Li. AEDC-TR INFLUENCE OF INITIAL BOUNDARY LAYER ON THE TWO-DIMENSIONAL TURBULENT MIXING OF A SINGLE STREAM

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1 Li. «B ~»»» INFLUENCE OF INITIAL BOUNDARY LAYER ON THE TWO-DIMENSIONAL TURBULENT MIXING OF A SINGLE STREAM R. C. Bauer and R. J. Matz ARO, Inc. April 1971 Apprved fr public release; distributin unlimited. ENGINE TEST FACILITY ARNOLD ENGINEERING DEVELOPMENT CENTER AIR FORCE SYSTEMS COMMAND ARNOLD AIR FORCE STATION, TENNESSEE

2 mmm When U. S. Gvernment drawings specificatins, r ther data are used fr any purpse ther than a definitely related Gvernment prcurement peratin, the Gvernment thereby incurs n respnsibility nr any bligatin whatsever, and the fact that the Gvernment may have frmulated, furnished, r in any way supplied the said drawings, specificatins, r ther data, is nt t be regarded by implicatin r therwise, r in any manner licensing the hlder r any ther persn r crpratin, r cnveying any rights r permissin t manufacture, use, r sell any patented inventin that may in any way be related theret. Qualified users may btain cpies f this reprt frm the Defense Dcumentatin Center. References t named cmmercial prducts in this reprt are nt t be cnsidered in any sense as an endrsement f the prduct by the United States Air Frce r the Gvernment. fc

3 INFLUENCE OF INITIAL BOUNDARY LAYER ON THE TWO-DIMENSIONAL TURBULENT MIXING OF A SINGLE STREAM R. C. Bauer and R. J. Matz ARO, Inc. Apprved fr public release; distributin unlimited.

4 FOREWORD The wrk reprted herein was spnsred by Headquarters, Arnld Engineering Develpment Center (AEDC), Air Frce Systems Cmmand (AFSC), Arnld Air Frce Statin, Tennessee, in supprt f Prgram Element 62302F, Prject The results f research presented were btained by ARO, Inc. (a subsidiary f Sverdrup & Parcel and Assciates, Inc.), cntract peratr f AEDC, AFSC, under Cntract F C The exact numerical slutin utlined in this reprt was develped under ARO Prjects N. RW2608 and RW5711. Initial results frm the exact slutin were presented t the Nvember 1968 Prject Squid Wrkshp h Turbulent Transprt Prperties in Chicag, Illinis. The simplified apprximate slutin was develped under ARO Prjects N. RW5807 and RW5905. The\ manuscript fr this reprt was submitted fr publicatin n February 9, This technical reprt has been reviewed and is apprved. Eules L. Hively Harry L. Maynard Research and Develpment Divisin Clnel, USAF Directrate f Technlgy Directr f Technlgy li

5 ABSTRACT An integral methd is presented fr estimating the influence f an initial bundary layer n the develpment f a tw-dimensinal, isbaric, turbulent, free shear layer. The basic equatin is derived by applying the principle that, at any streamwise statin alng the free shear layer, the mmentum f the entrained flw equals the ttal axial turbulent shear frce acting alng the dividing streamline. This equatin is slved using a single parameter family f velcity prfiles derived by Krst and Prandtl's mixing length cncept fr turbulent shear stress. The thery invlves ne empirical cnstant which was evaluated using Tllmien's experimental data fr incmpressible, turbulent mixing. The thery is verified by cmparing with experimental data fr free-stream Mach numbers up t 6.4. iu

6 CONTENTS ABSTRACT NOMENCLATURE I. INTRODUCTION II. BASIC EQUATIONS III. NUMERICAL TECHNIQUES... IV. EXPERIMENTAL VERIFICATION V. CONCLUSIONS REFERENCES Page iii vi APPENDIXES ILLUSTRATIONS Figure General Mixing Zne 13 Typical Family f Velcity Prfiles 14 Cmparisn f Thery and Chapman's Data a. \jj = 2.0, M 8 = b. tfr»3.3, M, c. it = 4. 7, M,,, = d. i// = 9. 7, M w = Cmparisn f Thery and Hill's Data a. \fj = 2. 1, M«, = b. ijj = 4.6, M w = c. if/ = 8.4, M = d. ^ = 17.4, M 0B = e. < = 4.3, M a = f. 0 = 7.7, M. = g. ^ = 16.0, M 0 = Cmparisn f Thery and High-Speed Data frm Ref. 9 a. xjj = 2. 7, M r = b , M. = c. ii = 3.33, M. = d. ifj = 15.8, M w = Theretical Variatin f Dividing Streamline Frictin Cefficient 30

