canbeallowed for by introducing an effective nozzle radius (see part I,(referencel); for infinitely great distance from the nozzle, however,the size of the nozzle radius is of no influence.) The rati
46、o of thebreadth of the mixing region of temperature and velocity serving as abasis of the calculations is rigorously valid for the asymptote. Theabove value of K should therefore represent a safe value for asymptoticconditions.Experimental data for small temperature differences as treated hereare fe
47、w and limited to the case of outside air at rest U1 = 0).( Forgreater temperature differences (to be discussed in part B of thisreport), Pabsts comprehensivemeasurements (reference 2) are available.For the asymptotic velocity distributionReichardts measurements(reference7) are available. The comparison (fig. 9) shows practicableagreement up to the boundary zone where the divergence is fairly great.This difference is, as stated above, attributable to the nature of thecalculationmethod.The decrease of velocity and temperature along the jet sxis wasmeasured by