7 Figure Page 7. Theretical Variatin f Mixing Length Prprtinality Cnstant Cmparisn f Theretical Mixing Grwth a. With Chapman's Data 32 b. With Hill's Data 33 II. EQUATIONS FOR THE CHARACTERISTICS OF GENERAL, SINGLE-STREAM MDONG ZONES 34 NOMENCLATURE b Width f mixing zne (Fig. 1) C Crcc number Cf Cefficient f frictin, l P v l 2 F Ttal frce G Net mass flw I, J Integral quantities, defined as fllws: >7 1 J» = J T ^» J 3 = J - ^ r,aq k Rati f Prandtl's mixing length t mixing zne width 1 Prandtl's mixing length M Mach number p Base pressure and static pressure thrughut mixing zne and inviscid flw field R Gas cnstant r Radius frm axis f symmetry t inviscid jet bundary (Appendix H) VI

8 T Static temperature Tj. Ttal temperature u Velcity X Streamwise intrinsic'crdinate (Fig. 1) Y Transverse intrinsic crdinate (Fig. 1) y Transverse crdinate f initial bundary layer ß Dummy variable in Eq. (5) 7 Rati f specific heats 6 'Thickness f initial bundary layer (Fig. 1) <ry T) Nn-dimensinal mixing rdinate, -==- 0 Mmentum thickness f initial bundary layer, Eq. (13) r angle between crdinate reference line and axis f symmetry (Appendix II) p Density ex Lcal similarity parameter T Viscus shear stress 0 Velcity rati, UJO \jj Nn-dimensinal intrinsic streamwise distance, SUBSCRIPTS b Base regin (Fig. 1) D Dividing streamline FD Fully develped mixing 1 Inviscid i Initial bundary layer M Extremities f mixing regin (Fig. 1) m Inviscid jet bundary (Fig. II-1) P Psitin parameter t Ttal r stagnatin cnditin vii

9 V With mixing w Wall (Fig. 1) 00 Free stream 3 Statin 3 cnditins (Fig. II-1) Vlll

10 SECTION I INTRODUCTION The influence f an initial bundary layer n the develpment f a turbulent free shear layer is an imprtant factr in the analysis f separated flws as well as in the basic study f turbulent mixing prcesses. Histrically, the initial bundary layer is treated by the "displaced rigin" technique which accunts fr the increased size f the far field mixing zne, but des nt predict the evlutin f the velcity prfile. Hill (Ref. 1) has refined the displaced rigin technique and is able t predict the evlutin f the velcity prfile. He verifies his technique by cmparing with mre exact theretical results fr laminar mixing and experimental results fr turbulent mixing. Fr turbulent mixing, Hül assumes a linear variatin f eddy viscsity with distance and accunts fr cmpressibility effects with the Dnaldsn and Gray crrelatin. Lamb (Ref. 2) determines the evlutin f velcity prfile in a turbulent mixing zne by using the transverse mmentum equatin t determine the eddy viscsity. This technique has nt been extensively verified by experiment and will nt predict a laminar mixing prcess because use f the transverse mmentum equatin results in an verspecificatin f the prblem. The integral methd develped by Bauer (Ref. 3) has been extended by Wülbanks (Ref. 4) and applied t laminar mixing with an initial bundary layer. Wülbanks verifies his technique by cmparing with mre exact theretical results fr laminar mixing. In this reprt, the methd f Ref. 4 is applied t the case f turbulent mixing with an initial bundary layer. SECTION II BASIC EQUATIONS 1 A sketch f the type f mixing cnsidered is shwn in Fig. 1 (Appendix I). The cnservatin f mmentum cnditin is applied t the cntrl vlume shwn in Fig. 1 based n the fllwing assumptins: 1. Over the mixing zne length cnsidered (X), the average angle between the dividing streamline and the mixing zne centerline is small (<20 deg). 2. The mixing is tw-dimensinal, isbaric, and cnstant cmpsitin.

11 The mmentum equatin fr this cntrl vlume is f 'D dx = f P» 2 dy (!) Equatin (1) is written in terms f the intrinsic crdinate system intrduced by Chapman and Krst in Ref. 5. The intrinsic crdinate system results frm far-reaching simplificatins f the equatin f mtin which is reduced t the frm f the heat cnductin equatin fr nn-steady, ne-dimensinal heat flw. Slutins t this highly simplified equatin f mtin must be interpreted in terms f a crdinate system (intrinsic) which is shifted relative t the usual inviscid flw crdinate system. The lcatin f the intrinsic crdinate system can be determined by applying the cnservatin f ttal mmentum cnditin (Appendix II). By using the perfect gas law, p = p p RT RT <. ( - c ~* 2 ) where u * = and and by using the transfrmatin, -m ffy V = where a is the lcal similarity parameter and is a functin f bth C, and X, substitutin int Eq. (1) yields In the abve transfrmatin, a is the lcal similarity parameter as intrduced by Chapman and Krst (Ref. 5). The lcal similarity parameter is a functin f bth C^ and X and asympttically appraches the well-knwn Gertler parameter as X-*-».

12 with If a cefficient f frictin is defined as fllws C fd = ( 5 v 8 (l3) n J - C 2 G<> 2 and then substitutin int Eq. (2) yields 'p - T" = T (3)!' C D# = JL_ a - c,»)(l8v, D (4) Equatin (4) is the basic mmentum equatin and applies fr either laminar r turbulent mixing. T slve Eq. (4), it is necessary t specify Cfj, in terms f X and r?p, and it is necessary t describe a family f velcity prfiles with rjp as a parameter. The desired family f velcity prfiles was derived by Chapman and Krst in Bef. 5. The equatin is as fllws: <6 = -i [1 + erf (, -, p )] ^ j [* (^)]^2 iß (5> rj-rjp Equatin (5) may be regarded as an interplatin equatin between the initial bundary layer ^if-p, ) and a fully develped mixing velcity pr- file given by 1/2(1 + erf 17) fr crrespnding variatins in rjp frm infinity t zer. A typical family f velcity prfiles is presented in Fig. 2.

13 In this reprt, nly turbulent mixing is cnsidered, and Prandü's mixing length hypthesis is used t define the cefficient f frictin (Cf_J. By definitin, c h = M (6) Prandtl's mixing length hypthesis, as it is applied in this reprt yields r in nn-dimensinal frm 2Ü 2 U-Ci) rp_ "p"-^' /a_\ fj6\ (7) The usual assumptin cncerning the mixing length irj is (Ref. 6) l» = kb (8) where k is a nn-dimensinal empirical parameter and b is an arbitrarily defined width f the mixing regin (Fig. 1). The crdinate % is defined by b = I* \ä (9) where»7^ is the crrespnding nn-dimensinal width f the mixing regin. Since the nn-dimensinal velcity prfile [Eq. (5j] is a functin f X, then rf will als be a functin f X. Substituting Eq. (9) int Eq. (8) yields D - 'b k ( ) (10) As in Ref. 3, k is assumed t be independent f free-stream Mach number, and in additin, it is further assumed that the prduct n^k is independent f X. The latter assumptin specifies the functinal relatin between k and X t be the inverse f the functinal relatin between 175 and X. These relatinships are predicted by the thery since the cmbined assumptins allw the prduct 77. k t be determined frm a single experiment. The validity f these assumptins will be established by cmparisn with experimental results.

14 The cefficient f frictin is btained by substituting Eqs. (10) and (7) int Eq. (6); therefre, ' D " 12-, (11) --C 2 < The ttal temperature distributin thrugh the mixing zne and initial bundary layer was determined frm the well-knwn Crcc relatin fr unity Prandti. number. Applicatin f the Crcc relatin assumes an instantaneus change in the ttal temperature distributin at the separatin pint, which f curse is nt true. Therefre, nly mderate ttal temperature differences are adequately apprximated by the Crcc relatin. The lcatin f the dividing streamline was determined by the methd discussed in Appendix II, which was develped fr the mre general case f axisymmetric mixing. The equatin is Jir^-r^J<-(7%)(f) (u. where 9 is the mmentum thickness f the initial bundary layer defined by the fllwing equatin: Sufficient infrmatin is nw available fr the numerical slutin f Eq. (4) fr a specified initial bundary layer. SECTION III NUMERICAL TECHNIQUES 3.1 EXACT SOLUTION The numerical technique used t slve Eq. (4) is the methd f finite differences based n the fllwing cnditins at tp = 0, Tjp = e and Cf_ = 0 Varius values f»7p were selected starting in the range frm 6.0 t 10.0 and decreasing t zer in increments f 0. 1 t This calculatin

15 yields rjp as a functin f q2/. The velcity prfile can then be determined at each value f ijj frm Eq. (5) and transfrmed int physical crdinates by the fllwing equatin: V P S It is interesting t nte that the similarity parameter (a) is nt required in the transfrmatin t the physical plane except fr the case f fully develped mixing which ccurs when \(J = and np = 0; The variatin f the lcal similarity parameter (a) with ^ can be determined frm Eq. (3). This numerical technique was prgrammed n an IBM 360/50 cmputer, and a typical calculatin time is apprximately 1 min fr each value f rjp cnsidered r abut 15 min fr a cmplete mixing prcess. 3.2 APPROXIMATE SOLUTION Experience with the exact numerical slutin has led t the develpment f an apprximate slutin which can be very useful in slving practical engineering prblems. The apprximate slutin is btained by integrating the left hand side f Eq. (4) by parts. This yields the fllwing ill Y 1 ' d * (14) The integral n the right-hand side f Eq. (14) can be neglected fr the fllwing tw reasns: dc fd 1. Experience with the exact slutin shws that,.» 0 fr <p > 10. v 2. Fr \}J< 10, the exact slutin underestimates the value f the integral appearing n the left-hand side f Eqs. (4) and (14) relative t its true physical value because f the bundary cnditin Cf_ = 0 and ^ = 0. This bundary cnditin implies that, at the separatin pint, the shear stress varies discntinuusly frm that prduced by the initial bundary layer t zer. This is physically unrealistic, and therefre, the errr prduced by neglecting the integral n the right-hand side f Eq. (14) results in a mre realistic estimate f the integral fr ttal shear.

16 Based n these arguments, Eq. (4) can be written in the fllwing apprximate frm: c,,*- «^CJ>, D (15) With Eq. (15), it is pssible t calculate the prperties f the mixing prcess at any given statin (<//) withut the need fr calculating the cmplete mixing prcess up t that statin. The lcal similarity parameter (a) can then be determined frm Eq. (3). A typical calculatin time n an IBM 360/50 cmputer is less than ne minute. SECTION IV EXPERIMENTAL VERIFICATION The thery develped in the previus sectin invlves ne cnstant (rjtjk) t be determined frm experiment. The experiment selected is that f Tllmien (Ref. 7) fr incmpressible (C«, 2 * 0), fully develped, turbulent mixing. Tllmien experimentally determines a value f 12 fr the similarity parameter (a) which results in a value f fr rj^k. By using this cnstant, the thery is cmpared with available twdimensinal data in Figs. 3, 4, and 5. T facilitate the cmparisn f velcity prfile shape and size, the transverse lcatin f the theretical prfiles was determined by matching their half-velcity pint with the experimental. Hwever, it is pssible t theretically lcate the velcity prfiles by the methd presented in Appendix II. The thery, evaluated either by the exact r apprximate methd, is shwn t predict reasnable well bth the evlutin f velcity prfile shape and the size f the mixing zne fr free-stream Mach numbers up t As anticipated, the apprximate methd is mre realistic than the exact methd fr values f 4> < 10. A basic assumptin in the develpment f the apprximate methd is that dcf D Tj = 0 fr ijj > 10. This assumptin is verified by the plt f Cf versus \jj presented in Fig. 6. The basic assumptin in this analysis is that the prduct (n^k) is a cnstant which is independent f bth free-stream Mach number and mixing distance (X). Based n this assumptin, it is pssible t determine the variatin with X f the cnstant f prprtinality tic) between

17 the mixing length (JL) and a selected mixing width (b). A typical variatin f k with ^ is presented in Fig. 7 fr a mixing zne width defined by the velcity ratis and The influence f an initial bundary layer n the width f a turbulent mixing zne is presented in Pigs. 8a and b fr incmpressible flw (Chapman's data, Ref. 5) and fr supersnic flw (Hill's data, Ref. 1). Theretical estimates are included based n the exact and apprximate slutin techniques and fr fully develped turbulent mixing withut an initial bundary layer. Again, the apprximate slutin is shwn t agree better with experiment than with the exact slutin. 1 SECTION V CONCLUSIONS Analysis f the develpment f a tw-dimensinal, isbaric, cnstant cmpsitin, cmpressible, free turbulent shear layer frm an initial bundary layer leads t the fllwing cnclusins: 1. The velcity prfile becmes essentially fully develped in a streamwise distance crrespnding t abut ten initial bundary layer thicknesses. 2. The near field lcal mixing rate varies significantly in the first ten initial bundary layer thicknesses. The experimental data cnsidered in this reprt are t limited t establish the far field r fully develped mixing rate. 3. The thery presented in this reprt has been verified fr the near field mixing prcess. 4. The analytical technique presented in this reprt becmes impractical when applied t mixing prcesses that are strngly influenced by axisymmetric effects. The cmplexity intrduced by axisymmetric effects is primarily in cnnectin with the determinatin f the lcatin f the dividing streamline. A typical example is presented in Appendix II. Fr the mre cmplex mixing prcesses, numerical techniques (Ref. 10) are believed t be superir in bth phenmenlgical representatin and cmputatin time.

18 REFERENCES 1. Hül, W. G. "Initial Develpment f Cmpressible Turbulent Free Shear Layers. " Ph. D Thesis, Rutgers, May Lamb, J. P. "An Apprximate Thery fr Develping Turbulent Free Shear Layers. " Jurnal f Basic Engineering, September 1967, pp Bauer, R. C. "An Analysis f Tw-Dimensinal Laminar and Turbulent Cmpressible Mixing. " AIAA Jurnal, Vl. 4, N. 3, March 1966, pp Willbanks, C. E. "An Integral Analysis f the Cmpressible Laminar Free Shear Layer. " Furth Annual Sutheastern Seminar n Thermal Sciences, May 20-21, Chapman, A. J., Krst, H. H. "Free Jet Bundary with Cnsideratin f Initial Bundary Layer. " Secnd U. S. Natinal Cngress f Applied Mechanics, June 14-18, Hinze, J. O. Turbulence. McGraw-Hill, New Yrk, TILmien, W. "Calculatin f Turbulent Expansin Prcesses. " NACA TM-1085, translated by J. Vanier, Chapman, A. J. "Mixing Characteristics f a Free Jet Bundary with Cnsideratin f Initial Bundary Layer Cnfiguratin. " Ph. D Thesis, University f Illinis, Lamb, J. P. and Bass, R. L. "Sme Crrelatins f Thery and Experiment fr Develping Turbulent Free Shear Layers. " Paper N. 68 FE-9 Presented at the ASME Fluids Engineering Cnference, Philadelphia, Pennsylvania, May 6-8, Lee, S. C. and Harsha, P. T. "The Use f Turbulent Kinetic Energy in Free Mixing Studies. " AIAA 2nd Fluid and Plasma Dynamic Cnference, June Zumwalt, G. W. "Analytical and Experimental Study f the Axially Symmetric Supersnic Base Pressure Prblem. " Ph. D Thesis, University f Illinis, Bauer, R. C. "Characteristics f Axisymmetric and Tw- Dimensinal Isenergetic Jet Mixing Znes. " AEDC-TDR (AD426116), December

19 APPENDIXES I. ILLUSTRATIONS II. EQUATIONS FOR THE CHARACTERISTICS OF GENERAL, SINGLE-STREAM MIXING ZONES 11

20 Cntrl Vlume Fig. 1 General Mixing Zne a n

21 7} Fig. 2 Typical Family f Velcity Prfiles 14

22 0.10 l mm « CD c Q. "t MM t a: a? 3 > 0 t a> c c I/) a_ CO a. r Initial Bundary \ Layer Exact Slutin Apprximate Slutin -3 _J L a. tf» - 2.0, M = 0.19 Fig. 3 Cmparisn f Thery and Chapman's Data 15

23 0.15 i- H I^-l.O M 0.10 O z O 03 M Ö mm JY 0> > -t ' 0.05 t a> Qi c a» > *^ as a: 1 *-» CO 0 Q_ "<5 ^ ^ - t as: O a. is '"XD t Oil 0-1 y/v \ X \-Exact Slutin i I I i b. ^ = 3.3, NU = 0.19 Fig. 3 Cntinued

24 0.15 ^ E c E "_J _OJ ZJ M CD O N "^ N O <t> > ed c > 0) i2 C a: HIM c "en 0 MM O. Q- CO 15 Q a CO u OS c. ^ 4.7, M = 0.19 Fig. 3 Cntinued 17

25 0.20 r i :i mm _a> E / Ö CD > CO CD 1/1 CO M M O CD > r CD c t/i O a. IB CO V - - ' / V ^^ - Exact and Apprximate Slutins / i d. (// = 9.7, M. = 0.19 Fig. 3 Cncluded 18

26 " ^-1.0 \ in. > > 1 TO c ~ i Apprximate Slutin Q'/l r Exact f / \ Slutin // -y O^ """ i^* ^-Initial Layer s Bundary / / " a. tf/- 2.1, M Fig. 4 Cmparisn f Thery and Hill's Data 19

27 b. V - 4.6, «L = 2.1 Fig. 4 Cntinued 20

28 > TO O a> TO c Q_ c. $ = 8.4, M. = 2.1 Fig. 4 Cntinued 21

29 l 6 = 0.46 In > CD > 2 t 0.8 m 0.6 Q_ 0.4 Exact and Apprximate Slutins d. \j/ = 17.4, M = 2.1 Fig. 4 Cntinued 22

30 'W 1.0 CD 6 =0.49 in. < c Apprximate Slutin ^ 9' > O > C O Q- Bundary e. \j/ = 4.3, M. = 2.5 Fig. 4 Cntinued 23

31 AEDCTR > O > O a. f. ^ = 7.7, M B = 2.5 Fig. 4 Cntinued 0 24

32 2.0 'w =0.49 in. 1.0 > tu O a> I CD t O 0.5 Exact and Apprximate Slutins / i i g. ^ = 16.0, M. = 2.5 Fig. 4 Cncluded 25

33 AEDCTR =0.3 in 2.9 ± TO OS 2.7 Initial Bundary Layer _ L 0 2J a. \j/ = 2.7, M Fig. 5 Cmparisn f Thery and High-Speed Data frm Ref. 9 26

34 I»L X 6 -> 0.3 in. \ d 1 Exact and Apprximate Thery -^ /I /l /l S re ^s*^ s^ / / v 1 nitial Bundary Layer II b. i// = 13.3, WL = 4.0 Fig. 5 Cntinued 27

35 J*-as 6 =0.3 in. C 2.9 Initial Bundary Layer- Q_ = 2.8 TO CO Exact Slutin 2.6 -,'\_ Apprximate Slutin 2.5 & c. $ = 3.33, M. = 6.4 Fig. 5 Cntinued 28

36 3.3 i 1 1 r = = 0.3 in Initial Bundary Layer 2.8 ± Exact Slutin Apprximate Slutin d. V = 15.8, M = 6.4 Fig. 5 Cncluded 29

37 > -i 3) k77 b Incmpressible _z t Apprximate Values fr ty-c M « j. * Fig. 6 Theretical Variatin f Dividing Streamline Frictin Cefficient

38 AEDC-TR-7W Apprximate Value 1 i fr Hi-*t> -I 0.06 ~ >^ i D = kb f 1 i i i Fig. 7 Theretical Variatin f Mixing Length Prprtinality Cnstant 31

39 2.0 T i i i i i i i i i i r Chapman's Data (Ref. 5) M«, = 0.19 b Apprximate Slutin Exact Slutin GO t Fully Develped Mixing 0 Nte: b is based n 0 between 0.2 and ,, 1 I 0 10 j L 15 * a. With Chapman's Data Fig. 8 Cmparisn f Theretical Mixing Grwth

40 2.0 T i i i I i r i i i i i i i i i r Exact Slutin b Hill's Data (Ref. 1) fr M m Apprximate Slutin Fully Develped Mixing r'~ 0 FD = Nte: b is based n 0 between 0.2 and i i i i I i i i i J I b. With Hill's Data Fig. 8 Cncluded 3)

41 APPENDIX II EQUATIONS FOR THE CHARACTERISTICS OF GENERAL, SINGLE-STREAM MIXING ZONES Applicatin f the Chapman-Krst flw mdel t base flw prblems requires details f the assciated mixing zne and its lcatin with respect t the crrespnding inviscid jet. The mixing zne is lcated by the requirement that the mmentum flux in the actual viscus case is identical t the mmentum flux f the crrespnding inviscid jet. With a specificatin f the prperty variatins in the mixing zne, cntinuity requirements can be used t lcate the dividing streamline between the jet flw and the fluid entrained frm the surrundings. Zumwalt (Ref. 11) first derived the relatinships fr the axisymmetric mixing zne characteristics in cnjunctin with his analysis f the base pressure prblem. These relatinships were rederived by the authrs frm a mre general apprach, and the results have been reapplied t the cases treated by Zumwalt. ANALYSIS The present analysis is simply based n the dividing streamline and crrespnding inviscid jet bundary definitins and the realizatin that these are lcal characteristics relating the actual viscus case t a crrespnding inviscid jet. Cncepts inherent t the analysis can be demnstrated by cnsidering a general cntrl vlume that is everywhere exterir t the mixing zne except at the pint (statin 3, Fig. II-1, fr example) in questin. If steady cnditins prevail and base regin velcities are negligible, it is evident that frces n the upstream prtin f the cntrl vlume in the actual case must be identical t the upstream frces n the same cntrl vlume superimpsed n the crrespnding inviscid jet. Therefre, at the pint where mixing zne characteristics are desired, YM3,YM3 F 3i - F 3V (n-i) '-Y M3 I-YM3 Applicatin f mass cnservatin t the same cntrl vlume fr the actual viscus case and fr a crrespnding inviscid jet yields Y M3 Y M3 r G 3I. = G 3V (II-2) Y m3 " Y D3 34

42 APPLICATIONS Zumwalt's mst general analysis invlves an "externally expanding" flw field similar t that shwn in Fig. n-1 withut an initial bundary layer. Pressure gradients were cnsidered in the streamwise directin, but unifrm transverse static pressure cnditins were assumed t exist at each statin. Applicatin f the present analysis t this flw field yields " f "3V U 3V 2TT r, + (Y - Y m3 ) cs 0. + r, + (Y- Y m3+ dy) cs 0 3 -^L ~ Y M3 J L J (n-3) + 2ff p» (^ir) ['»' (YM3 + Ym3) cs 0s + r ' + (YM3 " Ym3) cs 0i J and Y M3 M3 G 3 V I = f p 3V u 3V 2ir Tr, + (Y - Y m3 ) cs r s + (Y - Y m3 + dy) cs 0,1 -^L m_ 4 ) l IY D3 J L J Y D3 fr the actual viscus case and P i Y F M3, n (YMS -Y m3 \r "1 3J " P31U31 2ff ^ 3 Jjr, * r, + (Y M 3 -Ym^csÖj -YM3 (n-5) + 2n p > V~T/ U 1 " (YM3 + Ym3) cs ö» + r» + (YM3 " Ym3) cs Ö»J and G 3I I " 3 = p 3, u 3I 2TT ( YM3 2" Ym3 ) [r, + r, + (Y M3 - Y m3 ) cs 0,] <n _ 6) Y n>3 fr the crrespnding inviscid jet. Substituting int Eqs. (JI-1) and (II-2) and nn-dimensinalizing the results yield ( M3 -B,)* + 2(l-C SI *)[(I 1 ) I, M3 - (I^B, - 2(1-C 3I *)[(J 1 ) 7?M3 - (j^j = (- kj and (n-7) 11-3 ~ * + X cs 0, (II-8) 35

43 where B, = ' " ["'SS- U^M3 + ( V?M3_ Zumwalt's results, which are based n a mre restrictive cntrl vlume specificatin, are (B -, y + 2<j-c sl ')[<g, 7M8 - a 1 ) J7 jb- 2 (i-c 3I»)[(j 1 ) 77M3 -cwj - (-^j and where ' S ~ U + (n-9) X cs fl 3 (H-10) B = < J^D3 ' HjHjj» ~ (Jfl)!^»«. - (J^M 3 ] Althugh the equatin frms are similar, Zumwalt's results differ frm the present analysis because upstream flw cnditins, base pressure, and specific heat rati are invlved in B, whereas Bj is independent f these parameters. The effect f this difference n the dividing streamline velcity rati, fr example, and the relatinship t the twdimensinal result are illustrated in Fig. II-2. Nte that 0D3 cmputed by the present analysis appraches the tw-dimensinal value as the gemetric parameter appraches infinity, which is intuitively expected. Fr the special case f unifrm flw, static pressure is als cnstant in the streamwise directin, and C21 = C31. Under these cnditins, B = Bi, and Zumwalt's results agree with the present analysis. Equatins (II-7) and (II-8) are als applicable t the prblem f internally expanding jets (i. e., cnverging-diverging nzzle exit flws) if the sign f r3 is changed. Under these circumstances, unifrm flw cnditins exist alng the jet bundary, and the present analysis is nce again in agreement with Zumwalt. 36

44 Cntrl Vlume 00 Y m 3(lnviscid Jet Bundary) Yp3 (Dividing Streamline) Crdinate Reference Line 3D Fig. M-1 General Mixing Zne

45 1000 Zumwalt's Results Eqs. (11-9) and (11-10) C 3I /C 2I I All CM CO 100 a» E r i- Q. a> E a> O 10 Present Analysis, - Eqs. (11-7) and (11-8) (I ndependent f - y and C T ) N \ \ 1.0 Tw-Dimensinal Value (Ref. 12) D Fig Variatin f Dividing Streamline Velcity Rati 38

46 UNCLASSIFIED Security Classificatin DOCUMENT CONTROL DATA -R&D (Security clatrlllcatin f Ml; bdy f abstract and indexing anntatin must be entered when the verall reprt la claaellled) I. ORIGINATING ACTIVITY (Crprate authr) Arnld Engineering Develpment Center, ARO, Inc., Operating Cntractr, Arnld Air Frce Statin, Tennessee a. REPORT SECURITY CLASSIFICATION UNCLASSIFIED 2b. GROUP 3. REPORT TITLE INFLUENCE OF INITIAL BOUNDARY LAYER ON THE TWO-DIMENSIONAL TURBULENT MIXING OF A SINGLE STREAM 4. DESCRIPTIVE NOTES (Type t reprt and Inclusive datea) July 1, 1969, t June 30, 1970 Final Reprt 8- AUTHORIS) (Firat name, middle Initial, laat name) R. C. Bauer and R. J. Matz, ARO, Inc. N/A «. REPORT DATE April a. CONTRACT OR GRANT NO. F ^ PROJECT NO c- Prgram Element 62302F 7a. TOTAL NO. Or PAGES 46»a. ORIGINATOR'S REPORT NUMBER(S) AEDC-TR NO. OF REPS 9b. OTHER REPORT NOISI (Any ther numbers that may be assigned thla reprt) ARO-ETF-TR DISTRIBUTION STATEMENT Apprved fr public release; distributin unlimited. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Arnld Engineering Develpment Available in DDC Center (XON), Arnld Air Frce Statin, Tennessee ABSTRACT An integral methd is presented fr estimating the influence f an initial bundary layer n the develpment f a tw-dimensinal, isbaric, turbulent, free shear layer. The basic equatin is derived by applying the principle that, at any streamwise statin alng the free shear layer, the mmentum f the entrained flw equals the ttal axial turbulent shear frce acting alng the dividing streamline. This equatin is slved using a single parameter family f velcity prfiles derived by Krst and Prandtl's mixing length cncept fr turbulent shear stress. The thery invlves ne empirical cnstant which was evaluated using Tllmien's experimental data fr incmpressible, turbulent mixing. The thery is verified by cmparing with experimental data fr free-stream Mach numbers up t 6.4 DD FOR M 1 NOV UNCLASSIFIED Security Classificatin

47 UNCLASSIFIED Security Classificatin 14. KKV WORDS ROLE LINK 8 LINK C free shear layer develpment bundary layer separatin mixing bundary layer flw tw-dimensinal flw turbulent flw laminar flw shear flw UNCLASSIFIED Security Classificatin